A STABILITY RESULT FOR THE SOLUTIONS @F A CERTAIN

SYSTEM DF THIRD-ORDER DIFFERENTIAL EQUATIONS

A . ~ . A . Abo~-E£-E£~

The paper determines sufficient conditions under which all solutions

of (1.1) tend to zero as t + ~ .

INTRODUCTION AND STATEMENT OF THE RESULT

We consider the real non-linear third-order vector differential

equation

~" + F ( X , X ) ~ +G(X) + H(X)= P ( t , X , X , X ) ( 1 . 1 )

where X c R n, F is an nxn-matrix function, G: R n ÷ Rn,H:R n ~ R n and

p:R+xRnxRnxR n + R n.

Let the non-linear functions F,G,H and P be continuous and constru-

cted, such that the uniqueness theorem is valid and the solutions are

continuously dependent on the initial conditions.

The equation (1.1) represents a system of real third-order differen-

tial equations of the form:

... n

x i + ~ fik(Xl,..,×n;~l, ~ "" . ~n)+hi(x k=l ''' n)Xk+gi(xl'''' l''''Xn)

=Pi(t,xl,..,Xn;~l,..,~n;Rl,..,Xn) (i=l,2,..,n

~fik Moreover, l e t t he derivatives

~xj

~fik (j=l,2,..,n) exist; furthermore and

~xj

~fik ~gi ~h , - - , .......... and - -

a~ j @~j axj

Bh. i are continuous.

Bxj

Special cases of the differential equation (i.I) have been treated

in Abou-E1-Ela [1] ,[2] ; Ezeilo [3],[5]; Ezeilo & Tejumola [4] and

Others. This paper generalizes Ezeilo [5; theorem 3] for the case

A=F(X,X) and also gives an n-dimensional extension for Ezeilo [3] .This

work extends further a result given by the author in [2] where P is not

necessarily identically zero.

Using Y=X and Z=Y the differential equation (1.1) will be trans-

formed to the equivalent system

= Y~

= Z,

= -F(X,Y)Z-G(Y)-H(X)÷P(E,X,Y,Z).

We need the following notations and definitions:

(1 .z )

I.Xi(A) (i=l,2,..,n) are the eigenvslues of the nxn-matrix.

2.<X,Y> corresponding to any pair X,Y of vectors in R n is the usual n

scalar product i~l xiYi'II x If2 = <x,x> for arbitrary X in R n.

3.The matrix A is said to be negative-definite, when <AX,X> < o for

all nonzero X in R n

4. The Jacobian matrices JG(Y),JH(X),J(F(X,Y)YIX) and J(F(X,Y)YIY)

are given by:

8g i ~h i JG(Y) = ( - ~ j ) , JH ( X ) = ( ~--x-T. ) ,

J n n 8fik

J(F(X,Y)YIX)= (~x ~ fikYk )= ( ~ ~ Yk )' j k=l k=l j

n n J(F(X Y)YIY)= (--~--~ Z fikYk)=F(X,y)+ ( ~ fik yk) "

' ~Yj k=l k=l ~Y---~ Now let us formulate the foIlowing conditions:

(i) FiX,Y) is symmetric and ki(F(X,Y))~ 61 > o(i=l,2,...,n).

(ii) J(F(X,Y)YIY) is symmetric and J(F(X,Y)YIX) is negative-definite.

iii) G(O)=O, JG(Y) is symmetric and Xi(JG(Y)) ~ 62 > o (i=l,2,..,n).

iv) H(O)=O, JH(X) is symmetric and 65 3 Xi(JH(X)) ~ 63 > o

(i=l,2,..,n).

v) 6 1 6 2 - 6 5 > o .

(vi) JG(Y) and JH(X') commute with JH(X) for all X,X',Y ¢ R n.

(vii) There exist constants ~(o ~ ~ < i), 6o ~ o and continuous

functions @l(t),O2(t) ; such that for all t ~ o and every solut-

ion X,Y,Z of (I.2) the following inequality

II P(t,X,Y,Z)II ~ 81(t)+e2(t)(ll Y 112+11ZII ~ )½a+6o(II YII 2 +II zIl 2 )½

(1 .3) is valid; furthermore Ol(t),O2(t) are positive and satisfy

max el(t) < ~ and ~ 8i(t)dt < ~ (i=1,2). o<t<~ 0

Our a~m is to prove the foilowing

THEOREM: Under the additional assumptions (i)-(vii) there exists a

posilive constant A o

such that if 8 ° ~ A 0

(1 .4 )

, whose magnitude depends only on 61,62 and 6''3,

then every solution X(t) of (i.i) satisfies

3

II x ( t ) I I + o, II ~ ( t ) I I + o and II R ( t ) I I + o as t + ~. (1 .5 )

Remark: In the case F(X,Y)E A, i t follows from the assumptions ( i ) and

(ii):

A is symmetric and li(A) ~ 61 > o (i=l,2,..,n).

The assumptions (i)-(vii) are then exactly those of Ezeilo [5;theorem 5].

A FUNCTION T(X,Y,Z)

The proof of the theorem depends on a scaler differentiable compar-

ison function ~(X,Y,Z). This function and its total time derivative

satisfy fundamental inequalities. We define ~ as follows

~(X,Y,Z)= ofl<H(oX),X > do + ~of~ G(oY),Y > do

+ ofl< F(X, oY)Y,Y>do+ ½ ~ < Z,Z>

+ ~ < Y,H(X)> + <Y,Z> (2.1)

where 62 ~ - i > 6 > 6; 1 .

Let (X(t),Y(t),Z(t))be an arbitrary solution of (1.2), we define then

@(t)= ~(X(t) , Y ( t ) , Z ( t ) ) . (2.2)

The following two lemmas are important for the proof of the theorem

LEMMA 1: There exists a posit ive constant 67, such that

T(X,Y,Z) Z 67(11 xlP +11 YIP +II zll 2) (2. ; )

is valid for every solution of (1.2).

LEMHA 2: There exists a positive constant Ao=8o(~1,62,6~) such that if

6 o _< 8 o then for t _> 0

~(t) d - ~8P2(t)+ 69 {81( t )p ( t )+ez( t )pa+ l ( t ) } (2.4)

where P2(t)~ II Y(t)II 2 +II Z(t)II 2 and ~8,~9 are posit ive constants,

dependent on ~i,~2, ~ .

For the proofs of the two lemmas see Abou-E1-Ela [ i ] .

SOME PRELIMINARY RESULTS

We shall require the following two algebraic results

LEMMA 3: Let A be a reel symmetric nxn-matrix. If A a ~ xi(A) ~ 6 a > o

(i=l,2,..,n), then

Aa il x IP z <AX,X>z 6a II X II= ( 3 . 1 ) for any X c R n.

LEMMA 4: Let A,B be any two real nxn commuting symmetric matrices, then

the eigenvalues of AB are all real; and if ~ Z ~(A) ~ ~a > o ,

Z ki(B) Z ~b > o (i=l,2,..,n), then

AaAb Z Xi(AB)~ ~a 6b" (3.2)

For the proofs of the two lemmas see Abou-E1-Ela [2].

The actual proof of the theorem depends on the following two proper-

t i es of H

LEMMA 5:

then

If H(O)=O, JH(X) symmetric and

]I H(X)-H(X' ) II _<~ II x-x, II

for a l l X,X' in R n.

Proof: See Ezeilo [5;Lemma 9]

LEHMA 6" If H(O)=O, JH(X) symmetric and commutes with JH(X') for all

X,X' in R n. If Xi(JH(X)) > 6 3 > o , then

II H(X) II > ~3 II x II

Xi(JH(X)) <~' (i=1,2 . n)

(3.3)

for all X ¢ R n.

Proof: Since

8 < H(OlX),H(olX)> = 2 < JH(OlX)X, H(olX) > 8a 1

by integrating both sides from oi=o to ~i=i , and because of H(O)=O,

we obtain

1 II H(X)II = =ol 2 < JH(OlX)X,H(alX) > da I

but H(~IX) = ° f l " ~ 2 H ( a l ° 2 X ) d a 2

1 = o f a 1 J H ( a l e 2 X ) X da 2

therefore 1

<JH(OlX)X,H(~IX) >= ~< JH(alX)X,OlJH(alO2X)X > do 2 o

consequently we get

II H(X)II 2 =2oS~f I ~i < JH(~IX)X'JH(°IO2X)X > do2 do1

11

= 2 o f o f < 3 H ( a l o 2 X ) ~ H ( a l X ) X , X > a 1 do 2 da 1. ( 3 . 5 )

(3.4)

Since JH(OlX), JH(OlO2 X) commute and are symmetric and because of

_ R n Xi(JH(X)) > 63 for any X c , then it follows

X i ( J H ( O l O 2 X ) J H ( O l X ) ) ~ 632 ( 3 . 6 )

by Lemma 4.

Hence, according to Lemma 3, we have from (3.5) and (3.6)

II H(X)II ~ _> ~3211 x ii ~ .

THE PROOF OF THE THEOREM

L e t ( X ( t ) , Y ( t ) , Z ( t ) ) be any s o l u t i o n o f the e q u i v a l e n t sys tem ( 1 . 2 ) .

We s h a l l show t h a t t he r e s u l t ( 1 . 5 ) o f t he theorem h o l d s f o r any s o l u t -

in (1.3) satisfies 6o < & where ion X(t) of (1.1), if the constant 6 ° -- o

4 o is the constant in Lemma 2.

The proof will be in two stages: first we show that II Y(t)II + o and

II z(t) ll ÷o as t ÷ ~ , and then, as a consequence we verify that

l l x ( t ) ll + o as t +

From Lemma 2 we have seen t h a t 9 ( t ) s a t i s f i e s the i n e q u a l i t y ( 2 . 4 )

f o r t > o . By i n t e g r a t i n g bo th s i d e s o f ( 2 , 8 ) we o b t a i n t he i n e q u a l i t y

t t 9 ( t ) 2 4 ( o ) - 88 f p2 (T )dT+ 69 f { e t ( ~ ) p ( ~ ) + e 2 ( ~ ) p ~ + l ( ~ ) } d T ( 4 . 1 )

o o

From ( 2 . 3 ) i n Lemma i and ( 2 . 2 ) we c o n c l u d e t h a t

9 ( t ) 2 o for t ~ o . ( 4 . 2 )

Combining (4.1) and (4.2) leads to

t t ~8 f P 2 ( ~ ) d T ~ 4 ( 0 ) + 6 9 ~ { 8 . 1 ( T ) p ( T ) + e 2 ( T ) p a + I ( T ) } dT. ( 4 . 3 )

o O

The a u t h o r p r o v e d i n [ 1 ] a boundedness t heo rem f o r t he s o l u t i o n s

o f ( i . 1 ) w i t h t he same h y p o t h e s e s e x c e p t f o r e i , e 2 s a t i s f y i n g

7 ~ e K / ( 1 - ~ ) ( t ) d t < ~ ( 4 . 4 ) o I 8i(t)dt < ~ and o 2

where i 2 ~ 2 2.

But since (1.4) implies (4.4) the boundedness result holds also here

under the new strong conditions on el,e 2. Hence there is a finite

positive constant &l such that

II x ( t ) l l = +ll Y ( t ) I P + I I z ( t ) l l 2 ~ A~, for t Z o .

Thus in particular, p(t) ~ &l for t Z o

(4.3)

o ; ~ p 2 ( T ) d T <

(4 .5 )

and t h e n i t f o l l o w s f r om

(4 .6 )

by (1.4).

d 2 ( t ) = 2 < Y, Z > + 2 < Z, ~ > Since Ttp

=2 < Y,Z> -2 < Z,F(X,Y)Z + G(Y)+H(X)-P(t,X,Y,Z)> we obtain

by using (4.5), (3.1) and the hypotheses on F,G,H and P

dp2(t) is bounded as + t (4.7)

It is quite an elementary matter (see for example [6;P. 273]) to show

from (4.6) and (4.7) that

p 2 ( t ) + 0 as t +

t h a t i s , i n v i e w o f t h e d e f i n i t i o n o f p

I[ Y ( t ) H + o and U z ( t ) I I + o a6 t + ~ . ( 4 . 8 )

Thus we h a v e t h e f i r s t p a r t o f o u r r e s u l t .

To c o m p l e t e t h e p r o o f o f ( i , 5 ) i t r e m a i n s t o v e r i f y t h a t

tl X ( t ) ] ] ÷ o as t ~ ~ ( 4 . 9 )

I t w i l l be s u f f i c i e n t t o p r o v e t h a t

II H ( X ( t ) ) l l ÷ o as t ÷ ~ ( 4 . 1 0 )

s i n c e , by Lemma 6 , 1 1 H ( X ( t ) ) I I ~ 63 I[ X ( t ) l l , so t h a t ( 4 . 1 0 ) n e c e s s a r i l y

i m p l i e s ( 4 . 9 ) .

The m e t h o d o f t h e p r o o f o f ( 4 . 1 0 ) i s d e r i v e d f r om an a d a p t a t i o n o f

an i d e a i n [ 7 ; §2 7 .

Now by ( 1 . 2 )

Z= - F ( X , Y ) Z - G ( Y ) - H ( X ) + P ( t , X , Y , Z )

and on i n t e g r a t i n g b o t h s i d e s f r om t t o t + l ( t > o) we have t + l

H ( X ( t ) ) = - [ Z ( t + I ) - Z ( t ) ] - f F ( X ( % ) , Y ( 7 ) ) Z ( % ) d % - t

t + l t + l - I G(Y(T) )d% + f { H ( X ( t ) ) - H ( X ( T ) ) } d~+

t t

t + l + I P( T,X(T) ,Y(T) ,Z(T) ) tiT.

t I t is evident from (4.8) that

(4.ii)

l l z ( t + i )

Now s i n c e t + l

11 f F(X(~) ,Y(T))Z(~)d~l l 2{ f t t

- Z ( t ) II ÷ 0 as t ÷ ~o.

t + l II F ( X ( T ) , Y ( T ) ) Z ( T ) I I 2 t iT} ½

< max II F ( X ( T ) , Y ( T ) ) Z ( T ) I I t<~<t+l

max II F(X,Y)II . max llz(~)II II x II 2 41Y ll~a~ tE~t+l

= ~ 0 max II z ( ~ ) II , t<T< t + l

t + l it follows l[ f F(X(T),Y(T))Z(~)d~ll ÷o as t ÷ ~ ,

t

by (4.8).

According to (4.8) also and the

satisfies G(O)=O, we conclude that

t+l I I I G ( Y ( T ) ) d T l l ÷ 0 as t ÷ ~ .

t

( 4 . t 2 )

(4.13)

fact that G is continuous in Y and

(4.i4)

By Lemma 5

bu t

II H ( X ( t ) ) - H ( X ( T ) ) H ~ 6~ II X ( t ) - X ( T ) II

t+l t+l II X ( t + l ) - X ( t ) l l = II I Y(T)dT I I 2{ I II Y(~ ) I I 2

t t dT} ½

then from

II x(t)- x ( ~ ) I I _<

we have t + l

Ilf t

therefore

< max II Ye)ll, t<T_< t + l

max f lY(s)I I < msx llY(s)ll ,

t<s<T t < s < t + l

{ H ( X ( t ) ) - H ( X ( T ) ) } d~lIda~ max t<T< t + l

t + l I I f { H ( X ( t ) ) - H ( X ( T ) ) } dT II -~ o

t

II y ( ~ ) I I ,

as t + ~. (4.15)

We have a t last from ( 1 .3 )

t + l t + l II S P(T,X(T),Y(T),Z(T))dTII2±I

t t { e l ( T ) + e 2 ( T ) p ~ ( ~ ) + 6 p (~ ) }2 d~

0

t + l ±

t

But , i n v iew of ( 1 . 4 ) and ( 4 . 5 ) , there are cons tan ts 811,612 such t h a t

e i ( t ) ~ $ i i ( i =1 ,2 ) , p2(t) ! 612 , t ! o,

Consequently

t + l t + l ~ (T) } +~2-2(T) ]dT II P(T,X(T),Y(T),Z(T))NTII2~ 3 ; [811{01 (T)+ 81202 0 p

t

t+l 5 613 7 [ e I (T )+O2(T )+p2 (T ) ] dT,

t

where 613 = 3 max (611,611612 , o

Because of the integrability conditions on 0 i and p2in (1.4) and (4.6)

respectively, it is obvious that the last integral tends to zero as

Hence t + l

i [ I P(T,X(T),Y(T),Z(~))dTII + o as t ÷ =. (4.16) t

From (4.11) and owing to (4.12)-(4.16) it follows (4.10) and the proof

of the theorem is now complete.

REFERENCES

[l]Abou-Ei-Ela, A.M.A., VII. Internationale Konferenz Ober Nichtlineare

Schwingungen,Band I, I. pp. 17-23. Abh. Akad. Uiss., Akademie-Verlag

Berlin 1977.

[2]Abou-Ei-Ela, A.M.A., Math. Nachr. 81(1978), 201-208.

[ 3 ]Eze i l o , J.O.C., Ann. mat. Pura AppI. , IV Vol. 66 (1964),233-249.

[ 4 ]Eze i l o , 3.0.C. & Tejumola, H.O., Ann. mat. Pura Appl . , IV Vol. 74

(1966), 283-316.

[5 ]Eze i lo , J.O.C., Journal of math. Analysis and appl . , 18 (1967),

395-416.

[6]hefschetz,S.; Differential equations: geometric theory (interscience,

New York, 1957).

[7]Levin 3.J. & Nohel 3.A., Arch. Rational Mech. Anal. 5 (1960),194-211.

A LOCAL EXISTENCE THEOREM FOR THE QUASILINEAR WAVE

EQUATION WITH INITIAL VALUES OF BOUNDED VARIATION

H.D. Alber

i. Introduction

We consider the quasilinear hyperbolic initial value problem

u t : f(u) x x E ~, t ~ O, (i.I)

u(x,0) = U(x), (1.2)

with u = (Ul,U 2) : ~ × ~+o ÷ ~2, U = (UI,U 2) : ~ ÷ ~2 f(u)': (u2,~(ui)).

We assume~.that ~ E C2(~, ~) satisfies ~'(~) ) 0 for all T £ ~, which

implies that the system is hyperbolic. This system is genuinly non-

linear if a"(<) $ 0 for all ~ £ ~, and we assume without restriction of

generality that ~"(T) > O. For i:i,2 the i-Riemann invariants

R i : R 2 + ~ are defined by

_ . , )112 RL(UL,U2 ) = u2 ~1 ~ (~ d~, (1.3)

O

R2(Ul,U2 ) = U2 + ~i c,(T)i/2 d~, (1.4) O

and we assume that the following condition holds:

Condition C. Every level curve of R i intersects with any level curve of

R 2 •

Under these assumptions the following result holds:

Theorem i.i: If the initial data U have compact support and are of

bounded variation, then there exists T > 0 such that a weak solution u

of (1.1), (1.2) exists in the domain ~ x [O,T) with

var u(.,t) ~ C , 0 ! t < T.

This weak solution satisfies the entropy condition.

A bounded measurable function u : ~ × [O,T) ÷ ~2 is called weak solutiol

of (1.1), (1.2) if

F ~ (u¢ t - f(U)~x)dX dt + ~ U(x)~(x,O)dx : 0 O -~ -~

for all smooth functions ¢ vanishing on the line t : T. A function

n : ~2 ÷ ~ is called entropy, if there exists a function q : ~2 ÷

10

such that for every continuously differentiable solution u of (i.l) the

additional conservation law

n(u(x,t)) t + q(u(x,t)) x = 0

holds. We say that a weak solution u of (1.1), (1.2) satisfies the

entropy condition, if for every convex entropy the relation

nt + qx ~ 0 (1.5)

holds in the distributional sense, cf. [6]. An example for a convex

entropy is the energy density

1 2 ~i n(Ul,U 2) : ~ u 2 + c(~)dT, q(Ul,U 2) : ~ U 2 q(Ul).

O

Condition C) is necessary and sufficient for the result in theorem I.i

to hold. For, if this condition is not satisfied, then it is possible

to construct initial data U for which a local solution satisfying the

entropy condition does not exist.

Glimm and Lax proved in [4] that a global weak solution of (i.i),

(1.2) exists under essentially the above assumptions for c, provided

that U has bounded variation and, in addition, small oscillation. This

means that U(x+) - U(x-)il must be less than a given small constant for

all x E ~. Recently DiPerna proved in [2] that to all bounded initial

data a global weak solution exists if ~" < 0 for x < 0 and ~" > 0 for

x > O. The proof uses the results on compensated compactness in [7].

This method can be applied if a-priori L -estimates are known, and for

the systems considered by DiPerna these estimates follow from the exist-

ence of invariant subregions. However, under the above assumptions the

system (1.1) does not have invariant subregions, in general. We there-

fore cannot show in this simple way that a-priori estimates hold. In-

stead, for the proof of theorem i.i we proceed as follows. A sequence

{u(n)}n~ of approximate solutions to problem (1.1), (1.2) is construc-

ted out of solutions to the Riemann initial value problem, which entirely

consist out of shock waves. Of course, we have to admit shocks violating

the entropy condition (1.5). However, we construct u (n) such that the

strength of the shocks violating the entropy condition tands to zero for

large n, and approximate in this way rarefaction waves. It will be seen

that the construction is similar to the one given by DiPerna in [I]. We

prove that these approximate solutions are uniformly of bounded varia-

tion, and therefore contain a subsequence converging to a solution of

(1.1), (i.2). The proof of the theorem is too long to reproduce it here

completely. So we only give a sketch. ~n section 2 we introduce special

solutions of the Riemann initial value problem and define the approxi-

mating sequence, and in section 3 we give some of the essential steps

11

in deriving the a-priori estimates. The necessary estimates for the

shock interactions are collected at the end of the paper without proof.

2. The approximate solutions

We construct the approximate solutions by solving the Riemann ini-

tial value problem by piecewise constant functions. A piecewise con-

s~ant weak solution of (1.1) must satisfy the Rankine-Hugoniot relation

s(u~v) : - (f(~) f(v)), (2.I)

where u = (ul,u2) , v -- (Vl,V2) are the constant values of the solution

on either side of the curve of discontinuity x = x(t), and where

s = dx{dt is the speed of the discontinuity. It follows that

u2-v 2 s

Ul-V I ( 2 . 2 )

( u l - v 1 ) ] 1 / 2 . D e f i n e the functions and that u 2 : v 2 ± [(~(ul) - ~(vl) )

Civ, C2v : ~ ~ ~ by

I v 2 - [(c(m - ~(v i) (T-vi)] 1/2 , T < v I

Clv(T) = (2.3a v 2 + [(d(T - ~(vi) (T-vl)] 1/2 , • > v I

i ]i/2 v 2 + [(~(~ - ~(v i) (~-v i) , T < v i

C2v(~) = (2.3b v 2 - [(~(T) - ~(Vl))(~-Vl)] 1/2 , • > v 1.

We call the discontinuity a i-wave, if s is negative, otherwise a

2-wave. Thus, the discontinuity is a i-wave if u lies on the graph of

Clv , and a 2-wave if u lies on the graph of C2v. If v is the state on

the left of a i-wave or the state on the right of a 2-wave, then, with

the terminology of Lax [5], the discontinuity is a shock if u I > Vl,

otherwise a "rarefaction discontinuity". This follows from lemma A1 (i

in the appendix and from the assumption s"(T) > 0. We now define a weak

solution u of the Riemann initial value problem of (1.1) to the initial

values

i I U , X < 0

u(x,O) : ( 2 . 4 ) r

U } x>O.

For i=1,2, n a nonnegative integer, and v C ~2 first define the func-

tion c~ : ~ ~ ~ as follows: Let iv

12

0 • = C . , CiV 1v

For n > 0 let

o c n v : v and iv(T) : Civ(T) , v I ! <' (2.5)

If v m E i~ 2 has been defined, then let

c n Civm( i v ( .~ ) : .~) , ~T +1 < .~ <_ v~, ( 2 . 6 )

where v m+l , m+l m+l, IR2 = ~v$ , v 2 ) E is the point on the graph of s such

that the arclength of this graph in between v m and v m+l is ivm equal

n into pieces of arc- to 1/n. These points subdivide ~he graph of Civ

length 1/n. We now define solutions of (1.1), (2.4).

In lemma A2 it is shown that the graphs of c n and c k intersect i r

in exactly one point. Let lu 2u 1 m-1 . m m-i i i

w ° : u I, let w 4...,w wlth w I < w I <...< w I < u I be the points sub- r n . 1 m m . . dividing g aph o i in between u and w , w being the poznt of inter-

- ~lu- m+l j-i ._, m m+l j-i r section, an~ ±e~ w ,..., w wm~n w I < w I <...< w I < u I be

the points subdividing graph c k Let w j : u r Then define the solu- r"

2u tion of (i.i), (2.4) by

U I , x < S I t

i U(x,t) : w , s i t < x < si+ I t , 1 <_ i < j (2.7)

r u , s. t < x

$

whe re

i i-1 w 2 - w 2

si = - ~ ~ - i ~ 1 ! i <_ j. (2.8) w I - w I

Lemma A1 (i) together with the assumption ~"(~) > 0 implies that

s i < si+ I. (2.6) and (2.8) imply that u satisfies the Rankine-Hugoniot

condition (2.1), and therefore is a weak solution of (1.1).

Definition 2.1: We call the weak solution u of (1.1), (2.4) construc-

ted above resolution of the discontinuity [ul,u r] of type (n,k).

Now we are ready to construct the sequence {u(n)}. Choose a sequence

{U (n)} of step functions U (n) : ~ ~ ~2 each one having finitely many

steps, such that the following properties are satisfied:

supp U (n) ~ {x E ~ [ dist(x,supp U) S 1} (2.9)

IU (n) - U l ~ o ( 2 . 10 )

13

var U (n) ~ var U, (2,11)

where U denotes the given initial data. We choose U (n) such that all

the intervals on which U (n) is constant contain more than one point,

which is possible, since we can assume that U is continuous from the

left. We next use the following lemma.

Lemma 2.2: If h : ~ ~ ~2 is a function of bounded variation then to

every ~ > 0 and every x E ~ there exists 6 > 0 with va~h L(x-6,x)],

varthl (x,x+6)] < c. The proof is obvious. (2.10) and (2.11) imply that

for every interval I we have var(u(n)I~l) ~ var(U~l). From lemma 2.2 it

thus follows that to every ~ > 0 arid to every x E ~ there exists 6 > 0

with

var[u(n)j(x-6,x)], vartu(n)1(x,x+6)] < ~ (2.12)

for all n sufficiently large.

Next we define a set {tk}k> O and simultaneously the function u (n)

on the domains ~ × [tk,tk+ i) by induction. Let t o : O, and solve the

initial value problem

u(~ ) : f(u(n))x

u(n)(x,0) : u(n)(x)

by resolution of the discontinuities of U (n) of type (n,n), and continue

the solution u(n)(x,t) as long as the discontinuities of u (n) do not

intersect. Let t I be the time where the first such intersection occurs.

Thus, u (n) is defined on R × [to,tl).

Assume that times t <...< t have been defined, and that u (n) is o m

defined on ~ x [O,tm). The line ~ x {t m] is intersected by discontin-

uities of u (n) at finitely many points (xl,tm),...,(xk,tm) with

x I < Xl+ I. Assume that at the point (xj,t) there intersect the v lines

of discontinuity

x : hi(t) : si(t-tm) + xj , tm_ 1 i t < t m , i:l,,...,v, (2.13)

with s i > si+ i and v h i. We first continue the functions

li : [tm-l'tm) ~ ~ defined by (2.13) to the interval [tm_l,tm]. Choose

> 0 small enough such that

< ~(tm-tm_l) (2.14)

and

1 i [Xl-~,Xl+~] c ([(Xl_l+Xl), [(Xl+Xl+i)) , l=l,...,k,

where we define x 0

= - ~, Xk+ 1 = ~. Now choose Xi(tm) C (xj-~,xj+~). If

14

~i(tm) is defined, then let ~i+l(tm) = ~i(tm) if i+1 is even, else

choose ~i+l(tm) E (xj-~,xj+~) with ~i+l(tm) > hi(tm). The functions h i

are called segments of discontinuity. If li(tm) = ~i+l(tm), then the

point (hi(tm),t m) is called point of interaction.

Now let v ° v+l ,..., v be the constant values of u (n) in between the

discontinuities along the lines x : hi(t), where

u(n)(x,t) : v i , h i(t) < x < hi+l(t) , i:l~..., v-l.

Define the function v on the interval

i 1 [[(Xj_l+X j), [(xj+xj+l)] by

o I v x < ll(tm)

v(x) : v i li(tm) < x < li+l(tm) , i=l .... iv.

v+l v ~v(t) < X

If this is done for every j, then v is defined on all of ~ and is dis-

continues only at the points k(tm) , where h is a segment of discontin-

uity defined on [tm_l,tm]. Now continue u (n) as solution of (1.1) to

the initial values

u(n)(x,t m) : v(x)

as follows: If v is discontinues at x : ~, then choose the type of the

resolution of the discontinuity at ~ according to the

Condition R. If (~,t m) is the point of interaction of two segments of

discontinuity k,~, then

a choose type (0,n), if the discontinuities along x : k(t) and

x : ~(t) are both l-waves.

b Choose type (n,O), if these discontinuities are both 2-waves.

c Choose type (0,0), if one discontinuity is a 1-wave, the other one

a 2-wave.

If (<,tm) is not a point of interaction, then use resolution of type

(0,0).

This construction implies that u(n)(x,tm +) differs from u(n)(x,tm -)

only on the intervals (xj-~,xj+~), j:l,...,k. Therefore we can choose

small enough such that

! 2 -m (2 15) lu(n)(x,tm+) - u(n)(x,tm-)i dx S n

Continue u (n) until two discontinuities intersect, and let tm+ I be the

15

time where this happens. If such intersections do not occur, then de-

fine tm+ i : ~.

In this way the function u (n) is defined now on the domain

× [O,T n) with

: sup t m. (2.16) Tn m>O

In section 3 it is shown that T > O exists with t > T. In the remainder n

of this section we introduce some definitions and notations. These de-

finitions depend on n, but we do not denote this dependence explicity.

Let L i be the set of all segments of discontinuity with 9t ~ < O,

and let L 2 be the set of all segments of discontinuity with ~t ~ > O.

Let L : L 1 U L 2. We say that XI,~2 E L interact, if Xl $ X2 and if

XI,X2 are both defined on the same interval [tk,tk+ i] with

~l(tk+l) : ~2(tk+l). We say that ~ C L issues from the point (x,tk) ,

if ~ is defined on [tk,tk+i] with x = X(tk). Let C ~ ~. We say that

~ L can be connected to a, if there exist ~o,,..,Xm ~ L with

~j : [tj,tj+ i] ~ ~, ~o(O) ~ ~, ~j(tj+ i) = ~j+l(tj+i), and with X m = ~.

By Li(~ ) we denote the set of all ~ ~ L i which cannot be connected to

R~. For i:i,2 let ~i(~) be the set of all subsets ~ = {~m}m~l ~ Li(~)

defined by the following properties:

a) I is a set of nonnegative integers with O ~ I.

b) Either ~o issues from the line ~ x {O}, or there exist interacting

XI,~2 ~ Lj, j $ i, such that ao issues from the point of interaction

of ~i and ~2"

c) If a m is defined on [tk,tk+l] , then by condition R) above there

exists at most one X £ Li(a ) defined on [tk+l,tk+ 2] with

~(tk+ 1) = ~m(tk+l). If such X exists, then m+l ~ I and ~m+l = ~" If

such X does not exist, then I = (O,l,...,m).

Let ~(~) = rl(a) u r2(~) and r i = ri(l~). We say that ~,B E r(~) inter-

act, if there exist a m E ~, ~j E 6 which interact. The point of inter-

action of a m and ~j is also called point of interaction of ~ and B.

Note that to every couple ~,B E r(a) there exists at most one point of

interaction, since ~ and ~ coincide from the point of interaction on.

It is clear that ~ E r(~), which is by definition a subset of L, also

defines a function ~ E F(~), which is by definition a subset of L, also

defines a function ~ : [tk,t m] ~ l~ for suitable t k < tm, and we use

both meanings interchangeably. Let a(~) : {(~,~) £r(~) × L(~) I ~ E ~}.

If ~,~ E F(~) interact, then define

< B : (~,~m) C a(~), ~ > ~ = (~,~m+l) E A(~), (2.17)

16

where ~ E ~ interacts with B. C ~. m ~ r J

For A E L i let u ± and u be the constant values of u (n) on the left

and right hand sides of the straight line x = ~(t). Then either

u I, u r E graph c if i=i, or u I, u r E graph c if i=2. The arc-

length 6(A) of i~althe graph between u I and u r 2ur'is called strength

of the discontinuity along ~.

3 A priori estimates

Theorem 3.1: There exist constants T, 0, V 2, V 3, C > O, independent

of n, such that for all n the function u (n) is defined on ~ x CO,T) and

satisfies

lu(n)(x,t)I : < p , (x,t) E ~ × [O,T) (3.i

var[u(n)(.,t)] ~ V 2 , 0 < t < T (3.2

7 ° lu(n)(x,t) - u(n)(x,t')Idx ~ V3(t-t') + K , 0 i t' S t < T(3.3

It is shown in E3] that (3.1) - (3.3) imply the existence of a subse-

quence of (u (n)} converging to u E Li(~ x (O,T)) with lu(x,t) I < 0,

var[u(-,t)] S V 2, It follows from (2.15) that u is a solution of (1.1 ,

(1.2). It thus remains to prove theorem 3.1. In this proof we can as-

sume without restriction of generality that a constant M > 0 exists with

M-2 ~'(~) ~ ~ M 2 , ~ E ~. 4 (3.4)

For, the constant o of (3.1) will be chosen in (3.14a), and this choice

is independent of the behaviour of ~(T) for large values of I~I. As we

assumed that ~'(~) > O, ~"(~) > 0, we therefore can modify ~(~) for

large values of T such that (3.4) is satisfied. Theorem (3.1) is a con-

sequence of the following local result.

Le~na 3.2: For every x o E ~ there exist constants 8, 0, V 2, V 3 > O,

independent of n, such that for all n the points of interaction do not

accumulate in the triangle

D(Xo~) = ((x,t) E ~ × ~+ I Xo-6+Mt < x < Xo+6-Mt} O

where M is the constant defined in (3.4). Also

lu(n)(x,t)I S 0 , (x,t) E D(Xo,~) N (~ × [O,Tn)) , (3.5)

17

var[u(n)("'t) l(Xo-~+Mt' Xo - - n +~-Mt)] < V 2 , 0 < t < T (3.6)

Xo~-Mt u(n)( u(n) i I x,t) - ' (x,t')Idx <_ ~ + V3(t-t') ,

Xo-6+Mt (3.7)

O < t' < t < T n

Note that if X E L is defined on [tk,tk+l] , then (2.8), (2.23), lemma

A1 (i) and (3.4) imply

i 2M -I < I~ t l(t) I < [ M t k < t < tk+ l

(2.14) and the definition of l(tk+l) thus imply

M-i(t-tk ) < IX(t) - X(tk) < M(t-tk) , t k <_ t ~ tk+l, (3.8)

for all sufficiently large n. This inequality and (2.9) show that u (n)

vanishes on the set D : {(x,t E 19 × 19 + I dist(x,supp U) > i+Mt}. O

Using this fact it is clear that theorem 3.2 follows from lermma 3.2 by

choosing finitely many of the sets D, D(Xo,~) covering a strip 19 × [O,T),

Lennna 3.2, in turn, is a consequence of the following basic result.

Lemma 3.3: For every x ° E 19 exist constants 6, p, V2, V 3 > O, inde-

pendent of n, and for every n ~ ~ a function ~ : a(a) ~ R~, where

a = (Xo-~, Xo+~), such that

0(I) < E ~(a~) , for all x E L(~) (3.9)

with XE~

i 9(X) ~ ~ V 3 , if the discontinuity along x E L(~) (3.10)

is a rarefaction wave.

sup ~(~,x) S V 2 (3.11) ~Er(a) xE~

l~lcx)l, lur(1)l ! P , for all X E L(~). (3.12)

Here ul(1) and ur(1) denote the constant values of u (n) on the left and

right hand side of graph i.

Observe that (3.5) is an immediate corollary of (3.12), since (3.8)

implies that graph X N D(Xo,{) = ~ for all X [ L~L(~). For the proof of

(3.6) note that (3.9) implies

var[u (n) +~-Mt)] < Z 8(~) < Z E ~(~,X) (',t)l(Xo-~+Mt, x o _ ,

with ~C~

for t k < t ! tk+i, where the summation is over all ~ E L(~) defined on

[tk,tk+l]. For such that xl,l 2 the sets {~ C r(a) I x I E a} and

18

{~ E F(~) I ~2 E ~} are disjoint if ~1 ~ ~2" and therefore (3.6) follows

from (3.11). Finally, (3.7) follows from (3.6) just as in [3]. It thus

remains to prove lemma 3.3.

Sketch of the proof of lemma 3.3: I) For every n E ~ let F(~) be the

set of all functions p : a(~) ~ ~+. Of course, F(~) depends on n, but o

we do not denote this dependence explicitly. We define an operator

B : F(~) ~ F(~) and show that B possesses a fixed point ~ satisfying

(3.9) and (3.11) for ~ sufficiently small, with the constants V 2 and 6

independent of n. Note that (2.10) implies lim u(n)(x±) = U(Xo±). It n-~

follows from this that if the resolution of the discontinuity

[U(Xo-), U(Xo+)] contains an i-shock wave, then ri(a) contains an

i-shock ~i issuing f~om (Xo,O) for all n sufficiently large. Similarly,

if the resolution of [U(Xo-) , U(x ° +)] contains i-rarefaction waves,

then Fi(a) contains i-rarefaction waves issuing from (Xo,O) for all n

sufficiently large. We consider here the case that the resolution of

[U(Xo-) , U(Xo+)] consists of a i- and a 2-shock wave, which implies

that for all n sufficiently large c I 6 FI(~) and ~2 6 r2(~) exist

issuing from (Xo,O). In this sketch we restrict ourselves to the proof

of the existence of ~.

For the definition of B note that by (2.10) and the fact that var U

is finite there exists a compact set W ~ ~2 with u<n)(~)~ W for all n.

According to lemma A2 in the appendix there exists a constant V* with

l (w,u) + v*l vl (3 3)

for all u,v [ W. Choose p sufficiently large such that

{w E m 2 [dist(w,W) ! V 2} ~ {w E m21 lwl i P}, (3.14a)

where

V 2 : V*(var U+I) + i. (3.14b)

Also, let V be the larger one of the two constants belonging to the

compact set

{ IwI ! ~} × [0,v 2] × [0,v 2] ~4,

whose existence is assured by lemma A3 (ii) and by lemma A4 (iii), and

let V i < I be the constant belonging to the compact set

{ lwl ! p} × [O,V 2] ~ m3

according to lemma A4 (iv).

We define B as follows. Let ~ : {~m}mEl C F(~). We first define

19

Bp(~,~o). There are two possibilities.

1.) If ~ issues from 2 × {0}, then

Sp(~,~ o) : e(~o). (3.15)

2.) Assume that ~ E ri(2) issues from the point of interaction of

kl,k 2 E Lj(2), j • i. Let { l,...,a m} ~_ ri(~) be the set of all the

discontinuities issuing from the point of interaction of kl and ~2"

This set contains e, of course. Assume that

I~ t ~I <'''< I~ t ~iI l~tkll > l~tk21 0 '

and let m o be the largest nonnegative integer with m ° < m and with

mo/m S A, where

p(6,11)][ Z p(y,~2)] , if 12 ¢ sj y ~ r ( ~ ) wifih 12E ¥

v[ z s e t (a ) with Xf£g

A :

Vj. ~ s~r (a ) with XlEB

Then define

i i ~ 1/n

Bp(~ ,~o ) : I A-m°/n

0

p(B,kl) , if ~2 e ~j.

(3.16)

, l < l < m - O

, i : m +I (3.17) O

, m +2 < i < m. O - -

In the situation above we call 1 m ri ( • if ,...a E ~) reflected at sj

k 2 E ~j. The point of interaction of 1 1 and 1 2 is also called point of

reflection. NOw we define Bp(a,al+l) if Bp(~,~l) is known. If no inter-

action takes place in between ~i and ~i+i' then let

Bp(e,el+l) : Bp(e,el). (3.18)

If B1~...,6 m E F(~) interact with ~ at the same point of interaction,

then let

m

Bp(~ > 81) : ff [i + Vp(8 j <~)]Bp(~ < Bl) , (3.19) j:1

where the symbols <,> are defined in (2.17). Note that Bp(~,I) is

uniquely defined by (3.16) - (3.19) for all (~,k) E A(2), since X E L(2)

cannot be connected to ~2.

II) We next show that B is a contraction mapping if the interval

• < i F(~) is : (Xo-6 , Xo+~) is choosen sufficiently small Let O < ~o

a complete metric space with the metric

d ( p , q ) : Z sup ~ ( t ) I p ( a , t ) q ( ~ , k ) I , ( 3 .20 )

20

where

I ~o ' if k E s I U a 2

~(k) : I , if sl(t) <_ k(t) ! s2(t) for all t, but k ¢¢i

-i ~o , else

The proof is based on the following four inequalities for p,q E F(2)

and ~ : {~m]mEl E F(~):

eVd (p ,0 ~(~m)BP(~,~ m) <_ )~(ao)BP(~,~o)

~(c~ m) Bp(~,c~ m) - Bq(c~,~m) ] <_

< Vd(p ,q ) eV[d(p 'O)+d(O 'P) ]~(ao)BP(c~ ,c~ o) +

+ eVd(O'q)~:(C~o )],.Bp(c~,~ O) - Bq(a,C~o)

IBp(~x,C~o) - Bq(c~,C~o) I <_ V 1 d ( p , q ) X

reflected at s i U s 2

X ~r(~) a not reflected at al U ~2

~(Cto)lBp(c~,C~o) - Sq(~,C~o) I <_

U c 2 .

3.21)

3.22

3.23

3.24

where ai : {~im}mEll" (3.23) -(3.25) with q : O now imply together with

(3.13) and the fact that ~(~o ) : i if ~ is reflected at ai U ¢ 2

Y

not reflected at s~ U

g2

K(C~o)BP(~,aO) + X ~(O,o)B~(a,~, )] issuing from

(a~{Xo} x {0}

2 eVd(p'O)[ X ~(~o)BP(~'~o ) + ~o .X BP(si,aio) +

reflected l:i at Sl U ~2

2 d(Bp,O) : X sup ~(k)Bp(~,k) + ~ X sup BP(si,k)

~£r(2) kE~ o i:l kE¢. i

~*~1,e2

< V[d(p ,O) + d ( O , q ) ] d ( p , q )

T o g e t h e r w i t h t h e e s t i m a t e

X !Bp(~,~o) I < V*[var(u(n)i(Xo-~,Xo)) + var(u(n)I(Xo,Xo+~))],

issuing from (2~{Xo}) × {0} (3.25

which is an immediate consequence of (3.13) and (3.15), it follows from

these inequalities that B is a contraction mapping for 6 sufficiently

small. For, (3.21) yields

21

_ (P'O){v I v*Iu(n)(xo +) - u(n)(xo-)t + d(Bp,O) < e Vd d(p,O) + <o (3.26)

+ V d(p,O) 2 + V*G~i[var(u(n)l(xo-~,Xo)) + var(u(n) I(Xo,Xo+6))]}.

Note that by (2.11)

Iu(n)(xo +) - u(n)(xo-)I ! var U (n) ~ C (3.27)

for every n E ~. Since V I <l we can choose K 1 >0 such that VI+KIV* C ~ I

for this constant C. We choose now K > 0 sufficiently small such that

N : eVK(v I + KIV* C + VK)K < K. (3.28)

Finally, by (2.12) we can choose ~ > 0 small enough such that

- 1 VK n ) N + V* G ° e [ v a r ( U ( l ( X o - 6 , X o ) ) + v a r ( u ( n ) l ( X o , X o + 6 ) ) ] ~ K ( 3 . 2 8 )

f o r a l l s ~ f f i c i e n t l y l a r g e n . We s e t G o : K1K a n d c h o o s e K s t i l l s m a l l e r

< 1 I t t h e n f o l l o w s f o r t h i s c h o i c e o f G o , i f n e c e s s a r y , s u c h t h a t G o _ .

K, and ~ from (3.26) (3.29) that for all p E F(2) with d(p,O) S K

we have d(Bp,O) ! K, which shows that the operator B maps the closed

det F K = {p E F(2)Id(p,O ) i K} into itself. In the same way we conclude

from (3.22) - (3.24) that for p,q E F K

d(B~,Bq). <_ e2VKv{vIK + G oV*IU(n)(xo+) - u(n)(xo-) I + VK 2 +

+ V* G-l[var(u(n)l(xo-~'Xo ) ) O + var(u(n) l(xo'Xo +6))]}d(p'q)

eVK(v1 + 2VK)d(p,q). +

Since V I < i, we obtain with the same choice of K I as above, but by

further decreasing K and 6 if necessary, that d(Bp,Bq) < n d(p,q) with

a suitable constant n < I, which shows that B possesses a fixed point

E Fk, hence d(~,O) ~ K. Since the choice of ~o,K, and ~ is independent

of n, (3.11 follows from (3.20). To complete the proof, inequalities

(3.21) - (3.24), (3.9), (3.10), and (3.12) must be shown. (3.21) -

(3.24) follow from the definition of B in (3.15) - (3.19), and (3.9),

(3.i0), and (3.12) follow from the choice of ~,V,V 2 in (3.13) and (3.14).

The complete proof will be given elsewhere.

~ppendix

Lemma AI: Let v = (Vl,V 2) and u = (Ul,Civ(Ul)) be the constant values

on either side of the discontinuity curve x = x(t) of an i-discontinuity.

(i) There exists T between v I and u I such that

22

Clv(Ul ) - Civ(Vl ) i )1/2, s : : (-1) o'(T

u I - v 1

where s : s(ul,v I) = dx/dt is the speed of the discontinuity.

(ii) (-i) i-1 ~Ul S(Ul,V I) < 0

(iii) e' _1)i-i I/2 iv(~) : ( a'(~)

)i/2 i/2 i ~'(~ I~-Vll Io(~) [ r ......

Io( ) <vl) I Iz2 + - ~,(~)

1 ~ ' ( T ) )I] ~js(~,vl)I + ~(~,v i

o(vl)1 i/2 i/2iT_viiiT~2i] : (_1) i-1

(iv) Ic' )i/2 iv(~)I i °'(~

Lemma A2 (i): For v = (Vl,V 2) C E2 let nlv, n2v : ~ + R be defined by

1/2 ~ )1/2 nlv(~) = v 2 + f ~'(s) ds, n2v(T) = v 2 - f ~'(s ds.

v I v 1

Then to every compact set [r,v I] there exists a constant C, independent

of u, with v

i(Civ)n I YC v( I i n n

for every r ! x ~ v I with the exception of the isolated points where the

derivative of c9 does not exist. iV

(ii) For every u,v 6 ~2 and every nonnegative integers m,n the graphs

of clun and c2vm intersect at exactly one point w.

(iii) For every compact set W c ~2 there exists a constant V, inde-

pendent of n,m, with

m l~(w,u) + 12(w,v) ~ vlu-vI,

m v), respectively, denote the for every u,v 6 W, where l~(w,u) and 12(w , m

n between u and w, and the arclength of graph C2v arclength of graph elu

between v and w, respectively.

Lemma A3: Let ~ £ LI, ~2 £ L 2 interact, assume that the discontinuity

at the point of interaction is resolved by resolution of type (0,0),

and let ~i 6 L1, ~2 £ L2 issue from the point of interaction.

(i) Then the discontinuity along ~i is a rarefaction wave if and only

23

if the discontinuity along ~[ is a rarefaction wave, for i=1,2.

(ii) Let u-be the constant state in between ~ and ~2. Then there

exists to every compact set W x [O,sl] x [O,a2] c_ ~2 x ~+o x ~+o a con-

stant V > 0 with

e(X 1 ) ! (I+Ve(X~))8(I~)

e(X 2) i ( /+ve(x l ) )e(x ~)

for a l l ( u - , e ( x ~ ) , e ( ~ ) ) E w x [o,~ 1] x [o ,~2 ] .

Lemma A4: Let x~, l~ 6 L i interact, and assume that the discontinuity

at the point of interaction is resolved by resolution of type (O,n) if

i:l, and by resolution of type (n,O) if i=2. Let X ELi, xl,...,xm 6 Lj,

j ¢ i, issue from the point of interaction with la t lm I <...< la t ~i I .

(i) Then the discontinuities along ~[ and X2 cannot both be rarefaction

waves.

(ii) The discontinuity along ~ is a shock.

(iii) To every compact set W x [0,Cl] x [O,s2] ~_ ~2 x ~+o × ~+o there

exists a constant V > O, independent of n, such that for every

(u-,e(x[),e(x[)) E w x [O,E1] x [O,e2]

e(l) < e(ll) + 8<12) + V min(l,e(l~) + e(12))e(l~)e(l ~)

if both discontinuities along l~ and l~ are shocks,

e(l) ! e(x[) + V min(l,e(x~) + 8<12))8<X~)e<12)

if the discontinuity along l~ is a rarefaction wave,

m X e(l I) 5 V min(i,e(l~) + 8(~2))8(1~)8(1~)

i=i

and

e(x I) = ! 1:1 ..... m-l, e(X m) < !. n' - n

<iv) Assume that I~ t ~[I > I~ t 121" Then to every compact set +

W x [ 0 , ~ 2 ] c ~ 2 x No t h e r e e x i s t c o n s t a n t s 0 < V i < 1 , e l > O, i n d e -

p e n d e n t of n, such that for all (u-,e(x~)) EW x [O,a2]

m

e(xl) i v 1 e(x 1) i f e(x~) i c i. l=l

24

References

[1] DiPERNA, R.J.: Global existence of solutions to nonlinear hyper- bolic systems of conservation laws, J. Differential Equations 20 (1976), 187-212.

[2] DiPERNA, R.J.: Convergence of approximate solutions to conser- vation laws, Arch. Rational Mech. Anal. 83 (1983), 27-70.

[3] GL!MM, J.: Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965), 697-715.

[4] GLIMM, J., LAX, P.D.: Decay of solutions of systems of nonlinear hyperbolic conservation laws, Memoires of the Amer.Math.Soc. 101 (1970).

[5] LAX, P.D.: Hyperbolic systems of conservation laws II. Comm. Pure Appl. Math. 10 (1957), 537-566.

[6] LAX, P.D.: Shock waves and entropy, in Contributions to Nonlinear Functional Analysis, ed. E.A. Zarantonello, Academic Press, 603-634 (1971).

[7] TARTAR, L.: Compensated compactness and applications to partial differential equations, in Research Notes in Mathernatics, Non- linear analysis and mechanics: Heriot-Watt Symposium, Vol. 4, ed. R.J. Knops, Pitmann Press (1979).

BIFURCATION PROBLEMS ASSOCIATED WITH NONLINEAR WAVE PROPAGATION

Stuart S. Antman

i. Introduction

In this paper we study three seemingly innocuous problems associated

with nonlinear wave propagation, which exhibit complicated and fascin-

ating behavior.

The first problem is yet another example, albeit a technologically

important one, of a bifurcation problem for the steady motion of a

string. (For an extensive list of references on this subject, cf.

[5,3].) Of particular interest for us is the work of Stuart [14], who

modelled the pro~ess by which thread is manufactured as a problem for

a whirling inexter~sible string with nonstandard boundary conditions.

His beautiful analysis furnished the first true application of

Rabin0witz's global bifurcation theory [ii].

Stuart neglected the fact that in the manufacturing process the

string is drawn as well as whirled. In Section 2, we analyze this tricky

problem, showing that the qualitative behavior of its solutions differ

markedly from those of Stuart's problem.

If the string is extensible, then the solutions may have shocks.

One of the many difficulties we face is that there is no clear-cut

doctrine for imposing shock conditions in solid mechanics.

In Section 3, we show that a very pretty fourth-order quasilinear

hyperbolic system describing shearing motions of an elastic layer,

possesses a countable family of standing waves, which we can easily

analyze by global bifurcation theory.

To initiate the study of these hyperbolic equations with an eye

toward finding shock conditions, we examine travelling waves for a

related sixth-order quasilinear parabolic system, which describes the

most general dissipative mechanism within a certain class of such

mechanisms. We have already alluded to the importance of finding

such shock (or entropy)conditions.

2. The Whirling and Drawing of Strings

Let {i,~,k} be a fixed right-handed orthonormal basis for Euclid-

ean 3-space. We ~tudy the steady motion under gravity of the part of

a uniform string between 0 and ak when it is; fed through an inlet

26

at O and withdrawn through an outlet at ak at a constant rate

y > 0 of natural (undeformed) length per unit time while it is

simultaneously whirling about the k-axis with constant angular velocity

~. Let s denote the natural length of the string between the material

point at the inlet and the position of a material point at a given

actual arc length along the string. (s is a spatial or Eulerian

coordinate.) Let ~(s) be the stretch at s, i.e., the local ratio

of deformed to natural length. As a consequence of the steadiness of

motion, the position of the material point at s has the form

Ul(S) [cos mt!+sin ~t~] +u2(s)[-sin wti+cos ~t~] +z(s)k. 2.1)

We set

u = u I + iu 2 . 2.2)

Then

= ~u,12+(z,) 2 23)

We seek motions in which u and z are absolutely continuous (so that

the string has length and has a tangent almost everywhere). Let p A

be the constant mass density per unit natural length of the string,

n(s) be the tension at s, and g be the acceleration of gravity.

Then a careful formulation shows that the motion of the string is

governed by the equations

n(s) u' [p~) y (s - 2i~yu' (s) + e2u(s) = 0, 2.4)

u(0) = 0 = u(£) , 2.5)

[ n(s)X2 1 pAw(s) z' = eg (b-s) , 2.6)

z(0) = 0, z(£) = a, 2.7)

where b is a constant of integration and c = ±i according as

gravity acts in the ±k-direction. Equation (2.4) just holds almost

everywhere in consequence of a bootstrap argument applied to the basic

integral equations of motion with u and z absolutely continuous.

(Note the presence of Coriolis and centrifugal accelerations in (2.4).)

The string is inextensible if

= i. (2.8)

27

In this case we have the boundary value problem (2.4)-(2.8) for u, z,

n. (n is the Lagrange multiplier corresponding to the constraint

(2.8).) The string is elastic if there is a function N on (0,~)

such that

n(s) = N(~ (s)) . (2.9)

In this case the substitution of (2.9) into (2.4)-(2.7) reduces it to

a system for u, z. We require that

N'(v) >0,N(1) = 0,N(~) ÷~ as w ~,N(~) ÷-~ as ~ ÷ 0. (2.10)

It is an easy exercise to show that if X > 0, then the only

solution of the boundary value problems corresponding to the string

lying in a rotating vertical plane is the trivial solution, defined by

u = 0. (In contrast, every steady solution of Stuart's problem must

be planar [14].)

By means of the contraction mapping principle, we can show that the

zeros of a nontrivial u on [0,£]~{b} are simple and isolated. This

fact enables us to define functions v and ~ continuously on

[0,~]\{b} so that

u = ve i~ . (2.11)

v need not have one sign. (Equation (2.11) enables us to factor out

the rotational invariance of our problem.) Let m denote the coeffi-

cient of z' in (2.6). Then (2.11) reduces (2.4), (2.5) to

m~' = ~X, (2.12)

(mv')' + w2(l+x2/m)v = 0 a.e. on [0,Z]\{b}, (2.13)

v(0) = 0 = v(~). (2.14)

We now study inextensible strings under the further requirement

that m be continuous. (This requirement excludes pathological solu-

tions.) For nontrivial solutions, it is necessary that a < ~, which

means that trivial solutions are folded. Equations (2.3), (2.8), (2.6)

y&eld specific representations for z' and m in terms of u' for the

three inclusive cases: (i) z' has one sign on [0,Z] (which must

be positive since a > 0), (ii) b E (0,Z) and z' > 0 on [0,b),

z' < 0 on (b,Z], (iii) b 6 (0,Z) and z' < 0 on [0,b), z' > 0

on (b,Z]. Now let us set w = mv' Then a lengthy computation shows

28

that (2.13) is equivalent to the system

V' = w/m, w' = -~2(l+y2/m)v a.e. on [O,i]\{b}, (2.15)

m(s)

with

q H ~ 1 in case (i) when b ~ Z

-i in case (i) when b s 0

Moreover,

z'(s) = ~g(b-s)/Im 1

so that (2.7) implies that

g [(b-s)/Imllds = qa, 0

= /g21b_ sl2÷ 2 2v(s)2+w(s)2 (2.16)

and in case (ii),

and in case (iii). (2.17)

(2.18)

(2.19)

the solution of which yields b as a functional of v and w. The form

of m shows that m cannot vanish for nontrivial solutions. Hence

nontrivial solutions are classical solutions.

Our intention is to obtain a detailed qualitative picture of all

nontrivial bifurcating solutions sheets in (u,~,y)-space in terms of

the nodal properties of u, which are inherited from those of v

through (2.11). To study the bifurcation process, we linearize (2.15),

(2.16) about a trivial solution (determined from (2.18), (2.19)). This

linearization is equivalent to the Sturm-Liouville problem of

2( g IS--(Z+~]a)/21V')'+~O2[eT]g+y2[S--(Z+T]a)/21--1]V = 0 (2.20)

subject to (2.14). Independent solutions of (2.20) are Bessel functions

of imaginary order. For $(s) H Is-(i+na)/2 I small these functions

behave like sines and cosines of logarithms of 6(s). They are there-

fore bounded, but discontinuous, having an infinite number of zeros.

They accordingly have no nodal properties to bequeath to nontrivial

solution sheets. On the other hand, the nonlinear problem has the

property that if v has a double zero, then v = 0. This implies

that connected sets of nontrivial solutions can change their nodal

properties only at trivial solutions.

To reconcile these facts we consider a family of regularized prob-

lems. We first study the variant of (2.19) obtained by replacing m

with the expression obtained from (2.16) by adding k -2 to the radicand.

29

Here k is a positive integer. Let B k, which depends on v, w, y~,

denote the unique solution of (2.19), which is readily shown to exist.

Substitute this 8 k for b in the modified m, calling the resulting

expression m k. The modification of (2.14), (2.15) obtained by replac-

ing m with m k can easily be converted to integral equations that,

together with their linearizations, involve compact operators. Now

the linearized problem can be shown to possess a countable infinity

of analytic eigencurves G~(k), j = 0,i,... in the (~,X)-plane. To J

each eigencurve there corresponds an eigenfunction v. (k), depending 3

on ~ and ~, having exactly j zeros on (0,Z), each of which is

simple. The global bifurcation theory of [2] implies that bifurcating

from each eigencurve Gj(k) is a connected family Kj(k) of non-

trivial solution pairs (v,(ul,y)) each point of which has Lebesgue

dimension 2. Moreover, on Ks(k) near Gi(k) v has the same nodal

properties as the eigenfunction v (k). Since the only solutions of 3

our modified problems having double zeros are trivial solutions, the

now standard argument of [9] enables us to deduce that everywhere on

K.(k), v has exactly k + 2 zeros on [0,Z], each of which is 3

simple.

To see what happens as k + ~, we can use Sturmian Theory to get

very strong estimates and use connectivity results of [i] or [15] to

show that the exact problem (with k = ~) has a countable family

{K.,j=O,I .... } of connected sets of nontrivial solution pairs. Each 3

point of Kj has Lebesgue dimension at least 2. Everywhere on Kj,

v has exactly k + 2 zeros on [O,Z], each of which is simple.

Each K. bifurcates from the lines w = O, Y = 0 of the (~,X)-plane. 3

(For other applications of such connectivity arguments cf. [3,4].)

A sketch of the bifurcation diagram is shown in Fig. 2.21. For

I llv,wll /~° Kl

2

~2

~2

Fig. 2.21. Schematic illustration of bifurcating sheets for £n =-I.

30

y fixed and positive all nontrivial branches bifurcate from e = 0,

which is the boundary of the continuous spectrum of the linearized

problem. The same remark holds with the roles of y and ~ switched.

For elastic strings, (2.4)-(2.7), (2.9) may admit discontinuous

solutions corresponding to stationary shocks. Away from shocks, the

bifurcation analysis described above can be carried over to elastic

strings. Indeed, elastic strings permit solutions with £ < a, which

behave better than those with ~ > a. Where shocks occur, there is a

choice of shock conditions coming from different dissipative mechanisms

such as viscosity and heat conduction. Such conditions are inconsistent

with each other. Even if a particular shock condition is adopted, it

is not evident that the resulting governing equations can be treated by

the available global bifurcation theories. Moreover, for functions N

that are nowhere affine one can always find a ~ such that the trivial

solution must be discontinuous. 2

In the degenerate case that N(~) = e pAm (for 9 greater

than some positive number) we can solve (2.4)-(2.7) explicitly. We

obtain parabolic eigencurves

kz 2 = ~ ( 2_. 2), ~ 4{ 2 , k = !,±2 .... (2.22)

and corresponding eigenfunctions

k~s. .k~ys. (2.23) u = sin (-~-)exp it ~£ ) .

The behavior of the solutions u is typical of that for all problems~

lul preserves nodal properties, but the number of nodes of u I and

u 2 on any solution sheet change with ~ and y. This nodal structure

for our fourth order system (2.4) is novel.

The results of this section are based on [8].

3. Standing Shear Waves

Let the reference configuration of an incompressible isotropic

nonlinearly elastic body be the infinite layer {(x,y,z): 0 ~ z s 1}. We

study shearing motions of this layer that take the material point

(x,y,z) to the position (x+u(x,t),y+v(y,t),z). Let the density of

any material point in the layer be independent of x and y. We

denote it by p(z). If the prescribed normal pressure on the faces

z = 0,i is independent of x and y, then the equations of motion

for the layer reduce to

31

[~(u2+v2'Z)Uz]zz Z = P(z)utt' (3.1a)

[~ (u2+V2z z' z)v z] z = p (z)vtt, (3. ib)

where ~ is a constitutive function, which we assume to be twice

continuously differentiable and satisfy

2[ Y~ (2y ~(72,z) > 0 and ,z)] > 0 V7 >- 0, z E [0,1],

y~(72,z) ÷ ~ as 7 ÷ ~. (3.2)

Note that (3.1) has an elegant complex form for w = u + iv. On each

face z = 0,i we require that either

u = 0 = v (3.3)

so that the face is fixed, or

u = 0 = v (3.4) z z

so that the shear force is zero.

We seek standing wave solutions of (3.1) of the form

u(z,t) = f(z)cos ~t, v(z,t) = f(z)sin ~t, (3.5)

so that (3.1) reduces to

[~(f2,z)f ] + ~2p(z)f = 0. (3.6) z z z

This equation is subject to the boundary conditions coming from (3.3),

(3.4) that for z = 0,i

either f = 0 or f = 0. (3.7) z

To be specific suppose that

f(0) = 0 = f (i). (3.8) z

Now (3.2) implies that 7~ 7~(72,z) has an inverse ~(.,z) with

~(-h,z) = -~(h,z). We can accordingly convert (3.6),(3,8) to the form

f f(z) ~ (h(~) ,~)d~ h(z) 2 1 = , = p(~)f(~)d~- (3.9a,b)

0 z

32

We seek solution pairs (~2,(f,h)) of (3.9) with f, h 6 C0([0,1]).

Since the integrals in (3,9) define compact operators on [C0([0,1])] 2,

we can apply the global bifurcation theory of [9,10,11]. Supplementing

this with some careful estimates based on comparison and oscillation

obtain: Let {~} be the eigenvalues of the linearization theory we

2 < 2 <...) . Bifurcating from of~(3.6), (3.7) {whence 0 ~ ~0 ~I

(~,(0,0)) are two connected branches of nontrivial solution pairs

(~,±(fk,hk)) of (3.9). Each such branch never intersects any other

nontrivial branch and never returns to the trivial branch. On each

such branch (fk,hk) is unbounded. On the k th branch fk has

exactly k zeros on (0,1) each of which is simple, and ~ ~ 0. If

there is a number M such that ~(72,z) ~ M, then ~ is bounded

above on each branch and if there is a number m ~ 0 such that

m ~ ~(X2,z), then ~ is bounded below on each branch. If ~(X2,z)+~

as X ÷ ~, then ~ is unbounded on each branch; if ~(X2,z) ~ 0 as

~, then ~ ÷ 0 as maxrfkl+ maxlhkl + ~ along each branch. X

Note that the availability of a countable family of standing waves

may be directly attributed to the two degrees of freedom of our system

and to the dependence of ~ on its invariant argument. One-dimensional

analogs of (3.1) do not admit any standing waves. The results described

here are based on [6]. Related problems for semilinear hyperbolic

equations are treated in [13].

4. Travellin@ Viscoelastic Wave

To study the hyperbolic system (3.1) it is useful to embed it in a

system with small dissipation. The shock structure of (3.1) can then

be related to certain travelling waves in the dissipative system (cf.

[12]). The traditional approach would be to modify (3.1) by adding

viscous terms wu to the left side of (3.1a) and vv to the left zzt zzt

side of (3.1b) . Here v denotes a positive constant. This approach

is motivated by gas dynamics, the main source of shock waves, and by

the successful mathematical analyses it has yielded in other settings.

But our material is a solid for which the example of a Newtonian fluid

is irrelevant. We wish to examine how the shock structure is affected

by a whole class of physically reasonable viscous mechanisms.

We assume that an incompressible isotropic homogeneous nonlinearly

viscoelastic material (of differential type with complexity i) occupies

the whole space. If this medium undergoes the shearing motions des-

cribed in Section 3 and if the pressure field at ~ is independent of

x and y, then the motion is governed by

33

[~ (~) Uz+~ (~) Uzt] z = PUtt' (4 .la)

[~ (~) Vz+V (~)Vzt] z = PVtt (4 .ib)

where ~ and ~ are given continuously differentiable constitutive

functions depending on the set of invariants

q - (~0,~l,h2) =- (u2+v2,2 ,u2 +v2 ). (4.2) z z (UzUzt+VzVzt) zt z~

We require that (4.1) be parabolic by requiring that ~R 2 ) (Uzt,Vzt)

(~Uz+~Uzt,~Vz+~Vzt) 6 IR 2 be strictly monotone. We further assume that

~(~0,0,0) > 0. (4.3)

Cf. (3.2).

Now we seek travelling wave solutions of (4.1) of the form u(z,t) =

u(z-ct), v(z,t) = v(z-ct). If we denote derivatives of u and v by

primes and if we set U = u', V = v', then the travelling wave equa-

tions for (4.1) admit integrals of the form

c~U' = (~-c2)U - a, c~V' = (~-c2)V - b (4.4a,b)

where a and b are constants of integration. The rotational invar-

iance of this problem permits us to take b = 0 and a ~ 0 without

loss of generality.

We can readily analyze the second order autonomous system (4.4).

In the nondegenerate case that b = 0 and a > 0, the singular points

of (4.4) lie on the U axis. Their location and type are determined

solely by the elastic response ~(-,0,0) and by a. (The singular

points are generically saddles and proper nodes with axes parallel to

the coordinate axes.) Our basic problem is to determine how the topology

of the phase portrait, and in particular, the nature of heteroclinic

orbits change with changes in the viscous response, i.e., with changes

in ~ and ~. We can get a partial answer to this question in the

most important case for the study of shock structure: If the viscous

effects are small and if for large values of I~ll and n 2, the func-

tions ~ and ~ behave like positive powers of Inll and h 2, then

we can rescale the problem in a way somewhat more complicated than the

traditional approach (described in [12]) and study a limit problem for

vanishing viscosity. In the generic case we find that the loci of

horizontal and vertical tangencies in the phase portraits of the limiting

problem are themselves determined solely by the elastic response. For

34

many problems these loci together with our information on singular

points determine the topology of the phase portrait. To this extent,

at least, the shock structure is independent of the viscous mechanism.

These results are based on [7].

Acknowledgement. The preparation of this paper was supported in part

by a grant from the National Science Foundation.

References

[i] J. C. Alexander, A primer on connectivity, in Proc. Conf. on Fixed Point Theory, 1980, edited by E. Fadell & G. Fournier, Springer Lecture Notes in Math 886, (1981) 455-483.

[2] J. C. Alexander & S. S. Antman, Global behavior of bifurcating multidimensional continua of solutions for multiparameter non- linear eigenvalue problems, Arch. Rational Mech. Anal. 76, (1981) 339-354.

[3] J. C. Alexander, S. S. Antman, & S.-T. Deng, Nonlinear eigenvalue problems for the whirling of heavy elastic strings II: New methods of global bifurcation theory. Proc. Roy. Soc. Edinburgh, 93A, (1983) 197-227.

[4] C. J. Amick & J. T. Toland, On solitary water-waves of finite amplitude, Arch. Rational Mech. Anal. 76, (1981) 9-95.

[5] S. S. Antman, Nonlinear eigenvalue problems for the whirling of heavy elastic strings. Proc. Roy. Soc. Edinburgh, 85A, (1980) 59-85.

[6] S. S. Antman & Guo Zhong-Heng, Large shearing oscillations of incompressible nonlinearly elastic bodies, J. Elasticity, (1984) to appear.

[7] S. S. Antman & R. Malek-Madani, Travelling waves in nonlinearly viscoelastic media, in preparation.

[8] S. S. Antman & M. Reeken, The whirling and drawing of strings, to appear.

[9] M. G. Crandall & P. H. Rabinowitz, Nonlinear Sturm-Liouville problems and topological degree, J. Math. Mech. 19, (1970) 1083- 1102.

[i0] M. G. Crandall & F. H. Rabinowitz, Bifurcation from simple eigen- values, J. Functional Anal. 8, (1971) 321-340.

[ii] P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Functional Anal. 7, (1971) 487-513.

[12] J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer, 1983.

[13] W. A. Strauss, Existence of solitary waves in higher dimensions, Commun. Math. Phys. 55, (1977) 149-162.

[14] C. A. Stuart, Spectral theory of rotating chains, Proc. Roy. Soc. Edinburgh 73A, (1975) 199-214.

[15] G. T. ~yburn, Topological Analysis, Rev. Edn., Princeton Univ. Press, 1964.

CONDITIONS FOR A LIMIT SET TO BE A SINGLETON

Bernd Aulbach •

I. Introduction

The concept of limit sets is fundamental for the qualitative analy-

sis of dynamical systems. In particular, for the study of the asymptotic

behavior of a solution x(t) of an autonomous differential or difference

equation some knowledge of the location of its limit set is essential.

The most desirable result in this context is that the limit set consists

of one point only because in this case x(t) converges to this point. From

basic properties of limit sets (see e.g. Hale [3]) it follows that in or-

der to show that a limit set is a singleton it suffices to find a limit

point Xo, say, with a neighborhood which contains no other limit point

but x O. It is the purpose of this paper to weaken this condition ° In

fact, we prove that under a certain stability assumption on the center

manifold through x ° it is enough to show that near x O there are no limit

points off the center manifold through Xo.

2. The Main Result

We first consider autonomous differential equations

x = f(x) (I)

where f is a C 2- mapping from a neighborhood of x = 0 6 ~n into ~n. We

look for a condition which guarantees that the limit set R of a solution

of (I) consists of a single point only. Obviously, a necessary condition

for this is that ~ contains a stationary solution x ° and that there ex-

ists an ~n_ neighborhood N of x such that ~ N N is contained in a local O

center manifold of system (1) through x o . In this paper we show that,

under an additional stability assumption, this condition is also suffi-

Sponsored by the Volkswagenwerk foundation•

36

cient for ~ to be a singleton. In fact, the additional hypothesis con-

cerns the flow on a center manifold through x ° which particularly im-

plies that this center manifold is uniquely determined (see the lemma

in the appendix).

Theorem I: Suppose that the positive (negative) limit set ~ of a solu-

tion of (I) contains a stationary solution x o which is Ljapunov stable

in both time directions with respect to the flow on the (uniquely de-

termined) center manifold C of system (I) through x o.

Then ~ is a singleton, in fact ~ = {Xo}, if and only if there ex-

ists an ~n_ neighborhood N of x o such that ~ N NcC.

Remarks: 1. Note that the stability assumption of Theorem I concerns

only the flow of system (I) on the center manifold C. As far as the

flow of (I) outside C is concerned we allow the Jacob±an fx(Xo) to have

both eigenvalues with negative and positive real parts.

2. If fx(Xo) has no eigenvalues with positive real parts then Theorem I

can be derived from the well known reduction principle for center mani-

folds (see e.g. Carr [2, Theorem 2]).

3. Simple examples of center manifolds having the stability property of

Theorem I are those which carry only stationary solutions near x o or on-

ly periodic solutions around x . o

Proof: Since the case of negative limit sets is analogous to the case

of positive limit sets and since the necessity of our condition ~ NNcC

is obvious it suffices to consider only the "if- part" of the proof for

positive limit sets. Without loss of generality we may choose a (y,z) -

coordinate system such that x ° corresponds to (y,z) = (0,0) and that the

system (I) gets the form

~ =A ± ± y + r (y,z)

z = A°z + r°(y,z) (2)

where Ai,A ° are matrices with eigenvalues off and on the imaginary axis,

respectively. The functions r ± and r ° are nonlinearities which vanish

together with their first order derivatives at (y,z) = (0,0). Next we al-

ter r ± and r ° outside an ~n_ ball B of radius p with center at 0 such P

that the nonlinearities RZ,R ° of the new system

= A±y + Ri(y,z)

z = A°z + R°(y,z) (3)

37

are globally Lipschitzian and R ± is bounded throughout ~n. By choosing

p small enough we may make the global Lipschitz constants for R ± and R °

arbitrarily small. This means that for a suitable choice ~ of p we may

apply Palmer [4, Theorem 6.1] which says that there exists a homeomor-

phism H of ~n which maps the solutions of (3) onto the solutions of

the partially linearized system

= A±Y (4)

= A°z + R°(s(z),z)

where y= s(z) represents the unique global center manifold of system

(3) through (0,0). Since in our case the center manifold is locally u-

nique (see the lemma in the appendix) the homeomorphism H maps the so-

lutions of system (2) in B5 onto the solutions of the system

A ± = Y (5) = A°z + R°(c(z),z)

in the neighborhood H(Bs)of (0,0) where y = c(z) is the representation

of the unique local center manifold of system (2) through (0,0). In or-

der to utilize this local topological equivalence between the systems

(2) and (5) we consider how the properties of the solution x(t) appear-

ing in Theorem 1 carry over to system (5). It is not clear beforehand,

however, whether or in what sense the transform

(y(t),z(t)) := H(x(t)) (6)

of x(t) under H is a solution of (5). This is because x(t) is allowed

to leave B5 again and again, hence (6) may leave the region H(BH) again

and again. However, the following can be said:

t ~ as ~ with (i) there exists a sequence , t

lim (y(t),z(t )) = (0,0) ,

(ii) the zero solution of

(7)

= A°z + R°(c(z),z) (8)

is stable both as t ~ and t ~-~,

(iii) there exists an n- dimensional ball B a (~H(B~)) with center 0

in ~n and radius ~ such that any positive limit point (y~,z ~) of (6)

in B satisfies

y~ = 0. (9)

As far as the solution property of the function (6) is concerned the

situation is as follows: for each sufficiently small v (w.l.o.g. for

each 9 a 0) there exists a maximal closed interval J~ := [T ,T ] with

38

-~<T <t <T ~ such that

(y(t),z(t)) is a solution of (5)

with (y(t),z(t)) 6 closureB on J v and

II(y(T ),z(T )II =a for all v>0t

II (y(T),z(T )HI = ~ for all ~ ~ 0.

In order to prove Theorem I we have to show that

lim (y(t) ,z (t)) t~

(10)

We distinguish two cases.

(11)

= (0,0). (12)

Case I: There exists some J~ which is unbounded to the right. Without

loss of generality we assume that J~= [0,~). The relation (10) then says

that y(t) is a bounded solution of the linear hyperbolic system y =A±y

on [0, ~) and this implies that

lim y(t) = 0. t~

Next we suppose to the contrary of the assertion (12) that z(t) does not

converge to zero as t ~. This means that there exists a positive E and

a sequence ~ such that lim ~ = ~ and

llz(T ) II Z ~ > 0 for all ~ ~ 0. (13) u

Let 6 = 6(E) be the modulus of stability of the trivial solution of sys-

tem (8). Because of (7) there exists a fixed instance t- such that v

llz(t 5) II < ~.

The stability concept implies then that

i)z(t) il < e for all t ~ t-.

This inequality contradicts (13) and we have proved the validity of

(12) in case I.

Next we show that the alternative to case I cannot occur.

Case 2: Each interval J is bounded to the right. Because of (11) there

exists a convergent subsequence (y(T~) ,z(Tv~)), ~ >~ 0, of (y(T) ,z(Tv))

with limit (y*,z*), say, on the a- sphere around (y,z) = (0,0). The pro-

perty (iii) implies then that

11 z * II = ~ .

Since lim z (Tv~) = z* we may choose an integer ~o such that

llz(Tv~) II >~ z for all u ~ Uo" (14)

8g

Let 6 = ~( ) be the modulus of stability associated with 5" Because of

(7) there exists a fixed ~ such that

JJz(t~) JJ < 6

and the stability of the trivial solution of (8) implies that

o lJz(t) JJ < ~ for all t a t~.

This, however, is a contradiction to (14). Thus, case 2 does not occur

and the proof of Theorem I is complete.

3. Difference Equations

For the sake of completeness we state in this section the discrete

counterpart of Theorem I although it follows easily from Aulbach [I,

Theorem 10.1]. Consider an autonomous system of difference equations

x(k+1) = f(x(k)) (15)

where f is a C3-mapping from a neighborhood of x = 0 6 ~ n into ~n The

lacking connectedness and the possible nonunique backward continuation

of the orbits of the discrete time system (15) cause problems which are

not present in the continuous time case. In order to take care of this

we have to impose additional hypotheses in comparison to Theorem I.

Theorem 2: Suppose that the positive (negative) limit set ~ of a solu-

tion x(k) of (15) contains a fixed point x o of f which is Ljapunov

stable in both time directions with respect to the flow on the (unique-

ly determined) center manifold C of system (15) through x O. Furthermore

suppose that the Jacobian fx(Xo) is nonsingular and that

x(k) is bounded as k~ (k~-~) ,

x(k+1)-x(k) ~0 as k~ (k~-~).

Then ~ is a singleton, in fact ~ = {Xo}, if and only if there exists

an ~n neighborhood N of x o such that ~ n NcC.

4. Appendix: Local Uniqueness of Center Manifolds

In this appendix we prove a simple lemma on the local uniqueness

of center manifolds whose understanding requires only a basic knowledge

of center manifold theory. For this we refer the reader to Carr [2].

40

In Section 2 we have used the fact that the center manifold under

consideration is uniquely determined. Since this uniqueness is not true

in general we show that under the stability assumptions of Section 2 it

in fact is true. Let us consider an autonomous differential equation

x = F(x) , (16)

F 6C2(~n,~ n) , which has a stationary solution X . Then there exists a o

local center manifold C through Xo, i.e. a locally invariant C I -mani-

fold containing X ° whose dimension equals the number (counted with mul-

tiplicity) of purely imaginary eigenvalues of the Jacobian Fx(Xo). Fur-

thermore, C is tangent at X ° to the generalized eigenspace associated

with the purely imaginary eigenvalues of Fx(Xo). If Fx(X o) has no eigen-

values on the imaginary axis we agree that the center manifold through

X ° is simply the point X ° itself.

Although the following result is known to experts we include a

short proof because this result is not explicitly stated in the liter-

ature.

Lemma: Suppose the stationary solution X O of (16) is Ljapunov stable

in both time directions with respect to the flow on a local center mani-

fold C through X o-

Then this center manifold is uniquely determined, i.e. there exists

an ~n_ neighborhood U of X such that there is no center manifold of O

system (16) through X O in U but C.

Proof: After having performed an affine change of coordinates x~ (y,z)

we may assume that X is the (y,z) - coordinate origin and that the sys- o

tem (16) appears in the form

A ± ± = y + r (y,z)

= Ao z + rO(y,z ) (17)

where the matrices A±,A ° have eigenvalues with nonzero and zero real

parts, respectively. Any center manifold of system (17) near (0,0) has

then a local representation of the form y = c(z) with c(0) = 0, Cz(0) = 0

and the flow on this manifold is described by the system

= A°z + r°(c(z),z).

Now we suppose that beside the center manifold y= s(z), say, with the

two- sided stable zero solution there exists another center manifold

41

y = r(z), s(z) ~ r(z), near z = 0. Then there exists a sequence of points

z , z ~ 0 as v ~, such that

s(z ) ~ r(z ) for all ~ > 0. (18)

For sufficiently large ~ the solution (y~(t),z (t)) of (17) starting

at (s(z),z ) for t=0, i.e.

(y~(0) ,Z (0)) = (S(Z) ,z ) , (19)

remains sufficiently close to (0,0) for all t 6 ~ because of the stabi-

lity of the flow on the manifold y=s(z) near (y,z) = (0,0). A basic pro-

perty of center manifolds says then that the solution (y~(t),z (t)) lies

on any center manifold of (17) through (0,0), thus in particular it lies

on y=r(z), i.e.

y~(t) = r(z (t)) for all t 6 ~R.

For t=0 we get with (19) s(z ) =r(z ) for sufficiently large ~ and this

contradicts (18). This completes the proof of the lemma.

5.

[1]

[2]

[3]

[4]

References

B. Aulbach. Continuous and Discrete Dynamics near Manifolds of

Equilibria. Lecture Notes in Mathematics 1058, Springer, Berlin,

Heidelberg, New York, Tokyo 1984.

J. Carr. Applications of Centre Manifold Theory. Springer, Berlin,

Heidelberg, New York, Tokyo 1981.

J.K. Hale. Ordinary Differential Equations. Wiley- Interscience,

New York, London, Sydney, Toronto 1969.

K.J. Palmer. Qualitative Behavior of a System of ODE near an Equi-

librium Point - A Generalization of the Hartman - Grobman Theorem.

Preprint 372, Institut fur Angewandte Mathematik, Universit~t Bonn

1980.

A HYDRODYNAMICAL MODEL OF THE SEA HARE'S PROPULSIVE MECHANISM

J. BELL

Abstract: A sea hare (Aplysia brasiliana) swims via a form of jet pro-

pulsion which involves folding over a pair of parapodia to form a fun-

nel and squeezing water out the posterior end of the funnel. We devel-

op a simple model of this swimming mechanism, then calculate the mean

thrust and instantaneous speed in a particular instance. We also dis-

cuss aspects of its mechanical efficiency and possible model extensions.

I. Introduction

Hydromechanical studies of marine animal locomotion have concen-

trated on analyzing the propulsive mechanisms of fish and birds [4],

and in particular fish undulatory body movement which is a primary mech-

anism of propulsion throughout the marine animal kingdom. But a com-

mon mode of swimming, particularly among mollusca, is propulsion by jet

reaction. This is the mechanism pertinent to swimming sea hares

(Aplysia, both brasiliana and fasciata) though the nature of the mech-

anism is quite different than that employed by squid [3].

Aplysia swimming consists of curling a pair of dorsally position-

ed parapodia inward, one overlapping the other to form a kind of funnel.

This movement starts at the animal's anterior end. As the parapodia

roll inward, the captured water mass is moved caudally, then the para-

podia begin to unroll, again initiated at the anterior end. These

parapodial oscillations seem to be controlled by neuronal oscillators

in the pedal ganglia [5]. A view of the swimming behavior is shown in

[1,3].

In the next two sections we formulate a (one-dimensional) model

of the jet-like propulsion mechanism, then in section 4 we consider a

special case where all calculations can be done without resorting to

numerical simulation. In section 5 we demonstrate the notion of effic-

iency for this model and discuss ways of extending our ideas to more

realistic models.

2. Modellin~ assumptions

As a first approximation we neglect all aspects of swimming be-

haviour except the thrusting aspect, which we consider is that part of

48

the activity cycle after the basic "funnel" has been formed by suffic-

iently overlapping parapodia. We do not consider any aspects of poss-

ible lift from the opening of the parapodia. Thus, the question of

rate of opening of the parapodia and the environmental factors that

contribute to this aspect will be totally neglected. In fact, the

basic shape of the animal is neglected although it would appear to be

important in any calculation of drag, wake vorticity, etc. Thus our

sea hare is represented by a funnel pushing the water mass. We view

the activity cycle as composed of a thrust epoch, of duration T 0, and

a "glide" epoch, of duration T - T O = Tg.

The funnel mechanism is fixed with respect to the coordinate sys-

tem and the water flow is in the positive x-direction. In essense the

"animal" is neutrally buoyant. In actuality the sea hare is slightly

more dense than his watery environment. We also assume that velocities

vary little over cross-sections and so our model is spacially one-

dimensional. We assume smooth flow at the trailing edge, as well as

no significant leading edge suction effects. We also assume the flow

remains attached everywhere to the body. These ideas are part of an

inviscid fluid assumption. For a typical ~ of length L = 20 cm.

and speed U = 10 cm/sec., the Reynolds number is 0(104), so we will neg-

lect viscous effects.

3. Velocity equations

Let u,T,D, and M be, respectively, the instantaneous velocity,

By Newton's second law, thrust, drag, and virtual mass of our animal.

T-D 0 < t < T O (thrust time regime)

(I) ~--£(Su) = -D T O < t < T (glide time regime)

Here M = m + • where m is the actual animal mass and m is the "added"

mass, given by m = kQV where V is the volume of water displaced, p is

the density, and k is a coefficient whose value depends on the body

(funnel) shape. M is really time dependent but, for simplicity, we

assume the fluctuations are small and consider M as being some averaged

virtual mass. Similarly we write T = T + T*(t) but we consider the time

periodic fluctuations away from the average thrust, namely T*(t), to be

small compared to T and will replace T by T in expression (I). Through-

out the paper a bar over a quantity will mean its average over the

thrust regime, that is

T 0

1 I f(t)dt f~o0 0

44

We also express the drag as D ~ pcv~/3u2/2 z K1 u2, where v b is submer-

ged body volume, and c is a constant which depends on body shape.

Siekmann [6] used a similar expression for drag in his study of squid

propulsion. It can be considered an average form drag.

With these assumptions (I) becomes

du [ ~ - K1u2 0 < t < t 0

(2) M ~ = I 2 < t < T -K1u T O

This has the solution

(3) u(t) =

tanh{/ TKI (t+tl)/M}, 0 < t < T O

u(T 0)

(t_t0), T 0 < t < T I + (KI/M) u(T 0)

where t I H (M/2 T~I)I/~[( ~ 1 + u(0))/( JTKI - u(0))],

and u(T 0) = / T/K I' tanh{ T/~I(T0+tl)/M}. In order for this model to

give a continuous u(t) over 0 < t < T, and hence over all t, we must i _ _

have u(T) = u(0) . This requires solution (3) to satisfy

(4) K l ( t - t 0) = M Umin u ~ a

F r o m s w i m m i n g d a t a [ 2 , 5 ] we c a n o b t a i n e s t i m a t e s f o r a l l t h e p a r a m e t -

e r s a p p e a r i n g i n ( 4 ) , w i t h t h e e x c e p t i o n o f c , t h e s h a p e p a r a m e t e r i n -

v o l v e d i n t h e d e f i n i t i o n o f K1, w h i c h m u s t b e d e t e r m i n e d b y a t o w i n g

e x p e r i m e n t .

To c o m p l e t e t h e m o d e l d e s c r i p t i o n o f t h e s e a h a r e ' s v e l o c i t y ,

g i v e n b y ( 3 ) , we m u s t c a l c u l a t e a n e s t i m a t e f o r T .

We do this by considering a funnel geometry where cross-sectional

variation in fluid speeds are neglected so that we have unidirectional

flow. Thus at any location x, 0 < x < L, A(x,t) is the cross-sectional

area at time t. We designate the upstream speed by U, the "jetstream"

speed by U*, and we scale the ambient, hence upstream, pressure to be

zero. Again, external flow perturbations are neglected. With u being

the velocity in the x direction, the conservation of mass equation is

(5) ~A ~ x ?-~ + (Au) = 0.

By balancing net forces in the x direction, the conservation of momen-

tum equation, where p(x,t) is the perturbation pressure, can be writ-

ten in the form

45

~u ~u (6) ~--~+ U~ = - ~X

AS it stands equations (5,6) do not constitute a closed system because

of the pressure, but we need only apply (6) at x = L (the exit) where

it is known and so no further relations need be developed.

Our job is to find u(L,t) = U* since the mean thrust is given by

(7) T = pA(L,t)U*(U*-U)

With A = ~@/~x for some 0 (x,t), substitute this into (5), and integrate

to obtain ;x x A

u :- (80/~t)/(3O/?x), O = a(t) + LA( ,t)dx ,

where a(t) is an arbitrary function. These are then combined to yield

(8) u(L,t) = U* = - (da/dt)/A(L,t).

Define q(t) - (da/dt)/A(L,t), then substituting the appropriate deriv-

atives of u into (6) and evaluating the equation at x = L, we obtain

the following Riccati equation for q:

(9) dq + [~A ] 2 {3A ] I ~p dt ~/A q - ~-~/A q = ~ 3--x '

where all the coefficients are evaluated at x = L. Because p is a smo-

oth perturbation pressure across the exit jet, the right hand side of

(9) vanishes so that

(10) dq + q ~-~/A q = 0.

dt [~'--J x=L x:L

Solving (I@) yields the velocity we need for calculating the mean thrust.

4. pumpin9 tube model

The simplest motion we can consider for our swimming tube (or fun-

nel) model for the sea hare's behavior and the only one not needing num-

erical simulations to determine parameter values is that of the whole

tube simultaneously contracting. That is, let

(11) A = A(t) = A 0 + A I sin ~t, IAII < A 0 .

Substituting (11) into (10), it is easy to see that q, hence U*, is pro-

portional to A, say U* = kA, where k is an integration constant. If we

let U*(0) = U, then k = U/A 0 so that U* = U(I + (AI/A 0) sin mt) . Sub-

stituting this into the expression for thrust, we obtain

T = PAl U2 sin ~t (I + (AI/A 0) sin ~t) 2 The period of motion is 2~/~

46

so we let ~T 0 = ~/2; that is, the thrust epoch lasts just long enough

for one contraction. Then

(12) 2 2 + + 4 2) .

= QAIU {~ (A1/A 0) ~(AI/A 0) ~ K0 U2

SO, for this model (3) becomes

(13) u(t) = I U ~ tanh[U ~1(t+tl)/M], 0 < t 0 T O

u(~ 0) I + (KI/M)u(T 0) (t-T 0 ) ' TO < t <

5. Discussion

So far we have developed one approach to computing mean thrust

and speed of a sea hare by considering a crude approximation to the fun-

nel behavior of the closing parapodia. One application is to determine

the mechanical efficiency as predicted by the model. If the kinetic

energy of a unit mass of water upstream (respectively, downstream) is

U2/2 (respectively, U'2/2), then the unit mass of water receives an en-

ergy increase of (U*2-U2)/2 in passing through the tube. Thus the rate

energy is given by E = AoU0(U*2-U2)/2. In Froude mom- of increase of

entum theory, if one derives E for an actuator disk, A 0 is taken as the

area of the disk and speed U 0 past the disk is taken as (U*+U)/2. With

~0 as in the last section, then, from (11), we obtain

E = AIU3COS ~t(l+~ cos ~t) 2, where ~ E AI/2A 0. If we use the pointed

brackets "<...>" to represent the mean over the whole activity cycle,

then

(14) <E> = 0AIUB~T(~),

where ~T(e) H ~+[(4+3e2)/4wT]sin~T+[~/2wT]sin2~t+[e2/12wT]sin3~t.

We define the system's efficiency by

(15) ~ ~ <T>U/<UT+E> = TU/(TU + <E>)

Substituting (12) and (14) into (15), we obtain ~=K(e)/(K(~)+~T(e)),

where K(~) H 2/~ + 2e + 16~2/3~ = K0/0A I. To compute q, ~ and T must o

be chosen which best fits data and expression (4). Since <E> > E,

then

(16) q < TU/(TU + E) = K(e)/(K(~) + b(e)), with b(e) E 2/~+~+4e2/3~.

For e small, K/(K+b) = °5+0(~). Since the inequality in (16) is very

rough, we suspect this bound on ~ is too large. In actuality the swim-

ming mechanism must be much less than 50% efficient because there is

great energy loss during the glide phase of the activity cycle when the

47

parapodia must extend itself. We have not accounted for this in (15),

of course, and so the right-hand side of (16) should be much larger

than the left-hand side.

A particular inadequacy of the pumping tube model is that it

gives a mean thrust which appears too low. Measuring thrust through

the use of a calibrated swim tank and strain gage experiment, vonder

Porten and Pinsker (private communication) have indicated we should

expect T = 0(2000), but the pumping tube model gives T = 0(700). There-

fore the next approximation is to retain (3), (4) and (10), but replace

(11) with some form of "traveling contraction tube" model, or other mo-

del which accounts for spatially varying contractions. Steven Childress

(private communication) has suggested the form A = ~R 2, where

R(x,t) = R 0 + sin ~t + b(x-x0)sin(~t+~)

This form allows a motion involving pitching and heaving as in models

of bird wings. We have also considered forms such as A(x,t)=f(x)g(s)

and A(x,t) = exp{g(s)}, with s = kx-~t, but all these forms require num-

erical computation to solve (10) to compute T which we haven't completed

yet.

A different approach entirely could be employed to calculate mo-

mentum by considering the strength of the shed vorticity in the wake

of the animal. This would allow us to calculate induced velocities and

power consumption by the animal. Induced power would be calculated as

the limit of the mean rate of increase of wake kinetic energy as time

progresses. Such a model might suggest ways in which the animal could

reduce power consumption per stroke and might indicate why some obser-

vations are made between rate of flapping, amplitude of motion, etc.

Such an approach is tractable if we again consider the animal

during funneling, and consider the posterior edge of the enclosed para-

podia as the "wing disk" which ejects a planar, small-cored vortex ring

per activity cycle. This would assume the period of flapping is long

compared to the time the generated vortex sheet rolls up into a hoop of

concentrated vorticity. If so, the ring then is carried astern on its

own self-convection and on the influence of the total velocity field of

the other rings in the wake. If we consider the flapping period suf-

ficiently long, then it might be reasonable to suppose the ring inter-

actions negligible. In any case, the velocity and energy of ring vor-

tices is a classical problem with known solutions.

Unfortunately we have no experimental evidence suggesting vortex

rings as a mechanism by which wake momentum is generated. The only

evidence experimentally for such a mechanism concerns the flight of

certain birds [4].

48

Acknowledgement: I wish to thank Ken yon der Porten and Harold Pinsker

(U.T. Medical Center, Galveston, Texas), for sending me original data

on swimming behavior and for various discussions on behavioral charact-

eristics of Aplysiarand to Steven Childress (Courant Institute, NYU,

New York) for helpful discussions concerning the modelling.

References

1. A. Bebbington and G.M. Hughes, Locomotion in Aplysia (Gastropoda, Opisthobranchia), Proc. Malac. Soc. Lond. 40 (1973), 399-405.

2. P.V. Hamilton and H.W. Ambrose, Swimming and orientation in Aplysia brasiliana (Mollusca: Gastropoda), Mar. Behav. Physiol. 3 (1875), 131-144.

3. E.R. Kandel, Behavioral Biolo~ of Aplysia, 1979.

4. M.J. Lighthill, Mathematical Biof!uiddynamics, SIAM Regional Conference Series in Applied Mathematics, vol. 17, SIAM, 1975.

5. K. yon der Porten, G. Redmann, B. Rothman and H. Pinsker, Neuro- ethological studies of freely swimming Aplysia brasiliana, J. Exp. Biol. 63 (1979), 1-13.

6. J. Siekmann, On a pulsating jet from the end of a tube with application to the propulsion of certain aquatic animals, J.F.M. 15 (1963), 399-418.

PERIODIC SOLUTIONS OF SOME SECOND ORDER NONLINEAR

DIFFE.RENTIAL EQUATIONS WITH PERIODIC COEFFICIENTS

I. BIHARI

In the present paper we give a more complete discussion of the prob-

lem studied in a previous paper [I] and we add some new results, too.

In the parts (A) and (B) equations of two different types will be dealt

with.

(A) First consider the second order nonlinear "factorized" differen-

tial equation with periodic coefficient

i d p(t+T)=p(t) ~t,T~R + /I/ y''+p(t)f(y)g(y')=0, tcR, dt' " "

In [23 the author has shown that one of the Sturmian comparison theorems

can be extended to equation /I/. Namely if

(i) f(y)E t(R) and f(y) is increasing and f(0)=0,

(ii) f(Y) and g~H, where H is the class of positive functions Y

h=h(u) EC(R) increasing for u<0, decreasing for u>0 and h£Lip(1) for

u£R,

(iii) PieC(R) and P1(t)<P2(t), tcR,

(iv) Yi are solutions of the equations

Yi' '+pi(t) f(yi)g(yi)=0' i=I,2 /2i/

with the initial conditions Yi(to)=Yo->0, yi(to)=Y~>0 then y1>Y2 (t>t O)

as long as Y2 does not vanish. If Y0~0, y~ 0, then y1<Y2 .

Lemma. Under conditions (i)-(ii) and

(iii') f(-y)=-f(y) ,g(-u)=g(u) , y,ueR,

the solutions of the equation

y''+k f(y)g(y')=0, k=const>0 /3/

are periodic with congruent half and quarter waves. - Here half wave means

the part of the graph of y=y(t) between two successive zeros and a quar-

ter wave is the part from a zero to the next/previous extremum.

'>0 which is not a restriction since /3/ is In fact, assume to=Yo=0,y °

autonomous. Let n (t)=-y(-t) (t>0), then y(t)=-n(-t), y' (t)= ~' (-t),

y'' (t)=-~'' (-t). Putting these in /3/ and changing t to -t we have

50

0 2'

t

Fig. I .

n'' (t)-k f(-u[t))g(q'(t))=0

or by (iii')

n'' (t)+k f(~(t)) g(q'(t))=0.

With respect to n (0)=y(0), n' (0)=y'(0) and the uniqueness we have

n(t)~y(t) and - being y oscillatory ~4]- the solutions are periodic

and the consecutive half waves are congruent and so are also the quarter

waves situated on both sides of a zero of y. The same is true concerning

the quarter waves on both sides of an extremum. Namely choose now as the

origin an extremum place. Then q(t)=y(-t) satisfies /3/, consequently

- as above n(t)~y(t) -. Equation /3/ can be integrated by quadratures.

Namely with the notations

• 10 t

I

Fig. 2.

Y u; v d v F(y) =/f (u)du, G(u)=

O O g (v)

C ( u ) _

c~ ] (u)

Fig. 3.

51

equation /3/ assumes the form

dG(~')÷k÷= 0 dt f

whence

c=G(y')+kF(y)=G(y~)+k F(Yo)>0 ' /4/

namely by definition F(y)>0, G(u)a0. - If to=Yo=0 and t I is the first

positive zero of y and ~ the first extremum for t>0, then c=G(y~)+kF(Yo)=

=G(0)+k F(b), b=y(~), Y(tl)=0, whence

e=G(y~)=kF(~) , ~o" '~>0. /5/

! Hence it is obvious that Yo and b determine each other mutually and

uniquely and are increasing/decreasing in the same time. From /4/

y '=G-I (k[F(~)-F(y) ]) , F(~) aF(y) , /6/

where

G_Iu)=]G_I[ (u) , Y'>0

2 (u), y'<0.

By (iii') G21(u)=-G11(u) and /6/ implies

Y du - I :~ t-t y=H (t-t ° ) .

H(y) G -I (k[E(~)-F (u) ]) o' o

The period of y is (provided to:Yo=O

Tk=Tk(Y~)=4H(~)=4I(0,~;k,~)

where

b du

I (a,b;k,~) =~. -I a G (k[F(~)-F(u) ])

171

and

, 1 8 1

t I

I I(0,y;k,~)=t, ts~-,

H(y)= tl I(0,~;k,~)-I(~,y;k,~)=t, ta~-.

Once k is fixed T k depends only on Yo Regard now the solution y of

52

/I/ and at the same time the solution Yi(i=l,2) of the equations

y~'+mif(yi)g(yl)=0, i=1,2 ~ m1=min p>0

[m2=max P

with the initial conditions

/9i/

' - '> (or<0) /10/ to=Yio=Yo=0, Yio-Y ° 0

The period of Yi is Ti(Y~,mi)=Ti=4 I(0,#i;mi,# i) and G(y~)=mi(F# i) i=I,2.

Obviously T 2<T I. - Then we give as the first step the following simple

theorem.

Theorem I. If p(t)Ec(R) and is positive periodic function of period

T and has sy~etrical monotonic half periods (*), then conditions (i)-

-(ii)-(iii') and

(v) TA =lim T~ ' ) < 2T < lim T2(Y~,m 2) I° y~0 1 (yO'ml y~ =T2~

involve the existence of four periodic solutions of /1/ with the period

2T. Two of them, y and 9 start from the maximum and minimum points of

p(t), respectively, finally -y and -9 are also such solutions.

Proof. By the extended Sturm theorem

Yl

T 2 t I T ~I " t

2 2

Fig. 4.

Y2< Y<Y 1 • 0<t-<~ -~2

/11/

Y<Yl ' 0<t-<tl

T 2 T I therefore, if Yo increases from zero to ~ , the interval (7-, 7-) - and

T (*) It is meant by this that if ml=P(T I) , then p(t)(TI<t<TI+~) is in-

T creasing, p(t) (TI+2<t<TI+T) is decreasing and the corresponding T

arcs are symmetrical to the ordinate t=T1+ ~.

53

inside the point t=t I - moving from left to right passes the point t=T ! by (v) precisely once. Thus there is a unique value Yo with the property

t (yi)=T. If we choose the coordinate system so that m1=P(0)=p(T), then I u T

m2=P(~), furthermore y(0)=y(T)=0, moreover y~=y' (0)=-y'(T), thus by the T

symmetry of p(t) to t=~ the graph of y is symmetrical to the same ordi- T

nate t=~. By shifting the arc of y(0~t~T) to right by T and reflecting

it to axis t - i.e. forming -y(t+T) - we get also a solution of /1/ which

together with the former arc is a period of the 2T periodic solution y=

=y(t). - In the same manner a 2T periodic solution 9 can let start from T

the maximum point t=~ of p(t) by choosing the initial slope ~y'~ appro-

priately. The four 2T periodic solutions looked for are y,9,-y,-@, the

ml ~ :

I ^ I

i Y ' ~ y 3T ,

t

Fig. 5.

existence of which was to be proved.

Now we give a complete description of the manifold of the periodic

solutions of /I/. Theorem I and its discussion were only the first step

to put the rather involved situation in its proper light.

Theorem 2. If

(vi) n TI~ T<n T2~ for some neZ, then there are solutions yn,~n (and

-yn,-@n ) of period T with n half waves which half waves cover the inter- T T vals (0,~) and (~,T) in such a way that y(~)=0 and 9(T)=0, respectively,

2 and if

TIo T2~ ~ii) (2n+l)-~- < T < (2n+I)--~-- for some of the values n=0,I,2,.., then

there are solutions nn,~ n (and -nn, -~n ) of period 2T with 2n+I quarter

waves covering the intervals (0,~), and (~,T), respectively, in such a L

way that n'(~)=~' (T)=0 /Fig. 6./. In the intervals where p(t) increases/

/decreases the lengthsof the half and quarter waves and the amplitudes

decrease/increase. (In the sequel property DI.)

54

=] i i 1

n=0 I

' I

, 2

n=0

Fig. 6.

t

Proof. The last assertion is an old result [4] and it must be kept in

mind. If in (vi) n=l then - by the same argument as in proof of Theorem , ( , , T

1 - there is an Yo with t I yo)=~ where again t1(Y~)=t I means the first

positive zero of y. This y(t) and its continuation gives the T periodic

solution in question. Now y~#-y' (), but of course yo-y (T).In this case

(n=l) it is not necessary to assume the monotonicity of the half periods

of p(t).

As to an arbitrary n the whole length z of the sequence of the n T,

first half waves of y is by property DI less than n~(which is the length

of the first n half waves of yl) and by (vi) this is less than T for

small enough,and also by (vi) £ is larger than T Yo ~ for Yo large enough.

Namely,the n-th half wave of y is longer than the n-th half wave of Y2

(~I what can be proved as follows. The function P(t)=G(y')+p(t)F(y) is

increasing in intervals where p(t)increases,since p(t)=P(O)+~F(y)dp(t) o

(see [4]) and being G(u) even and increasing for u>O the slope Jy'Jof

y at the zeros of y increases. Therefore the solution Y2 of (9 2 ) (Vig.7)

! starting from t I with the slope Yo has a shorter first half wave -

the half waves of Y2 are congruent with those of Y2 - than ~2 which

is the solution of the same equation starting from the same point

with the slope y' (tl) (NB. {y' (tl)i>ly' (to) I) and this is shorter

(e) Similar statement holds concerning the corresponding n-th quarter waves of y and Y2"

55

Y2

Fig. 7.

t

than the second half wave of y (starting also from tl), etc. Obviously,

Sturm's theorem was applied several times. - Denote the endpoint of the

above sequence by tln=tln(Y~). Then there are Yo with tl~ and also such T I l T with tI~_7, therefore there is an Yo with tln(Yo)= [. By the symmetry of

p(t) the solution y will have the period T and in details such as de-

scribed in Th.2. .T1o _

In the second case (which corresponds to (vii)) condition (2n+1;-~--<x

ensures that the whole length £ of the sequence consisting of the first T T

2n+I quarter waves (its endpoint t~_(yl)=t~_) is less than E, i.e. t1~ , T ~* u ,~ T

for Yo small enough and T<(2n+1)~ ~-- ensures t1~ for y~ large enough. , T y,

Again there is an Yo with tln=~, i.e. (z)=0 and by reflecting the sec-

tion (0,) of y to the ordinate t=~ (forming y(T-t)) and further reflect-

ing the section thus obtained to the point t=T (forming -y(T+t)) we get

the 2T periodiz solution required by Th.2.

Remark. If TIo< T < Im

(viii) (2no+1) 2 2J2 ~ '

then all the above periodic solutions exist with nsn o. Namely (viii) in-

volves (vi) and (vii) with nsn . o

(B) The case of th 9 half linear equation. This equation has the form

y''+p(t) f(y,y')=0 1121

where

f~C(RXR),f(lu,lV)= ~(u,v),f(0,v)=0,f( u,-v)= f(u,v),¥ ,u,v,

u-f(u,v)>O, U~0

and a similar Sturmian theorem (as in (A)) extends to this equation, too

[5]. First consider also now the equation

56

y''+k f(y,y')=0, k=const > 0.

Then u =y' (y~0) satisfies the Riccati like equation Y

u'+u2+k f(1,u)=0, y~0

/13/

/14/

whence

f dv =t, y~o /15/ u v2+k f(1,v)

namely lim u=+ ~ . Equation /15/ holds in intervals where y@0. Since t++0

lim u =-~ t*tl-O

S du =t I /16/ -~ u2+k f(1,u)

where t I is the first positive zero of y (which satisfies /10/). With

the same procedure as above it can be shown that every solution y of /13/

is periodic of period 2t I and the neighbouring quarter waves are congru-

ent. Regard now the equations

y' '+Ip(t) f (y,y') =0 1171

yi'+Imif(yi,yl)=0, i=1,2 118ii

and their solutions satisfying /10/. Then also now we have /11/ and in-

stead /7/

du Ti=T i (I m i) =2~

u2+Im, f (I ,u) 1

Obviously, T i decreases with increasing Im i and it is independent from

' Furthermore lim T.=0, lim T =~ Now the conditions (vi)-(vii) are YO" l+~ 1 l÷0 i " satisfied for every n, therefore all the above periodic solutions exist

for suitable I -s. However, the part of property DI concerning the ampli-

tudes holds only with some additional assumptions (see [67 p.416). Both

cases (A) and (B) can be thought of as a kind of boundary value problems ! with eigenvalues Yo and I , respectively. - To be sure that conditions

(vi) and (vii) can be satisfied regard the following examples.

Example I. Consider the equation

57

y'' ly'j n-1+p(t)yn=0, un:lul n sgn u, /19/

which was newly thoroughly investigated by A. ELBERT, [3]. In particular

for n=1 /19/ is linear. In the present case f(u,v)=unjv] l-n, f(1,u)=

=Ju~ 1-n. Now f~C(RXR), however /19/ possesses all the (good) properties

of equation /12/, moreover property DI, too. The Riccati like equation

is now

u'+u2+p(t) [ul l-n=0

and its integral for p(t)=k=const

oo

I du t, u2+k luJ 1-n

U

and its period

and

0o oo

Tk=2 u2+kluj 1-n u2+k u 1-n -~ 0

o~

T (I m i) :: 4 ] du

u24 imu 1-n 0 1

The formulae are evenmore simple if we take the modified Riccati like

equation which is fulfilled by the function u=( )n. This reads

n+ I

u'+n u n +n p(t)=0.

If p(t)=k, then

7 I du T :4 du n n+1 :t, k n n+1

u n O n u +k u +k

and

co °I T (1 m i) :n

0

du n+1

n u +Im. 1

° Now T =0,To= Therefore all the above periodic solutions exist and

property DI holds in full extent, too.

Example2. Consider the following factorized differential equation

58

I y' '+p(t)y 2 =0

1+s2y ,

and in the same time the equation

I y''+k y 1+S2y,2-=O, k=const>O.

2 y' 2 2 , ~ ~,2

Here f(y)=y,g(u)= 12 2 ,F(Y)=2~__,G(y,)=f u(1+s u lau= 2 1+~ u o

and by integration we have

y,2 2y,4 I~ 2 I. 2 I ,2 ~y,4

whence by solving the biquadratic equation

e2y,4+2y,2-2k(u2-y2)=0

+e2~44

/20/

/211

we have

, I /_i+/i+2k~2 Y =2 (2_y2).

Hence

1

~_~, 2,~ 2, m o o 1+ /+2ks ~ -u J

and by the substitution y=~ sin z

i ~ cy) ....... 7- ~'--O ~1+~+'2ke~2c°s z dz = t'

+2ks2(~2-u2) du=t ~2-u2

and in turn

! . + +2k~ ~ cos z Tk (Yo) --Tk (~) =~7~- o ~ dz

and

cos z Tm (y~)=Tm. (~i)- ~ I +2mi~2 i 2 2 1 l o

I 2 I ,2+ 21 ,4. (here ~mi~ i =~Yo e 4No ;'

whence

59

TI0_ 4 /~Tr = 2z

which is the period of the solutions of the linear equation arising from

• =~ ' ÷ ~ involves ~i ÷ ~ " Condi- /21/ when c =0, k=m I Now T2~ , since Yo

tions (vi)-(vii) will have the form

~n~, n=I,2,...

T ~I > [(2n+I)~, n=0,I,2 .....

respectively.

REFERENCES

[13 Bihari,I:

[2] Bihari,I:

[3] Elbert,A:

[4] Bihari,I:

[5] Bihari,I:

[6] Bihari,I:

On periodic solutions of certain second order ordinary

differential equations with periodic coefficients

Acta Math.Acad.Sci.Hung. 12 /1961/ 11-16.

Note to an extension of a Sturmian comparison theorem

/to appear/.

A half-linear second order differential equation

Colloquia Math.Soc. J~nos Bolyai, 30 Qual.Theory of Diff.

Equations, Szeged /Hungary/, 1979. pp.153-180.

Oscillation and monotonity theorems concerning non-linear

differential equations of the second order,

Acta Math.Acad. Sci.Hung. 9 /1958/ 83-104.

Ausdehnung d. Sturmschen Oscillations- und Vergleichss~tz~

Magyar Tud.Akad.Mat.Kut. Int.KSzlem~nyei 2 /1959/ 159-173.

On the second order half linear differential equation~

Studia Sci.Math.Hung. 3 /1968/ 411-437 /see p.416/.

WIENER OBSTACLES FOR THE BIHARMONIC OPERATOR

M. Biroli

§.i. Introduction and results .

Recently results on ponintwise continuity of the solution of a second

order elliptic variational inequalities have been given by J. Frehse ,

U. Mosco , (3)(4)(7) ; we observe that in this framework the criterium

on continuity of potentials can be deduced .

A generalization of the above results to the case of solutions of sec-

ond order parabolic variational inequalities has been given by M. Biroli,

U. Mosco , (2)

The aim of this paper is to extend to the case of the biharmonic operator

the above results , this extension has been obtained jointly with U.Mosco.

We introduce the biharmonic capacity of a compact set E with respect to

N an open domain G c R , G~ E , by

caPG(E) = inf { IG I A u I 2 dx ; ue C i (G) , u > Im in a

neighbourhood of E }

We observe that we have defined in such a way a Choquet capacity

Let F denote the fundamental solution of the biharmonie operator

r ( x ) = [2 ( N - 4 ) ( N - 2 ) ~ ] - l I x I 4 - N i f N >4 N

r ( x ) : (4 ~ ) - i l o g ( d [xl -z) i f N=4 N

where ~ = measN_ I ~B ( B is the ball with centre in the origin and N i i

radius 1 ) and d is a constant .

-N p-Ix) Let ~eC (B I) with I ~(x)dx = 1 , ~p(X) =P ~ ( , we define the

regularized fundamental solution Fp as F * P

Let g be a bounded open set in R N and ~: ~ ÷ R U {- ~] a quasi-every-

2 where defined function ; a function u aH (~ ) is a local solution of the

variational inequality relative to the biharmonic operator and to the

obstacle ~ if

(a) u ~ ~ q.e. in

61

(b) I n ~u A([v-u)~)dx ~0 V v oH2( ~ ) , v h ~, ~ c C~( n ) , ~ ~ 0

Consider x ¢ ~ and denote 0

E(Xo, ~ , e ,r) = E( e ,r) = { x e B(r;x O) , @ (x) ~ supB(r/2;Xo)~- e } (e~O)

~( e, r) = caPB(2r;XO ) E( e ,r)

N-2 ,~( e, r) A ( e , r) /~ r

N

where a N = caPB(2;O) B(I;O)

We define the Wiener modulus of the obstacle ~ as

~ (r,R) = inf{ ~ ; ~ 6 { m p ) dp /P ~ 1 ,~ ~0 } r

We consider the case 4~ N~ 7 and we prove the following results :

Th. 1 - Let u be a loca I, solution of the variational inequality rel-

ative to the biharmonic operator and to the obstacle ~ ; we

have

M(r) ~ K{ M(R) ~(r,R) B (R;Xo)~ + ~(r,R)A osc B }

where B ~(0,1) and

M(r)=(fB(r;XO)[ (V2u)2+(v u)21X-Xo 12] F xO dx)½+OSCB(r;XO)U

( ~u (V2u) is the vector of the first (second) derivatives 2 2

of u and (vu) ( ( v2u) ) is the square of its modulus ) .

Let: the assumptions of Th. I hold , then

OSCB(r~xo)U ~ K {(R Y + OSCB(R;xo ) ~ ) ~(r,R) ~ +

+ ~(r,R)A OSCB(R;Xo )

where Ye(0,1) .

From Th.l , Th. 2 the following corollaries can be easily deduced :

Coroll.l.- Suppose

lim ~ (r,R) = 0 , r +0

then u is continuous at x 0 ; moreover if

~(r,R) ~ K (r/R) ~ v E(0,1)

u is H~lder continuous at x 0

Coroll.2.- Let ~ be (H~lder) continuous at Xo, then u is (H~lder)

continuous at x 0

Let E ~E ~ ~ be a borel set ; considering in Coroll.l the carachteristic

function of E as obstacle , we obtain :

62

Coroll. 3. - Let u be the capacitary potential ( relative to the bi -

harmonic capacity ) of E ; if x 0 is an interior point of E ,

u is H~lder continuous at x 0

Consider x 0 ~ DE and suppose

lim I I ~-N ( EC~ r caPB(2r;XO r ~ 0 r ) B(r;XO)

) dr/r=+~

then u is continuous at x , moreover if 0

4-N (EC~ B(r;XO)) ~ c o > 0 r caPB(2r;XO )

u is H~lder continuous at x 0

Remark i. - We observe that for N < 4 a local solution of the obstacle

problem relative to the biharmonic operator and @ is H~ider

continuous being u in H2( ~ )

In §° 2. we give a sketch of the proof of Th. 1 , 2 , where , for sake

of simplicity , we suppose N > 4 .

We observe that the proof uses the same methods as in second order case ,

(2)(3) , and an inequality proved in (5)

§.2.Sketch of the proofs of Th. 1 , 2 .

We recall the following lemma , (5):

Lemma I. - Let ~ R N be a bounded open set and v eH~( ~ ) ~ C ~(n) ,

then for every point p E ~ ( and in the case N=4 for any d

satisfying d ~ diam(supp (v)) ) we have

v(p) 2 + c I [( v2v )2 + ( vv)21p_xl-2 I F p dx

< 2 I Av ~(v F p) dx

where FP(x) = F(x-p)

From the lemma i we have easily :

Lemma 2. - Let ~ be a bounded openset in RNand v eH~( ~ ) , then for

every point p ~ ~( and in the case N-4 for any d satisfying

d ~ diam(supp(v)) + p ) we have

2 )2 rp + ( v v) 2 (Ip-xl-2FP)~a (p)]dx v ~ ~ (p) + c ( p I[ v2v p p

< 2 I ~v A(v r p ) dx .

63

Remark 2. - (a) We observe thar in the lemma 2. we can consider also

the case u ~ C ( £ )

(b) We observe that the assumption 4 ~ N ~ 7 is used in

the proof of the lemma i. , if N=8 the lemma I. does not

hold .

Let B(R ;p) C ~ and q= q(x) such that 0

q eC (B(r/2;p) , 0

Ivn I ~ c r -l , 2

C o n s i d e r v eH ( ~ ) ; we h a v e

(2.1.) A(q2v ) A( n2v r p)

~=I in B(r/4;p) , O<q < 1

-2 Iv2n I -< cr , r _< R

0

Av A n 4 v £P )

then

2 v F p (2.2.) I [A(n2v) A(q2v F p) - Av A (q ) ] dx < p p =

< 8 $ ( Vv Vn ) q ( V( q2 r p ) V v ) d x +

+ I g~v2dx - I v A

P

p 2 gl n v ) dx

f A(u-d) A ( n4(u-d) r p ) dx < 0 P

then , ta~ing into account the lemma I. and (2.3.) ,

( 2 . 4 ) n 4 ( u - d ) 2 e p ( p ) + c I £ (V ( q 2 ( u _ d ) ) ) 2 P • ~ r dx +

P

+ c I£ ( v( n2(u-d)))2(ip-xl -2 P I" ). ~ (p) dx < p s

2 p) < 16 I (V(u-d) Vn) n( V(n r V(u-d) dx + -- ~ p

% ( ~ - d ) 2 d × 2 .ra (u -d ) ~ ( ~ 2 + 2 I£ q (u-d)) dx .

Taking the limit as P-~O , we obtain

2 p n 2 V r ) - p 2 A ( q ) div A ( q go = A r - 2 p) p p

-2 div ( A( q2 Fp ) n qn ) P

p £p 2 r p gl = 4 ( 2 ( qq ) + n Vh V )

P p

( In the following gO' gl are defined as in (2.3) replacing FPp by FP )

Let d _> ~ in B(r/a ;Xo) , frem the variational inequality we have

P P = (4 qVq Vv + v Aq 2 ) ( 2 qv V ( n 2 F pp ) + v A ( q2 F:)_

2 £p - Av [ 4v q (2 ( Vq) 2 + n Vq Vr p )

P P

where

( 2 . 3 . )

64

( 2 . 5 . ) ( u - d ) 2 ( p ) + c I n [ V 2 ( q 2 ( u - d ) ) 2 r p + V( q 2 ( u - d ) ) ) 2 1 p - x i

2 p 16 I q ( u - d ) Vq nV q r V ( u - d ) dx + 2 I go ( u - d ) £

2 - 2 I a (u-d) A(glq (u-d)) dx

we have

(V u) 2 ] p - x l 2 £P]

From (2.5.) , after some computations ,

(2.6.) (u-d)2(p) +I B(r/4;p) [( V 2 u) 2 r p +

-N 2 < C r I (u-d) -- i B(r/4;p)-B(r/8;p)

4-N + r I

B(r/4;p)-B(r/8;p)

dx +

[ ( V2u)2 -2 ( + r Vu)

- 2 r p ] dx

2 dx -

dx

] dx

Suppose now B(r;x O) C ~,XoC £ ; taking the supremum for p~ B(r/8;x O) we

obtain the following result :

Lemma 3. - Let u be a local solution of our variational inequality , the

following relation holds

(u-d) 2 ] dx + + IB(r/8;Xo)[ ( V2u)2r Xo

( Vu)21X_Xoi-2 Xo -N r dx ~C 2 r y (u-d) 2 dx + B(r;xo)-B(r 8;x O)

2 -2 2 [ ( v 2 u ) + r (v u) ] dx

, u ( x ) ~ d ) ~ 4 -1 n (E , r )

- 1 , u ( x ) £ d ) £ 4 A ( ~ , r )

suPB(r/8;x O)

I B(r/8;x 0 )

4-N + r ~(r;Xo)_B(r/8;Xo ) ^

t be now d such that

( x a E ( ~ , r c a P B ( 2 r ; x 0 )

c a P B ( 2 r ; X O ) ( x eE ( E , r

^

a n d c h o o s e d = d +

We i n d i c a t e - 1

Y( e , r ) = c N c a p E(e,r) ~ (B(r;Xo)-B(r/2;Xo))

From the lemma 3. , using the Poincar&'s inequality , (4) , we have

u )2 + ( V2u)2FXo 2 I -2 Xo + (v u ) X-Xol r ( 2 . 7 . ) (OSCB( r /8 ;Xo ) ~ ( r / 8 ; X o ) [

< Y(~ .r) -I I [ ( v2u)2 XO 2 -2 _ , r + (v u) IX-Xol r x ° ] B(2r;Xo)-B(r/8;x O)

Suppose

2 + C e 4

(2.8.) 2C a 2 u) 2 < ( o s c + 4

B(7/8;x 0 )

]d×

dx

65

+ I [ ( v u)arxo+ ( Vu)21X-Xo I-2 £ xO 2

B(~/S;x o )

then , using the hole filling trick , we have as in (i)

(2.9.) ( osc U) 2 + I [( V u)2F xO + ( VU) B(r/S;x 0 B(r/S;x 0 ) 2

(I+C5 Y( ~'r))-i /B (2r;x ) [ ( V2u)2 £ xO 0

(rA ~).

By the integration lemma in (4) , we obtain

(2.10.)

C 6

] dx

21X_XoI-2FxO]dx

x 2 -2 0

+ (v u) Ix-x01 r ] d x

x

( OSCB(r;XO ) u )2 +f B(r;XO)[(V2u)2FXo +(Vu)21X_Xoi-2F O] dx

R 0 x 0 exp(-B 1 Ir Y( ~,0 ) do /0 ) [IB(Ro,Xoj(V. 2 u)2F dx +

rXo 2 I ) ( V u ) 2 1-2 d x ] +C e B(R 0 ;x O X-Xo 7

for 0 <r ~ RO/16 , B(Ro;Xo

• e observe now that

R 0

f Y(~ ,0 r

then we can write (2.10) as

(2.11.) (OSCB(r;XO) u) 2

R o C exp (-g I 10 2 r

R dP /0 > C I 06 (e,p) dO /P C

8 r 9

+ I [(V2u)2 Xo Xo B(r;x ) F +(V u)21X-XoI-2r ] dx

0

6(e,p) dP/P )[ /B(Ro;Xo) (V2u)2 F xO

)2 iX_XoI-2 2 IB(Ro;Xo) ( Vu r xO dx] + Cye

Choosing E = ~(r,R )+n 0

,q >0, and taking into account that

r/R 0 ~ w(r,R 0 ) < r/R 0 V osc B R )~ -- ( o;Xo

dx +

we obtain , for n ÷0 , the result of Th.l. ; the result of Th.2 follows,

as in ( 7) , by an iteration method .

§.3. Generalizations and applications .

We will observe , at first , that the result can be generalized to more

general fourth order operators ; in particular we can consider operators

66

of the type

D 2 2 a D ij x x

1 j

where a.. is a constant coercive matrix. zj

Consider now the following linear stochastic equation

(3.1.) dy = Ay dt + o dw , y(O) = x t

N where A and are constant matrices NxN , ye R and w

t

normalized Wiener process ; we denote

( 3 .2 . ) J ( e ) f A t x = 0 f(y(t))exp(ct) dt + ×(y( t))e< ~

is a N-dimensional

exp(c t ) (cs R)

where T is the first exit instant of y(t) relative to the open set

N Q R , f ~ are given functions and f is convex , X8< ~ is the charac-

teristic function of the se~ { e<~ }

We seek for the minimum of the functional J ( e )+4 j ( e ) ; x x x

denote by u(x) the value of J corresponding to the minimum , by the x

same methods used in (I) we can prove that , formally , u(x) is the sol-

ution of a fourth order elliptic obstacle problem relative to @ and

our result gives informations on the continuity of u(x)

We observe that , roughly speaking the stochastic control problem , con-

sidered above , corresponds to minimize with respect to the cost J and x

to its rate of change in x ; this problem can be interesting in economics

and in medicine

REFERENCES

(I) A. Bensoussan , J.L. Lions - Applications des in@quations variation-

nelles en eontrSle stoehastique - Dunod , 1978

(2) M. Biroli , U. Moseo - Wiener obstacles : the parabolic case -

submitted to Indians Math. J.

(3) J. Frehse , U. Mosco - Irregular obstacles and quasi-variational

inequalities of the stochastic impulse control - Ann. Sc. Norm.

Sup. Pisa IV ,IX ,i, 1982 , 105-157 .

(4) J. Frehse , U. Mosco - Wiener obstacles - Coll@ge de France Sem-

(5

(6

(7

67

tv

inar on nonlinear partial differential equations , Pitman

1984.

W. Mazja - Einbettungss~tze fur Sobolewsche R~ume - Teubner, 1979.

W. Mazja - On the behavior near the boundary of solutions of the

Diriehlet problem for the biharmonic operator - Soviet Math.

Dokl. ,18, 4 , 1977 , 1152-1155 .

U. Mosco - Lecture at the " Summer Institute on Nonlinear Functional

Analysis and Applications " A.M.S. , July , 1983 .

Marco Biroli

Dipartimento di Matematica

Politecnico di Milano

Via Bonardi ,9

20133 MILANO ( Italy )

INVERSE METHODS FOR REFLECTOR IMAGING

Norman Bleistein

I n v e r s e p r o b l e m s i n t he p h y s i c a l and b i o l o g i c a l s c i e n c e s a r e a c u r r e n t s o u r c e

of ongoing research in applied mathematics and its related disciplines.

Mathematically, the objective of inverse methods may be viewed as the determination

of one or more parameters in the governing equation or system of equations of some

process. A closely related problem is the determination of the size or shape of a

scattering domain, which may be viewed as a domain with parameters which are

different from those in the host medium.

For a number of years, I and my associates have been engaged in a coordinated

research program in inverse problems, primarily with application to seismic and

ocean acoustic problems. However, our methods also have application to

nondestructive testing and other areas in which the objective is to image a

scattering domain from backscattered (or nearly backscattered) data produced from

controlled experiments. We model the unknown environment as an acoustic medium

c h a r a c t e r i z e d by v a r i a t i o n s i n t he s o u n d s p e e d .

Fundamental to our approach is that we formulate mathematical problems for the

determination of the shape functions which mathematically image the boundary of the

scattering domain(s). These boundaries are the reflectors in the domain which

produce the asymptotically dominant portion of the backscattered energy at high

frequency. Through the use of asymptotic analysis we obtain asymptotic solutions of

these equations, sufficiently accurate to image the reflectors.

Delineation of a shape requires distinction between those points which are on a

surface and those which are not. In one dimension, this would require simply the

identification of the boundary points of an interval. A point is well defined by a

Dirac delta function whose support (singularity) is at that point. Thus, in

simplest form, shape discrimination requires identification of the support of a

delta function.

In the next section, we describe an extension of this idea to higher

dimensions. We show how -- asymptotically, st least -- we can relate Fourier data

for a function characterizing a domain to Fourier data for a function characterizing

the boundary of the domain. The former function is the ~h~$~[!~$i~ [~$!9~ of

the domain and the latter we call the ~!~g~l~ ~$!9~ of a surface. It is this

latter function which is the basis of our imaging methods.

69

I n s e i s m i c e x p l o r a t i o n , the dominan t method p r e s e n t l y employed to image the

s u b s u r f a c e f r o m n e a r b a c k s c a t t e r e d d a t a i s c a l l e d wave e g u a t ~ o n ~!g~i~

[ C l a e r b o u t , 1971, S c h n e i d e r , 1978, S t o l t , 1978]. Th i s method i s based on the p r e m i s e

t h a t t h e e n s e m b l e o f b a c k s c a t t e r e d s i g n a l s i s i t s e l f a wave w h o s e p r o p a g a t i o n i s

governed by the wave e q u a t i o n w i t h ha lved p r o p a g a t i o n speed. The o b s e r v a t i o n s on the

upper s u r f a c e a re assumed to be the r e s p o n s e to a s i g n a l i n i t i a t e d a t the r e f l e c t o r s

a t t ime equa l to ze ro . One then images the s u b s u r f a c e by u s i n g the wave e q u a t i o n to

p r o p a g a t e t h e o b s e r v a t i o n s b a c k w a r d s i n t i m e and d o w n w a r d s i n s p a c e , s u b j e c t t o

Cauchy d a t a - - the o b s e r v a t i o n s a t the upper s u r f a c e . The method does n e t p r o v i d e a

means of q u a n t i f y i n g t]~e r e l a t i o n s h i p b e t w e e n the downward p r o p a g a t e d e n s e m b l e - w a v e

and t h e r e f l e c t i o n s t r e n g t h on t h e i n t e r f a c e s ,

On the o t h e r hand, ou r i n v e r s i o n method employs p e r t u r b a t i o n t h e o r y to deduce

a l i n e a r i n t e g r a l e q u a t i o n f o r t h e u n k n o w n p e r t u r b a t i o n o f t h e v e l o c i t y p r o f i l e .

Thus, the r e l a t i o n s h i p b e t w e e n v e l o c i t y changes in the s u b s u r f a c e and the o u t p u t of

t h e m e t h o d i s e x p l i c i t . The r e l a t i o n s h i p b e t w e e n m i g r a t i o n and i n v e r s i o n i s

d i s c u s s e d i n a n u m b e r of p a p e r s . See , f o r e x a m p l e , Coen [ 1 9 8 4 ] , B l e i s t e i n [ 1 9 8 4 ] ,

Mager [1983] and B l e i s t e i n and Cohen, [1982] .

Mathematical imaging of a reflecting surface is facilitated by introducing a

distribution whose support is on the reflecting surface. This function is called

the ~!~g~!~ ~ of the surface. It is a Diruc delta function of normal dis-

tance from the r e f l e c t i n g s u r f a c e . Thus, f o r each p o i n t on the s u r f a c e S, l e t o

deno t e arc l e n g t h a long a cu rve from S which i s normal to S. Then, the s i n g u l a r

f u n c t i o n may be deno ted by 5 ( o ) , We decompose a s u r f a c e w i t h c o r n e r s , say a

" p i e c e w i s e smooth" s u r f a c e , i n t o i t s smooth c o n s t i t u e n t s and d e f i n e ¥(x) s e p a r a t e l y

f o r each of t h o s e . We s h a l l assume t h a t the s u r f a c e of i n t e r e s t i s f i n i t e , b u t no t

n e c e s s a r i l y c l o s e d .

A more r i g o r o u s d ~ f i n i t i o n of the s i n g u l a r f u n c t i o n i s p r o v i d e d by d e f i n i n g i t s

e f f e c t under i n t e g r a t i D n wi th t e s t f u n c t i o n s . We s h a l l b e g i n by d e f i n i n g a g a i n the

s i n g u l a r f u n c t i o n , ¥ ( x ) , x = ( x ~ , x z , x a ) ' of a smooth s u r f a c e , S. T h e r e f o r e , l e t us

i n t r o d u c e the c l a s s of " i n f i n t e l y d i f f e r e n t i a b l e " f u n c t i o n s which v a n i s h o u t s i d e a

f i n i t e domain. For f(~i) be ing any such f u n c t i o n , we r e q u i r e t h a t

~_ f ( ~ ) ¥ ( ~ ) d V = I s f ( ~ ) d S . (1)

That i s , the e f f e c t of the s i n g u l a r f u n c t i o n on a volume i n t e g r a l i s to reduce i t to

a s u r f a c e i n t e g r a l ove r the s u p p o r t s u r f a c e of the s i n g u l a r f u n c t i o n .

70

Typically, in the imaging problems of interest, remote observations yield

information about the Fourier transform, 7(~), of the function 7(~)- That Fourier

transform is given by

7(k) = j" 7(_x)e- i -k ' -xdV

= ~ e - i k ' X d s , k = ( k l , k z , k 3 ) , S

In t h i s e q u a t i o n , we have o b t a i n e d t h e s e c o n d l i n e by e x p l o i t i n g ( 1 ) .

(2)

U s u a l l y , one can d e t e r m i n e d t h i s t r a n s f o r m in some l i m i t e d a p e r t u r e i n t h e k -

p l a n e . T h a t a p e r t u r e can be c h a r a c t e r i z e d by k, t h e m a g n i t u d e o f t h e k - v e c t o r ,

b e i n g l a r g e and t h e a n g u l a r r a n g e of t h e k - v e c t o r b e i n g l i m i t e d to l e s s t h a n t h e

u n i t s p h e r e . Of c o u r s e , by l a r g e k, we mean t h a t a t y p i c a l w a v e l e n g t h , 2 n / k , i n t h e

a p e r t u r e i s much s m a l l e r t h a n t h e t y p i c a l l e n g t h s c a l e s , i n p a r t i c u l a r t h e p r i n c i p a l

r a d i i o f c u r v a t u r e of t h e r e f l e c t o r , S.

h c l o s e l y r e l a t e d f u n c t i o n of i n t e r e s t i s c a l l e d t h e 9 h ~ [ ~ 9 ~ i ~ $ i ~ ~ £ $ ! ~ o f

a d o m a i n , D. T h i s f u n c t i o n , ~ ( x ) i s d e f i n e d by

P ( ~ ) =

w i t h F o u r i e r t r a n s f o r m g i v e n by

1 , x i n O ,

O, x_ n o t i n D

(3)

~(k)_ = i - r ( x ) e - _ i-k'-XdV

= ~ e - i-k'-XdV.

(4)

Let us now d e f i n e t h e s u r f a c e S to be t h e b o u n d a r y of t h e domain D and i n t r O -

duce 7(~) a s t h e s i n g u l a r f u n c t i o n o f t h i s b o u n d a r y s u r f a c e . One can show t h a t

[ B l e i s t e i n , 1984] a t h i g h f r e q u e n c y t h e a s y m p t o t i c e x p a n s i o n s o f t h e s e two f u n c t i o n s

a r e d o m i n a t e d by c o n t r i b u t i o n s a t s t a t i o n a r y p o i n t s and t h a t t h e s e c o n t r i b u t i o n s a r e

r e l a t e d by

71

~x(k) ~ ~ikrx(k). (5)

In this equation, the subscript x denotes a single stationary point contribution to

t h e F o u r i e r t r a n s f o r m . The s t a t i o n a r y p o i n t s a r e t h o s e p o i n t s on S a t which t h e

normal t o S and k a r e e i t h e r c o l i n e a r ( - ) o r (+) a n t i - c o l i n e a r . Tha t i s , t h e l e a d -

i ng o r d e r a s y m p t o t i c e x p a n s i o n of t h e e l e m e n t s of t h e F o u r i e r t r a n s f o r m of y ( k ) a r e

o b t a i n e d from the same t e r m s in ~ ( k ) by m u l t i p l i c a t i o n by ~ i k .

In some s e n s e , t h i s r e s u l t i s an e x t e n s i o n - - a s y m p t o t i c a l l y - - of t h e

c o r r e s p o n d e n c e b e t w e e n d i f f e r e n t i a t i o n i n t h e s p a t i a l domain and m u l t i p l i c a t i o n in

t h e F o u r i e r domain . However , i t i s now a p p l i e d t o a d i r e c t i o n a l d e r i v a t i v e i n a

d i r e c t i o n wh ich i s i t s e l f a f u n c t i o n of x.

~"nis r e s u l t has i m p o r t a n t i m p l i c a t i o n s as r e g a r d s i m a g i n g of domains from a p e r -

t u r e l i m i t e d F o u r i e r d~l ta . I t a l l o w s us t o e x t e n d to h i g h e r d i m e n s i o n s t h e o b s e r v a -

t i o n d i s o u s s e d i n t he i n t r o d u c t i o n f o r t h e d e l i n e a t i o n of t h e b o u n d a r y of an i n t e r -

v a l from b a n d l i m i t e d F o u r i e r d a t a . Tha t i s , g i v e n a b a n d l i m i t e d F o u r i e r t r a n s f o r m

of a c h a r a c t e r i s t i c f u n c t i o n , we need o n l y m u l t i p l y by an a p p r o p r i a t e s c a l e to

o b t a i n t he b a n d l i m i t e d F o u r i e r t r a n s f o r m of t he s i n g u l a r f u n c t i o n of i t s b o u n d a r y .

I n v e r s i o n of t h i s d a t a i m a g e s t h e b o u n d a r y of t h e domain of s u p p o r t of t h e

c h a r a c t e r i s t i c f u n c t i o n .

Le t us d e f i n e ~B(!~) as the a p e r t u r e l i m i t e d F o u r i e r t r a n s f o r m of 7(~) which i s

t a k e n t o be z e r o o u t s i d e t he a p e r t u r e i n wh ich v a l u e s of 7(~) a r e known. The

F o u r i e r i n v e r s i o n of t h i s f u n c t i o n y i e l d s a b a n d l i m i t e d v e r s i o n , 7B(x) , of t h e

s i n g u l a r f u n c t i o n . One can show t h a t t h i s i n v e r s i o n y i e l d s a b a n d l i m i t e d D i r a c

d e l t a f u n c t i o n of no rma l d i s t a n c e t o t h e s u r f a c e i n t h o s e p a r t s of t he s u r f a c e where

the s u r f a c e normal p o i n t s i n a d i r e c t i o n which i s i n t he a n g u l a r a p e r t u r e i n which

~B(k) i s n o n z e r o . For n o r m a l s o u t s i d e of t h i s a p e r t u r e , t h e i n v e r s e t r a n s f o r m

d e c a y s r a p i d l y t o z e r o w i t h i n c r e a s i n g d e v i a t i o n of t h e a n g l e of t h e no rma l from t h e

r a n g e i n t h e a p e r t u r e . See Mager and B l e i s t e i n [ 1 9 7 5 ] , Cohen and B l c i s t e i n [1979a]

and B l e i s t e i n [ 1 9 8 4 ] .

We s h a l l a p p l y t he t h e o r y of the p r e v i o u s s e c t i o n t o an i d e a l i z a t i o n of t h e

s e i s m i c e x p l o r a t i o n e x p e r i m e n t i n wh ich we assume t h a t we know t h e b a c k s c a t t e r

r e s p o n s e from a c o a s t i c p o i n t s o u r c e s s e t o f f a t e v e r y p o i n t ,

x l = ~x, x~ = ~2, x3 = O, on t he s u r f a c e of t he e a r t h . We assume t h a t t he t o t a l

f i e l d , u ( x ; ~ ) , i s a s o l u t i o n of t h e H e l m h o l t z e q u a t i o n , w i t h p o i n t s o u r c e a t

= (~x , ~ s , 0 ) . We i r~ t roduce a r e f e r e n c e v e l o c i t y , c o , and a R ~ ~ , a ( x ) ,

d e f i n e d by t h e e q u a t i o n ,

72

-I- = -~- [1 + a ( x ) ] . V ~ C03

We now d e c o m p o s e t h e t o t a l f i e l d i n t o an i n c i d e n t and s c a t t e r e d f i e l d ,

(6)

u(x;~) = ui(~; ~;~) + Us(~; ~; ~), (7)

in which u I ( x ; ~ ; ~ ) i s the r e s p o n s e to the s o u r c e in the u n p e r t u r b e d medium and

uS(x; ~; ~) i s the s c a t t e r e d f i e l d which i s the r e s p o n s e to the p e r t u r b a t i o n a ( x ) .

Under the a s s u m p t i o n t h a t a (~) i s s m a l l , we use p e r t u r b a t i o n t h e o r y to o b t a i n

the f o l l o w i n g i n t e g r a l e q u a t i o n to d e t e r m i n e a ( x ) :

Us(L; ~; ~o)= o~ " ~ J a(x%=._ ui~( - x; .~; ~) dr. (8)

This i s a l i n e a r i n t e g r a l e q u a t i o n r e l a t i n g the s u r f a c e v a l u e s of the b a c k s c a t t e r e d

f i e l d , US(L; ~; w), and a ( x ) , which i s now the on ly unknown in the e q u a t i o n . Th i s i s

a ~ h ~ ~ g ~ ~ g ~ ~f ~h~ ~ ! ~ ~ ! ~ f o r a ( x ) . When the k e r n e l of such an

i n t e g r a l e q u a t i o n - - i n t h i s c a s e , u i 2 ( x ; ~; ~ ) , - - i s such t h a t i t s modulus has a

bounded s q u a r e i n t e g r a l in a l l of i t s v a r i a b l e s , ~, ~, ~, t hen i t i s known t h a t the

s o l u t i o n to t h i s type of i n t e g r a l e q u a t i o n i s i ~ - R ~ i ~ i ~ , w i t h e i g e n v a l u e s t h a t

have a l i m i t p o i n t a t z e r o . However, t h i s k e r n e l i s no t s q u a r e i n t e g r a b l e in a l l of

i t s v a r i a b l e s and t h a t t h e o r y does no t a p p l y . I n d e e d , a p r o t o t y p i c a l one d imen-

s i o n a l ana log of (8) has as i t s k e r n e l the s q u a r e of the F o u r i e r k e r n e l , e x p { 2 i ~ z / c }

which i s known to have a l l of i t s (complex) e i g e n v a l u e s on a c i r c l e of n o n z e r o

r a d i u s . I f we t h i n k of the k e r n e l as a lways be ing the ray method g e n e r a l i z a t i o n of

t h i s o n e - d i m e n s i o n a l k e r n e l , then i t i s r e a s o n a b l e to e x p e c t t h a t the k e r n e l n e v e r

has a bounded squa re i n t e g r a l in a l l of i t s v a r i a b l e s .

For an a r b i t r a r y r e f e r e n c e speed , we canno t s o l v e (8) in c l o s e d form, a n a l y t i -

c a l l y . T h e r e f o r e , we f u r t h e r s p e c i a l i z e t h i s r e s u l t to the case in which c o i s a

c o n s t a n t . In t h i s e a s e , (8) t a k e s the form

2i~ lx-~ l /c = ~z Sa(x) a----= ...... de.

n s ( ~ ; ~; ~) ( 4 ~ ) ~ - l ~ - ~ 1 ~ (9)

We solve (9) for a(x) by Fourier methods. The result is

a(k) = - 8 ~ i c o 3 k s O ( k x " k2 ; ~ ) , k = ( k l , kz ' k s ) ,

I n t h i s e q u a t i o n ,

(~o)

73

_ r u c ( ~ ; ~ ; ~ ) ~ o) . . . . ] . (11,

w i t h O ( k l , k 2 ; ~) i t s t r a n s v e r s e F o u r i e r t r a n s f o r m w i t h r e s p e c t t o t h e s p a t i a l c o o r -

d i n a t e s . The t h i r d c o m p o n e n t o f t h e w a v e v e c t o r , k s , a p p e a r i n g i n ( 1 6 ) i s r e l a t e d

t o t h e t r a n s f o r m v a r i a b l e s , (k 1 , k z , o J ) , b y

k 3 =

s i g n t o ¢ ~ ; 7 ; : 2 - - - k : ; - - - - k : ~ , ~ ' / c o z _) k, 2 + k2",

i / k : ; - : - k : ; - - - : ; / c : : , ~ Z / c o ' < k 1' + k , ' .

(12)

F o u r i e r i n v e r s i o n o f ( I 0 ) o v e r r e a l v a l u e s o f k y i e l d s a ( x ) . H o w e v e r , f o r

f i e l d d a t a , t h e s o l u t i o n i s n o t q u i t e so s t r a i g h t f o r w a r d . A t y p i c a l f r e q u e n c y r a n g e

f o r f i e l d d a t a m i g h t b e 8 - 4 0 h z ; a t y p i c a l s o u n d s p e e d , 1 5 0 0 - 6 0 0 0 m / s ; a t y p i c a l

l e n g t h s c a l e t o t h e f i r s t s i g n i f i c a n t v a r i a t i o n s i n a ( x ) , 500m. F o r t h e s e v a l u e s ,

we o b t a i n a d i m e n s i o n l e s s p a r a m e t e r ,

4~fo H = - . . . . ~ 9 ( 1 3 )

c

a t m in imum f r e q u e n c y a n d maximum s o u n d s p e e d , l a r g e e n o u g h f o r a s y m p t o t i c s .

L e t u s a s s u m e , t h e r e f o r e , t h a t v ( x ) i s a p i e c e w i s e c o n s t a n t f u n c t i o n , so t h a t

t h e same i s t r u e f o r a ( x ) . I n t h i s c a s e , a ( x ) i s a sum o f c h a r a c t e r i s t i c f u n c t i o n s ,

e a c h n o n z e r o on some d o m a i n o f c o n s t a n t v ( ~ ) , a n d e a c h w e i g h t e d b y t h e c o n s t a n t

v a l u e o f a ( ~ ) o n t h a t d o m a i n . We t h e n a p p l y t h e m e t h o d o f t h e p r e v i o u s s e c t i o n t o

f i n d t h e b o u n d a r i e s b e t w e e n t h e r e g i o n s o f c o n s t a n t a ( ~ ) - - t h e r e f l e c t o r s i n t h e

s u b s u r f a c e . One c a n show f o r t h i s c a s e t h a t t h e a p p r o p r i a t e c h o i c e o f s i g n i n ( 5 )

i s s i g n ~ . T h u s , we d e f i n e

~(k) = 2ik (signk,) u(k) = ~!~u(k). (14) c o

The i n v e r s e t r a n s f o r m o f ~ ( k ) i s t h e n a n a r r a y o f s i n g u l a r f u n c t i o n s , e a c h w i t h s u p -

p o r t on a n i n t e r f a c e o f t h e p i e c e w i s e c o n s t a n t f u n c t i o n , a ( x ) . The w e i g h t i n g o f t h e

s i n g u l a r f u n c t i o n on e a c h i n t e r f a c e i s t h e l i n e a r i z e d u p w a r d j u m p i n a ( x ) a c r o s s t h e

i n t e r f a c e i n t h e d i r e c t i o n o f i n c r e a s i n g x 3 . I f we d e f i n e v+ t o be t h e v a l u e s o f

v ( x ) b e l o w ( a b o v e ) a n i n t e r f a c e a n d d e f i n e a+ c o r r e s p o n d i n g l y , t h e n o n e c a n show

t h a t

V+-V a - a + ~ 2 . . . . . , ( 1 5 )

c o

74

to leading order asymptotically. We note that to linear order in the perturbation,

this jump is just four times the normal reflection coefficient. The singular func-

tion theory predicts that the peak value of the band limited inversion is related to

the amplitude of the jump in a(g) with a known scale, namely the length of the

bandwidth divided by ~. More generally, if we use a filter in the frequency domain

other than just a characteristic function of the bandwidth, then the length of the

bandwidth is replaced by the area under the filter.

In summary t h e n , t he l o c a t i o n of t he peak of the b a n d l i m i t e d i n v e r s i o n of t he

singular function locates the reflector and the amplitude at the peak provides a

means of estimating the normal reflection coefficient.

One f u r t h e r m o d i f i c a t i o n of t h e s o l u t i o n i s n e c e s s a r y as a r e s u l t of t h e con-

s t r a i n t s of t he r e a l s e i s m i c e x p e r i m e n t , Most d a t a i s g a t h e r e d o v e r a l i n e a r a r r a y

on the surface of the earth rather than over an areal array. Consequently, it is

not possible to obtain a solution fo~ a function a(x) of three variables. The way

we account for this in our model is to assume that we have an areal array of data

but that the data is independent of ~2. This result in which we allow three dimen-

sional propagation over a cylindrical earth is known as the ~R-g~-R~-~!£ ~!~:

~!R~! case. We further simplify our solution by exploiting high frequency to

retain only the leading order asymptotic expansion of the Fourier inversion of ~(k).

This amounts to retaining only the leading order contribution in (11) and in calcu-

lating the integral in transverse wave number by the method of stationary phase.

This leads to the following asymptotic formula for the bandlimited array of scaled

singular functions of the reflectors in the two-and-one-half dimensional case:

16y3 d~1 -2i~p/c o +i (signe) ~/4

• i t U S ( ~ ; ~ I ; t ) d t e i ~ t , 9 2 = ( Y I - ~ I ) 2 + Y~.

(16)

I t i s t h i s f o r m u l a wh ich we use t o p r o c e s s a l i n e of s e i s m i c d a t a . The

i n t e g r a l i n t he t ime domain i s c a l c u l a t e d by u s i n g t he f a s t F o u r i e r t r a n s f o r m (FFT)

f o r each t r a c e . The d a t a i s t h e n m u l t i p l i e d by t he i n d i c a t e d s c a l e s i n t h e f r e -

quency domain . The d a t a i s a l s o m u l t i p l i e d by a f i l t e r , h e r e , t o d i s c a r d t h a t p a r t

of the d a t a f o r which the s i g n a l to n o i s e r a t i o f a l l s be low some t h r e s h h o l d . T y p i -

c a l l y , we use a f i l t e r which t a p e r s a t i t s b o u n d a r i e s i n o r d e r t o p r o d u c e a s m o o t h e r

o u t p u t ,

75

The i n v e r s e F o u r i e r t r a n s f o r m i s a l s o computed by u s i n g an FFT r o u t i n e . Th i s

p r o v i d e s d a t a f o r the ~I i n t e g r a l a t a u n i f o r m g r i d o f t ime p o i n t s . I n o r d e r to

d e t e r m i n e the o u t p u t a t the t ime 2 p / c o , we use q u a d r a t i c i n t e r p o l a t i o n . We c a l c u -

l a t e the s p a t i a l i n t e g r a l p o i n t w i s e f o r esch o u t p u t p o i n t , ( Y x , Y s ) . The domain of

i n t e g r a t i o n in ~x i s s u b j e c t e d to t h r e e c o n s t r a i n t s . F i r s t l y , we canno t i n t e g r a t e

beyond the r ange of the g i v e n t r a c e s of d a t a . S e c o n d l y , 2 p / c o must be l e s s t han the

maximum t ime a v a i l a b l e on each t r a c e . F i n a l l y , t h e r e i s a c o n s t r a i n t imposed by the

d i s c r e t e n e s s of the samples in s p a c e . I f A~ d e n o t e s the s p a c i n g be tween s amp le s ,

t hen i t i s n e c e s s a r y t h a t the maximum t r a n s v e r s e wave number of the i n t e g r a n d be

l i m i t e d by the N y q u i s t s p a t i a l f r e q u e n c y , I /2&~. The i n t e g r a n d i s a s u p e r p o s i t i o n

of waves w i t h p h a s e s 2~0 /co . The t r a n s v e r s e wave number i s t h e n g iven by the p a r -

t i a l d e r i v a t i v e of t h a t phase w i t h r e s p e c t to ~ , namely , 2 ~ ( ~ 1 - y l ) / P C o . I t i s

t h i s q u a n t i t y which must be c o n s t r a i n e d by the s p a t i a l N y q u i s t f r e q u e n c y .

A more e x t e n s i v e d i s c u s s i o n of the d e s i g n of t h i s computer code can be found in

B l e i s t e i n , Cohen and Hagin [1984] . We a l s o show in t h a t p a p e r , v i a a s y m p t o t i c

a n a l y s i s , t h a t the r e s u l t (16) r e m a i n s v a l i d f o r v a r i a t i o n s in a ( x ) much l a r g e r t h a n

one would e x p e c t on the b a s i s of i t s d e r i v a t i o n v i a p e r t u r b a t i o n t h e o r y . That p a p e r

a l s o c o n t a i n s more d e t a i l s of the the d e r i v a t i o n of the s o l u t i o n f o r m u l a . For sam-

p l e s of o u t p u t p roduced by t h i s method, see Cohen and B 1 e i s t e i n [1979a] and B1e i -

s t e i n and Cohen [1982] . For a c o m p a r i s o n wi th a l t e r n a t i v e a l g o r i t h m s of image p r o -

c e s s i n g , see Gray, [1984] . Th i s method, w i th a compute r a l g o r i t h m d e s i g n e d a long

the l i n e s d e s c r i b e d h e r e , i s shown to p r o c e s s s e i s m i c da t a f a s t e r than a comparab ly

program k - f m i g r a t i o n a l g o r i t h m a f t e r S t o l t [1978] . In a d d i t i o n , the i n v e r s i o n

method p r o v i d e s an e s t i m a t e of r e f l e c t i o ~ s t r e n g t h s a t the i n t e r f a c e s which the

m i g r a t i o n methods do no t p r o v i d e .

R e c e n t l y , we have deve loped e x t e n s i o n s of t h i s method to the case of a dep th

d e p e n d e n t background v e l o c i t y . I n t h i s c a s e , we s o l v e the i n t e g r a l e q u a t i o n (15)

o n l y a s y m p t o t i c a l l y , p r o d u c i n g the b a n d l i m i t e d F o u r i e r t r a n s f o r m s of t he a r r a y of

s i n g u l a r f u n c t i o n s d i r e c t l y . See B l e i s t e i n and Gray [1984] . Computer

i m p l e m e n t a t i o n of t h i s r e s u l t i s a l s o comparab le in speed to k - f m i g r a t i o n , even

though the fo rmer a l l o w s a dep th dependen t r e f e r e n c e v e l o c i t y and the l a t t e r does

n o t .

76

B l e i s t e i n , N., 1984, ~h~g~!~! ~lh~ ~ ~ ~ g ~ : Academic P r e s s , New York, to appear .

. . . . . . . . . . . Cohen, ~. K., and H a g i u , F. G., 1984, C o m p u t a t i o n a l and a s y m p t o t i c a s p e c t s of v e l o c i t y i n v e r s i o n : Geophys i c s , s u b m i t t e d .

. . . . . . . . . . . and Gray , S. H., 1984, An e x t e n s i o n of t h e Born i n v e r s i o n m e t h o d to a d e p t h d e p e n d e n t r e f e r e n c e p r o f i l e : 4 6 t h M e e t i n g of t h e E u r o p e a n A s s o c i a t i o n of E x p l o r a t i o n G e o p h y s i c i s t s , London.

C l a e r b o u t , 3. F. , 1971 , Toward a u n i f i e d t h e o r y of r e f l e c t o r m a p p i n g : ~ R h Y g ~ , 36, 3, p. 467-481.

Coon, S., 1983 , M i g r a t i o n , an i n v e r s e s o u r c e p r o b l e m : 53rd Annua l I n t e r n a t i o n a l Meet ing of the S o c i e t y of E x p l o r a t i o n G e o p h y s i c i s t s , Las Vegas.

Cohen, Y. K., and B 1 e i s t e i n , N., 1 9 7 9 s , V e l o c i t y i n v e r s i o n p r o c e d u r e f o r a c o u s t i c waves: ~ R ~ , 44, 6, p. 1077-1085.

. . . . . . . . . . . 1979b, T~e s i n g u l a r f u n c t i o n of a s u r f a c e and p h y s i c a l o p t i c s i n v e r s e s c a t t e r i n g : Wave ~ 2 ~ , ] , i , p. 153-161.

Gray , S. H., 1984, A c o m p a r i s o n of F-K m i g r a t i o n and Born i n v e r s i o n a l g o r i t h m s : p r e p r i n t .

H a g i n , F. G., and Cohen , J. K., 1984, R e f i n e m e n t s to t h e l i n e a r v e l o c i t y i n v e r s i o n t h e o r y : ~ 2 ~ X ~ £ , ~ , 2, p. 112-118.

Mager , R. D., 1983 , An a s y m p t o t i c a n a l y s i s of s e i s m i c i m a g i n g by w a v e f i e l d e x t r a p o l a t i o n : p r e p r i n t .

S c h n e i d e r , W. A., 1978, I n t e g r a l f o r m u l a t i o n f o r m i g r a t i o n i n two and t h r e e d i m e n s i o n s : ~ R ~ Y ~ , ~ , I , p. 49-76.

S t o l t , R. H., 1978, M i g r a t i o n by F o u r i e r t r a n s f o r m : ~ R h X ~ ! S ~ , ~ , 1, p. 23-48.

W o r l e y , S. C., and Cohen, J. K., 1984, S p a t i a l - t e m p o r a l s l i a s i n g f o r t h e wave e q u a t i o n : p r e p r i n t .

TRAVELLING WAVE FRONT SOLUTIONS OF A DIFFERENTIAL-DIFFERENCE

EQUATION ARISING IN THE MODELLING OF MYELINATED NERVE AXON

by N.F. Britton

Abstract

Most nerves In hlgher animals are myel lnated, that is they have an

insulat ing sheath made up of myelin cel ls. Between these cel ls are short

gaps, known as nodes of Ranvler. A simple model of myetinated nerve gives

an ord inary dif ferential equat ion for the potent ial at the n- th node in terms of

the potent ial at ad jacent nodes. Looking for t ravel l ing wave front solut ions of

such a system this becomes a d t f f e ren t l a l -d l f f e rence equat ion with both

advanced and retarded arguments. The analysis of such an equat ion is

undertaken in the ( u , u ' ) plane, and exhibits some interest ing di f ferences from

the cor respond ing unmyel lnated model , where the equat ion is simply an

ord inary di f ferent ial equat ion.

t . The Model

The model is that of Belt ( ] 9 8 ] ) . He cons idered the myelin to be perfect ly

Insulat ing, with leakage from the axon takln 9 p lace at the nodes between the

myel in cel ls , and assumed that the axon had a pertodtc structure. Let u n be

the potent ial at node n, i n the current and R the res is tance between node n - ]

and node n, and l (u n) the current and C the capac i tance between node n and

the ex t ra -ce l lu la r medium, as in Figure ] .

78

node myelin sheath

.... .... 1,

n e e axon

Un_ 1 i n Un [ n~l Un+1

- j JVV~" , R 1 ," ,, '~/k4~'/~ R . . . . . . . . . . . I

C ~ |(Un-1)+~ L C~ I(Un)I~ C-~, |(Un~l;I~ , t ..... l [

Fig. 1 The c i r cu i t mode l fo r a mye l tna ted axon.

The c u r r e n t I ( u n) gene ra l l y depends on one or more recove ry va r iab les , wh ich

con t ro l r ecove ry of the axon to the s teady s ta te a f te r the passage of a nerve

impu lse . For e x a m p l e , In the F t t z h u g h - N a g u m o c a r i c a t u r e of Hodgk ln -Hux tey

d y n a m i c s , we have

l ( u n) = - f ( u n) + w n

w h e r e w n is a r e c o v e r y va r iab le , and sa t is f ies

dwn

dt = e(Un - ~ n )

Is a smal l p a r a m e t e r , so that w n is a s low var iab le , In th is paper we shal l

c o n s i d e r ~ = 0, so tha t w n = O and l (Un) = - f ( u n ) . We a re t h e r e f o r e only

c o n s i d e r i n g the t r i gge r i ng of a s igna l , and wil l expec t to ob ta in a t rave l l ing

wave front, not a full t rave l l ing wave with eventua l re tu rn to the s teady s tate.

The equa t i ons c o r r e s p o n d i n g to the c i r cu i t in F igure ] a re

u n - ] - Un = R i n

dun dun i n - in+ 1 = C ~ + I (u n) = C ~ - f ( u n)

79

Sca l ing the va r iab les so that R = C = 1 we have

dun = f ( u n) + Un+ ] - 2u n + Un- I d t

Let us now def ine u ( x , t ) for x = n ~ Z by

u ( x . t ) = un ( t ) Ti~en the equa t i on b e c o m e s

u t = f ( u ) + u ( x - l , t ) - 2u + u ( x + i . t ) = f ( u ) + 62u ,

w h e r e we have de f ined

62u = u ( x - l . t ) - 2u + u ( x + l . t ) .

( 1 )

2, The Cauchy P rob lem

The Cauchy p rob lem cons is t s of the equa t i on ( ] ) tn Z x (0.co) t o g e t h e r

with ini t ia l cond i t i ons

u ( x . 0 ) = Uo(X) (2)

fo r x ~ Z. A so lu t ion u ( x , t ) of th is p rob lem is de f ined to be a b o u n d e d

f unc t i on u ( x . t ) wh i ch , c o n s i d e r e d as a func t ion of t for f ixed x, is in the c lass

C ] ( 0 , ~ ) n CO[0,a0). and which sat is f ies ( ] ) and ( 2 ) . A subso lu t lon u ( x , t ) Is

de f ined to be a bounded func t ion of the s a m e c lass wh ich sat is f ies

u t =- f ( u ) + s2u

in Z x (0.o0) and

u ( x , 0 ) ~ Uo(X)

I o r x e Z. Supe rso lu t l ons a re de f lned s imi la r ly . Then the fo l low ing t h e o r e m

ho lds (e . g. Wal ter ( t 9 7 0 ) ) .

C o m p a r i s o n T h e o r e m : tf ( I ) u and u a re s u b - and supe rso lu t t ons fo r the

Cauchy p rob lem (1) with ( 2 ) . and ( i t ) f Is L ipscht tz con t inuous , then

u -= ~ in Z x [ 0 .~ )

Note: E i ther of u, ~ may be a s o l u t i o n of the Cauchy p rob lem,

Co ro l l a r y l : If f ( 0 ) = f ( 1 ) = 0 and 0 =- u o =- l , then 0 and 1 a re s u b -

and s u p e r s o l u t i o n s of ( ] ) with ( 2 ) . so that any so lu t ion u sa t is f ies

O ~ - u ~ 1 ,

80

Corq! , !ary 2: If u 1 and u 2 a r e two s o l u t i o n s , then u 1 .~ u 2 and u 2 "~ u 1, so

tha t u ] -= u 2 and we have at m o s t o n e s o l u t i o n o f the C a u c h y p r o b l e m .

C o r o l l a r y 3: With the h y p o t h e s e s of c o r o l l a r y I t h e r e ex is ts a u n i q u e

s o l u t i o n o f the C a u c h y p r o b l e m .

The p r o o f f o l l ows by c o n s t r u c t i n g m o n o t o n e i n c r e a s i n g a n d d e c r e a s i n g

s e q u e n c e s (v n) a n d {w n) by

v ° = 0, Lv n + Kv n = f ( v n - l ) + Kv n -1

w ° = 1, Lw n - Kwn = f ( w n - l ) - Kw n - ]

w h e r e K is the L lpsch i t z c o n s t a n t f o r f. T h e s e s e q u e n c e s e a c h c o n v e r g e to a

s o l u t i o n o f the p r o b l e m , wh i ch mus t be u n i q u e by c o r o l l a r y 2.

3. T r a v e l l i n g Wave S o l u t ! o n s

By a n a l o g y with the pa r t i a l d i f f e r e n t i a l e q u a t i o n u t = f ( u ) + Uxx w h i c h a r i s e s

o r

CU" = f ( u ) + 62u ( 3 )

f o r Z ~ (-0o, oo). No te tha t a l t h o u g h x is d i s c r e t e , z is a c o n t i n u o u s va r i ab l e .

We sha l l c o n s i d e r the c a s e of e x c i t a b l e d y n a m i c s , so tha t the z e r o s o l u t i o n o f

the C a u c h y p r o b l e m , wh ich c o r r e s p o n d s to the r e s t i n g s ta te o f the n e r v e , is

s t a b l e to sma l l d i s t u r b a n c e s but t h e r e is a t h r e s h o l d a b o v e wh i ch the n e r v e

s w i t c h e s to the exc i t ed s ta te . Thus f is o f the f o rm Shown in F i g u r e 2, i . e .

( i ) f ~ C 1

( i i ) f ( O ) = f ( ] ) = 0

( i i i ) f ' ( O ) /_ O. f ' ( 1 ) /_ 0

( iv ) f L 0 in ( O , a ) , f ~ 0 in ( a , 1) .

f o r u n m y e l i n a t e d a x o n s , we look fo r s o l u t i o n s o f the f o r m

u ( x . t ) = u ( z ) = u ( x + c t )

w h e r e c is a c o n s t a n t ( t h e wave s p e e d ) , We o b t a i n

du c ~ = f ( u ) + u ( z - I ) - 2 u + u ( z + l )

81

+ - O >

U

Fig. 2 The fo rm of f ( u ) fo r exc i tab le k inet ics

The usual f o rm c h o s e n for f is f ( u ) = u ( u - a ) ( ] - u ) .

Note tha t ( u , u ' ) = ( 0 , 0 ) and ( u , u ' ) = ( ] , 0 ) a re both so lu t ions of ( 3 ) .

We look fo r t rave l l i ng wave f ront so lu t i ons , that ts so lu t ions such that

u - 0 as z - -o0 "~ (4)

J U - 1 a s Z ~ o o

We c o n s i d e r such so lu t ions to be the l imi t o f so lu t ions of the Cauchy p r o b l e m .

so tha t 0 ~ u ++ I fo r al l z. The t rave l l ing wave f ron t p r o b l e m (3 ) and (4) may

be c o m p a r e d with the c o r r e s p o n d i n g equa t ion

cu" = f ( u ) + u"

with the cond i t i ons (4 ) .

b e c o m e s

In the la t ter case we de f ine v = u'

U t = V

v' = - f ( u ) + cv

o r phase , and c o n s i d e r t r a j e c t o r i e s in the ( u , v ) ,

t r a j ec to r y at any po in t ( u , v ) is g iven by

p lane.

(5)

so that (5)

The s lope of a

82

dv v" f ( u ) + cv

du u" V

and is t h e r e f o r e d e t e r m i n e d so le ly by the po in t i tsel f . It fo l lows tha t cond i t i ons

u=u o, v=v o at Z=Zo are suf f i c ien t to d e t e r m i n e a t ra jec to ry . In the d i f f e r e n t i a l -

d i f f e rence equa t ion case the s lope of the t r a j ec to r y is g iven by

dv v' u " (f' (u ) u '+u ' ( z - ' l ) - 2 u ' + u ' (z+ ' l ) } / c

du u' u" u'

f ' ( u ) v + v ( z - l ) - 2 v + v ( z + l )

cv

and depends not only on u and v at the g iven po int but a lso on v at o the r

po in ts , and it is not c l ea r that cond i t i ons U=Uo , v=v o at z=z o a re su f f i c ien t to

d e t e r m i n e a t ra jec to ry . However . it is c l ea r f rom (3 ) tha t cond i t i ons over an

in terva l of length 2 mus t be su f f i c ien t ( i f they a re c o n s i s t e n t wi th the equa t i on )

to d e t e r m i n e a so lu t ion . If we a re look ing fo r a so lu t ion wh ich tends to a

c r i t i ca l po in t as z - -o0 ( o r as z - o0) we may g ive such c o n s i s t e n t cond i t i ons to

any requ i red d e g r e e of a c c u r a c y by stat ing that u en te rs the c r i t i ca l po in t a long

an e l g e n v e c t o r . This is the a p p r o a c h we shal l take.

A s z - -®, u - 0, and we I l near l se the equa t ion (3) abou t u=0 to ob ta in

cu" = f ' ( O ) u + u ( z - t ) - 2 u + u ( z + l ) .

Look ing for so lu t ions of the fo rm U=Uo e'~z we ob ta in

CA = f ' (O) + e - x - 2 + e '~

o r

cA - f ' (O) = 2 ( c o s h X - ] )

83

, /2,oo.h ,,

x. oi ~,+ "^

Fig. 3 The e tgenva lues of the l i near i sed equa t ion abou t (0 , 0)

it is c lear f rom F igure 3 that this equa t ion has one posi t ive and one negat ive

root , reca l l i ng that f ' ( 0 ) L 0. so that the or tg ln is a sadd le point . S imi la r ly ,

s ince f ' ( ] ) L O, (1 ,O) IS sadd le point , and the phase p lane Is as shown be low

in F igure 4.

U' H

c)

. . . . . . . m

Fig. 4 The phase p lane for (3) showing the ha l f -s t r ip H = { ( u , u ' ) t 0 L U L I , U ' . ~ 0 }

84

We c o n s i d e r t h e b e h a v t o u r o f t h e t r a j e c t o r y T ( c ) , w h i c h t e n d s to ( 0 , 0 ) as

z - -0o, as z - 00. We w ish to s h o w t h a t t h e r e Is a u n i q u e va l ue of c , c = c ~',

s u c h tha t T ( c ) - ( 1 , 0 ) as z - co. so t ha t we h a v e a t r a v e l l i n g w a v e f r o n t f o r

c = c " . We f i r s t s h o w tha t T ( c ) l e a v e s t h e h a l f - s t r i p H = { ( u , u ' ) 1 0 / - u z l , u ' ~ 0 }

w h e r e u = l , u'-k0 if c Is s u f f i c i e n t l y l a r g e , and w h e r e 0 / -u / - ] , u '=0 If c Is

s u f f i c i e n t l y s m a l l . If t he s o l u t i o n is c o n t i n u o u s l y d e p e n d e n t on c , w h i c h m a y be

s h o w n to be the case , th is i m p l i e s e x i s t e n c e of a w a v e f r o n t .

I f c Is l a r g e , we a s s u m e cu ' >> f ( u ) , ( w h i c h m a y be p r o v e d r i g o r o u s l y ) , so

t ha t ( 3 ) m a y be a p p r o x i m a t e d by

c u ' = 62u ,

Th ts has s o l u t i o n u = Uoe "k°z , w h e r e x o is t h e p o s i t i v e s o l u t i o n o f

c;~ o = e - x ° - 2 + e ~ ° = 2 ( c o s h x o - l ) .

T h u s t h e t r a j e c t o r y l e a v e s the h a l f - s t r i p H w h e r e u = l and u'=~, o ", 0,

I f c Is s m a l l , we a s s u m e c u ' << f ( u ) , ( w h i c h m a y a g a i n be p r o v e d

r i g o r o u s l y ) , so t h a t ( 3 ) m a y be a p p r o x i m a t e d by

0 = f ( u ) + 62u

We n o w le t

f { u ) = u ( u - a ) ( 1 - u )

a n d a s s u m e tha t

J 0 du .& 0 f ( u)

o r e q u i v a l e n t l y , tha t

a z . 1 / 2

Th i s a s s u m p t i o n is a l s o n e c e s s a r y in tt'~e d i f f e r e n t i a l e q u a t i o n c a s e to e n s u r e

t h a i c ~ 0. I . e . t h a t t h e t r a v e l l i n g w a v e f r o n t is a w a v e o f e x c i t a t i o n . We sha l l

a l s o a s s u m e t h e f o l l o w i n g t h e o r e m .

85

NO Cross ln .q T h e o r e m : Let u 1 and u 2 sa t i s fy

C lU 1' = f l ( U l ) + 82Ul

c2u 2 ' = f 2 ( u 2 ) ~- 62u2

w h e r e c 2 ~= c 1, f 2 ( u ) =- f l ( u ) V u ~ ( 0 , 1 ) . T h e n Ul c a n n o t c r o s s f r o m b e l o w

to a b o v e u 2 at a po in t (Uo, U o ' ) ~ H as z i n c r e a s e s , e x c e p t tha t it may d o so

o n c e fo r a f in i te z - i n t e r v a l .

We sha l l p r o v e th is t h e o r e m la te r . We app l y it wi th c I = c 2 = 0,

f l ( u ) = u ( u - a ) ( 1 - u ) ,

a /- 1 / 2 . a n d

f 2 ( u ) = u ( u - l / 2 ) ( t - u ) .

T h e n f 2 ( u ) "¢ f l ( u ) ¥ u ~ ( 0 . 1 ) . M o r e o v e r

f 2 ( u 2 ) + 82u2 = 0

has a s y m m e t i c wave f r o n t so lu t i on , The p r o o f of th is f o l l ows by no t l ng tha t the

t r a j e c t o r i e s l eav ing ( 0 . 0 ) and ( 1 , 0 ) in th is c a s e a r e r e f l e c t i o n s of e a c h o t h e r

In the l i ne u = 1 / 2 , and s i n c e they c a n n o t c r o s s e a c h o t h e r e x c e p t o n c e f o r a

f i n i t e z - i n t e r v a l t hey mus t m e e t s m o o t h l y to p r o v i d e a s y m m e t r i c so l u t i on . It is

t h e n e a s y to s h o w by c o n s i d e r i n g the e t g e n v e c t o r s fo r u, at ( 0 , 0 ) a n d ( 1 , 0 )

tha t we have the s i t ua t i on i l l us t ra ted .

U j

=

0 1 "u

Fig. 5 T h e s y m m e t r i c w a v e f r o n t s o l u t i o n u, a n d the e l g e n v e c t o r s f o r u I at

( 0 , 0 ) ana ( 1 , 0 )

86

The no c ross ing t h e o r e m then imp l ies that the t r a j ec to ry leav ing the o r ig in must

pass out of the ha l f - s t r i p H whe re u' = 0 and 0 /_ u /- 1, as requ i red .

We have thus shown ex i s tence of a t rave l l i ng wave f ront . The no c ross ing

t h e o r e m a lso g ives un iqueness of the wave speed c =, af ter c o n s i d e r a t i o n of the

e l g e n v e c t o r s at ( 0 , 0 ) and ( 1 , 0 ) for d i f fe ren t va lues of c.

It r ema ins to p rove the no c ross ing t heo rem. Let us assume the con t ra ry ,

so that we have the s i tuat ion of F igure 6.

IJ

u¢

I ! !

UO ~u

Fig. 6 u] c ross ing f rom be low to above u 2 as z i n c r e a s e s

S ince u o ' is pos i t i ve , then u' is pos i t ive c l ose to the c ross ing po in t , and we

may de f ine the g r a d i e n t s P l ( U ) of u 1 and P2(U) of u 2. Then c r o s s i n g as

d e s c r i b e d imp l ies P2 - P l is as shown in F igure 7,

87

P 2 " P l /

%=

r U

Fig . 7 T h e d i f f e r e n c e P2 - P ] in t he g r a d i e n t s o f u 2 a n d u l as f u n c t i o n s of u

in t h e s i t u a t i o n o f F ig . 6

F o r s i m p l i c i t y we t a k e P2' - P l ' L 0 a t u = u o. T h e n at t h e c r o s s i n g p o i n t

du 2 ' d U l " dP2 du 2 d P l d u l

dz dz du dz du dz = (P2" - P l ' ) U o ' L 0

so tha t u 2" - u 1' is as s h o w n in F i g u r e 8, a n d u 2 - u 1 as s h o w n "in F i g u r e

9.

u2'-u )

Z

Fig . 8 T h e d i f f e r e n c e u 2 ' - u 1' in t he d e r i v a t i v e s of u 2 and Ul w i th r e s p e c t to

z in t h e s i t u a t i o n o f F ig . 6

U2"U 1

88

r z

Fig. 9 The d i f f e r e n c e u 2 - u I as a f u n c t i o n o f z in the s i t ua t i on o f Fig. 6

T h e r e a r e now two a l t e r n a t i v e s : e i t h e r ( a ) u 2 - u 1 L 0 V Z ~ z o :

o r ( b ) 3 Zo ~ Zo such that u 2 - u 1 = 0, Let us a s s u m e tha t ( a ) ho lds ,

u 2 ( z o - 1 ) L U l ( Z o ) - l ) . u 2 ( z o + l ) L U l ( Z o + l ) , and at z o

C l U t ' ( z o ) = C lU o" = f l ( U o ) + U l ( Z o - 1 ) - 2 U o + U t ( Z o + l )

c 2 u 2 ' ( Z o) = c2u o" = f 2 ( U o ) + U 2 ( Z o - 1 ) - 2 U o + U 2 ( Z o + l )

so that

T h e n

0 ,~ ( c 2 - c l ) u o ' - ( f 2 ( u o ) - f l ( u o ) )

= u 2 ( z o - 1 ) - U l ( Z o - 1 ) + u 2 ( z o + l ) - u 2 ( Z o - 1 ) /_ 0

and we have a r r i v e d at a c o n t r a d i c t i o n . Thus (b ) ho lds . A s s u m e f i r s t tha t

Zo --~ Zo, a n d r e d e f i n e Zo to be the f i rs t po in t a b o v e z o w h e r e u 2 = u 1. T h e n

u 2 ( z o ) = U l (Z 'o ) = Uo and u ' 2 ( z ' o ) - U ' l ( Z ' o ) >" 0. It f o l l o w s tha t

p 2 ( ~ o ) - p l ( U ' o ) ~= 0, i . e . u 2 is a b o v e u 1. so tha t t h e r e mus t have b e e n a

r e c r o s s i n g of t he t r a j e c t o r i e s . ( E v e n if e q u a l i t y h o l d s at Uo, it c a n be s h o w n

that p 2 ( u o - e ) - p ] ( u o - ~ ) -~ 0 f o r ~ su f f i c i en t l y s m a l l ) . Th is poss ib i l i t y a l l ows

u I to be a b o v e u 2 fo r a f i n i te z - l n t e r v a l . A s i m i l a r a r g u m e n t h o l d s if ~o /- Zo,

so tha t to COmplete the p r o o f we must show that u I c a n n o t c r o s s u 2 fo r a

~,econd t ime, i , e , we c a n n o t h a v e the s i t ua t i on s h o w n in F i g u r e 10.

U'

89

I I I I | | ...... , ,,, .........

U 0 UD

Fig, 10 R e r e c r o s s i n g of the t r a j ec to r i es

U

Let us de f ine zi to be the va lue of z at wh ich u i r e a c h e s D. for i=1 .2 .

Then

z, = ~,o + ~ uD du

.Ju o Pl(U)

But s ince P2(U) ~ P l ( U ) on the in terva l then

z l -~z 2

and the z - i n t e r va l (Zo,Z 1) which u] takes to t rave l f rom A to D is l onge r than

the z - i n t e r va l (Zo ,Z 2) wh ich u 2 takes, But a s im i la r a r g u m e n t , reve rs ing the

s ign of z, shows that u 2 takes a l o n g e r in terva l to t rave l f rom D to A than u 1

does . S ince the in te rva l taken does not d e p e n d on the d i r ec t i on of t rave l , th is

is a con t rad i c t i on and so the s i tua t ion d e p i c t e d in F igure 10 canno t occu r . We

have thus c o m p l e t e d the proo f of the no c r o s s i n g t h e o r e m and h e n c e the proo f

of ex i s t ence of a un ique t rave l l i ng wave f ront for ou r mode l of mye l l na ted nerve

axons .

Re fe rences

J. Be l l , Some th resho ld resu l ts for mode l s of mye l i na ted ne rves , Math.

B iosct . 54. ] 8 ] - ] 9 0 , ] 9 8 ]

W, W,-~Iter, DItlerenlial and Integral lnequalilie~, 1 9 7 0 ,

THE NON-PARAMETRIC INFLUENCE OF LIGHT

ON MAMMALIAN CIRCADIAN RHYTHMS

Gail A. Carpenter

I. The DiSCovery of Long-term After-effects

In 1960, Pittendrigh noted that the circadian period of an animal,

kept in the dark and free from external time cues, could vary as a func-

tion of the lighting regime to which the animal had previously been ex-

posed (Pittendrigh, 1960). This phenomenon was termed "after-effects."

Pittendrigh cited as examples hamsters which, after entrainment to a

23-hour day, had shorter subsequent periods in the dark (DD) than did

hamsters entrained to a 25-hour day; and mice which, after exposure to

constant light (LL) , had long circadian periods in the dark. After-ef-

fects can persist for weeks. In some cases, a median circadian period

is re-established after a long-term transient (Pittendrigh, 1960, Fig-

ure 8); in other cases, after-effects appear to be stable (Pittendrigh,

1974, Figure I).

2. The Reintroduction of Long-term After-effects

Perhaps because of the long time periods involved, and perhaps be-

cause they called into question the very notion of "the" circadian peri-

od, after-effects did not play an important role in circadian rhythm

research until years after their discovery. Pittendrigh (1974, p.441)

wrote: "They are more widespread than the current literature suggests;

they are not accounted for by any of the several mathematical models so

far published; and they must be reckoned with in the mechanism of en-

trainment." Pittendrigh re-emphasized the importance of after-effects,

presented new after-effect experiments, and outlined a two-oscillator

theory to help explain after-effect results. During the past ten years,

theoretical and mathematical investigations have begun to reckon with

long-term after-effects. These recent results will now be compared.

* Supported in part by the National Science Foundation (NSF MCS-82- 07778) and the Office of Naval Research (ONR-N00014-83-K0337).

Acknowledgements: Thanks to Cynthia Suchta for her valuable assistance in the preparation of the manuscript and illustrations.

91

3. Three Types of Models: Tw0 0scillators, Multipl e Oscillators, and

Gated Pacemakers

The two-oscillator model described by Pittendrigh (1974) was fur-

ther elaborated by Pittendrigh and Daan (1976b) and then given concrete

realizations in a discrete version (Daan and Berde, 1978) and a dynami-

cal systems version (Kawato and Suzuki, 1980). A more general dynamical

systems version was also studied by Pavlidis (1978). In each of these

versions, two systems, each capable of oscillating independently, are

coupled in such a way that the activity of one oscillator resets the

phase of the other. Enright (1980) developed coupled stochastic system

models in which many oscillating units are indirectly coupled through

the output of a master "discriminator" which receives input from each

oscillating unit. The ordinary differential equation gated pacemaker

model (Carpenter and Grossberg, 1983a, 1983b, 1984a, 1984b, 1984c) des-

cribes a unit whose oscillations depend upon the dynamics of slowly ac-

cumulating transmitters. In a gating, or mass action, step, transmitters

are depleted when the pacemaker sends feedback signals to itself.

All three of these types of models have been used to simulate many

aspects of circadian rhythm data. The present discussion will focus on

their analyses of after-effects.

4. Defininq the Domain of a Model

Circadian rhythms have been studied at levels ranging from cell

fragments (Schweiger and Schweiger, 1965) to humans (Wever, 1979). When

evaluating the success of a model, it is important to keep in mind which

organisms, as well as which types of experiments, are under considera-

tion. For example, the phenomenon of desynchronization of the activity-

rest cycle from the temperature cycle in humans is the subject of the

two-oscillator model analysis of Kronauer, Czeisler, Pilato, Moore-Ede,

and Weitzman (1982). This desynchronization has been observed in humans

and a few other species. Kronauer et al. take as their model two coupled

van der Pol oscillators, one of which represents the activity-rest pace-

maker. After-effects on the period of the activity-rest cycle are thus

outside the scope of this model. In contrast, Pittendrigh's two-oscilla-

tor model represents only the activity-rest pacemaker.

There is abundant evidence pointing to the mammalian suprachiasma-

tic nuclei (SCN) as a complete circadian pacemaker or, at least, as a

central element of the mechanism which generates circadian rhythmicity

(Hedberg and Moore-Ede, 1983; Inouye and Kawamura, 1979; Moore and Eich-

let, 1972; Stephan and Zucker, 1972). In non-mammalian species, such as

92

birds, the pineal organ seems to be a more central element (Menaker,

1974). Another difference between birds and mammals is in the location

of photoreceptive elements. Mammals process light inputs through the

retina, which projects directly to the SCN, whereas some birds can pro-

cess light signals even after blinding (Menaker, 1968). Thus the ways

in which light influences circadian rhythms of birds may be expected to

differ from the ways in which light influences the circadian rhythms of

mammals. In addition, the physiological interpretation of a model of

avian circadian rhythms would differ from the physiological interpreta-

tion of a model of mammalian circadian rhythms.

An extensive set of experimental results on mammalian after-effects

is contained in the study of Pittendrigh and Daan (1976a) of nocturnal

rodents. Daan and Berde (1978) focus their analysis on these after-ef-

fects and on other circadian properties of nocturnal mammals. Pavlidis

(1978) considers after-effects generally as long-term transients follow-

ing desynchronization of weakly coupled oscillators, but does not exa-

mine how different light regimes generate different after-effects. Kawato

and Suzuki (1980) do a bifurcation analysis of split rhythms, but do not

study after-effects.

Gated pacemaker model analysis has focused primarily on data from

mammals, both diurnal and nocturnal. This focus on mammals avoids, for

example, the possibly confounding effects of extraretinal light inputs

in birds.

In contrast, Enright's scholarly book includes discussions and simu-

lations which range across a wide variety of mammalian and non-mammalian

species. He can thus state, for example, that Aschoff's rule (Aschoff,

1960), which predicts the effect of constant light level on circadian pe-

riod, is "observed in essentially all nocturnal and diurnal vertebrates"

(Enright, 1980, p.103). In fact, most diurnal mammals fail to obey Asch-

off's rule (Aschoff, 1979). In the light of such generalizations, it is

notable that Enright's concluding physiological interpretation for his

model is applied to birds. If, indeed, Enright's discriminator is a bet-

ter model of the pineal organ in birds than of the circadian rhythm sys-

tem in mammals, then his difficulty in simulating key mammalian after-

effect data (Enright, 1980, p.195) might be traced to a non-mammalian

model domain.

Enright's model, and particularly his analysis of Aschoff's rule,

are reviewed in detail elsewhere (Carpenter and Grossberg, 1984a). In

order to focus attention on the interesting and paradoxical nature of

mammalian after-effect data, the two-oscillator and gated pacemaker mo-

93

dels will be the primary subjects of the remainder of this article.

5. How Does an Increase in Daily Light Duration Affect the pacemaker?

Two examples of after-effect experiments on nocturnal mammals il-

lustrate a key paradox whose resolution points to major differences be-

tween the two-oscillator and the gated pacemaker models. In the first

type of experiment, an increase in the duration of the daily light input

typically decreases the subsequent free-running period in the dark.

These experiments, which examine the after-effects of photoperiod, have

light regimes in which light is turned on and off with a period of 24

hours. In this case, even if light is on for 18 hours each day, the noc-

turnal mammal has 6 hours of darkness in which to be active. In the sec-

ond type of experiment, an increase in the daily light input typically

increases the subsequent period in the dark. These experiments have a

constant light regime in which the nocturnal mammal has no choice but to

be active in the ever-present light, which it would normally try to

avoid. The experiments will be described in more detail in Section 7.

Experiments on the after-effects of photoperiod and constant light

create a paradox which could not be resolved by assuming, for example,

that free-running period in the dark is a monotonic function of daily

light duration in the preceding regime. Two-oscillator models resolve

the paradox by assuming that a non-parametric, or on-off, light input

influences the circadian pacemaker via a different mechanism than does

a parametric, or constant, light input. In contrast, the gated pacema-

ker model resolves the paradox by computing an average of pacemaker out-

put, which, in a nocturnal model, tends to be greater during the dark

portion of a non-parametric light regime than during a parametric light

regime.

6. Double Plots

The long time intervals involved in after-effect experiments and

simulations require a special format for the compact presentation of re-

sults. Figure 1 illustrates how a 24-hour portion of a model solution

and light regime can be represented as a narrow histogram. A typical mo-

del solution as a function of time (Figure la) is hereby compressed to a

single line (Figure ib), which forms part of a double plot. The height

of the solution, above an "activity threshold," is translated into five

line widths in the double plot. Times during which the solution is large

are represented as horizontal lines of width 5. Times during which the

model light is on are drawn as open regions in the plot. This method of

94

lal

Ibl

O

2 l ! I " x , ............. 1 l l , \ /,,,, \,,,

I I L I G H T t t

I i

A C T I V I T Y

, , , , ,

R E S T

2 4 H O U R S I I I I I I I I

! i i I I

I . . . . . . ! ......... I . . . . . . . ! 0 i I , , 2 4 H O U R S

i I I I

Figure i: Representation of a model solution as a function of time (a) and as one row in a double plot (b).

95

representing the daily activity profile retains the approximate solution

profile as seen, for example, in the bimodal pattern on days 1-90 of Fig-

ure 2 (Section 7) as well as in the intense unimodal pattern on days 91-

150.

Long-term trends are brought out when rows such as the one in Figure

ib are stacked to form a column. The left column of Figure 2, for exam-

ple, represents a model solution computed for 180 days. It would have ta-

ken pages to plot the solution as a function of time, and the long-term

trends, which occur very slowly, would be nearly invisible. Patterns

which cross the 24-hour line are more easily seen when two identical col-

umns are placed side-by-side with the right column shifted up one line

higher than the left column. Each horizontal line in a column represents

a 24-hour day, so each horizontal line across a double plot represents a

48-hour time span. A circadian period of exactly 24 hours is seen as

dark bars which are aligned in a vertical column, as during days 5-60 in

Figure 2. A circadian period greater than 24 hours is seen as a drift

toward the right, as on days 60-90 of Figure 2. A circadian period less

than 24 hours is seen as a drift toward the left, as on days 150-180 of

Figure 2.

7. After-effects of Photoperiod and Constant Light

A typical photoperiod after-effect experiment, simulated using the

gated pacemaker model, is illustrated in Figure 2. For the first 60 days,

light input is on for 1 hour of each 24-hour day (LD 1:23), and the mod-

el's period is entrained to the 24-hour light cycle. During the next 30

days of free-run in the dark, the period is greater than 24 hours. On

days 90-120, light input is on for 18 hours of each 24-hour day (LD 18:6)

and, during the final 30 days of free-run in the dark, the period is less

than 24 hours. By themselves, the photoperiod after-effect experiments

might suggest that the increased duration of light input during an LD 18:

6 regime causes a decrease in period. However, the experiment simulated

in Figure 3 suggests the opposite conclusion.

Figure 3 illustrates both Aschoff's rule and the circadian rule for

nocturnal mammals (Aschoff, 1960, 1979) and the after-effects of constant

light (LL) . On days 1-60, light input is on for 12 hours of each 24-hour

day (LD 12:12). During the next 30 days (DD) the period is greater than

24 hours and is, in fact, about the same as after LD 1:23 in Figure 2.

During the constant light regime on days 90-150, the period is increased

further (Aschoff's rule) and the level of activity is much reduced (the

circadian rule). During the final 30 days in the dark, the period is ~rea-

96

O

HOURS l

2~

- - , 1 , ; ' , ' , . . . .

I q8

I I =1

r

r

I'

LD 1:23

DD =_-=

LD 18:6

DD

Figure 2: Gated pacemaker model simulation of photoperiod after-effects.

ter than it was following LD 12:12.

8. The Non-parametric Influence of Liqht

Since two-oscillator models use distinct mechanlsms to process non-

parametric (LD) and parametric (LL) light regimes, the nature of these

two light regimes will now be examined. An essential assumption of two-

oscillator models is that an external parametric light regime is also

registered as an internal parametric input. Thus Pittendrigh and Daan

9 7

0 3 ) , -

E 3

HOURS I

2 q I

q8

/ m

J E

/

: : = ~ = = = = = =

.~': ,. ~I ~? . . . . . D D

, , r

, - ' - ~ _ ::,,~T=,=,=~=~=,i,i, ~ ~: I I' ii i i i . . . . . . ,,,,,,,,i,,,,I,Ii, I

~t ,,',,,',',

LD 12:12

LL

DD

Figure 3: Gated pacemaker model simulation of the after-effects of con- stant light. Model parameters and light levels are the same as in Figure 2.

(1976a, pp.242-243) write: "By definition we must conclude that the

lengthening of [period] in constant illumination is due to a parametric

effect on the pacemaker: no change in external conditions occurs through-

out its cycle. The after-effect of photoperiod is surprising only if we

assume that the parametric action of a long light pulse (photoperiod) is

its dominant effect. In Drosophila pseudoobscura the characteristically

98

different effect of each photoperiod on the circadian pacemaker can be

accounted for by the interaction of the two non-parametric effects due

to the transitions at the beginning and end of each photoperiod (Pitten-

drigh and Minis, 1964)...[T]he after-effect of photoperiod on our rodent

pacemakers is similarly attributable to the interaction of non-parametric

effects at the beginning and end of the photoperiod."

Although constant light is "by definition" parametric at the source,

it does not necessarily have a "parametric effect on the pacemaker." Dur-

ing LL, an animal periodically goes to sleep and wakes up as part of its

circadian cycle. When the animal goes to sleep, eye closure, burying the

head, or retreat to a dark nest can decrease or eliminate the registered

light input. Similarly, when the animal wakes up, eye opening can in-

crease the effective light input. In this sense, even the parametric con-

stant light paradigm is experienced as a non-parametric paradigm by the

nervous system. Moreover, Terman and Terman (1983) have demonstrated that

sensitivity to light oscillates with a circadian rhythm in the rat, and

that this rhythm persists after the SCN is removed. Thus even if external

light intensity remains constant, the animal's internal sensitivity to

light need not. In sum, both parametric and non-parametric light regimes

are experienced non-parametrically at mammalian central pacemakers. Two-

oscillator models assume that only non-parametric light regimes are ex-

perienced non-parametrically.

Conversely, two-oscillator models assume that a constant light re-

gime causes a significant change in one or more of a model's parameters,

but that a light-dark regime causes no such change, even if the light is

on for most of the day.

The gated pacemaker model makes no assumptions about different mech-

anisms whereby parametric and non-parametric light regimes influence the

central pacemaker. The extent to which the non-parametric nature of a

parametric light regime affects circadian rhythms is investigated using

a light attenuation parameter @. If L(t) describes the light input func-

tion for an awake model subject, then the actual light input function is

given by:

L(t) if awake J(t) = , (i)

eL(t) if asleep

where 0 < @ < I. The "waking" and "sleeping" states are dependent on the

pacemaker output. If e = 0, the model animal is completely insensitive

to light when asleep. This case corresponds to an animal sleeping in a

99

dark nest or otherwise self-selecting its light-dark cycle. I9 @ = i,

the model animal is equally sensitive to liaht throughout the day. There

is no attenuation of the light input during sleep. If 0 < @ < i, the

model animal's sensitivity to light is diminished when asleep, but a

bright light input still reaches the pacemaker. This case corresponds

to simple eye closure or to modulation of photoreceptor sensitivity by a

circadian pacemaker. When 0 < e < i, all model light inputs are non-para-

metric. In addition, any model light input which directly affects the

gated pacemaker does so in real time, regardless of whether the external

light source is parametric or non-parametric.

The light attenuation factor @ plays an important role in the gated

pacemaker analysis of certain aspects of circadian rhythms. For example,

both diurnal and nocturnal models in a certain class obey Aschoff's rule

when e = 1 (Carpenter and Grossberg, 1984a) . This case 9 = 1 may corres-

pond to that of non-mammalian species with extraretinal photoreceptors

which remain sensitive during sleep. When 9 decreases to 0 in the gated

pacemaker, model nocturnal animals continue to obey Aschoff's rule, while

model diurnal animals disobey the rule. The gated pacemaker analysis of

light attenuation during sleep thus suggests why diurnal mammals often

disobey Aschoff's rule (Aschoff, 1979).

Gated pacemaker analysis of after-effects is not critically depen-

dent on the value of 9, which is set equal to .5 in Figures 2 and 3.

Rather, model after-effects are due to an internal gain control process

which computes a time-average, or long-term memory, of pacemaker activity

levels. An LD 18:6 light regime (Figure 2), during which the nocturnal

model animal is vigorously active for 6 hours a day, causes an increase

in the slow gain term, and a subsequent decrease in the free-running peri-

od in the dark. A constant light regime (Figure 3), during which the noc-

turnal model animal is only weakly active for a portion of each day, cau-

ses a gradual decrease in the slow gain term, and a subsequent increase

in the free-running period in the dark.

Various alternative gain control designs and exceptional cases can

also be explored (Carpenter and Grossberg, 1984b) . For example, the same

gain control processes which yield inconsistent after-effects on period

are also shown to cause the slow onset of split rhythms. Both the incon-

sistent after-effects and split rhythms are observed together in the gol-

den hamster (Pittendrigh and Daan, 1976a, 1976b) .

9. A TwO-oscillator Model

An explicit two-oscillator nocturnal model is given by Daan and Berde

100

(1978), who interpret as after-effects the transients observed when the

two weakly coupled oscillators reestablish a stable phase relationship

after perturbation.

The Daan and Berde model clearly distinguishes the separate mecha-

nisms for processing parametric and non-parametric light inputs: in a con-

stant light regime, light level alters model parameters; in a non-parame-

tric light regime, light onset or offset resets the phase of the constitu-

ent oscillators. The model is defined in terms of the period, phase, and

instantaneous phase shifts of the two oscillators. Oscillator onset times

are defined recursively.

Consistent with Pittendrigh's notation, the two oscillators are la-

beled E (evening) and M (morning), although either E or M can occur first

on a given day. In the uncoupled state, E has a natural period T E and M

has a natural period T M. Their difference is denoted by D:

D = T E T M. (2)

Hypotheses [1]-[5] below define the dynamics of the Daan and Berde

two-oscillator model.

[I] Fixed average period: The average period of the two oscillators

is set identically equal to 24 hours. Thus

T E = 24 + D/2 hours (3)

and

T M = 24 - D/2 hours. (4)

[2] Osci!lator onset triggers an activity bout: The onset of each

oscillation (E or M) triggers an activity bout of fixed amplitude and dur-

ation. In most of the simulations, this activity bout lasts 5 hours. If,

say, an M onset triggers an activity bout 2 hours after an E onset, then

that activity cycle lasts 7 hours.

[3] Co uP!ing between the oscillators: The onset of the M oscillator

instantaneously resets the phase of the E oscillator. That is, an M onset

advances or delays the subsequent E onset. The amount by which the phase

of E is reset is a sinusoidal function of the time between the previous E

onset and the M onset. Similarly, the onset of the E oscillator resets the

phase of the M oscillator.

More precisely, suppose that an E onset occurs at time E o. Then the

subsequent E onset will occur at time E ° + XE' unless an M onset occurs

sooner. If an M onset occurs at time E + m, with o

m < T E, (5)

then the subsequent E onset will occur at time

E ° + T E + A sin2~(m + ~E ) , (6) E T E

101

where the parameters A E

example, when

2~(m + ~E ) TE = ~ ,

then the M onset at time E O

hours. When

2~(m + ~E ) = ~,

T E

then the M onset at time E O

hours.

and ~E represent coupling strength and phase. For

+ m maximally delays the phase of E, by A E

(7)

onset occurs at time

• 2~ E 1 + AzESln~E(E 1

and the next M onset

M 1 + AzMSin~(M 1

where AZE , AZM, ~ZE'

light regimes cause

Daan and Berde

- Z - CZE ) (i0)

occurs at time:

- z - ~ZM ), (11)

and ~ZM are amplitude and phase parameters. On-off

no changes in model parameters.

do not uniquely specify when "Zeitgeber events" occur

Similarly, an E onset which occurs e hours after an M onset resets

the phase of the M oscillator by

AMsin2~ (e + (9) ~M ) T M

hours, where A M and ~M represent coupling strength and phase.

In the Daan and Berde simulations A E and A M equal approximately 1

hour. In most of the simulations, ~E = 15 hours and ~M = 21.25 hours.

Parameters are chosen so that the system obeys Aschoff's rule for noctur-

nal animals.

[4] Parametric light regimes: If external illumination is constant

(LL), then either the difference between the nocturnal periods (T E - T M)

or the difference between the coupling strengths (A E - AM) is an increa-

sing function of light intensity.

[5] Non-parametric light regimes: Onset and/or offset of an external

light source (a "Zeitgeber event") instantaneously phase resets both os-

cillators according to rules similar to those in [3]. The amount by which

the phase of the E (M) oscillator is reset by a Zeitgeber event is a sin-

usoidal function of the difference between the time, Z, of the event and

the time, E 1 (MI) , at which the next E (M) onset would have occurred with-

out the Zeitgeber event. When a Zeitgeber event occurs, then, the next E

(8)

+ m maximally advances the phase of E, by A E

102

during each light-dark cycle. For example, in a typical experiment demon-

strating the after-effects of short and long periods of the entraining

stimulus, an LD ii:ii light regime (period 22 hours) is compared with an

LD 13:13 light regime (period 26 hours). In the simulations, a Zeitgeber

event occurs only at the offset of the model light input. On the other

hand, in the simulation of photoperiod after-effects, when a model LD 6:

18 light regime is compared with a model LD 12:12 light regime, Zeitgeber

events occur both at light onset and at light offset.

i0. The Gated Pacemaker Model

The gated pacemaker model describes the dynamics of "on" and "off"

subunits which mutually inhibit one another. In a model SCN, the units can

be interpreted as on-cell/off-cell modules, populations of which are dis-

tributed throughout each SCN. In alternative interpretations, the on/off

units can be interpreted as intracellular entities, such as competing

chemical concentrations or excitatory and inhibitory membrane channels. In

any case, each element of the model has a physical interpretation, so that

dynamic processes correspond to physiological predictions.

The following processes define the dynamics of the gated pacemaker

model (Figure 4).

[i] Slowly accumulating transmitter substances are depleted, or habi-

tuated, by gating the release of feedback signals according to a

mass action law.

[2] The feedback signals are organized as an on-center off-surround,

or competitive, anatomy.

[3] Both on-cells and off-cells are tonically aroused.

[4] Light excites the on-cells of a diurnal model and the off-cells

of a nocturnal model.

[5] The on-cells drive observable activity, such as wheel-turning,

in both the diurnal model and the nocturnal model.

[6] On-cell activity gives rise to a fatigue signal that excites the

off-cells in both the diurnal model and the nocturnal model. The

fatigue signal is a time-average of the on-cell output signal on

an ultradian time scale of about 4 hours.

[7] On-cell activity gives rise to a slowly varying gain control sig-

nal that excites the on-cells in both the diurnal model and the

nocturnal model. The gain control signal is a time-average of the

output signal on a time scale of months.

103

{a} A C T I V I T Y -~"

LIGHT A R O U S A L

O N - C E L

} FATIGUE

OFF-CELLS

b) A C T I V I T Y - ~ \

O N - C E L L S -- "! F - C E L L S

AROUSAL LIGHT

Figure 4: Gated pacemaker circuits of diurnal (a) and nocturnal (b) mod- els. In both circuits, on-cells and off-cells excite themselves via posi- tive feedback, inhibit each other via negative feedback, and are tonical- ly aroused. Light excites on-cells in the diurnal circuit and off-cells in the nocturnal circuit. Activation of on-cells or suppression of off- cells energizes wakefulness and activity. Fatigue builds up during the wakeful state and excites off-cells in both diurnal and nocturnal cir- cuits. A conditionable slow gain control process (not shown) activates on-cells in both diurnal and nocturnal circuits.

104

Processes [1]-[7] can be studied as a hierarchy of models. Processes

[1]-[5] define a four-dimensional basic pacemaker whose circadian dynamics

include the characteristic phase response curves of nocturnal and diurnal

animals, the suppression of oscillations at high light intensity, a clock-

like stability of period except near parameter values where oscillation

ceases, and a parameter range in which complex oscillation patterns occur

(Carpenter and Grossberg, 1983b) . The addition of a fatigue signal [6]

yields Aschoff's rule and the circadian rule for nocturnal and diurnal

mammals as well as an explanation of why diurnal mammals frequently dis-

obey Aschoff's rule (Carpenter and Grossberg, 1984a). Finally, the com-

plete gated pacemaker models a variety of after-effects as well as split

rhythms and SCN ablation results (Carpenter and Grossberg, 1984b).

The general model equations for a nocturnal gated pacemaker are de-

fined as follows.

NOCTURNAL MODEL

ON-POTENTIAL

dx 1 dt .... AXl + (B-Xl) [I + f(xl)Zl +Sy] - (Xl+C)g(x2)" (12n)

OFF-POTENTIAL

dx 2 dt - -Ax2 + (B-x2) [I + f(x2)z2 +F+J(t)] - (x2+C)g(xl)' (13n)

ON-GATE

dz 1 dt - D(E - z I) - Hf(Xl)Z I, (14)

OFF-GATE

dz 2 dt - D(E - z 2) - Hf(x2)z 2, (15)

FATIGUE

dF _ (16) dt -KF + h(Xl),

GAIN CONTROL

dy = -Uy + Vf(x I) (17) dt °

Variable x I in equation (12n) is the potential of an on-cell (population)

v I. Variable x 2 in equation (13n) is the potential of an off-cell (popula-

tion) v 2. Both x I and x 2 obey membrane equations (Hodgkin and Huxley,

1952). In (12n) and (13n) , the parameter -A in the terms -Ax I and -Ax 2 de-

termines the fast decay rate of the potentials x I and x 2. Also in (12n)

and (13n), term I represents the constant arousal level that equally ex-

cites v I and v 2. In (12n), the transmitter substance z I gates the nonnega-

105

tive feedback signal f(xl) from v I to itself. Term f(xl)z I is proportion-

al to the rate at which transmitter is released from the feedback pathway

from v I to itself, thereby re-exciting x I. Term Sy describes the effect of

the gain control process y on v I. Term S is a signal that is gated by y,

thereby generating a net excitatory input Sy at the on-cells v I. The off-

cells v 2 inhibit the on-cells v I via the nonnegative signal g(x 2) in term

-(Xl+C)g(x2). Equation (13n) is the same as equation (12n) , except that

the indices 1 and 2 are interchanged; both the light input J(t) and the

fatigue signal F excite v 2 but not Vl; and the slow gain control process

excites v I but not v 2. Light input is defined using the attenuation factor

@ of equation (I) (Section 8).

Equations (14) and (15) define the transmitter processes z I and z 2.

In (14), the transmitter z I accumulates to its maximal level E at a slow

constant rate D via the term D(E - Zl). This slow accumulation process is

balanced by the release of z I at rate Hf(Xl)Z I, leading to the excitation

of x I in equation {12n). A similar combination of slow accumulation and

gated release defines the dynamics of transmitter z 2 in (15).

The endogenous interactions between potentials x I and x 2 and trans-

mitters z I and z 2 define a clock-like pacemaker (Carpenter and Grossberg,

1983b). This pacemaker has a stable period in the dark that varies inverse-

ly with the transmitter accumulation rate. Any genetic or prenatal factor

capable of fixing this accumulation parameter can specify the period of

the clock in the dark. The remaining processes F and y modulate the beha-

vioral patterns that are generated by the pacemaker, as during split rhy-

thms and long-term after-effects, but are not the source of the pacema-

ker's clock-like properties. Both F and y average indices of pacemaker ac-

tivity, but are not independent oscillators.

The fatigue signal F in (16) is a time-average of h(Xl), which in-

creases with on-cell activity x I. Speaking intuitively, an increase in x 1

and a decrease in x 2 arouse neural circuits that support the awake state.

Fatigue builds up as a function of increasing metabolic activity during

the awake state, including but not restricted to overt action. Fatigue,

in this sense, can thus build up in an alert but physically restrained

animal. Since F excites the off-cells v 2 in (13n) , it tends to inhibit

the arousal generated by the pacemaker. The decay rate K of the fatigue

signal F is assumed to be ultradian. In particular, A > K > D so that the

potentials x I and x 2 react faster than the fatigue signal F, which in turn

reacts faster than the pacemaker gates z I and z 2.

The slow gain control process y in (17) is also a time-average, but

on a time scale that is much slower than the circadian time scale. Pro-

cess y averages term Vf(x I) at an averaging rate U. Then Sy in (12n) acts

106

as an excitatory input to the on-cells v I. Term Sy in equation (12n) com-

bined with equation (17) formally define a long-term memory trace y

(Grossberg, 1968, 1969, 1982a) . In all the simulations, terms S, U, and

V are chosen to be constant, or to vary as a function of light or on-

cell activity (Carpenter and Grossberg, 1984b) .

The diurnal model differs from the nocturnal model only in the equa-

tions (12d) and (13d) that define its on-cell and off-cell potentials. In

particular, light input J(t) excites the on-cells but not the off-cells of

the diurnal model. By contrast, the fatigue input F excites off-cells in

both the diurnal and the nocturnal models, and the slow gain input y ex-

cites on-cells in both the diurnal and the nocturnal models. The diurnal

model equations are listed below.

DIURNAL MODEL

ON-POTENTIAL

dx 1 dt - -AXl + (B-Xl) [I + f(xl)Zl+J(t)+SY] - (Xl+C)g(x2) ' (12d)

OFF-POTENTIAL

dx 2 dt - -Ax2 + (B-x2) [I + f(x2)z2 +F] - (x2+C)g(xl)' (13d)

ON-GATE

dz 1 dt - D(E - z I) - Hf(Xl)Z I, (14)

OFF-GATE

dz 2 dt - D(E - z 2) - Hf(x2)z 2, (15)

FATIGUE

dF d-~ = -KF + h(Xl), (16)

GAIN CONTROL

d-l[ -Uy + vf ) (17) dt = (Xl -

The models in equations (12)-(17) are completely defined by a choice

of the signal functions f, g, and h; the light input J(t) ; the signals S,

U, and V; and the parameters. In the simulations shown in Figures 2 and 3,

the signal functions f(w) and g(w) are chosen to be threshold-linear func-

tions of activity w:

f(w) = g(w) = max(w,0) o (18)

The signal function h(w) in (16) is defined by

h(w) = M max[f(w) - N,0]. (19)

107

The definition of h(w) can be interpreted as follows. With f(xl(t)) as

the output signal of the pacemaker, behavioral activity is triggered when

f(xl(t)) exceeds the positive threshold N (Figure i). The function

h(Xl(t)) defined by (19) then provides an index of unrestrained behavior-

al activity.

REFERENCES

Aschoff, J., Exogenous and endogenous components in circadian rhythms. Cold Spring Ha rb0r Symp. Quart. Biol., 25, (1960), 11-28.

Aschoff, J., Influences of internal and external factors on the period measured in constant conditions. Z. Tierpsychol., 49, (1979), 225-249.

Carpenter, G.A. and Grossberg, S., Dynamic models of neural systems: Pro- pagated signals, photoreceptor transduction, and circadian rhythms. In J.P.E. Hodgson (Ed.) , Oscillations in mathematical biology. Berlin, Hei- delberg, New York: Springer-Verlag, 1983(a) , pp. I02-196.

Carpenter, G.A. and Grossberg, S., A neural theory of circadian rhythms: The gated pacemaker. Biol. Cyber., 4_88, (1983), 35-59 (b) .

Carpenter, G.A. and Grossberg, S., A neural theory of circadian rhythms: Aschoff's rule in diurnal and nocturnal mammals. Am. J. Physiol., 1984(a).

Carpenter, G.A. and Grossberg, S., A neural theory of circadian rhythms: Split rhythms, after-effects, and motivational interactions, 1984(b).

Carpenter, G.A. and Grossberg, S., Neural dynamics of circadian rhythms: The hypothalamic pacemaker. In C. DeLisi and J. Eisenfeld (Eds.), Biome- dical systems modelling. Amsterdam: North-Holland, 1984(c) .

Daan, S. and Berde, C., Two coupled oscillators: Simulations of the cir- cadian pacemaker in mammalian activity rhythms. J. Theor. Biol., 70, (1978), 297-313.

Enright, J.T., T h e timing of sleep and wakefulness. Berlin, Heidelberg, New York: Springer-Verlag, 1980.

Grossberg, S., Some physiological and biochemical consequences of psycho- logical postulates. Proc. Natl. Acad. Sci., 60, (1968), 758-765.

Grossberg, S., On the production and release of chemical transmitters and related topics in cellular control. J. Theor. Biol., 22, (1969), 325-364.

Grossberg, S., Studies of mind and brain: Neural principles of learning, perception, dey~igp~ent, cognition, and ~otor control. Boston: Reidel Press, 1982.

Hedberg, T.G. and Moore-Ede, M.C., Circadian rhythmicity in multiple-unit activity of rat hypothalamic slice. Neurosci. Abstr., 9, (1983), 1068.

Hodgkin, A.L. and Huxley, A.F., A quantitative description of membrane current and its applications to conduction and excitation in nerve. J. Physiol., 117, (1952), 500-544.

108

Inouye, S.T. and Kawamura, H., Persistence of circadian rhythmicity in a mammalian hypothalamic "island" containing the suprachiasmatic nucleus. Proc. Natl. Acad. Sci., 76, (1979), 5962-5966.

Kawato, M. and Suzuki, R., Two coupled neural oscillators as a model of the circadian pacemaker. J. Theor. Biol., 86, (1980), 547-575.

Kronauer, R.E., Czeisler, C.A., Pilato, S.F., Moore-Ede, M.C., and Weitz- man, E.D., Mathematical model of the human circadian system with two in- teracting oscillators. Amer. J. Physiol. , 242, (1982), R3-RI7.

Menaker, M., Extraretinal light perception in the sparrow, I: Entrainment of the biological clock. Proc. Natl. Acad. Sci., 59, (1968), 414-421.

Menaker, M., Aspects of the physiology of circadian rhythmicity in the vertebrate central nervous system. In C.S. Pittendrigh (Ed.), Circadian __°scillati°ns and organization in nervous s~stems. Cambridge, 5~: MIT Press, 1974, pp.479-489.

Moore, R.Y. and Eichler, V.B., Loss of a circadian adrenal corticosterone rhythm following suprachiasmatic lesions in the rat. Brain Res., 42, (1972), 201-206.

Pavlidis, T., Qualitative similarities between the behavior of coupled oscillators and circadian rhythms. Bull. Math. Biol., 40, (1978), 675-692.

Pittendrigh, C.S., Circadian rhythms and the circadian organization of living systems. Cold Spring Harbor Symp. Quan t. Biol., 25, (1960), 159- 185.

Pittendrigh, C.S., Circadian oscillations in cells and the circadian orga- nization of multicellular systems. In C.S. Pittendrigh (Ed.), Circadian oscillations and organization in nervous systems. Cambridge, MA: MIT Press, 1974, pp.437-458.

Pittendrigh, C.S. and Daan, S., A functional analysis of circadian pace- makers in nocturnal rodents, I: The stability and lability of spontaneous frequency. J. Comp. Ph~siol., 106, (1976), 223-252 (a).

Pittendrigh, C.S. and Daan, S., A functional analysis of circadian pace- makers in nocturnal rodents, V: Pacemaker structure: A clock for all sea- sons. J. Comp. Physiol., 106, (1976), 333-355 (b).

Pittendrigh, C.S. and Minis, D.H., The entrainment of circadian oscilla- tions by light and their role as photoperiodic clocks. Amer. Nat., 98, (1964), 261-294.

Schweiger, H.G. and Schweiger, E., The role of the nucleus in a cytoplas- mic diurnal rhythm. In J. Aschoff (Ed.), Circadian clocks. Amsterdam: North-Holland, 1965, pp.195-197.

Stephan, F.Y. and Zucker, I., Circadian rhythms in drinking behavior and locomotor activity of rats are eliminated by hypothalamic lesions. Proc. Natl. Acad. Sci., 6_99, (1972), 1583-1586.

Terman, M. and Terman, J., Circadian rhythm of luminance detectability in the rat: Independence from SCN pacemaker. Neurosci. Abstr., 9, (1983), 1071.

Wever, R.A., The circadian system of man: Results of experiments under temporal isolation. Berlin, Heidelberg, New York: Springer-Verlag, 1979.

ON BIFURCATION FOR NONDIFFERENTIABLE PERTURBATIONS OF SELFADJOINT OPERATORS

Raffaele Chiappinel l i

Let H be a real Hi lber t space with norm I f ' i f , l e t T:D(T)÷H be a se l fad jo in t ope

rator with discrete spectrum o(T), and le t F:H÷H be a continuous map sending bounded

sets into bounded sets and such that F(N)=O. Consider the eigenvalue problem:

( l ) Tu + F(u) = ~u , u~D(T)

I t is well known that , i f F is Fr~chet d i f fe ren t iab le at 0 and F'(O)=O, then

any eigenvalue of T of odd m u l t i p l i c i t y is a bi furcat ion point of ( I ) .

In the appl icat ions, when H=L2(~) (~ bounded open set in ~n), T is some d i f feren

t i a l operator and F is the Nemytskii operator induced by some real-valued function f ,

the assumption F'(O)=O is only sa t is f ied when f~O (e.g. Vainberg L g_I).

We prove that , i f the d i f f e r e n t i a b i l i t y condition F'(O)=O is replaced by:

I iF(u) l [ ~ k I lu [ [

for some k~O and a l l u~H, and i f ~u is an eigenvalue of T of odd m u l t i p l i c i t y with

iso lat ion distance (Kato L 4 ] ) d is t (Xo, ~(T)I{Xo}) >2k, then global bi furcat ion

takes place in the s t r i p [~o -k , ~o+k]xH.

For the proof, we make use of the coincidence degree of Gaines-Mawhin L 3 ] ;

our argument is based on a generalization of the classical formula re la t ing the index

(at zero) of an admissible map to the spectrum of i t s Fr~chet der iv~iveat 0 (e.g.

L 3 ] , L 7 ] ) .

This resul t can be used to prove properties of "almost discrete spectrum" for

semilinear Sturm-Liouvi l le problems such as:

-u" + sin u = xu, O<x<=

u(o) = u(~) : O.

For related results about b i furcat ion for nondif ferent iable mappings, we refer

the reader to L 8 ] and the references therein, where d i f fe ren t methods are used.

110

Statemeqt and proof of the result.

Given Xoe~ and ~>0, we set J(~o,~)::{x~m:l~-~oI<=} and ~(~o,~)={~R:l~-Xol~}. Moreover, we denote with S the closure in IRxH of the set

{ ( k , u ) ~ x H : Tu + F(u) : xu, u#O}

Theorem I - Assume that there exists k~O such that llFeu)ll~Ilull for al~ ~H, and

let k o be an eigenvalue of T of odd multiplicity, such that dist (ho, s(T) l{ho})>2k.

Let C O be the component (maximal connected subset) of SU(J(ho, k) x {0}) which contains

J(Xo, k) x {0}. Then C o is an unbounded subset of J(Xo, k) x H.

In other words, ( I ) has a "continuum" of eigenfunctions of a rb i t ra ry norm with

corresponding eigenvalues within k of ~o; we see therefore that the "size" k of the

perturbation F gives, in a sense, a d i rect measure of the deformation of the eigenspa

ce ker (T-~ol) under the action of F.

Proof. Let ~ = d is t (Xo,o(T)I{~o})-2k>O. I t is easy to check that, for any

~ J(Xo,k+~) ~(~o,k), we have

d is t (~,~(T))>N.

Now take ~,~ with

Xo-(k+~)<~_<~o-k , ~o+k<~<~o+(k+~).

The basic step is the fol lowing. For convenience we shall denote with i(T+F-XI)

the coincidence index at zero of the couple (T, x I -F) , when defined (L.3_~),

Lem~a 2 - Let ~j~ be as above. Then the indices

i (T+F-X_I) , i (T+F-~I)

are defined and unequal.

Proof. For a l l t~LO, l ] and a l l ucD(T) we have

IITu+tF(u)-3ullZIITu-~u -tllF(u)IIz

~llTu-&ull-IIF(u)II~c(3 Ilull-kllu

where c(~) = inf{llTu-~uil:u~D(T),Ilull=l}= (T-#I)-Zll -z. Since T is se l fad jo in t ,

then c(~) = dist(~,~(T)) and therefore

IITu+tF(u)-~ull ~ d (~ ) I l u l l ( t~LO,I~,u~D(T))

where d(~) = c(~)-k>O because of the choice of ~. This shows f i r s t l y (on taking t= l )

111

that Tu+F(u)=~u has no nontr iv ia l solut ion, so that in par t icu lar u=O Ks an isolated

solution and the index i(T+F-~I) is defined; and secondly (by homotopy invariance)

that

i(T+F-~I) = i (T-~ l ) .

The same argument proves that

i(T+F-~I) = i (T -~ l ) .

On the other hand,

i (T -~ l ) = ( - I ) u i (T-TI )

where u is the sum of the m u l t i p l i c i t i e s of the eigenvalues of T lying between ~ and

(L 3 ] ) . The resul t now fol lows.

Now le t C o be as in the Theorem. The proof of the lemma above shows that ( I ) has

no nontr iv ia l solution i f ~ J(~o,k+~)IJ(~o,k); therefore, since C o is a connected

subset of SU(J(~o,k)x{O}), i t must l i e in J(ko,k)xH.

The fact that C o is unbounded follows now as in the proof of Rabinowitz'global

bi furcat ion theorem L 7 ~ by means of straightforward adaptations. See also Laloux

L5J

Applications to ordinary d i f fe ren t ia l equations.

As a typical example, consider the problem

I -u" + sin u : ~u~ O<x<~ (2) u(o) : u(~) : 0

On sett ing H=L2(o,~)=:L 2, D(T)={ucH:u',u"~H and u(o)=u(~)=O}, Tu=-u" for ueD(T),

F(u)=sin u for ucH, we see that (2) is a problem of the type ( I ) .

Here we have o(T) = {~n=n2:n=l,2 . . . . } , each eigenvalue ~n having m u l t i p l i c i t y

one, and since c lear ly F(u)=sin u sa t is f ies l l F ( u ) l l ~ l l u l l for a l l ucH ( i .e . k=l in

th is case), we see that the condition

dist(Xn,o(T)I{~n} ) > 2k

is sa t is f ied for a l l ne~.

Moreover, the condition dist(~,o(T)>k, under which (as shown by the proof of

Lemma 2) ( I ) has no nont r iv ia l solut ion, becomes here k~En2-1, n2+l~ (n=l,2 . . . . ).

112

Therefore there is b i furcat ion ( in L 2) from each "eigeninterval" 6 2 - I , n 2 + l ] ;

moreover, equation (2) i t s e l f and the continuous embedding D(T) --+C1(Eo,~ -])=: C 1

2 lu" 2 and C I equipped with i t s (with D(T) equipped with the graph norm I Iul IL2+I I IL2

usual norm) show that global bifurcation actually takes place in C I.

Making use of standard arguments (e.g. Rabinowitz L7]) together with Lemma 4

given below, we also see that the "eigenfunctions" of (2) possess the right nodal

properties and we get the following description for the solution set of (2).

Theorem 3 - (i) Problem (2) has no nontrivial solution if

~[n2-1, n2+1] (n=1,2 .... );

(ii) for each neIN, there exist in /RxC 1 an unbounded connected set C n

of solutions (k,u) of (2) which lies in [n2-1, n2+1] x a I and meets

the line { (~,0) :heiR} of the trivial solutions. Moreover, if (k, u) aC n

with u~O, then u has exactly n-1 nodal zeros on ]O,'cE.

Remark l - The above results obviously extend to the more general problem:

(3) Lu+f(x,u)=~u , O<x<~

B.C.

where L is a regular Sturm-Liouvi l le operator of the second order, B.C. denotes a set

of separated boundary conditions at the endpoints 0 and x, and f is a real-valued

continuous function on [-O,~_~x ~R satisfying:

I f ( x ,u ) l < klul

for a l l xcEO,~ ] and a l l uc]R.

I f (~n) denote the eigenvalues of L, then for n large enough we have

(4) C n c ~ _ x n - k, X n + k_j x C z

(see L I ] for de ta i l s ) .

Remark 2 - The above class of problems contains as a very special case the l inear

problem :

I Lu + q(x)u = ~u

B . c . '

In this case we have k=I Iq l Ico; c n is simply the s t ra ight l ine {X'n } x E n, where x' n

is the n-th eigenvalue of L+q and E n is the associated eigenspace, and (4) reduces to

saying that IX'n-XnI<k for a l l n, a well known fact in perturbation theory of sel fa-

d jo int d i f f e ren t i a l operators which can be derived, e .g . , by Courant's minimax prin c i

113

ple L2J.

Remark 3 - Under the general condit ion I f ( k ,u ) I sk l u l , no bet ter location of C n seems

to be possible than that given by (4). rhis can be seen from the extreme special cases

f(x,u)=O and f(x,u)=ku: we have ~'n=~n , ~'n=~n+k (n=l,2 . . . . ) respect ively.

We conclude with a tecnical resul t complementing Lemma 2.7 in Rabinowitz L 7 J .

For n=l,2 . . . . set

Sn = {uEC1:u sat is f ies B.C. and u has exactly

n-I simple zeros in ]O, :~E} .

Lemma 4. Let L, hn and f be as in Remark 1 and suppose moreover that f is differen-

tiable w.r.t, u at u=O. Then there exists an E>O such that if (~,u) is a solution of

(3) with O<[lUl[c1<~ and [~-~nI~k, then ucS n,

Proof. Suppose not: then there exists a sequence (~ j ,u j ) inRxC z of solut ions of (3)

with l lu j l lc1~O, uj#O, l~j-~nl~k and ujIS n for al l j .

We can assume that ~j+~o so that [~o-~n[~k. Let u j : = u j / I [ u j [ I c I and set

f ( x ,u ) /u , u#O g(x,u)=

f 'u(X,O) , u=O.

Then g is continuous on [ O , ~ ] x ~ and (~ j ,u j ) sa t is fy

L~j 4 g(uj) ~j = ~j uj

B.C.

for al l je~ . Since (uj) is bounded and L has compact resolvent, we can assume that

u j converges in C 1 to Uo#O. By the cont inui ty of g~ we also have g(.,uj)-~g(.,O)=f~(.,O)

in c ° ( m 0 , ~ ] ) , and therefore u o sa t is f ies : Luo+f~(x,O)Uo=~oU o. In other words, u o is

an eigenfunction of the l inear problem

Lv + f 'u(X,O)v = ~v (s)

B.C.

corresponding to the eigenvalue ~=~o"

Let ( ~ ) be the eigenvalues of the perturbed operator L+f~(x,O); since If~(x,O)l~k

for a l l x ~ [ O , ~ ] , we have Ix~ - ~nI~k for a l l n=l,2 . . . . and since moreover the in-

tervals m~n-k, ~n+k] are d is jo in t for n large enough (because ~n+1-~n++~ as n-~o,

see e.g. L6 i ) , the re lat ion I~o-~nl~k implies that ~o=~.

Therefore, by well known propert ies of eigenfunctions of Sturm-Liouvi l le ope ra to rs ,

114

Uo~S n and so uj~S n for j large (recal l that S n is open in CI). Contradiction.

Remark 5 - The conclusion of Lemma 4 s t i l l holds true when f is no longer assumed to

be d i f fe ren t iab le at u=O. To see th is , set (e.g.) g(x,O)=O; then i t is easy to check

that (despite d iscont inui ty of g at u=O) the map x÷g(x,u(x)) is measurable on L o , ~ ]

for any measurable u=u(x). Let (uj) be as before; then, since Ig(x,u) l~k for a l l

( x , u ) ~ E o , ~ x ~ , the sequence (g ( - ,u j ) ) is bounded in L 2 and therefore i t can be as-

sumed to converge weakly to some w~L 2 with lw(x)I~k for a.a. x ~ L o , ~ ] . Now the con-

clusion follows on considering weak solutions of (5), with w(x) replacing f~(x,O).

References

I . R. Chiappinel l i , On eigenvalues and bi furcat ion for nonlinear Sturm-Liouvi l le ope- rators, to appear on Bol l . Un. Mat. I t a l . .

2. R. Courant and D. H i lber t , "Methods of Mathematical Physics", vol. I ( Interscience, New York, 1953).

3. R.E. Gaines and J.L. Mawhin, "Coincidence degree and nonlinear d i f f e ren t i a l equa- t ions", Lecture Notes in Math. 568, Springer, Ber l in, 1977.

4. T. Kato, "Perturbation theory for l inear operators", Springer, Ber l in , 1970.

5. B. Laloux, Indice de coincidence et b i furcat ions, Th~se, Univ. Catholique de Louvain, 1974.

6. B.M. Levitan and I.S. Sargsjan, "Introduction to spectral theory", Transl. Math. Monographs 39, AMS, Providence, 1975.

7. P.H. Rabinowitz, Some aspects of nonlinear eigenvalue problems, RockY Mt. J. Math. 3 (1973), 167-202.

8. K. Schmitt and H.L. Smith, On eigenvalue problems for nondif ferent iable mappings, J. D i f fe ren t ia l E~uations 33 (1979), 294-319.

l| v - 9. M.M. Vainberg, ar la t ional methods for the study of nonlinear operators", Holden Day, San Francisco, 1964.

ON SURFACES DEFINED BY ORDINARY DIFFERENTIAL EQUATIONS: A NEW APPROACH TO HILBERT'S 16TH PROBLEM

Chin Yuan-Shun (Qin Yuanxun)

Part I

Part II

Part III

Contents

The Problem, Methods and Results

Surfaces Defined by Ordinary Differential Equations

An Example of the Results

Part I

The Problem, Methods and Results

i.i The Problem

D. Hilbert's famous "Mathematische Probleme" of 1900 [i] attract-

ed wide interest among mathematicians and gave tremendous stimulation

to different branches of mathematics. The results and further

developments of these problems, except one, are summarized in the two

volumes [23. The exception is the 16-th problem. Nothing is mention-

ed in [2] about this problem except for a statement of the original

problem ([2] p.23) as follows:

16 The Problem of the Topology of Algebraic Curves and Surfaces.

The maximum number of closed and separated branches that a plane

algebraic curve of the n-th order can have has been determined by

Harnack. There arises the further question as to the relative

position of the branches in the plane. As to the 6-th order, I have

satisfied myself, by a complicated process, that of the eleven

branohes which occur according to Harnack, by no means all can be

external to one another. In fact one branch must exist such that one

of the eleven branches exis~ in the exterior of this branch while the

remaining nine exist in its interior, or conversely. A thorough in-

vestigation of the relative position of the separate branches when

their number is maximal seems to me to be of very general interest,

and not less so the corresponding investigation as to the number, form

and position of the sheets of algebraic surfaces in space. Even now

it is not known what is the maximum nu~er of sheets which a surface

of the 4-th order in three dimensional space can really have.

116

In connection with this pure algebraic problem, I wish to bring

forward a question which, it seems to me, may be attacked by the method

of continuous variation of coefficients, and whose answer is of

corresponding value for the topology of families of curves defined by

differential equations. This is the question concerning the maximum

number and position of Poincar4's boundary cycles (cycles limites) for

a differential equation of the first order and degree of the form

Y dy= n dx X '

n

where X and Y are rational integral functions of the n-th degree n n

in x and y. Written in homogeneous form this equation is

X(YSZ/~t- Z~y/~t) + Y(Z~X/~t- X~Z/~t)

+ Z(X~Y/3t - Y~X/~t) = 0,

where X, Y and Z are rational integral functions, homogeneous of the

n-th degree in x, y and z and where the latter are to be determined

as functions of the parameter t.

The problem concerning differential equations can be restated as

follows:

Find the maximum number N(n) of limit cycles and their relative

positions for the system of ordinary differential equations

d~Xdt = Xn(X'Y) , d~dt = Yn (x'y) (En)

where X (x,y) and Y (x,y) are polynomials of the n-th degree in n n

the real variables x,y.

By"limit cycle" is meant an isolated closed trajectory existing

as the limit of a neighbouring spiral trajectory.

1.2 The methods

The concepts and methods for treating limit cycles were initiated

by H. Poincar~ [3] and are summarized in [4]. H. Poincar~ [3] has

pointed out

"Des que les principes du calcul infinitesimal furent estabilis,

l'analyste se trouva en face de trois probl~mes:

Resolution des equations algebriques;

Integration des differentielles alg~briques;

Integration des equations differentielles.

L'histoire de ces problemes est la m~me ....... "

117

In analogy with Gauss's fundamental theorem of algebraic equa-

tions the solution for the problem of the maximum number of limit

cycles should be sought in the complex domain. Of course this leads

to further complications. I.G. Petrovskii et al [51 and NoNo Molchanov

studied the problem in the complex domain but th4ir results are in-

correct even for n = 2. Nevertheless this approach deserves further

attention.

We begin again with a study in the complex domain. The system

under consideration is a natural extension of the real system

du _ U(u,x) dx at ' d-{ = x(u,x) (E)

to the complex system

dw _ W(w,z) dz dT ' d-T = Z (w,z) (E*)

where w = u + iv, z = x + iy, T = t + iT

and W(w,z) = U(w,z), Z(w,z) = X(w,z).

The geometrical figures of the general solutions for (E*) :

F(w,z) = constant,

are 2-dimensional manifolds in real 4-dimensional (u,v,x,y) - space.

We shall call them solution surfaces.

It is natural to now turn to the study of the topology of surfaces

defined by ordinary differential equations. The real curves defined

by the system (E) are just the intersections of the surfaces of (E*)

with the plane v = y = 0.

As an illustrative example we may take the simplest equation

exhibiting a limit cycle considered by H. Poincar~ :

dw _ - i), dT -z + w(w 2 + z 2

dz - i). dT - w + z(w 2 + z 2

The general solution can be written as follows:

(w 2 + z 2 - i) (w + iz)-l+i(w - iz) -l-i = constant.

There are three special solution surfaces, namely

w 2 + z 2 - 1 = 0, w + iz = 0, w - iz = 0

and three singularities

(w:z:l) = (0,0,I), (l,i,0) and (l,-i,0).

Through each singularity there are two special solution surfaces. On

each special solution surface there are two singularities. The topo-

logy of the family of solution surfaces is completely determined by

these singularities.

118

1.3 An example of the results

The following theorem is an example of specific results.

Theorem. The maximum number of limit cycles for the real quadratic

system (E 2) is 4 with (1,3) structure. In notational form we have

N(2) = 4 with structure (i) + (i m 1 m i) .

The meaning of the notation for structure is self evident as one limit

cycle and one series of three nested cycles.

Part II

Surfaces Defined b[ Ordinary Differential Equations

2.1 ~ y about a singular point of first order [7]

Definition.

A solution surface F is called an isolated limit surface if

there exists another solution surface F 1 such that

FI ~ F but F / F I

Consider the nonsingular system

dw ew+Sz ~I dz ~w + 6z with ~ 0

and define A = -11/12, where

characteristic equation

I ~- X B x = 0.

Theorem 2.1

(i)

(2)

(3)

11 and 12 are the roots of the

If ImA ~ 0, then all solution surfaces pass through the origin

and there exist two and only two isolated limit solution surfaces.

This type of singularity is said to be of "Focus Type".

If ImA = 0 but ReA < 0, then all solution surfaces pass

through the origin, but there exists no isolated limit solution

surface. This type of singularity is said to be of "Node Type".

If ImA = 0 but ReA > 0 then there exist two and only two

solution surfaces passing through the origin and there exists no

isolated limit solution surface. This type of singularity is

said to be of"Saddle-Centre Type".

119

The proof of Theorem 2.1 is given in [73.

One can verify the results directly from the general solution of

the following form

F 1 F 2 = constant .

On the basis of this theorem and the results on the analytic theory of

ordinary differential equations [123 one gets

Theorem 2.2

For the nonsingular system with nonlinear terms:-

dw = ~w + Bz + ......

dz yw + 6z + ...... '

with ~ ~ ~ 0, in the neighbourhood of the origin we have

(i) the conclusion of Theorem 2.1 for a focus type singularity still

holds,

(2) the conclusion of Theorem 2.1 for a node type singularity still

holds provided A ~ -i.

(3) for the case ImA = 0 but ReA > 0 there are two solution

surfaces passing through the origin.

This last case is very complicated to analyse and should be classified

further as shown in the next section.

2.2 Focus of hi@her order

The focus of higher order is a special type of the case

ReA which is of particular interest.

Theorem 2.3

ImA = 0,

Consider the real system:-

dx d-~ = -y + X(x,y),

dy = x + Y(x,y), dt

where X(x,y) and Y(x,y) are terms of higher order. Then there

exist two functions

~(x,y) = (x+iy) + F2(x,y) + F3(x,y) +...

M(x,y) = i + C2(#~) + C4(¢~)2 +...

such that

d~ _ Me d~ -- dt ' ~ = ~M,

120

where f denotes a change in sign of the imaginary part of coeffic-

ients.

(i) If all real parts of C2n vanish, i.e. C2n = 0~n=l,2 .... ) then

one obtains the solution corresponding to a real centre as

Poincar4 has done: namely

~ = constant.

(ii) If Re(C2n) ~ 0 for some n then the first nonvanishing Re(C2n)

gives the order of the real focus as n.

The general solution in the neighbourhood of the focus can be written

in the following complex form:

[~] ire (C2n) i ~! < }_~ Exp 1 + [ d2k(%~) k = constant.

(~)n kkl _

2.3 Connections of singular points with solution surf_aces in the large

The connections of singular points with solution surfaces are

crucial parts of the toplogy of the family of solution surfaces. For

the connections the following theorem is proved in [73.

Theorem 2.4 (Rooted Theorem)

Each solution surface of (E*) with analytic right hand sides

either connects with certain singular points of (E*) or extends to

infinity.

There is no corresponding relation in the real domain as shown by

the following counter example:

dx d--[ = -Y' = x

2 2 with a series of circles, x + y = constant ~ 0 as solution curves

neither passing through the singular point (0,0) nor tending to infin-

ity in the real (x,y) domain.

We are interested, in particular, in the solution surfaces pass-

ing through the real limit cycles of (E).

According to the rooted theorem concerning the solution surfaces of

(E*), there are also two types of limit cycles of (E) ; those of (E)

connecting with finite singular points of (E*) through solution

surfaces of (E*) and those of (E) extending to infinity through solu-

tion surfaces of (E*) without passing any finite singular points of

(E*). The type considered by H. Poincar6 is an example of the latter.

121

The first type was first considered by the author [8] as the following

system

dx _ dt -y(ax + by + c) + (x 2 + y2 _ I),

dy x(ax + by + c) b 2 2 b 2 2 dt = , < a + < c ,

with the unique real limit cycle of (E) passing through the two finite

complex singular points of (E*) namely

(- ac _+ ib/~ a2 L b-~ - bc ~ ia/cZ - aZ - bZ 1

a 2 + b 2 ' a2 + b 2 "

A general theorem about this type of limit cycle is the following.

Theorem 2.5

If a real limit cycle of a real system (E) connects with a focus

of the complex system (E*) through a solution surface F of (E*),

then F is an isolated limit solution surface attached with this

focus of (E*).

The proof is based on the isolated limit property of limit

cycle

This theorem shows that the number of real limit cycles of (E)

connecting with foci of (E*) through solution surfaces of (E*) is

controlled by the number of isolated limit solution surfaces attached

with the foci of (E*). As an example, the solution surfaces

2 z 2 w + - 1 = 0

of the system

dw _ dT -z (aw + bz + c) + (w 2 + z 2 - i),

dz d--T = w(aw + bz + c),

passing through the real limit cycle

2 x 2 u + - 1 = 0

is also an isolated limit surface attached with the two complex foci

whose coordinates have been given above.

Now we turn to the other type of limit cycles. There is the

potential to produce a series of nested limit cycles from a focus of

high order. In fact Bautin [93 was the first to establish the poss-

ibility of the appearance of three nest limit cycles from a focus of

order 3 of (E 2) . The author and Pu [101 gave a method and constructed

a specific example as follows:

122

dx 2 d-~ = Ix - y + (2+6)xy + y ,

d_Z = x + ly + x 2 - (5+s)xy - y2 dt

with three nested limit cycles jumping out successively from the focus

as the three parameters s, 6 and I varied from 0 to

S = 10 -102, 6 = 10 -466 , I = 10 -1428

We now give a general theorem about this type of limit cycles.

Theorem 2.6

By suitably perturbing the coefficients of (E) with a focus of

order n (n>l) the general solution can be changed to the standard °rml n

where L~ = 0 (j = 1,2 ..... n) are equations of the limit cycles jump- 3

ing out from the focus of high order, ¢i = 0 and ~I = 0 are the

two isolated limit solution surfaces of the focus now of zero order,

Hj~j+ 1 < 0 (j = l,...,n-l). The right hand side is regular near the

focus.

Hence the order of the focus limits the number of limit cycles

jumping out from the critical focus.

Now we are ready to discuss some specific problems.

Part III

An Example of Results

Theorem

The maximal number of limit cycles of (E 2) is 4 with (1,3) struc-

ture. That is in notational form N(2) = 4 with structure (i) +

( i ~ i ~ i ) .

This theorem will be proved via a series of steps using

successively variation of the coefficients of (E 2) and considering the

changing of the topological structure of the family of solution

surfaces of (E~), and getting results for (E 2) in the real domain.

123

3.1 The system !E~} with A focus of order an_ d a uni_ que ~ingularr ]~oint at__infinity

I shall first recall a theorem proved by the author, Shi and Tsai

[4].

Theorem

Assume that (E 2) possesses a critical focus of order three and a

unique singular point at infinity, then (E 2) possesses another non-

critical focus and no other singular points exist. Around the non-

critical focus there is an odd number ~ (at least one) of limit

cycles. Around the critical focus of order 3 there is an even

number B (possibly zero) of limit cycles. The total number N of

limit cycles is given by

N = ~ + 8 = odd number a i.

In the real domain I now give the exact number of this theorem as

follows.

Theorem 3.1

Assume that a real system (E 2) possesses a real critical focus of

order 3 and a unique real singular point at infinity. Then (E 2)

possesses another real non-critical focus and no other real singular

points exist. Around the real non-critical focus there is one limit

cycle of multiplicity i, i.e. a : I.

Around the real critical focus of order 3 there is no limit cycle, i.e.

8 = 0.

The total number N of the limit cycles is

N = ~ + 8 = i.

The proof of theorem 3.1 is based on the following theorem relat-

ing to the complex domain.

Theorem 3.2

The assumptions are the same as for theorem 3.1. Then there exist

four finite singular points for the complex system (E~) , namely the

real critical focus PI' the non-critical focus P2 and two complex

foci P3 and P4 in complex conjugate positions. All solution

surfaces of (El) pass through P3 or P4"

For simplicity of the proof we use the reduced form of (E 2) as in

[4].

124

du _ 2 dt -x + £u 2 + 5aux + x ,

dx d-~ = u(l + au + (3£ + 5)x),

(3.1)

where

inequali ties.

a > 0, ~ < 0,

2a 2 + 2 + £ ~ 0,

and

500a 4 + (25 (2£ + 5)

a and £ are two real parameters satisfying the following

a2(5£ + 6) - 3(I + 1)2(£ + 2) # 0

(3.2

(3.3

2 + 90(2£+5) - 27)a 2 + 4(2£ +5) 3 < 0 (3.4

We shall now consider the complex system

aw = -z + Zw 2 + 5awz + z 2 dT

dz dT - w(l + aw + (31 +5)z)

(3.1)*

In order to establish theorem 3.2 we go one step further by varying

(a,i), but retaining the inequalities (3.3) (3.4), such that a tends to

zero. From (3.4) one gets, for a = 0,

2~ + 5 < 0. (3.5)

Now we consider the system (3.1)* with a = 0, i.e.

d_~w = -z + £w 2 + z 2 dT

dz dT- w(l + (3i + 5)z),

(3.6)*

subject to the condition (3.5).

2 On using w as a variable (3.6)* can be integrated explicitly

to give

~l(W,Z = Constant(~2(w,z) ) (2£/(3Z+5)) (3.7)

where

¢l(W,Z _ w 2 _ 1 5z2 + 4(£+3)z 2(£+3) (3 8) (2£+5) (£+5) + £(2£+5) (£+5)

and ~2(w,z H 1 + (3£+5)Z (3.9)

Note:- for Z = 0, -5/3, -5/2, -5, the general solutions contain terms

with logarithmic or exponential functions and should be integrated

separately.

Now it is easy to see that there are two real centres PI(0,0)

and P2(0,1). Furthermore there are two finite complex singular points

P3,4: (-+i/ 3(£+2)/~(3~+5)2), -I/(3Z+5)),

125

with i = -11/I 2 = -2£/(31+5).

So Im(A) = 0 and Re(A) < 0 for 2~ + 5 < 0. These two singular

points are of nodal type. All solution surfaces pass through the

intersection of the two solution surfaces %1 = 0 and %2 = 0, and

so they pass through the two complex nodes.

For simplicity we denote the four singularities by

(C,C,n,n) ,

where C denotes a real centre and n denotes a complex node.

Now by varying (a,£), under the conditions (3.2), (3.3) and (3.4),

the real centres change to real foci and the complex nodes change to

complex foci or in notational form

(C,C,n,n) => (F,F,f,f,) .

That is, we get a system with two finite real foci and two finite

complex foci.

It is of interest to note that in the neighbourhood of n and f

all solution surfaces pass the singular points. Thus when we vary

(a,£) such that n changes to f, in the neighbourhood of the complex

singular points, the property that all solution surfaces pass through

the complex singular points remains unchanged.

Outside the neighbourhood of the complex singular points, the

solution surfaces vary continuously with continuous variations of

(a,£). Thus the property that all solution surfaces pass through the

complex singular points remains unchanged. This proves theorem 3.2.

Now we return to the proof of theorem 3.1. The coordinates of the

two complex loci P3(w3,z 3) and P4(w4,z4) of the system (3.1)* are

3a(2£ + 3) ± i~(3£ + 5) (3(£(£ + 2) - 3a2)))

w3'4 = (3£ +5) 2 - a2(15£ + 24)

3a 2 - £(3£ + 5) ~ ia~3(£(£ + 2) - 3a2))

z3'4 = £(3£ + 5) 2 - a2(15£ + 24)

According to theorem 2.5 the real limit cycles of the system

(3.1) connect with the complex focus P3 through the isolated limit

solution surfaces attached to the complex focus P3" There are only

two isolated limit solution surfaces attached to P3 and so the total

number of real limit cycles of the real system (3.1) is then limited

by the number of isolated limit solution surfaces attached to the com-

plex focus P3 of the complex system (3.1)*, i.e.

N = ~ + B S 2

126

with both N and ~ odd and B even. Hence one gets precisely

N = i, ~ = 1 and B = 0.

Theorem 3.1 is thus proved. This limit cycle is denoted by L 0 = 0.

L 0 passes through P3 and P4' P2 is inside L 0 and Pl is

outside L 0. We can construct the general solution of (3.1)* in the

following form

(~i/~l)iCo expl-- ! ..... 311 +. ~1%~0~3~ -~2 5~ 3(~i~1) ..... -0 -3 ~ 32 ~2=J

= constant

where 4. = 0 (j = 2,3,4) are isolated limiting solution surfaces ]

at Pj, L 0 = 0 is one at P3 and P4 and ~3 = v4"

Now we consider the singular points at infinity. In the same way

as Poincar4 we put

w = i/~, z = ~I~ and investigate the singular points on the plane ~ = 0. There are

three singular points denoted by P5' P6 and P7" The set of seven

singular points of the system (3.6)* can be represented by

(C,C,n,n,S,n,n)

and the system (3.1)* by

(F,F,f,f,S,f,f).

Theorem 3.3

If the system (E2)* possesses the set of singular points

(F,F,f,f,S,f,f) then for the corresponding real system (E 2) the

maximum number of limit cycles N(2) = 4 with the structure

(i ~ i ~ i) + (i).

Proof

Without loss of generality we may start from the following

standard form

dw _ 2 dT -z - lw - £w 2 + (5a-6)wz + nz ,

dz d--t = w(l +aw + (-3£ +5n+~(£-n) - ~)z ,

with six parameters £, n, a, 6, e and I.

Starting with the case

a = ~ = s = I = 0, n > 0, -3Z + 5n < 0,

one gets the set of singular points

(3.7)*

127

(C,C,n,n,S,n,n).

Keeping n and Z fixed and varying a > 0 one gets the set of

singular points

(F,F,f,f,S,f,f)

with a critical focus of order 3 at the origin and all solution sur-

faces pass through the finite complex foci as proved in theorem 3.2

Varying a, 8, e and 1 successively such that

>> n >> a >> ~ >> e >> I > 0 ,

the general solution can then by expressed by theorem 2.6 as

[(~I)~I$~I]LIPlL2~2L3~3[L0P0~3~3~4~4][(~2)~2(~2 )~2]

= constant G(w,z),

where G(w,z) is regular in w and z and G(w,z) # 0 for finite

w and z, with one limit cycle around the non critical finite focus

P2 and three nested limit cycles Lj = 0 (j = 1,2,3) around the non

critical focus PI" Hence one gets the structure

(i ~ i ~ i) + (i).

By varying Z, n, a, 6, C and ~ arbitrarily but retaining the set of

singular points

(F,F,f,f,S,f,f)

the unique limit cycle around P2 remains but some or all of the three

real nested limit cycles L 1 = 0, L 2 = 0 and L 3 = 0 may disappear

either through the coincidence of a pair of them separating as a pair

of solution surfaces without real foci or through the singular point

PI" Hence the total number N cannot be increased and theorem 3.3

is then proved.

Now we relax conditions (3.2) and (3.4) for the system (3.1).

Starting again with the system (3.6)* we get a strong rooted

theorem.

Theorem 3.4

For the system (3.6)* each solution surface passes through some

of the seven singular points.

Proof

We already have an explicit expression for the general solution

and so we can verify the theorem directly. In fact we get more in-

formation as the following table shows:

128

Set of Singular Points Control Set

i < -5/2 C,C,n,n,S,n,n (P3P4)

-5/2 < Z < -2 C,C,n,n,S,N,S (P3P4)

-2 < ~ < -5/3 C,S,N,N,S,N,S (P3P4)

-5/3 < ~ < 0 C,S,S,S,N,N,N (P5P6P7)

0 < ~ C,S,n,n,N,S,N (P3P4)

Now we go one step further and consider the two parameters (a,Z).

We shall consider the critical lines for the changing of the singular

points.

P2 changes from F to N on 25a 2 + 12(Z+2) = 0.

P2 coincides with P3 or P4 on i = -2.

P3 and P4 coincide on 3a 2 - Z(Z+2) = 0.

P3 or P4 goes to infinity to coincide with the singular point at

infinity on the line (15Z+24)a 2 - Z(3~+5) 2 = 0o

P6 and P7 coincide on the line defined by (3.4).

These lines divide the parameter plane (a2,Z) into ten regions with

different types of sets of singular points. The ten regions on the

parameter plane (a2,Z) are labelled from (1) to (X) by the order of

increasing Z and then of a 2. The following is the table of the

sets of the ten regions.

No. of Re@ion Set of Sinqular Points

I F,F,f,f,S,f,f

II F,F,f,f,S,N,S

III F,N,f,f,S,N,S

IV F,N,N,S,S,N,S

V F,N,S,S,N,N,S

VI F,S,N,N,S,N,S

VII F,S,S,N,N,N,S

VIII F,S,S,S,N,N,N

IX F,S,N,S,N,S,N

X F,S,f,f,N,S,N

One can see at once that except in Cases I and II, P2 is not a focus

and so there is no limit cycle around P2" Case I has been proved in

theorem 3.1. Thus only Case II needs to be studied.

129

Relaxing the condition (3.4) a little as done in [ii] we still

get N = i. However near the line 25a 2 + 12(Z+2) = 0, the limit

cycle may make contact with the singular points at infinity P6 and

P7 and cease to be a closed curve in the ordinary sense. Therefore

for Case II, N s 1 and there is also no limit cycle around the

critical focus PI" In summary we have the following theorem.

Theorem 3.5

For the system (3.1)*, N s I. If N = i, the limit cycle is

around the non-critical focus. Each solution surface passes through

some singular points of the system (3.1)*

Based on theorem 3.5 and by similar reasoning along the lines of

the proof of theorem 3.3 the final conclusion is reached as follows.

Theorem

For (E2), N(2) = 4 with the structure (i ~ 1 z l) + (i).

Furthermore one can realise all possibilities as the following

theorem shows.

Theorem 3.6

For (E 2) all possible combinations of

8 ~ 3 can be realised.

The proof is simply given by specific numbers for the system

_ dw = 2 dT -z + lw + £w 2 + a(5+s)wz + nz ,

_ d__zz = w + Iz + aw 2 + (3£+5n+8(£+n)+6)wz, dT

as the following table shows

~,8) with ~ s 1 and

N e B n £ a 6 ~ 1

0 0 0 1 -3 1 0 0 0

1 0 1 1 -3 1 10 -113 0 0

2 0 2 1 -3 1 10 -113 10 -508 0

3 0 3 1 -3 1 10 -113 10 -508 10 -1556

1 1 0 1 -6 1 0 0 0

2 1 1 1 -6 1 -10 -108 0 0

3 1 2 1 -6 1 -10 -108 -10 -487 0

4 1 3 1 -6 1 -10 -108 -10 -487 -10 -1496

The numerical values were calculated by Qin Jen-Sui [133 with a

programme designed for this special purpose.

130

All possible relative positions of limit cycles for (E 2) are

now realised.

Note: Mr Qin Chao-Bin has devised a method to calculate the complex

solution by computer [14]. Hence one can trace the solution surfaces

in four dimensional space to provide intuitions and to check results.

References

[i] D. Hilbert, Mathematische Probleme, Gottingen Nachrichten (1900) 253-297.

[2] Mathematical Developments Arising from Hilbert Problems, Vol. I and II, Proceedings of Symposia Pure Mathematics, Vol. 28 (1974).

[3] H. Poincar4, Sur les courbes definies par des 6quations differentielles (1881-1886) OEUVRES de Henri Poincar6 I.

[4] Chin Yuan-Shun (Qin Yuanxun), Shi Song-Ling, Tsai Sui-Lin, On limit cycles of planar quadratic system, Scientia Sinica (1982) Series A, Vol. 25, 41-50.

[5] I.G. Petrovskii and E.M. Landis, On the number of limit cycles of the equation dy/dx = P(x,y)/Q(x,y), where P and Q are poly- nomials of the second degree, Mat. Sb.N.S. 37 (79) (1955), 209-250 (in Russian); Amer. Math. Soc. Transl. (2) i0 (1958) 177-221.

[6] N.N. Molchanov, The use of the theory of continuous groups of transformations in investigating the solutions of ordinary differential equations, Dokl. Akad Nauk SSSR(N,S,) 112 (1957) 998-1001 (in Russian).

[7] Chin Yuan-Shun, Qualitative Theory of Ordinary Differential Equations in Complex Domain, I.II.III. (in Chinese) Research and Applications of Mathematics, Institute of Applied Mathematics, Academia Sinica, No. 4 (1979), 17-33; No. 5 (1979) 18-48; No. 1 (1980) 15-36. Also see Journal of Northwest University No. 3 (1982) 1-18.

[ 8 ] Chin Yuan-Shun, Uber den Differentialgleichungen

dy/dx = ~ a..xlyl/ ~ bijxly3 i+j~z 13 i+j~z

mit algebraishen Grenzzyklen zweiter Ordnung. Science Record N.S.I.2. (1957) Academia Sinica.

[ 9 ] N.N. Bautin, On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or centre type, Mat. Sb.N.S. 30 (72) (1952) 181-196 (in Russian); Amer. Math. Soc. Transl. No. i00 (1954), and in Amer. Math. Soc. Transl. (i) 5 (1962) 396-413.

[i0] Chin Yuan-Shun, Pu Fu-Chung, Concrete example of three limit cycles appearing in the neighbourhood of a singular point of a quadratic system dx/dt = P, dy/dt = Q. Mathematica Sinica Vol.9 (1959) 213-226.

131

Ell] Chin Yuan-Shun (Qin Yuanxun) So Guan-Jan, Du Xun-Fu, On limit cycles of planar quadratic system II, Scientia Sinica (1983) Series A Vol. 26 (1983) 1025-1038.

[12] L. Bieberbach, Theorie Der Gewohlichen Differential Gleichungen (1953) .

[133 Qin Jen-Sui, Limit Cycle Producing Program System, (1983) (To be published).

[14] Qin, Chao-Bin, numerical method for calculation of two dimensional surfaces in four dimensional space in connection with solution of ordinary differential equations in complex domain, (1981) (To be published).

SELF-ADJOINTNESS FOR GENERAL ELLIPTIC OPERATORS WITH SOBOLEV-TYPE COEFFICIENTS

Dung Xuan Nguyen

I. Introduction.

Let T be a general elliptic operator of the form

T = Z (-i) lai~ea B(x) ~8 + q(x), on L2 ~Rn).

o<l~t,l~l~m

Here m is an integer greater or equal to i, e and 8 are multi-indices,

= (~i ..... an)' De = (~i) "'" (~)n

n

Throughout the whole paper we assume the following:

(i) Uniform strong ellipticity: there exists E 0 > 0 such that

(x)~B) > E01~J 2m Re ( Z £ a ~ _

l~l--l~l=m for all x,~ 6~ n,

(ii) a B(x) is complex-valued with a 8(x) = aBe(x),

(iii) q(x) is real-valued.

When m = i, the coefficients a 8(x) are put in the form of a matrix

(aik(X)). In view of this, we shall refer to (a 8(x)) as a "general-

ized" matrix.

The purpose of this paper is to establish minimal conditions on

the differentiability of the coefficients of T, in order for T to be

self-adjoint on H2mdRn). Roughly speaking, a~8(x) need only have dis-

tributional derivatives in some appropriate LP-spaces. The main results

are in theorem 2 and its corollaries. Theorem 1 gives a key a priori

estimate of the type given by Browder in [2]. We shall make frequent

use of the Sobolev imbedding theorem which can be found in [i]. The

notations we use for Sobolev spaces and their norms are standard and

can also be found in [i]. (The space --WK'2dR n) is customarily denoted

by Hk~Rn), and its norm is denoted by I J "J JK-)

We wish to express our gratitude to Professor T. Kato for his

133

valuable suggestions and his constant encouragement.

II. A Priori Estimate.

In this section we establish an inequality of the type given by

Browder in [2] (theorem 2). This is formally G~rding's inequality for

T 2, since T is symmetric in our case. Because our assumptions on T are

different, we need to modify Browder's argument somewhat, in order to

prove our desired estimite. Throughout the whole paper, unless speci-

fied otherwise, T will satisfy the following general assumptions:

(A.I) 2(2m - I) < n.

(A.2) For I~ + BI : 2m, a ~(x) is uniformly continuous and bounded

on ~ n. And if 0 < IYI < max(lal, IBI), then the distribu-

tional derivative 3Ya (x) 6 Ls(~ n) for some s > T~ (in • > n

the case m > i, we can admlt S _ ~-~)"

(A.3) For 2 ~ i~ + 81 < 2m, IYI ~ max(lel, 181), ~ ~' the distributional

S y a B n derivative (x) 6 LS(~ n) for some s h 2m_l~+Bl+Iy I"

n (A.4) q(x) 6 LS(R n) for some s > ~ (if 2(2m-i) < n, we can n

admit s ~ ~-/T).

Let x 0 6 R n bel Ifixed" For la+81 = 2m, let a O = ~ a B(x0), and

T O = Z (-i) l~l~a° ~. le+Bl=2m ~B For 0 ~ t ! i, let

T t = T O + t(T-T 0) = Z (-i) l~l~a t ~B

0<l~i,lSl!m ~(x) + q(x),

whe re

a~8(x) =

o for I~+8 t a 8(x) + (l-t)a B ,

t a B(x) , for [e+8

= 2m

< 2m.

We first show that T t with domain D(T t) = H2m(~ n) takes values in L2(~n).

Proposition 1. There exists a positive constant C independent from t,

such that I IT t ul I 0 ~ C I I u 112m for all u 6 H2m(~n).

Proof. It suffices to prove the proposition for T. By Leibnitz's rule,

we have formally

134

TU = Z (-l) l~la ~e+Su l~+sl=2m

g (-i) Ic~IZy(~(~Ya 8)

I a+B I=2m 0<y:i~

E 2<_la+S I <2m

O<y<a

(-i) I~ £yG($Ya B)zc~+8-Yu + qu. ( l)

Consider the terms with I~+81=2m. By (A.2), a S is bounded, so

we have

lla~Ba~+Su 11o E lla~ B 11 ~ lla~+~u 11o L

(2}

I I%s II ~ Ilull2m- (2 L

Also from (A.2), if 0 < y ~ ~, then 3Ya B 6 LS~ n) with s > ~. This,

½ 1 1 1 > 1 I~I > 0 In this case, together with (A.I), imply ~ r - 2 s 2 - n -- "

the Sobolev imbedding theorem says that HIYI(~ n) is imbedded in Lr(~n).

This means there exists a positive constant C(Iyl,r,n) such that

l l~+S-¥u It r - - ' IHIy I < C ( t y I r , n ) Ila~+S-Yu I (3) L

c I lul l~m, because I~+s-Yl+lYl=l~+Bl = 2m. (3) '

(We notice that if m > 1 and s ~ ~, then we would have

1 1 1 1 1 1 ~_> r= ~- ~ > ~ - > 0.

The Sobolev imbedding theorem is still applicable, and (3) and (3)'

still hold.)

H61der's inequality now gives

It(aYahs )a~+B-Yu Iio £ lla~a~BtlLslla~+B-YullLr (4)

I ILS Ilu112m, by (3 '. (4) '

NOW let us consider the terms In (i) with 2 ~ le+8I < 2m and Y ~ e.

~ya~ B n By (A.3), we have £ LS(~ n) with s > 2m_ ~+Bi+y I This, together

1 1 1 1 1 2m~le+~-Y > 0 The Sobolev with (A.I), yield ~ ~ ? - 2 s ~ 2 - n

135

imbedding theorem then implies that H2m-le+B-YI(~ n) is imbedded in

Lr(~n), i.e., there exists a positive constant C(~,~,y,r,n) such that

II a~+s-~u llLr ! C(~,8,y,r,n) II~+B-Yu l]2m_l~+s_~l (5)

c I u [12m. (5) '

H61der's inequality gives

l l (a~aB) a~+S-~ull o ~ l laYas IILs lla~+S-~u llLr (6)

c l i a~a~t lL s II u ll2m, by (5)' (6) '

n Finally, by (A.4), q 6 Ls(IR n) with s > ~ . This and (A.I) give

1 1 1 1 1 2m-i > 0 Thus H2m-I(IR n) is imbedded in Lr(IR n) . 2 >--?= 2- s > 2 n -- " This fact and Holder's inequality yield

llqull 0 ~ IIqIILSlIUlILr (7)

Constant I lql IL s I lu112m_l. (7)'

n 1 > 1 2m-i > 0 So (We notice that if 2(2m-i) < n and s ~ 2-~-i' then F -- 2 n "

H2m-l(~ n) is still imbedded in Lr(Bn), and (7) and (7)' still hold.)

Putting (2)', (4)', (6)' and (7)' back into (i), we see that if

u 6 H2m(Rn), then Tu 6 L2(~ n) and I ITul I 0 ~ Constant I lu112m. •

We shall use the following lemmas which are special cases of re-

sults given by Browder in [2]. We rewrite them here in order to facil-

itate future references.

Lemma i. (Partition of unity; see Lemma 2 of [2]).

There exist an integer R and constants K > 0, 60 > 0, such that

given 6 with 0 < 6 < 60, there exist a countable open covering {N k} of 2m

~n and a family of functions {~k } with qk £ C0 (Nk) which satisfy the

following conditions:

a) At most R distinct members of the covering {N k} have a non-

empty intersection, while each N k has diameter diam(N k) < 6.

b) For every x 6 ~n ~k(X ) =~in~ , 0 ~ ~ 1 and (x) = i. k

c) For every e and 8 with I~I < 2m, IBI < 2m, we have

136

co

[ IDa~k (x) 121DB~k(X)I 2 <_ K26 -21~+BI k = l

< oo

Lemma 2 (see Lemma 3 of [2]).

Let {n k} be as in Lemma i. There exist constants K.,0 j j J 2m,

depending only on K and R of Lemma i, such that for all3u £ H2m(~n),

we have for 0 < j < 2m

IJ~Jlj _ j k

zIL~ultj < ~ z 6-2(j-i) tluLti < ~.

k -- 3 i! j

Lemma 3 (see Lemma 8 of [2]).

Let B 0 be the unit ball in ~n. Suppose A 0 = ~ a°3 ~ is a homo- i~]=2m

genous uniformly elliptic differential operator with constant coeffi-

cients and ellipticity constant c O . Then there exists a constant

K(c 0) ~ 0 such that if u 6 H2m(B0 ) has compact support in B 0, then

u < K(c 0) i IA0ui iL2(B0) i I i iH2m(B0 ) _

Remark. Because A 0 is homogenous and has constant coefficients, Lemma

3 still holds if we replace B 0 by B(x,@), a ball centered at x with

radius 6.

Proposition 2. There exist positive constants 60,K depending only on

n,m,E0, 113~ 811LS where s is given by assumptions (A.2) and (A.3), the

moduli of continuity of a S with I~+BI = 2m, and IIql IL s where s is

2 given by assumption (A.4), such that if {~k,Nk} is the partition of

unity given in Lemma 1 with diam(N k) < 60, then for 0 j t J 1

2 ~ 12 < K(I iTt(n~u) [ 12 + i i~ul i2 ), I lqkUl I H2m(Nk ) -- L2(Nk ) L2(Nk )

for all u 6 H2m(~n).

2 Proof. Let {~k,Nk} be the partition of unity given in Lemma i.

N k c B k = B(Xk,6), a ball centered at x k with radius 6. Let

T k = ~ (-i) l~I~a t (Xk) 3B , and u k 2 ie+Bl:2m ~ = ~k u.

Suppose

137

In view of the remark that follows Lemma 3, we have

IIUkll 2 < Constant H2m (Bk) --

tTkUkll 2 L2(Bk )

IIt(aeB(x)-a s(Xk))5~+SUkl 12 _~ Constant l~+~{=2m L2(Bk)

Z £T~l It(3Ya B) a~+8-TUkl I 2 [a+Sl=2m L2(Bk ) 0<y£~

Z 2< I <y,+8 I <2m £Ta ] t(~Ya B) 3a+8-TUkl 12L2(Bk)

+ t lqukl t 2 + I ITtUkl { 2 )- L 2 (Bk) L 2 (Bk)

(8)

Let s > 0 be arbitrary. Because a S is uniformly continuous for Ia+BI = 2m,

there exists 61> 0, such that if Ix-yI< 6 i then Z (x)-a B(y) 12 {a+Bl:2mla B <a.

Thus if 6 < 61, then

E I I (a B(x ) ) ~e+BUk] 12 < sl lUkl 1 2 , (9) [~+8]=2m -aaB(Xk) L2(Sk ) H2m(Bk )

Let us consider all the a B's such that 3Ta B 6LS(~ n) with s# ~. Then

there exists 62 > 0, such that if 6 < 62 then I lSTa BI 12 < s. LS(Bk )

Thus by (4)' and (6)', we have, whenever 6 < 62:

I I (3Ya B) 3~+S-YUkl I 2 < s Constant L 2 (Bk) -- I lUkII2H2m(Bk) (lO)

In the case s = ~, we have

I I (STa 8) Se+B-~ukl 12 < A115~+8-TUk[ IL2(Bk) L 2 (B k ) --

, A : max I IZYa 811 a,B,Y

oo

L

< Allukll ~ -- H2m-i (Bk)

, because l~+B-Yl!2m-i

<A(~ttuklt 2 -- H2m (B k) + C(~)llUklt2L2(Bk) ) , (ii)

138

where C(£) is a positive constant depending only on ~ (this is a stand-

ard fact; see for example Lemma 6 of [2]).

Finally, by (7)', we have

Itqukll 2 < Constant Iqll 2 Itukl L 2 (Bk)-- L s (Bk)

2

H 2m-I (Bk)

< Constant (~llukll 2 + C(~)llukll 2 ). -- H2m (Bk) L 2 (Bk)

12)

Putting (9), (i0), (ii) and (12) back into (8), and since 0 ~ t ~ i, we

get

I lUkl 12 < Constant (tl lUkl i 2 + c Constant i lUkl i2H2m(Bk ) H2m(Bk )- H2m(Bk )

+ c(~)I lUkl ILe(Bk) + I ITtUkll 2 L 2 (Bk)

whenever ~ < min(61,62). Here, the generic term "Constant" does not

depend on k, 6 nor s. Also min(61,62) does not depend on k. Since c

is arbitrary, we can choose ~ small enough to absorb the terms involv-

ing llUk112H2m(Bk ) on the right hand side, into the left hand side.

Proposition 2 is thus proved, m

We now state and prove the key a priori estimate.

Theorem 1. There exists a positive constant C depending only on

n,m,E0, I l~Ya 811LS where s is given in assumptions (A.2) and (A.3), the

moduli of continuity of a B(x) with I~+81 = 2m, and I lql IL s where s is

given in assumption (A.4), such that for 0 < t < 1

I l u l I 22m < c < l t T t u l l ~ + I l u l l 2 f o r a l l u 6 H2m(~ n) _ 0 ) , -

Proof, Let {~k,Nk be the partition of unity given by Lemma 1. Let

diam(N k) < 60, where 60 is given by proposition 2. Let u 6 H2m(~n).

By Lemma 2 and Proposition 2, we have

l luI t22m < Constant E( I InkTtU t 12 + % + i lqn~ul 12 -- k L 2 (N k) L 2 (N k )

+ I In~ul 12 ), (13) L 2 (Nk)

139

where

Ak= o<l~l~l~l!m o<x!~+~-y ~ x ' ~ + ~ - ~

I1( ax (n~))(3Yat~)a°~+s-~-Xu II 2 L2(Nk)"

Using Leibnitz's rule and Lemma lc, we get

Zl~l(n~) I ! Constant. k

This and the fact that 0 < t < 1 immediately imply

Z A k ~ Constant ~ Z k o<l~l, lSl<m O<X!~+S-~

If (~Ya B) 3~+B-Y-lu I 12 L 2 (i~ n )

Constant Z Z I I~Ya BI I 2 < -- L s (Bn) I I 3°~+8-Y-Iul 12Lr([Rn ),

(14)

in view of (2),(4) and (6).

From the Sobolev imbedding theorem, we have (as in (3) and (5))

113~+B-Y-Iul 12 < constant I I~R+8-Y-Iul 12 Lr(an) -- H2m-l~+8-yl

Constant (£I lu1122m+C(c) I lull~), for any £ > 0 (15)

and for some C(e) > 0, because 2m-If I < 2m-l. I

Combining (15) with (14) yields

Z A k ~ Constant (El lu] 122m + C(c) I luI I~), k

(16)

where the constant factor does not dependent on t nor ~.

Now from Lemma 2, we get

and

lq~ Ttul 12 < Constant I ITtul I 2 L 2(Nk) -- L 2(~n)'

In~ul 12 < Constant I lul 2 L 2(Bk) -- L 2(~n)"

2 lqq~ul 12L2 (Bk) _< Constant I lq ILS k ~ I In~ul I 2m-l' by (7)'

Constant I lu112m_l ~ Constant (e[ luIl2m + c(e) I lul I0).

(17)

(i8)

(19)

140

Putting (16),(17),(18) and (19) back into (13), we obtain

llull 2 2 122= 2 2 2m ~ Constant (llTtull 0 + ~Iiul + c(c){luIF 0 + lluli0),

where the constant factor is independent from t and ~. Since ~ > 0 is

arbitrary, we can choose it to be small enough, so that we can absorb

the term El lu1122m on the right hand side into the left hand side of

the above inequality. Theorem 1 is thus proved. •

III. Self-Ad~ointness.

It is evident that T O is self-adjoint on H2m(~n), and Tt, with

0 ~ t ~ 1 and domain D(Tt) = H2m(~n), is symmetric. The self-adjointness

of T t will be shown to propagate in a finite number of steps from T0(t=0)

to T(t=l).

Theorem 2. Suppose T satisfies the general assumptions (A.I) - (A.4).

Then T is self-adjoint on H2m~n).

Proof. Let 0 < t',t < i. Let u 6 H2m(~n). Then

I I (Tt,-Tt)ul I 0 = It'-tl I I (T-T0)ul I 0

It'-tl Constant i lu112m, by proposition i,

! It'-tl C(I ITtul I 0 + I lul I0), by Theorem i.

Here the constant C does not depend on t nor t'. So (Tt,-T t) is T t- 1

bounded with bound less than i, provided It'-tl <~. By the Kato-Rellich

perturbation theorem Tt~ =T t + (Tt,-T t) is self-adjoint on H2m(~n), if

T t is. Since T O is self-adjoint on H2m(~n), T is self-adjoint also,

after applying the perturbation theorem a finite number of times, m

We extend condition (A.4) to

(A.4)' q = ql + q2' where ql satisfies (A.4) and q2 6 L~(~n).

Corollary i. Suppose T satisfies (A.I) - (A.3) and (A.4)', then T is

self-adjoint on H2m(~n).

Proof. This is obvious in view of the Kato-Rellich perturbation

theorem. •

Let M be the set of constant generalized matrices (c B) that are

symmetric and satisfy the ellipticity condition (i). Let wk'P(~ n) + M

denote the set of functions of the form a B(x) = ba~(x) + cab , where

b B 6 wk'P(~ n) and cab 6 M.

141

Co rol!ary 2. Suppose T satisfies (A.I), (A.4)' and

n n (B) a 6 6 wk'P(~ n) + M, where k = max(I~l,I8 I) and ~ < p ! k-l"

Then T is s e l f - a d j o i n t on H2m(Rn).

n n Proof. We will show that (B) implies (A.2) and (A.3). Suppose [ <p < k-l"

Then, from the Sobolev imbedding theorem, a 8 is bounded and H@ider

continuous, and for IYI > i, wk-IyI'P~9 n) is imbedded in LS(~ n) where

0 < 1 _ 1 k-IYl < Y~i < 2m-I~+6-Yl. Thus (A.2) and (A.3) are sat- s p n n n

isfied.

n wl'm(~ n) is exactly the Suppose p = ~-~. If k = i, then p = ~.

space of bounded Lipschitz continuous functions whose distributional

derivatives are in L~(~n). If k > 1 and 171 = i, then W k-I7 'P~R n) can n

be imbedded in Ls(~ n) with s = n + 1 > n = ~ 2m_l~+B_yi. If k > 1

1 1 and IYI > I, then ~,p(~n) can be imbedded in LS(~ n) with 0 [ =

_ k-IYl < [Yl < 2m-l~+8-Yl So in any case, (A.2) and (A.3) are satis- n n -- n

fied. Corollary 2 then follows immediately from Corollary i.

Corollary 3. Suppose a B(x) = b B(x) + c 8 , where b B(x) E HS(~ n) with

s > k-i + ~, k = max(l~l, IB I) and cab 6 M, and suppose (A.I} and (A.4)'

are satisfied, then T is self-adjoint on H2m(~n).

Proof. For s > k-i + ~, HS(p n) can be imbedded in wk'P(~ n) with n n k < P ~ k - l " So C o r o l l a r y 3 f o l l o w s i m m e d i a t e l y f rom C o r o l l a r y 2.

Remark. Because functions in 9~,p(~n) for appropriate k,p, vanish at

infinity, it is necessary to introduce the set M in corollaries 2,3,

in order to ensure the ellipticity condition (i).

REFERENCES

I. R. A. Adams, Sobolev Spaces (New York: Academic Press, 1975).

2. F. E. Browder, "On the Spectral Theory of Elliptic Operators. I," Math. Ann. 142(1961), 22-130.

ON A NEHARI'S CONSTANT

A. Elbert

We are concerned with the half-linear differential equation of the

form

y" + p(t)f(y,y') = 0 , (i)

where the coefficient p(t) is piece-wise continuous on [a,~) with

-~ < a < ~ and the function f(y,z) has the following properties:

f(y,z) is continuous on ~ = R × (R\{0}) ;

yf(y,z) > 0 if yz ~ 0

(i)

(ii)

(iii) f(ly,lz) = ~f(y,z) for ~ e R +, (y,z) • ~ ;

~ dT < ~ r ~ dT < ~ (iv) 1 + Tf(T,I) ' -~ 1 + Tf(T,--I)

(V) lira Tf(T,-+I) = ~.

In a forthcoming paper we shall generalize the Sturmian theorems

known for the linear differential equations (i.e. when f(y,z) H y) for

(i). A preliminary report was made at the EQUADIFF Conference held in

Bratislawa in 1982 [2]. The systematical study, including the Sturmian

comparison theorems, of the half-linear differential equations goes

back to I. Bihari [i]. We must remark here that I. Bihari did not

assume a condition like (iv) but we can show that (iv) follows from

the continuity of the function f(y,z) in the neighbourhood of the

point (i,0).

In [33 we have studied the differential equation

Y"IY'I n-I + p(t) lylnsgny (n > 0 , real)

which is a special case of (I) with f(y,z) = lyl n sgn y .Izl l-n. In

what follows we fix the function f(y,z).

According to one of the Sturmian theorems the zeros of two

linearly independent solutions of (i) are interlacing. This property

makes it possible to classify the coefficient p(t) either into

oscillatory or nonoscillatory classes. In this paper we shall be

concerned with the nonoscillatory differential equations. If a

solution is nonoscillatory then it is either ultimately positive or

ultimately negative. Since in general we have not the relation

f(-y,-z) = -f(y,z), therefore if y(t) is a solution of (i) then the

143

function -y(t) is in general not a solution of (I).

Let the function p(t) be given. We say that an interval

I c [a,~) is a (+)-disconjugacy interval for the differential equa-

tion (i) if every ultimately positive solution of (i) has at most one

zero on I. (In our investigations the function y(t) { 0 , held on

some interval, is excluded from the set of solutions.) If I is a

disconjugacy interval then we can say also that the differential

equation (i) is disconjugate on I.

By our assumption the differential equation (i) is nonoscillatory

hence an ultimately positive solution y(t) has finitely many zeros.

Let b denote the largest one among those. Then the interval Fb,~)

is a (+)-disconjugacy interval for (i).

It is clear that if an interval I is a (+)-disconjugacy interval

for (i) then any other subinterval I' c I is also a (+)-disconjugacy

interval. Hence for the nonoscillatory differential equation there is

a largest interval and we may suppose that (a,~) is (+)-disconjugacy

interval. The definitions and the statement above can be formulated

in the same way for the ultimately (-)-solutions, too. The (+)-

disconjugacy interval and the (-)-disconjugacy interval of (i) may be

in general different.

For the linear differential equation

y" + py = 0 (2)

under the restrictions

0 ~ p(t) ~ p ~ = const > 0 (3)

Z. Nehari [4] proved that if (2) is disconjugate on [0,~) then there

is a constant c o ~ {½, 33/~ 2 '-½} such that the inequality

f p(t)dt ~ c0/~ (4) 0

holds.

Here we shall prove the following theorem.

THEOREM. Let the interval [a, ~) be (e) -disconjugacy interval for

the differential equation (i), where e = + or - and the coefficient

p(t) satisfies the restriction (3). Then the relation

~ dT fp(t)dt ~ ~ /

l+~eTf (eT,e) 0 0

holds. The inequality is sharp in the sense that the value on the

right hand side can not be replaced by a smaller one.

144

REMARK. In the linear case f(y,z) - y we have by THEOREM that

dE f

z---

0 l+~T

i.e. the Nehari's constant in (4) is ~/2.

PROOF. Let the functions ~(t) and f(y,z) be defined by

1 ~(t) = ~ p(t)

(t) = ~ f(y,z)

It is clear that the function f(y,z) satisfies the restrictions

(i)-(v) above and 0 -< ~(t) _< 1 . Instead of (I) we shall consider

the differential equation

y" + p(t)f(y,y') = 0 (5)

The two equations, (5) and (2),are equivalent in the sense that if

y(t) is a solution of (I) then it is also of (5) and vice versa.

Let s = s(~) be the solution of the differential equation

s" + f(s,s') = 0 (6)

with the initial conditions ~(0) = 0 , ~' (0) = i. The solution §(~)

can be uniquely defined for all % c R. Let

oo co

~ = ~ dT ~i ~ + f dT ~½ I+Tf(T,I) ' = ~½ 0 l+Tf(r,--l)

0 0 ~ dT

= dT ' ~2 + ~ W3A Wi + ~ l+~f(T,-l) = ~3A i+~f(T,i) --co --oo

Then ~(%) = 0 if ~ = 0 , ~l(m°d 92 ) and

s(~) > 0 , s' (~) > 0 if % E ~ E (0,{½) (rood ~2 )

~(~) > 0 , ~' (~) < 0 if ~ - ~ { (W½,~I) (rood ~2 ) (7)

§(¢) < 0 , ~' (~) < 0 if ~ = ~ { (~l,~3/2)(mod ~2 )

{(}) < 0 , ~' (~) > 0 if ~ - ~ ~ (~3/z,9~(mod ~2 )

The function ~(~) plays the same role for the half-linear

differential equation (5) as the sine function for the linear differ-

ential equations.

By the aid of the functions ~(~), ~' (~) we define the generalized

Prufer transformation for the half-linear differential equation (5).

Let %(t), 0(t) be defined by

145

y(t) = p(t)s(~) , y' (t) = p<t)S'(~) (8)

Then the functions satisfy the system of differential equations

~' = i-G(~) + ~(t)~(~) , (9)

p' = p(l-p) ~ G(%) ~(~)

where G(~) = ~(~)f(~(~),~'(~))/[s'2(¢.)+~(%)f(~(%),~' (~)) ]. The func-

tion G(¢) is continuous and 0 s G(%) ~ i.

Now we consider the case s=+. Let y(t) be the solution of (5)

satisfying the initial conditions y(a) = 0, y'(a) = i. We claim now

that y(t) > 0 on (a,~) and y' (t) ~ 0. The first statement follows

from the definition of (+)-disconjugacy. The second one needs some

considerations. Since by (ii) y" = -~,(t)f(y,y') s 0, hence the

function y' (t) is decreasing. Hence for T > t a a we have

T y(T) = y(t) + [ y' (T)dT ~ y(t) + y' (t) (T-t) ,

t

hence

y' (t) > lim inf [(T) > 0 T÷~

Let #(t), p(t) be the polar functions belonging to the solution y(t).

We may suppose % (a) = 0. By (7) we have

0 ~ ~(t) s ~½ . (i0)

On the other hand by (9) the relations 0 ~ ~(t) ~ 1 and 0 ~ G(~) ~ 1

imply the inequalities

~(t) ~ ~' (t) ~ i .

Hence the function ~(t) is nondecreasing.

obtain for T > a

T ] ~dt ~ ~(T) a

and consequently by (i0)

7 i ~ d T a ~(t)dt ~ ~½ = I+Tf(T,I) '

or

7a Z ~ dE p(t)dt -< ~ =

I+uTf(T,I)

(ii)

Then by integrating we

as we stated.

146

The sharpness of this inequality can be demonstrated by the example

y" + P0(t)f(y,y') = 0 (12)

where

P0(t) = {i if t ~ 9½

if t > 9½

Now clearly ~ = i, and the function

Y0(t) = - z½ (9½) if t > 9½

is a solution of (12) and the interval (0,~) is a (+)-disconjugacy

interval. Hence

f P0(t) = f dT = u~½. 0 0

The proof of the case (-)-disconjugacy goes in similar manner.

The only difference is that y(t) is eventually negative on [0,~)

and

y(t) ~ 0 , y' (t) ~ 0 , 9~ < ~ < W2"

Then (Ii) holds again and

~ ~ dT 0 dT f ~(t)dt ~ W2 - ~ = f 1 - Tf(-T,-I) = ~ 1 + Tf(T,I) ' a 0 -~

which completes the proof.

In the proof of our THEOREM the most useful relation is the

inequality (ii) which was deduced by making use of the relations

0 ~ ~(t) ~ 1 and 0 ~ G(~) ~ 1 .

Let y(t) be a solution of (i) such that y(t 0) = 0 , y' (t o ) > 0 ,

y'(t 6) = 0 and y'(t) > 0 on t o ~ t ~ t;. An integration of (Ii)

yields

t' f0 p(t)dt ~ ~[~(t;) - ~(t0)~ = ~½ (13)

t o

Similarly on (t6,t I) where t I is the next zero to t 0 we have

~i p(t)dt ~ (91 - 9½)~ .

t 6 A combination of (13), (14) gives the following result.

COROLLARY. Let y(t) be a solution of (i) such that t O , t I are

the consecutive zeros of y(t) and y(t) > 0 o__n (t0,tl) I__f (3)

147

holds then

~ip(t)dt ~ ~I "

t o In the linear case we find the following inequality

~I p(t)dt ~ Zl/~ ,

t o

where t0, t I are consecutive zeros of the solution y(t)

REFERENCES

[13 I. Bihari, Ausdehnung der Sturmschen Oszillations- und Vergleich- ssitze auf die L~sungen gewisser nichtlinearer Differentialglei- chungen zweiter Ordnung, Pub l. Math. Inst. of Hung. Acad. Sci. 2 (1957) 159-173.

[2] ~. Elbert, Qualitative properties of the half-linear second order differential equations, Publ. of Computing And Automatization Institute of Hung. Acad. Sci. 26 (1982), 27-33.

[3] ~. Elbert, A half-linear second order differential equation, Colloq. Mathematica Soc. J. Bolyai, 30, Qualitative theory of diff. eqs., Szeged.

[4] Z. Nehari, Oscillation criteria for second-order linear differen- tial equations, Trans. Amer. Math. Soc. 85 (1957), 428-445.

ASYMPTOTICS OF EIGENVALUES FOR SOME "NON-DEFINITE" ELLIPTIC PROBLEMS

J. FLECKINGER-PELLE

Let ~ be an open set i n ~ n . (~ is not necessarly bounded) , °

( i ) Let g be a real bounded funct ion which changes sign in ~.

(2) We suppose tha t gc Ls(~) w i th s > n/2.

(3) Let q be a real func t ion in Ln/2(~) ; we suppose tha t q is bounded below (by a

non necessarly pos i t i ve constant) .

Moreover, when ~ is unbounded, we suppose that q(x) tends to +~ at i n f i n i t y .

We study the eigenvalues ( i . e . the complex numbers ~ associated wi th non zero so lu-

t ion u of the fo l l ow ing problem (P) (which is considered in the v~ r i a t i ona l sense) :

Lu ~ (-A+q) u = ~gu in (p) L u = 0 on ~ (the boundary of ~) .

This problem is ca l led "non -de f i n i t e " because the operator L = -&+q is not necessa-

r l y pos i t i ve (q is al lowed to take negative va lues) , and g changes sign.

Such equations can appear, f o r example, in laser theory I l l ] p i n mul t i -parameter Sturm

theory [2 ] . . . . .

The f i r s t paper concerned wi th the "non -de f i n i t e " problems seems to be due to

Richardson (1918) [19] . He mentioned tha t non real eigenvalues may occur.

Indeed the fo l l ow ing simple problem ' 4 - u " - (9~2/16)u = ~gu in (~2)

I u(O) = u(2) = 0

{ + i x c ( 0 , 1 ) w i th g(x) = has two complex eigenvalues ~ # t i 4,3628 . . . . [16 ] . - x ~ ( 1 , 2 )

Jn th is paper we are concerned wi th asymptotics of pos i t i ve eigenvalues.

Obviously, analogous resu l ts f o r negative eigenvalues can be deduced by w r i t t i n g ~g = ( - ~ ) ( - g ) .

More prec ise ly ,we w i l l obta in lower-and upperbounds f o r the eigenvalues and asympto-

t i c s f o r N+(~,L,g,Q) the number o f pos i t i ve eigenvalues less than ~}under su i tab le

assumptions, we w i l l prove :

i ) when ~ is bounded, or when ~ is unbounded wi th { g~/2 < ~ then J

I (Xg+) n/2 as X + +~o (4) N+(X'L 'g '~) ~ (2~)-n ~n ~ '

where ~n is the volum of the un i t ba l l in ~n and g+ = max(g,O) ;

when is unbounded and { gy2 i i ) J m

t49

(5) N+(X,L,g,~) ~ (2~)_ n Wn r ~ (kg_q)~/2 ~ , as k ÷ +~o ~+

where ~+ = { x ~ / g(x) > 0}.

We notice that when g(x) ~ 1 th is formulae are "well known". When ~ is bounded,(4)

is the usual "Weyl Gourant" formula [4] ; when ~ is unbounded with f i n i t e Lebesgue

measure, (4) has been proved by Clark [3 ] .

When ~ = ~ n (5) is the usual "de Wett Handl formula" for the Schr~dinger operator

[18,21] ; th is formula has been extended to unbounded domains in [6,20] .

For " l e f t de f in i te " problems (L is pos i t i ve ) , such estimates are proved in

[1 ,9 ,10 ,14 , ]7 ] .

When L is posi t ive and degenerate, in the one dimensional case, analogous asymptotics

are establihed in [13,22].

Estimate (4) has been conjectured by J~rgens (1964) [12] , in the one dimensional

case, for non de f in i te problems, and Mingarel l i (1983) [15] proved i t under some more

r es t r i c t i ve assumptions on the number of zeros of g. More references on non de f in i te

problems can be found in M ingare l l i ' s survey paper (1984) [16].

Throughout the paper we w i l l make the fo l lowing assumptions :

(6) I f ~ is a real eigenvalue, the associated eigenfunction ~ i s real .

This is always possible by considering the real part and the imaginary part of the

eigenfunct ions, which are eigenfunctions too.

We w i l l denote ( , ) the inner product in L2(Q), and by il il the associated norm.

Let us denote by ~+ = { x ~ / g(x) > 0}, ~ = I x ~ / g(x) < O} and

~o = { x ~ / g(x) = 0}.

We suppose that

(7) I~E > o, l~°f_ > o and

where I I denotes the Lebesgue measure and ~o the i n t e r i o r of ~.

i . Estimates for bounded sets

We suppose throughout th is section that d)~is bounded.

A. Recalls on "non-def in i te" problems

Let us f i r s t recal l some results of [7 ,8 ] .

We suppose that (1), (3), (6) and (7) are sa t i s f ied and that

(8) g is continuous.

150

Proposition 1 : There e ~ t s an at most countable s~ t of d ~ t i n e t e igenvalu~ .

Proposition 2 : There e z ~ t s an ~ most f i n i t e number of d ~ t i n c t non r~al ~ g e n v a l u ~ .

P~oposition 3 : There e ~ t s an a t most f i n i t e number of d i s t i n c t pos i t i ve eigenva-

l u ~ such tha t a s so~a ted ~Lgenfunctio~ are in

D+~ : {uc H~(~) / (gu,u) 0}.

+ ~ j t h In the fo l low ing , we w i l l denote by ~ j (L ,g , ) the eigenvalue (repeated according

to m u l t i p l i c i t y ) such that associated e igenfunct ion is in

D+ = {u~H~(Q) / (gu,u) > 0}.

I t fo l lows from proposi t ion 3 that the asymptotics, when ~ tends to +~ , of

N+( I ,L ,g,~) and of M+(~,L,g,~) = ~ i are the same. X~(L,g,a)~t

B. Recalls on " l e f t de f i n i t e "~ rob lems

We suppose that hypothesis (1) to (3) and (7) are sa t i s f i ed .

Let us choose a pos i t i ve and continuous funct ion h, def ined on Q, such that :

3m > O, VucH~(Q) a(u,u) = (Vu,Vu)+ ((q+h) u,u) ~II U II 2 •

Then the pos i t i ve eigenvalues l~(L+h,g,~) of the var ia t iona l problem :

(S) I (L+h) u ~ ( -~q+h) u = Xgu in

( u = 0 on ~

are character ized by the max-min p r inc ip le ( [ 5 ,9 ,10 ,17 ,23 . . . ] ;

(g) i+ ] -1 [ j (L+h,g,~) = sup i n f a ~ Ej u ~ Ej

where Ej is a j dimensional subspace in H~(~).

I t fo l lows immediatly that :

+(L+h,g,~) /s non i n ~ e ~ i n g when g i n c r e ~ e s . Proposition 4 : The eigenvalue 10

Proposi~on 5 : The eigenvalue i~.(L+h,g,~) is non increasing when ~ i n c r e o ~ .

N+ i (Xg+)n/2 Proposition 6 : ( l ,L+h,g ,~) ~ (2~) "n ~n as I ÷ +~ , J

where N+( l ,L+h,g,~) = ~ 1. I j (L+h ,g ,~ ) -< I

151

C. "Right d e f i n i t e " problems

(10) We suppose that g is a p o s i t i v e and continuous func t ion in ~ such tha t

there ex is ts y > 0 and g(x) >- y > O.

We consider the va r i a t i ona l eigenvalue problem (P) where q s a t i s f i e s (3).

We deduce from (10) tha t there ex is ts k > 0 such tha t :

( (L+kg)u,u) -> IlUll 2 fo r a l l uOH~(f~).

Problem (P) can be w r i t t e n :

(L+kg)u - (-A+q+kg)u = (X+k) gu in (P')

u = 0 on ~ ;

th is problem is " d e f i n i t e " on both sides and the usual resu l ts f o r the eigenvalues

can be appl ied.

The eigenvalues ~j(L+kg,g,Q) are p o s i t i v e , tending to +~ as j tends to +~jand

character ized by the usual "max-min" p r i n c i p l e [4 ,18 . . . . ] ; moreover the j t h

eigenvalue of (P) is such tha t :

(11) ~ j (L ,g ,~ ) = ~ j (L+kg,g,~) - k, VjE]N.

Therefore :

Proposition 7 : When ~ is bounded and when (3) and (I0) are sat is f ied , the j th

eigenvalue of (P}, denote by )~j(L,g,~) /s characterized by :

= Sup i n f L u ~ _ k. kj (L,g,Q) Ej u±E.

We not ice tha t there ex is ts an at most f i n i t e number o f negative eigenvalues and

therefore N(2,,L,g,31) = ~ 1 and N+(~,L,g,~) have the same asymptotics whenX ~j (L ,g ,Q)~<~

tends to +~ ; t h i s est imate is given by (4) .

Moreover, we know by [5 ] t ha t :

Proposition 8 : The s m ~ t eig~nvalue )~l(L,g,~) (which is not nec~sa~ly positive)

is simple and the ~ s o ~ a t e d eigenfunction does not change sign.

152

D. The "non -de f l n l t e case

We suppose that (1) , (3) and (6) to (8) are s a t i s f i e d .

Let us choose k > O, big enough so tha t :

( (L+k)u,u) ~ [jUH 2 fo r a l l u~H~(~) .

Problem (P) can be w r i t t e n as

(L+k)u ~ X(g+k/~) u in (P") ( u = 0 on 3~.

In the fo l l ow ing we w i l l consider the eigenvalues (pos i t i ve on negat ive) +

~j(L+k,g+k/~,~) o f the va r i a t i ona l problem (Q) def ined fo r a l l ~E~ :

(L+k)u = ~(g+k/~)u in ~ ; (Q)

u = 0 on ~ .

+ The f i xed points p = ~j (L+k,g+k/~,~) are the eigenvalues of (P).

Problem (Q) is " l e f t d e f i n i t e " and we can apply resu l ts of sect ion B. For a l l

f i xed p > O, there ex is ts an i n f i n i t e sequence of pos i t i ve eigenvalues

~ (L+k ,g+k /p ,~ ) character ized by (9) , and we deduce from proposi t ions (4) and (5)

tha t , f o r a l l c > O, f o r a l l ~ > 0 :

+(L+k,g+k/p,~) < ~ j (L+k,g+k/p,~+) . (12) ~j(L+k,g++c+k/~,Q) ~ ~j -

We not ice tha t g+ being pos i t i ve on ~, and g pos i t i ve on ~+, the upper and the lower

bound in (12) are the eigenvalues of two " d e f i n i t e " D i r i c h l e t problems :

(q l ) (L+k) u = X(g++c+k/~)u in f~,

(Q2) (L+k) u = X(g+k/~)u in ~+.

For a l l j , the " f i xed po in ts" of upper and lower bounds in (12) are the eigenvalues

of the " r i g h t d e f i n i t e " D i r i c h l e t problems :

(R1) Lu = ~(g++c)u in ~,

(R2) Lu = ~ gu in Q+.

I t fo l lows from propos i t ion (7) tha t fo r a l l j E

~j (L,g++c,~) ~ ~j(L,g,~+) ;

1 5 3

hence, denoting by X](L,g++c,~) [Resp. X;(L,g,~+)] the j th posit ive eigenvalue of

(RI) [Resp. (R2)], we deduce from (12) that4for al l ~ > 0 and for al l j c ~ :

( 1 3 ) X;(L,g++c,~) -< X;(L,g,£) -< k;(L,g,a+).

We note that X+(L,g,~+)~ = %j+j(L,g,~+) with JE]N.

By letting c tends to zero, and by (4), we prove that k;(L,g,f~+) and X+(L,g+,~)j have the same asymptotics when j tends to 4-= ; i t follows that (4) holds for N+(X,L,g,~) when g is continuous.

When g is in Ls(~) with s > n/2, we choose ~e C°(~) such that :

" glIkn/2(~ )

We deduce from the HSlder and the Sobolev inequali t ies that there exists a posit ive

constant ~ such that

f lull.<. li - ll o I t follows now ,by considering ~+ in place of ~+,that:

Theorem 7 : I f hypo thes~ (7) to (3), (6) and (7) a r e s a t i s f i e d and i f ~ i s bounded,

then the es t imate (4) hold~ :

I (Xg+) n/2 as X + +oo. N+(X'L'g'Q) % (2~)-n mn

Let us denote by ~+(L,g,~) the smallest eigenvalue such that associated eigenfunction is in D+ ;

X+(L,g,~) = in f (Lu,u) u~HI(~)

(gu,u) : l

X+(L,g,~) is not necessarly posit ive.

Theorem 2 : I f h y p o t h ~ i s ( I ) t o (3), (6) and (7) a r e s a t ~ f i e d and i f J ~ i s bounded

then ~+ (L,g,~) ~ a "pr inc ipal eigenvalue", i . e . the associated

~ g e n f u n c t i o n do~ not change s ign .

Proof of theorem 2 : Let us dencte by ~(~]) the f i r s t eigenvalue of (Q) ; ~(~) is

continuous and monotonic in ~ and, for al l I~

(14) ~l(L,g++~+k/~,~) -< ~(I~) -< Xl(L,g++k/~,C~+).

154

Theorem (2) and the existence of ~+(L,g,~) can be deduced from (14) and from the

existence of ~I(L,g,Q~) and of XI(L,g++~,~) f o r a l l e > 0.

2. Estimates fo r unbounded sets

(15) We now suppose that ~ is unbounded.

(16) The po ten t ia l q is cont inuous, bounded below, and tends to +~ at i n f i n i t y .

(17) We suppose that g is continuous and bounded in ~.

We consider a p a r t i t i o n o f A n i n to non over lapping and d i s j o i n t cubes (Q~)~EZ n

wi th sides q and centers x~.

For any pos i t i ve number ~, l e t us denote by

~+~ = {x c ~+ / q ( x )< ~g(x) } ; I = {~ c Z n / Q~ c ~+~} and J = {~ c Z n / Q~ n ~+~ # ¢}.

(18) we suppose tha t f o r any pos i t i ve number X big enough, O+X is Lebesgue measu-

rable and

l i m [ ~ g~/2] -1 [ Z g~/2]= 0# where g~ : g ( x ) . n÷0 ~ ~ J ~ J \ I

Moreover there ex is ts k' > 0 s a t i s f y i n g ~+~]~ k'[a+X/2]~ with [~] = f gn/2.

(19) There ex is ts~ pos i t i ve numbers eo, a I , a 2 such that g and q can be

extended to ~ = { x c A n / d i s t ( x , ~ ) < ~o } and fo r a l l cc (0,Eo), there

ex is ts q > 0 such that

Ix-yJ < n x ~ , ~ > lq+( x ) - q + ( y ) l ~ e(q+(x)+e I)

Ig+( x ) - g + ( y ) l ~ E(g+(x)+c2)"

For example q(x) = (1+Ix12) r - 2 w i th r > 1 and g(x) = (1+x2) -1 - 1 def ined on A n

sa t i s f y (15) to (19).

Let us denote by V(Q) the completion of ~o(Q) w i th respect to the norm

llul = (I (Ivu(x)t2 + l (x)ilu(×)12) dx)l/2

We consider the variational eigenvalue problem (P) with uE V(Q).

I t follows from (16) that the imbedding of V(~) into L2(~) is compact,

We prove as in [7,8], that :

Proposit/~n 9 : There cx i s t s an at most countable s e t of d i s ~ n ~ t ~ igenvalu~.

155

Proposition 10 : There e~ists an a t most fi~Cte number of d i s t i n c t non real

eigenvalues .

Proposition 11 : Then e x i t s an at most f i n i t e number of d i s t i n c t pos i t i ve

~igenvalu~ such tha t ~ s o ~ a t e d e igen func t io~ ~ e in

O~ = { u ~ V ( ~ t / (gu, ul ~ 0} .

We w i l l denote, as in part 1, by X~(L,g,Q) the j th eigenvalue such that associated d

eigenfunction is in D+ = {ueV(Q) / (gu,u) > 0}.

We deduce from [9,103, that for " l e f t de f in i te " problems, the "max-min" pr inc ip le (9)

holds when replacing H~(Q) by V(~) ; propositions 4 and 5 hold too and estimates (4)

and (5) can be applied.

For " r ight de f in i te " problems we establ ish the "max-min" pr inc ip le as in

proposition 7.

We then prove as in part 1}by use of the estimate for the " l e f t de f in i te " case~[9] :

Theorem 3 : I f hypothesis (I) to (3), (6), (7) and (15) to (19) are s a t i s f i e d , then

i) if I g /2 < -n I (xg+)°/2

i i ) i f # g ~ / 2 = ~ N+(X,L ,g ,~) ` (2~)-n mn{ (Xg_q)~/2 as X ÷ +~. ~+

References

[1] M.S. BIRMAN, MZ SOLOMYAK,Siber. Math. J. Vol. 20, n° l , 1979, p. 1.

[2] P.A. BINDING, PJ BROWNE, "Multiparameter Sturm theory" (to appear).

[3] C. CLARK, Bul l . AMS, 1966 ; 72, p. 709.

[4] R. COURA~T, D. HILBERT, "Methods of mathematical physics", Intersciences, 1953.

[5] D.G. de FIGUEIREDO, Lectures Notes in Maths, 957 (Springer-Verlag), 1982, p. 34.

[6] J. FLECKINGER, Proc. Roy. Soc. Edinburgh, 89 A, 1981, p. 355.

[7] J. FLECKINGER, AB. MINGARELLI, Maths Studies n°92 (North-Holland), p. 219.

[8] J. FLECKINGER, Proc. "Journ#es d'Analyse non Lin#aire" publi IRMA V. 5, f .2 ,

L i l l e 1983.

[9] J. FLECKINGER, M. EL FETNASSl, Proc. "Wor~hop on spectral theory of SL d i f f e -

rent ia l operators", Argonne Nat Labo. ( I l l ) , 1984 ; or CRAS, Paris 1984

(to appear).

156

[10] J. FLECKINGER, M. LAPIDUS, "Eigenvalues of e l l i p t i c boundary value problems

with an indef in i te weight funct ion", prepr int , 1984.

[11] J. HEADING, J. Phys. A, 15, 1982, p. 2355.

[12] K. J~RGENS, "Spectral theory of 2 nd order ODE", Mat. Inst . Aarhus Univ., 1964.

[13] H G. KAPER, QG. LERKKERKERKER, MAN KAM KWONG, A. ZETTL, Proc. Roy. Soc.,

Edinburgh, 1984, to appear.

[14] M. LAPIDUS, Note CRAS, Paris, s#rie I , t . 298, 1984, p. 265.

[15] A.B. MINGARELLI, "Lectures Notes in Maths", 1032 (Springer-Verlag), 1983, P. 375.

[16] A.B. MINGARELLI, Proc. "Workshop on spectral theory of SL d i f f e ren t i a l

operators", Argonne Nat Lab. ( I l l ) , 1984.

[17] A. PLEIJEL, Arkiv. Mat. Astr. Fys. 29 B, n°7, 1942, p. 1.

[18] M. REED, B. SIMON, "Methods of mathematical physics", t . 4, Acad. Press, 1978.

[19] RGD RICHARDSON, Amer. J. Maths 40, 1918, p. 283.

[20] GV. ROZENBLJUM, Math. sb 22, 1974, p. 349.

[21] E. TITCHMARSH, "Eigenfunction expansions", t . 2, Oxford Univ. Press, 1958.

[22] EJM VELING, Integr. Equ. and op. th . , 1984, to appear.

[23] H.F. WEINBERGER, Reg. Conf. Series in Appl. Maths, v. 15, SIAM, 1974.

OPTIMAL CONTROL OF A SYSTEM GOVERNED

BY HYPERBOLIC OPERATOR

I.M. Gali, H.A. E1-Saify, and S.A. E1-Zahabi

ABSTRACT

I.M. Gali et al have considered a distributed control problem for

hyperbolic operator with an infinite number of variables [6]. Also

they established the solvability of the mixed problem for nonlinear

infinite order hyperbolic equations [7]. The authors in [9] have

obtained the set of inequalities defining an optimal control of a

system governed by infinite tensor product of elliptic operators A k.

In the present paper, a distributed control problem for the hyper- ~2

bolic operator ~ + Q A k is considered. ~t 1

The necessary and sufficient condition for the control to be

optimal is obtained, and the set of inequalities that characterize

this condition is also obtained.

SOME FUNCTIONS SPACES

The following construction of a weighted infinite tensor product

of Hilbert spaces with a given stabilizing sequence will be used

further [2,3,8]. Suppose (Hk)k= 1 is a sequence of Hilbert spaces, k ~ (k)

e = (e)k=l (e eH k) be a fixed sequence of unit vectors and

6 = (6k k=l )~ (6k > 0) is a fixed numerical sequence (a weight) . In each ,e(k))~ ~k) (k) H k we consider an orthonormal basis L j j=l such that e = e

We form a formal product

e = e~l(1) ~ e(2)~2 ~ "'" (~ = (ej)]= I) '

where dl' @2' ... = 1,2, ... and moreover, en+l = ~n+2 :'''= 1

beginning from some number n depending on a; let ~(~) be the

minimal n = 1,2 .... possessing this property. Let A be the count-

able set of all vectors indices ~ of this kind.

We define the weighted infinite tensor product

= Q H k of the Hilbert spaces H k with stabilizing sequence He'6 k=l;e,6

e and weight 6 as the Hilbert space spanned by the basis (6 ~ ~(~)e~)~A'

which assumed orthonormalized by definition. Thus, the element of

158

He 6 has the form f = [ fsa , where ' sEA

[ Ifel 2~9(e) = II fll~e,~ < " ; seA

(f'g)He,6 -s~Afsgs @V(s)

This definition does not depend on the choice of the basis (e~n))j= 1

(n) (n) = {esAlu(s) = n} (n = 1 2 .... ). @uch that e I = e ). We set A n

These sets are pairwise nonintersecting and their union is A.

If ~ = I, i.e., ~k = 1 (k = 1,2,...), then He, 1 coincides with

a separable subspace of the complete Neumann product of the spaces H k.

In particular(k) if H k = L2(R',d~k(Xk)) (~k(R') = i) and e = (e(k))k=l' ~

where e (x k) = i, then

He, 1 = L2(R~,d~l(Xl ) ~ d~2(x2) ~ ...) (R ~ = R'×R'×...)

With the aid of a well-known procedure [1,2], it is possible to

construct the following chain of spaces:

Hk, - ~ H k = H k (k = 1,2,...) -- _ ,+

We assume that in each Hk, +, the unit vector e (k) is chosen such

that II e(k) IIHk = I, it is possible to construct the following chain

of spaces:

Q H k ~ Q = H 8-1 H+,e ~ = ~ Hk,+ -1Hk, - - , e ' k=l;e,6 k=l;e,l k=l;e,~

where

~ - 1 - 1 ~ (k) = (~k)k=l ; II e II = 1 H k ,-

Therefore, the negative spaces are well defined.

Analogous to the above chain we have a chain of the form:

i _ W k - (RI); (k = 1,2 .... ) Wk(RI ) c (R I) c Wk Z

where W~(R I) are Sobolev spaces constructed by the completion of the

class C~(R I) of infinitely differentiable functions of compact

support with respect to the scalar product

n = ~ (D%,OSvl ; l~l = [ ~i (U,V) £ [~ <Z L2 (RI) 1

The differentiation is taken in the sense generalized functions on R'

WkZ(RI) are the duals of W~k(RI).

159

For such a construction we must form an appropriate manner the

chain

WZ(R ~1 = ~ W~(R') ~ L2(R~,d~) = Q Wk(R') k= l ;e ,~ k=l

k = l ; e r ~ -1

In the following we shall use a chain of the form:

L2(0,T;W~(R~)) S L2(0,T;L2(R~,dp)) S L2(0,T;W0i(R~))

£(R ~) = {ulueWZ(R~), D~u = 0 on FIll < ~-1, £>1} w 0

-z and W 0 is its dual [2,8].

L2(0,T,W~(R~)) denotes the space of measurable function t + f(t)

of 30,T[ and the variable t denotes "time". We assume ts]0,T[,

T < ~ with the Lebesgue measure dt on ]0,T[ such that

T ( f l l f(t) l~W~ dt) ½ = lifll £ ~ < + ~ 0 v(R~) L2(0'T'W0(R ))

The spaces considered in this paper are assumed to be real.

The following notation will be used later:

Q R ~ = x ]0,T[ , ~ = F x ]0,T[

is the lateral boundary of Q, F is the boundary of R ~.

FORMULATION OF THE PROBLEM

Let us consider the elliptic operators

(Ak(t)u) (x) = -(D2u) (x) + qk(x,t)u(x) (k=l,2 .... ) (2.1)

1 (R') into where Ak(t) are bounded self-adjoint that map W0, k W-I 0,k(R'), and qk(x,t) is a real valued function which is bounded and

measurable on R 1 such that

co

Oqk(x,t) -> C > 0, C is constant. k=l

Construct the infinite tensor product B = O A k acting from

W 01(R~) onto W01- (R ~) [9,10]. k=l

To set our problem, we introduce the following bilinear form:

(t;u,v) = (B(t)u,v))L2(R ~) (2.2)

160

It can be proved that the coerciveness of such bilinear form [4,5],

i.e.,

~(t;u,u) 2 cll ul~w21 , c > 0 (R ~)

"0 u,veW~(R ~) the function t + ~(t;u,v) For all

differentiable on ]0,T[ and

(2.3)

is continuously

z(t;u,v) = ~(t;v,u) (2.4)

Under the hypotheses of (2.3) and (2.4) if f = f(x,t), Y0(X) and

Yl(X) are given in L 2(Q) , W 1 (R ~) , L 2(R ~) , respectively, then there

exists a unique element y that satisfies [6,73,

Y' ~Y/~K ' ~Y/~teL2(Q)

~2y/~t2 + B(t)y = f in Q

with y/Z = 0

and with initial condition

y(x,0) = Y0(X), Zy/~t(x,0) = yl(K) in R ~

We may now formulate our control ~roblem. Let L 2(Q) be the space

of controls. For f,y0,y I given with feL2(Q), Y0sWl(R~), YlaL2(R~ andif

(2.3) and (2.4) hold then for control usL2(Q) the stateof the system

y(u), which depends on x,t is denoted by y(x,t;u) and is given by

the solution of

Z2y/~x2(u) + B(t)y(u) = f + u in Q

y(u) I z = o

y(x,0;u) = Y0(X), ~y/~t(x,0;u) = Yl(X) in R ~

Y(u)sL 2(0,T;L 2(R ~)) = L 2(Q) , ~y/~tsL 2(Q)

The observation equation Z(u) is given by

Z (u) = y(u)

Finally, we are given NsL(L 2(Q) ,L 2(Q)) , N is Hermitian positive

definite, i.e.,

(Nu'UlL2(Q) -> Y[[ ull2P (e) Y > 0 (2.5)

With every control u we associate the cost function

J(u) = II y(u) - Zdl ! L2(Q ) + (Nu,U)L2(Q)

161

where Z d is a given element in L2(Q).

Our problem is to find inf J(v), VeUad where the set of admissible

controls Uad is a closed convex subset of L 2(Q) .

Under the given consideration, we may apply the theorem of Lions

[ii] and [6,7] to obtain:

Theorem

Let us assume that (2.3), (2.4) and (2.5) hold and the cost

function is given by (2.6). The optimal control u is then character-

ized by the following system of partial differential equations and

inequalities:

~2/~t2y(u) + B(t)y(u) = f + u in Q

y(u) = 0 on F~

co

y(x,0;U) = Y0(X), (~y/St) (x,0;u) = Yl(X) in R ,

82/$t2p(u) + B(t)P(u) = y(u) - Z d in Q,

P (u) = 0 on E

o0

P(x,T;u) = 0, (SP/Zt) (x,T;u) = 0 on R

and

f (P(u) + Nu)(v-u)dp(x)dt -> 0 Q

(P(u) + Nu,v-u)L2(Q ) -> 0

with

y(u) ,P (u) cL 2 (Q)

(~y/3t) (u), (~P/~t) (u)gL2(Q) .

for all VSUad,

for all VSUad

OUTLINE OF THE PROOF

As in the proof of the theorem in [~], the control UeUad is

optimal if and only if

J' (u) (v-u) >- 0 for all VeUad,

that i s

(y(u)-Zd, y(v)-y(u))L2(Q ) + (Nu,v-u)L2(Q) -> 0

The above equation may be written as:

T

0(Y(U)-Zd y(v) - y(u))dt + (Nu,v-u)L2(Q) >- 0

We folnnally introduce the adjoint state P (u) by

(2.6)

162

~2

~t 2 --P(u) + B(t)P(u) = y(u) - Z d in Q

P (u) = 0 on Z (2.7)

P(x,T;u) = 0, ($P/~t) (x,T;u) = 0 on R,

and from theorem 1 [6], equation (2.7) admits a unique solution which

satisfies

8P (u) eL 2 (Q) P(u) cL 2 (Q) , Z-~

We shall now transform (2.6) as follows. We scalar multiply both sides

of (2.7) by y(v) - y(u), which gives us

T ~2 (--~ P(u) + B(t)P(u), y(v) - y(u))dt

= (y(u) - Z d, y(v) - y(u)) L 2 (Q)

We now apply Green's formula to the left-hand side of the above

equation. We note that if ~£L2(Q) , ~'~L2(Q), ~"£L2(0,T;W0~(R~)) and

if ~ has the same properties then

T f (¢",9)dt = (~'(T),~#(T)) - (~' (0),~(0)) - (~(T),~' (T)) 0 T

+ (@(0),~' (0)) + f(~,~")dt 0

From this we deduce that

(y(u) - Zd,Y(V) - y(u)) L2(Q)

T 22 22

= ~(P(u), (~t 2 + B(t))y(v) - ( t 2- + B(t))y(u)dt

T

= 0/(P(u) ,v-u)dt = (P(u),v-u)L2(Q )

Then, (6) may be written as:

(P(u) + Nu,v-u)L2(Q )

which completes the proof.

0 for all V£Uad

163

REFERENCES

i. Yu.M. Berezanskii, Self-adjoint operators on spaces of functio~of infinitely many variables, Naukova D~mka, Kiev. 1979 (Russian) (This book is under translation by the Amer. Math. Soc., Trans. Math. Monographs).

2. Yu.M. Berezanskii and I.M. Gali, Positive definite functions of infinitely many variables in a layer, Ukrainian Math. Z. 24, No. 4 (1972).

3. Yu.M. Berezanskii, I.M. Gali, and V.A. Zuk, On positive definite functions of infinitely many variables, Soviet Math. Dokl., Vol. 13, No. 2, (1972), 314-317.

4. I.M. Gali, Optimal control of systems governed by elliptic operators of infinite order, Ordinary and Partial Differential Equations Proceedings, Dundee, Scotland, 1982, Springer-Verlag Series, Lecture Notes in Mathematics 964, pp. 263-272.

5. I.M. Gali and H.A. E1-Saify, Optimal control of a system governed by a self-adjoint elliptic operator with an infinite number of variables, Proceedings of the International Conference on Functional-Differential Systems and Related Topics, II, Warsaw, Poland, May 1981, 126-133.

6. I.M. Gali and H.A. E1-Saify, Optimal control of a system governed by hyperbolic operator with an infinite number of variables, J. of Mathematical Analysis and Applications, Vol. 85, No. i, January 1982, pp. 24-30.

7. I.M. Gali and H.A. Ei-Saify, Control of systems governed by infinite order equations of hyperbolic type, Proceedings of Optimization Days, 11-13 May 1983, Montreal, Canada.

8. I.M. Gali and A.M. Zabel, Criteria for the nuclearity of spaces of functions of infinitely many variables, J. of Functional Analysis, Vol. 53, No. i, August 1983, 16-21.

9. I.M. Gali and S.A. EI-Zahabi, The infinite tensor product of operators and its relation with functional spaces, The 8th Conference on Operator Theory, June 6-16, (1983), Timisoara- Herculane, Romania.

i0. I.M. Gali and S.A. E1-Zahabi, Optimal control of a system governed by infinite tensor product of elliptic operators. Submitted in Optimization Days, May 2-4, 1984, Montreal, Quebec, Canada.

ii. J.L. Lions, Optimal control of system governed by partial differential equations, Springer-Verlag Series, New York Band 170, (].971).

12. J.L. Lions and E. Magenes, Nonhomogeneous boundary value problems and applications, Vols. I & ii, Springer-Verlag, New York, (1972).

A GLOBAL STUDY OF A RAMILTONIAN SYSTEM

WITH MULTI TURNING POINTS

Harry Gingold I) and P o - F a n g Hsieh 2)

I. INTRODUCTION.

(l.i)

Consider an n-dimensional differential system

, d i E V' = . - - |Ho(X) + ~HI(X,~)]V,_ = d-~

where H0(x) is an analytic Hermitian matrix with no two eigenvalues

identical on I = [a,b], and HI(X,E) is in the class of CI(I × Sc )

with S = (O,c]. Here c is a positive constant, a may be - ~ and C

may be + =. By a theorem due to F. Rellich (eg. see F. Rellich [9]

G. Strang [I0]), there exists a unitary matrix U(x) analytic on I

that

(I.2)

where

Let

( 1 . 3 )

T h e n ,

( 1 , 4 )

Dl(X) = U-l(X)Ho(X)U(x ) = diag{kl(X) , k2(x), -.-, kn(X)} ,

{kj(x)lj = I, 2, ... , n} are real analytic eigenvalues of

y = U-l(x)V .

satisfies a differential equation

i ~ Y' = [Dl(x) + E RI(X,~)]Y

with

(1.5) RI(x,E) - u-l(x)H1(x,~)U(x) - i u-l(x)U'(x).

Let

b

or

such

He(X).

I) The work of this author is partially supported by a Senate Research Grant,

West Virginia University.

2) The work of this author is partially supported by a Faculty Research Fellow-

ship, Western Michigan University.

165

(1.6)

and

Ro(X,E) = diag RI(X,E),

D(x,e) = Dl(X) + eRo(X,e) ,

( 1 .7 ) Ro(X,E) = diag{r O, r O, .-.

Then, r.. -" 0 and (I.4) becomes 33

R(x,E) = RI(X,E) - R0(x,~),

r~}, R(x ,e ) = ( r j k ) , j , k = 1 , 2 , ~ . ' , n .

(1.8) i ~ Y' = [D(x,e) + ER(x,E)]Y.

Assume that for j # k, J, k = i, 2, ... , n:

( 1 . 9 ) k j ( x ) - kk(X) ~ O, fo r x E I ;

(1 .10) r j k (X ,C) a re bounded f o r x £ I , ~ E Sc,"

b 0 / ~ [ r j k ( X , e ) [ d x a re un i fo rmly bounded f o r Sc; (1 .11) f a [ r j ( x , ~ ) [ d x and ~ E

(1.12) f~Irjk(X,e)[dx are uniformly bounded for ~ E S c.

We shall prove the following

Theorem Under the assumptions (1.9) - (1.12), there exists an

n b_[ n matrix P(x,£) in the class CI(I x S ), (0 < ~ ~ e), A C

P(~,E) = O for some e E I, and np(x,e)ll = o(£d), with d > O,

uniforml Z o__nn I a_ss E ~ O + such that the transformation

(1.13) ¥ = Z(I n + P(x,~)), In: n b_z n identity matrix,

reduces (1.8) to

(1.14) i c Z' © D(x,e)Z.

The points x 0 E I such that kj(x 0) = kk(X 0) for some j,k = 1,2, ".. , n

(j ÷ k), are called the turning points of (1.8).

An immediate result of the theorem is that the system (l.l) has a fundamental

solution

166

(1.15) V(x) ~ g(x) exp{-i -i ix D(t ~)dt}(l n , + P(x,c))

which is uniformly valid on I. In fact, as I contains multi-turning points

of (i.i), the fundmental solution (1.15) itself is a "central connection formula"

on entire I (which is called "two point connection formula" by H. Turrittin [ii]

when only two singularities are present). As pointed out by H. Turritin [II] and

J.A.M. McHugh [6], the lateral connection formulas (or called sectorial connection

formulas in [Ii]) follows from the central connection formulas. Thus it is

essential to have (i.15) for the global study of (I.I). Furthermore, (1.15)

not only gives the asymptotic approximation of the solutions of (1.1) for

~ O +, it also provides the double asymptotic expansions when a and/or b is

infinite.

It is noteworthy that multiplying the transformation matrix I + P(x,E) n

from the right in (1.13) gives a simpler equation for P than otherwise.

A variation of the method in this paper can be applied to prove an adiabatic

approximation theorem in quantum mechanics for an n-dimensional Hamiltonian

system with degenerate energy levels (eg. see R.L. Liboff [5] and A. Messiah [7]).

This will generalize the result of K.O. Friedrichs [I] and H. Gingold [2] and will

appear in a forthcoming paper. In fact, T. Kato [4] provided a rigorous proof of

the adiabatic approximation theorem for the general setting when the energy levels

are not degenerate.

2. PRELIMINARY REDUCTION. From equations (1.8), (1.13) and (1.14) we have

(2.1) iP' = Z-IRz(I + P), P(a,E) = O. n

This is a simpler equation than that obtained by reversing the right hand side of

(1.13). The equation (2.1) can be written as

(2.2) x P(x,e) ~ -ifa Z-IR(t'e)Z(In + P(t,¢))dt.

Put

(2.3) LP = -i I~ Z -I R Z P dr,

then, (2.2) is expressible as

(2.4) P = PO + LP

PO = L I n ,

or

(2.5) P = PO + LP0 + L2P = LIn + L21n + L2P"

From (i.14), since

(2.6)

Thus,

(2.7)

167

D(x,e) is diagonal, we have

Z(x,s,E) = exp{-i -i f : D(t,e)dt}.

L2p = f : Z ( x , s , E ) - l R ( s , c ) Z ( x , s , ~ ) { f : Z ( x , t , ~ ) - l R ( t , e ) Z ( x , t , e ) P ( t , a ) d t } d s ,

or, by changing the order of integration,

x x -I (2.8) L2p = fa{ft z (x,s,¢)R(s,~)Z(x,s,E)ds}Z-l(x,t,E)R(t,E)Z(x,t,e)P(t,~)dt.

Put

(2.9) D(x,e) = diag{dl(X,E) , d2(x,~) , . . " , dn(X,E)},

where, by (1.6) and (1.7)

(2 .10) d j (x ,E) " kj(X) + e r~ (x ,E) , j " 1, 2, " ' " , n.

Put

(2.11) L2p = (Ajk) arid P = (Pjk), j, k = i, 2, ... , n.

By ( 1 . 7 ) , (2 .6) and ( 2 . 9 ) , we have

(2 .12) ( Z - l ( x , s , ¢ ) R ( s , E ) Z ( x , s , ¢ ) ) j k

r j k exp{i~ -1 f [ d . ( D , e ) - d k ( n , e ) ] d n } , i f j ~ k, x

O, if j ~k.

Then,

(2.13) Ajk = n n

x ( x fa ]~ [ 7. ft r jh (S

$-I h=l ,E) exp{i -I f : ( d j ( ~ , E ) _ d h ( n , $ ) ) d n } d s )

rh~(t,e) exp{i E-Ifii(dh(~,E)-d~(n,E))dn}]p~k(t,¢)dt

In order to prove this theorem, we have to establish

168

(2.14) [j lL2pll; ! L(e) IIIPI~

for a suitable norm ~I III of a matrix, where L(E) is a quantity which

depends only on E and tends to O as e ~ O +. For this purpose, we intro-

duce an alternative argument to the stationary phase method in §3 and use it to

establish (2.14) in §4.

3. AN ALTERNATIVE TO STATIONARY PHASE METHOD. Consider

(3.1) J(a,b): = f~ r(s,~) exp{i -i /: p(~,c)d~}ds

where r(x,E) is in CI(I × Sc ),

satisfy the following conditions:

(i) p(x,O) vanishes at some points of I, but not identically zero on I;

(ii) there exists a positive constant gl such that

3.2) IP'(X'e) I N gl for x E I, e E Sc;

(iii) there exist two positive constants m I and m 2 such that

3.3) Jr(x,e) l ~ m I for x E I, E E S c,

3.4) f~Ir(s,E)Ids and f~ Ir'(s,E)Ids S m 2 for c E S c.

p(x,C) is real analytic in I × S and c

The zeros of p(x,O) are the "turning" points of the integral J(a,b). In order

to prove the theorem, we need

Lemma If p(s,£) is independent of E and expressible as

(3.5) p(s,e) =- p(s) = [ H (s-a.) O]~(s) j--1 d

where a ~ al < a2 < "'" < am-I < am E b (equalities hold only when

a # - m, b # ~), and vj(j = I, 2, .-. , m) are positive integers.

Under the assumptions (3.2), (3.3), (3.4) and

(3.6) 0 < g2 ~ l~(s)l, for s E I ,

with a positive constant g2' there exist positive constants

KI, d I and c I (0 < c I S c) such that

d 1 (3.7) IJ(a,b) l ~ K 1 ~ for e E S

c I

169

The proof of this lemma is given in several steps and will appear in a

forthcoming paper [3]. Essentially, the interval [a,b] is divided into two

m ~J + ], = disjoint sets I 1 and 12, with I 1 = Uj~I[~ j - 6jl, 6j2 ~Jk

8jk s~J , (j = i, 2, ... , m; k = 1, 2) for 8jk = 0 or 1 and suitably

chosen positive constants ~j, and then the integrals (3.1) over both I 1

and 12 are shown to behounded by positive powers of ¢. The method used is more

general than the traditional stationary phase method (eg. see F.W.J. Olver [8])

and similar to that employed by H. Gingold [2]. A similar lemma for turning points

at infinity can he proved also.

4. PROOF OF THEOREM. Let

(4.1)

where

Djk(S,~): = d.(s,e)-dk(S,S) = + E qjk (s) qjk(S, c)

qjk(S) has the ,expression

mjk (jk)

(4.2) qjk(S) = [ n (s - ~!jk))'i ]~jk(S ) i=l 1

( j , k = t , 2 , - . - , n ; j#k)

A with mjk and ~[jk) positive integers, ~(Jk)i E I and qjk

A (4.3) 0 < gjk ~ lqJk(S)l

satisfying

for s E I and gjk a positive constant. Let

(4.4) b E-I s

JJk (a'b) = /a rjk(S'~) expel /a ~Jk (~'~)dq}ds"

Then, hy Lemma and the fact that rjk(S,E)exp{i/~jk(D,c)dD} satisfies (3.3)

and (3.4), there exists a function Gjk(S) such that

(4.5) IJjk(a,t) I E Gjk(¢) , for all t E I,

where Gjk(e) tends to O as £ ~ O +.

Let IIP(t,e)II be a suitable norm of P and

(4.6) I I I P I ; ~ - supllP(t,g)II.

Then, by (2.13) and (4.5), we have

170

(4.7) IAjkl n n Z f Z

~=I h=l Gjh(e) I / : l r h~ ( t , e ) exp{i e -1 /~ Ah~(n,e)dD}Idtt] . l]lP}l

n

<- [ E Gjh(~) ~h~]IltPltl,

where ~h~ are suitable positive constants. Let

(4.8) L(g) = n

max { ~ Gjh(E) ~h~}. l~j~n ~,h=l

Then, we have

(4.9)

where L(g) tends to

Similarly, if we let

(4.10)

then we have

Il lL2pll l ~ L(e) l l ]PIII

O as s ~ O ÷.

~(e) = ll(Gjk(e))ll,

(4.11) lllPolll ~ ]IlL Inlll S ~(E)

where ~(e) tends to O as e ~ O +.

Furthermore, if we choose c 2 such that

(4.12) O < L(¢) < 1

for e E Sc2, then (2.5) defines a contraction mapping, and

(4.13) lllPlll ~ ll]Po]ll + Ille 2 Inlll + llIL2plIl

~(s) + L(e)Illnl I + L(E)IIIPIII .

Therefore,

(4.14) ~(e) + L(e) }Jlnl I

llPlll N I - L(C)

for x E I, s 6 S^, where ~ = min{cl, c2}. Thus, Theorem is proved. c

171

REFERENCES

I. Friedrich, K.O., Special Topics in Analysis. Lecture Notes, New York

University, New York, 1953.

2. Gingold, H., An asymptotic decomposition method applied to multi turning

point problems, to appear in SIAM J. Math. Anal.

3. Gingold, H. and Hsieh, P.F., Global simplification of a Hamiltonian system

with multi turning points, in preparation.

4. Kato, T., On the adiabatic theorem of quantum mechanics, J. Phys. Soc. Japan,

5(1955), 435-439.

5. Liboff, R.L., Introductory Quantum Mechanichs, Holden-Day, San Francisco,

1980.

6. McHugh, J.A.M., An historical survey of ordinary linear differential

equations with a large parameter and turning points, Arch. History Exact.

Sci., 7(1971), 277-324.

7. Messiah, A., Quantum Mechanics, Vol. II, Interscience, New York, 1961.

8. Olver, F.W.J., Asymptotics and Special Functions, Academic Press, New

York, 1974.

9. Rellich, F., StSrungstheorie der Spektralzerlegung, I, Mitteilung. Math.

Ann., 113(1936), 600-619.

I0. Strang, G., Linear Algebra and its Applications, Academic Press, New York,

1976.

ii. Turrittin, H. L., Solvable related equations pertaining to turning point

problems, Asymptotic Solutions of Differential Equations and Their

Applications, edited by C. H. Wilcox, Wiley, New York, 27-52, 1964.

SOME FURTHER RESULTS ON OSCILLATORY BEHAVIOR

OF SOLUTIONS OF NTH ORDER DELAY DIFFERENTIAL EQUATIONS

S.R. GRACE

B.S. LALLI

ABSTRACT. Some new integral conditions for the oscillation of the

nonlinear nth order delay differential equation

m

x (n) (t) + [ qi(t)fi(x[ai(t)]) = 0 , n even , i=l

are established.

I. INTRODUCTION

The purpose of this paper is to establish some new oscillation

criteria for the nonlinear delay equation

m

x(n) (t) + [ qi(t)fi(x[ai(t)]) = 0 , n even , i=l

(z)

where qi ' gi: [to' ~) ÷ R = (-~, ~) and fi: R + R are

continuous, qi(t) is nonnegative and not identically zero on

any ray [t*, ~) , t* ~ t O , 0 < gi(t) ~ t , ai(t ) + ~ as

t ÷ ~ and xfi(~) > 0 for x ~ 0 , i = i, 2, .... m .

Most of the results in the literature concerning equation (i)

and/or related equations require that the functions fi' i = I, 2, ..., m

be monotonic, see [1-5, 9-11]. In [6, 7] Mahfoud relaxed the condi-

tions of monotonicity on fi by assuming that fi' i = i, 2 ..... m

belong to the class Cp(R) of functions of bounded variation on

finite intervals not containing zero.

173

In this paper we extend and improve some of the known results

for equation (1) when n = 2 to the case when n > 2 , and relax

the condition of monotonicity on f. , i = i, 2 .... , m by 1

allowing f. to belong to the class C (R) We also discuss i p

the case when f. , i = i, 2, .... m , is neither differentiable 1

nor monotonic.

Without further mention we will assume throughout that every

solution x(t) of equation (i) that is under consideration is

continuable to the right and is nontrivial, i.e. x(t) is

defined on some ray IT x, ~) and sup {Ix(t) I : t ~ T} > 0 for

every T > T Such a solution will be called oscillatory if -- x

its set of zeros is unbounded, and will be called nonoscillatory

otherwise.

As in [7] the following notations will be used throughout

this paper:

R = (-~, -el U [a, ~) if e > 0 e

: (-~, 0) U (0, ~) if e = 0

C(R) = {f: R + R I f is continuous and xf(x) > 0

and

for

x ~ 0 },

CI(Ra) = {f e C(R) [ f is continuously differentiable in Re} ,

Cp(Re) = {f e C(R) I f is of bounded variation on every

[a, b] c R e) .

The following three lemmas will be needed in the sequel. The

first lemma can be found in [7] and the other two lemmas appeared

in [i, 8].

LEMMA i. Suppose e > 0 and f e C(R) Then f e Cp(R e) if

= ÷ (0, ~) , and only if f(x) g(x)h(x) for all x £ R e , where g: R e

nondecreasing on (-~, -a] and nonincreasing on [e, ~) , and

h: R + R and nondecreasing in R e e

174

Remark. The function h in Lemma 1 is called a nondecreasing

component of f while g is called a positive component of f

LEMMA 2. Let u be a positive and n-times differentiable function

on an interval [to, ~) If u (n) is of constant sign and not

identically zero on any interval of the form It*, ~) , t* ~ t O ,

then there exist a t u ~ t o and an integer £ , 0 ~ £ ~ n ,

with n + £ even for u (n) nonnegative or n + £ odd for u (n)

nonpositive and such that

£ > 0 implies u(k) (t) > 0

and

£ < n - 1 implies

for every t > t4~(k = 0, 1 ..... £ - i)

(-l)£+ku (k) (t) > 0 for every t > t u

(k = £, £ + I, .... n - i)

LE~4A 3. If the function u is as in Lemma 2 and

u (n-l) (t)u (n) (t) _< 0 for every t _> t u ,

then for every I , 0 < I < 1 , we have

1 1 n-i

u[It] > [~ -I~- t n-I l u (n-l) (t) I for all large t . 2 n-I (n - I) !

We need the following theorem (for proof see [5]).

THEOREM A. If

f sn-l( T (s))ds : ~ i=lqi

(2)

then every bounded solution of equation (1) is oscillatory.

For convenience of notation we let

h(x) = min hi(x) , where h i l<i<m

is the nondecreasing component of

fi ' i = i, 2 ..... m and ~(t) = min ~i(t) l<i<m

We shall assume that

0 < ~(t) < t , a(t) > 0 and ~(t) + ~ as t + ~ , t>to > 0 .

175

2. OSCILLATION CRITERIA

THEOREM i. Let fie CI(R ) , e ~ 1 and let gi' hi be a pair

of continuously differentiable components of fi ' i = i, 2 .... , m .

Suppose that

h' (x) > 8 > 0 for x ~ 0 , (' = ~x ) , (3)

t f T lim sup t 1-£ f (t - u) £-3 u r t - u) 2 t+~ t o i=l qi (u) gi (kon-i (u) (4)

[(t- u) r - (£ - l)]21d u . . . . . . ] = oo I - 22n-3(n - I) ! 8 n_2(u) .(u )

for every k > k 0 ~ l,some integer £ > 3 and some constant

r < n - 1 . Then every solution of equation (i) is oscillatory.

Proof. Let x(t)

say x(t) > 0 for t ~ t I ~ t O

that x[ai(t) ] > 0 for t ~ t 2 ,

there exists a t 3 ~ t 2 so that

x(t) > 0 and x (n-l) (t) > 0

Hence x(t) ~ c as t + ~ , 0 < c <

t ~ t 3 , there exists a constant k I > 0

for t ~ t 3 . Integrate this inequality

to t and choose t 4 ~ t 3 and k 2 ~ 1

for t ~ t 4 Choose t 5 ~ t 4 • so that

o(t) ~ t 4 for t ~ t 5 , then

be a nonoscillatory solution of equation (i),

Then there exists t 2 ~ t I so

i = i, 2 .... , m . By Lemma 2,

for t > t 3 .

Since x (n) (t) < 0 for

such that x (n-l) (t) < k 1

(n - l)-times from t 3

so that x(t) < k2tn-i

a i (t) > t 4 and

_ k n-i x[ai(t)] < 2 i (t) , t >_ t 5 and i = i, 2 ..... m . (5)

Suppose 0 < c < ~ As (4) implies (2), the conclusion of the

theorem follows from Theorem A. Suppose c = ~ , there exists

t 6 sufficiently large so that

X[ai(t)] _> x[O(t)] _> e for t _> t 6 and i = i, 2, .... m .

Since

176

fi(x) = gi(x)h i(x) for x >

we have

and i = i, 2, ..., m ,

for t ~ t 6

Define

fi(x[ai(t)] ) = gi(x[ai(t)])hi(x[ai(t)]) n-i

> gi (k2ai (t))h(x[~(t) ])

and i = i, 2, .... m .

Then w(t) satisfies

+

w(t) = trx (n-l) (t)

h(x[la(t) ])

m [ (t)fi(x[ai(t)])

w(t) < -t r i=lqi -- 1 h(x[~o(t)])

rtr-lx (n-l) (t)

h(x[½o(t)])

-l$(t)trh,(x[lo(t)]) x (n-l) (t)x[la(t)]

h2(x[½o(t)])

1 Next, applying Lemma 3 for u = x and I = ~ we conclude that

there exists t 7 sufficiently large so that

22-2n (n-l) x[ a(t)] ~ (n - i) ! an-2(t)x (t) for t ~ t 7 .

Using (3), (6) and (7) we obtain

m

w(t) < "tr i=l[ qi(t)gi(k2an-li (t)) + rtr-lx (n-l) (t)

h(x[la(t) ])

2 l-2n 8tran-2 { x (n-l) (t) .>2 (n - i) ' (t)g(t) - 1

• h(x[~(t)])

Thus

t t m ~_l(u))d u / (t u)£-lw(u)du ~ ~ (t u)£-i r - - - u ~ qi(u) gi(k2a

t 7 t 7 i=l

t < > + f (t u) 9..-1 r-i x (n-l) (u)

- ru du t 7 h(x[la(u) ])

t 21-2n n-2 - f (n - i) ! B(t - u)£-lura (u) a(u)

t 7

lh x(n-l) (u))2 du

(x [21-- o(u) ] )

(6)

(7)

(8)

177

Since

t t f (t - u)£-lj(u)du = (£ - l) /(t - u)£-2w(u)du - ~(t7)(t - t 7)

t 7 t 7

we have

t r T /k an-i ( ~)£-i t I-£ f (t - u)£-lu qi(u)gi\ 2 i (u))du < ~(t 7) 1 -

t 7 i=l --

t £-2ur[(t- u)r- (£- i)](x(n-1) (u)) du + t l-£t7f (t- u) ~ h(x[lo(u)])

,n 2''n ( x,'n '>'u'l 2 ~- li! B ( t - u)£-luron-2(u)o(u) x[~a~ui]) / t 7 h(

du

£-i t

(t - u)£-3ur[(t - u) r (£ - i)] t + t l - £ f 3 - 2 n

t 7 2 Bo n-2 (u) d(u) (n - i) !

I { 2 l-2n - tl-~ ~ ~(n -- ~ ,

t 7 "

£-3 (t - u) 2 ur[(t - u) r

u 2 1 _ 2 n

2 (n - 1 ) ! B u r a n - 2 ( u ) J ( u )

x (n-l) (u) B(t - u) Z-luran-2(u)J(u) \h(x[lo(u)])

£ - 1)]] 2

1/2 du

[ r ] + tl-£ ~ 22n-3(n - i) ! (t - u)£-3ur (t - u) u - (£ - i) 2

t7 B o n-2(u) J(u) du

+ ~(t7) as t + ~ ,

which contradicts (4). Thus our proof is complete.

COROLLARY i. Let condition . (4) in Theorem 1 be replaced by:

t £_lurC T q (kan-l(u))) du : lim sup t I-£ l(t - u) (u)g i t+~ t O \i=l i

(9)

2

du

178

lim tl-£ ~(t - u)£-3ur[ r ] t÷~ to j(u) on_2(u ) (t - u) u - (£ - I) 2du < m ,

for every k > 1 , some integer £ > 3 , and some constant

r < n - 1 , then the conclusion of Theorem 1 holds.

The following example is illustrative.

(10)

EXAMPLE I. Consider the equation

x(t) + t x[ct] = 0 ,

1 + x2[ct] t > 0 , 0 < c < 1 ,

If we choose r = 0 , then this equation is oscillatory by

Corollary I, whereas none of the criteria given in [2], [9], [i0],

and [ii] is applicable.

REMARK. It is easy to see that condition (4) is weaker than condi-

tions (9) and (i0). We can easily check that (4) can be applied

to ensure the oscillation of all solutions of the equation

x'(t) + t-2x(t) = 0 , while Corollary 1 as well as the results

in [2], [9], [i0], and [ii] are not applicable.

f, # l

In the following theorem, we do not require that the functions

i = i, 2, ..., m , be differentiable.

f (x) THEOREM 2. Let l > c. > 0 for x ~ 0 ,

x -- 1

Suppose that

i = i, 2, ..., m .

t £_3ur[ lim sup t I-£ f (t - u) (t - u)

t+~ t o

m 2

ciq i (u) i=l

- 22n-3(n - i) ! [(t- u) r - (i - 1)] 2 ]du =

] co I

O n-2 (U) J(U)

for some integer £ > 3 and some constant r < n - 1 . Then

every solution of equation (i) is oscillatory.

(11)

179

Proof. Let x(t)

and assume that

proof of Theorem 1

x(t) > 0 and x (n-l) (t) > 0

Define

Then

be a nonoscillatory solution of equation (i)

x(t) > 0 for t ~ t I ~ t O . As seen in the

trx (n-l) (t)

w(t) = x[[l o(t) ]

for t > t 3 .

m f(x[°i(t)]) + rtr-i x (n-l) (t) w(t) = -t r X qi (t)

i=l x[[o(t) ] x[ o(t) ]

trx (n-l) (t)x[ 1 g(t)]o(t)

2x 2 [}g(t) ]

For 1

I = [ , by Lemma 3, we choose t 4

2 2-2n 0n_ 2 (t)x (n-l) • 1 xt[r (t)] > (n - i) !

sufficiently large so that

(t) for t ~ t 4

Thus

m rtr_ 1 x (n-l) re(t) < -t r [ eiqi(t) + (t)

-- i=l x [}0(t) ]

1-2n 2 (n - i)!

/x(n-1) (t)h2 trgn-2 (t)° (t)~x [} o(t) ] ]

The rest of the proof is similar to that of Theorem 1 and hence is

omitted.

THEOreM 3. Let fi ~ CI(Ra) ' a ~ 1 and let gi and h i be a

pair of continuously differentiable component of fi ' i = i, 2,

..., m , and that condition (3) ihold. Suppose that

t [ r m 2 2n-3(n

lim sup f LU i i qi 6 t÷ ~ to __[ (u)g i(k~i-!(u)) - - i)!

for every k > 1 and some constant r < n - 1 .

solution of equation (i) is oscillatory.

2 r-2 ] r u du =

0 n - 2 (u) 0 (u)

(12)

Then every

180

Proof. Let x(t) be a nonoscillatory solution of equation (i),

say x(t) > 0 for t ~ t I ~ t O . Following the proof of Theorem i,

we get

m -trllq i (t) gi (~ an ~'(t) < ~ -l(t)) + 22n-3(n - i)! r2tr-2

"= BO n-2 (t) $(t)

r/2 l-2n o n-2 ] - , , ~ 6 _ (t) o-(t)11/2 t 2 t r / w(t) _ 121-2n I [, 2 ~ ~ $ On-2 (t) O (t) 11/2

tr / J

m . r2tr_ 2 < -tr [ qi(t)gi(k2on-l(t)) + 22n-3(n - i)! ~on_2 (t) $(t) -- i=l

Thus

t[ r m (k2°n-l(u 22n_3( n r2ur-2 ] < - f u i__[ (u)g i )) - - i)' n ~ - |du ~(t) _ m(t 7) t7 lq i " 6~ - (u)O'(u)J

or

t I m 2 r-2 ] / u r [ qi(u)gi(k2on-l(u)) - 22n-3(n - i) ! r u du

t7[ i=l Bc n-2 (u) o(u)

< w(t7) - w(t) < w(t 7) < ~ ,

a contradiction to (12). This completes the proof.

The following thorem is immediate and we omit the proof.

THEOREM 4. fi(x)

> C. > 0 for x ~ 0 , Let x -- 1 i = i, 2, ..., m .

Suppose

lim sup (u) - - i) ! n-2 du = t +~ t o (u) ~(u)

(13)

Then every solution of equation (i) is oscillatory.

REMARK. The results of this paper are presented in a form which

is essentially new. The function fi and its components gi and

h i , i = i, 2, ..., m have been chosen without any restriction

such as lim inf Ifi(x) I > 0 or gi bounded away from zero or s÷±~

h i bounded non-decreasing, i = I, 2 ..... m , [see Mahfoud [7]].

We also do not impose any assumptions on qi ' i = i, 2, ..., m

similar to conditions (9) and (14) in [6].

181

The following examples are illustrative.

EXAMPLE 2. The equation

l-n t t x (n) (t) + t x[~]exp(sinx [ ]) = 0 , n even , t > 0 ,

is oscillatory by Theorem 4 for r = n - 2 whereas none of the

known criteria in [i], [6-8] leads to this conclusion.

EXAMPLE 3. The equation

x (n) (t) + kt-nx[ct] = 0 , n even , t > 0 , and 0 < c < 1 ,

is oscillatory by Theorem 3 for r = n - 1 and k > cl-n22n-3(n - i)!

We believe that the oscillatory behavior of this equation does not

appear to be deducible from other known criteria.

REFERENCES

i. M.K. Grammatikopoulos, Y.G. Sfacas and V.A. Staikos, Oscilla- tory properties of strongly superlinear differential equations with deviating arguments, J. Math. Anal. Appl., 67 (1979), 171-187,

2. I.V. Kamenev, Integral criterion for oscillation of linear differential equation of second order, Mat. Zametki, 23 (1978), 249-251.

3. I.V. Kamenev, Some specific nonlinear oscillation theorems, Matem. Zam. i0 (1971), 129-136 (Russian).

4. I.V. Kamenev, Oscillation criteria related to averaging of solutions of ordinary differential equations of second order, Differencial'nye Uravnenija, i0 (1974), 246-252, (Russian).

5. A.G. Kartsatos, Recent results on oscillation of solutions of forced and perturbed nonlinear differential equations of even order, in "Stability of Dynamical Systems Theory and Applications", Lect. Notes in Pure and Appl. Math. Vol., 28 (1977), 17-7 .

6. W.E. Mahfoud, Characterization of oscillation of the delay

equation x~n) (t) + a(t) f(x[q(t)]) = 0 , J. Diff. Eqns., J %

2S (1978), 437-451.

7. W.E. Mahfoud, Oscillation and asymptotic behavior of solutions of nth order nonlinear delay differential equations, J. Diff. Eqns., 24 (1977), 75-98.

8. V.A. staikos, Basic results on oscillation for differential equations with deviating arguments, Hiroshima Math. J., 10 (1980), 495-516.

182

9.

i0.

A. Wintner, A criterion of oscillatory stability, Quart. Appl. Math., 7 (1949), 115-

C.C. Yeh, An oscillation criterion for second order differen- tial equations with damped term, Proc. Amer. Math. Soc., 84 (1982), 397-402.

A MODEL FOR A MYELINATED NERVE AXON

P. Grindrod

INTRODUCTION

A myelinated nerve axon consists of a long cylindrical cell mem-

brane surrounded by an insulating sheath of myelin. At about one

mil!imetre intervals there are small gaps, called nodes of Ranvier.

At the nodes the axon membrane is selectively permeable to the charged

ions within the axoplasm and external ionic fluid, and the trans-

membrane potential is controlled in a similar manner to that of the

simpler unmyelinated axon.

Between the nodes the myelin sheath prevents ionic transport

through the membrane, except possibly for some leakage from the

axoplasm into small pockets of plasma held within the sheath.

In our model we allow the nodes to have some finite longitudinal

length. This approach differs from that of Bell and Cosner [23 who

assume the nodes to be point sources of possible excitation. However

we are able to see how the nerve-axon may become degenerate as the

node length becomes small when compared to that of the myelinated

segments.

Figure 1 shows a schematic picture of the axon in longitudinal

section. By considering the equivalent electric circuits for a

myelinated and unmyelinate interval we obtain the following diffusion

equations for the transmembrane potential v(x,t) (see [4]) ;

- Z for x in a myelinated interval (la) CmVt = Vxx R

= + I for x in a node (ib) cNv t Vxx

Here c m, c N are the capacitances per unit length for the myelinated

and unmyelinated membrane respectively; R >> 0 denotes the resist-

ivity of the myelinated-membrane; and I(v) is the ionic membrane

current density. Ideally we would take Hodgkin Huxley kinetics to

describe I. However here we will take the much simpler form;

I = f(v) = v(l-v) (v-a) , where a c (0,%). (i.e. we are assuming

FitzHugh-Nagumo kinetics and ignoring the recovery process, which

follows any excitation.)

184

NODE

MYELIN

11

ME NB RANE AXOP L A S M

Figure 1

A wave of excitation propagating down the axon corresponds to a

shift in the potential v away from the resting equilibrium (v=0) up

to some positive excited range.

We patch the equations (la) and (ib) by applying the following

continuity conditions at the myelin-node junctions, say at x = x 0

v(x0t) I + =_ lim{v(x 0 + ~,t) - v(x 0 - c,t)} = 0 (2a) e÷0

Vx(X0t) 1 +_ = lim{Vx(X 0 + e,t) - Vx(X 0 - £,t)} = 0 (25) ~÷0

Thus once we specify the positions and widths of the nodes we may use

(la) , (Ib) , (2a) and (2b) to obtain a model of a myelinated axon.

Central to the theory of what follows is the following Comparison

Principle which we state here for convenience.

Comparison Theorem.

{X i } ÷ -~ as Let i=-~ be a sequence in ~ such that x i

i ÷ -~ and x. + +~ as i ÷ ~. 1

Let g be a continuous function: [0,i] ÷ IR such that g(0) =

g(1) = 0.

Suppose u(x,t) and v(x,t) are continuously differentiable

functions: IR × ~+ ÷ [0,i], satisfying

CnU t - Uxx - g(u) z CNV t - Vxx - g(v)

for x c (Xi;Xi+l) , i - even, t a 0

185

CmU t - Uxx - u/R a CmV t - Vxx - v/R

for x c (Xi,Xi+l), i - odd, t h 0

where c m, CN, R > 0.

Then u(x,0) ~ v(x,0), for all x ~ ~ implies

u(x,t) ~ v(x,t), for all x c ~ , t a 0

The proof of this theorem follows in the same way as that of

Aronson and Weinberger, [i], and uses the continuity of vt, u t, v x,

u at x = X. -~ s i s ~. x 1

§i Superthreshold steady-state solutions

Consider the following model for an infinitely long uniform axon.

: + f(u) x e (0 8) mod (i) t > i] CnU t Uxx , -

CmU t : u - u/R x c (8 i) mod (i) t > xx

u(x t)I + , _ : Ux(X,t) I +_ : 0 for x ~ 0, 8 mod (i)

(3)

Here R >> 0, i >> 8 > 0 (8 is the ratio of node length/internode

length), c n, c m > 0, and f(u) : u(l-u) (u-a) for some a ~ (0,i/2)

Suppose u(x,0) is given and takes values in [0,i]. Then the

Comparison Principle applied to (3) ensures that 1 > u(x,t) ~ 0 for

all x ~ ~ , t a 0. If u(x,0) is constant then by symmetry u(x,t) =

u(x+l,t), for all x c ]19, t ~ 0. Thus u can only converge to

1-periodic steady-state solutions, v(x) say. Clearly v H 0 is one

such solution and we wish to determine whether there are any others.

In particular we would like to know whether a super-threshold steady-

state exists, that is one which satisfies v(x+l) : v(x) for all

x ~ ]19 and i ~ v(x) > a for x ~ [0,0]mod(1) . (i.e. v is excited

by ionic currents at each node.)

On the otherhand, if such a solution does not exist then our

solution u of (3) satisfying constant initial condition, will

decay to zero. In this case the nerve-axon could not support any

propagating waves of action potential and would be degenerate. The

next Theorem gives a sufficient condition for this to happen.

Theorem 1 For each R > 0, fixed, there exists 80 > 0 such that

for 0 < ~ < 80, no super threshold period (i) steady-state solutions

for (3) exist.

Moreover a lower bound 8 x to 80 is determined as the unique

186

solution of m8 a tanh((l-8)/2/R) = 0 where m : max f(u) . 2 /~ u~(0,1)

The proof of Theorem i involves superimposing the phase planes of

the problems y" + f(y) : 0, y" - y/R = 0, and using a "Time-map"

argument to show that, no closed loop {(w(x) ,w(x)) jx ~ [0,i]} can

exist and satisfying

Q' = -f(w), w' = Q

~' : w/R, w' =

for x c (0,e)

for x c (Q,I)

(see [4] for a full proof and further discussion).

Our next theorem gives a simple condition under which the zero

solution of (3) is globally stable.

Theorem 2 Suppose without loss of generality that c N : c m = i, and

U0(X) ~ L2(IR) satisfies 0 ~ u0(x) ~ 1 for all x e ~ . Let u(x,t)

denote the solution of (3) satisfying u(x,0) = u0(x) .

Define s = sup [f(u)/u] u~[0,1]

(l-e) es > 0. and suppose ~ R

-qt Then JJ u(.,t)IIL2 s ce

Sketch of Proof (see [4] for further details.)

Let w(x,t) satisfy

- a(x)w for x ~ ]19, t >- 0 w t = Wxx

and

w(x,0) = Uo(X) where

~(x) = -s for x c (0,8)mod(1)

= I/R for x ~ (e,l)mod(1) .

Then the Comparison Principle implies

w(x,t) a u(x,t) a 0 for all x ~ ~ , t ~ 0

So we prove the result for w.

Define D = {v ~ L2[IR ]iv' (x)

L 2 and -v" + ~v E (IR)}.

exists and is absolutely continuous

For v ~ D define

LV = - -- + a(x) v. d x 2

Then i) L is self adjoint

ii) The spectrum of L is real, continuous and

I ~ o(L) ~ i ~ ~ > 0. (see [3]).

187

Using i) and ii), together with the theory of Sectorial Operators (see

Henry [5], for example), we have

II w(-,t)IIL2 -< constant e -qt

as required. D

§2 Non-existence of propagatio__nn

Here we show that if any of the internode lengths exceed a given

value, then the model cannot admit a propagating superthreshold wave

of excitation.

Consider the half line problem

CmU t = Uxx u/R x ~ (Xi,Xi+l), i - even

CNU t = Uxx + f(u) x ~ (Xi,Xi+ I) , i - odd

when {X i} is a sequence in ~ satisfying

X 0 = 0, limX i = ~ i+~

(4a)

4b)

At x = X. we apply the continuity conditions 1

u(Xi,t) I + : Ux(Xi,t)I +_ = 0 4c)

Finally we impose the initlal-boundary data

u(O,t) = 1 , t -> ()

u(x,O) : h(x) , for x _> 0

4d)

(4e)

Steady-state solutions of (4a-4d) must be given by

where

q" = ~/R x E (Xi,Xi+ I) i - even

q" = -f(q) x 6 (Xi,Xi+ I) i - odd

q(xilI_ + = q Ixil I_ +

u(x,t) = q(x)

(5)

and q (0) = 1.

Lemma 3 S u p p o s e X l / / R ,and R a r e s u f f i c i e n t l y l a r g e s o t h a t

-f(q)/q > I/R for q e (0,exp{-Xl//R}). Then there exists a solution ,

q (x) o f (5) s a t i s f y i n g

q '(X) < 0 for all x ~ ]R and lim q (x) = 0 . X+OO

188

Sketch of Proof (see [4] for more details)

Superimpose the phase planes for

ql = P I

and q' = p ,

For 7 = (i,~)

q(x)

p, = -f(q)

p' = q/R

T , ~ ~ 0, let ¥.x be given by (q(x), q' (x)

satisfies (5) together with q(0) = I, q' (0 = ~.

Let L 1 = {~.XlI~ = (l,n) for some n ~ 0}.

(6a)

(65)

where

P L 1

(0,0) ~/ Ca,0)

W/ / ! )

1 . w

/ /

/

q = 1 .)

q

Figure 2

Our hypothesis implies that L 1 is as shown in Figure 2,

may define W to be the closed set in ~2 shown shaded. t

Let S O = ~W n {q=l}.

And define S i = {7 • S01~.[0,Xi+13cW}, for i ~ 0. (Here

{~.xlx • I}.)

Then a) S i # @ for i = 0,i ....

b) S. is closed for i = 0,I,... 1

c) Si+ 1 c S i for i = 0,i,...

(The hypotheses are important here and imply that the flow

by (6a) takes orbits starting on p = - ~-q to the left of

and we

~.I =

defined

189

L 1 n {(q,P) IP : - lq} immediately out of W.)

Thus S = i~0Si ~ @. So we choose y e S and it is easy to show

that (q (x),q*~x)) = y.x is the required solution. D

Using this result together with the comparison principle we obtain

Theorem 4 If -f(q)/q > I/R for q ~ (0,exp{-Xl//R}) and 0 ~ h(x)

q (x). Then the solution u of (4a-e) satisfies

0 < u(x,t) _< q*(x) for all (x,t) c ]19 + × IR +

(see [4] for a further discussion).

§3 Propagation of wave-front type solutions

Here we make some remarks about the possible propagation of excit-

ed potentials in a non-degenerate axon. We assume R is very large

and approximate (3) by

+ e(x) f(u) (7)

for x e (0,e)mod 1

for x ~ (e,l)mod i.

Notice that u - 1 is always a superthreshold solution to (7). Here

we are interested in solutions which behave like travelling wave-

fronts, raising the potential from zero up to the excited state u = 1

as it propagates along the axon in time; that is, a solution of the

form u(x,t) = ~(z,t) where

i z = x + ct for some wave speed c > 0.

ii ~(z,t) : ~(z,t+p) for all z ~ IR, t c IR for some period of

oscillation p, >0.

lim g(z,t) = 1 , . , z -~oo

lll for all t c [0,p] lira @(z,t) = 0 z+-oo

In [43 we prove existence of such solutions to (7) where e is re-

placed by some small one-periodic perturbation of a positive constant.

This result, together with numerical investigation leads us to make

the following conjecture; if e(x) is some one-periodic non-negative

function, with a finite number of discontinuities on F0,1] and

1 ~(x)dx = ~ > 0, then (7) admits a solution u(x,t) = @(z,t) ;o

U t = UXX

where

(x) = i

0

190

u(z )

1.0

0.8

0.6

0.4

0.2

0.0

' 5

/

/ . ! . , J

10 15

Z

Figure 3

' 2'o

1.000

0.995

0.990

u(z) 0.985

0.980

0.975

0.970 12.0

J

%,

i | i

12.5 1~.0 1~.5 ' 14.0

Z

F i g u r e 4

191

satisfying (i)-(iii) above, where

c : /~ c , p = 1/c

and c is a positive constant depending upon f (c = (i-2a)//2 ,

if f : u(l-u) (u-a) for some a c (0,½)) .

Solutions of this form have been obtained numerically for a

variety of periodic functions e. Figures 3 and 4 depict such a

solution when f(u) : (l-u) (u-.l)u, and ~(x) = 32 for x ~ (0,.25)

mod 1 and =0 otherwise. In these figures we have changed to the

(z,t) variables so that the solution appears as a standing oscilla-

tory front.

REFERENCES

[1] D.G. Aronson and H.F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, in partial differential equations and related toDics, Lecture Notes in Mathematics, Vol. 446, Springer Verlag, 1975.

[2] J. Bell and C. Cosner, Threshold conditions for a diffusive model of a myelinated axon, J. Math. Biol. i_88, 1983.

[3] M.S.P. Eastham, The spectral theory of periodic differential equations, Scottish Academic Press, 1973.

[4] P. Grindrod and B.D. Sleeman, A model of a myelinated nerve axon with discrete finite nodes: threshold behaviour and propagation (in preparation).

[5] D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics Voi.840, Springer-Verlag, 1981.

SPATIAL PATTERNING OF THE SPRUCE BUDWORM

IN THE PRESENCE OF DEFOLIATION

Guo Ben Yu and B. D. Sleeman

I. Introduction

The spruce budworm is a defoliating insect whose host is the balsam

fir. Ludwig, Jones and Holling (1978) modelled a spatially uniform

budworm population by using a scaled ordinary differential equation

with some parameters which describe the foliage density and the

interaction between the budworm and its predators. Spatial effects of

dispersion in an infinite strip were considered by Ludwig, Aronson and

Weiberger (1979). In particular they considered the possibility of

preventing an outbreak of budworm population by a barrier. Wilson

(1982) considered the possibility of preventing an outbreak by spraying.

Recently Guo Ben Yu, Mitchell, Sleeman (1983) and Guo Ben Yu, Sleeman,

Mitchell (1983) considered similar problems for circular and rectangular

regions respectively.

In this paper we analyse a variation of a model proposed by Murray

(1983). The steady state solution is analysed in section III. We

determine the critical size of a patch in order to prevent an outbreak

in sections IV and V while we consider the asymptotic behaviour of the

budworm population in section VI.

II. The Model Problem

Let u (x,t) be the scaled density of the budworm population and

assume that the domain to be considered is the infinite strip

characterised by "il < x < Z2 ' where Z i > 0 , i = 1,2 . The

boundary conditions are

~u ~x 0 for x = -Z 1 12,t > 0

We suppose that the domain is divided into two parts in which the

defoliation in the second part (0<x<Z2) is considered more serious

than in the first part (-Zl<X<0) Furthermore we assume that there

are neither population sources nor sinks on the boundary between the

193

two parts of the domain (see Fig i).

Consider the following model

~u - 0

3x fl f2 ~u ~= 0

x

~2

Figure 1

~u 3 2u - + fl (u) ~t 3x 2

~u ~ 2u - + f2 (u) 3t ~x 2

-i I < x < 0, t > 0 ,

0 < x < ~2' t > 0 ,

~u ~=0,

~u u and ~ are continuous,

for x = -ZI,Z2, t > 0

for x = 0, t _> 0 ,

(i)

where

u(x,0) = ~(x) ,

2 fl(u) = ru (i- u__) u

qi l+u 2 '

-£i -< x -< £2 '

i = 1,2 .

The parameters r > 0 and qi > 0 where ql < q2 describe the

conditions of the forest and the interaction between the budworm and

its predator.

The corresponding steady state problem is

d2v~ + fl (v) = 0 -£i < x < 0 dx 2 '

--d2v + f (v) = 0 0 < x < Z 2 dx 2 2 '

(2)

dv ~-{= 0 ,

~v v and ~-~ are continuous,

for x = -il,Z2 l

for x = 0 .

We suppose that r and qi are such that

194

(i) fl (v) has only one positive root u I (see Fig 2),

(ii) f2(v) has three positive roots ordered as (see Fig 3)

~i) (2) < uJ3) u -< u 2 -

fl (v)

o ~v

f2 (v)

Figure 2

0 u2 u2

V

Figure 3

Let

and

Vlm = max v (x) -~I -<x-< 0

V2m = max v (x) , 0-<x-< £2

then it can be proved that

Vlm -< u 1

and

V2m < u~l)- or u~ 2) <- V2m -< u~3)

For fixed values of r and qi,Vlm and V2m depend on i I and £2

We shall determine the critical values of i I and £2 for which

V2m <- u~ I) In the terminology of Ludwig, Aronson and Weinberger

(1979) this corresponds to the prevention of an outbreak of the budworm

population. The corresponding solution is called an endemic state.

195

III. The Steady State Solutions

Let v (i I) : a, v (£2) = b, v (0) = v 0 and

v 2 3 r

= ]0fi(v)d v rv2 rV3q i v + arctgv , i = 1,2 Fi(v) o

The graph of F 1 (v) against v is shown in Fig 4.

F 1 (v)

4

0 u I

~v

Figure 4

NOW let q2 (r) be the critical value of q2 for which

F 2(uj i)) =F 2(u~ 3))

If q2 < q2 (r) or q2 = q2 (r) , then the graphs of F2(v)

are as shown in Fig 5 and Fig 6 respectively. If

graph of F 2 (v) is. as shown in Fig 7 where

Fm(uJl) ) = F2(~ 2)

F 2 (v)

I I

' l l \ i I I (i) _ (2) _ . (3) '

0 u 2 u 2 u 2

D v

against v

> - q2 (r) , then the q2

F 2 (v)

. v 0

Figure 5 q2 < q2 (r) Figure 6 q2 = q2 (r)

196

I F (v)

~V

Figure 7 q2 > q2(r)

We now look for the solutions of (2). We first have

1 dv 2 ~(~-~) + F 2 (v) = F 2 (b) , 0 < x _< ~2 (3

We suppose that the maximum value of v(x) in the interval (0,~ 2)

occurs at the point x = x, i.e.

V2m = v (x) , 0 < x < i 2

then

1 dv 2 ~(~) = F 2 (V2m) - F2(v) , for v -< V2m (4

from which it follows, on using (3) that

F 2(b) = F 2(v2m) (5

If q2 < q2 (r) , then V2m < u~ I) From (5) and the monotonicity of

F 2 (v) (for 0 -< v -< u~ I)) , we have b = V2m If q2 = q2(r) , the

situation is similar. If q2 > q2 (r) , then there are three

possibilities

(i) V2m > ~2 ' then by (5) and the monotonicity of F2(v) (for

u 2 -< v -< u ) , we have also b = V2m

(ii) V2m < uJ I) and b -< V2m , then we still have b = V2m .

(iii) V2m < u~ I) and b > V2m , then u~l) < b < ~2 or b > u~ 3)

But the latter is not possible according to the continuity of v(x)

Thus b = v 2m "

We now establish the monotonicity of v (x) for 0 -< x -< £2 " We

first suppose that v(x) has a minimum v2, = v(x) where 0 < x < "£2 "

We have

197

i (dv. 2 F 2 (v2.) - F 2(v) = ~ ~) -> 0 , for v >- v2,

and so

Jl) (2) u < v2, < u 2

Thus from Fig 8 we see that

u < v2, < v (x) < u2 < ~2 ' for 0 < x < 12

where

F2(v2,) = F2(u 2) .

from which we conclude that u~ l) < < v V2m u 2 which is impossible.

Thus we conclude that v(x) is a nondecreasing function of x for

0 ~ x ~ ~2 "

F 2 (v)

Jl )v2 * (3) 0 u . u u 2 (2) 2

u 2 Figure 8

~v

On integrating (3) , we obtain

1 [b dv

Z2 --/~ ]v 0 /F 2(b)-F(v) (6)

Similarly we have

i(dv) 2 2 dx + F 1 (v) = F 1 (a) , for - %1 < x < 0

and so

_ I I a dv

il -/2 v 0 /F 1 (a)-F 1 (v)

By the continuity of du__ at x = 0 , it follows that dx

F2(v 0) - F I (v 0) = F2(b) - F 1 (a)

(7)

(8)

Now for the fixed values of "%1 and '%2 ' the values a, b and v 0

are determined by (6) - (8) while the solution of (2) is determined by

198

and

£2 -x = ~ r]b dv , 0 < x < £2 ' (9)

/2 v (x) /F2(b)-F2(v )

il+x = 1 [ ja dv ' -Z1 < x < 0 , (10)

/[ v(x) /F 1 (a)-F 1 (v)

IV. The Critical Value of Z2 for Preventing an Outbreak of Budworm

To begin with we consider the following related problem

i d2w (w) = 0 0 < x < 2~ 2 d~ + F 2 ,

(11) w(0) : w(2~2) : V 0 .

(i) where v 0 = u 2

By an argument similar to that of Ludwig, Aronson and Weinberger

(1979) we obtain

w m : max w(x) = w(gv2) 0-<x- < 212

and so for x = "%2 '

dw -- 0 .

dx

We have

1 ,dw) 2 ~[~ : F2(w m) - F 2 (w) , for v 0 < w < w m

If q2 -< q2 (r) , then 0 < w m < u If q2 > q2 (r)

0 < Wm < uJ 1) or ~2 < Wm < uJ 3) The graph of "%2

* (v ) is the critical value of shown in Fig 9 where "%2 0

refuge for the budworm, i.e. if %2 < i2 (v0)' then (ii) has only the

trivial solution. Now ~2(v0) is the critical value of ~2 for which

*(v0) < ~2 < ~2(v0 ) ' then (ii) has there is a budworm outbreak. If "%2

only a small and stable solution w (I) (x,v 0) which is taken to be an

endemic state. If .% > ~2(v0) then (11) has three positive solutions

ordered as

w (I) (x,v0) -< w (2) (x,v0) < w (3) (x,v0)

where w (I) (x,v 0) corresponds to an endemic state, w (3) (x,v 0) is a

large and stable solution and corresponds to an outbreak state,

w (2) (x,v 0) is an unstable solution.

we have

against w m is

Z2 securing the

199

~2 (v0)

~ (v 0

1) v ~ J3) u 2 u 2 (v 0 ) u

,w

Figure 9

Now let w(x,v 0) be the solution of (ii) corresponding to ~2(v0)

andhave the maximum value u 2 (v0) . We now define the value ~2 (v0)

for the fixed value of v 0 Indeed we have

w(x) = v(x) , for 0 < x < ~2

As shown before, we have b = V2m Let b = u 2 (v 0) then for b = b ,

dZ 2 (b) 0 .

db

Now we have

di 2 (b) _ 1 iLm ( 1 .)_ __i I b f2(b)

db /[ v÷b /F 2 (b)-F 2(v) 2/5 v 0 (F 2 (b)-F 2 (v)) 3/2

(12)

A computation shows that

-i [b f2 (b)dv = -_II [b f2 (b)

2/2 3v 0 (m2(b)-F2(v)) 3/2- /2 ]v 0 f2 (v)

1 - - d ( )

/F 2 (b) -F 2 (v)

-i f2 (b) f2 (b) - -- lira( ) +

/2 v÷b f2 (v)/F2 (b)-F2 (v) /~. f

1 [b f2 (b)f~ (v)dv

3 2(v)/F2(b)_F2 (v) /7 v 0 f2

2 (v0)/F2 (b)-F 2 (v 0)

(13)

200

By substituting (13) into (12) we find that

db /[ f2(v0)/F2(b)_F2 (v0) / [ Jv0f~(v)/#2[S:% (V)

Thus b is a root of the following equation

i g f~ (v)dv

= 0 1 - f2 (v0) 2(v)/F2 (~)_F2 (v)

/F2'(b)-F2 (v0) v0 f2

(14)

and %2 (v0) is defined as

i___ [b dv

%2 (V0) = /5 ]v 0 /F 2 (b)-F 2 (V)

In particular, if v 0 = 0 , then

(15)

* * 7[ Z2 (v0) : ~2 (0) = ~ and the value

~2(0) is identical to that of Ludwig, Aronson and Weinberger (1979).

If v 0 > 0 , then a comparison principle argument shows that

u 2 (0) < u 2 (v o)

and so

"~2 (Vo) < '~2 (o)

We know that the solution of (i) is identical to the solution of (ii)

for 0 ~ x s Z2 and v 0 > 0 . Thus if Z2 ~ ~2 (0) l

V2m > u

i.e. in this case we can not prevent a budworm outbreak for any value

of Z1

V. The Critical Value of %1 for Preventing an Outbreak of Budworm

Now we suppose that the value of

We look for the critical value of Z1

(recall that b : v m is an increasing function of v 0)

to (14) and (15), we define the values b and v 0 via

f ' (v)dv j - 0 ,

1 _ f (v0) f2(v)~Fo(~)_F2(v ) = ~ ./F2 (~)_F 2 (v0) 2 v0

1 ]b dv

£2 = ~-~ v0 /F 2 (~)_F 2(v~

Z2 is fixed and Z2 < ~2 (0)

such that 0 < V2m < u 2 (i)

According

(16)

201

we know that ~2 (v0) is a decreasing function of

so if v 0 < v 0 , then V2m < uJ I)

v 0 (see Fig i0) and

~2 (v0)

~2 (~o) ~ (v o)

'b ~ (v o)

~2

~2

/

<

]

(t7 o )

u~ I) {2

h (~o) I I I I I I I

(3) u 2

Figure I0 v 0 < v 0

| V m

For the critical value of v 0 , the corresponding value of a ,

denoted by a , satisfies

F 1 (a) = F 1 (v 0) + F 2 (b) - F 2 (v 0)

and the corresponding value of £i ' say ~i ' satisfies

~1- i I a dv

/ ~ v o / F 1 ( a ) - F 1 (v)

The value il determined by (18) is the critical value of

prevention of a budworm outbreak.

(17)

(18)

i I for the

202

VI. Asymptotic Behaviour

In this section we consider the asymptotic behaviour of the

solutions to (I). Let Q(x) be a twice continuously differentiable dv

function except at the point x = 0 while ~ is continuous at x = 0.

A function ~(x) is defined as a supersolution of (i) if

d2v (i) fl (v) > 0

dx 2 - '

d2v f (v) > 0 , (ii) 2

dx 2

dv (iii) dx 0 ,

-£i < x < 0 ,

0 < x < £2'

x = -£i ' £2 "

Similarly a function v(x) is a subsolution of (i) if

d2v

(i) - - fl (V) < 0 -£i < X < 0 dx 2 -- -

d2v

(ii) f2(v) < 0 0 < x < £2 dx 2

dv

(iii) dx - 0 x = -£i,£2

we have the following results.

Proposition 1 Let v(x) be a supersolution and let u(x,t) be a

solution of (i) with u(x,0) = ~(x) , then for each x,u(x,t) is a

nonincreasing function of t . If moreover u(x,t) is bounded from

below then

lira u (x,t) = v I (x) t+~

where v I (x) is the largest solution of (2) for which v I (x) -< v(x)

Proposition 2 Let v(x) be a subsolution and let u (x,t) be a

solution of (i) with u(x,0) = v(x) , then for each x,u(x,t) is a

nondecreasing function of t . If moreover u (x,t) is bounded from

above then

lim u(x,t) = v2(x) t÷oo

where v 2 (x) is the smallest solution of (2) for which v 2 (x) > v(x)

Indeed the solution of (i) is identical to w(x,t) for -£i -< x -< £2

where w(x,t) is the solution of the following equation

203

Sw ~2w fl (w) = 0

St ~x 2 -£i < x < 0, t > 0 ,

~w ~2w

~t 2 ~x

f2(w) : 0 , 0 < x < ~2, t > 0 ,

w(x,t) = b(t) , x = Z2, t -> 0 ,

w(x,t) = a(t) , x = -il, t >- 0 ,

~w w and ~ are continuous, x = 0, t >_ 0 .

The conditions in proposition 1 (or 2) imply w(x,t) _< v(x) (or

w(x,t) _> v(x)) for x : -ZI,Z 2 which can be proved by an argument

similar to that of Ludwig, Aronson and Weinberger (1979).

Now we suppose that ~i and Z2 are chosen such that (2) has only

one endemic solution say v0(x). Then v0(x) can be taken to be both

the supersolution and subsolution of (i). By using the comparison

principle and proposition i and 2, we conclude that for all initial

values u(x,0) and each x

u (x,t) ÷ v 0 (x), as t÷~ .

References

[i] Aronson, D.G., Weinberger, H.F. Nonlinear diffusion in population genetics, combustion and nerve propagation, partial differential equations and related topics, Lecture Notes in Mathematics, Vol. 446, Berlin, Springer-Verlag, 1975.

[2] Aronson, D.G., Weinberger, H.F. Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math. 30, 33-76, 1978.

[3] Greenbank, D.O., Schaefer, G.W., Rainey, F.R.S. Spruce budworm (lepidoptera: tortricadae) moth flight and dispersal, New understanding from canopy observations and aircraft, Memoirs of the Entomological Society of Canada, No. ii0, 1980.

[4] Guo Ben-Yu, Mitchell, A.R., Sleeman, B.D. Spatial patterning of the spruce budworm in a circular region, UDDM Report DE 83:5, 1983.

[5] Guo Ben-Yu, Sleeman, B.D., Mitchell, A.R. Spatial effects in a two-dimensional model for the budworm-balsam fir ecosystem, 1983.

[6] Ludwig, D., Aronson, D.G., Weinberger, H.F. Spatial patterning of the spruce budworm, J. Math. Biology 8, 259-263, 1979.

[7] Ludwig, D., Jones, D.D., Holling, C.S. Qualitative analysis of insect outbreak systems, the spruce budworm and forest, J. Anita. Ecol. 47, 315-332, 1978.

[8] Murray, J.D. Private communication to Professor Mitchell, A.R., 1983.

[9] Wilson, A. Spatial effects of a model for the budworm balsam fir ecosystem, preprint, 1981.

VECTOR MODELS FOR INFECTIOUS DISEASES

K. P. Hadeler, THbingen

Abstract:

A model for infectious diseases designed by K. Dietz and the author

classifies the host population according to age and the number of para-

sites carried. Parasites are transmitted by vectors. The infection rate

is a function of the average parasite load. The model can be formulated

in terms of a differential equation for a generating function or of a

Volterra integral equation for the infection rate. There is a trivial

stationary state corresponding to a noninfected population and a branch

of infected states which bifurcates at a critical level of the trans-

mission rate. This branch may exhibit backward bifurcation and hystere-

sis. The change of stability along this branch is investigated. In a

more elaborate model the birth and death rate of the host as well as

the transmission function depend on the total population size of the

host.

Introduction.

In [5] K. Dietz and the author have presented a model for parasitic

infectious diseases which takes account of the age structure of the

host population, the small population numbers of parasites, and the

transmission by vectors. The model has been designed to describe di-

seases such as onchocerciasis. Other models of a similar character,

though less detailed in the a priori assumptions, have been given by

[I],[13]. Stochastic processes which describe schistomiasis have been

discussed in [14,15] and [2]. For classical models for epidemic disea-

sis, with or without age structure, see e.g. [3],[4],[17],[18].

The model has the form of a first order partial differential equation

for a generating function which classifies the host population accord-

ing to age and individual parasite load. The differential equation con-

tains nonlinearities with functionals of the unknown solution which

describe the transmission of new parasites. In [5] the initial value

problem has been transformed into a single nonlinear Volterra integral

equation for the parasite acquisition rate ~. In [8] the authors have

given a proof of existence and uniqueness for the initial value problem

which takes into account the properties of generating functions. The

model has been extended to the case where parasites multiply in the

205

host, and the connection to birth and death processes with killing has

been established [12],[16].

Similar to the classical model of Kermack and McKendrick for endemic

diseases the present equation exhibits a bifurcation phenomenon with

an obvious biological interpretation: If the contact rate or the trans-

mission ratelrespectively , exceeds a critical value, then the trivial

stationary solution, corresponding to a population without parasites,

looses stability, and a branch of stationary solutions with positive

infection level bifurcates. In the present model the existence of a

branch of nontrivial stationary solutions has been shown [5],[10]. In

[5] and [6] it has been proved that the trivial solution is stable up

to the point of bifurcation and looses its stability afterwards. If

the transmission function f is monotone and concave in the sense of

Krasnoselskij then the branch of nontrivial solutions is monotone [5].

However, without the concavity condition, but f still monotone, the

branch may bifurcate backwards or may have several turning points [9].

In such cases there are several nontrivial stationary solutions. The

question arises, which of these stationary solutions are stable. Of

course similar problems occur in many bifurcation problems, e.g. in

the standard nonlinear two-point boundary value problems. In the pre-

sent case it appears, that the bifurcation diagram determines complete-

ly the stability of the stationary solutions. Change of stability occurs

exactly at the turning points, the arcs with positive slope correspond

to stable solutions, those with negative slopes describe unstable solu-

tions.

The model can be extended to include situations, where the birth rate

and death rate of ho~3ts as well as the transmission function depend on

the total size of the host population. In this more general situation

only the stationary solutions are discussed.

The model.

The host population is described by the usual equation for a population

with age structure. The parasite population is governed by a birth and

death process with killing of the host (in the sense of [12] or earlier

[16]). Let ~ ~ 0 and a > 0 be the birth rate and death rate of the para-

sites within the host. It should be underlined that most helminthic

parasites do not multiply within the host. (For such parasites one has

= 0.) Let ~(a) and b(a) be the mortality and fertility of a host of

age a in the absence of parasites. Let ~ > 0 be the differential morta-

206

lity and ~[0,I] the factor of decrease in fertility due to the presence

of one parasite. Then, for a host of age a carrying r parasites, the

mortality is ~(a) +~r and the fertility is b(a)w r. The additive resp.

multiplicative dependence on the parasite number reflects the indepen-

dent action of the parasites upon the host, it also allows the use of

generating functions and first order differential equations.

Let ~ = ~(t) be the acquisition rate of new parasites. From the view-

point of parasites within a host ~ is the immigration rate of the birth

and death process. It is assumed that ~(t) is a nonlinear function of

the average parasite load, ~(t) = Bf(w(t)) . Here f with f(0) = 0,

f(u) > 0 for u > 0, is normalized by f' (0) = I, and B is a parameter.

One can interpret this parameter as a contact rate between hosts and

vectors, whereas the transmission mechanism is incorporated into the

function f.

Let n(t,a,r) be the density of hosts carrying r parasites, and u the

generating function

u(t,a,z) = ~-- n(t,a,r) zr. ( 1 . 1 ) r=O

The function u satisfies the differential equation

u t +u a +g(z)u z = [~(t) (z-l) -- ~(a)]u~ (1.2)

where

g(z) = (a.+o+~)z -o-?z 2 (1.3)

and

~(t) = ~f(w(t)) , (1.4)

co

fu (t,a,1)da Z

w ( t ) = o ( 1 . 5 ) co

/u(t,a,1)da O

The initial and side conditions are

u(0,a,z) = Uo(a,z ) ,

u(t,0,z) Ib(a)u(t,a,~)da. o

(I .6)

(I .7)

207

Notice that u(t,a,w) is a power series in ~, and the term counting r

hosts with r parasites is multiplied by

In a simplified version the Lotka birth law (1.7) is replaced by a

condition which prescribes the number of newborns

u(t,0,z) = N(t). ( 1 . 8 )

In the following we shall be mainly concerned with the initial value

problem (1.2) (1.4) (1.6) (1.8) . In [ 5 ], [ 8 ],[10] the following approach

has been choosen: Assume the function ~(t) is known. Then one can solve

the initial boundary value problem using the method of characteristics.

The solutions thus obtained, which depends explicitely one, can be in-

troduced into equation (1.4). This step yields an integral equation

for the function ~(t). In the case of the boundary condition (1.7) one

solves the initial-value problem (1.2) (1.6) (1.8) for prescribed ~(t)

and N(t) and inserts the solution u into the equations (1.4) ,(1.7).

Thus one obtains a system of two coupled integral equations for the

functions ~ and N.

The solutions operator G(t,z) of the Riccati equation z = g(z) plays

an essential role (see [10]). G(t,Zo) is the solution with initial

condition z(0) = z evaluated at t. The explicit representation is as o

follows. Let Zl,Z 2 be the roots of the polynomial g(z) , where

z I ~ I h z 2, and let

= I ? ( Z l - Z 2 ) ' ~ > 0

[ ~+0 , £ =0 (1 .9 )

Then -Mt z I (z-z 2) + z2(zl-z)e

G(t,z) = , ~ > 0 (z-z2) + (Zl-Z)e-)~t

G(t,z) = I-(1-z)e ~(t - ~ (1-e xt) , £ = 0

( 1 . 1 0 )

The following functions will be used

F (s) = G(-s,~) - I ( 1 . 1 1 )

Tz(S) = Gz(-S,I )

a Qw(a) = - f Fm(s)ds

o

( 1 . 1 2 )

( 1 . 1 3 )

208

We put

F: = F1 ' Q: = QI #

a

q(a) = f F z(s)ds. o

(i .14)

I .15)

A simple calculation shows

F'(s) = -~Fz(S) for ? ~ 0 , 1.16)

Q'(a) = ~q (a) for ~ _-> 0 • 1.17)

Furthermore define

a

M(a) = / #(s)ds. o

( 1 . 1 8 )

The average parasite load has the explicit representation

W = ~ S

t t / F (s-t)~(s)ds - M(a)

~ = f e t - a N d a

o t / F (s-t)~(s)ds -M(a) + M(a-t)

+ fe O Uo(a-t ,G(-t,1))da t

t t f F (s-t)~(s)ds - M(a) t

= f e t-a N f F z (s-t)~(s)dsda o t-a

t / F (s-t)~(s) ds-M (a) +M (a-t)

+ / e ° t

t / F(s-t)~(s)ds-M(a)+M(a-t)

oo

+ f e ° t

(I .19)

(1.20a)

(1.20b) t

Uo(a-t,G(-t,1)) / Fz(S-t)~(s)dsda O

u (a-t,G (-t, I) ) G(-t,1)da oz

The equations (1.4),(1.19) ,(1.20) represent an integral equation for

the function ~ =~(t).

~(t) =~(~)(t), (I .21)

~(~) (t) = ~f(w(t))

209

In [8] [10] it has been shown that this equation has a unique continuous

n o n n e g a t i v e g l o b a l s o l u t i o n ~ p r o v i d e d t h e f u n c t i o n f i s s u b l i n e a r a n d

the initial datum u is a generating function satisfying the following: o

i) 0 < fu (a,1)da <co. 0 o

ii) There is a constant p such that for all a ~ 0

Uoz(a,1) <p Uo(a,1).

Stationary solutions.

The stationary solutions of the initial boundary value problem corre-

spond to the constant solutions of the integral equation. Such constant

satisfy an equation

= Bf (W(~)~) (2.1)

where

W(~) = 11 (~)/Io(~), (2.2)

io(~) = I e-Q (a)~-M (a) da , (2.3) O

1 1 ~) = /e-Q(a)~-M(a)q(a)da, (2.4) o

If ~ is a solution of equation (2.1) then the corresponding solution

of the initial boundary problem is

u(a,z) = N e-Q(a~-S(a). (2.5)

In view of f(0) = 0 there is a trivial branch ~ = 0 and a branch of

nontrivial solutions with the explicit representation ~ = ~/f(W(~)~) .

Using f' (0) = I one finds that the latter bifurcates from the trivial

solution at

co oo

B o = Ie-M(a)q(a)da/ /e-M(a)da o o

(2.6)

In [5],[10] it has been found that the conditions

df d [ f~)] < 0 (2 7) d--u > 0 , d--u --

are sufficient to guarantee that the nontrivial branch is monotone.

210

On the other hand, in [9] it has been shown, that the direction of

bifurcation at ~ = D ° is given by

d f~ =0 2 ~ j ~-0

The stationary problem.

As in many other bifurcation problems (in particular for two point

boundary value problems) related to dynamical systems one may conjec-

ture that the turning points of the bifurcation diagram correspond

to the changes of stability. On the other hand physical intuition may

be misleading, the chosen parametrization may not be related to the

dynamical behavior at all. In the present case the parameter ~ can be

interpreted as a contact rate, and this fact lends some support to the

conjecture.

First, from (2.1) one obtains an explicit formula for the direction

of the nontrivial stationary branch parametrized by ~ o,

d_~ = ~ I - Bf' (W(~)~) (~) Io (~)~ d~ ~ [f (W(~)~] 2 [io(~)]2

- 11 (~) Io(~) ~ + Io(~)I I (~) 3.1)

where ' denotes differentiation with respect to ~.

Consider a fixed value ~ = ~ and let u(a,z) be the corresponding sta-

tionary distribution of hosts according to age and parasite load.

In the integral equation (1.21) introduce a perturbation of the initial

datum v(a,z) and let

(t) :~ ÷ ~(t)

be the solution of (1.21) corresponding to the initial datum

u(a,z) = u(a,z) + v(a,z)

If one linearizes with respect to ~ and v one obtains a Volterra inte-

gral equation of convolution type for the function ~,

~(t) = (~) (t) + S(t) (3.2)

t ~)~t) = f k(t-a) ~ (a)da (3.3)

o

211

The kernel k has the product decomposition

k(t) = Bf' (W(~)~) ko(t). [Io (~) ]2

(3.4)

All information about the function f is contained in the first factor,

whereas the kernel ko(t) depends only on the parameter ~. After

several transformations invoking the properties of the Riccati

solution operator G and the functions derived from it, one obtains

the following form

~k (t) = IP(a)da • F(t) o

t (3.5)

+ IP(a)da • [ I P(a)daF' (t)-P(t)F(t)] o t

where F(t) = -F (t) .

If one could show that k were a nonnegative function, then the o

stability problem were relatively simple. But for a large class of

kernels just the opposite is true.

Assume that p is a positive constant. Define

K(t) = / P(a)da • F(t) (3.6) t

It follows immediately that

~ko(t) = K(t) + Io(~)K' (t) (3.7)

Thus in the case ~ = const, the kernel K is the sum of a nonnegative

function and a multiple of its derivative.

For ~ > 0 the kernel k ° is never positive. One can show that the

function K' (t)/K(t) is strictly decreasing and that k ° has exactly

one sign change.

The stationary solution is linearly stable if the characteristic

equation

/ k (t) e-~tdt = I o

has no roots with Re~ ~ 0

(3.8)

212

One can derive from (3.1) that the condition d~/d~> 0 is equivalent

with the condition

f k(t)dt < I. (3.9) O

For Re~ 0 define

F(~) = f k (t)e-~tdt (3.10) O

O

One has

F(0) = f k (t)dt = f K(t)dt > 0 (3.11) O

o o

Introduce the following property

Property S: If Re~ 0, ImF(~) : 0 then

ReF(~) ~ F(0).

It is sufficient to check the property for ~ on the imaginary axis.

Numerical computations show that the curve {F(iy), y ~ ~ } quickly

spirals to zero and satisfies (even stronger than Property S)

IF(iy) I { F(0) . However, due to the sign change in k O, it seems

difficult to prove Property S analytically.

With Property S follows: If d~/d~> 0 then equation (3.8) has no

roots with Re% ~ 0, and the stationary solution is linearly stable.

On the other hand, if d~/d~< 0, then ~ k(t)dt > I, and the stationary O

solution is clearly unstable.

Influence of total population size.

Following the idea of Gurtin and MacCamy, one can assume that the

birth rate b and the death rate ~ depend not only on age but also

on total population size

P(t) = f u t,a,1)da o

Then the differential equation reads

ut + Ua +g(Z)Uz - [(z-1)~(t) - ~(a,P) ]u = 0

The initial datum has the form

u(0,a,z) = u (a,z) o

(4.1)

(4.2)

(4.3)

213

Again, the side condition can be imposed in the form of prescribed

neonatals (which are assumed to be noninfected)

u(t,0,z) = N(t)

or the neonatals are specified by a Lotka birth law

N(t) = f b(a,P)u(t,a,~)da o

where the meaning of the parameter ~ has been explained earlier.

(4.4)

(4.5)

Also the transmission function is allowed to depend on total population

size

~(t) = ~f(P(t) ,w(t))

(4.6) I Uz (t,a,1)da o

w(t) =

u(t,a,1)da O

Following the same approach as before, one can reduce the problem to

two coupled Volterra integral equations for the functions ~ and P in

the case of the boundary condition (4.4), and to three such equations

for the functions ~, P, N in the case of the boundary condition

(4.4) (4.5). We shall not print these equations here, but immediately

proceed to the stationary preblem for the case of the boundary

condition (4.4) (4.5). Then N, P,~ are constants which satisfy

three equations. If we exclude the trivial case N = 0, these equations

read

7 b(a,P) e-M(a,P)e-Q~(a)~d a = I O

= Bf (P,w)

co

f e-Q(a)~-M(a,P)q(a)da

W = O

o~

I e-Q(a)~-M(a,P)da

o

P = N f e-Q(a)~-M(a'P)da O

where

M(a,P) a

/ ~(s,P)ds o

Assume the natural conditions that the birth rate b(a,P)

and the death rate ~(a,P)

4.7)

4.8)

4.9)

(4.10)

(4.11)

is decreasing

is increasing in P. Then the function

214

b(a,P)exp(-M(a,P)) is decreasing in P. In fact, this last property

is sufficient for the subsequent discussion.

Define the function

h(P,~) = I b(a,P)e -M(a'P) -~(a)~da (4.12) O

If h(0,0) < I then the equations have no solution except N = 0,

P = 0 (there are no hosts). If h(0,0) > I then in the quadrant P, 0,

~ 0 there is a curve ~ = ¢(P) which separates the domains h > I

and h < I. Along this curve one has ¢'(P) ~ 0 and h(P,~) = I. This curve

starts at some point P = 0, ~ = ~o where ~o is the solution of the

equation h(0,~) = I. Typically the curve ends at some point (Po,0),

where h (P ,0) = I. This point describes an equilibrium where the host o

population stabilizes in the absence of parasites. However, P may o

not exist. Then in the absence of parasites, or even with low infection

levels, the host population increases to infinity.

For given P and ~, = ~(P) there is a corresponding B given by

f (P,w(P,9)) (4.13)

Thus (if Po is finite) there is a branch P : Po' ~ = 0, B ~ 0 of sta-

tionary solutions which describes the noninfected population and a

branch parametrized by P, namely P, ~ = ¢(P) , and ~ as in (4.13) which

describes infected populations.

This branch is now inspected in more detail. For P ÷ 0, ~ ÷ ~o from

(4.13) follows

qQo --~81 : (4.14)

f (0,w(0,~o))

Thus, following this branch to P ÷ 0, the population size goes to zero,

the acquisition rate ~ approaches a maximal value, and the host popu-

lation is finally killed by the parasite.

Assume PO < ~. If P ÷ Po' ~ ÷ O, then

I B ÷ 8 o = fw(Po,0)- "

f e-M(a'Po)da

0

ie M(a'P°) q (a) da

o

215

The branch bifurcates from W~ = 0 at B = D o.

Now assume Po = ~" Since % is decreasing, the limit ~ = lim % (P)

exists. P÷~

The expression for 8 as given by (4.13) may not have a limit for P ÷ ~,

+~ . In the D,P-plane the nontrivial branch extends to large values

of P. For small values of D the parasites do not control the host. For

large values of D they keep the host population bounded or even at zero

level.

< ~. In general the nontrivial branch is not monotone. However, Let Po =

assume Z and b as before and furthermore that f is monotone with re-

spect to P and concave with respect to w,

8f(P,w) ~f(P,w) ~ (f(P,w) 1 ~p > 0 ~W > 0 -- < 0

' ' ~w w

Then one can show that along the nontrivial branch holds dD/dP < 0,

i.e. to each D there is at most one nontrivial stationary solution.

The stability properties haw~ not been investigated. The biological

interpretation suggests the following: In the absence of parasites

the population is in stable equilibrium Po" This equilibrium P = Po'

= the branch of infected popula- 0 remains stable for small D. At D °

tions bifurcates, and (Po,0) looses its stability. Along this branch

the parasite acquisition rate ~ increases and the population size P

decreases. At D = D I the population size P becomes zero, for D > D I

P = 0 is the only stable situation. Here ~ is not defined.

%

%

216

References:

I. Anderson, R.M., May, R.M. Population biology of infectious disea- ses I. Nature 280, 361-367 (1979).

2. Bailey, N.T.J. The mathematical theory of infectious diseases and its applications, 2nd ed. London, Griffin (1975).

3. Busenberg, S., Iannelli, M. Separable models in age-dependent popu- lation dynamics, J. Math. Biol. to appear.

4. Diekmann, O. Thresholds and travelling waves for the geographical spread of infection. J. Math. Biol. 6, 109-130 (1978).

5. Hadeler, K.P., Dietz, K., Nonlinear hyperbolic partial differential equations for the dynamics of parasite populations. Comp. and Math. with Appl. 9, 415-430 Pergamon Press (1983).

6. Hadeler, K.P., An integral equation for helminthic infections: Stability of the non-infected population. In: Trends in Theor. Pract. Nonl. Diff. Equ. p. 231-240 V. Lakshmikantham Ed., Lecture Notes in Pure Appl. Math. 90, M. Dekker (1984).

7. Hadeler, K.P., Integral equations with discrete parasites: Hosts with a Lotka birth law. In: Conf. Proc. Autumn Course on Math. Ecology, Trieste 1982, S. Levin, T. Hallam Eds., Lect. Notes in Biomathematics 54 (1984).

8. Hadeler, K.P., Dietz, K., An integral equation for helminthic in- fectious: Global existence of solutions. In: Recent Trends in Mathematics, Conf. Proc. Reinhardsbrunn, Teubner-Verlag, Leipzig (1982).

9. Hadeler, K.P., Hysteresis in a model for parasitic infection., In: Conf. Num. Math. for Bifurcation Problems, Dortmund 1983, H. Mit- telmann, T. K~pper, H. Weber, Eds., Birkh~user ISNM 70 (1984).

10 Hadeler, K.P., Dietz, K., Population dynamics of killing parasites which reproduce in the host., J. Math. Biol. to appear.

11 Hadeler, K.P., A transmission model for parasitic diseases: Stabi- lity of the infected states., J. Math. Biol. to appear.

12 Karlin, S., Tavar&, S., Linear birth and death processes with killing., J. App. Prob. 19, 477-487 (1982).

13 May, R.M. Anderson, R.M., Population biology of infectious disea- ses II, Nature 280, 455-461 (1979).

14 N~sell, I., Hirsch, W.M., A mathematical model of some helminthic infections. Comm. Pure Appl. Math. 25, 459-477 (1972).

15 N~sell, I., Mating models for schistosomes, J. Math. Biol. 6, 21-35 (1978).

16 Puri, P.S., A method for studying the integral functionals of stochastic processes with applications III., Proc. Sixth Berkeley Symp. Math. Stat. Prob. Vol. III, 481-500, UCLA Press (1972).

17. Thieme, H.R., A model for the spatial spread of an epidemic., J. Math. Biol. 4, 337-351 (1977).

18. Webb, G.F., Theory of nonlinear age-dependent population dynamics, Mscr. 330pp. Nashville (1983).

MULTI-DIMENSIONAL REACTION-CONVECTION-DIFFUSION EQUATIONS

F. A. Howes

i. INTRODUCTION

In this note we consider initial-boundary value problems for the general scalar

singularly perturbed parabolic equation

N N

(i.I) ut + i=l~ ai(x't'U)Ux~ i + b(x,t,u)~ = ¢i~lUxixi ,

in a cylindrical domain ~ c ~N X [0,T] , as the positive perturbation parameter ¢

tends to zero. Our approach is to describe solutions of (I.i) satisfying given

initial data on the base of ~, in terms of certain solutions of the associated

hyperbolic equation

N

(1.2) U t + Zlai(x,t,U)U x + b(x,t,U) = 0

which are supplemented by appropriate boundary and interior layer terms in various

subdomains of ~. Such equations arise frequently as pointwise statements of inte-

gral conservation laws, and they are referred to in the literature as reaction-

convection-diffusion equations (of. [7], [8], [2], [3], r4] and the references con-

tained therein). In this setting the s~mllness of ~ signifies that the effects of

reaction (represented by the b-term) and convection (represented by the gradient

term) formally dominate the effects of diffusion or dissipation as measured by the

size of ~. Thus it seems natural to use solutions of the reaction-convection equa-

tion (1.2) to describe solutions of (l.1) in ~, except in boundary and interior

layer subdomains where second derivatives of u become large as ~ -~ 0 +. This pro-

cedure is reminiscent of the (formal) approach to the study of uniform viscous flow

at high Reynolds number past a streamlined body, which consists in solving the

hyperbolic Euler equations for an inviseid fluid and supplementing the resulting

potential flow in a boundary layer region close to the body surface with a solution

of the Prandtl equations.

2. AN EXAMPLE

In anticipation of some of our results on the behavior of solutions of (I°i),

let us consider the linear problem

218

u t + u x = CUxx, 0 < x < I, 0 < t < T,

(2.1) u(x,0) = ~(x), 0 < x < i,

u(O,t) = A(t), u(l,t) = B(t), 0 < t < T,

with the smoothness conditions A(n)(0) = ~(n)(o) and B(n)(0) = ~(n)(1). Even though

the exact solution is available, we proceed formally by setting ¢ equal to zero

and looking for a solution of the reduced problem

U t + U = O, U(x,0) = ~(x), U(0,t) = A(t). x

= 0 is w(x,t) = W(x-t) the Since the general solution of the equation w t + w x

solution of this reduced problem is

~A(t-x), x ~ t, U(x,t)

L~(x-t), x 2 t.

The function U thus satisfies the given data along portions of the parabolic bound-

ary where the characteristic curves, x - t ~ eonst., of w t + w = 0 enter the rec- x

tangle ~, namely along t = 0 and x = O. It turns out that along such "inflow"

portions of the boundary it is not possible to supplement the reduced solution with

any type of boundary layer correction term. The reduced solution U is continuous

by virtue of the smoothness condition. If, in addition, A'(O) = -~(0), then U is

continuously differentiable in ~, since along t = x, (BU/bx) + = -A'(0) = ~'(0) =

(bU/Bx)- and (BU/Bt) + = A'(0) = -~'(0) = (bU/Bt)'. Thus U is smooth and satisfies

all of the initial-boundary data with the exception of the data along x = I. It is

here that U must be supplemented with a boundary layer term; indeed, a little

analysis shows that the solution of (2.1) satisfies

u(x,t,,) = U(x,t) + O(IB(t)-U(l,t)lexp[-(l-x)/¢])+O(¢ )

for (x,t) in ~.

Suppose now that A'(0) # -~'(0), which is usually the case, since we require

only that A'(0) = ~'(0) for the smoothness of the data. Then the function U is

not differentiable along t = x, and we must supplement it there with an interior

layer term which removes this irregularity, in addition to the boundary layer term

along x = i. It is not difficult to see that in this case the solution of (2.1)

satisfies (of. [5])

u(x,t,e) = U(x,t) + O(c{IA ' (0)+~' (0)lexp[-Ix-tl/c{])

+ O(IB(t)-u(l,t)lexp[-(l-x)/¢]) + O(¢)

219

for (x,t) in ~. We note that the thickness of the interior layer is of order e~,

owing to the fact that the line t = x is characteristic; however, the effect of the

layer is weak in the sense that the teN~ ¢~l.lexp[..] tends to zero uniformly as

¢ ~ 0 +.

3. THE GENERAL PROBLEM

Motivated by this discussion, we turn now to a consideration of the initial-

boundary value problem

(P)

u t + a(x,t,u).Vu + b(x,lt,u) = ¢vZu, (x,t) in ~,

x in u(x,0) = q~(j~),

u(x,t) = A(x,t), (x,t) on ST,

2 where ~ =: (x I ..... XN)' Z =: (~/bXl ..... ~/bx N) and V = Z'Z is the Laplacian. In

addition, ~(~,t,u) =: (al(x,t,u) ..... aN(x,t,u)) , ~ =: ~× (0,T) for ~ a bounded

open set in ~N whose boundary 5Q is a smooth (N-l)-dimensional manifold, and

S T =: ~q× [0,T] is the lateral portion of the parabolic boundary of ~. The functions

ai, b, ~ and A have the necessary smoothness for values of (~,t,u) in the domain

where U is a certain solution of the reduced equation U t + ~.VU + b = 0 and d is

a smooth positive function such that for t in [0,T], d(x,t) ~ IA(~,t)-U(~,t)l + 6

for ~ in ~f~6/2 (=: [~ in ~: dist(~,bQ) < 6/2}) and d(x,t) ~ 6 for ~ in ~\5~6.

The correct solution U of the reduced equation which we anticipate approxi-

mates the solution of (P) (in ~) away from any boundary and interior layers is de-

termined, as in the last section, by asking that it satisfy the prescribed data

along portions of the parabolic boundary where its characteristic (base) curves

enter ~. (The characteristics are solutions of the system d~/ds = ~(~,t,U),

dt/ds = I.) Clearly we must require that U(~,0) = ~(~), since the characteristics

enter ~ through the base O X [0}. In order to determine where along the lateral

boundary S T the characteristics enter ~, let us assume that

Q =: Ix: F(x) < 0},

for a smooth real-valued function F; for instance, we may take F(x) =:-dist(x,5~)

if 5f~ is sufficiently smooth. Then 5Q =- F-I(0), vF(x) is the outer normal at x on

~Q (provided, of course, that v F(x) ~ O), and ~ -n [(x,t): A(x,t) < 0} for ~(x,t) =:

tF(x). Therefore the correct reduced solution U must satisfy also the boundary

data in the subset of S T where

220

~(x,t) =: ~(x,t,U(x,t)).DA < 0,

for ~(x,t,w) =: (a(x,t,w),l) and D =: (V,5/St), that is, for all

on 5f~ where

t in (O,T) and x

a(x,t,U(x,t)).vF(x) < 0.

These conditions are nothing more than straightforward generalizations of the inflow

conditions given above for the one-dimensional problem (2.1)o Accordingly, along

portions of S T where 9 > 0 we anticipate the occurrence of a boundary layer of width

O(~), since here the characteristics of U leave ~ nontangentially (cf. (2.1)).

Along portions of S T where ~ = 0 we anticipate the occurrence of a boundary layer of

width ~(¢2), since here the boundary is itself a characteristic (cf. [5]).

In order to make these ideas precise, let us assume that the reduced problem

(Q)

~(~,t,U).DU + b(~,t,U) = O,

U(~,t) = given data

(x,t) in ~,

for (~,t) in the subset of the parabolic boundary where ~(~,t) < 0, has a smooth solu-

tion U = U(~,t)° In a small neighborhood of the outflow portion of S T we assume that

the characteristics of U exit nontangentlally, that is, there exists a positive

constant k such that

(3.1) ~(a,t,u) D~(x,t) _> k[IDA]] 2

for (~,t,u) in the domain

R =: F 8 X [u: [u-U(x,t) I < r(t)].

Here r 6 is the set of points in ~ whose distances from F (=: [(~,t) on ST:

#(xx,t) ~ 0})are at most a small number 6 > 0 and r(t) =: 6 + maxlA(x,t)-U(x,t)l. S ~

Finally let us introduce the function T

H(x,t,u) =: ~(x,t,u).DU(x,t) + b(x,t,u),

and let us assume that there exists a positive constant m such that

(3.2) Hu(x,t,u) _> m > 0

for (~,t,u) in the domain ~ defined above. (This is not a restriction since the

change of variable v =: u exp[-Kt], for K a positive constant, converts (P) into

an analogous problem for which (3.2) obtains.) Using these ideas we can now state

the

221

Theorem. Assume that th___ee reduced problem (Q) has a smooth solution U = U(~,t) and

that conditions (3.1) an__~d (3.2) obtain. Then there exists an ¢0 > 0 such that the

problem (P) has a smooth solution u = u(~,t,¢) whenever 0 < ¢ j ¢0" In addition, we

have that

lu(x,t,¢)-U(x,t) I < I~, for (x,t) in ~\F 6

and

lu(x,t,¢)-U(x,t)l _< K exp[klA(X,t)/¢] + L6,

for (~,t) i__~n F 6 where K =: maxlA(~,t)-U(~,t)l , 0 < k I

S T

constant de~ending on U.

< k, and L is a known positive

The preof follows by noting that the existence of such a solution of (P) is

guaranteed %y a theorem of Amann [i], provided we can construct a lower bounding

function ~ and an upper bounding function w satisfying ~ ~ w , ~(~,0) ! ~(x)

~(~,0) in ~, ~ ~ A ~ ~ on ST, and ¢V2~ ~ g(~,t,~,D~), ¢V2~ ~ G(~,t,w,~W) in ~, for

G =: ~.Du + b. To this end, we define for (~,t) in ~ and ¢ > 0

_~(x,t,¢) = U(x,t) - K exp[klA(X,t)/¢] - I4

and

~(~,t,¢) = U(~,t) + K exp[klA(~,t)/¢] + I~,

2U l where L =: Im~x (x,t) +l]/m. Then it is not difficult to show that ~ and w n

satisfy the required inequalities for ¢ sufficiently small. The theorem of Amann

guarantees the existence of a solution u of (P) such that ~ < u < w in ~.

Suppose now that the inequality (3.1) only holds in the weaker sense that

(3.3) ~(x,t,u).DA(x,t) _> 0

for (~,t,u) in the domain ~. Then we anticipate that the boundary layer subdomains

of S T have thickness of order c ~. In order to see this we can define the bounding

functions w =: U - K exp[mlA(~,t)/¢~] - Lc and ~ =: U + K exp[mlA(x,t)/¢~] + Lc,

where 0 < m I < m ~, and proceed as in the proof of the Theorem. Such reasoning also

helps us understand what happens when the solution U of (Q) is not smooth along an

(N-l)-dimensional manifold ~ =: f-l(0), for a smooth real-valued function f = f(~,t)

such that ll!~fll ~ 1 on E. We assume that ~ can be written as ~i U ~ U N2' where

~I =: [(~,t): f(~,t) < 0} N ~ and ~2 =: {(~,t): f(~,t) > 0] N ~. In addition, we

assume that the reduced equation has two solutions U = Ui(~,t) (i=1,2) defined and

smooth on ~[ U Z which satisfy the given data at points (~,t) on the parabolic boundary

222

of ~ where ~(x,t,Ui(x,t)).DA(x,t) < 0o Then if U I = U 2 but 5Ul/bn # 5U2/bn on Z,

for 5/5n =: D.(Df) the normal derivative along E, the functions U I and U 2 must be

supplemented by an interior layer term near Z of the form (cf. Example (2.1))

w(x,t,¢) =: l(¢/ml)e~ exp[-If(x,t)l(ml/~)9],

where ~ =: 5U2/~n - 5Ul/bn along ~2 and 0 < m I < m. The function w smooths out the

irregularity of the composite reduced solution

~Ul(X,t), f(~,t) _< O, Uo(x,t) l

LU2(x,t), f(x,t) > 0,

since D(Ui+w)oDf = ~(bUl/~n + 5U2/~n ) on Zo Finally we must assume that (cf° (3.3))

(3.4) ~(x,t,Ul(X,t))-Df(x,t) _> 0

6 for (x,t) in ~I (=: ~I N [(x,t): dist(x,t;Z) < 6}), and that

(3°5) ~(x,t,U2(x,t)).Df(x,t) < 0

6 for (x,t) in ~2" These two inequalities imply that 52 can support a layer for both

~I (whose outer normal along Z is Df) and ~2 (whose outer normal along E is -Df)o We

can, in fact, allow the inner products in (3.4) and (3.5) to have any sign, provided

that (l(x,t,Ui(x,t)).Df(x,t) = C>(If(x,t)l ) in If. 6, since we then have that

(aoDf)exp[-Ifl/¢ ~] = O(¢m[Ifl/¢2]exp[-Ifl/¢e]) = d>(¢ =) in ~i" This nonlinear theory is illustrated best by a problem for Burgers' equation

(cf. [6; Chap° 4]), namely

(3°6)

u t + uu x = CUxx, (x,t) in ~ =: (0,i) X (O,T),

u(x,O) = ~(x), x in [0,I],

u(O,t) ~ I, u(l,t) ~ 2, t in [0,T]

The initial value problem w t + ww x = 0, w(x,0) = I + x, has the simple wave solution

w(x,t) = (l+x)/(l+t), defined for t ~ x. Since u(O,t) > 0 the characteristics of this

function also enter K along x = 0, and so the theory tells us to consider the con-

tinuous reduced solution

U(x,t) =: I I, t ~ x,

(l+x)/(l+t), t ~ x.

Along the outflow boundary x = I there is a boundary layer of width O(¢), since

(u,l).(l,0) = u > 0 for all values of u between u(l,t) m 2 and U(l,t); cf° (3.1).

Along the characteristic t = x, however, U is not differentiable, and so we must

223

supplement it with an interior layer term there. Letting f(x,t) =: (x-t)/v~ and

~(x,t) =: (U(x,t),l)-Df = (U(x,t)-l)/v~2, we see that ~ m 0 for t > x and that

~(x,t) = 6/[v'2(x+l)] + O(82 ) for 0 ! x-t ! 6. Consequently, the solution of (3.6)

satisfies for (x,t) in

u(x,t,¢) = U(x,t) + O(12-U(l,t)lexp[-(l-x)/¢])

I + O(¢~(~2 /(l+x))exp[-Ix-tl/¢~]) + O(~(x,t,¢)),

where ~ = O(¢ ~) for (x,t) in a neighborhood of t = x and ~ = 0(¢) elsewhere.

ACKNOWLEDGMENT

It is a pleasure to thank the typist, Mrs. Ida Mae Zalac, for her fine

secretarial work.

REFERENCES

I. H. Amann, Periodic Solutions of Semilinear Parabolic Equations, in Nonlinear Analysis , ed. by L. Cesari et al., Academic Press, New York, 1978, pp. 1-29.

2. C. Bardos, A. Y. Le Roux and J. C. Nedelec, First Order Quasilinear Equations with Boundary Conditions, Cormm. Partial Diffo Eqns. 4(1979), 1017-1034.

3. C.M. Dafermos, Asymptotic Behavior of Solutions of Hyperbolic Balance Laws, in Bifurcation Phenomena, ed. by C. Bardos et al., Reidel, Holland, 1980, pp. 521-533.

4. P.C. Fife, Propagating Fronts in Reactive Media, in Nonlinear Problems, ed. by A. Bishop et al., North-Holland, Amsterdam, 1982, ppo 267-285.

5. F.A. Howes, Perturbed Boundary Value Problems Whose Reduced Solutions are Non- smooth, Indiana U. Math. J. 30(1981), 267-280.

6. J. Kevorkian and J. D. Cole, Perturbation Methods in Applied Mathematics, Springer-Verlag, New York, 1981.

7. P. Do Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, CBMS Series in Appl. Math°, vol. Ii, SIAM, Philadelphia, 1973.

8. J.D. Murray, Singular Perturbations of a Class of Nonlinear Hyperbolic and Parabolic Equations, J. Math. and Physics 47(1968), 111-133o

ON PERIODIC SOLUTIONS OF FORCED SECOND ORDER

DIFFERENTIAL EQUATIONS WITH A DEVIATING ARGUMENT

R. Iannacci

and

M.N. Nkashama

ABSTRACT: Using classical Leray-Schauder~s techniques and coincidence

degree, we prove the existence of periodic solutions for forced second

order delay-differential equations under nonuniform nonresonance condi-

tions with respect to the spectrum of the linear ordinary-differential

equation with periodicity conditions. Our approach allows us to derive

some uniqueness result.

Key words and phrases: Delay, periodic solutions, Caratheodory's

conditions, Leray-Schauder's techniques, Mawhin's coincidence degree.

AMS (MOS) Mathematics subject Classification (1980):

34B30, 34C25, 47HI0, 47H15.

i. INTRODUCTION

This paper is devoted to the study of existence and uniqueness

results for 2J-periodic solutions of the second order delay-differential

equation

x"(t) + g(t,x(t-T)) = e(t) ( i . i )

x(0) - X(2Z) = x' (0) - x' (2~) = 0

with a fixed delay T E [0,2z[, where e : [0,2z] ÷ IR is integrable,

g : [0,2z] × JR÷ IR satisfies the Caratheodory's conditions and the

unknown function x : [0,2z] + ~R is defined for 0 _< t < T by the

equality x(t-T) = x(2~ + (t--T)) (i.e. in the standard way for the

periodic case).

In extending a result due to W.S. Loud, D.E. Leach has established

the following (see [6]) : If g(t,x) - g(x) (i.e. g is autonomous),

g continuously differentiable, if for some integer m,

2 2 m < p -< g' (x) -< q < (re+l)

holds for all x e JR, then the differential equation (I.i) with T = 0

225

has at least one solution. A.C. Lazer and D.E. Leach have proved that

under the above conditions, the differential equation (i.i) with T = 0

has a unique solution (see [6] Theorem 3.1.)

Recently, J. Mawhin (see [73 theorem II.2) has proved that if g

satisfies Caratheodory's conditions and for some integer m, the

inequalities:

2 -i m ~ a(t) ~ lim inf x-l.g(t,x) ~ lim sup x .g(t,x) ~ b(t) ~ (m+l) 2

Ixl Ixl

hold uniformly for a.e. t e [0,2~3 where a,b e LI(0,2~) are such

that m 2 < a(t) and b(t) < (m+l) 2 on subsets of [0,2~3 of positive

measure then the equation (i.i) with T = 0 has at least one solution

for each e e LI(0,2~). Some uniqueness result can be derived from

Mawhin's approach.

In this paper we extend the above results to the case of delay-

differential equations (i.i). One feature of our approach is that no

relationship between the delay and the period is necessary. In order

to obtain the required a priori estimates, we follow a technique due

to J. Mawhin and J.R. Ward [83.

We recall that the case of effective delay (i.e. T ~ 0) is quite T Q,

different from the case without delay. Indeed if T ~ 0 and ~ c

then the eigenvalues of the problem

x"(t) + Ix(tiT) = 0

x(0) - x(2~) = x'(0) - x' (2~) = 0

are not contained only in the non negative part of the real axis (see

Remark 1 in [23).

In the sequel, we shall use notations from papers [23 or [83 for

spaces, weak and strong convergence.

2. MAIN RESULTS

Let g : [0,2~3 x ~ ÷ ~ be a Caratheodory's function i.e. g(.,x)

is measurable on [0,2~3 for each x e ~ , g(t,.) is continuous on

for a.e. t e [0,2~] and for each r > 0 there exists ~ £ LI(0,2~) r

such that Ig(t,x) l ~ ~r(t) for a.e. t e [0,2z] and all x e [-r,r].

We shall prove the following result for (i.i):

226

THEOREM 1 (Existence): Assume that the inequalities

-i -i a(t) _< liminf x .g(t,x) _< limsup x .g(t,x) -< b(t) (2.1)

Ix] ÷+~ Ixl ÷+~

hold uniformly for a.e. t c [0,2~3, where a,b ~ LI(0,2~) satisfy the

following conditions for some m ~ ~N :

2 2 m -< a(t) -< b(t) -< (m+l) for a.e. t ~ [0,273 (2.2)

with m 2 < a(t) and b(t) < (m+l) 2 on subsets of [0,273 of positive

measure.

• L 1 Then: the problem (i i) has at least one solution for each e ~ (0,27).

TO prove theorem i, we need some useful lemmas:

LEMMA i: Let m ~ IN and let p E LI(0,27) be such that for a.e.

t e [0,2~3, m 2 -< p(t) -< (m+l) 2 with strict inequalities on subsets of

[0,273 of positive measure. Then the equation:

X"(t) + p(t)x(t-T) = 0 with T e [0,2~], T fixed.

X(0) - x(2z) = x' (0) - x' (2z) = 0

has only the trivial solution.

Proof: Let x ~ w2'l(0,2z) be a solution of the equation above.

x has the Fourier series:

x(t) = a 0 + ~ a n cos nt + b n sin nt. n=l

m Let us consider x(t) = a 0 + ~ a n cos nt + b n sin nt and

n=l

~(t) =

Then

a n cos nt + b n n=m+l

sin nt so that x(t) = £(t) + x(t).

Easy computations show that

(x" (t) + p(t)x(t-T)) (x(t-T) - x(t)) =

= (x"(t) + x"(t)) (x(t-T) -x(t)) +

+ p(t)[-½x2(t) - ½x2(t-T) + ½x2(t-T) +

+½(x(t-T) + x(t-T) - x(t))23.

Using the identity -ab = ½a 2 + ½b 2 - ½(a+b) 2 and the orthogonality in

L2(0,27) of x and x, one gets:

27 0 = f (x"(t) + x"(t) + p(t) x(t-T))(x (t-T) - x(t))dt =

0

227

Since

2~ 2~ = 0/2~(x' (t))2dt - 0 f (x' (t)) (x' (t-T))dt +

+ f P(t)[-½x2(t) - ½x2(t-T) + ½x2(t-T) ]dt + 0 2z

+ f p(t)[½(x(t-T) + x(t-T) - x(t))2]dt 0

27 ~ n2 b2 = f (x'(t))2dt + (a2n + n) (- cos nT) + 0 n=l 2~

+ f P(t)[-½x2(t) - ½x2(t-T) + ½x2(t-T) ]dt + 0 2z f p(t)[½(x(t-T) + x(t-T) - x(t))m]dt 0 2~ m f (x' (tl)2dt - [ n2(a2n + b2ln + 0 n=l 2~ / P(t)[-½x2(t) - ½x2(t-T) + ½x2(t-T) ]dt 0

+ Ix(t-r) + (x(t-z) - x(t)) ]2dt 0 2z ~ 2~ 2z _ ~2 f (X' (t))2dt - f (x' (t-T))2at + / (P~2t) Ex 2(t-T .~ (t-T) 0 2~ 0 0

- x2(t)]dt + f (~2t)[x(t-T) + (x(t-T) - x(t))]2dt 0

2~ 27 ½/ [ (x' (t)) 2 - p(t)x2(t)]dt + ½f [(x' (t-T)) 2 p(t)x2(t-T) ]at 0 0 2~ 27

½f [p(t)x2(t-T) - (x' (t-T))2]dt - ½f (x' (t-T))2dt 0 0 2~

+ ½/ p(t) Ix(t-T) + (x(t-T) - x(t)) ]2dt. 0

p (t) :-> m 2 for a.e. t ~ [0,2~]/ the last two terms imply that:

27 2~ -½f (x' (t-T))2dt + ½f p(t) Ix(t-T) + (x(t-T) - x(t))]2dt

0 0 2~ 2 27 2 2 2z ~

-> -½f (x' (t-T))2dt + % f x (t-T)dt + % f (x(t-T) - x(t))2dt 0 0 0 2z 2~

+ m2f i~(t-T)x(t-T)dt - m2f x(t-T)x(t)dt -> 0 0 0

Since the last two terms are zero by orthogonality of x and

and the sum

Therefore

X

of the first two terms is nonegative by Parseval equality.

2~ 2z 0 -> ½4 [(x' (t))2 - P(t)x2(t)]dt + ½f [(x'(t-T))2 - P(t)x2(t'T)]dt

2~ x2 0 + !~f0[P(t) (t-T) - (x'(t-T))2]dt .

228

It follows from Lemmas II.2 and If.3 in E73 that x = O and the proof

is complete.

LEMMA 2. Let a,b be elements of LI(0,2~) satis~ing assumptions of

Theorem 1.

Then there exists e = e(a,b)> 0 and ~ = ~(a,b) > 0 such that for all

p • Ll(0,2z) satisfying

a(t) - e ~ p(t) ~ b(t) + e for a.e. t • [0,2zl

one ha

2~ I I x'' (t) + p(t)x(t-T) ldt a 6 Ixl

1 0 C

W2,1 for all x • (0,2z).

Proof: Assume that the assertion is not true.

sequence (Xn) c W2'I(0,2~) with IXnIcl = 1

LI(0,2~) with

Then, one can find a

and a sequence (pn) c

_ 1 - for all n • a(t) ~ ~ Pn(t) ~ b(t) + i * n

and a.e. t c [0,2z]

such that

2~ 1 f Ix"(t) + Pn(t)Xn(t-T) fat < -- . 0 n n

By means of the same arguments used in [8] (Lemma 4), one has, going

if necessary to subsequences that

x + x in C[0,2z] n

x' + x' in C[0,2~] n

Pn = p in L 1 (0,2~)

with x • W2'I(0,2~) , IXlc I = I,

a(t) -< p(t) -< b(t) for a.e.

x"(t) + p(t)x(t-T) = 0

t • E0,2~3 and

for a.e. t • [0,2~3.

It follows from Lemma 1 that x = 0 which is a contradiction to

IXlc I = 1 and the proof is complete.

We are now in position to prove the theorem i,

Proof of Theorem i.

Let e > 0 and ~ > 0 be given by Lemma 2. Then, by (2.1) we

can find a real number r = r(e) > 0 such that for a.e. t ~ [0,2~3

and all x • ~ with Ixl a r, we have

229

-i a(t) - ~ _< x .g(t,x) -< b(t) + E .

Like in the proof of Theorem 1 of [8], we write the equation in (i.i)

in the form

x"(t) + x(t,x(t-T))x(t-T) + h(t,x(t-T)) = e(t)

where

a(t) - E -< ~(t,x) _< b(t) + e (2.3)

lh(t,x) I -< Yr(t)

for a.e. t ~ [0,2z], all x ~ ]R and some Yr e LI(0'2~)" By the

same degree argument used in the proof of Theorem 1 of [8], our result

will be proved if we show that the set of possible solutions of the

family of equations

x"(t) + [(l-l)b(t) + /y(t,x(t-T))] x(t-T) + lh(t,x(t-~)) = he(t)

x(0) - x(2~) = x' (0) - x' (2~) = 0 (2.4)

is a priori bounded in CI[0,2~] independently of i £ [0,i]. If x

is a solution of (2.4), then using (2.3) and Lemma 2 with p(t) = (i-I)

b(t) + l~(t,x(t-T)), we obtain

2~ 0 = f Ix"(t) + E(l-l)b(t) + 17(t,x(t-T)) ] x(t-T) +

0

+ lh(t,x(t-~)) - he(t) Idt _> ~IXIc I - IYrlLl + (JelL I)

i.e.

IXlcl -< 6-1(IXrlLI + felL I) = R = R(a,h,y r)

and the proof is complete by Theorem 1.2 in [7] with z = 0.

THEOREM 2 (Existence and uniqueness):

that

a(t) -< ~(t,x) - g(t,y) _< b(t) x - y

for a.e. t ~ EO,2~] and all x 7{ y

Let g : E0,2~] x ~ ÷ ~ be such

(2.5)

in ~ with a and b as in

Theorem 1. Then the problem (1.1) has a unique solution for each

e ~ LI(0,2~).

Proof: It follows from (2.5) that the conditions (2.1) hold. Thus,

the existence follows from Theorem i. If now x and y are solutions

of (i.i), then, considering v = x - y, v will be a solution of the

problem

v"(t) + g(t,v+y) - g(t,y) = 0 (2.6)

230

Let us set

[v-l[g(t,v+y) - g(t,y)] ,

f (t,v) [a(t) , if v = 0 .

Then, (2.6) can be written in the form

v"(t) + f(t,v)v = 0

with

a(t) -< f(t,v) -< b(t)

for a.e. t c [0,2~] and all v c JR.

that v = 0

if v ~ 0

(2.7)

By Lenm~a i, we easily deduce

i.e. x = y and the proof is complete.

3. FURTHER RESULTS AND REMARKS

i) When m = 0, more general results have been given by the authors

in [4] for the existence of 2~-periodic solutions for Li4nard equations

of the form

x"(t) + f(x(t))x' (t) + g(t,x(t-T)) = e(t)

with arbitrary continuous f, where some crossing of the eigenvalue

zero is allowed. These results have been extended to Li~nard-type

systems of the form

x"(t) + [~t grad F(x(t))] + g(t,xl(t-Tl),...,Xn(t-Tn)) = e(t)

where g : [0,2~] x ~n ÷ ~n satisfies Caratheordory conditons,

e : [0,273 ÷ ]R n is integrable, F : IR n +IR is in C2(~ n ,~) and

TI,...,T n are fixed distinct delays in [0,2z[. [See G. Conti and the

authors (to appear).] The resonant situation is also considered in that

paper.

2) Our results of Section 2 are in the line of those by W. Layton [5]

for conservative systems. In contrast to that paper we allow touching

of eigenvalues on subsets of positive measure. This is due to the

fact that our nonlinear is nonautonomous. Moreover some crossing of

eigenvalues is also allowed since it follows from the proof of Theorems

1 and 2 that their conclusions still hold if conditions (2.1) are

replaced by

a(t) - ~ < lim inf x -I -I - .g(t,x) ~ lim supx .g(t,x) ~ b(t) + Txl Ixl

with 0 ~ D < s, c given by Lemma 2.

231

3) In the case of effective delay (i.e. T # 0), it could happen that

the orthogonal system of eigenfunctions associated to the eigenvalues

(see Remark 1 in F2]) of the linear problem

x"(t) + hx(t-T) = 0, x(0) - x(2~) = x' (0) = x' (2~) = 0 (3.1)

complete in w2'l[0,2z] (Take for example T = ~). This is due is not

to the fact that there is no relationship between the delay and the

period. Thus, it is not possible to hope to get results similar to

those of Section 2 by following our approach and comparing the non-

linearity to the eigenvalues of the problem (3.1) since our approach

relies heavily on the fact that the system of orthogonal eigenfunctions

is complete in w2'l[0,2z].

Fortunately, using Schauder fixed point theorem, one can easily prove

the following existence result:

Let ~ : E0,2~] × IR ÷JR be a bounded continuous function and let ~ c IR

be different from the eigenvalues of the linear problem (3.1).(See

Remark 1 in [2]).

Then the problem

x"(t) + zx(t-T) + t(t,x(t-T)) = e(t) (3.2)

x(0) - x(2 z) = x' (0) - x' (2z) = 0

has at least one solution for each given

e ~ L2(0,2~).

4) In the equation (i.i), one may consider the second member to be

e : E0,2~] × ~ × IR × ~ × IR ÷ IR: (t,x,y,z,v) + e(t,x,y,z,v)

bounded and continuous so that (i.i) becomes:

x" (t) + g(t,x(t-T)) = e(t,x(t) ,x(t-T) ,x' (t) ,x' (t-T)) .

X(0) - X(2~) = X' (0) - X' (2z) = 0.

REFERENCES

[13

[2]

H. Berestycki, D.G. de Figueiredo, Double resonance in semi- linear elliptic problems, Comm. in Partial Diff. Eq., 6(1), (1981) 91-120.

E. De Pascale, R. Iannacci, Periodic solutions of a generalized Li~nard equation with delay, Proceedings of the Int. Conf. (Equadiff 82) W~rzburg 1982, Lecture Notes in Math., 1017, Springer-Verlag, Berlin (1983) 148-156.

232

[3] L.E. El'Sgol'Ts, S.B. Norkin, Introduction to the theory and application of differential equations with deviating arguments, Academic Press, New York, 1973.

[4] R. Iannacci, M.N. Nkashama, Nonresonance conditions for periodic solutions of forced Li4nard and Duffing equations with delay, To appear.

[53 W. Layton, Periodic solutions of nonlinear delay equations, Journ. of Math. An. and Appl. 77, (1980) 198-204.

[63 A.C. Lazer, D.E. Leach, Bounded perturbations of forced harmonic oscillators at resonance, Annali di Mat. pura ed Appl. (IV), Vol. LXXXII (1969) 49-68.

[7] J. Mawhin, Compacit~, monotonie et convexit~ dans l'4tude de problemes aux limites semi-lin~aires, S4m. Anal. Moderne No. 19, Universit4 de Sherbrooke, Quebec,1981.

[8] J. Mawhin, J.R. Ward, Jr., Nonuniform nonresonance conditions at the two first eigenvalues for periodic solutions of forced Li4nard and Duffing equations, Rocky Mountain Journ. Math., Vol. 12, No. 4 (1982) 643-654.

[9] M.N. Nkashama, M. Willem, Periodic solutions of the boundary value problem for the nonlinear heat equation, Bull. of the Australian Math. Soc. (to appear).

[103 R. Reissig, Continua of periodic solutions of the Li4nard equation, in Constructive Methods for Nonlinear Boundary Value Problems and Nonlinear Oscillations, ISNM 48, Basel (1979) 126-133.

THE THEORY OF QUASI PERIODIC MOTIONS

Lin Zhen-sheng

I. Introduction

In this paper we prove the existence of a quasi periodic solution

of a nonlinear differential system in the critical case. In order to

indicate the significance of this work we state several previously

known results.

Firstly it concerns the work of Kolmogorov and Arnold. They have

proved the existence of a quasi periodic solution of the analytic

Hamiltonian system

d x d--t = Hy(x,y,e),

dy = _Hx(x,y,¢ ) dt

(z)

where x = col. (Xl,X 2 ..... Xm),y = col. (yl,y 2 ..... ym ) , H(x,y,E)

analytic in all arguments, 2~-periodic in Xl,X2,...,x m, and

H(x,y 0) = H(y). They established the following theorem.

is

THEOREM (Kolmogorov). Suppose that w = (Wl,W2,...,w m) satisfies the

inequality

m

I j=[ikjw j

m >- K(w) ( Z Ikjl )-(m+l) •

j= l (2)

where K(w) 0, the integer vector k = (kl,k 2 ..... k m) # 0, Hy(y 0) =

w, and det. Hy(Y0 ) ~ 0, when s = 0. Then there are analytic

functions u(@) and v(@) , 8 = (81,82 ..... @in) such that

x = w + u(0),

Y = Y0 + v(@)

represent an invariant torus. The flows of (i) on the torus are given

by

d@ dt - w,

i.e. the system (i) has a quasi periodic solution. The quasi periodic

234

function f(t) means that it can be expressed as f(t) = F(wlt,w2t,

...,Wmt), where F(Ul,U2,...,Um) is 2~-periodic in Ul,U2,...,u m.

In the 1960s Bogoliubov established the following theorem.

THEOREM (Bogoliubov) . Suppose that the analytic differential system

dx dt

- w + I + gf(x,y,g,l)

d_~ = my + gg(x,y,g I) dt

(3)

satisfies the inequality (2), and 9 = diag{~l,~ 2 ..... ~n } with Re~j

< 0. Then there is a function i = l(g) so that (3) admits a quasi

periodic solution expressed as

x = wt + c + cu(wt+c,g) ,

y : gv(wt+c,g),

where c is a vector parameter.

Moser has shown the validity of such results in case where

f(x,y,e,l) and g(x,y,g,l) are differentiable. Indeed he goes

further and proves the following theorem.

THEOREM (Moser). Suppose that w and ~ = diag{~l,~2,...,~ n}

satisfy the conditions

m m lij_~ikjw j - 9k >- K(w,9) ( [ Ikjl )-(m+l) ,

- j=1 m m

I i [ k.w ~k + ~ I -> K(w,~) ( [ Ikjl) -(m+l) j:l 3 ] s j:l

(4) I

(4) 2

where K(w,9) > 0, and the integer vector k = (kl,k 2 .... ,k m) # 0

Then there are functions i : l(g),p : p(g) and M = M(g) analytic

in g, vanishing at g = 0, and satisfying

9*~(g) = 0, M(e)~* - ~*M(e) = 0,

where ~* is the adjoint matrix of D such that the differential

system

dx d--t : w + l(s) + gf(x,y,s,l(g)),

d-z = ~y + p(s) + M(s)y + sg(x,y,s,l(c)) dt

235

where f(x,y,e,l) and g(x,y,e,l) are analytic in all arguments, and

2~-periodic in x l,x 2,...,x m, admits a quasi periodic solution

For the same problem as (3) we only require the functions

f(x,y,s,l) and g(x,y,E,l) be in C (T) with T z 2 (m+2), and w

and ~ satisfy ~j # 0 and the inequalities (2) and (4)1, to prove

the existence of a quasi periodic solution of (3).

2. The statement of the result

THEOREM. Suppose that the functions f(x,y,£) and g(x,y,s) are

continuous in all arguments, 2~-periodic in Xl,X2,...,x m, and belong

to C (T) with T a 2(m+2) in x and y, and w and ~ satisfy

~j ~ 0 and the inequalities (2) and (4) 1 . Then there is a function

I = l(e) vanishing at s = 0, such that the system (3) admits a

quasi periodic solution, i.e. the system (3) admits a center integral

manifold

y = r(x,s),

z = q(w(s-t) + x,E),

where r(x,s) is 2z-periodic in x and q(w(s-t) + x,e) is the

solution of the differential system

dz ds

- w + l(s) + ef(z,r(z,8),e)

where J(x,s) = q(x,8) - x is 2z-periodic in x. At the same time

x = q(w(t-t 0) + x0,s)

y = r(q(w(t-t0) + x0,e) ,e)

is the quasi periodic solution of (3) for all x 0 and t O .

3. Preliminary Results

In order to prove the theorem we give the following lemmas.

LEMMA i. Suppose that the function r(x) is in the class C (T) with

T a 2(m+2), 2z-periodic in x, II DSr(x)II s M 0, II sll = 0,1,2 ..... T,

the Fourier series of r(x) is

r(x) : ~ akexp(i(k,x)),

236

and the constant vector w satisfies the inequaZity (2). Then there

is a constant K 0 dependent on w only, such tllat

[ I] ak(k,w)-lll s K0M 0. k#0

Proof For proof of this Lemma we refer to [6] (lemma 1 in §4, Ch.l)

or [7].

Let us consider the differential system

dx d--[ = w + i + sf(x,s), (5)

where f(x,c) is continuous in x and c and belongs to C (Y) with

T z 2(m+2), 2~-periodic in x, and the constant vector w satisfies

the inequality (2).

LEMMA 2. Suppose that (5) satisfies the above conditions. Then there

is a I = l(e) with l(s) = 0(s), such that the system (5) admits

the solution

x : q(w(t-t 0) + x0,S)

where J(x,£) = q(x,s) - x is 2~-periodic in x.

Proof By the method of successive approximation we take

10(s) = 0, q0(u,s) = u,

It is evident that q0(u,s) is in the class C (Y) for all u.

Suppose that qj(u,s) has been determined such that

Jj(u,e) : qj(u,s) - u

which is in the class C (T) and 2~-periodic in u and the mean value

M(Jj) is equal to zero, i.e.,

2~ 2~ M(Jj) = (2~) -m / ... S Jj(u,e)du = 0.

0 0

Then take lj+l(S) = -sM(f(qj(u,s),s)) and

Jj+l(w(t-t 0) + x0,s): e/(sf(qj(w(t-t0) + x0,s) ,e) + lj+l(S))dt,

indefinite integral of £f(qj(w(t-t 0) + x0,E),e) + lj+l(g) with

237

M(Jj+ I) = 0, i.e., qj+l(U,S) = u + Jj+l(U,6).

We shall prove that Jj+l(U,E) is a quasi periodic function with

the same properties as Jj(u,s). Now f(qj(u,s),s) is in the class

C (T) with T > 2(m+2), and w satisfies the inequality (2), so

Jj+l(w(t-t0) + x0,e) is a quasi periodic function in the class C (T) .

Therefore we obtain two sequences of functions {Ij(E)} and {Jj(u,e)}.

By the construction of qj(u,s) we have

q'j+l(w(t-t0) + x0,e) : sf(qj(w(t-t0) + x0,e),g) + lj+l(£), (6)

! where qj+l is the derivative of qj+l with respect to t.

Now we shall prove that there is a positive constant SO,

independent of j = 0,1,...,T such that

s (x,~)ll < 2, k 0 1,2 11sll : 1,2 ..,T, [l Oxq = , . . . . . . . (7)

when 0 ~ ~ ~ g0" It is indeed true for k = 0. By mathematical

induction we assume that (7) is true for k = 0,1,...,j, then prove

that it is held for k = j + 1 too. S Under the condition II Dxqj(x,E) II ~ 2, it is evident that there

is a constant M* independent of j = 0,1,2,..., such that

llmSf<qj(x,~),~)ll < M*, llsll: 0,1,2 ..... ~

We expand f(qj(x,e),e) into Fourier series, then

qj+l(w(t-t 0) + x0,e),e) = w(t-t0) + x 0 + ~ [ - i(k,w) -I k~0 aj+ik(e) exp(i(k, (t-t0)w + x0))

with

II (k, w)-la k~0 j+k (e) II _< KoM* , ~ k a .... (s) (k,w) -I k~0 r j±im II -< k0M*'

where k = (kl,k2,...,k m) , as a consequence of Lemma i. Next we take

the constant vectors w(1)= w,w (2) ..... w (m) satisfying (2) , with

~et. (w (I) , . . . . ,w(m)) I > C > 0, and the functions

(w (r) II Jj+l (t-t0) + x0'g)II = £ [ -i(k,w)-i k~0 aj+ik (~)exp(i(k'w(r) (t_t0)

+ x0)) r = 1,2,...,m, so that

II J~+l(w(r) (t-t0) + x 0 c)II -< s [ (k,w(r)) (k,w) -I , aj+ik(E) II < SKoM*K I, k#0

where I (k,w (r)) (k,w)-ll <_ ml, i.e.

238

w(r) II z m7 Jj+l(X,~) i II -< KoM*K 1 , r = 1,2 ..... m.

Then to solve these linear equations we have the constant K 2 inde-

pendent of j = 0,i,..., such that

II mS x Jj+l (x,s)II < ~K2 < ~ or II D s ' x qj+l (x's) II < 2, (8)

when II sll = i Similarly we can prove that (8) holds for II sll = 2,3,...,T. Therefore we have the constant s 0 independent of

j = 0,1,2 .... , such that

II Dsx q j ( x , E ) I I < 2 , w h e n o _< s -< a 0 ,

and then it is easy to conclude that

II DS(f(qj (x,s), s) - f(qj_l(X, s) ,~))II < K311 mS(qj (x'~)x

- qj_l(X,8)) I] , (9)

the constant K 3 being independent of j = 0,1,2,...

From (6), (8) we have

i.e.

(r) mS I[ DSx (Z ~i(qj+l-qj)wi I] -< K 4 S S~ll x(qj-qj_l ) II,

II S l l = 0 , 1 , 2 . . . . . T - 1 , r : 1 , 2 . . . . . m.

zll D s x(qj+l-qj) II < K5EII Ds < DS x(qj-qj_l II - ½Eli x(qj-qj_l ) lJ ,

where the constants K 4 and K 5 are independent of j = 0,i,2, ....

Therefore the sequence of functions qj (x,s) converges to the limit

function q(x,e) uniformly. By the definition of I. (s) it is 3

evident that the sequence i. (s) tends to the limit l(s). In short 3

the equality

d dtq(w(t-t0) + x0,s) = w + I(8) + sf(q(w(t-t 0) + x0,s),s )

holds. Lemma 2 is proved completely.

239

4. T__he proof of Theorem

Let us consider the matrix 9(q) = diag{~l(q) ,~2(q) .... ,~m(q) }

with

9~, if Re~ ~ 0, aj(q) = 3

- + ~. with q > 0, otherwise. 3

We may assume that Reg.(q) < 0, j = 1,2 ..... k, and ReQ (q) > 0, 3 s

s = k+l, ... ,n. Now we prove that the differential system

dx d--{ = w + I + ~f(x,y,s)

d_~ = Q(~)y + eg(x,y ~) dt

(I0)

admits a center integral manifold.

Take r0(x,s,q) = 0 and suppose that rn(X,s,q) has been deter-

mined. By Lemma 2 we have z = qn+l(W(S-t) + x,e,q) and In+l(E,q)

to be the solution of the differential system

dz d--s = w + In+l(e,~) + sf(Z,rn(Z,S,q),s )

with Jn+l(u,c,~) = qn+l(u,s,q) - u which is 2z-periodic in u, and

the mean value of Jn+l : 0. Then there is

t rn+l(X,£,q) = s(f Yl(t-s) f Y2(t-s))gn(S)ds, (ii)

-~ t

where Yl(t) = diag{exp(dl(q)t) ..... exp(gk(q)t),0 ..... 0};

Y2(t) = diag{0 ..... 0, exp(dk+l(q)t) ..... exp(gn(q)t) };

gn(S) = g(qn+l(s),rn(qn+l(S),e,q),c), qn+l(S) = qn+l(W(s-t) + x,s,q).

It is easy to prove that rn+l(x,s,q) is indeed independent of

we omit the details here.

By the above procedure we obtain the sequences of functions

t;

{rn(X,S,q)}, {qn(X,C,q)} and {in(S,q)} ,

then give the proof of the convergence of these sequences.

I t i s w e l l known t h a t q n ( X , e n) and rn (X,C q) b e l o n g to C (T) ! i i

for all x, and so there is a positive number ~0 such that

240

II Ds rn(X,g,~) I < 1 II Ds' x - ' x qn (x's'~) [I _< 2, (12)

n : 0,1,2 ..... I sll : 0,1,2 ..... T, 0 _< ~ < gO' I; s' II : 1,2 ..... T.

It is evident that inequality (12) holds for n = 0. By mathematical

induction we assume (12) holds for n : 0,1,...,j, then prove that it

also holds for n : j+l.

By the same procedure as in the proof of Lemma 2 we have a positive

number e 0 independent of j = 0,1,2,... such that

s' x qj+l (x'e'q) II ~ 2, when 0 m ~ < g6, mi D rj(x,g,n)el s i,

In s' II ~ 1

We expand the integrand of rn+l(X,g,q) in (ii) into a Fourier

series, as t

rj+l(X,C,~) = g(/ Yl(t-s) - / Y2 (t-s))~cj+ik(g)exp(i(k,w(s-t)+x))ds -~ t

: ~(diag{i(k,w) - ~i(~) , ... ,i(k,w) - ~n(n)}) -Icj+Ik(x,¢),

so that

II rj+iri s s K0M j, 0 s ~ s s~

where the meaning of K 0 is the same as that in Lemma i, and

Mj = supir D s • xg(qj+l(X,~,~) ,rj(qj+l(X,e,q) ,g,~),s) II • xts

By the definition of M., the assumption of mathematical induction

(12) and that II ms' 3 x qj+l (x'g'~) II < 2, there is a constant s~

independent of j = 0,1,2,... such that

]l Ds rj II < 1 0 < g < e" x ÷i ' - - 0"

By the definition of Ij(E,~) we have

27 2~

lj(s,~) = s (2~)-mf0 "'" 0 / f(qj,rj_l (qj , ~,~) , g)du ,

where qj = qj(u,g,~) . From this formula the uniform convergence of

qj and rj implies the existence of the limit of lj(g,n) • Since

[I qj+l(U,g,~) - qj(u,g,n)II : HI Jj+l(U,S,~) - Jj(u,g,~) II

is 2z-periodic in u, it follows that

241

I] qj+l (w(s-t) + x,e,~) - qj(w(s-t) + x,s,~)JJ < II qj+l - qjll

: SupjJ qj+l(U,S,~) - qj(u,s,n)II • u

On the other hand the set

S = {ulu - w(s-t (mod 2~)}

is dense everywhere in the unit cube because of the inequality (2),

which implies that

SupJJ DS(qJ+s-t x l(W(S-t) + x,s,~) qj(w(s-t) + x,e,~)) II (13)

= SupJ I DSx(qj+l(X,S,q) - qj(x,~,~)) J x

From (13) and the following equalities

qj+l - qj : /(£fj - efj-i + lj+l(c'n) - lj(s,n) ds

D s' ~' x (qj+l-qj) = s/D (fj+l-fj)ds Jl s' J] ~ s,

where fj = f(qj+l'r(qj+l '~'~) ' qj+l = qj+l (w(s-t) + x,s,~), the same

procedure as in the proof of Lemma 2 shows we have constants K' and

t 0''' independent of j = 0,1,2,..., such that

Ell Ds (qj+l -q)x 11 -< eKoE J[ Ds (fj-fjx -1)II

< KoK'CE((I+P)11DS(qj+l-qj ) [J + lJ Ds x(rj-rj_l ) Jl ) (14)

½zll DSx(rj-rj_])If, o <_ ~ _< ~ 6 ' .

Similarly there is a constant K" independent of j = 0,1,2,...,

such that

EJl DSx(rj+l-rj) JJ <sK"ZIJ DS(g(qj+l'rj ' E ) x - g(qj,rj_l,C) s (15)

<- ½Eli Ds(r'-r ) II , 0 < ~ -< E0'~0 =min(~0'~'£~ ') • x 3 j-i

Evidently the inequalities (14) and (15) give the uniform convergence

of the sequences {rn(X,e,~)} , {qn(X,S,q)} and {In(S,~)}.

Suppose that

lim rn(X,S,~) : r(x,s,n), lim qn(X,S,q) = q(x,t,n) ,

lim in(S,n) = l(e,n) n+~

242

Then I = l(s,q), y = r(x,s,q) and z = q(w(s-t) + x,s,q is the

center integral manifold of (i0), and

I = l(s,q), x : q(w(t-t 0) + x0,c,q), y = r(q(w(t-t 0 +x0,E,n),e,n)

is the quasi periodic ~solution of (i0) for all x 0 and t O .

The functions q(x,s,q) - x and r(x,6,q) are periodic and

differentiable in x, so that these functions are uniformly bounded

and equicontinuous, when q tends to zero, and there is a sequence of

qn tending to zero such that

lira q(w(s-t) + x,~,n n) : q(w(s-t) + x,s), n+~

lira r(x,e,q n) = r(x,s) , n-~o

lim l(s,q n) = l(s). n+oo

uniformly in any compact set (w(s-t) ,x,e) . Therefore

I = l(c),y = r(x,s) and z : q(w(s-t) + x,c)

in the center integral manifold of (3), and

I = I(E), x = q(w(t-t 0) + x0,c), y : r(q(w(t-t 0) + x0,~),s)

is the quasi periodic solution of (3) for all x 0 and t O . Theorem

is proved completely.

References

[13 Arnold, V.I., Small Denominators and Problems of Stability of Motion in Classical and Celestial Mechanics, Uspekhi Nauk USSR 18 (1963), 91-192.

[2] Bogoliubov, N.N., On Quasi Periodic Solutions in Nonlinear Problems of Mechanics, Publ. Akad. Nauk Ukrain, SSR, Karev (1963).

[3] Moser, J., On the Theory of Quasi Periodic Motions, SIAM Rev., 8 (1966), 145-172.

[4] Moser, J., Convergent series Expansions of Quasi Periodic Motions, Math. Ann. 169 (1967), 136-176.

[5] Moser, J., A Rapidly Convergent Iteration Method and Nonlinear Partial Differential Equations, International Mathematical Summer Center, Varenna, Italy, 1964.

[6] Lin Zhen-Sheng, The Almost Periodic Differential Equations and Integral Manifolds, Shanghai Scientific and Technological Litera- ture Publishing House, Shanghai (1984) (Chinese).

[7] Lin Zhen-Sheng, The Floquet Theory of Quasi Periodic Linear System, AppI. Math. and Mech. (English Edition) V.3 (1982), 365-381.

STABILITY CRITERIA FOR

LINEAR INTEGRO-DIFFERENTIAL EQUATIONS

W.E. Mahfoud

Abstract

We consider a system of Volterra equations of the form

= A(t)x + f~ C(t,s)x(s)ds x'

in which A(t) and C(t,s) are continuous n x n matrices. Necessary

and sufficient conditions for stability of the zero solution are given.

Introduction

We consider the system

X' = A(t)x + f~ C(t,s)x(s)ds (i)

where A(t) is an n × n matrix continuous for 0 ~ t < ~ , and C(t,s)

is an n x n matrix continuous for 0 ~ s ~ t <

We discuss stability and instability of the zero solution of (i)

via constructing Liapunov functionals for the system

t X' = Lx + Ll(t)x + f0Cl(t,s)x(s)ds + (d/dt) H(t,s)x(s)ds (2)

where L, LI, CI, and H are n x n matrices, L(t) is continuous for

0 ~ t < ~ and Cl(t,s) and H(t,s) are continuous for 0 ~ s ~ t < ~.

For each t O a 0 and each continuous function ~ : [0,t 0] ÷ R n,

a solution of (i) is a continuous function x : [0,~) ÷ R n, denoted

usually by x(t,t0, ~) or simply x(t), which satisfies (i) for t at o

and such that x(t,t0, ~) = ~(t) for 0 ~ t :5 t o . Stability defini-

tions as well as existence, uniqueness, and continuation of solution

can be found in Driver [5].

If B is any positive definite n x n matrix, then there is a

positive constant k such that

klxl 2 ~ xTBx for all x e R n. (3)

244

Stability

Let D be an n x n symmetric matrix satisfying

LTD + DL = -I

For a detailed discussion of the existence of such a matrix

[i] and [3].

Let

P = sup f01Cl(t,s)Ids, t_>0

J = s u p foIH(t,s)Ids, t_>0

m = sup Ir1(t) l, t_>0

and

(4)

D, see

(5)

(6)

(7)

~(t,s) = ]~[(l+J) ICl(u,s) I + (ILl + P + m) IH(u,s) l]du (8)

assuming of course that ¢(t,s) exists for 0 ~ s ~ t < ~.

THEOREM i. Let D, P, J, m, and # be defined by (4) - (8) and

suppose that for some constant e,

(i) J < 1

and

(ii) IDI[P + 2m + J(m + ILl) + ~(t,t) ] ~ ~ < I.

In addition, suppose there is a continuous function h: [0,~) + [0,m)

such that IH(t,s) I ~ h(t-s) and h(u) + 0 as u + ~. Then the zero

solution of (2) is stable if and only if D is positive definite.

PROOF. Let

v t,xi.)) = ix- s)xIslds)TDIx- Sb(t s)x s)dsl

, /g,(t,sl IxIs112as

and show that the derivative of V(t,x(.)) along the solution

x(t) = x(t,t0, ~) of (2) satisfies V' (t,x(.)) ~ -~Ixl 2, ~ > 0. Now,

if D is positive definite, then one can easily argue that x = 0 is

stable. If D is not positive definite, then there is a nonzero x 0 T

arbitrarily near zero such that x0Dx 0 ~ 0 It follows from this and

V' (t,x(.)) ~ -~Ixl 2 that the solution x(t,0,x 0) remains bounded away

from zero for all sufficiently large t. This completes the proof.

Most stability results for (i) require that A(t) = A = constant.

In this case, Theorem I, when applied to the system

245

x' = Ax + [~c(t,s)x(s)ds (9)

with

ATD + DA = -I (i0)

yields the following result

THEOREM 2. Suppose (I0)holds and there is a constant e such that

IDI[ IC(t,s) Ids + ftlC(u,t) ldu] s ~ < i.

Then the zero solution of! (9) is stable if and only if

definite.

This is [3, Theorem 8].

and is in %

If A(t) is not constant, and /tA(v) dv exists LI[0,~),

we let

and

Cl(t,s) = C(t,s) - A(~t-s)

D is positive

H(t) = -ftA(v)dv

SO that (I) takes the fo]na

t Io x ' = - H ( O ) x + A ( t ) x + f o C l ( t , s ) x ( s ) d s + ( d / d t ) H ( t - s ) x ( s ) d s .

In this case

= sup ftlc(t,s) - A(t-s)Ids , (il) P t_>O

J = fOl[tA(s)dsldt , (12)

M = suplA(t) l, (13) t_>0

and D satisfies

/o[At(t)D + DA(t) ]dt = -I (14)

THEOREM 3. Let P, J, m, and D be defined by (ii) - (14) and suppose

there is a constant ~ such that

CO

IDl{2[m + J(m + If0A(t)dtl)]

+ (i +J)[P + /tlC(u,t) - A(u-t)Idu]} -< ~ < i. (15)

Then the zero solution of (i) is stable if and only if D is positive

definite.

246

PROOF. Letting L = -H(0) : -/oA(V) dv, L 1 (t) = A(t) and observing

Iof that ft IH(u't) Idu = H(v) Idv = J, we see that (ii) of Theorem 1 co

reduces to (15). Since (]4) implies that 21D I I/0A(t)dtl -> n, then

it follows from this and (15) that J < i/n _< i. Taking h(t) = IH(t) I,

the result follows from Theorem i.

In the special case where C(t,s) = C(t-s), Equation (i) reduces to

x' = A(t)x + /oC(t-s)x(s)ds (16)

and

P = /oIC(v) - A(v) Idv (17)

THEOREM 4. Let J, m, D, and P be defined by (11)-(14) and (17)

respectively, and suppose that

(m+P) (J+l) + JI/oA(t)dt I < I/(21DI). (18)

Then the zero solution of (16) is asymptotically stable if and only if

D is positive definite.

PROOF. As C(t,s) = C(t-s) , then

/01C(u't) - A(u-t) Fdu = IC(v) - A(v) Idv : P.

Thus (18) implies (15) and the stability or instability of (16) follows

from Theorem 3. To show asymptotic stability, we observe from (17) and

(18) that C(t) is in LI[0, ~] and hence, by (16), x' is bounded. ! ! ! !

Since (Ix(t) [ 2) = (x T) x + xTx , we have (x(t) I 2) bounded. But,

the Liapunov functional considered in the proof of Theorem 1 yields

V' (t,x(.)) s -@Ixl 2, @ > 0, along any solution of (16). Thus, Ix(t) I 2

is in LI[0,~] and hence x(t) ÷ 0 as t + ~. This completes the

proof.

COROLLARY I. Let J, m, and P be defined as above and suppose that

n = 1 and

(m+P) (J+l) + (J-l)I/oA(t)dtl < 0.

Then the zero solution of (16) is asymptotically stable if and only if

f0A(t)dt < 0.

EXAMPLE i. Consider

2/t -~(t-s) x' = (-½)~2e-~tx-(½)~ roe x(s)ds, 0 < ~ < I/3 .

247

Here, C(t) = A(t) = -~2e-~t/2. Thus, P = 0, m = ~2/2, J = 1/2, and

fO A(t)dt = e/2 < 0. Hence,

(re+P) (J+l) + (J-l) IfoA(t)dt I = ~ (3e -i) < 0.

and, by Corollary i, the zero solution is asymptotically stable.

Now, we present another interesting application of Theorem i.

Consider the scalar equation

x' = AX + ft[k(t-s) + C(t,s)]x(s)ds (19)

where A is a constant, C(t,s) is continuous for 0 -< s -< t < ~, and

k : [0, ~) ÷ (-~,~) is differentiable with k' in LI[0,~).

We differentiate (19) to obtain

x" = Ax' + k(0)x + ftk' (t-s)x(s)ds + (d/dt)/tc(t,s)x(s)ds.

Let

x' = y

y ' = k(0)x + Ay + /O k' (t-s)x(s)ds + (d/dt)foC(t,S)X(s)ds.

Then we have the system

Z' = LZ + foC1 (t-s)z(s)ds + (d/dt)/oH(t,S)z(s)ds

where

z = [y] Cl(t)= [k ,0 0] [ 0 :] ' (t) , H(t,s) = C(t,s) '

and

Let

and

(2O)

d ~

J = s u p fglc(t,s)Ids . (22) te0

If A < 0 and k(0) < 0, then L is stable and there is a symmetric

positive definite matrix D such that

LTD + DL = -I (23)

p = f[Ik'(v) Idv (21)

248

THEOREM 5. Let L, P, J, and D be defined by (20)-(23). If A < 0

and k(0) < 0, and there is a constant ~ such that

(i) J < 1

and

(ii) IDI[P(J+I) + P + ILIJ + (P + ILI)ftlC(u,t)Idu] -< ~ < i,

then the zero solution of (19) is stable.

PROOF. If we observe that the condition IH(t,s) I -< h(t-s) is need-

ed only to prove the converse of Theorem i, then Theorem 5 is an

immediate consequence of Theorem i.

REMARK. There is no integrability condition on the kernel in (19).

Thus, if we take k(t) = k = constant and C(t,s) = C(t-s), then (19)

reduces to

x' = Ax + /0[k + C(t-s)]x(s)ds . (24

In this case P = 0,

co

J = /0 fc(v)Idv , (25

and Theorem 5 reduces to

COROLLARY 2. Let L, D, and J be defined by (20), (23), and (25).

If A < 0, k < 0, and

21LIIDIJ < i, (26

then the zero solution of (24) is stable.

EXAMPLE 2. Consider the equation

= -x + f0[-e+8(t-s+l)-2]x(s)ds x'

where e and ~ are positive constants.

Then L = [0 ':I and a simple calculation will show %

that D = [~ b] with a = (~ + i + i/~)/2,

b = i/(2~), and c = (i + i/e)/2. Thus, by corollary 2, the zero

solution is stable if 8( 2 + 4e + 8 + 8/e) < i.

For further application of Theorem 1 we consider the convolution

system

x' = Ax + f0c(t-s)x(s)ds (27)

where A is a constant n × n matrix and C is an n x n matrix

continuous for 0 -< t < co.

249

Let

C(t) = C 1 (t) + C 2(t)

where Ci, i = i, 2, are continuous on [0,~) .

We assume that Cl(t) and ftC2(v) dv are Ll-functions and let

H(t) = -f~C2(v)dv. Then (27) takes the form

x' = Lx + f~Cl(t-s)x(s)ds + (d/dt)f~H(t-S>X(S)ds (28)

where

foC2 L = A + (v)dv.

In this case

and

co

P = f0 f Cl(t) I dt, co co

J : f 0 I ftC2(v)dvldt,

LTD + DL = -I,

~(t,t) = (I+J)P + (P+ILI)J.

(29)

(30)

(31)

(32)

THEOREM 6. Let L, P, J, and D be defined by (29)-(32).

(i) if P(J+I) + JILl < I/(21D[) ' then the zero solution of

(27) is stable if and only if D is positive definite.

(ii) If, in addition, C(t) is in LJ[0,~) , j = 1 or 2, then

the zero solution of (27) is asymptotically stable if

and only if D is positive definite.

(iii) If, furthermore, J ~ 0 and f~C(v)dv is in Lq[0,=),

0 < ~ < 2, then the zero solution of (27) is uniformly

asymptotically stable if and only if D is positive definite.

PROOF. (i) follows from Theorem 6 and (ii) can be proved as Theorem

4. To prove (iii), we translate the argument t by t o in (27) and

use the variation of parameters formula to conclude that x(t+t 0) ÷ 0

uniformly in t o and this is uniform asymptotic stability.

In [4] Grossman and Miller gave a characterization of the uniform

asymptotic stability of the zero solution of (27) in terms of the ^

location of the zeros of det(s-A-C(s)) where C(s) denotes the

Laplace transform of C(t).

250

THEOREM (Grossman-Miller). Suppose C ~ LI[0, ~) . Then the zero

solution of (27) is uniformly asymptotically stable if and only if det

(s-A-C(s)) = 0 for Re s >- 0.

When n = 1 and C(t) -> 0, it is known that s - A - C(s) = 0

foC (t) for ReS -> 0 is equivalent to the condition A + dt < 0.

Obviously, such an inequality is much easier to verify than to locate ^

the zeros of the function s - A - C(s). In fact, apart from kernels

such as C(t) = ke -~t ~ > 0, the determination of zeros can indeed be

a problem, especially when A > 0. It is also known that when

C(t) -< 0 or C(t) changes sign the condition A + fo C(t)dt < 0 is

not sufficient for uniform asymptotic stability and therefore a

stronger size condition on C(t) must be required. See Brauer [23,

Jordan [63, and Burton and Mahfoud [33. In this case, Theorem 6

provides us with a practical stability criterion under mild conditions.

EXAMPLE 3. Consider the equation

1 t -2 x . x' = -~x - ~ f0(t-s+l) (s)ds

Here, the kernel is of the form C(t) = -k(~t+l) -p, k > 0 and ~ > 0.

An effective decomposition of such a kernel may be obtained as follows:

Let

C 2 (t) = C(t) if 0 _< t -< ~,~ z 0

= -y(st + i) -q if t >-

with y = k(~B+l) q-p and q > max(2,p).

oo

We choose 8 and q so that J < i, L = A + /0C2(v)dv < 0, and

P(J+I) + (J-l) IL I < 0.

Thus, by choosing ~ = 6 and q = 4 for Example 3, we have

L = -32/315, P = 2/63, and J = (42 in 7-17)/126 < i. Hence, all

conditions of Theorem 6 are satisfied and the zero solution is un

uniformly asymptotically stable.

References

[13 E.A. Barbashin, The construction of Lyapunov functions, Different- ial Equations 4 (1968), 1097-1112. (This is the translation of Differentsial'nye Uravneniya 4 (1968), 2127-2158.

[23 F. Brauer, Asymptotic stability of a class of integro-differential equations, J. Differential Equations 28 (1978), 180-188.

251

[3] T.A. Burton and W.E. Mahfoud, Stability criteria for Volterra equations, Trans. Amer. Math. Soc. 279 (1983), 143-174.

[43 S.I. Grossman and R.K. Miller, Nonlinear Volterra integro- differential systems with Ll-kernels, J. Differential Equations 13 (1973), 551-566.

[53 R.D. Driver, Existence and stability of solutions of a delay- differential system, Arch. Rational Mech. Anal. i0 (1962), 401-426.

[63 G.S. Jordan, Asymptotic stability of a class of integro-different- ial systems, J. Differential Equations 31 (1979), 350-365.

A MECHANICAL MODEL FOR BIOLOGICAL PATTERN FORMATION: A NONLINEAR

BIFURCATION ANALYSIS

P.K. Maini, J.D. Murray and G.F. Oster

Abstract

We present a mechanical model for cell aggregation in embryonic

development. The model is based on the large traction forces exerted

by fibroblast cells which deform the extracellular matrix (ECM) on

which they move. It is shown that the subsequent changes in the cell

environment can combine to produce pattern. A linear analysis is

carried out for this model. This reveals a wide spectrum of different

types of dispersion relations. A non-linear bifurcation analysis is

presented for a simple version of the field equations: a non-standard

element is required. Biological applications are briefly discussed.

I. INTRODUCTION

A central question in developmental biology is the process by

which geometrical patterns emerge during embryogenesis. Several models

have been proposed to describe the mechanisms of development of bio-

logical form (morphogenesis). Turing (1952) showed how a system of

reacting and diffusing chemicals (morphogens) could produce pattern

due to instability of a homogeneous equilibrium state. Such reaction-

diffusion models have been widely studied since (e.g. Gierer and

Meinhardt (1972), Thomas (1975), Murray (1977, 1981), Meinhardt (1982)).

A somewhat simpler scheme is the gradient model (Wolpert (1969)),

which supposes the existence of a group of morphogen secreting cells

(e.g. Saunders and Gasseling (1968)). The morphogen diffuses away from

the source setting up a stationary gradient. Cells differentiate when

the concentration of the morphogen reaches a certain threshold value

(Smith and Wolpert (1981), Wolpert and Hornbruch (1981), Smith et al

(1978), Tickle (1981)).

253

The central principle underlying these models is the setting up

of a chemical pre-pattern to which the cells respond. However, such

models have certain problems: (i) except in rather special cases, the

morphogens remain, as yet, unidentified; (ii) the mechanism by which

cells respond to the concentration of morphogen is vague and, in general,

must be exquisitely sensitive.

In this paper we present a mechanism for spatially patterning

populations of mesenchymal cells which is based on the following well

documented mechanical properties (Harris et al (1981)) : I) Cells spread

and migrate within a substratum consisting of a fibrous extraeellular

matrix (ECM). 2) They generate large contractile forces which deform

the ECM.

Section 2 contains a brief resume of the model equations; the

reader is referred to the paper by Oster, Murray and Harris (1983) (and

references therein) for full details. Section 3 contains a linear

analysis of the model. This gives rise to an abundance of dispersion

relations, suggesting a richness in the pattern forming abilities of

the model. In Section 4 a non-linear bifurcation analysis is presented

on a simplified version of the model. This gives certain predictions

for the amplitude of the heterogeneous cell density. The biological

applications to long-standing problems of feather germ formation and

wound healing are discussed at the end of the paper.

2. CELL TRACTION MODEL MECHANISM

The model is based on the three field variables

n(x,t) = density of mesenchymal cells at position x and time t

p(x,t) = density of ECM at position x and time t

u(x,t) = displacement at time t of a material point in the matrix

which was initially at x.

The equation for cell movement is

~n = V. (D1(s)Vn-D~(s)V3nz - ~[nV(p+~'V2)]-n~)p + rn(m-n) ~-~ = = ~% ( 1 )

random dispersal haptotaxis convection mitosis

where c = I/2[Vu + Vu T] is the linear strain tensor. We motivate each

of these terms in turn.

Random dispersal. We model this with a Fickian flux: _J : -DI(a)Vn

where DI( [)_ is a strain dependent (i.e. matrix directed) diffusion

coefficient. However, at large cell densities, Fick's Law is inadequate

because it does not take into account non-local effects. These effects are

254

important here because mesenchymal cells can detect non-local concen-

trations via long cell protuberances (filopodia) . Thus cell mov~,~nt also

depends on the average cell concentration in the immediate surrounding.

This average may be modelled by including a higher order term, D2(~)V3n,

in the flux expression, where D2(~) is the long range diffusion coeffi-

cient: for simplicity we take DI (~) and D2(~) to be constant in this

paper. Thus the random dispersal flux is modelled as

3 ~random =-DIVn + D2V n.

Note that the sign accompanying D 2 is plus. This implies that D 2, like

DI, is dispersive.

Haptotaxis. Cells actively move by attaching their filopodia to certain

specialized adhesive sites on the extracellular matrix. They tend to

move up a gradient in adhesive sites (haptotaxis) because the filopodia

have a better grip where there are more adhesive sites and will thus

drag the cell up the gradient (Harris (1973)). Assuming the adhesive

sites are uniformly situated throughout the matrix, we have

~ h a p t o t a x i s = anV(o + a ' V 2 p )

where the V2p(~,t) term takes into account long range interactions (c.f.

long range diffusion).

Convection. Cells may move passively due to the movement of the matrix.

We model this in the usual way by the term

~convection = n~u ~t

Hence the total flux is:

3 J = -DIVn + D2V n + ~[nV(Q + a'V2p)] + n~u -

Mitosis. We assume cells proliferate until a limiting density N is

reached, according to the logistic growth law

mitotic rate = rn(N-n)

Eqn (I) gives the conservation law for density:

~n _ D]V2n_ D2 ~4 ~t n- V.(anV(p + a'V2p)) - ?(n~) + rn(N-n)

St

rate of change of cell density = net flux + cell division

Mechanical Balance Equation.

We are dealing with systems in the realm of low Reynolds number

(Purcell (1977)) so that the viscous and elastic forces dominate

255

inertial terms (i.e. motion of cells instantly ceases when the applied

forces are turned off). Therefore, the restoring forces in the ECM

are in equilibrium with the cell contractile forces. We model the ECM

as a simple visco-elastic material with a stress tensor

o = ~ ~ + ~ ~ + E (~ + v 8I) (2) =m I~ 2~-~ :I (1+v) : I-2v --

viscous elastic

where E is the Young's modulus, ~i,u2 the shear and bulk viscosities,

the Poisson ratio, 8 = V.u the dilatation (this measures the increase

in volume due to applied forces) and I the unit tensor.

E measures the passive elastic modulus of the ECM, which we assume

is initially isotropic. However, the contractile forces exerted by the

cells align the collagen fibrils and thus increase the strength of the

ECM. To account for this strain alignment in the simplest way (without

introducing additional elastic constants), we set E : E(0), where E(@)

is some monotonically increasing function of 8. The Poisson ratio

accounts for the transverse compression of the ECM when it is stretched.

We assume the stress due to cell traction has the form

o . : 7 (n)n[0 + BV20]I (3) =cell-matrix

T where z(n) = i--~2 is the traCtion per cell (e.g. dyne/cm/cell) . This

says that as the cell density increases, the traction exerted by each cell

decreases, that is, as the cells pack together their traction decreases

due to contact inhibition (Trinkaus (1984)). Because the cells exert

contractile forces via attachments to the ECM, traction also depends on

the local matrix density 0(x,t) (short range traction) and non-local

(average) matrix density V2p(x,t) (long range traction). The parameter

B measures the strength of the long range traction. Note that in an

environment of large cell density and low matrix density, the cells

would exert contractile forces by attaching their filopodia to other

cells.

as

Hence, we model the cell-cell contribution to active traction

The equation for mechanical equilibrium is

V.g + pF : 0 (5)

F o r where =° = :m ~ + _~cell-matrix and _F accounts for body forces.

example, if the ECM is attached elastically to an external substratum

then

F = --SU.

2

~cell-cell = T(n)[n + ~'V n]. (4)

256

The conservation equation for matrix material has the form

= -V.(p~) + S(n,u,p) (6) ~t

convection secretion

We shall assume in this model that secretion of matrix is negli-

gible on the time scale of pattern formation (i.e. S(n,~,p) = 0).

Equations (I), (5) and (6) constitute the field equations describing

mesenchymal morphogenesis.

3. LINEAR ANALYSIS

Before linearizing, we non-dimensionalise the system. This reduces

the number of parameters, making the mathematics slightly simpler. More

importantly, the non-dimensional parameter groupings are biologically

significant since they show which processes have equivalent effects.

The non-dimensionalised system is presented in Appendix (a) . The

equations admit the non-trivial steady state

n = p = I, u = 0. (7)

Linearizing about this steady state and substituting solutions of a(k2)t+ik.x

the form e - --, where k is a wave vector, gives rise to various

types of dispersion relations o(k2), depending on the parameter values.

(See Appendix (b) for the full dispersion relationship). We consider

various ~(k 2) behaviours and discuss the isolation of modes.

Types of ~(k 2) behaviours. From Appendix (b) all of the dispersion

relations have the form

~(k 2) = -b(k2)±A2(k2)-4~k2c(k2) (8)

2~k 2

where b(k 2) and c(k 2) are polynomials in k 2 and involve the parameters

of the model.

The homogeneous steady state will go unstable if R£(u(k2)) > 0 for

some k 2. We look for diffusion-driven instabilities wherein the homo-

geneous equilibrium is stable to non-spatial variations. With all

parameters non-zero, it is possible to have the behaviour shown in

Fig. 1(a) . Since R~(~(k2)) > 0 ¢=~b(k 2) < 0 or c(k 2) < 0 it may be

possible (depending on the parameters satisfying certain relationships)

to have the ~ behaviour illustrated in Fig. 1(b) .

257

FIG. I .

Ca) Two possible dispersion behaviours.

(b) In (b), the relative

values of the maxima depend on the parameter values.

We now examine the effect on the dispersion relationship of setting

certain parameters equal to zero. This helps us to understand how

various parameters affect the model and we can see if it is possible for

simpler models to mimic the behaviour of the more complicated system.

Setting ~ = 0 gives ~(k 2) = -c(k2) /b(k 2) hence we have the possi-

bility of infinite linear growth rates for certain non-zero modes k.,

where blk ) = 0. Consider, for instance, the model in which only s,r,

B and T are non-zero. In this case,

b(k 2) = 8Tk 4 + (I 2T)k 2 -- + S

(9) c(k 2) = r[BTk 4 + (I -T)k 2 + s]

Fig. 2 illustrates the behaviour of o(k 2) as T increases, while

Fig. 3 shows the dispersion relation with s = 0 as well.

a(k } '

- F - F

l¢Cs/pc ) I !

- r

c(~:) j

- r

I i

I I

(c)r>r¢.

I

(e) c > q .

258

~(~)' <j - r - - - ~ 4-

V (d)

h~= (2c-1)+-J(~z-1)~-- 4~#sr - 2 # r

&= 1+s# +,/(]+s,~f-I_ 2

% : l+2Sf3+~(1+2S/~)%- 1

FIG. 2. Dispersion relation for the model with ~,8,r and s non-zero.

Notice the character of the predicted growth rate as T increases (a)-(e).

- r

~: imcr~as~.~ 8 ca) ~ < I / 2 .

~P

c ~crzasin~[

: f - - - - ,/

(b) I/2< r< 1.

r7 S (c1 c ~ 1,

c~mg

FIG. 3. Dispersion relationship in Fig. 2 with s = 0. As T increases

through the bifurcation value I/2, there is a discontinuous change in

d(k2).

When viscosity is reintroduced the growth rate is rendered finite

(Fig. 4). The central issue in the linear analysis is which of the

259

FIG. 4. Dispersion relation with ~,s,7 and B non-zero. The uniform

steady state becomes unstable as T increases.

modes k± in Fig. 2(c) grows fastest, as this is presumably the only one

observed. This, together with the biological significance of such

behaviour, is currently under investigation. Setting different para-

meters zero gives rise to a varied collection of ~(k 2) behaviours

(Murray and Oster (1984)).

Isolating modes. With fixed geometry, we show that, in order to isolate

modes, it is necessary to vary at least two parameters.

Consider the one-dimensional domain [0,1] with boundary conditions

nx Px 0, u = 0 on x 0,1 The eigenfunctions are then

~(kZ)t (10) = b sin kx e

c cos kx

where k = nw and a,b,c are arbitrary constants and the unstable modes

have R£a(k 2) > 0.

We can try to isolate these modes by continuously varying one

parameter (that is, as the parameter changes, the first mode goes

unstable by itself, then the second mode by itself, etc.). To illus-

trate the argument, consider the model wherein b,s,T and 8 are non-zero.

(We discuss this model in more detail in section 4.) Let us increase

T while keeping the other parameters fixed. The dispersion relation is

~(k 2) = -b(k2)/ ~k 2 where b(k 2) : 5~k 4 + (I -2T)k 2 + s and g(k 2) > 0

if and only if b(k 2) < 0. The parabolas b(k 2) intersect at k 2 = 0 and

in one non-zero fixed point for

b(k2,T1 ) : b(k2,T2 ) ~ k 2 : 0 or k 2 : 2/B.

260

Clearly one can isolate the first mode by increasing 7 (Fig.5(a)) . If

we try to isolate mode 2, we note that the parabolas b(k2,T) for T in-

creasing must intersect at the fixed point A (see Fig.5(a)) and thus go

parabola b(k2,Tl ) . Hence mode 1 will remain unstable and below the

it is impossible to isolate the second mode. The difficulty is that A

is fixed. For a general b(k2), it may be possible to vary the point

of intersection by varying more than one parameter and thus isolate many

modes (Fig.5(b)). It is clear how to generalise this simple argument

for the full dispersion relationship. Note that a necessary condition

for isolating modes by varying only one parameter is that it must occur

quadratically in the dispersion relation.

~ b(¢;H

'

(a.~ (b}

FIG.5.

most.

detail.)

Isolating modes. (a) Varying T alone will isolate one mode at

(b) Varying two parameters for a general b(k2). (see text for

The above discussion illustrates the wide and varied behaviour that

such a model may exhibit. It can be shown that the corresponding three

species reaction-diffusion system can only give rise (for purely real

o(k2)) to a dispersion relation of the form illustrated in Fig. 4. This

suggests that the model here has a much richer class of pattern forming

abilities than reaction-diffusion models.

4. NON-LINEAR ANALYSIS

Consider the simple model in which cell movement is by convection

only and the ECM is tethered to an underlying layer (e.g. the subdermal

layer in feather tract formation). The one-dimensional equations for

this model are

261

Sn ~x ~u ~-~ + (nT{) = 0

~3u ~2u + T~x [n(p + B~2P)] - sup = 0 P x2------ ~ + 8x--- ~ ~x--~

( 1 1 )

~P + 8-~ ( au ~--~ p-5-~) = 0

where we have approximated ~ (n) by %n.

The dispersion relation is

(BTk4+ (1-2T)k2+s) o(k 2) = 0 or o(k 2) = -

k 2

which is similar to that in Fig. 4: here the bifurcation

T = I/2{(I + sS) + [(I + s6) 2 -I] I/2 } c

and we have incorporated p into the time scale.

We perform a nonlinear bifurcation analysis similar to that in

Lara and Murray (1983) as % exceeds its critical value T C

2 = T + s 6 , 0 < ~ << 1 , 6 = ±1

C

(12)

by setting

(13)

We look for solutions of (11) of the form

n(x,T,s) = I + j=IE sJ{Aj(e,T)cos JkcX + D (E,T)sin3 3k cx}

u(x,T,s) = Z cJ{B (s,T)sin JkcX + E (£,T)cos JkcX} j:1 ] ]

p(x,t,8) = I + Z 8J{c (e,T)cos JkcX + Fj (s,T)sin JkcX} (14) j=1 J

where A (e,T) = Z A~(T)s i etc and T = ~2t. ] i:0 ]

To motivate the stretching of the time scale, (~onsider the behaviour

of ~(k 2) for a small variation about the critical traction T . A c

Taylor expansion of ~ about k 2 tires C -

e ( k c , ~ ) = e c) + 2 ~o 4 %-~ + 0( ) (15)

~0 k 2 ,T C C

and the exponential growth term in the solution becomes e 0(82)t,

suggesting a slow time variable T = s2t.

We motivate the form of solution (14) as :[ollows. Assume

n = I + ~83n (x,t,T) j ]

u = ~sJu (x,t,T) (16) j J

p = I + Ee]p (x,t,T) ] J

262

Equating powers of £, we have from (11)

9 t(n 1+Sxu 1) : 0

9xxtUl + 3xxUl + ~cgx(n I + pl ) + 7c~3xxxn I - su I : 0

~t(P1+~xul ) = 0.

(17)

The solution of (17) is

) • } n I /alj(k j) o(k2)t +ikjx u I = Z I bl j (kj) e 3 Pl J \c1j (kj)

(18)

where alj(kj I °(k2)t + ikjx blj(kj e 3 are the eigenfunctions of (17).

• Clj (kj)

The uniform steady statE: becomes unstable only through the eigen-

function of waver_umber kc, that is,

o(k~l > 0,~{k~l < 0 V k 2 k 2 c n n c

so, over a long time scale, we expect this eigenfunction to dominate.

Thus the asymptotic form of ~18) is (Matkowsky (1970))

nl) <alc(T'kc)> ik x Ul ~ blc (T'kc) e c (19)

Pl Clc (T'kc)

where the exponential term has been zncorporated into the T dependence

of alc,blc,Clc. Continuing this calculation at every step, the t dependence can be

ignored in the subsequent analysis, and all calculations carried out

asymptotically for large time. Hence we look for solutions of the form

{14} .

To lowest order in s, we have

+ k °1) T : 0

0 0,F~} For the remaining and a similar set of equations for {D1,E I _ cal-

culation we shall only consider {Aj,Bj,Cj} to simplify the analysis.

(The analysis can be repeated exactly for {Dj,Ej,Fj}.)

Order 82 terms give

263

(A02 2kcB02) T 0 0 : + + kcA1BI T 0

2kcTcA ~ 2 c 0 2k T (I- 4~k2)C2 : + (4k + s)B2 + c c

_ kcTc(1 _k2B) 0 0 s 0 0 c A l C l - 7 B l C l

02 1 0 0 (C 20 + 2kcB )T + kC BIT

3 and order 8 terms give

0 0 -k2cB1T

k T c c 2 c 0 0 s - 0 0 0 0

+ --~ (Sk -I)A2C I - ~ [C.IB 2 -BiC 2] = 0

(21)

(22)

Standard nonlinear analysis simply requires successive suppression of

secular terms. With the structure of our equations this is not

sufficient to determine the amplitude equations. It turns out that we

must use an integrated form of the conservation equations. Integrating

the first and third of (20) we have three simultaneous equations for 0 0 and 0

A I ,B I C 1

0 + kcB~ = y~ A I

0 k28)Ci kc~cA01 + {k~ + slB I +kc~c(1- 0 = 0 (231

0 0 kcB ~ + C I = 7 3

0 0 where YI and 73 are constants. This system is degenerate and has non-

trivial solution if and only iJ-

(k2c(i Tc ) + s)y0 + 2 0 - kc<c7 1 = 0 (24)

i.e. there is a constraint on the initial conditions. As we have two

conservation equations in the system (I I) we obviously expect some

constraints on initial perturbations.

Integrating the first and third of (21) gives

0 2 I A20 + 2kcB ~ = AI /2 + 71 (25)

O 2 0 + 2kcB ~ C I /2 I C2 = + Y3

I I where 71 and y1 ~ are constants.

Note0 that y = AI0 (0) + kcB (0). If we assume initial perturbations to be 01(2),

0 = 0. Moreover, assuming initial perturbations to be 0 (s 3) inplies YI = then 71 = Y3 I Y3 = 0. (Making these assumptions is not necessary but they help make the analysis

simpler. ) 0 0 0 0 0 0

We can solve the system~ (23) and (25) for B I,C I,~,B 2 and C 2 in terms of A I

and substituting into (22) we have the Landau equation

dA~/dT = 6XA01 + YA~ 3

264

O where X : 2 - ~k S = (2T c

+ I)/(2~ c )

and Y = (14BsT + 24T - 63Bs - 12)/72s~ C C

0 V? Notice that the coefficient of A I is simply 3@

we expect from the linear analysis, k 2 ctTc

The possible behaviours for (26) are sunmlarised in the Table.

TABLE. Behaviour near the bifurcation when = ,

Y as in (26).

(26)

that is, what

Y < 0

0 evolves to / ~ 6>0 A I

0 tends to 0 6 < 0 A I

Y > 0

0 A I goes unbounded

0 there is a threshold in A I

0 __~ 0 ÷ 0 A I (0,0) < / ~ A I

0 / X ~ 0 A I (0,0) > ~ A I ÷

If we are in the parameter space P (Fig. 6) the cell density

evolves to the bounded steady state

I + ¢/X/-~-~ cos(kcX) (27)

&

> $

FIG. 6. Parameter space, P, in which Y > 0 and the homogeneous steady

state evolves to the heterogeneous solution (27).

If we had kept in the initial constants, we would have finished

up with a perturbed version of the above Landau equation, namely

02 03 0 + ZoAI + y AI (28) dA0/dT = C O + ~(X + X0)A I

I 0 I

where C0,X 0 and Z 0 are functions of yi,Yi,i = 1,3. Thus the homogeneous

steady state would evolve to a heterogeneous steady state dependent on

initial perturbations. Since we are dealing with small perturbations,

however, these variations will also be small.

265

5. BIOLOGICAL APPLICATIONS

Formation of skin organs. In the early stages of skin organ development

(hair, teeth, feathers, scales) dermal cells aggregate to form a

regular spatial pattern. These aggregations (papillae), in association

with overlapping arrays of columnar epidermal cells (placodes), lead to

the formation of skin organ primordia (e.g. Rawles (1963), Wessels

(1965)).

It is found that feather primordia develop in a hexagonal pattern

within well-defined regions of chicken skin (pterylae) . The primordia

do not develop synchronously, however. In the posterior part of the

spinal pteryla, for example, an initial row of feather primordia forms

along the dorsal midline (Stuart and Moscona (1967), Davidson (1983))

and successive rows form on either side of this initial row.

We now apply our model to this with the following scenario: the

columnar condensation forms first as dermal cells along the dorsal mid-

line break up into isolated clumps. This could be triggered by the

increasing traction of cells (or, equivalently, other parameters involved

in the dimensionless traction parameter). This parameter evolution may

be due, for example, to cell maturity: cells "age" into the unstable

regime in parameter space wherein the homogeneous steady state becomes

unstable and evolves into a heterogeneous steady state (c.f. Section 4).

The tractions produced by these aggregates strain the matrix and the

secondary row of papillae form at loci midway between the primary

papillae, where the strain is a local minimum. This recruits other

cells and thereby forms a hexagonal pattern. Fig. 7 illustrates the

situation. (Numerical and analytical studies are underway to find these

two-dimensional patterns).

® ® @ @

@ ® @

@ @ @ @

@ @ @

(i) T < T (ii) T > T c c iii)

FIG. 7. Idealised section of chick pteryla. As traction increases, the

uniform cell density (i) becomes unstable and forms aggregations (ii).

This row of aggregates causes condensation along a neighbouring row at

266

interdigitating points. (iii) shows how this could give rise to hexa-

gonal structure.

Wound healing. During wound healing, cells migrate towards the damaged

site and the fibroblasts exert large contractile forces to pull the

wound closed (Trinkaus (1984), Fig. 12.1). This gives rise to gross

disfiguring particularly after skin graft subsequent to severe burns.

The model equations are currently being studied with this application

in mind. It gives rise to a formidable free boundary problem. However

the potential practical rewards justify an in-depth study.

6. DISCUSSION

We have pr(sented a model for cell aggregation, based on well

documented mechanical properties of cells and extracellular matrix. We

have illustrated how cell traction on an elastic substratum can produce

various aggregation patterns. No directed cell migration is necessary,

although if cells are motile this will merely enhance the tendency to

form patterns. Thus the model illustrates how different mechanical

properties of the cells can lead to cell pattern. The predictions of

the model can be (and are being) tested experimentally as, in principle,

all the parameters are measurable.

Mechanical models can thus lead to a greater understanding, bio-

logically, of the phenomenon of development. For the mathematician and

numerical analyst, the models provide an interesting and formidable

class of problems to be investigated.

Acknowledgements: PKM wishes to thank the Department of Education of

Northern Ireland for a postgraduate studentship. GFO would like to

acknowledge support from the Science and Engineering Research Council

of Great Britain (Grant GR/C/63595) for a visit to the Centre for

Mathematical Biology in Oxford.

267

APPENDIX

(a) Let L, T O be typical length and time scales respectively and let

P0 be a typical matrix density. With the following dimensionless

quantities

u DIT0/L 2 : n/N, ~ = P/P0' ~ = ~' ~ : t/T0' ~ : x/L, DI = ' ~ = IN2'

= 6/L2' s = sPoL2(I+v)/E' D2 D2T0/L4' & = ~PoT0/L2' ~' = ~'/L,

r = rNT0' ~I = Z(I+v)/ToE' ~2 : #2(1+v)/T0 E' { = TNP0(I+v)/E

the system (I), (5), (6) becomes (dropping tildes)

~tSn _ DIV2n _D274n _ ~V. [nV(p+~'V2p)] - V. [n~] + rn(1-n)

V. [#I~ + #2~-t :I + (s= + ~8I)= + - -

~P ~u + v (#--) = o

~t

Tn {p + ~V2p}I] = sup 1+in 2 = --

where v - 1 - 2 v "

Notice that by taking different values for TO, we can work on

different time scales, e.g. we could work on the haptotactic time scale

by setting T O = L2~p0 (Oster, Murray and Harris (1983)).

(b) Linearizing about the steady state n = p = I, u = 0 by setting

n = I + n, p = I + ~, u = u (where the tilde variables are small vari-

ations from the steady state), we find the dispersion relation, in the

usual way, by looking for solutions

1] ~ e o ( k 2 ) t + i k . x

I n one d i m e n s i o n , t h i s g i v e s r i s e t o t h e r e l a t i o n s h i p

o ( k 2) ( ~ k 2 o 2 ( k 2) + b ( k 2 ) o ( k 2) + c ( k 2 ) ) = 0

where

b(k 2) = ~D2k6 + ( ~ B / ( I + X ) + ~D1)k 4 + (1+~r-2T/(1+X)2)k 2 + s,

2 8D2k 8 c(k ) = T/(I+I) + (T/(I+I) (8D I - D 2 + e~' (1-2X/(1+X)) + D2)k6

268

+IT/(1+1) (rB - D I- ~(I-21/(i+I)) + sD 2 + Dl]k4 + (SDl+r-rT/(l+l))k2

+ rs

where ~ = ~1 + ~ 2 ' a n d we h a v e n o r m a l i s e d ~ , T a n d s b y d i v i d i n g b y

(i+0).

References

Davidson, D. (1983) . The mechanism of feather pattern development in the chick, i. The time determination of feather position. J.Embryol. Exp. Morph., 74, 245-259.

Gierer, A. & Meinhardt, H. (1972). A theory of biological pattern formation. Kybernetik, 12, 30-39.

Harris, A.K. (1973) Behaviour of cultured cells on substrata of variable adhesiveness. Exp. Cell Res., 77, 285-297.

Harris, A.K., Stopak, D. & Wild, P. (1981). Fibreblast traction as a Mechanism for collagen morphogenesis. Nature, 290, 249-251.

Lara Ochoa, F. & Murray, J.D. (1983). A non-linear analysis for spatial structure in a reaction diffusion model. Bull.Math.Biol. 45, 917-930.

Matkowsky, B.J. (1970). A simple non-linear dynamic stability problem. Bull. Amer. Math. Soc., 76, 620-625.

Meinhardt, H. (1982). Models of biological pattern formation. Academic Press, London.

Murray, J.D. (1977). Lectures on non-linear differential equation models in biology. Oxford University Press: Oxford.

Murray, J.D. (1981). On pattern formation mechanisms for Lepidopteran wing patterns and mammalian coat markings. Phil. Trans. Roy. Soc. Lond. B295, 473-496.

Murray, J.D. & Oster, G.F. (1984). Generation of biological pattern and form. I.M.A.J. Maths. Appl. to Biol. & Med., i,

Oster, G.F., Murray J.D. & Harris, A.K. (1983). Mechanical aspects of mesenchymal morphogenesis. J. Embryol. Exp. Morph. 78, 83-125.

Purcell, E.M. (1977). Life at low Reynolds number. Amer. J. of Phys., 45, 3-11.

Rawles, M. (1963). Tissue interactions in scale and feather development as studied in dermal-epidermal recombinations. J. Embryol. Exp. Morph., ll, 765-789.

Saunders, J.W. & Gasseling, M.T. (1968). Ectodermal-mesenchymal inter- actions in the origin of limb symmetry. In epithelial-mesenchymal interactions (ed. R. Fleischmajer & R.E. Billingham) , pp.78-97.

Smith, J.C., Tickle, C. & Wolpert, L. (1978). Attenuation of positional signalling in the chick limb by high doses of y-radiation. Nature, 272, 612-613.

Smith, J.C. & Wolpert, L. (1981). Pattern formation along the antero- posterior axis of the chick wing: the increase in width following a

269

polarizing region graft and the effect of X-irradiation. J. Embryol. Exp. Morph., 63, 127-144.

Stuart, E.S. & Moscona, A.A. (1967). Embryonic morphogenesis: Role of fibrous lattice in the development of feathers and feather patterns. Science, 157, 947-948.

Thomas, D. (1975). In analysis and control of immobilized enzyme systems, (Thomas, D. & Kernevez, J.P. eds.) New York: Springer-Verlag, pp. 115-150.

Tickle, C. (1981). The number of polarizing region cells required to specify additional digits in the developing chick wing. Nature, 289, 295-298.

Trinkaus, J.P. (1984). Cells into organs: The forces that shape the embryo. (Prentice-Hall).

Turing, M.A. (1952). The chemical basis of morphogenesis. Phil. Trans. Roy. Soc. Lond., B237, 37-73.

Wessells, N.K. (1965). Morphology and proliferation during early feather development. Dev. Biol., 12, 131-153.

Wolpert, L. (1969). Positional information and the spatial pattern of cellular differentiation. J. Theor. Biol., 25, 1-47.

Wolpert, L. & Hornbruch, A. (1981). Positional signalling along the anteroposterior axis of the chick wing. The effect of multiple polarizing region grafts. J. Embryo. Exp. Morph., 63, 145-159.

ALMOST SURELY NON-LINEAR SOLUTIONS OF

STOCHASTIC LINEAR DELAY EQUATIONS

S.E.A. Mohammed

§ 1. INTRODUCT ION

In this note we give a simple example of a linear stochastic

delay differential equation whose solution field has a version which

is almost surely a non-linear function of the initial state.

§2. THE STOCHASTIC LINEAR DELAY EQUATION

Our objective is to study the dependence of solutions of the

one-dimensional stochastic delay differential equation (SDDE):

dx(t) = x(t-r)dw(t) t > 0 ] (i)

x(t) = ~(t) t E I-r,0] ] upon the continuous initial path n:[-r,0] ÷ I~. The lag r -> 0 and the

SDDE (i) is driven by white noise viz. Brownian motion w on a complete +

filtered probability space (~,F, (Ft)t_>0,P) • Indeed w:~ x ~ ÷ I~ is

a measurable process such that w(0,.) = 0 a.s., w(t,-) is Ft-measur-

able for all t >- 0 and the system {w(t,-): t >- 0} is Gaussian with

Ew(t,-) = 0, Ew(t,-)w(s,-) = rain (t,s) (2)

for all t,s >- 0 (McKean [6], Hida [43). The differentials dx(t) and

dw(t) in (i) are defined through the Ito stochastic integral

x ( t ) = x ( 0 ) + x ( u - r ) d w ( u ) , t -> 0 (3) o

D e n o t e b y C t h e B a n a e h s p a c e o f a l l c o n t i n u o u s p a t h s n : [ - r , 0 ] -*-N

f u r n i s h e d w i t h t h e s u p r e m u m n o r m

i ln i l c = sup { l n ( s ) i: s ~ E - r , O ] } .

Let B(C) stand for the Borel o-algebra of C.

Note that when r = 0, (I) becomes the ordinary linear stochastic

differential equation

271

dx(t) = x(t)dw(t) t > 0

x(0) = v ~ I~

(4)

with the unique solution

Vx(t) = v e w(t)-½t t >- 0 (5)

It is clear that the right hand side of (5) gives a measurable ver-

sion X:~ + × ~ x i2 ÷ ~ of the solution field {Vx(t) :t ~ 0, v ( ~} with

the property that X is a.s. linear in the initial data v i.e. for

each t a 0 and a.a. e c ~ the map

X(t,-,~):~ -- >

v -- > X(t,v,~) ~ ve w(t)-½t

is linear.

On the other hand, if the lag r is strictly positive, a solution

of (i) in closed-form is not available to us. However, a unique

(Ft)ta0-adapted solution ~x of (i) through n ~ C may be obtained via

successive Ito integration over steps of length r, viz.

x(t) = j(0) + n(u-r)dw(u), 0 s t s r, o ;t

x(t) = x(r) + x(u-r)dw(u), r ~ t ~ 2r, r

etc.

§3. POSITIVE DELAY

For positive lag (r > 0), we prove that the solution field

{nx(t):0 < t s r, ~ c C] has a measurable version which is a.s. a

non-linear function of the initial path ~ c C. The proof draws upon

well-known results on sample function behaviour of Gaussian fields

parametrized by function spaces (Dudley [2], [3]).

Theorem:

Suppose the delay r > 0. Then the solution field {~x(t):0 -< t-<r,

n e C} of the SDDE (i) has a ($[0,r] @ ~(C) @ F, B~))-measurable

version X:[0,r] × C x ~ + IR so that for each t c (0,r] and a.a. ~ c ~,

the map X(t,',m):C + l~ is non-linear. The Borel G-algebras of [0,r]

and I~ are denoted by B[0,r] and BaR) respectively.

272

Proof:

The proof breaks up into three steps.

Step 1 :

First we show that for each t ~ (0,r] and any measurable version

X:[0,r] x C x ~ ~ IR of the solution field there is a set ~t e F of

full P-measure such that, for all w c ~ X(t,.,~):C + lR is not locally

bounded. For simplicity - and without loss of generality - take

t = r = i. Define the field Y:C x ~ + IR by

1 ? Y(~,-) = X(I,~,') ~ (0) = I n(u-l)dw(u) a.s.

J o

Since Brownian motion is a Gaussian system, then so is the field Y

(hida [43, pp. 31-34). For any e > 0 we need only show that the set

~ {~:~ ~ ~, sup [LY(~,~)I: n ~ c, JlnlLc

k~ is P-null. Define the sequence {~ }k=l in C by

k n (s) = e sin 2zks, s c [-i,0~,

for each integer k a 1. As w has independent increments and the co-

variance property (2), then for any two integers k I, k 2 e 1 we have

k I k 2 [i k I k 2 EY(~ ,-) Y(~ ,.) = J ~ (u-l)D (u-l)du

o

= ½e 2 6klk2

where 6klk2 is the Kronecker delta. Therefore the Gaussian family

{y( k,.): k a i} are mutually independent with mean zero and variance

Ok (k,.)[ 2

for all k a 1 (Hida [43, pp. 34-35).

Now Jlnkllc = e for each k a i, so ~e

measurable set

is a subset of the F-

{~: ~ ¢ ~, sup IY(Nk,~) ] < ~}.

k_>l

From the independence of the y(k,.), the latter event has probability

273

P[ ~ n {~: ~ ~ a, IY(nk,~)l < N}]

N>_I k>_l

K = lim lim

N+~ K+~ k= i P{w. ~ ~ n, ly(nk,w)[ < N}

K I l_!_ IN e -y2/2°2 dy = lim lim ~ -- qk -N

N~ K+~ k=l / 2~

N

limlim( eX2x)Kd o £

i [N/E _x 2 because -- 1 e dx < i for each integer N >- i.

S i n c e t h e p r o b a b i l i t y s p a c e (~,F,P) i s c o m p l e t e , t h e n ~e c F a n d

P(~e) = 0. To conclude the proof of our assertion set ~i = n

for the required set of full P-measure. n=l

Step 2:

Secondly, we show that the solution field of (i) has an (Ft)rata 0-

adapted and a (B([0,r]) ~ B(C) ® F, B~R))-measurable version

X;[0,r] x C x ~ ÷ ~. To see this, view the solution field as a con-

tinuous map

Z:[0,r] xC ÷ L°(S,IR)

(t,q) ~ > ~x(t)

(Mohammed [73, pp. 158-163). Since [0,r] × C is a complete separable

metric space, it follows from a general result of Cohn [13 and Hoffman-

J~rgensen [5] that Z has the required version X. Alternatively, the

integrand in the stochastic integral

f t

Z(t,q) - q(0) = ~(u-r)dw(u) o

is B(C)-measurable in n, so it follows from a result of Sticker and

Yor ([9]) that Z has a measurable version X:[0,r] x C x ~ ÷ ~ as

required.

Step 3:

Finally, suppose X is the measurable version of the solution

field obtained via Step 2. We contend that this version satisfies

the conclusion of the theorem. Fix 0 < t ~ r. Let

274

~L = {w: ~ ~ ~, X(t,-,~) is linear},

= {~: ~ E ~, X(t,-,~) is continuous} c

and ~t be defined as in Step i. Then clearly ~c c ~\~t and

P(~\~t ) = 0. By completeness of the probability space,

~c c F and P(~c ) = 0. According to Douady's theorem (Schwartz [8],

pp. 155-160), a Borel-measurable linear map C + ~ is continuous. So

~L c ~c' because X is measurable. Using completeness of the probabi-

lity space once more gives ~L e F and P(~L ) = 0. This finishes the

proof of the theorem. D

§4. DELAYED DIFFUSION WITH LINEAR DRIFT

Consider the linear SDDE

dx(t) = x(t)dt + x(t-r)dw(t)

x(t) = ~(t)

with a linear drift.

to } t e I-r,0]

Its solution field is given by

(6)

= ~(0)et + et f |t ~(u-r)e -u dw(u), 0 ~ t s r (7) nx(t) J o

where q e C. Using the last formula, an easy modification of the

proof of our theorem in §3 shows that the field {e-t[~x(t)]:0~t~r,~£C)

has a measurable version [0,r3 x C x ~ + ~ which is a.s. non-linear

in ~ for each t £ (0,r3 when r > 0. It therefore follows that, for

positive delay r, the solution field of (6) has the same property

also.

§5. ACKNOWLEDGEMENTS

The author wishes to thank Peter Baxendale and David Elworthy

for helpful suggestions and stimulating conversations, and also to

thank Terri Moss for the typing.

REFERENCES

[i] Cohn, D.L., Measurable Choice of Limit Points and the Existence

of Separable and Measurable Processes, Z. Wahr. verw. Geb. 22

(1972), 161-165.

275

[2] Dudley, R.M., The Sizes of Compact Subsets of Hilbert Space

and Continuity of Gaussian Processes, J. Functional Analysis 1

(1967), 290-330.

[33 Dudley, R.M., Sample Functions of the Gaussian Process, Ann.

Prob. 1 (1973), 66-103.

[4] Hida, T., Brownian Motion, Springer-Verlag, New York-Heidelberg-

Berlin (1980).

[5] Hoffmann-J~rgensen, J., Existence of Measurable Modifications

of Stochastic Processes, Z. Wahr. verw. Geb. 25 (1973), 205-207.

E6~ McKean, H.P., Stochastic Integrals, Academic Press, New York

(1969).

[7] Mohammed, S.E.A., Stochastic Functional Differential Equations,

Research Notes in Mathematics,Pitman Publishing Ltd., London

(1984).

[8] Schwartz, L., Radon Measures on Arbitrary Topological Spaces

and Cylindrical Measures, Tata Institute of Fundamental Research,

Oxford University Press (1973).

[93 Stricker, C. and Yor, M., Calcul Stochastique d~pendant d'un

param~tre, Z. Wahr. verw. Geb. 45 (1978), i09-133.

PROPERTIES OF THE SET OF GLOBAL SOLUTIONS FOR TIlE CAUCHY

PROBLEMS IN A LOCALLY CONVEX TOPOLOGICAL VECTOR SPACE

Pedro Morales *

i. Introduction

In this note we present first a generalization of Lemma 2.2 of [5] for a Fr6chet • , ~ w .

space. Using the modified version of the Phillips integral [12] given by Millionsclkov

[i0], we are able to describe the topological properties of the set of global solu-

tions of the Cauchy Problem (CP) y'=f(t, y), Y(t0)=x 0 where J is an interval not

necessarily bounded of the real line containing to, X is a complete Hausdorff

locally convex topological vector space, x0cX and f is a function from J x X

to X. In the case where J=[O, a] with a>O, X is a Fr6chet space and f satisfies

the Carath6odory conditions, we show then that the set of global solutions of (CP)

is an R6-set. This permits us unify several results due to Knight [8], Phillips [12],

Pulvirenti [13] and Szufla ([16], [17]). In the general case, with J=[0, ~), we

show that the Hukuhara-Kneser property holds for (CP) under the hypotheses, for the

existence of a global solution, established by Millmonscikov [i0].

2. Preliminaries

Consider the class M of all metrizable spaces. A topological space S is

called an absolute retract for metrizable spaces, in symbols SEAR(M), if S~I and,

for each closed subset F of a metrizable space M, every continuous function h;F÷S

admits a continuous extension h:M÷S [2 , p. 87]. It follows from [6 , Theorem 4.1,

p. 357] that, if S is a convex subset of a locally convex topological vector space,

then SeAR(H). A topological space S is called an absolute retract, in symbols

SEAR, if S is compact and SEAR(H) [2, p. I00]. It follows from [2 , property 2.1,

p. i01] that, if SEAR and h:S+S ' is a homeomorphism, then S'cAR. A topological

space is called an R6-set if SeM and S is homeomorphic to the intersection of

a decreasing sequence in AR[I]. One of the most important properties of this kind

of spaces is the following: Every R6-set is aoyclic [9, p. ii0]. In particular,

if S is an R6-set , then S is non-empty, compact and connected.

Let J and X be topological spaces. The set of all continuous functions from

J to X will be denoted by C(J, X). If X is a uniform space, the symbols ~u' Tc

denote the topology of uniform convergence, the topology of uniform convergence on

compacta, respectively, on every non-empty subset of C(J, X). If X is a Hausdorff

topological vector space, the symbol Cb(J , X) denotes the vector space of all bounded

continuous functions from J to X. In this case, it is well-known that (C(J, X),~ c)

~resp. Cb(J, X), Tu) ) is a Hausdorff topological vector space whose topology has

(*) This research was partially supported by a grant from the Natural Sciences and Engineering Council of Canada•

277

as local base the collection of all sets of the form N(K, U)={y~C(J, X): y [K] ~ U}

(reap. N(U)={YCCb(J, X): y[J] ! U}), where K is a compact subset of J and U is

a neighbourhood of 0 in X.

If x is an element of a metric space and ~>0, the symbol B (x)(resp. B (x))

denotes the open ball (reap. the closed ball) of center x and radius ~.

The following Lepta generalizes Lemma 2.2 of [5]:

2.1 Lemma. Let J be a bounded and convex subset of a normed space with norm If° If,

let X be a Fr~chet space, let Y=(Cb(J ~ X), T u) and let F:Y÷Y. Assume that F

satisfies the following conditions:

(i) F is continuous.

(ii) There exist t0~J and x0cX such that F(y)(t0)=x 0 for all y~Y.

(iii) For every ~>0, the relations YI' Y2 ~Y and yllJ = y21J imply

F(Yl) IJ c = F(Y2)IJ , where Jc = JnB (t0).

(iv) For every neighbourhood U of 0 in X, there exists ~>0 such that

s, t~J and IIs-tll<~ imply F(y)(s) - F(y)(t) ~ U for all y e Y.

(v) If (yn) is a sequence in Y such that Yn - F(Yn)+0' then (yn) has a

convergent subsequence.

Then the set Fix(F) of all fixed points of F is an R6-set.

Proof. Since X is a Fr~ch~;t space, Y is also a Frgchet space. Then its topology

can be induced by a complete invariant metric d such that the open balls Be(y) are

convex. Then, by Lemma 2.1 of [5], the set S=Fix(F) is non-empty and compact. More-

over, by Lemma 1 of [18], there exists a sequence (Fn) of continuous mappings from

Y to Y such that each I-F n is a homeomorphism of Y and lira Fn(Y)=F(y) unifor- n-~

mly in y~Y. Put T=I-F anti Tn=I-F n for all n=l, 2, 3, ... Then we may suppose

(by choosing a subsequence of (F n) if necessary) that d(Tn(Y)' T(Y))<I~n for all

y~Y and all n=l, 2, 3, ... Hence d(Tn(Y), 0) _< 1 for all yes and all n=l, 2,

3, ..., and therefore, for each n, the sets Tn[S] and Qn = co (THIS]) are con-

tained in the closed ball BI(0). Since Tn[S] is compact, it follows from theorem

n

3.25 of [14] that Qn is compact. So QnCAR. Since T n is a homeomorphism,

R=TnIEQ n] ~ AR.

It is clear that S c__ ~i for all n=l, 2, 3 .... So S c_ n~=l k~__n Rk=lim n inf R n.

Let now y~limnSU p R n = n~__~l i O=n R k. Then there exists a snbsequence, again denoted

(R n) for simplicity, such that y~R n for all n=l, 2, 3, ... Thus Tn(Y)~Qn and

therefore Tn(Y) c B1 (0). Since d(Tn(Y) , T(y)) < 1 , it follows that T(y) = 0.

n So y~S, and therefore S=lira R . Let: V be a neighbourhood of S. Suppose that

n n

there exists a positive integer k such that n>k implies R n _~ V. For every

278

n=k+l, k*2, k+3, ... choose WnCRn-V. Let Yn=Wn+k for all n=l, 2, 3, ... Since 1

d(T(Yn), 0) -< n÷k for all n=l, 2, 3, ..., it follows that yn-P(yn)+O. Then, by

the condition (v), there exist a subsequence of (yn), again denoted (yn) for sim-

plicity, and an element ycY such that yn+ y. So lira T(Yn)=T(y) , and therefore n-~Co

T(y)--0. Then y c S c_ V. This is a contradiction because Yn ~ V for all n=l, 2,

3, ... Consequently, for each neighbourhood V of S there exists a subsequence

of (Rn) contained in V. By Lemma 5 of E3], it follows that S is an R~-set.

2.2 Lemma. (See E4, Lemma 6]). Let J = [0, ~) be an interval of the real line

~, let I = E0, t o ] be a compact interval of IR, let X be a Hausdorff topological

vector space and let H: (C(I, X), ~u ) ÷ (C(J, X), ~c ) be the mapping defined by the

formula fy(t) if t E I

H(y) (t) = ~ LY(t0) if t c J\I.

If S c__ C(I, X) is connected, then HIs] is also connected.

Proof. It is easy to show that if ycC(l, X), then H(y) cC(J, X). It remains to

prove that H is continuous. Let Y0~C(I, X), let K be a compact subset of J

and let U be a neighbourhood of 0 in X. Let y~C(l, X) be such that ycy0*N(U).

Then (y0-y) (t)~U for all tel. Let tcKo But K=(Knl)u(K\I). If tal, then

(H(Yo) -H(y))(t) = (y0-Y)(t)cU. If t ~ I, then (H(Y0) -H(y))(t) = (y0-Y)(t0)~U.

So H(y) cH(y 0) +N(K, U) .

3. The Phillips-Million~ikov Intesral

Let X be a complete Hausdorff locally convex topological vector space and let

P be a separating family of semi-norms on X generating its topology. Let L be

the o-algebra of Lebesgue measurable sets in fl~, let ~ be the Lebesgue measure on

and let E E L. A function from E to X is countably-valued if it assummes

at most a countable set of values in X each on a Lebesgue measurable subset E i

of E.

We say that a function y:E-~X is Phlllips-Milllonsclkov intesrable if there

exists a net (Y~)~cD of functions from E to X with the following properties:

i) Each y~ is countably-valued.

2) For every p~P, Ip (y)-- ~. p(y (t~))~(E~)<+~, where E i~ are the subsets

of E on which y~IE i is constant and t i e E i.

3) For every pep and every ~>0, there exists ~0 e D such that ~', ~" e D

and ~'' ~" -> ~0 imply Ip (y~, - y ,,) < c.

f) For every ~>0 there exists E c L such that E c E, ~(E ) < E and E --

(Y~)~eD converges uniformly to y on E - Ec.

Since X is complete and Hausdorff, the properties i) - 3) imply that

~(E~)y~(t~) ~ X for all ~D and the corresponding net (~ ~(E~)y~(t~))~D i i

279

converges to a well-defined element I(y ) of X. Moreover, using the property 4)

it can be shown that, if (z) c D is another net of functions from E to X satis-

fying the conditions i) - 4), then I(y )=I(z ). By definition lim ~ ~(E~)y~(t~) i

is called the Ph1111ps-M1111onsclkov integral of y o_n_n E and it is denoted by

f E y(t) dt. We indicate some important properties of this integral:

f

i. If y: E÷X is integrable, then, for all P~P, r p(y(t))dt <+~ and J E

p(f y(t)dt)-< f p(y(t))dt. E E

2. Let y: E+X be integrable. If E = i~l Ei with E i { L and E i n E.= ~.

if i~j, then y(t)dt y(t)dt.

E -i E i

3. Let y: E÷X be integrable. Then, for every neighbourhood U of 0 in

X, there exists 6 > 0 such that E' { L, E' _c E and ~(E') < 6 imply ~ y(t)dt~U. J E'

a, b ~ ~, ay I + by 2 is 4. Let YI' Y2: E÷X be integrable. Then, for every

integrable and

f (aYl + bY2)(t)dt = a f Yl(t)dt + b f Y2(t)dt. E E E

5. Let E be compact and let y: E~X be a bounded function. If, for every

4>0, there exists E ~ L such that E • E, ~(E c) < ~ and y E\E is continuous, C £ -- C

t h e n y i s i n t e g r a b l e a n d y ( t ) d t ~ ~ (E) e o ( y [ E ] ) . E

6. I f y : E÷X i s i n t e g r a b l e and c o n t i n u o u s a t t O ~ E, t h e n

where ~(E')

lim I I y(t)dt = Y(t0), ~(E~+O ~(E') E'

t o c E '

denotes the diameter of E'.

Henceforth, when E is an interval of ~ of extremities a and b with

f a<b, the integral y(t)dt will be denoted by y(t) dt. E a

• v ~ . Finally we note that the Phillips-Milllonsclkov integral substantially generali-

zes, when X is a Banaeh space, the Bochner and the Pettis integral with respect

to the Lebesgue measure on ~.

4. Applications to the Cauch~ Problem

Let J be an interval of ~ non necessarily bounded, let t0~J , let X

a complete Hausdorff locally convex topological vector space, and let x0eX.

be

Consider the Cauchy Problem:

(CP) y' = f(t, y), y(t 0) = x 0

280

where f: JxX÷X.

Let I be a subinterval of J containing t o . We say that (CP) has a solution

on I if there exists a differentiable function y: I~X such that Y(t0)=x 0

and y'(t)=f(t, y(t)) for all t~I. Every solution on J of (CP) is called a

~lobal solution. The set of global solutions of (CP) will be denoted by Sol(f).

Assume that the function s~f(s, y(s)) is integrable on every compact interval

[to, t] of ~ for every ycC(J, X). (According to property 5 and Lemma 2 of [i0],

this is the case, in particular, if f is continuous). It is easy to show then

that (CP) is equivalent to the integral equation

(IE) y(t) = x 0 f(s, y(s))ds, t~J.

t O

Then Sol(f) is exactly the set of all fixed points of the integral operator

defined by the formula

(1) F ( y ) ( t ) = x 0 * f ( s , y ( s ) ) d s f o r a l l y e C(J , X)

t O

Cauchy Problem

(CPn_I) y'=f(t, y), Y(t0+n-l)=Yn_l(t0+n-i )

for all n=2, 3~ 4, ... and with Y(to)=X 0 if n=l.

Define the function y: J÷X by the formula

l Yl(t) if t c [0, t0+l]

y(t) lYn(t) if tc[t0en-i , t0+n] , n=2, 3, ...

It is clear that y is a global solution of (CP).

4.2 Theorem. Let J=[0, a] with a>0, let X be a Fr~chet space and assume

that the function f: JxX+X satisfies the following conditions:

and all t£J.

4.1 Lemma. (See E4] and El9]). Let J:E0, ~). If the Cauchy Problem (CP) has

a solution on every compact interval [0, a] of ~, then every solution of (CP)

on [0, t0+l] can be continued to a global solution.

Proof. We note first that J = n__Ol [0, t0Tn]. Let Yl be a solution of (CP) on

[0, t0*l]. By hypothesis the Cauchy Problem

(CP I) y'--f(t, y), Y(t0+l ) - Yl(t0+l)

has a solution Y2 on [0, t0+2]. Again, by hypothesis, the Cauchy Problem

(CP2) y'=f(t, y), Y(t0+2 ) = Y2(t0t2)

has a solution Y3 on [0, t0+3~. By induction we can construct a sequence (yn)

of X-valued functions such that each Yn is a solution on [0, t0+n] of the

281

i) For every xcX, f(,, x) is measurable.

2) For every tcJ, f(t, • ) is continuous.

3) f[J×X] is a compact subset of X.

Then the set of global solutions of (CP) is an R6-set in (C(J, X), Tu).

Proof. From Lemma 9.2 and the first paragraph of the proof of Theorem 9.5 of [12],

it follows that the function s*f(s, y(s)) is integrable on every compact subinter-

val of J for every ycC(J, X). Then we can define the integral operator

f t F ( y ) ( t ) = x 0 + f ( s , y ( s ) ) T s

t O

f o r a l l y~C(J , X) and t e J . From p r o p e r t y 3 o f t h e P h i l l i p s - M i l l i o n ~ i k o v i n t e g r a l ,

i t f o l l o w s t h a t F i s a mapp ing f rom C ( J , X) t o C ( J , X) w h i c h s a t i s f i e s t h e

c o n d i t i o n ( i v ) of Lemma 2 . 1 . The c o n d i t i o n s ( i i ) and ( i i i ) o f Lemma 2 . 1 a r e s a t i s -

f i e d t r i v i a l l y by P. A p p l y i n g t h e T h e o r e m s 6 . 1 and 6 .2 and u s i n g t h e a r g u m e n t s [12 ]

of t h e p r o o f o f Theorem 9 . 5 , i t can be shown t h a t F i s c o n t i n u o u s . To f i n i s h

t h e p r o o f , i t s u f f i c e s to show t h a t F s a t i s f i e s t h e c o n d i t i o n (v) o f Lemma 2 . 1 .

Le t (yn) be a s e q u e n c e i n ( C ( J , X), ~u ) s u c h t h a t Yn - F(Yn) ÷ 0. Le t t c J .

S i n c e f ( t , Y n ( t ) ) ~ f [ J x X ] and f [ J x X ] i s c o m p a c t , t h e r e e x i s t a s u b s e q u e n c e

(Ykn) o f (yn) and an e l e m e n t u ( t ) cX s u c h t h a t f ( t , Y k n ( t ) ) ~ u ( t ) . From Theorem

6 . 2 ° f t [ 1 2 ] r i t f o l l o w s th~tr u i s i n t e g r a b l e on e v e r y compac t s u b i n t e r v a l of J

and ] f ( s , y. ) ) d s ÷ [ u ( s ) d s = y ( t ) . So y~C(K, X) and F(y k ) ( t ) ÷ Xo*Y(t) ~n )t O n t o

for all t~J. But from the property 3 of the Phillips-Million~ikov integral it

follows that {F(y k ): n=l, 2, 3, ...} is equicontinuous. So (F(y k )) T -conver- n

ges to x0+Y [7,Theorem 15, p. 232]. Since Ykn - F(Ykn) ~ O, we getnthat (Ykn)

~u-C°nverges to x0*Y.

Henceforth, J will be the half-line [0, ~) of ~. Moreover, the symbol

denotes a continuous function from J×J to J such that, for every tcJ, the

function ~(t,- ) is increasing. We say that w is a ma~orant of f with respect

to p~P if p(f(t, x)) ~ ~(t, p(x)) for all (t, x) ~ J×X .

4.3 Theorem. Assume that the function

i) f is continuous

2) For every compact subset K of J, f[K×X] is contained in some compact

subset of X.

3) f has a majorant m with respect to peP.

4) For every r0>0 the real differential inequality.

6 e ~(t, u), u(t 0) - r 0

has a solution on J.

f: J×X÷X satisfies the following conditions:

282

Then the set of global solutions of (CP) is a non-empty, compact and connected

subset of (C(J, X), ~c)-

Proof. Let S=Sol(f). Accordin~ to Theorem 5 of [ ], S~. Let Y=(C(J, X), ~c ).

f t By Lemma 3 of [ 1 0 ] , t h e s e t { f ( s , y ( s ) ) d s : ycY} i s r e l a t i v e l y compac t and

t O

A ( y ) ( t ) = f ( s , y ( s ) ) d s

t o

i s a c o n t i n u o u s m a p p i n g f rom ¥ t o Y. So t h e i n t e g r a l o p e r a t o r d e f i n e d by (1) i s

a continuous mapping from Y to Y such that F[Y] is compact. Since

S=Fix(F) ! F[Y] and Fix(F) is closed and non-empty, S is compact.

To show that S is connected, consider two arbitrary non-empty closed sets S'

and S" such that S=S'uS". For every n=l, 2, 3, ... consider the non empty

subsets of Y: S~={ycY: (3u)(uES', YI[0, t0+n]=uI[0 , t0+n] and y(t)=u(t0+n ) if

t>t0+n)} and S~={yeY: (3v)(veS", yI[0, t0+n]=vl[0 , t0+n] and y(t)=v(t0+n ) if

t>t0+n)}. Put Snnn=S'uS". It follows from Lemma 4.1 that the set SnI[O , t0+n] =

{yI[0, t0+n]: yeS n} coincides with the set of solutions of (CP) on [0, t0+n].

By the hypotheses i) and 2) it is clear that integral operator

G ( y ) ( t ) = x 0 + f ( s , y ( s ) ) d s

t O

w h e r e y e t ( [ 0 , t 0 + n ] , X) and t e [ 0 , t 0 + n ] s a t i s f i e s t h e h y p o t h e s e s o f Theorem 2

of [ 1 8 ] . T h e r e f o r e S n l [ 0 , t 0 + n ] i s a c o n n e c t e d s e t a n d , by Lemma 2 . 2 , S n i s a l s o

c o n n e c t e d i n Y. S i n c e S ' and S" a r e c o m p a c t i t f o l l o w s t h a t S~ and S"n a r e

c l o s e d , s o S'n n S ~ . Le t Yn e S'n n S"n. Then t h e r e e x i s t u n e S ' and

VneS" s u c h t h a t U n l [ 0 , t0+n] = Ynl[0 , t0+n] ~ Vnl[0 , t0+n] , Y n ( t ) = U n ( t 0 + n ) =

Vn( t0+n) i f t > t 0 + n . S i n c e S ' i s compac t t h e r e e x i s t s a s u b n e t ( u ' ) _ o f

(Un)n~ N which converges to an element ueS'. Then there exists a function v_

N': D4N such that u -UN,(~ ) and, for every ne~, there exists ~0eD such that

~eD and a~ 0 imply N'(~)en. Let v'~ = VN,(~ ) for all ~eD. Then (v)~e D' is

a net in S" and, since S" is compact, there exists a subnet (v~)BeD, of

(v') _ which converges to an element veS". Then there exists a function

N": D'÷D such that v~=v~,,(B ) and, for every ~eD, there exists B0eD ' such that

BeD' and BeB 0 imply N"(~)e~. So vB-VN,N,,(B) and therefore (vB) BeD, is a

subnet of (Vn)n~ which converges to v. Put uB-UN,N,,(p). Then (u~)BeD, is

a subnet of (u')~ ~eD and therefore a subnet of (Un)n~ N. Since (u')~ ~eD converges

to u, it follows that (u~)BeD, converges to u. To finish the proof it suffices

to show that u=v. Let K be a compact subset of J and let U be a neighbourhood

of 0 in X. Choose a symmetric neighbourhood V of 0 in X such that V+VcU.

We can choose $~eD' such that ~eD' and ~eB~ imply UN,N,,(B)-ueN(K, V) and

VN,N,,($)-veN(K , V). Let neON such that K L [0, t0+n]. Choose B0eD" ' such that

283

~£D' and B~B~ imply N'N"(B) an. Let 80{D ' be such that ~0~B~ and 60_80> " .

Let t~k. 8o t~t0+n~to+N'N"(B0) and therefore UN,N,,( ~ )(t)=YN,N,,( B )(t) =

VN,N,,(BO ) (t) . Thus u(t)-v(t):(u(t)-UN,N,,(~0 ) (t))*(VN,N,,(B0) (t)-v(t)9~V+V~ U . O

Then u-v~N(K, U) and therefore u:v.

4.4 Remark. For X:R n, two interesting special cases of the existence part of

Theorem 4.3 were considered by Stokes [115] and Wintner [20].

REFERENCES

i. N. ARONSZAJN, Le correspondant topologique de l'unicit6 dans la th6orie des

6quations diff6rentielles, Ann. of Math. 43 (1942), 730-738.

2. K. BORSUK, Theory of Retracts, Polish Scientific Publishers, Warszawa (1967).

3. F.E. BROWDER and C.P. GUPTA, Topological Degree and Nonlinear Mappings of

Analytical type in Banach Spaces, J. Math. Anal. Appl. 26 (1969), 390-402.

4. A.I. BULGAKOV, Properties of Sets of Solutions of Differential Inclusions,

Differential Equations 12 (1977), 683-687.

5. J. DUBOIS and P. MORALES, On the Hukuhara-Kneser Property for some Cauchy

Problems in Locally Convex Topological Vector Spaces, Proc. 1982 Dundee Conf.

on Ordinary and Partial Differential Equations, Lect. Notes Math. 964, Springer-

Verlag, New York (1982), 162-170.

6. J. DUGUNDJI, An extension of Tietze's Theorem, Pacific J. Math. 1 (1951),

353-367.

7. J. KELLEY, General Topology, D. Van Nostrand Company, Inc. New York (1965).

8. W.J. KNIGHT, Solutions of Differential Equations in B-Spaces, Duke Math. J. 41

((1974), 437-442.

9. J.M. LASRY and R. ROBERT, Analyse non lin6aire multivoque, Cahier de Math. de

la d~cision No. 7611, Paris (1978).

i0. V.M. MILLIONSCIKOV, A contribution to the Theory of Differential Equations dx d--~ : f(x, t) in Locally Convex Spaces, Soviet Math. Dokl. 1 (1960), 288-291.

Ii. R.S. PALAIS, Critical Point Theory and the Minimax Principle, Proc. Sympos.

Pure Math., Vol. 15, Amer. Math. Soc., Providence (1970), 185-212.

12. R.S. PHILLIPS, Integration in a Convex Linear Topological Space, Trans. Amer.

Math. Soc. 47 (1940), 114-145.

13. G. PULVIRENTI, Equazioni Differenziali in uno spazio di Banach. Teorema di

esistenza e struttura del pennello delle soluzioni in ipotesi di Carath6odory,

Ann. Mat. Pura Appl. 56 (1961), 281-300.

284

14. W. RUDIN, Functional Analysis, McGraw-Hill Book Company, New York (1973).

15. A. STOKES, The application of a Fixed-Point Theorem to a variety of Nonlinear

Stability Problems, Proc. Nat. Acad. Sci. U.S.A. 45 (1959), 231-235.

16. S. SZUFLA, Solution Sets of Nonlinear Equations, Bull. Acad. Polon. Sci., S~r.

Sci. Math. Astronom. Physics 21 (1973), 971-976.

17. S. SZUFLA, Some properties of the Solutions Set of Ordinary Differential Equa-

tions, Bull. Acad. Polon. Sci., Sgr. Sci. Math. Astronom. Phys. 22 (1974), 675-678.

18. S. SZUFLA, Sets of Fixed Points of Nonlinear Mappings in Function Spaces, Funkcial.

Ekvac. 22 (1979), 121-126.

19. AoA. TOLSTONOGOV, Comparison Theorems for Differential Inclusions in a Locally

Convex Space. I. Existence of Solutions, Differential Equations 17 (1981),

443-449.

20. A.A. TOLSTONOGOV, Comparison Theorems for Differential Inclusions in a Locally

Convex Space, II. Properties of Solutions, Differential Equations 17 (1981),

648-654.

21. A° WINTNER, The Non-Local Existence Problem of Ordinary Differential Equations,

Amer. J. Math. 67 (1945), 277-284.

BOUNDARY VALUE PROBLEMS FOR FORCED NONLINEAR EQUATIONS AT RESONANCE

Pierpaolo Omari and Fabio Zanolin

I . INTRODUCTION

Many boundary value problems (BVPs) fo r forced non l inear o rd inary d i f f e r e n t i a l

equations (as wel l as p a r t i a l d i f f e r e n t i a l equat ions) at resonance can be formulated

in an abs t rac t s e t t i n g as

(1.1) Lx = Nx + e,,

where L is a l i n e a r d i f f e r e n t i a l opera tor w i th n o n t r i v i a l kerne l , N is a non l inear

Nemytzki opera to r and e represents a fo rc ing term. In th i s paper, using Coinciden-

ce Degree theory [7 ], an existence theorem f o r equation (1.1) is proved in the case

N =AG (A l i n e a r and G non l i nea r ) . Then, app l i ca t i ons are given to the pe r iod i c

BVP associated to o rd inary d i f f e r e n t i a l systems of the type

(1.2) Lx = Agox + e ( t ) .

In th i s way, previous resu l t s f o r ( 1 .2 ) , ( [ 8 ] , [ 9 ] ) , dea l ing w i th L ( f o rma l l y )

symmetric or skew-symmet r i c , are extended to more general kinds of d i f f e r e n t i a l ope-

ra to rs .

2. AN ABSTRACT EXISTENCE RESULT

Let X and Z be two real Banach spaces w i th norms l - I x

ve ly . We deal w i th the problem of the ex is tence of a so lu t i on

equat ion in Z,

(2.1) Lx = AGx + e.

Henceforth, the f o l l ow ing assumptions are considered.

( i ) L: domL c X -÷ Z is a l~inear Fredholm mapping of index zero [ 7, Ch . l , ~ I ] .

For any pa i r (P,Q) of continuous p ro jec to rs P: X ÷ X, Q: Z + Z, such tha t ImP =

k e r L , kerQ = ImL (so tha t X = kerL m k e r P , Z = ImL m ImQ ) , we denote by

Kp: ImL ÷ domL r~ kerP the (a lgeb ra i c ) inverse of the r e s t r i c t i o n of L to domL

n k e r P . Then, the genera l ized inverse of L is def ined by K = Kp,Q:= Kp(l - Q):

Z ÷ domL n k e r P . Moreover, l e t A: ImQ ÷ kerL be any l i n e a r isomorphism.

and l . l z , r espec t i -

x ~ X to the abs t rac t

286

(~3L) A: Z -+ Z is a cont inuous l i n e a r mapping such t h a t AQ = QA and

ImQ ÷ ImQ is a ( l i n e a r ) isomorphism.

Let us suppose a lso t h a t KA : Z ÷ domL n ke rP is con t inuous .

AIIm Q

(~LL) G: X -÷ Z is a ( p o s s i b l y n o n l i n e a r ) mapping such t h a t A G : X + Z is L -

comp le te l y cont inuous ( [ 7, C h . l , §4 ] ) .

(iv) e e Im L .

Let us assume there e x i s t s a rea l H i l b e r t space Y, w i th i nne r p roduc t ( ° , - )

norm I ' [ y = ( ' , ' ) ½ , such t h a t X c Y c Z, a l g e b r a i c a l l y and t o p o l o g i c a l l y , w i t h and

Y dense in Z. Fu r the r , l e t a (con t inuous) b i l i n e a r p a i r i n g < . , . >: X × Z ÷ ~ be

de f ined such t h a t :

<x,y> = ( x , y ) , f o r a l l x E X, y • Y ,

I<×,=>l < Ixlx. I=1 z , f o r a l l x c X, z ~ Z ;

in t h i s s i t u a t i o n , the t r i p l e (Z,Y,X) is sa id to be in normal p o s i t i o n ( [ I ] ) .

We a lso suppose

(v) < u ,v> = 0, f o r a l l u ~ ke rL , v ~ ImL .

Now we d e f i n e , f o r s = + I ,

2 s := max { 0 , sup s( KAv , v ) / i V i y

O#vEY•ImL

observe that, whenever KA = A K,

2 } s = max ( 0 , sup s( Lw , Aw)/[LwIy

0 #Lw~Y

Anyhow, maX{ms : s =_+I } < " I "IIKAIyII

Le t , f o r each k, r c~R , +

W(k,r ) := { x ~ domL : i L x i z

F i n a l l y , f o r e ~ ImL , we set

d(e) := min Iu + Ke Ix . u e ker L

, w i th K A Iy

} ;

(w e domL n k e r P ) .

: Y + Y .

> r } k, QGx = 0, .IPXlx I

In the s ta tement o f the f o l l o w i n g theorem, we use the conven t ions : IzI = +~

z • z \ Y, 0.~ = 0, and i nf 0 = + ~

f o r

THEOREM I . Let ( i ) , ( i i ) , ( ~ ) , (iv) and (v) ho ld . Moreover, l e t us suppose:

(HI) t he re are a, b, c EIR such t h a t , f o r some s c { - 1 , I } , +

( j s ) s ( x , G x ) i> aImxI~ + b l G x l z - c, f o r every x ~ d o m L ,

287

ho lds , w i t h a > ~ and b > d ( e ) ; S

(Hf) f o r each k C ~R l i m i n f IGxl = + ~ - + ' Z '

r++~ xeW(k,r)

(H3) t h e r e i s r > 0 such t h a t Q Gu # 0 f o r e a c h

k e r L , 0 )

u e k e r L ,

l U l z > r , and dB (A q G , B ( O , ~ ) ~ O,

w h e r e d B i s t h e B r o u w e r d e g r e e .

Then the e q u a t i o n (2 .1 ) has a t l e a s t one s o l u t i o n x ~ d o m L .

We r e f e r to {HI) as a growth r e s t r i c t i o n and to (H2) as a s ign c o n d i t i o n .

P roo f : We app l y the Mawhin 's G e n e r a l i z e d C o n t i n u a t i o n Theorem [ 7, I V . 1 3 ] .

A c c o r d i n g l y , we are l o o k i n g f o r a cons tan t R > 0 such t h a t

(2 .2 ) Lx # XNx, f o r ( x ,~ ) • domL ~ f r ~ x ] 0 , I [ ,

where Nx := A G x + e and ~ = B(O,R) ( t he open ba l l in X, cen te red a t O , w i t h

rad ius R),

(2 .3 ) QNx # O, f o r x d k e r L n f r ~ ,

( 2 .4 ) d B ( J Q N , ~ n k e r L , O) # O,

w i t h J: ImQ ÷ k e r L any l i n e a r isomorph ism.

F i r s t o f a l l , we observe t h a t (by the assumpt ions on the spaces X, Y, Z) the

b i l i n e a r system (X,Z) i s a r i g h t dual system [ 5, §15] and t h e r e f o r e a ( c a n o n i c a l ) +

l i n e a r embedding o f X i n t o Z* ( t he a l g e b r a i c dual o f Z) i s d e f i n e d by x ~ x ,

x+ (z ) = < x , z ) , f o r eve ry z ~ Z. I f we c o n s i d e r a t the same t ime the canon i ca l em-

bedding o f Z i n t o i t s a l g e b r a i c b idua l Z * * , g i ven by z F-~z ++ z++(z * ) = z * ( z )

++(x + ) f o r eve ry z * • Z* , and r e c a l l t h a t X • Z, we have t h a t z = <x,z> , f o r each

x E X and z • Z. Then, by c o n s i d e r i n g the o p e r a t o r K: Z ~ X c Z* , we have t h a t

the l i n e a r con juga te K* o f K is d e f i n e d [ 5, §17 ] , K*: Z** ÷ Z, and

(2 .5 ) (K*w) (v ) = w(Kv) = <Kv,w) , f o r each v , w • Z.

Now, l e t us suppose t h a t ( x , ~ ) • domL x ] 0 , I [ is a s o l u t i o n o f ( 2 . 2 ) , i . e . ,

( 2 .6 ) L x = XAGx + ~e .

P r o j e c t i n g e q u a t i o n (2 .6 ) onto I m L , by Q, we ge t , f o r e • ImL = ke rQ and ( i i } ,

( 2 .7 ) X Q A G x =XAQGx = 0 = QGx .

T h e r e f o r e , Gx • I m L .

We app l y to both the s ides o f (2 .6 ) the l i n e a r f u n c t i o n a l sK*(Gx) and, r e c a l l i n g

( 2 . 5 ) , we o b t a i n

s< K Lx,Gx> = ~s< KAGx,Gx> + ~s<Ke ,Gx> = ~s<KAGx,Gx) + ~s< u + Ke,Gx> ,

f o r each u ~ k e r L (by (v] and ( 2 . 7 ) ) .

288

Since KLx = x - Px, and < Px, Gx > = 0 (by (v) and ( 2 . 7 ) ) , using the d e f i n i -

t i on of d(e) , we have

(2.8) s<x ,Gx> < ~s<KAGx,Gx ) + d(e) IGxlz .

Two s i tua t i ons fo r (HI) - (js) are now examined ( fo r a f i xed s = + I ).

I. Let a = 0 in ( j s ) . Then ~s = 0 and so, s (KAv,v ) = s<KAv,v > < 0, fo r

each v e y n ImL. As Y is dense in Z, by the con t i nu i t y of KA and of the b i -

l i n e a r pa i r i ng , we have that the i nequa l i t y s (KAv,v > < 0 holds fo r each v ~ ImL.

Therefore, from (2 .8) , using ( js) , with a = 0, and I > 0 , we get

b l G x l z - c -<< d(e)Imxl z .

Thus a bound (independent of x and ~) to the so lu t ions of (2.2) is obtained:

(2.9) IGxlz < c i : = c/( b - d(e) ) .

2. Let a > 0 in ( js). Then, fo r any ( f i xed) x E X, we get alGxl~ ~<s< x,Gx )

+ c <+oo and hence G: X+Y, i . e . ImG c y and (by (2.7)) Gx c y n ImL. blGxl z

From (2 .8) , using (js) and the d e f i n i t i o n of ~s we have ( r e c a l l i n g also i E ]0 , I [ )

alGxl~ + blGxlz - c < ~slGXl 2 + d(e) IGxlz .

Hence (2.9) is proved again, because a ~>m S

Let us apply the operator K to both the sides of (2.6) . By (2.9) and the con-

t i n u i t y of KA, we get

= ~< II KAll IGXlz + IKel ( 2 . 1 0 ) Ix - Px I× I K L x I x ~< I K A G x I x + I K e l x x

~<c2:= II KAII .c I + IKeIx -

On the other hand, from equation (2.6) and the con t i nu i t y of A, we get

(2.11) ILx lz ~< JIAl r lGxlz + le lz <c3 := I IA l r . c I + le lz

Then (2.11) , (2 .7) , (2.9) and the hypothesis (H2) (wi th k = c 3 ) , provide the ex i -

stence of an r I > 0 such that x ~{ W(c3,r I) fo r each (x,~) ~ domL × ] 0 , I [ ,

so lu t ion of (2 .6) , that is

d r (2.12) IPxlx I

F i n a l l y , by (2.10) and (2.12) , a bound fo r Ixl is obtained: X

d R : = r + c 2 . (2.13) I x l x I I

Then x ~ domL n f rB(O,R) , fo r every R> R I , and

At l a s t , (2.3) and (2.4) fo l l ow d i r e c t l y from (H3).

(2.2) is proved.

Namely, QNx = QAGx = AQGx=O,

) - I , JQN = AQG. i f and only i f QGx = O, and, fo r the l i n e a r isomorphism J:= A(AII mQ

Then the above quoted Mawhin's Theorem may be appl ied taking ~= B(O,r) , r > r , R I .

289

We not ice tha t the est imate (2.13) holds fo r ~ = I too. Hence (by the L - c o m p l e t e

con t i nu i t y of N), the so lu t i on set of (2.1) is compact in ×.

3. REMARKS AND COROLLARIES

From the proof of Theorem I i t is eas i l y seen tha t the growth r e s t r i c t i o n (HI)

can be subs t i t u ted by the f o l l ow ing one %

(HI) there ex i s t s a p o s i t i v e constant c such t h a t , f o r some s c { - I , I } , I

(js) s ( x , G x ) > aslGXl + d(e) lGXlz

holds, f o r every x EdomL such tha t GxE ImL and I G X l z > C I Q

%

Let us observe tha t (HI) is s a t i s f i e d by any bounded map, G.

Assumptions of the type (js) were considered in [ 2 ] , •6], [ 3 ] , [ 4 ] w i th ap-

p l i c a t i o n s to boundary value problems f o r ODEs and PDEs. The main fea tu re of our

r esu l t cons is ts in a l l ow ing a > la in the growth r e s t r i c t i o n (HI), instead of the S

more c lass i ca l s t r i c t i n e q u a l i t y . Many var ian ts to Theorem I can be obtained by su i -

tab le (mi ld ) mod i f i ca t i ons both of the growth r e s t r i c t i o n (HI) and of the sign con-

d i t i o n (H2). In p a r t i c u l a r , we can take, in place of the constant c in ( js) , a

func t ion c ( x ) , c: domL ÷ ]R . Then, s t renghten ing (H2) and assuming an appropr ia - +

te quasiboundedness hypothesis on c, a r esu l t cover ing some s i t ua t i ons examined in

[2] can be obtained ( s e e [ 8 ] , [ 9 ] ) . Concerning the sign and the degree assumptions,

a less awkward cond i t i on ( s u f f i c i e n t f o r (H2)- (H3)) can be given:

(H4) f o r each k E R , l i m i n f <Ux ,Gx > +

r + +~ x • W(k,r)

where U: X + × is a continuous linear mapping such that

kerL ÷ kerL is a (linear) isomorphism.

COROLLARY I . Let ( i ) , ( i i ) , ( i i i ) , (iv) and (v) hold.

= + c o

UP = PU and U l k e r L :

Moreover, l e t us suppose

(HI)- (js) and (H4}. Then the equat ion (2.1) has at l eas t one so lu t i on x E domL.

Proof: We f o l l o w the proof of Theorem I as f a r as step (2 .11) . Then, by (H4)

and the same argument used in [ 9 , Th. I } , we get (2.13) and (H3). Hence the thes is

is reached.

We po in t out a lso tha t in the app l i ca t i ons some geomet r i ca l l y meaningful assumptions,

ensuring both (H2) and (H3), can be der ived (see, f o r ins tance, [ 1 2 ] , [ 8 ] ).

F i n a l l y , we observe tha t a uniqueness cond i t i on can be produced too. Indeed, we have

PROPOSITION I . Let us assume ( i ) , ( i i ) and (v), and suppose

(H5) f o r some s = _+I , s < x I - x 2 , G x I - Gx 2) > e s IGxl - Gx 2 Iy , 2

290

holds f o r every x I , x 2 E domL , w i t h x I # x 2.

Then the equat ion (2 .1) has a t most one s o l u t i o n in domL.

The p roo f i s ob ta ined by c o n t r a d i c t i o n on the equa t ion L(x I

us ing the same es t ima tes as in Theorem I and observ ing t h a t (by

ve (see, e .g . [ 9 ] ) .

At l a s t , we remark t h a t an analogue o f Theorem 1 can be s ta ted f o r the a b s t r a c t

L i~nard equa t ion

(3 .1) LX = FX + Gx + e,

w i t h L, G and e as in Sect ion 2 and F: X + ImL , L - c o m p l e t e l y con t inuous .

This ex i s tence r e s u l t , when is a p p l i e d to the p e r i o d i c BVP assoc ia ted to a m - d i m e n -

s iona l L i~nard system, y i e l d s a (sharp) ex tens ion o f a p rev ious theorem by Re iss ig

[ 11 ] , concern ing the s c a l a r case. The i n t e r e s t e d reader is r e f e r r e d to [ 10 ] f o r

the p roo f .

- x 2) = A(Gx I - Gx2),

( H S ) ) G i s i n j e c t i -

4. APPLICATIONS

We present now some applications of Theorem I to differentia] equations. Al-

though we restrict ourselves to the periodic BVP for ODEs, we point out that our ab-

stract r e s u l t a lso a p p l i e s to PDEs ( f o l l o w i n g , f o r i n s tance , [ 2 ] ) .

Let C be the rea l vec to r space o f the cont inuous and p - p e r i o d i c (p > O) P

x: A ÷ A m . For x E C , l e t . . I x l f unc t i ons P

l e t IXlq = ( foP I x ( t ) l q a t ] l / q , where I '1

ced by the inner p roduc t ( ' 1 " ) . We denote by

= suP t I x ( t ) I and, f o r I ~ q < ~,

i s the euc l i dean norm in A m indu-

L q ( I ~ q < ~ ) the comple t ion P

o f (C , I ' l ) and se t L q = { x E L q : f~ x = 0 } . As in the p rev ious s e c t i o n s , P q P , # P

R is the se t o f the n o n - n e g a t i v e r e a l s . F u r t h e r , we put w := 2~/p. +

Let us cons ide r the Banach spaces X:= (Cp , I ' I ~ ) , Y:= ( L2p , I ' 1 2 ) ' Z : =

(L ! P , 1.11 ) which are in normal p o s i t i o n w i t h respec t to the b i l i n e a r p a i r i n g

( u , v ) = fO p ( u ( t ) l v ( t ) ) d t .

As a f i r s t example, we deal w i t h the s c a l a r (m = 1) Bu f f i ng equa t ion

(4 .1) x" + c x ' + g ( t , x ) = h ( t ) , c e A .

We set Lx := - x" - c x ' , dotaL := { x E C : x is o f c lass C 1, w i t h P L 1

(AC = a b s o l u t e l y con t i nuous ) , ke rL = ]~, ImL = ; Az := z ; P , #

w i t h g: [ 0 , p ] × A ÷JR s a t i s f y i n g the Caratheodory assumpt ions

x ' AC }

Gx := g ( o , x ) ,

[ 7 , Ch . I , §4]

291

and extended to ~2 by p - p e r i o d i c i t y on the f i r s t v a r i a b l e ; e : = - h ( . ) , w i th

h E L I P , #

I t is rou t ine to check tha t the hypotheses ( i ) - (w) are s a t i s f i e d . The constants

C~s (s = -+ I ) , and IJKA iyll are eas i l y computed by Four ier expansion and standard

i n e q u a l i t i e s , so tha t we have

= O, ~ I = I / ( W 2 + c 2 ) ' IIKAIyII = I /~(~2 + c2)~ CL_ I

Estimates f o r d(e) can be eas i l y der ived too; f o r ins tance, d(e) ~< ( p / 1 2 ) l h l l

Then Theorem I y i e l ds the f o l l ow ing existence r e s u l t :

Let us suppose there is a number d > 0 such t h a t , f o r each Ix I > d and fo r

a.e. t e [ O , p ] ,

e i t h e r (k] g ( t , x ) . x i> O, and (#k) ] g ( t , x ) l ~< ( 2 + c2 ) i x l _ B ( i x l ) '

(w i th B: R ÷~ continuous and such tha t l i m B(r) = + ~ ) , +

or (k ') g ( t , x ) ' x ~< O,

holds. Then equat ion (4.1) has at l eas t one p - p e r i o d i c so lu t i on ( in the genera l ized

sense [7 , Ch. V l ] ).

Indeed, i f (k) and (kk) are assumed, then, fo r a = m and fo r a f i xed I 2

b > d (e ) , g ( t , x ) . x = E g ( t , x ) l . l x E >~ ( 2 + c 2 ) - 1 1 g ( t , x ) [ + (~ + c 2 ) - I B ( i x l ) .

~> m11g( t ,x ) I 2 + b l g ( t , x ) l , holds, f o r a.e. t and every x Ig(t,x)I wi th I x l >

max {d,sup#-1(b(m2+c2))} . Whereas, i f (k ' ) is considered, then, fo r a = m = 0 and - I

f o r a f i xed b > d ( e ) , - g ( t , x ) . x = I x ] . I g ( t , x ) I i> 0 + b l g ( t , x ) I holds, f o r a.e.

t and every x w i th Ixl > max{ d , b } . In both cases, (HI) is f u l f i l l e d , using

the Carath#odory hypotheses on g ( p r e c i s e l y , f o r IxI-<k, Ig(t,x)I ~< Y k ( t ) , Yk e Llp)•

The sign cond i t i on (if2) and the degree assumption (H3) fo l lows from (#) , or ( # ' ) ,

w i th s t r i c t i n e q u a l i t i e s . The general s i t u a t i o n is reached by standard pe r tu rba t i on

arguments.

A s i m i l a r framework can be employed fo r studying the sca la r Duff ing equation

w i th delay

(4.2) x II + c x ' + g ( x ( t - [ ) ) = h ( t ) , c , ~ ~ ~ ,

tak ing L and e as above, A z ( . ) = A z ( . ) := z ( - - T), and Gx := gox, w i th g: T

~R ÷ R cont inuous. For the constants m = ~ (T) , a uniform est imate ( independent S S

of T) can be eas i l y der ived:

s(T) ~< I I I K A IyII ~ I/~(~ 2 + c2) ~ , ( f o r each ~ e R, and s = + I ) I

Accord ing ly , apply ing Theorem I , we have

292

Le t us suppose t h e r e is a number d > 0 such t h a t , f o r some s = ± I and each

Ixl > d, sg(x).x >~ 0 and Ig(x) l ~< ~(w2 + c2)½1xl - m(Ix I) (with B: ~ +tR +

con t i nuous and such t h a t l i m ~ ( r ) = + ~ ) ho ld . Then equa t i on (4 .2 ) has a t I? - ) - + c o

l e a s t one p - p e r i o d i c s o l u t i o n (wha teve r ~ c R may be) .

In p a r t i c u l a r , the growth c o n d i t i o n on g i s s a t i s f i e d i f

l i m sup l g ( r ) [ / l r l < m(w 2 + c2) ½.

I ~ I ~ + ~

We p o i n t ou t t h a t t h i s assumpt ion i s sharp (a t l e a s t f o r a c lass o f equa t i ons c o n t a i n -

ing the l i n e a r ones ) , in o r d e r to ge t the e x i s t e n c e o f

f o r a l l de lays % e ~ . Namely, f o r any f i x e d c • A , - I

a r c t a n ( c / w ) , e x i s t s such t h a t the l i n e a r e q u a t i o n

x " ( t ) + c x ' ( t ) + v x ( t - ~) = c o s ( m t ) ,

p - p e r i o d i c s o l u t i o n s to (4 .2 )

a d e l a y ~ • A , m = m(c) =

V = co(m 2 + ~2)½,

has no p - p e r i o d i c s o l u t i o n . Indeed (Halanay a l t e r n a t i v e ) , the l i n e a r homogeneous

con juga te e q u a t i o n x " ( t ) - c x ' ( t ) + v x ( t + %) = 0, w i t h ~ = %(c) , admi ts x ( t ) =

cos(mt) as a p - p e r i o d i c s o l u t i o n .

As a f u r t h e r example , we c o n s i d e r the second o r d e r s c a l a r (m = I ) d i f f e r e n t i a l

e q u a t i o n

(4 .3 ) x" + k 2 2 w x + g (x ) = h ( t ) ,

w i t h k a ( f i x e d ) p o s i t i v e i n t e g e r .

We set Lx := - x" - k2w 2 x , domL := { x ~ C : x is o f c lass C I , w i t h x ' AC } , P

k e r L = sp { ~ , ~ } ~ ~2 , w i t h ~ ( t ) := cos ( k~ t ) and ~ ( t ) := s i n ( k m t ) , ImL =

{ x ~ L I : < ~ , x > = < ~ , x > = 0 ) ; Az := z . We assume g : ~ ÷ R is con t i nuous

P L 1 • h ( - ) + and h ~ . We d e f i n e Gx := gox - ( ~ , x > ~ < ~ , x ) ~ , e := - P

( ~ , x ) ~ ( . ) + ( ~, x > ~ ( . ) . Then the hypotheses ( i ) - (u) are s a t i s f i e d .

Through F o u r i e r expans ion , and s p e c t r a l p r o p e r t i e s o f the symmetr ic o p e r a t o r s L

and K, we have

~-1 = 1 / (2 k - 1)m 2, ~1 = 1 / (2 k + 1)m 2, II K a I y I[ = 1 / (2 k - 1)m 2.

Hence, by a p p l y i n g Theorem 1, we ge t

Let n , ~ : R ÷ ~ be two con t i nuous f u n c t i o n s such t h a t l im r 2 q ( r ) = + ~ +

and l i m c ( r ) = + ~ . Let us suppose t h a t , f o r some s = ± I ,

(Z4} s g ( x ) . x ~ q ( I x l ) and ( £ I ) l g ( x ) l ~ 12 ks + l l m 2 l x l - ~ ( l x l )

h o l d , f o r eve ry x c A. Then e q u a t i o n (4 .3 ) has a t l e a s t one p - p e r i o d i c s o l u t i o n

( c l a s s i c a l i f h i s c o n t i n u o u s ) .

293

Indeed, (HI} - ( js) fo l lows from (1 s) and (~JLs), wh i le (H2) and (H3) come from (1 s)

using the same argument as in the proof of Propos i t ion I in [ 9 ] .

The above statement re f ines a previous r e s u l t [ 9, Cor. I ] , where only the case

s = - I was considered in the growth r e s t r i c t i o n ( g ~ ) . s

As a f i n a l example, we deal w i th a class of hiQher order d i f f e r e n t i a l systems

(m > I ) of the type

(4.4) x (n) + B g ( t , x ) = h ( t ) , ( x (n) := dn /d t n ) , J ~

wi th n > I and B a m×m constant nonsingular ma t r i x whose transpose is denoted

by B T. Now we set L x : = - x (n ) , domL : = { x • C : x is of class C n-1 w i th P

(n - I ) A c x } , kerL = R m, ImL = L I ; Az := Bz ( . ) ; Gx := g ( - , x ) , w i th g: [ 0 , p ] x P,#

R m÷ R m s a t i s f y i n g the Caratheodory assumptions and extended to R m+1 by p - p e r i o d i -

c i t y on the f i r s t v a r i a b l e ; e := - h ( . ) , w i th h • L I P,#

Again, a l l the hypotheses ( i ) - (v) are v e r i f i e d and the f o l l o w i n g est imates fo r the

constants m , (s = _+I) S

~ 11½(B-BT)I I /~ n, S

n n

= m B = M B m-1 /m ' ~I /~ , i f

where m B := Imin { 0 , o . }I and M B := max { 0 , o } , mzn max

the minimum and the maximum of the eigenvalues of the mat r i x

par t of B). Bounds fo r d(e) are eas i l y found in terms of

According to these pos i t i ons , we have , ~m

Let us suppose g v e r i f i e s f o r every x • and f o r a.e. t e [0 ,p ] ,

( r ) s(xlg(t,x)) i> alg(t,x)I 2 + b l g ( t , x ) I - c ( t ) , s = _+1,

wi th a >i ~ , b > d(e) and c • L 1 ( [O ,p ] , ~ ) . S +

Moreover, l e t there e x i s t a constant 6 > O and a nonsingular mat r i x U, such tha t

(rr) l im i n f (Ux lg ( t , x ) ) / I x l 6 ~> £ ( t ) , un i fo rmly w i th respect to t , ixi ÷+

wi th /o p £ > O.

Then equat ion (4.4) has at l eas t one p - p e r i o d i c so lu t i on ( in the genera l ized sense).

Indeed, Coro l l a ry I app l ies since (r s) and ( r r ) imply , r e s p e c t i v e l y , (HI) - ( js) and

(H4). (See the proof of the in ference (kB) ~ (w) ~ (d) in [ 8 ] ) .

Ac tua l l y , more general statements concerning (4.4) can be d i r e c t l y der ived from Theo-

rem I ( [ 8 ] ). See[ 8 ] a lso fo r a de ta i l ed d iscussion of the growth r e s t r i c t i o n s

are eas i l y computed

(s = ± I ) , i f n is odd,

n ~ 2 (mod 4) , m-1 = MB/Wn ml = mB/ran' i f n ~ 0 (mod 4)

w i th ~ . and o m l n m a x

½(B + B T) ( the symmetric

Lhl I

294

and the sign conditions which are involved.

Uniqueness results can be produced, for a l l the previous examples, by means of

Proposition I.

Applications to boundary value problems for nonlinear par t ia l d i f f e ren t i a l equa-

tions w i l l appear elsewhere.

REFERENCES

[ I ] . H. AMANN, Existence theorems for equations of Hammerstein type, Appl. Anal., 2 (1973), 385 - 397.

[ 2]. H. BREZIS and L. NIRENBERG, Characterizations of the ranges of some nonlinear operators and appl icat ions to boundary value problems, Ann. Scuola Norm. Sup. Pisa, (Rend. CI. Sci . ) , (4) 5 (1978), 225 - 326.

=

[ 3]. C.P. GUPTA, On functional equations of Fredholm and Hammerstein type with appl i - cations to existence of periodic solutions of certain ordinary d i f f e ren t i a l equa- t ions, J. Integral Equations, 3 (1981), 21 - 41.

=

[ 4]. C.P. GUPTA, Perturbations of second order l inear e l l i p t i c problems by unbounded non l inear i t ies , J. Nonlinear Anal., TMA, 6 (1982), 919 - 933.

[ 5]. H.G. HEUSER, Functional Analysis, Wiley In tersc i . Publ., New York, 1982.

[ 6]. J. MAWHIN, Landesman - Lazer's type problems for nonlinear equations, Conf. Sem. Mat. Univ. Bari, 147, 1977.

[ 7]. J. MAWHIN, Topological degree methods in nonlinear boundary value problems, Reg. Conf. Ser. in Math., CBMS, n. 40, Amer. Math. Soc., Providence, R. I . , 1979.

[ 8]. P. OMARI and F. ZANOLIN, On forced nonlinear osc i l la t ions in n - t h order d i f f e - ren t ia l systems with geometric condit ions, J. Nonlinear Anal., TMA, (to appear).

[ 9]. P. OMARI and F. ZANOLIN, Existence results for forced nonlinear periodic BVPs at resonance, Ann. Mat. Pura Appl., (to appear).

[ I0 ] . P. OMARI and F. ZANOLIN, Sharp nonresonance conditions for per iod ica l l y pertur- bed Li~nard systems (to appear).

[11]. R. REISSIG, Schwingungssatze fur die verallgemeinerte Li~nardsche D i f f e ren t i a l - gleichung, Abh. Math. Sem. Univ. Hamburg, 44 (1975), 45 - 51.

= =

[12]. F. ZANOLIN, On forced nonlinear osc i l l a t ions for some second order d e l a y - d i f f e - rent ia l systems, in Evolution equations and the i r appl icat ions (Proc. Conf. D i f f . Equations and Appl., Retzhof, 1981), F. Kappel and W. Schappacher, Ed., Res. Notes in Math., n. 68, Pitman, Boston, 1982; 295 - 313.

= =

PERIODIC-SOLUTIONS OF PRESCRIBED PERIOD

FOR HAMILTONIAN SYSTEMS

Paul H. Rabinowitz

A Hamiltonian system of ordinary differential equations has the

form

dp ~H (HS) d--t H 6 = -~(P'q) H -H (p,q) , q = Hp(p,q) q

An important property of such systems is that if z(t) = (p(t) ,q(t))

is a solution, then H(z(t)) H constant, i.e. the "energy" H is an

integral of the motion. During the past few years, there has been a

considerable amount of progress in proving the existence of periodic

solutions of such systems. Several questions have been studied for

(}IS) - see e.g. [i]. In this talk we will focus on the following one:

Given an energy surface, i.e. if e.g. H-I(1) is prescribed, what sort

of geometrical assumptions on this set imply that it must contain a

periodic orbit for (HS). This is a global problem and essentially all

of the progress that has been made on this question involves the use

of global methods from the calculus of variations. Successful approach-

es have come from three main directions: (i) geodesic methods from

geometry; (ii) direct methods from minimax theory, and (iii) methods

from convex analysis and optimization theory.

We will give a brief survey of some of the main results that have

been obtained for the above question and then discuss some recent joint

work of Benci and Rabinowitz. The first work we know of concerning

periodic solutions of (HS) of prescribed energy is due to Seifert [2]

who proved the following.

Theorem 1 :

(V I)

n

Suppose H(p,q) = ~ a.. (q)piPj + V(q) where i,j=l 13

D --- {q e ]Rnlv(q) < i} is diffeomorphic to the unit ball in IR n

and Vq(q) ~ 0 on ~D (i.e. ~ is a manifold)

and

(KI) aij e C 2(~R) and (aij (q))

all q e 9.

is a positive definite matrix for

Then there exist two points QI,Q2 e ~D, T > 0, and a solution (p,q)

296

of (HS) such that (p(0),q(0)) = (0,Q I) and (p (T) ,q (T) ) = (0,Q2).

Observing that H is even in p, it follows that if the func-

tions p(t),q(t) are extended beyond [0,T] as respectively odd and

even about their end points, then the extended function is a 2T periodic

solution of (HS). Roughly speaking, Seifert obtained the solution

(p,q) as a geodesic for a Riemannian metric associated with the quad-

ratic kinetic energy term in H.

Thirty years later, Seifert's ideas were picked up and extended

by Weinstein [3] who showed:

Theorem 2: Suppose H(p,q) = K(p,q) + V(q) where V satisfies (V I)

and

(K 2) K e C 2, K(0,q) = 0, and K is even and strictly convex in p.

Then the conclusions of Theorem 1 hold.

Weinstein used the kinetic energy term to get a Finsler metric

and obtained the solution as a geodesic in this metric. He then went

on to prove the following beautiful geometrical result via an ingenious

reduction to Theorem 2.

Theorem 3: Suppose H ¢ C2~R2n~R) and H-I(1) is a manifold which

bounds a compact strictly convex region. Then (HS) possesses a peri-

odic solution on H-I(1).

Subsequently a fairly elementary proof of Theorem 3 was given by

F. Clarke [4]. Using the convexity of H, he made a Legendre trans-

formation converting (HS) to an equivalent system of equations. This

new system was then formulated as a variational problem for which a

solution could be obtained as a minimum.

Simultaneous to Weinstein's work, we also studied (HS) and proved

Theorem 4 [5]: Suppose H e claR2n~R) and H-I(1) is a manifold

bounding a compact starshaped region. Then (HS) has a periodic solu-

tion on H-I(1).

The proof of Theorem 4 uses minimax methods from the calculus of

variations. The original proof used finite dimensional approximations

to an infinite dimensional variational formulation of (HS) together

with appropriate estimates to pass to a limit. Nowadays as will be seen

from the sketch of the proof of Theorem 7 below, a more direct proof can

be given.

297

Motivated by [4], the arguments of [5] were used to prove:

Theorem 5 [6] :

(V l) ,

(K 3 )

Suppose H(p,q) = K(p,q) + V(q) where V satisfies

K e C2aR2n~R), K(0,q) = 0, and p • Kp(p,q) > 0 for p ~ 0

(i.e. {p e ]RnlK(p,q) = constant} bounds a starshaped region for

fixed q)

Then H-l(1) contains a periodic solution.

In (K3) (and later) p • Kp denotes the inner product between

these two vectors.

The geometrical ideas of Seifert and Weinstein were pushed one

step further by Gluck and Ziller [7] who showed

Theorem 6:

and

Suppose H(p,q) = K(p,q) + V(q) where K satisfies (K 2)

(V 2) ~ -- {q e ]RnIv(q) < I} is compact with V # 0 on 9D and - q

v e C2(~R).

Then the conclusions of Theorem 1 hold.

Independently of (7), the set up of Theorem 6 with (K 2) replaced

by (K I) was studied by Hayashi [8] and Benci [9].

Recently, jointly with V. Benci, we have obtained the following

result [i0]:

Theorem 7: Suppose H e C2~R2n~R) and H-I(1) bounds a compact

neighborhood of 0 with VH # 0 on H-I(1). If p • H > 0 for

p ~ 0 near H-I(1), then (HS) has a periodic solution p H -I on (i).

The proof of Theorem 7 will be sketched below. The details can

be found in [i0]. To begin, note that the period, T, of the solution

we seek is a priori unknown. It is convenient however to treat (HS)

in a class of functions having a fixed period. Therefore by making the

change of time variable, t + ~-it where ~ = (2z)-iT, (HS) becomes

(8) ~ :-~H , ~ = ~H q P

Now we seek 2x periodic functions (p,q) such that H(p(t),q(t)) ~ 1

and I ~ 0 satisfying (8). Inverting our transformation, this leads

to a 27 periodic solution of (HS).

Let z(t) = (p(t),q(t)). Formally (8) can be interpreted as the

298

Euler equation for the problem of finding critical points of the func-

tional

2~ (9) A(z) E I p(t) • q(t)dt

0

subject to the constraint

2~ (10) 1 2--{ f H(z(t))dt = 1

0

where z lies in an appropriate class of 27 periodic functions. The

parameter, I, appears as a Lagrange multiplier due to the constraint

(I0). In addition to satisfying (8), if z is a solution of the vari-

ational problem associated with (9)-(10), then H(z(t)) E i. Indeed

since (8) is a Hamiltonian system, H(z(t)) ~ constant and (i0) then

implies the constant is i.

To find a critical point of A subject to (i), this problem must

be formulated more precisely. For various technical reasons the func-

tion H must be redefined away from H-I(1) so that the modified H,

which we denote by H, satisfies

(ii) p • Hp(Z) > 0 for all p # 0

and certain other technical conditions which can be found in [I0]. By

construction H-I(1) = H-I(1) and H ~ H near this set so 2~ peri- ----i

odic solutions of the modified problem on H (i) will satisfy (8).

Condition (ii) allows us to decompose H into the sum of potential and

kinetic energy terms. Indeed set V(q) ~ H(0,q) and K(p,q) ~ H(p,q)

- V(q). Then V satisfies (V 2) and K satisfies (K2). Actually

one more technical modification of V which will be omitted is required

[10].

Now the class of functions in which A is treated can be intro-

duced. Let L2(SI~R n) denote the set of n-tuples of 2~ periodic

functions which are square integrable and WI'2(SI~R n) the subset of

L2(SI~R n) of functions which have a square integrable derivative. The

usual Hilbert space norms will be employed in these spaces. Set

WI, 2 E E {z=(p,q) Ip e L2(SI~Rn), q e (SI~Rn)}

Then A e C (E~R).

The space E can be decomposed into three mutually orthogonal

subspaces which span E and play an important role in obtaining a

critical point of A subject to (i0). For z e E, let

1 2z 0 p0z H ~ ~ z(t)dt E [z] = ([p], [q]) E z

299

d L 2 and pOE -= E 0. Let D - dt " If p e (sluR n) and [p] = 0 then

D-ip exists and belongs to WI'2(SI~Rn). For z e E, we can write

^ ^ ^

Z = [Z] + z where z = (p,q) and [z] = 0. Set

_-,- , .(ql .,. z--'-

and E ± E P~E. It is easy to verify that p0,p+,p- are projectors of

+ E into E and E = E 0 @ E + 8 E-. Moreover if z = z 0 + z + z e E,

1 2 - 2) ¢12) A(~) = ~(11z+I1 - IIz II

Thus E ~ are respectively the subspaces of E on which the quadratic

form A is positive definite and negative definite and E 0 is the null

space of E. We also let Pl(p,q) E p and P2(p,q) E q.

Next for z e E, set

27 1

~(z) = ~ ~[ I H(z (t))dt 0

and M E ~-i(i). Thus M is just the set of points in E satisfying

our constraint (i0). The properties of ~ (as obtained from K and

V) imply that I~ is a C I'I manifold in E, i.e. M is a manifold

and ~ is continuously Frechet differentiable with a Lipschitz con-

tinuous Frechet derivative. Moreover M is the boundary of a neigh-

borhood of 0 in E and is a bounded set in L2(SI~R2n). These facts

are easy to see for the special case of H(z) = Izl 2

Consider AIM. Equation (12) shows that A is a highly indefinite

functional on M and in particular is not bounded from above or below

even though M is bounded in L2(SI~R2n). (Think again in terms of

the special case H(z) = IzI2). Thus obtaining a critical point of

AIM is a subtle matter. We will use a minimax argument to find such

a critical point. We do not have the time here to explain in detail

how minimax methods work, however some brief remarks may be helpful.

See also [11-12]. Roughly speaking one has to show that the functional

I (here AIM) under study is suitably "compact" and that there exists

a family of sets which are invariant under the negative gradient (or

the gradient) flow associated with the Frechet derivative I', of I.

By this flow we mean the solution of the differential equation

dD _ TIE' (n) dt

300

Then if c is defined by

n(0,z) = z

(13) c E inf sup I(u) BeS u e B

for the negative gradient flow case or by

(14) c E sup inf I(u) B e S u e B

for the gradient flow case, and if c is finite, then c is a criti-

cal value of I.

Perhaps the simplest nontrivial example of a critical point ob-

tained by such ideas is given by the Mountain Pass Theorem. Below B P

denotes a ball of radius p about 0.

Theorem 15 [13] Suppose E is a real Banach space, I e cl(EflR) and

I satisfies the Palais-Smale condition. If I(0) = 0 and I satis-

fies

(I I )

and

there are constants p,~ > 0 such that II~ B ~ P

(I 2) there exists e e E\B such that I(e) < 0 , p

then I possesses a critical value c > ~. Moreover c can be char-

acterized as

(16) c = inf max I(g(t)) heF te [0,i]

where

F = {g e C([0,1],E)Ig(0) = 0, g(1) = e} .

Remarks 17: (i) Comparing (16) to (13), we see B = g([0,1]) and

S = {g([0,1])Ig e F}.

(ii) The Palais-Smale condition or (PS) for short is the compact-

ness condition for the functional we referred to above. We say that

I satisfies (PS) if any sequence (u m) such that I(u m) is bounded

and I' (u m) ÷ 0 possesses a convergent subsequence. Actually in appli-

301

cations one can do with weaker versions of (PS), e.g. with c as de-

fined by (16), it suffices that whenever I(u m) ÷ c and I' (u m) + 0,

then (u m) has a convergent subsequence.

(iii) The theorem states that if 0 and e are separated by a

"mountain range" via (Ii)-(I 2) and (PS) holds, then I has a critical

value which can be obtained as the inf of I over all paths joining

0 and e.

Returning to AIM, it turns out that this functional satisfies a

suitable version of (PS). However to give a minimax characterization

of a critical value is not so simple. Let M + ~ M N E + and M- ~ M N

(E- ~ E 0 8 L +) where L + is a two dimensional subspace of E + such

z(t) e L + implies that z(t+8) e L + for all 0 e [0,2z] (i.e. L +

is a two dimensionial subspace of E + invariant under such transla-

tions).

Define

(18) ~ ~ inf+ A(z) zeM

and

(19) ~ 5 sup_ A(z) zeM

It is clear that e < ~, ~ > 0 since M is the boundary of a neigh-

borhood of 0 in E, and using (12) it is not too hard to show that

< ~ if L + is chosen appropriately.

Now set

F ~ {h e C(M,M) lh satisfies 10 - 40 }

where

10 If for 0 e [0,2~] and z e E, (Toz) (t) H z(t+0), then h

commutes with T O for all 8 e [0, 2~].

20

30

40

h(z) = z if A(z) 9' [0,~ + i]

h maps bounded sets to bounded sets

P+h(z) = 8+(z)z + + B+(z) where B + e C(M, [i,60))

60 = 60(h) > 1 and P2 B+ is compact.

with

The following "intersection theorem" holds for F.

Proposition 20:

Define

(21)

Since the identity map belongs to F,

wise by Proposition 20, for any h e F,

302

If h e F, then h(M +) A M- ~ ~.

c = sup inf+ A(h(z)) heF zeM

(21) and (18) show c > ~. Like-

(22) inf+ A(h(z)) ! A(w) ! sup_ A(z) zeM zeM

where w e h(M +) A M-. Since this is true for all h e F, c < ~ via

(19) and (21).

The estimates just established for c and properties of F can

then be employed to show c is a critical value of AIM. To complete

the proof of Theorem 7, it must be shown that a critical point of AIM

is a classical solution of (8) but this is not very difficult to do [i0].

Remarks 23: (i) A more refined version of Theorem 7 holds under the

milder smoothness condition H e C IQR2n~R) [i0].

(ii) An interesting open question is whether Theorem 7 is true

or is false if the requirement that p • Hp(Z) > 0 is eliminated. The

result - positive or negative - would be interesting already if H-I(1)

is diffeomorphic to the unit ball in ~2n.

Lastly we mention that there have been some recent results con-

cerning the number of geometrically distinct periodic solutions of (HS)

on H-I(1), mainly when H is convex. See e.g. [14]-[15].

References

[i] RABINOWITZ, P. H., Periodic solutions of Hamiltonian systems: a survey, SIAM J. Math. Anal. 13 (1982), 343-352.

[2] SEIFERT, H., Periodische Bewe-gungen mechanischen Systeme, Math. Z. 51 (1948), 197-216.

[3] WEINSTEIN, A., Periodic orbits for convex Hamiltonian systems, Ann. Math. 108 (1978), 507-518.

[4] CLARKE, F., A classical variational principle for periodic Hamil- tonian trajectories, Proc. Am. Math. Soc. 76 (1979), 186-188.

[5] RABINOWITZ, P. H., Periodic solutions of Ha-miltonian systems, Commun. Pure Appl. Math. 31 (1978), 157-184.

[6] RABINOWITZ, P. H., Periodi-c solutions of a Hamiltonian system on a prescribed energy surface, J. Differ. Equations 33 (1979), 363- 352.

[7] Gluck, H. and W. Ziller, Existence of periodic motions of conser- vative systems, Seminar on Minimal Submanifolds, Princeton Univer- sity Press, 1983, 65-98.

[8] HAYASHI, K., Periodic solution of classical Hamiltonian systems, Tokyo J. Math. 6 (1983), 473-486.

303

[9] BENCI, V., Closed geodesics for the Jacobi metric and periodic solutions of prescribed energy of natural Hamiltonian systems, preprint.

[10] BENCI, V. and P. H. RABINOWITZ, Periodic solutions of prescribed energy for a class of Hamiltonian systems, to appear.

[ii] PALAIS, R. S., Critical Point theory and the minimax principle, Proc. Symp. Pure Math. 15, American Math. Soc., Providence, R.I. (1970) 185-212.

[12] RABINOWITZ, P. H., Variational methods for nonlinear eigenvalue problems, Eigenvalues of Nonlinear Problems (G. Prodi, editor), Edizioni Cremonese, Rome (1974), 139-195.

[13] AMBROSETTI, A. and P. H. RABINOWITZ, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349-381.

[14] EKELAND, I. and J. M. LASRY, On the number of periodic trajectories for a Hamiltonian flow on a convex energy surface, Ann. Math. 112 (1980), 283-319.

[15] EKELAND, I., Une theorie de Morse pour les systemes hamiltoniens convexes, Annales de L'Institute Henri Poincare, Analyse nonlin- eaire l, (1984) 19-78.

This research was sponsored in part by the National Science Foun- dation under Grant No. MCS-8110556 and by the U. S. Army under contract no. DAAG29-80-C-0041. Reproduction in while or in part is permitted for any purpose of the U.S. Government.

BURSTING OSCILLATIONS IN AN EXCITABLE MEMBRANE MODEL

John Rinzel*

I. Introduction.

Various nerve, muscle, and secretory cells exhibit complex electrical activity

which has been observed experimentally by using intracellular electrodes to monitor

the dynamics of the potential across the cell membrane. Such activity may include

single spikes (time scale, msec.) in response to brief stimuli, repetitive spiking

for a maintained input, and repetitive bursts of spikes (time scale, sec) which may

be endogenous and modulated by chemical (e.g. hormonal) or electrical stimuli. Pan-

creatic B-cells respond with periodic bursting in the presence of glucose (3,13) and

this activity is correlated with their release of insulin (18). Figure I illustrates

computed solutions of a theoretical model (4) for such electrical behavior. The mathe-

matical model (based upon a biophysical model (2)) is an adapted and expanded version

of the classical Hodgkin-Huxley (11) description of nerve excitability and involves

five first-order nonlinear ordinary differential equations. The time course of mem-

brane potential V(Fig. I, upper) exhibits spikes of roughly constant size (30-40mV)

which appear to ride on a plateau potential of approximately -40 mV. Following each

"active phase" of spiking is a "silent phase" where V slowly increases. The intra-

cellular free calcium concentration Ca (Fig. I, lower) slowly increases (on the aver-

age) during the active phase and slowly decreases during the silent phase. The dynam-

ics of Ca determine the time scale of the bursts.

In this paper we present an analysis and qualitative viewpoint of bursting for

the Chay-Keizer (C-K) theoretical model. We exploit the slow behavior of Ca by first

considering Ca as a parameter and studying its influence on the faster spike-gener-

ating subsystem. Such spike generation dynamics are first illustrated (Section 2)

for a simplified model of excitable membrane activity with Ca fixed. This two-vari-

able, reduced HH, model yields single spike and repetitive spike activity such as

seen in the active phase of bursting. In some parameter ranges it exhibits bistabi-

lity in which V may rest at a lower stable steady state or oscillate stably around an

upper (unstable) steady state. This latter behavior is also in the repertoire of the

four-variable HH subsystem in the C-K model and it corresponds to the silent and active

phases. Next we append to the excitation subsystem the slow dynamics of Ca to account

for bursting. In Section 3 we describe a special case of bursting: slow wave activity

*Part of this work was performed while the author was a visitor at the Center for Mathematical Biology, University of Oxford (supported by SERC Grant GR/C/6359.5).

10

> E

-30 >

305

-70 " J 0

t (~ec) 8

.45

.35 U

.25 0 ~ 8

t ( s e c ) Figure I. Periodic bursting response to glucose of theoretical Chay-Keizer model(4). Time profiles of membrane potential V and intracellular calcium concentration Ca. Pa ~ rameters of this five-variable model are as in Table I in (4); here, kca = 0.04 ms -I and temperature is adjusted to 18°C (see text).

in which the active phase has no spikes. This corresponds to a bistable subsystem dy-

namics at high temperature in which case the upper and lower steady states for V are

both stable. As temperature is decreased, repetitive activity reappears in the active

phase and this is described in Section 4. Our analysis and the insight which we gain

is based on the (numerical) application of perturbation methods and bifurcation theory

to treat Ca as a parameter. Our approach facilitates interpretations of several exper-

imental observations and numerical results for the theoretical model. For example, we

see easily why the spike amplitudes and the plateau potentials of the active phase are

relatively independent of glucose concentration. Further, our viewpoint reveals how

the spike frequency must decrease dramatically near the end of a burst (a magnified

time base in Fig. I would make this more obvious).

We remark here that in this paper we will not consider the effect of spatially

distributed differences in membrane properties or variables. One should imagine a

cell to be isopotential. We expect this is an accurate description since B-cells are

306

quite small (physically and electrically). Furthermore, although the functional unit

of the pancreas (an islet) is multicellular (hundreds of electrically coupled cells)

we assume the cells are identical and act synchronously so the activity of one cell

reflects that of others in the islet (as experiments (8,13) seem to suggest).

2. A simple model for excitability and spikes.

Electrical behavior of excitable membranes is due to ionic and capacitive cur-

rents. The concentration differences of ions across the membrane provide the driving

forces for the ion flows. Membrane channels or pores (typically ion-selective) with

V-dependent properties are the pathways for the ionic fluxes. In many cases these

ionic currents are represented as ohmic with V and time (t) dependent conductances:

gj is the conductance for the jth ion species. Thus membrane current Im(t) (positive

for outward flow) is expressed as

(I) Im(t) : C m V + ~ gj (V,W) (V-Vj) J

where C m is membrane capacity (~1~F/cm2), Vj is the (Nernst-Planck) equilibrium poten-

tial for species "j", assumed constant in this treatment, and V is membrane potential.

Here, W is a vector of auxiliary variables used to describe the dynamics of the vari-

ous channel conductance states; these channels do not usually respond instantaneously

to V. Each component of W typically satisfies a first order nonlinear differential

equation with coupling to V but to no other W-component (c.f. eqn. (3)). The most

widely known example is the four-variable Hodgkin-Huxley or HH model (11) for squid

giant axon in which the principal ionic currents are for sodium (inward~ ~ith VNa = 55

mV) and potassium (outward with V K : -72 mV). In this case W has three components:

W = (m, h, n) where m represents sodium activation, h is sodium inactivation, and n is

for potassium activation (the K+-channel does not display identifiable inactivation as

such). For standard HH parameters, the rest state is dominated by gK so Vrest ~ -60

mV and is close to V K.

In some parameter ranges many features of the HH dynamics are reproduced (nearly

quantitatively) by a two-variable reduced model (15). In this simplification (moti-

vated by FitzHugh's classic work (11)) one exploits the relatively rapid time scale

of m and represents it as an instantaneous function of V. That is, from the equation

: (m~(V)-m)/~m(V) we set m E m=(V) where m®(V) is a monotone increasing function

of V and saturates with m~ ÷ I as V + ~ (similarly m= ~ 0 as V ~ -~). A further

reduction follows by lumping h and n (which have similar time scales) into a single

recovery variable W. Here, "recovery" denotes the functional role played by W which

acts after the rapid spike upstroke to recover, or return, V to its rest state on a

slower time scale. This model takes the following form (see (15)):

307

(2) CmV = -gNa m~(V)(I-W)(V-VNa) - gk(W/S)4(V-Vk )

- ~L(V-VL) + Iapp(t)

(3) w = ~(w®(v)-w)/~(v)

where gNa, gk, gL take their HH constant values, S is a positive constant (see (15)),

W=(V) behaves qualitatively as m=(V), ~(V) = 5 exp[-(V+10)2/552] +I, Iapp represents

an externally applied current, and @ is a temperature correction factor (in HH appli-

cations, @ = exp[(logQ10) • (Temp-6.3)/10] where QI0 = 3 for squid). (Note, in eqns.

(2-3) V represents deviation of membrane potential from its rest value for Iapp = 0.)

The excitation dynamics of this model may be viewed in the V-W phase plane (Fig.

2A). The nullclines are shown dashed and their single intersection corresponds to the

membrane's stable rest state. A brief, adequate, applied current leads to the impulse-

llke response. For other parameter regimes (e.g. adequate, constant Iap p) the unique

singular point may lie on the middle branch of the V-nullcline and be unstable with

a surrounding stable periodic solution (repetitive firing). In yet other cases the

nullclines can have three intersections. For parameters in this latter range the

lower V state is stable, the middle state is a saddle and the upper state may be sta-

ble or unstable. Figure 2B shows the V time course for an example in which the lower

state is stable and the upper is unstable and is surrounded by a stable oscillation.

A B

) -

(:3 .5 L) Ld rw

)-

i I

-25 50 125 V, NEt~BRF~E POTENTIF~_ (mY}

5(

>

. . . . . . . . . I [ . . . . . . . . . . . . .

6 0

Figure 2. Solutions to two-varlable simplified Hodgkin-Huxley model (cf. eqns.(2-3)) as derived in (15). Left panel shows phase plane with (dashed) nullcllnes, V=O (cubic- like) and W=0 (Sigmoidal), whose intersection corresponds to unique stable steady state. Solid curve is response to brief pulse of current Iapp(t);dotted curve is response of full (four-variable) Hodgkin-Huxley model. Inset compares time courses of V. Right panel is for parameter values (see (15)) such that membrane has stable (low-V) steady state or stable (high-V) oscillation. Brief current pulses (upper dashed curve) cause switching between these two states.

308

A brief current pulse can switch the membrane between the two stable modes of behavior;

either a steady state or a repetitive firing state. This type of bistability forms

the qualitative basis for the bursting of Fig. I (which is for the full HH-like exci-

tation model of Chay and Keizer).

3. A model for slow waves (active phase without spikes).

To model electrical activity of pancreatic ~-cells, Chay and Keizer considered

electrophysiological data to adapt the four variable HH-type equations to the bio-

physical model of Atwater, et al. (2). In their description (which they consider a

minimal one) m and h model calcium conductance rather than sodium conductance since

inward current across the B-cell membrane appears to be carried predominantly by cal-

cium ions. The model further incorporates an additional potassium selective, but non-

voltage dependent, channel which is activated (instantaneously) by intracellular cal-

cium. Its conductance gK-Ca is proportional to Ca£/(1+Ca £) where ~ is a positive con-

stant denoting the number of calcium ions which must bind in order to activate this

channel (£:I for most of the examples in (4)). For this model then Iio n = Iion(V,W,

Ca) where W = (m,h,n). We will not need the explicit f~rm of the equations here; they

may be found in (4).

In contrast to the classical HH model where intra- and extracellular ionic con-

centrations are assumed constant, here Ca is considered to vary with time. Since Ca

is typically quite low O(~M) and since B-cells are small the calcium influx during

the active phase can cause a substantial change in Ca (cf. Fig. I) and therefore in

gK-ca" For the dynamics of calcium exchange Chay and Keizer use

(4) Ca = e(~ICa - kcaCa)

where Ica = gCa m3h(V-Vca), ~ is a factor containing Faraday's constant and the cell's

surface area to volume ratio, and the term kcaCa is a minimal description for the dis-

appearance of intracellular free calcium (in their view, primarily due to mitochron-

drial uptake). The time scale set by e(e=O(10-2)) is slow relative to that of the

other variables; e represents the ratio of free ionized calcium to total calcium in-

side the cell. The rate constant kCa reflects a metabolic process which is glucose

dependent (kCa increases with glucose concentration).

Our analysis, which is described qualitatively (although based upon numerical re-

suits for the C-K equations), begins by treating the slowly varying quantity Ca as a

parameter in the (V,m,h,n) subsystem. From the global bifurcation structure of this

subsystem (which I have calculated numerically using AUTO (7)) and from the slow Ca-

dynamics we thus describe and interpret rather simply the solution behavior of the

full model.

These ideas are illustrated first for the simple case in which no spikes occur .

during the active phase. We generate such behavior in the model by increasing tempe-

309

rature which increases (identically) each of the rates m, h, n (as in eqn. (3)). The

temperature factor was not shown explicitly in (4) however the computations reported

there were for 20°C assuming QIO = 3. Here we also use QIO = 3 (even though the QIO

for B-cell spike dynamics experimentally appears to be lower). We do not draw physio-

logical conclusions about the temperature variation per se but use it only to illus-

trate better the dynamical structure which underlies the bursting behavior. With

sufficient temperature increase, the variables m, h, n respond to V so rapidly that

we may assume they attain their "steady state" values instantaneously, i.e. we make

the replacements m = m~(V), h = h~(V), n = n~(V). In this case we have only two

two dynamical equations

(5) CmV : - Iss(V,Ca)

(6) Ca = e(~Ica - kcaCa)

(in which we have set Iapp : 0). The function Iss(V, Ca) versus V is called the

steady state current-voltage relation. It satisfies Iss = Iio n (V, W~(V), Ca) where

the argument W~(V) indicates that each of the auxiliary variables m, h, n is evalu-

ated at steady state; the relation Iss versus V is plotted in Fig. 3A for various

values of Ca.

A value of V for which Iss = O corresponds to a "rest potential" of the membrane.

The curves in Fig. 3A reveal that for high Ca there is only one rest potential; it is

near V K since the membrane conductance is dominated by gK-Ca. Conversely, this con-

ductance is low when Ca is low and then there is a unique rest potential which is much

higher than V K. The multiple rest potentials plotted versus Ca form the Z-shaped

curve in Fig. 3B. This "Z-curve" is of course the V-nullcline of the equations (5)-

(6). At high temperature the upper and lower branches correspond to stable steady-

states of (5). The middle branch represents an unstable solution and so it is shown

dashed. One can now predict the response behavior by observing that for this value of

kCa the Ca-nullcline intersects the V-nullcline on the latter's middle branch. Be-

cause Ca is a slow variable the attracting periodic orbit is a relaxation oscillation

(indicated schematically by the heavy closed curve in Fig. 3B). It corresponds to a

slow wave (time scale of sec.) which does not generate spikes during the active phase

(this case is an analog to Fig. 7 of (4)).

This response may be interpreted biophysically as follows. During the active

phase Ca increases and thereby activates gK-Ca" Eventually this conductance dominates.

Then the plateau potential is lost and V falls to near V K. During the ensuing silent

phase Ica is small (because m~ is small) and Ca < 0 as calcium is taken up by the mito-

chondria. With this decrease in Ca goes the decrease of gK-Ca so that V rises slowly

until gK-ca becomes too small to balance the growing calcium conductance; hence there

is no longer a lower steady state and the active phase is reentered.

A

~9 ¢,o

-4 ' -70 -70 -45 -20

V (mY)

310

B -20

~ - 4 5

0 t 2 ca ( / ~ )

Figure 3. Dependence upon Ca of steady state current-voltage relation (left panel) for four-variable HH excitable subsystem of Chay-Keizer model. From lowermost to uppermost curve, Ca=O, l, 2, 3 (~M). Right panel shows Ca-V phase plane and slow wave response (heavy closed curve) for eqns. (5-6) which approximate Chay-Keizer model at high temperature (25oC, here). Ca-nullcline shown with long dashes. V-null- cline is Z-shaped (obtained from zero-current crossings in left panel). The Z-curve also represents steady state V versus Ca for four-variable HH subsystem of Chay-Keizer model; upper and lower states are stable; middle state is unstable, a saddle.

The preceding qualitative analysis is similar to that of Plant (14) for the slow

wave in a model of an invertebrate neuron's bursting behavior. In that case, the

spikes (largely sodium driven) were eliminated theoretically by setting gNa : 0 which

corresponds experimentally to applying a sodium channel blocker. For the B-cell prep-

aration (in which calcium is crucial for the spikes and for the plateau behavior) it

has not yet been shown how to reliably eliminate spiking yet retain a slow wave oscil-

lation. In the next section we extend our analysis to lower temperature and this takes

us beyond the special case of the slow wave.

4. Bifurcation analysis for bursting (active phase with spikes).

We continue by again treating Ca as a parameter and considering the dynamics of

the (V, m, h, n) - subsystem as we now lower the temperature. This four-variable HH

model shares many qualitative features with the simpler two-variable excitability

model of Section 2. We will represent succinctly the Ca-dependence of its solution

behavior by exhibiting the numerically determined (via AUTO (7)) global bifurcation

diagram.

Notice first that the Z-shaped, steady state curve is independent of temperature;

311

this is because the temperature factor merely multiplies the rates m, h, n. On the

other hand, stability of the upper branch solution changes as temperature decreases.

For temperature just below a critical value the upper steady state is unstable for a

small interval of Ca values, Ca I < Ca < Ca 2, and it is surrounded by a small stable

oscillation (which appears through supercritlcal Hopf bifurcation). This instability

is indicated by dashes in Fig. 4A; the maximum and minimum values of V associated with

the limit cycle are also plotted. This solution behavior for Ca I < Ca < Ca 2, i.e.,

coexistence of a low-V stable steady state and a high-V stable oscillation, corresponds

to that shown in Fig. 2B for the simplified V-W model. As the temperature is lowered

further the branch of periodic solutions makes contact with the intermediate steady

state (a saddle) and then splits into two separate (left and right) branches (Fig. 4B).

Each branch has one end point where the oscillation shrinks to zero amplitude (Hopf

bifurcation) around the upper steady state. At the other end point the periodic solu-

tion coalesces with the saddle point and, here, the periodic orbit has infinite period;

it is a homoclinic orbit. For still lower temperature, the right branch of periodic

solutions migrates toward and disappears into the Z-curve knee (this bifurcation is

described in (12)) leaving only the left branch of periodic solutions (Fig. 4C).

A B C -20 -20 r -20

- 7 0 ' ' ' - 7 0 ' ' ' ' - 7 0 ~ ' '

0 1 2 0 1 2 0 1 2

Figure 4. Dependence upon Ca of solution behavior (bifurcatiion diagram) of four- variable HH subsystem of Chay-Keizer model. As temperature decreases (left to right) the high-V steady state (upper branch of Z-curve) becomes unstable (dashes) for some Ca-range. This unstable state, for some Ca, is "surrounded" by a stable oscillation (maximum and minimum V values indicated by solid curves above and below dashes). When oscillation coalesces with saddle point (middle branch of Z-curve) it has infinite period and then (for slightly increased or decreased Ca) it disappears. All solution branches (periodics are shown schematically here) were determined numerically using AUTO (7).

The mechanism for bursting now becomes intuitively clear when we consider the bi-

furcation diagram of Fig. 4C along with the slow dynamics for Ca. Firstly, the exci-

tation subsystem exhibits, over a range of Ca, coexistence between a low-V stable

steady state and a high-V stable oscillation. Secondly, suppose Ca > 0 when the sub-

312

system resides in the periodic state (of repetitive spiking) and suppose Ca < 0 when

the subsystem is in the steady state. In this case, the slow dynamics of Ca will cause

Ca to sweep back and forth through the interval of coexistence as the subsystem

switches between spiking (active phase) and near steady state behavior (silent phase).

Figure 5A illustrates schematically such a burst pattern in the Ca-V plane. The pre-

sumptive Ca = 0 curve is also shown. While our argument is conceptually qualitative

it yields a quantitative prediction; Fig. 5B shows that the burst solution of Fig. I

projects appropriately onto the associated (numerically computed) bifurcation diagram

of the C-K model.

A -20 B o

-4s >

- 7 0

~ ' ACTIVE

t:--tkfJ

PHASE I I

0 1 2

~= -30 >

- 6 0 .25 .35 .45

Ca (/,~)

Figure 5. Compact representation of full Chay-Keizer dynamics. Ca is slow variable which couples to and modulates excitation subsystem (whose dynamics are summarized by bifurcation diagram similar to Fig. 4; here, for temperature 20°C). Left panel: closed trajectory corresponds (schematically here) to periodic bursting and long dashed curve represents approximate Ca-nullcline (see text). During active phase, excitation subsystem is in repetitive spiking mode and Ca (On average) increases; during silent phase, subsystem is in low-V pseudo-steady state and Ca decreases. Right panel: projection of solution in Fig. I onto numerically determined subsystem bifurcation diagram, temperature 18oC.

From the above representation of bursting behavior we would predict that adequate

voltage perturbations could switch the response from the active phase to the silent

phase or vice versa. This corresponds to the experimentally observed phase-resetting

of the rhythm induced by brief current pulses applied to an islet (6). We further

conclude immediately that the spike frequency should decrease dramatically near the

end of the active phase (because the trajectory passes close to the saddle point as

the subsystem periodic solution becomes homoclinic); this is also seen in experimental

data (I).

313

Before relating our analysis to experimental results on the effect of glucose we

discuss in more detail how to account for the increases and decreases of Ca during the

different phases of a burst pattern. To do this thoroughly we should examine, in the

five-dimenslonal phase space (V, m, h, n, Ca), the surface Ca=O and its location rel ~

ative to where lie the active and silent phase trajectories. From (3) this Ca null

surface is given by

(7) Ca = k~ gCa mBh(V-VCa ) •

However, this comparison may be approximated as follows. First, during the silent

phase we know that m ~ m~(V) and h ~ h~(V) so that (7) becomes

(8) Ca • kC~ gCa m~(V) h.(V)(V-VCa)

which projects as a curve in the Ca-V plane. Next consider the active phase. There

also, since m is fast, we estimate m ~ m~(V). Furthermore, during the relatively long

(0(10 msec)) interspike phases when the spike trajectory passes near the saddle we also

have h ~ h~(V). Thus, again, the curve (8) indicates approximately where Ca changes

sign. (Note, we have assessed the effect of the h variation during the upstroke-down-

stroke of a spike. This correction smears the curve into a finite-width band which

proves to be narrow in the cases we have examined.) The Ca = 0 curve shown in Fig. 5

represents the approximation (8) and thus accounts for the net increase of Ca during

the active phase spikes and fall of Ca (cellular uptake) during the silent phase.

We can now interpret qualitatively the experimentally observed effect of changes

in glucose concentration (3,13) in terms of this model. Observe first, since kCa

(the glucose sensitive parameter) appears only in the Ca-dynamics, that it does not

influence the isolated subsystem behavior (i.e., bifurcation diagram). Consequently

the spike heights and plateau potentials (identified as the locus of spike minima)

are not affected by changes in kca. If the glucose concentration is increased ex-

perimentally then the active phase duration is increased while the silent phase gets

shorter (3). This corresponds to increasing kca which roughly moves the curve Ca = 0

upward so that Ca is less positive during the active phase (hence it takes longer to

reach the end point of the periodic solution branch); correspondingly, Ca is more neg-

ative during the silent phase which thereby is shortened. If kCa is decreased suffi-

ciently (lower, long-dashed curve of Fig. 6) then the Ca = 0 curve intersects the

lower steady state branch of the bifurcation diagram. This intersection corresponds

to a stable steady state of the full system: the rest state of the B-cell in the

absence of glucose. At the other extreme of substantially increasing kCa the Ca = 0

curve moves up (upper, long-dashed curve in Fig. 6) to intersect the periodic branch

in which case bursting gives way to continuous periodic spiking. This may be under-

stood as follows. The trajectory of the continuous spike pattern falls below the

Ca = 0 curve when V is near its minimum. This removal of calcium just balances the

314

influx which occurs during the upstroke-downstroke phases of the spike. Thus there is

no net change in Ca from one spike to the next and the system never leaves the active

phase. (Note, in the case where kCa is just large enough to get continuous spiking

then ICaI<<1 during the calcium removal phase but, since the trajectory passes near

the saddle in this case, the duration of the removal phase is long enough to balance

appropriately the influx.)

10

-7( ' ' ' 0 1 Z 3

Ca ( /~)

Figure 6. Approximate Ca-nullcline (long dashes): its dependence upon kCa and its relation to subsystem dynamics (for temperature 20°C). Lowermost to uppermost cases, (kCa = 0.005, 0.03, 0.06 ms -I) correspond to increasing glucose concentration and to stable steady state, periodic bursting, and continuous spiking, respectively.

5. Discussion.

We have described a theoretical treatment of bursting electrical activity of in-

sulin-secreting 8-cells of the pancreas. Our analysis of the Chay-Keizer model ex-

ploits the crucial role of a biophysical variable (intracellular calcium concentration)

whose net changes are on a slow time scale and whose behavior controls an excitation

subsystem which can exhibit either periodic spiking or stable steady state behavior.

We first treat the control variable as a bifurcation parameter to describe fully the

subsystem dynamics. Then the control variable's dynamics are (interactively) coupled

to the subsystem's bifurcation diagram to understand the responses (bursting, contin-

uous spiking, etc.) of the full system. We believe that this qualitative viewpoint,

based on identifying an appropriate controlling variable, is applicable to other dyna-

mical systems which exhibit bursting. As a fairly direct application we intend to

consider quantitative biophysical models for various neuronal bursting pacemakers.

Hindmarsh and Rose (10) study, from a qualitative viewpoint similar to ours, an idea-

lized (less biophysical) model for neuronal bursting.

315

We previously employed a similar approach to a model of the Belousov-Zhabotinskii

chemical system which can also exhibit bursting phenomena (16). In that case however

the subsystem's multistability originates through a subcritical Hopf bifurcation; for

a range of the control variable the subsystem has a stable steady state which is "sur-

rounded" by two periodic orbits, one stable and one unstable. The controlling vari-

able was coupled interactively so that it swept back and forth across the hysteresis

zone associated with this coexistence. Also in that application even though, during

the spiking phase of a burst, the control variable's time scale was similar to that

of the subsystem our approach was insightful because each spike produced only small

changes in the control variable.

To apply our conceptual framework one requires a compact description of the sub-

system dynamics in terms of its bifurcation diagram. Since this requires not just

local but global information we employed numerical branch tracking methods which Doedel

(7) incorporated into usable software. With a numerical approach one may explore

models with sufficient detail to contain appropriately identifiable physical variables

and parameters. Of course one hopes that for some simplified models certain of the

bifurcation information might be obtained analytically.

There are a number of unanswered and interesting mathematical questions which

arise for the dynamical phenomena which we have described. By exploiting the slow-

ness of Ca we have described qualitatively a singular perturbation treatment. To

carry this out explicitly, say for a simple model problem, may lead to some new metho-

dologies. In this class of problems the trajectory leaves the slow manifold of the

active state when a subsystem periodic orbit coalesces with a steady state of saddle

type. This is in contrast to the more standard case in which the trajectory leaves

the slow manifold at a fold, i.e. at a turning point where two steady state branches

coalesce, as at the end of the silent phase. Other questions relate to the parametric

dependence of how the transition is made (i) from the rest state to the first appear-

ance of bursting, (ii) from a burst pattern with N spikes to one with N+I spikes,

and (iii) from the bursting mode to the continuous spiking mode.

We have begun to explore the continuous spiking to bursting transition and find

evidence of chaotic behavior in some cases (5). As kca is lowered from a value for

the continuous (periodic) spiking response we find a sequence of period-doublings and

then a chaotic spiking regime. For lower kca we enter a regime of chaotic bursting

before arriving at the case of periodic bursting. These observations of aperiodic

behavior were for a low temperature setting in which case the homoclinic terminal of

the periodic branch occurs near the left knee of the steady state Z-curve in the sub-

system bifurcation diagram. It is here that the time scale of the subsystem trajec-

tory, as it goes from the active to the silent phase, becomes comparable to e, the

time scale of Ca. This feature we expect may be exploitable for an analytic treatment

of chaos. As in many studies of deterministic chaos, various aspects may be illus-

316

trated in terms of a one-variable discrete dynamics. The controlling influence of Ca

in the C-K model makes it a particularly meaningful variable. In (5) we develop and

study a single-humped map C n ÷ Cn+ I where C n denotes the calcium concentration at the

upstroke of a spike (say where V = -45 mV). The map is obtained numerically from the

differential equation system (as was done in (17) for a chemical system which exhibits

bursting). This map generates sequences (periodic and, in some cases, aperiodic) which

agree well with the behavior of the full continuous model.

Acknowledgements The author thanks Drs. T. Chay and I. Atwater for encouragement and helpful discussion through various stages of this work.

I. Atwater, I., and P. M. Beigelman. 1976. Dynamic characteristics of electrical activity in pancreatic B-cells. Effects of calcium and magnesium. J. Physiol. (Paris) 72:769-786. 2. Atwater, I., C. M. Dawson, A. Scott, G. Eddlestone, and E. Rojas. 1980. The nature of the oscillatory behavior in electrical activity for pancreatic B-cell. In: Biochemistry Biophysics of the Pancreatic-E-Cell, Georg Thieme Verlag, New York. pp. 100-107. 3. Beigelman, P. M., B. Ribalet, and I. Atwater. 1977. Electrical activity of mouse pancreatic beta-cells II. Effects of glucose and arginine. J. Physiol. (Paris) 73:201-217. 4. Chay, T. R., and J. Keizer. 1983. Minimal model for membrane oscillations in the pancreatic ~-cell. Biophys. J. 42:181-190. 5. Chay, T. R., and J. Rinzel. Bursting, beating, and chaos in an excitable membrane model. Biophys. J. in press. 6. Cook, D. L., W. E. Crill, and D. Porte. 1980. Plateau potentials in pancreatic islet cells are voltage-dependent action potentials. Nature 286:404-406. 7. Doedel, E. J. 1981. AUTO; A program for the automatic bifurcation analysis of autonomous systems, (Proc. 10th Manitoba Conf. on Num. Math. and Comput., Winnipeg, Canada), Cong. Num. 30:265-284. 8. Eddlestone, G. T., A. Goncalves, J. A. Bangham, and E. Rojas. 1984. Electrical coupling between cells in islets of Langerhans from mouse. J. Membrane Biol. 77:1-14. 9. FitzHugh, R. 1961. Impulses and physiological states in models of nerve mem- brane. Biophys. J. 1:445-466. 10. Hindmarsh, J. L., and R. M. Rose. 1984. A model of neuronal bursting using three coupled first order differential equations. Proc. R. Soc. Lond. B 221:87-102. 11. Hodgkin, A. L., and A. F. Huxley. 1952. A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. (Lond) 117:500-544. 12. Kopell, N., and L. N. Howard. 1975. Bifurcations and trajectories joining cri- tical points. Adv. in Math. 18:306-358. 13. Melssner, H. P. 1976. Electrical characteristics of the beta-cells in pancreatic creatic islets. J. Physiol. (Paris) 72:757-767. 14. Plant, R. E. 1978. The effects of calcium ++ on bursting neurons. Biophys. J. 21:217-237. 15. Rinzel, J. Excitation dynamics: insights from simplified membrane models. (pre-print). 16. Rinzel, J., and W. C. Troy. 1982. Bursting phenomena in a simplified Oregonator flow system model. J. Chem. Phys. 76:1775-1789. 17. Rinzel, J., and W. C. Troy. 1983. A one-variable map analysis of bursting in the Belousov-Zhabotinskii reaction. In: Oscillations in Mathematical Biology (ed., J. P. E. Hodgson), Lect. Notes in Biomath. 51, Springer, pp. 1-23. 18. Scott, A. M., I. Atwater, and E. Rojas. 1981. A method for the simultaneous measurement of insulin release and B-cell membrane potential in single mouse islet of Langerhans. Diabetologla 21:470-475.

SEMILINEAR SECOND ORDER EVOLUTION EQUATIONS AND REGULAR BOUNDARY CONDITIONS

Herbert C. Sager

INTRODUCTION

The existence of solutions in a weak sense of

x" = (A + B(t,x,x' ))x - f(t,x,x' ) (O.i)

£ (x) = ~i~(O) + ei2x' (O) + 8i~(T) + Bi~' (T) = O (i = 1,2)(0.2) l

is established, in a Hilbert space H , where t E J = [O,T],eij,Bij ~ ~,

and ' = d/dt . It is supposed that the unbounded linear operator A is

such that A ½ generates an analytic and compact semigroup on H . The

perturbation B is a bounded linear operator and the function f may

have asymptotically sublinear growth.

Method: Linearization and Schauder's fixed point theorem.

Much has been written on the Cauchy problem for the nonlinear, and

on boundary value problems for the linear second order evolution equation.

But results on boundary value problems for nonlinear second order evolution

equations are scarce. Bruck [6] obtains existence and uniqueness for

u" ~ Au + f(t) , u(O) = x , u(T) : y , for a maximal, monotone, multi-

valued operator A. Becker [5] obtains results on abstract Hammerstein

equations which he applies to x" + A2x = B(t,x)x + f(t,x) and x(O) =

x(T) , x'(O) : x'(T) to prove the existence of a mild solution. In his

case B(t,x) is a uniformly bounded linear operator and f(t,x) is a

bounded function. They do not depend on x'

Organization of the paper: In Section 1 a Green's function is con-

structed for the problem x" = (A + eI)x - f(t) , ii(x) : O(i = 1,2) ,

for e h O sufficiently large. In Section 2 a Green's function is con-

structed for x" = (A + B(t))x - f(t) , £i(x) = O(i = 1,2) and it is

shown that the unique weak solution is bounded uniformly for B(" ) in a

set 5 which is compact in the weak operator topology in L2(j,H). Sec-

tion 3 contains the main result and in Section 4 we briefly describe an

application.

Notation: The bounded linear operators in H will be denoted by

H). For the domain of a closed, linear operator A we write D(A)

For J : [O,T] in ~ we denote by L2(j,H) the space of (equivalence

classes of) functions from J to H which are integrable in the sense

318

of Bochner and satisfy f _ ]jllx(t) [12dt < ~ • The inner product in H will

be denoted by ( , ) and that in L2(j,H) by [ , ] , where [x(t) ,

y(t)] : Ij(x(t), y(t))dt .

A detailed version of this paper will eventually appear in ~0 NonZingaa

A a a ~ a i ~ T~A.

1. The Green's Function for the Unperturbed Equation

Here we consider

x" = (A + @I)x - f(t) , (I.i)

f E L2(j,H) , together with the system of boundary conditions (0.2), in

a Hilbert space H . It will be shown in Theorem 1.4 that there exists

a unimue weak solution of (i.i) which satisfies the boundary conditions

(0.2), provided that they are r~ula~ and @ h 0 is sufficiently large.

In this section we will follow Krein [8; p.249-269]. However since we

are working in L2(j,H) the notion of a weak solution will be introduced.

Because of it extra care needs to be exercised when incorporating the

boundary conditions.

Throughout we will suppose that the operator A satisfies

(A.I) A is a closed, densely defined linear operator on a Hilbert space

H with the property that for all i h O

II (A + II)-lll A M(I + ~ )-i

(A.2) A -I is a completely continuous linear operator.

Remark. Fractional powers of A can be defined [8; p.llO, 119, 264].

Let $ ~ O. The operator A + $I satisfies (A.I) and (A.2). Therefore

-(A + eI) ½ generates an analytic semigroup Ve(t) . (1.2)

In the sequel we will abbreviate A + @I to A e or A + $

(A$)-½ is compact and with it also Ve(t) , t > O . (1.3)

The following facts can be proved by virtue of [2; (5.3)], [8; (5.8)

p.l12], [i; p.85].

[I V@(t) rl £ C , uniformly for t E [O,T] and @ ~> O . (1.4)

II A@½11 £ M(@ + 1) -½ ; [I Ve(T)II A C(@ + i) -~ (1.5)

A* Concerning the adjoint of A we have

A* = A ½. . A ½. (1.6)

* (A-½ * (A ½. z E O(A ) if and only if z : ) y for y ~ O ) (1.7)

For the definition of a weak solution of the first order evolution

equation x' = Ax - f(t) we refer to Ball [3].

Definition I.I. A function u ~ C(J,H) is a ~aa~ ~ol~ion of

x" : Ax - f(t) ,

f c L2(j,H), if and only if

319

(a) u'(t) exis~ on (O,T) and A-½U'(") belongs to C(J,H)

(8) (u('),z)' is absolutely continuous on J for any z ¢ O(A*)

(Y) For any z ~ O(A*) and for almost all t e J

(u(t),z)" = (u(t),A*z) - (f(t),z)

Theorem i.i. £ve~ ~ea~ ~olution o[ (i.i) in or the [o~n

(i.8)

-lit -½ u(t) = V # ( t ) x ° + 2 V e ( t - ~ ) A e f ( ~ ) d c ~ + ( 1 . 9 )

O

- I V@ (~ -t/A@ f(~)d~ , (i.iO) V@(T t)YT + 2-1 T _½ t

mhe~e Xo ' YT E H . Fzz~tke~mone eve~ ru~ction o~ the ~o~m (1.9), (i.io)

i ~ a ~ e a ~ 4 o l a t i o a o~ ( i . i ) .

The proof is modelled after Krein [8; p.250] making use of the result in

[3] on weak solutions of first order equations. It required substantial

technical alterations. We mention that u' (-) ~ C(J,H) if and only if

Xo' YT E D(A~)

Now the boundary conditions (0.2) are taken into account. Since for

a weak solution u, u' (o) and u' (T) may not be defined we first consider -½

A@ Li(u) = O (i = 1,2) and x" = (A + @I)x (i.ii)

We are looking for conditions under which this boundary value problem

has only the zero solution. Substituting (1.9), (i.iO) into (i.ii) and

solving for x ° and YT yields

D(A@)x ° = O and D(A@)y T : 0 ,

similarly to [8; p.252]. D(A e) is the operator determinant

D(A e) : LII LI2 (1.12)

L21 L22

-9 -½ where Lij are linear combinations of I, A 4 , Ve(T), A@ Ve(T)

Theorem 1.2. J~ D(A e) in iave~tible then ~eno ia the onl~ ~ea~ ~olulio~

o[ x" = (A + @I)x which 4atiariea 0.2). (The co~ve~ne need not hold./

The proof is similar to [8; p.254].

If the coefficients e., 8 . of £ (x) satisfy the conditions in [8; 13 13 1

p.259] then we call the boundary conditions ~zegulan. From now on we assume

(A.3) The boundary conditions £. (x) = O (i = 1,2) are regular. i

Theorem 1.3. Jr £i(x) = O (i : 1,2) ane re,ulna and i~ @ ~ 0 is au[-

[ i c i e n t l ~ la~e then D(A@) i ~ inve~tible.

~oo[. D(A ) can be multiplied out, [8; p.256, 259-260] and it can only

have the form D(A#) = Cl(I - Re1) or c2A#½(I - Re2) or c3A@l(I - Re3),

where c i are constants and Rei are polynomials in A@ ½ and Ve(T)

without a constant term. By (1.5), @ ~ O can be chosen so large that

320

l[Reil[ < i i : 1,2,3. For such e D(As) is invertible [] .

We return to (i.i) and seek to determine Xo' YT E H in (1.9), (i.iO)

to ensure a unique weak solution of (i.i) and (0.2).

Theorem 1.4. Suppose ~ha~ (A.I) - (A.3) hold a~d thc£ e ~ 0 £a ~u~-

£icie~l~ la~e. YAe~ £he boanda~ value p~oblem (i.i), (0.2) has a ~/qae

~ea~ solu£ioa ~o~ eve~ f e L2(j,H). J~ is ~ivea b~

T u(t) = ~G @(t, a) f(a)da t

~Ae~e Ge(t,o) is Zhe ~eea'a £aac£ioa o£ (1.13).

P~oo~ oa~l£me. The uniqueness follows from Theorem 1.2. Now set

2-1V~(t - a)A e for t ~ a

Vo(t , o) = _ire _½ (o t)A~ for t ~ a

A weak solution of (i.i) is of the form (1.9), (i.iO). By substituting

it into the boundary conditions of (i. Ii) and arguing similarly to [8;

p.259, 260], we arrive at

G~(t, o) = V~(t)h~½Ql(0 ) _ V~(T - t)ie½Q2(o) + V ° t,0) , (1.13)

where Qi(o) are linear operator functions uniformly bounded in 0 and

involving Lij, D(A$) -I and Vo(T,a). One interesting aspect is that

both Xo' YT turn out to belong to D(A~), so that u' e C(J,H). There-

fore u does not only satisfy (i.ii) but also (0.2) ~.

Theorem 1 5. The i~e~al ope~a~o~ I T . oG~(t,a) - do is compacl in L2(j,H).

P~oo~ outline. Since G#(t,0) is compact for almost any t,0, and square

integrable, we can apply a theorem due to Laptev [9] ~.

2. The Perturbed Equation

The Green's function GB(t,s) is constructed for

x" = (A + B(t))x - f(t) (2.1)

with boundary conditions (0.2). Because of Proposition 2.1 we assume

that H is a separable Hilbert space.

Conditions on B(t): Let ~(J,H) denote the set of strongly measurable

maps B : J + B(H). Let B(-) ~ ~(J,H) and liB(t) II ~ N a.e. for t ~ J,

then we define B : L2(j,H) ÷ L2(j,H) by ~x : B(t)x(t).

Proposition 2.1. Becker [4]. ffo~ a~ P > O a~d # h O ~he ae£

S : {BIB(-) z @(J,H) and liB(t) - eIll £ m for a.e. t ~ J}

is compac~ in ~he wea~ ~opolo~ £~ B(L2(j,H)).

Therefore we make these assumptions about B(t).

(A.4) B(-) E m(a,H) and liB(t) eI[l £ P for a.e. t ~ a and some

321

P>O,@~O.

Theorem 2.2 Assume (A.I) (A.4) aad that g ~ 0 is saf[icientl~ large.

Assume ~arthe~ thai

x" = (A + B(t))x (2.2)

has onl~ the weak sola~ioa u(t) ~ O ~hich satiz~ie~ tAe bouada~

c o n d i t i o n s (0.2). [hen the liaean o p e ~ a t o a

T @

= JoG@(t, a)(B(a ) - @I) - da F B

in compact and -i belongs to the nenolvent set o[ F B.

~aoo~ outliae. Since Ge(t,a)(B(a) - @I) is compact for a.e. (t,a),

the compactness follows as in Theorem 1.5. If -i ~ p(F B) then -i is

an eigenvalue. In such a case there exists v c L2(j,H) , and it turns

out that v is a weak solution of the problem (2.2), (0.2). This is

against our assumption ~.

Theorem 2.3. Assume tAat the A~potheses o~ Theorem 2.2 aae sails[ted

~oa ever~ B e 3 (aad that @ ~ 0 i4 4af[icientl~ la~e so thal G@

exists). [Aen the opeaaton4 in S(L2(j,H)) -i

(F B + I)

a ~ e uaiIo~mi~ bouaded foa B ¢ 5.

~oo~. Suppose (F B + I) -I are unbounded in B. Then there exists a

sequence (Bn) in S and (x n ) in L2(j,H) , llXnll = 1 such that

Yn = (FB + I)~In diverges. Set Zn : yn/llynl[ and Wn = Xn/[lYnIl then n

T

Wn(t ) - Zn(t ) = ]oGe(t, a) (Bn(O) - @I)Zn(a)da

Since (Bn(t) - @I)Zn(t) is a bounded sequence in L2(j,H) , we can find

a subsequence which converges weakly to b(-) say. Thus

T Wm(t) - Zm(t) ÷ I G@(t, a)b(a)da = -z , (m + ~) ,

O strongly, i.e. z ÷ z in L2(j,H).

m Since (Bm) is a sequence in a weakly compact set in B(L2(j,H)) there

(Bj ¢ S. From this and exists a subsequence ) weakly converging to Bo

the above it follows that r T

wj(t) - zj (t) ÷ ]oGe(t,o)(Bo (o) - @I)z(a)da = -z(t)

strongly (j ÷ ~) , and it turns out that z is a weak solution of (2.2),

(0.2), [Iz[l = i. This is against our assumption, since Bo E 5 ~.

Remark. By an argument similar to [6, p.161] we can show that

x" = (A + B(t))x and x(O) = O = x(T)

has only the zero solution, provided that B(-) e B(J,H) is strongly

H~ider continuous and that A + B(t) is accretive for every t ~ (O,T).

-i Set GB(t,a)x = (F B + I) G@(t,a)x

322

Theorem 2.4. A.Jaume £ha£ /he h~poZhe~eo o~ Theorem 2.2 hold ~o~ ~ ~ 5.

Then fo~z an~ f c L2(j,H) ~T

u(t) = ]oGB(t,a)f(a)da (2.3)

ia ~he anigae ~ea~ aola/iorL o~ 2he ~oandcz~ value p~oblem (2.1), (0.2).

.7n add/lion T ( CB(t 0)f(old0 (V B + z)-lI~e , : (t,a)f(a)do

~O a,td the~e£o~e T

Io GB(t,a) - do

i_.~ compac/ in L2(j,H) and boarLded an££o~ml~ £o6 ~3 ~ 5.

~)aoof oalline. Theorems 2.2, 2.3 and some technicalities.

3. The Nonlinear Equation.

Here we prove the existence of a weak solution of

x" : (A + B(t,x,x'))x - f(t,x,x') (3.1)

which satisfies Li(x) : O (i : 1,2) of (0.2).

We work in the Hilbert space H of functions f E L2(j,H) whose first

distributional derivative belongs to L2(j,H),

H = H I'2 : {f ~ L2(j,H) If has a distributional derivative in L2(j,H)}

and ~f ~]

[f'g]H : [f'g]L 2 + [~' t L 2

Convergence result:

Theorem 3.1. Aa~ame (A.I) - (A.4) and let @ ~ 0 be ~afficien~l~ large.

A~sa~e /Aak [oh eve~ ~ E 5 ,

x" = (A + B(t))x and i.(x) = O (i : 1,2) (3.2) 1

haa o~1~ / h e ~ea~ n o t a t i o n u ( t ) ~ O. Y£ (Bn) i n 5 aonve~gea ~ e a ~ l ~ t o

e B(L2(j,H)) and i~ (f ) ZnL2(j,H) conve~ge~ ~ea~l~ ~o f e L2(j,H), n

/hen ~he mea~ aola2iona u o£ n

x" = (A + B (t))x - f (t) n n

~hich aa~ia[~ (0.2) convenge a2~on~l~ in H /o u , /he ~ea~ ao~a/ion

o[ (2.1), (0.2).

In the proof we need Theorem 1.7, the fact that o~Ge(t,a) - da is

compact in L2(j,H) and the weak compactness of 5.

(B) Assumptions on B(t,x,x'):

B(t,x,y) < B(H) for all (x,y) c H 2 and a.e. t E J.

B(t,x,y) is strongly measurable in t for each (x,y) ¢ H 2 and

strongly continuous in (x,y) , for a.e. t E J.

B(-,~(-) , ~' (-)) E 5 for every t ~ H .

(F) Assumptions on f(t,x,x'):

' 3 2 3

(x,y) ~ H 2 f(t,x,y) is measurable in t for each and continuous

in (x,y) for a.e. t ~ J. f(.,n(.),p(.)) ~ L2(j,H) for any

n,p E L2(j,H). For every sequence (x n) in H with llXnll £ n one

has

limn ~inf~ n-lll f(t,Xn,X~) liE2 = O

Remark. As in [7; p.20-26] it follows that the operator (n(-),p(-)) ÷

f(',n('),p(-)) is continuous and transforms bounded sets into bounded

sets.

Here is then the main result

Theorem 3.2. A~ame lha£ (A.I) - (A.4), (B), (F) hold a~d le~ @ ~ 0

be a a ~ i a i e ~ £ 1 ~ la~e. f a r , h e r auppoae ~ha t ~or ever~ B ~ 5 the homo-

~eneoaa problem (3.2) admi~a onl~ £he mea~ aolalion u(t) ~ O. The~ (3.1)

ha~ a ~ea~ ~oia~ioa u(') ~hiah salia~iea £he boaadar~ aoadilio~ (0.2)

aad T f

u(t) = ]oGf(t, ~)[f(a,u,u ' ) (B(s,u,u') - @I)u(S)]d0.

~oo~ oa£line. For t E H , x" = (A + B(t, t, ~' ))x - f(t, t, t' ) together

with (0.2) has a unique weak solution u, by Theorem 2.4. This defines

a map W : H ÷ H , W(t) = u , which is compact and continuous and for

which there exists a ball K n such that W(Kn) c K n By Schauder's

theorem W has a fixed point which is the desired solution.

4. Example ref. Tanabe [lO; p.77-87].

Let H : L2(9) , where ~ is a bounded region of class C m in ~n , and

denote by Hm(~) the Sobolev space of order m.

In ~ let

T(x,D) = El al ~ 2p a (x)D a

be properly elliptic. In ~q let

B~(x,D)~ = sl 81 ~ , < m. b4 8(x)mj ' j = 1 ..... p, mj < 2p = 3

and assume that these boundary differential operators are normal.

Define the operator T

D(T) : {u E H2p(~) IBj(x,m)u(x ) = O on ~,j = 1 ..... p}

and

(Tu)(x) : T(x,D)u(x) for u E D(T)

and assume that the half-line arg(l) : O is a ray of minimal growth of

the resolvent of -T, [iO; p.82]. By adding if necessary a number k ~ O,

A = T + kI

satisfies (A.I) and (A.2). The operator in Theorem 1.5 is bounded by

C(@ + i) -½. Therefore we set

324

S = {B(') E B(j,L2(e)) I {IB(t)- 8Ill <h <(e + I)½/C, for a.e. t c J} e

and obtain the following result: ~2 u ~u ~u

- [T(x,m) + B(t,u,-~) + kI]u(t,x) -f(t,x;u,-~) ~t 2

has a strong solution which satisfies

~u au ~ilU(O,x) + ei2~-~(O,x) + 8ilU(T,x) + 8i2]-~(T,x) = 0

for a.e. x E C and i = 1,2 , provided that f(t,u,v) satisfies (F),

B(t,u,v)x satisfies (B)(5 = 5e) and they are H61dercontinuous in all

their variables.

References

i. A.V. BALAKRISHNAN, "Applied Functional Analysis", Springer-Verlag, New York, 1976.

2. A.V. BALAKRISHNAN, Fractional powers of closed operators and the semigroups generated by them, ~aci~ia J. Math., iO (1960), 419-437.

3. J.M. BALL, Strongly continuous semigroups, weak solutions, and the variation of constants formula, Paoa. Ame~. Maih. 5oc. 63 (1977), 370-374.

4. R.I. BECKER, Periodic solutions of semilinear equations of evolution of compact type, J. Math. Aaal. App,. 82 (1981), 33-48.

5. R.I. BECKER, Existence of solutions of Hammerstein equations of com- pact type, ~oc. Cash. ~A£1. Soc. (1981).

6. R.E. BRUCK, Periodic forcing of solutions of a boundary value problem for a second order differential equation in Hilbert space, ~. ~alh. Anal. AppI. 76 (1980), 150-173.

7. M.A. KRASNOSELSKII, "Topological Methods in the Theory of Nonlinear Integral Equations", Pergamon, Oxford, 1964.

8. S.G. KREIN, "Linear Differential Equations in Banach Spaces", Trans- lations Vol. 29, Amer. Math. Soc., Providence, Rhode Island, 1971.

9. G.I. LAPTEV, Eigenvalue problems for second-order differential equations in Banach and Hilbert spaces, Oi~[ere~cial'n~e U~awnenZ~a 2 No. 9 (1966), 1151-1160.

i0. H. TANABE, "Equations of Evolution", Monographs and studies in mathematics 6, Pitman, Belmont, U.S.A., 1979.

Symmetry-Breaking for Semilinear Elliptic Equations

Joel Smoller I and Arthur Wasserman

§l. Introduction. We are concerned with the bifurcation of symmetric

solutions of semilinear elliptic equations into non-symmetric ones.

Specifically, our frame-work is as follows. We consider solutions of

the equation

Au(x) + f(u(x)) = 0 , x ~ B (i)

with linear boundary conditions

~u(x) + Bdu(x)/dn = 0 , x a ~B (2)

Here B is an n-ball, d/dn denotes differentiation in the normal

direction on 3B , and ~ and B are constants. In this note we

shall only consider Dirichlet (~=I,B=0) , or Neumann (~=i,~=I)

boundary conditions; the general case is discussed elsewhere, [6].

Since our domain B is an n-ball, this problem admits, generally

speaking, solutions which depend only on the radius r = Ixl We

think of these radial solutions as the symmetric ones. They satisfy

the ordinary differential equation

u"(r) + n-lu' (r) + f(u(r)) = 0 , 0 < r < R , (3) r

together with the boundary conditions 2

u' (0) = 0 , ~u(R) + ~u' (R) = 0 . (4)

]o

2.

Research supported in part by the N.S.F.; (MCS 80-02337)

The condition u' (0) = 0 is a symmetry condition; it is forced upon us if we want our solution to be smooth at the origin.

326

The equation (3) can be written as a first-order (non-autonomous)

system

u' = v , v' =-n-iv- f(u) , (5) r

and in this notation, the boundary condition (4) becomes

v(0) : 0 , ~u(R) + 6v(R) : 0 (6)

Thus solutions of (5),(6) can be thought of as orbit segments of (5)

which "begin" on the plane u = r = 0 , and "terminate" on the plane

~u + By = 0 , r = R ; see figure 1 where the positive r-axis is to

be thought of as coming out of

II The plane

~u + 6v = 0 r = R

Figure i

> u

the paper. Now a natural parameter in this set-up is the value of

u at r = 0 . Then if u(0) = p (see figure i), we write the

solution as u(r,p) , where u(0,p) = p . Thus, in these terms, if

for some value ~ , a solution u(-,p) of (3),(4) bifurcates into a

non-radial solution of (i), (2), then we say that the symmetry breaks

a_tt u(-,~) ; for short, we sometimes say that the symmetry breaks at

In this note, we shall describe some new results for the symmetry-

breaking problem; complete details will appear elsewhere; (see [5] and [6])

§2. General Remarks. If u(.,p) is a radial solution of (3),(4),

and if the symmetry breaks at ~ , then, of course, ~ is a bifurcation

point, so an easy application of the implicit function theorem shows

327

that zero must lie in the spectrum of the linearized equations; that

is, there must be a solution v ~ 0 of the equation

Av(x) + f' (u(Ixl ,>)v(x) = 0 , x B , (7)

together with the boundary conditions

~v(x) + Bdv(x)/dn = 0 , x e ~B . (8)

Now we can write v

[2], as

in terms of its spherical harmonic decomposition,

v(r,9) = -- ~LA~T(r)~N(e) , N:0

(9)

where 0 < r < R , (where R is the radius of B) , and 9 s S n-1 ,

(n-1)-sphere . Here the %N are linear combinations of eigen-

IN = -N(N+n-2) (I0)

NOW consider the expression (9). If AN / 0 , for some N h i ,

then the kernel of the linearized operator has a component in the non-

radial direction, and this is a necessary condition in order that

there be symmetry breaking. Accordingly, we make the following def-

inition.

Definition. a) We say that the symmetry breaks infinitesimally at ~ ,

if A N / 0 , for some N ~ 1 .

b) We say that the symmetry breaks at ~ , if u(.,~) bi-

furcates into a non-radial solution.

satisfy

where the cN are constants, and JN is a monotonically increasing, 1

q u a d r a t i c f u n c t i o n o f N, [1] The c o r r e s p o n d i n g e i g e n v a l u e s , X N

JN

~0 = i , ~N(6) = Ci}i(8) , if N >_ i ,

i=l

the

functions of the Laplacian on the (n-l)-sphere, and are of the form

328

At this point, it is convenient to distinguish between the various

boundary conditions: (i) Dirichlet (e=l,B=0) , and (ii) Neumann (e=0,

B=I). In both of these cases we shall consider only the "simplest"

type of radial solutions; that is, radial solutions (u(r,p),v(r,p)) ,

0 < r < R , (see (5)) for which the angular rotation 0(r) , 0 < r < R

is -~/2 for the Dirichlet problem, and -~ for the Neumann problem.

(Here 8(r) is defined by tan 0(r) = v(r,p)/u(r,p).) Equivalently,

in these two cases we consider solutions u(r,p) , with u < 0 . r

The bifurcation problem for more general radial solutions will be

discussed elsewhere, [6].

§3. Dirichlet Boundary Conditions. In this case ~ = 1 , B = 0 in

(2) , and we only consider positive radial solutions. From the Gidas,

Ni, Nirenberg theorem, [3], we Know that all positive solutions of

the Dirichlet problem for (i) are actually decreasing radial functions.

Thus if u is a positive solution of (1),(2), (~=i, ~=0) , then

u=u(r,p) , for some p > 0 , and also

Ur(r, p) < 0 0 < r < R . (Ii)

Using this latter fact, it is shown in [4], that if ~ is a bi-

furcation point, then any solution of the linearized equations, (7),

(8), (~=I,S=0) , is of the form

v(r,%) = a0(r) + al(r)¢l(%) (12)

That is, all of the higher modes must vanish, and the kernel of the

linearized equations can have, at most, dimension Jl + 1 £ n + 1 .

' (9) ~ 0 where T D is the Furthermore, a 0 ~ 0 if and only if T D

Dirichlet "time" map; ie

TD(p) = min{r > 0 : u(r,p) = 0} .

Moreover, we have that a I ~ 0 if and only if

329

Ur(~(~) ,p) # 0 (13)

Thus, if the symmetry breaks at p , it is necessary that u(.,p) be

a solution of both the Dirichlet and Neumann problems in the ball

0 ~ r ! ~(P) As an easy consequence of (13), one has that

f(0) < 0 . (14)

Thus, the symmetry can never break if (14) is violated.

The condition (14) is fairly interesting in its own right in

the sense that all of the known existence theorems for the Dirichlet

problem for (i) require that f(0) > 0 . This is due to the fact

that the standard methods for obtaining solutions are of a variational

nature, and as a consequence, the solutions so obtained, inherit cer-

tain stability properties. But, as was shown in [4], if (14) holds,

there are classes of functions f whose corresponding solutions of

the Dirichlet problem cannot be stable. This point is discussed in

[4], where existence theorems are also proved under the condition (14).

Thus, if

f(0) < 0 , (f(u)/u)' > 0 , and f"(u) < 0

(so for example, if f(u) = u - e -u) , then it is shown in [4], that

there are numbers R 2 > R 1 > 0 , such that positive solutions of the

Dirichlet problem for (i) exist if and only if R 1 < R 4 R 2 • These

solutions are all unique, and the symmetry breaks infinissmally if and

only if R = R 2 . Furthermore examples are given in [4] where the

symmetry actually breaks. In such cases, the bifurcation diagram,

near the point of bifurcation, has the form of a cone consisting of an

n-manifold of non-radial solutions, attached, at the bifurcation

point, to a 1-manifold of radial solutions.

Thus, to summarize, for positive solutions of the Dirichlet

problem for (i), the symmetry can break only in the simplest possible

way, in that the kernel of the linearized operator has the form (ii),

330

where only the lowest non-radial mode can be non-zero. In this case,

the bifurcating non-radial solutions form an n-dimensional mani-

fold.

We shall see now that the situation is quite different for other

boundary conditions.

§4. Neumann Boundary Conditions. We consider equation (1), together

with the boundary conditions (2), where now we assume that ~ = 0 ,

8 = 1 . We shall discuss here the problem of infinitesimal symmetry

breaking for the class of radial solutions which satisfy Ur(.,p) ~ 0 .

Thus, let u(r,p) be such a solution of the Neumann problem for (i),

and let T N denote the Neumann "time" map, defined by

TN(p) = min{r > 0 : Ur(r,p) = 0} .

In these terms, u(r,p) satisfies the boundary conditions

Ur(0,p) : 0 = Ur(TN(p) ,p) (15)

Now if the symmetry breaks infinitesLmally at p , it is easy to show

that there must be an integer K > 0 such that AK(r) ~ 0 , where A K

is a solution (see (9)) of the equation

l K AK + --~-- Kn-IA' + (f (u(r,p)) + ~)AKr 0 , 0 < r < R (16)

together with the boundary conditions

AK(0) = 0 = A K' (R) -- 0 . (17)

Here R = TN(p) , and IK satisfies (i0) with N = K . This follows

from using the representation (9) in (7). That there are profound dif-

ferences between the Dirichlet and Neumann problems can be seen very

quickly; in fact, we have the following result, which shows that for

the Neumann problem, the symmetry must break in a more complex way.

Proposition i.

(16), (17).

331

For K = 1 , there are no non-trivial solutions of

Proof. If we differentiate (3) with respect to r, set w = u r '

and multiply the resulting equation by r n-I , we obtain

ii rn-I (rn-lw')' + (f' (u) +--~) w = 0 , (18) r

where we have used the fact that l I = n - 1 ; see (i0).

from (16) we get

11 (rn-iAl)' + (f' (u) +--~)rn-iA 1 = 0 . (19)

r

NOW multiply (18) by A 1 , (19) by w , subtract one equation from

the other, and integrate the resulting expression from r = 0 to

r = R , where, as above, R = TN(p) This gives

w' (R)AI(R) = 0 . (20)

Now using (3), we find

w' (r) + f(u(R)) = 0 ,

Similarly,

Proposition 2. Let u(-,p) be a radial solution of (1), which sat-

isfies the Neumann boundary conditions (2), (e=0,B=l) Referring to

any solution v of the linearized equations (see (9)) , there can be

at most one integer K > 0 for which A K ~ 0

and since f(u(R)) # 0 , we see that w' (R) ~ 0 . Then from (20),

AI(R) = 0 , and since A~(R) = 0 , it follows that Al(r) ~ 0 ,

0 < r < R .

Thus for the Neumann problem, if the symmetry breaks, it must

break with a bigger dimensional kernel, ie., the dimension of the

manifold of bifurcating solutions must be larger than the correspond-

ing dimension for the Dirichlet problem. On the other hand, the

following proposition is valid; (see [6] for the proof):

332

This result is important for symmetry-breaking because it implies that

the non-symmetric part of the kernel of the linearized equations is

irreducible; such a fact is necessary in order that the known bi-

furcation theorems be applicable (see [7], for example).

Again referring to (9), the question arises, given a radial sol-

ution of the Neumann problem for (i), which "mode" survives?; ie.,

which A K is non-zero? That this question has no simple answer, can

be seen from an example constructed in [6], which we now very briefly

describe here.

Let f(u) be a smooth function defined on

following three conditions: i) f(u) = u if

only on 0 < u < c , where f(c) = 0 , and c

IR which satisfies the

u < i; ii) f(u) > 0

is chosen so close to

1 so as to have F(c) - F(1) small; here F is any primative of f;

and iii) f' < 1 . For this f , we can show that the domain of the

time map T N is the open interval (0,c) ; in particular we have that

the domain of T N is a connected set. Next, for p s dom(T N) , and

K a non-negative integer, we define a function q[ by

IK q~(r) = f' (u(r,p)) + --~ , 0 < r < TN(p)

r

Next, define a space of functions

#p = {~ ~ C2[0,TN(P) ]:

by P

¢(0) = ¢' (0) = %' (TN(P)) = 0} ,

and let L~ be the operator on %p into C[0,TN(P) ] , defined by

nKP¢ = ¢,, + n-i PK r ~' + q ~ •

Observe that if we can find a function ¢ e P , ¢ ~ 0 , for which

L~¢ = 0 , then this ¢ can serve as the desired non-zero A K , and

hence we will have shown that the symmetry breaks infinitesimally (in

the K-th mode), on the radial solution u(.,p)

In order to carry this program out, we use the variational

characterization of the principal eigenvalue of L~ ; namely, the

principal eigenvalue ~ of L~ ; satisfies the relation

333

pK p = sup <LKP¢,¢> ,

P II *II ~=l

where

n - 1 r

K 0 >_ 2,, such that if K >_ K 0 , there are points PK and

d o m a i n o f T N f o r w h i c h b o t h r--

PK sup{<L K ~,~>: ~ e~K} > 0 ,

<.,.> is a weighted inner product on ~ , with the weight P

Now in this setting we can show that there is an integer

in the

and

qk sup{<L K %,~>: % C¢~K} < 0 .

Then using the fact that dom(TN) is connected, it follows easily

that there is a point PK e dom(TN) for which

PK sup{<LK ~'¢>: ¢ e~Pk} = 0 .

Then the corresponding eigenfunction AK ' is the desired non-zero

PK element in the kernel of the linearized operator L K , (about the

radial solution u(.,pK)) The integer K 0 is the smallest one

which satisfies

T 2 < -IK0 ,

where IK0 is the K0-th eigenvalue of ~ on S n-1 (see (10)) , t

and T is the radius of the smallest ball for which (3), (4) (e=0,B=l)

has a non-zero solution. Hence, to summerize, for this function f

(see [6]) we have that:

There is an integer K 0 > 1 such that if K ~ ~,

K ~ K 0 , there is a point PK E (0,c) and a function

PK A~ ~ 0 , such that L K AK(r) = 0 , on 0 < r < T(PK) ,

and AK(0) = A~(0) = A~(T(PK)) = 0

334

Thus the symmetry breaks infinitesimally at PK and the non-radial

component of the corresponding eigen-space for the linearized operator

about u(-,pK) , has dimension JK = , a rapidly increasing

function of K. The bifurcation diagram, near the (famil V of decreasing)

radial solutions can be deoicted~ as in f:[g~re 2.

dim JK ~ ~ \/// I

Figure 2 . . . . ~ ~ w ,

Pl P2 " " " PK C

branch of monotone radial

solutions P

That is, along this branch of radial solutions, there is a countable

number of bifurcation points {PK } ' PK ÷ C , where the svmmetry breaks,

and at each such point the dimension of the set of bifurcating non-radial

solutions is greater-than that of the previous point. This result

for the Neumann problem stands in sharp contrast to the rather

rigid way that the symmetry can break for positive solutions of the

Dirichlet problem.

References

i.

2.

3.

4.

5.

6.

7.

Berger, M., P. Gauduchon, and E. Mazet, Le Spectre d'une Variet6 Riemanniene, Springer Lecture Notes in Math., (194), Springer- Verlag: Berlin, 1971.

Courant R., and D. Hilbert, Methods of Mathematical Physics, Vol. I, Wiley-Interscience: New York, 1962.

Gidas, B., W. N. Ni, and L. Nirenberg, Symmetry of positive sol- utions of nonlinear elliptic equations in IR n , Comm. Math. Phys., 68, (1979), 209-243.

Smoller, J., and A. Wasserman, Existence, uniqueness, and non- degeneracy of positive solutions of semilinear elliptic equations, Comm. Math. Phys., (to appear).

Smoller, J., A. Wasserman, Symmetry-breaking for positive solutions of semilinear elliptic equations, Arch. Rat. Mech. Anal., (to appear).

Smoller, J., and A. Wasserman, Symmetry-breakinq for solutions of semilinear elliptic equations with general boundary conditions, (to appear).

Vanderbauwhede, A., Local Bifurcation and Symmetry, Research Notes in Math., (75), Pitman: Boston, 1982.

ON A BOUNDARY VALUE PROBLEM

ASSOCIATED WITH SOME

DIFFERENCE-DIFFERENTIAL EQUATIONS

R.P. Soni and K. Soni

ABSTRACT

The ex i s tence and uniqueness o f the cont inuous s o l u t i o n o f the f o l l o w i n g bound-

ary va lue problem is d iscussed.

Y x x t ( X , t ) = f ( t ) y ( x , t ) + ~ ( t ) ~ (x ) ,

y ( O , t ) = g ( t )

Y x ( O , t ) = h ( t )

y ( x ,O) = d O + xd I + k (x ) ,

where d O = g(O) , d I = h(O) and k(O) = k ' (O) = 0 . This p rov ides the s o l u t i o n to

the system o f equat ions o f the type

#n ( t ) = f ( t ) #n_2( t ) + # ( t ) , n = 2,3 . . . . . t > 0 ,

~ o ( t ) = g ( t ) , t ~ 0 ,

@l ( t ) = h ( t ) , t ~ 0 .

In the spec ia l case f ( t ) = t - b , @(t) = 0 , g ( t ) = t ¥ and h ( t ) = (y + l ) - I t Y+ I ,

y > 0 , the s o l u t i o n y ( x , t ) o f the above boundary va lue problem is the genera t i ng

f u n c t i o n f o r the system of po lynomia ls developed by Soni and Sleeman.

I . INTRODUCTION

Recent ly Soni and Sleeman developed a sequence o f po lynomia ls Pn( t ) which are

c h a r a c t e r i z e d by the d i f f e r e n c e - d i f f e r e n t i a l equa t ion

( I . I ) ( t Y m n ( t ) ) ' = ( t - b) tY Pn -2 ( t ) , n = 2 , 3 , . . . .

These po lynomia ls are o f p a r t i c u l a r i n t e r e s t in the computa t ion o f un i fo rm e r r o r

bounds f o r the asympto t i c expansions o f the i n t e g r a l s of the type

336

(1.2) l ( x , b ) = I t¥ q ( t ) e - x ( t 2 / 2 - b t ) dt , x ÷ ~ ,

when b is al lowed to vary in some neighborhood of the o r i g i n . I f we replace

t Y Pn(t) by Cn(t) and consider the d i f f e r e n c e - d i f f e r e n t i a l equation

(1.3) ¢'n(t) = f ( t ) ¢n_m(t) + a ( t ) Cn_ 2 , n = 2,3 . . . . .

then fo rma l l y , a generat ing funct ion G(x, t ) def ined by

oo

(1.4) G(x,L) = ~ Cn(t) xn/n! n=O

sa t i s f i es the t h i r d order pa r t i a l d i f f e r e n t i a l equation

Gxxt(X, t ) = f ( t ) G(x, t ) + a ( t ) ~(x) (1.5)

where

(I .6) eo

9(x) = ~ c xn/n! n=O n

This provides the mot ivat ion for studying the above p a r t i a l d i f f e r e n t i a l equation.

Under appropr ia te boundary cond i t ions , the system (1.3) has a unique so lu t ion when-

ever (1.5) has a unique so lu t ion which is ana ly t i c at x = 0 .

2. MAIN RESULTS

We prove the fo l l ow ing .

THEOREM I . Suppose that y ( x , t ) is def ined in [0,~) × [0 ,6 ] , 6 > 0 and

( i ) Y' Yx' Y t ' Yxx' Yxt ' Yxxt are continuous in [0,~) × [0 ,6 ] ,

( i i ) ~(x) , k" (x) are continuous and of f i n i t e exponent ia l order in

( i i i ) f ( t ) , a ( t ) are continuous in [0 ,6] .

Then there is exac t ly one funct ion which is of f i n i t e exponent ial order in

formly in 0 < t < 6) and sa t i s f i es

[ 0 , ~ ) ,

x ( u n i -

(2.1)

3 3 7

Yxx t (X , t ) = f ( t ) y ( x , t ) + a ( t ) ~(x) , x ~ 0 , 0 < t < 5 ,

y (O, t ) = g ( t ) ,

Yx(O,t) : h(t)

y ( x , O ) = d O + xd 1 + k (x

I t is given by

(2.2) y ( x , t ) = g ( t ) + xh( t ) + f t 0

+ ft 0

+ I t 0

rX

f t

F( t ) : f (u ) du 0

, k ( O ) = k ' ( O ) = 0 .

where

(2.3)

and

(2.4)

f (u) R2(x, F( t ) - F(u)) g(u) du

f (u) R3(x , F( t ) - F(u)) h(u) du

a(u) ~(v) Rl(X - v, F( t ) - F(u)) dv du 0

k ' (u) Ro(x-u, F( t ) ) du ,

PROOF OF THEOREM I . We sketch the proof a f t e r some pre l im inary remarks. I f

y (x , t ) is un i formly bounded by a funct ion of f i n i t e exponent ia l order in x , then

under the stated assumptions, so are Yx' Yxx and Yxxt " This j u s t i f i e s the use

of the Laplace transform. By the condi t ion ( i ) , d O = g(O) and d I = h(O) . Fur-

thermore, g ( t ) and h( t ) are cont inuously d i f f e r e n t i a b l e in [0, ~] . The func-

t ions Rn(X,~) def ined by (2.4) are ana ly t i c funct ions of x and ~ . By the

residue theorem,

co

(2.5) Ro(X,~ ) = ~ (am/m!) x2m/(2m)! m:O

inc lud ing s = 0 .

C is any simple closed contour in the complex s-plane, p o s i t i v e l y or iented and

Rn(X,~) = (2~i) - I f s -n- I e xs+~/s2 ds , n = 0, I . . . . . C

338

and fo r n = 1,2 . . . . .

x (2.6) Rn(X'm) = ~ fo (x - u) n-I Ro(u,m)du o

We also need to observe tha t

~0 e-sx Rn(X,~)dx e ~/s2 , Re s > 0 s-n- I

and that a bound fo r Rn(X,m ) can be obtained eas i l y by (2 .4) .

To prove (2 .2) , apply the Laplace transform to the f i r s t and the las t equation

in (2.1) and use the boundary cond i t ions . I f

(2.7) Y (s , t ) = e -sx y ( x , t ) d x 0

and ~(s) , K(s) denote the corresponding Laplace transform of ~(x) and k(x)

respec t i ve l y , we obtain

(2.8)

(2.9)

Y t ( s , t ) - s -2 f ( t ) Y (s , t ) = s -2 h ' ( t ) + s - l g ' ( t ) + s -2 a ( t ) ~(s) ,

V(s,O) = s -2 h(O) + s - I g(O) + K(s) .

The so lu t ion Y(s , t ) of the above f i r s t order ord inary d i f f e r e n t i a l equation is

(2.10) Y (s , t ) = g ( t ) / s + h ( t ) / s 2

+ e ( F ( t ) - F ( u ) ) / s 2 f (u ) {h (u ) /s 4 + g(u) /s3}du

0

f' + e (F ( t ) -F (u ) ) / s2 a(u) # (s ) /s 2 du 0

+ e F ( t ) / s2 K(s) .

Now apply the Laplace invers ion theorem and the convolut ion theorem to obtain

y ( x , t ) . The uniqueness of the so lu t i on fo l lows from the uniqueness property of the

Laplace transform. The existence fo l lows by a s t ra igh t fo rward v e r i f i c a t i o n of the

so lu t ion . The computation is s impler i f we use the series expansion fo r

Rn(X,F(t) - F(u)) f o r t h i s purpose.

THEOREM 2. Suppose that the sequences {Cn}, {d n} are bounded and f ( t ) , a ( t ) ,

339

g ( t ) , h ( t ) are cont inuous in [ 0 , 6 ] . Then the system

(2.11)

' t = ( t ) + a ( t ) , n = 2,3 . . . . .

i ~n ( ) f ( t ) #n-2 Cn-2

# o ( t ) = g ( t ) ,

# l ( t ) = h ( t ) ,

#n(O) d n , n = 0 , I . . . . . g(O) = d O , h(O) = d I ,

has a unique s o l u t i o n .

(2 .12) (b2n+2 ( t ) = f t 0

I t i s g iven by

f ( u ) g(u) { F ( t ) - F (u ) }n /n ! du

n ft C2n_2k

k=O 0

n

+ ~ d2n_2k+ 2 k=O

a(u) { F ( t ) - F ( u ) } k l k ! du

F k ( t ) / k ! , n = 0 , I . . . . .

(2 .13) #2n+3( t ) = I t 0

n +Z

k=O

n +Z

k=O

f ( u ) h(u { F ( t ) - F (u ) }n /n ! du

a(u) {F(t) - F(u)}k/k! du C2n-2k+l 0

d2n_2k+ 3 F k ( t ) / k ! , n = O,l . . . .

PROOF OF THEOREM 2. We can show t h a t under the g iven c o n d i t i o n s , { # n ( t ) } is

bounded in 0 < t < 6 . Def ine

co

y ( x , t ) = X q~n ( t ) xn/n! n=O

co

(2 .14) ~ (x ) = X c n xn/n! n=O

co

k (x ) = ~ d n xn/n! . n=2

Then y ( x , t ) s a t i s f i e s (2 .1 ) . I t a l so s a t i s f i e s a l l the c o n d i t i o n s o f Theorem 1

except the d i f f e r e n t i a b i l i t y c o n d i t i o n on g ( t ) and h ( t ) . This c o n d i t i o n how-

ever , i s not necessary. By s t r a i g h t f o r w a r d v e r i f i c a t i o n ~ we know t h a t y ( x , t ) de-

f i ned by (2 .2 ) i s a s o l u t i o n o f the boundary va lue problem. Fur thermore, by using

the r e p r e s e n t a t i o n

340

oo

Rk(X,a) = ~ ( ~ n / n ! ) x2n+k / (2n + k) ! n=O

and the s e r i e s r e p r e s e n t a t i o n s ( 2 . 1 4 ) , we f i n d t h a t the r i g h t s ide in ( 2 .2 ) is a

conve rgen t power s e r i e s in x . Hence by the uniqueness o f the power s e r i e s expan-

s i o n , we o b t a i n (2 .12 ) and ( 2 . 1 3 ) . The uniqueness o f the s o l u t i o n f o l l o w s f rom the

f a c t t h a t the system

A'n(t) = f ( t ) An_m(t) , n = 2,3 . . . . .

A o ( t ) = A l ( t ) = 0 ,

An(O) = 0 , n = 0 , I

has o n l y the t r i v i a l s o l u t i o n .

3. APPLICATIONS

We g i ve some a p p l i c a t i o n s o f Theorem 2.

Example I . Let f ( t ) = t - b , c = d n n

e l ( t ) = t ~ + I / y + I , ¥ > 0 . Then

F ( t ) = ( t 2 - 2 b t ) / 2 .

By Theorem 2, f o r n = 0 , I . . . . .

¢2n+2( t ) = (2n n ! ) - I / t O

and

= 0 , n = 0 , I . . . . . Co ( t ) = t Y

uY(u - b) { t 2 - u 2 2 b ( t - u ) } n du

¢2n+3 ( t ) = (2 n n ! ) - 1 / t ( u Y + l h + l ) (u - b) { t 2 - u 2 - 2 b ( t - u ) } n du . 0

By us ing i n t e g r a t i o n by p a r t s , the l a s t i n t e g r a l can be reduced to the form g iven in

[ 4 , p. 4 ] where Cn( t ) = t ~ Pn ( t ) , y > - I .

Example 2. Cons ider the system

341

#~(t) : t #n_2(t) + cos t Cn_ 2 , n : 2,3 . . . . .

#o(t) = sin t

# l ( t ) = 0

#n(O) = d n , n = 2,3 . . . . .

By Theorem 2, [3, p. 69] and [3, p. I I ] ,

¢2n+2 = 2n-I/2 ~ tn+3/2 Jn+3/2 ( t )

n

Z C2n_2k k=O

2k-I/2 ~'~ tk+I/2 Jk+I/2 ( t )

n

k=O d2n_2k+ 2 t2k/(2 k k[)

= i 2k-I /2 ~ tk+I/2 /2 ( t ) #2n+3 k~O C2n-2k+l Jk+l

n

k=O d2n_2k+ 3 t2k/(2k k!)

REFERENCES

I. R. Bellman and K. Cooke, D i f fe rent ia l -D i f fe rence Equations, Academic Press, 1963.

2. P.W. Berg and J.L. McGregor, Elementary Part ial Di f ferent ia l Equations, Holden Day, 1966.

3. A. Erd~lyi, Wo Magnus, F. Oberhettinger and F. Tricomi, Tables of Integral Transforms, Volume I , McGraw-Hill, 1954.

4. K. Soni and B.D. Sleeman, On uniform asymptotic expansions and associated polynomials, UDDM Report DE 82: 4, 1982.

5. D.Vo Widder, The Laplace Transform, Princeton University Press, 1946.

A-proper maps and bifurcation theory

J.R.L. Webb and S.C. Welsh (%)

Introduction

Bifurcation theory is concerned with proving the existence of nontrivial

solutions of an equation of the form F(u, I) = O, depending on a parameter I,

when it is known that F(O, l) = O. It is often assumed that F is Frechet

differentiable so can be written

FCu, I) = u - T(1)u + R(I, u)

where T(l) is a bounded linear operator and R is of '%igher order".

A general type of result is that a branch of nontrivial solutions can only

emau~ate from a "characteristic value" of /he linearized problem, that is 10 such

that the null space N(I - T(10)) is not trivial. However such bifurcation does

not necessarily occur but often one can prove bifurcation from characteristic

values of odd multiplicity. This can be done by using a theory of topological

degree and showing that, as a function of the parameter I, the degree changes as I

moves through a characteristic value.

Rabinowitz [3] first showed how degree theory arguments can be used to prove

global bifumcation results, whereby, not only is it shown that bifurcation occurs,

but also information is given as to the global behaviour of the bifurcation branch.

In the present paper we wish to give global results when T(l) has the for~D

n

$=1

Results for problems of this kind have been given by Toland [6] who assumed each B. J

was compact. He employed a result of Krasnosel'skii [i] concerning the Leray-

Schauder degree of a mapping on a topological product (or direct sum) space.

We shall generalize Krasnosel'skii's result to another class of operators,

the Approximation-Proper (A-proper for short) mappings of Petryshyn [2]. This

extension leads directly to global bifurcation results.

The degree of A-proper maps on a direct sum space

The A-proper maps are defined in terms of projection schemes. We say

that F = {~, Qn} is an admissible scheme for the Banach space E if each Qn is a

linear projection operator with finite dimensional range X n = %(E) and Qn~ -+x

for each Z in E. A not-necessarily linear map f : E-+ E is said to be A-pmoper

(%)Research supported by a Science and Engineering Research Council Studentship.

'~ 343

(with respect to F) if each Qn f : %n --~ X n is continuous and whenever x n 6 X n

(n = i, 2, ...) is a bounded sequence such that, for a subsequence xk,

Q~(x k) -~ w,

there is a further subsequence {x } converging to a point x and f(x) = ~). m

Thus the A-proper maps are those for which one can seek solutions of the pro-

blem f(x) = w as strong limits of solutions of the associated finite dimensional

problems Qnf(~) : Q# Examples of A-proper maps are maps of the for~ I + k where k is compact.

There are many other examples such as A + g where A is accretive and g is condensing

(for example [7]). We refer to the survey article [2] for more infor~,ation.

We wish to generalize a result of Krasnosel'skii [i] on the degree of a map on

a direct sum of spaces. Thus let E, El, E 2 be Banach spaces with E = EI~ E2, with

compatible norms. Let U, UI, U 2 denote the respective open unit balls. The result

of Krasnosel'skii as specialized to linear operators is as follows

Lemma ([I]! ~.129)

Let T. : E.-+ E. (j = i, 2) be compact linear operators such that J J J

I - Tj : Ej.-~ E.j (j = i, 2) are homeomorphisms. If x = xl + x 2, zj E E., define

Tx = TIx I * T2x 2. Then the Leray-Schauder degrees are related by

dLS(I-T , U, O) = dLS(I-TI, UI, O) dzs(I-T2, U2, 0).

We shall prove a similar result when we replace the compactness requirement by

an A-proper one and we employ a generalized degree. We briefly review the def-

inition of degree for an A-proper map f.

Let D be a bounded, open set in E and let f : ~-~ E be A-proper. Suppose

f(z) ~ 0 for all x in ~D (the boundary of D). Then, it is readily shown that

~nf(x) ~ 0 for all x in ~D N Xand all sufficiently large n so the Brouwer degl, ees

d n = d(Qnf , U fl X, O) are defined for sufficiently large n. The degree of the

A-proper map f is then defined by

Deg(f, U, O) = set of all limit points of {tin} , including +~.

Although Deg is no longer a singleton (in general) the usual properties of degree

hold in a modified for~.

Now let E = E l ~) E 2 where E l is finite dimens~l. Suppose T : E-+ E is such

that I - T is A-proper (relative to F) and that E l and E 2 are both invariant under T.

Let T I, T 2 denote the restrictions of T to E l , E 2 respectively. Finally let P be

the projection of E onto E l, so that P is compact.

344

Lemma I. I - T 2 ; E2-+E 2 is A-proper relative to r', where we take projections

Qn' = (I - P)Qn and subspaces X v = Qn' (E).

Proof For simplicity, we denote all subsequences by the same fixed subscript.

Thus, suppose z n £ X n' is a bounded sequence with

As P is compact, P~nT2Xn -+ q n n

X -*v. Then we obtain u n E , and we can suppose Pun

By A-properness of T relative to F, it follows that un-+ u where

u - Tu = w + v - Tv - q. Hence also v = Pu and q = PT(u - u).

z --* m where m = u -V and n

x - (I - P)T(x) = w.

(say). Also m can be written as z = u n - Pun,

Thus

As x = (I - P)u, z £ E2, so that Tm £ E 2 also. Therefore

x - T2(z) = w, as required.

Lemma 2. Let E = E l ~ E 2 with E l finite dimensional and let % : E. -+ F j

(j : i, 2) be bounded linear operators. Suppose I - T 2 is A-proper relazive to a

scheme r2= {~, Pn}. For z = z I + z 2 let Tx = Tlz I + T2z 2. Then I - T is A-

proper relative to the scheme {E l e Y n, P' } where Pn'(=l + =2 ) = x I + Pn=2.

- : e + ~n' en £ El' Yn E Yn" Then we Proof Suppose z n Pn'Tzn --~ w, and write x n n

can suppose en--* e 6 E l and that ~len-+ Tle. Then ~n - PnT2~n-+ Tl e - e + W. As

T 2 is A-proper we obtain ~n-+ y £ E 2 and ~ - T2Y =Tle - e + w. This shows that

m -+x with z - Tz = ~. n

Now let E = E l C~)E 2 with E l finite dimensional, T : E-* E be such that I - T

is A-proper (relative to F) and with T. : E.--* E. (j = I, 2) as before. We J J J

suppose I - T. are homeomorphisms on the spaces in which they act. Then I - T J

is A-proper (1-elative to a scheme r2, which we denote {Yn' Pn })' using Lemmas 1 and

2. We remark that the degree Deg(l - T, U, O) is the set of all limit points of

the Leray-Schauder degrees

dLS(I - PnT, U, 0).

By Krasnosel'skii's result, this set coincides with the set

Now PnTl = Tl on El so the first term equals dLS(I - T 1 , U I, 0).

gives the degree, Deg(l - T2, U2, 0). We have themefore proved

The second term

345

Theorem i. Under the above hypotheses,

Deg(l - T, U, O) : dns(I - TI, U I, O) Deg(l - T 2, U2, 0).

This result is unsatisfactory as it stands since we begin with T being A-

proper relative to one scheme and prove a result for another scheme. We shall

now show that the degree above can be taken with respect to the original scheme.

Lemma 3. Let I - T be A-proper relative to {Xn, Qn} and let L be defined by

L(xl + x 2) = x I ÷ (I - P)Qnx2, where P is the projection of E onto E l . Then, if

(I - T)x ~ 0 for all x E aU, we have, for all n sufficiently large,

d ( I - Q n T , u n ~ , o) = d ( I - ~ T , un ~ , o).

Proof. Since x = ~ z ) = ~ ( E ) : ~ and ~ - Qn ~ maps ~ n X into X , properties of Brouwer degree show that

d ( ~ - Q ~ , u n ~ , o ) : d ( ~ - ~ , u n y , o ) .

Now let ~(y, t) : t~n~y + (1 - t ) P y , y E Un Yn' 0 < t ~ l .

We claim that H(y, t) ~ y, for all y E @U N Yn' if n is large enough.

not, there are sequences with

In fact, if

tnQnTY n + (i - tn)PnTY n - Yn = O.

We can write Yn : en + Un" en £ El' Un : (I - P)QnZn, z n £ E 2. The terms en,

Ten, PQnTUn all converge (for subsequences) so we have

- -+W (say) QnTUn u n

It follows that QnT(Qnzn ) - Qnzn --~i" By A-properness of T this finally yields

x --~x with Tx : x, a contradiction as x E aU. The homotopy property now implies n

the conclusion of the Lemma

Chan~e of de~ree and bifurcation n

We shall consider operators T(X) with T(X) = Z XJB" and will give results

similar to ones of Toland [6]. j:l J

The "characteristic values" of T are defined by

ch(g) = {~ E m : N(I - g(l)) # {0}},

where N(.) denotes the null space (kernel) of the operator.

Given a bounded linear operator T, the ascent of T is the smallest positive

integer p such that N(T p) = N(T p+ i), if such a p exists, otherwise it is ~.

The descent is the smallest integer q with R( T q) = R(T q+l). If the ascent and

346

descent are both finite, they are equal and X can be written as

x = ~(~5 @ R(T~.

This is often called the Riesz decomposition. Proofs of these facts can be found,

for example, in Taylor's book [5].

Our basic assumption on ch(T) is that ch(T) contains a smallest nonnegative

element I 0 and that this is isolated.

We make two alternative decompositions of The space E. In the first we

suppose E has a Riesz decomposition corresponding to I - T(k 0) so that

E : N 1 ~ RI,

where we suppose that

dim N 1 = dim N(I - T(10))P

is finite. This dimension is the (algebraic) multiplicity of 10 .

We also assume

(i) n is an odd integer and & is injective on NI,

(ii) I - T(1) is A-proper (relative to a scheme F) for all I ~n an interval

(a, b) containing [0, 10] ,

(iii) B. commutes with B. (i ! i, j in), z j

(iv) If [I - T(10)]u = 0 for u # O, then It - f(~)]u # 0 for all ~ # 10, B E m.

We then have

Theorem 2. Let E be a real Banach space with an admissible scheme F and let T(1)

satisfy (i) (ii) (iii) and (iv) above where l 0 is the smallest positive element of

ch(T) as prescribed earlier. Then there exists ~ > 0 such that for

I 6 (k o, k o + E),

Deg( l - T(1), U, O) : { ( - i ) v}

where ~ is the multiplicity of 10 and U is the open unit ball in E.

Proof. (We remark that this px'oof is practically identical to Toland's [6].)

We have E = N 1 ~ RI, as above. Assumption (iii) shows that 2(I) commutes

with T(10) and, therefore, N 1 and R 1 are both invariant under I - T(l). Let ~> l 0

be such that ~ is less than any other positive element of ch(T). As I - T(l-~ is

one to one and A-proper, it is a homeomorphism of X onto X. For z 6 R 1 let

H(x, t) : x - T(tA--)z, 0 < t < I. By hypothesis, H(.,t ) is A-proper for each t.

Moreover, if B(z, t) : 0 we would have t[E ch(T) and therefore t[ : 10 . This

would imply That z E N(I - T(10)) and since m E R 1 this gives m = O. Therefore,

I - T(~-~ is homotopic to I on R 1 and Deg(l - T(~-~, UI, O) : {i}, where U 1 is The

open unit ball in R I.

In NI we take the homotopy

n H(z, t ) : ( 2 t - l ) z - [ ~iti/n(2t -1)(n-()/"B.z.

i=l

I

347

As N 1 is finite dimensional we only use the degree of continuous maps.

topy is well defined since (-i) I/n is a real number for n odd. Suppose

H(X, t) : O for some X, t. If t : ½ we would have

so by (i) x : O.

Thus, by (iv), k 0

The homo-

½~ nB x = O, n

So t ~ ½ and we obtain

x - TCYCt/C2t - l))i/n)x : O.

= [(t/2t - i) I/n, unless x = O. As (t/2t - i) I/n lies in the

range (-~, O] U [I, ~), this is impossible. Therefore x : 0 and this shows that

dLs~± ~D, up, 0~ : dLS~Z, U R, 0~

which is well known to be (-i) ~. By Theorem i, we obtain the proof of Theorem 2.

Our second alternative is when we assume that

(v) (I - T(1))N(I - T(k0)) N R(I - T(10)) = {0}, for k # 10 .

It then follows that

E : N 2 ~E2, where N 2 : N(I - 2(10))

and E 2 = (I - T(kI))-IR(I - T(k0)) , for any II not in ch(C), which does not depend

on 11 .

Theorem 3. Assume that hypotheses (i) to (v) hold. Then the conclusions of

Theorem 2 hold with the modification that

= dim(N(l - T(10)).

Proof. Since T(1) commutes with T(l 0) it is easily verified that N 2 and E 2 are

invariant under T(l). The proof proceeds exactly as before.

Remark. Hypothesis (v) is a standard type employed in bifurcation theory but was

not considered in this context in Toland's work [6].

Another set of hypotheses is possible when X is a Hilbert space H.

Theorem 4. Suppose 10 is as before and hypotheses (ii) and (iii) hold. Suppose

also that N(I - T(k0))2 = N(I - T(10)) = ~, that (Bju, ~)_~> 0 for all u in N and n

that ~ (Bju, u) > 0 for u £ N, u # 0. Then the conclusion of Theorem 3 holds. j=l

Proof. We have H = N DR where R = R(I - T(A0)). As before

Deg(l - f([), UI, O) = {+ i}.

348

In N we use the homotopy

H(x, t) = (2t - l)x - tT(~)x.

The positivity assumptions assure that this homotopy has no zeros for X ~ 0 and so

the proof follows.

This proof is exactly the same as the proof of Theorem 1.25 of Toland [6]; we

refer there for further details.

The global bifurcation result applies to problems of the form

u - T(l)u + R(~, u) = O, where

I - T(1) is as above and R is continuous withl;R(l , u~ I lull -+ 0 as I I~II-+ 0 uni-

formly for I in bounded intervals. We assume that I - T(I) + R(I, .) is A-proper

fop all i in some interval (a, b) finite or infinite.

Theorem 5. Suppose T(~) satisfies the hypotheses of one of Theorems 2 - 5 and

suppose ~0 has odd multiplicity. Then bifurcation occurs at ~0" Moreover, the

branch of nontrivial solutions emanating from ~0 has at least one of the following

properties:

(i) It is unbounded.

(2) It contains a point (ll, O) with 11 6 ch(f), l I # l 0.

(3) I approaches the endpoints of the interval (a, b) on which I - T(1) is

A-proper.

The proof is by showing that Deg(l - T(I), U, O) changes as I moves through 10.

For I < ~0' I - T(~) is homotopic to I since l 0 is the smallest positive element of

ch(T), while for I > I0, the previous Theorems apply. The idea of this proof goes

back to Rabinowitz [3], who applied it when T(I) = AT with T compact. Stuart [4]

extended this to allow T to be a k-set contraction. Theorem 2.6 of Toland [6]

proves Theorem 5 when each B. is compact. J

We give an example of linear operators which fit into our framework but fall

outside that covered by Toland [6].

Let X be a Banach space and C : X-~ X be a compact linear map and let

Thus

where

T(A) = IC + A2C * k31

I - T(X) = (i - ~3)(I - ~(I)C), (X % i)

~(~) = XC1 + ~)I(i - X3),

and I - T(1) : -2C.

Therefore, for ~ % I, I - 2(I) is A-proper. We suppose that the smallest character-

istic value of C is ~ = 5/7 . This corresponds to ~ = ½. Moreover, by considering

349

the graph of ~(l), one sees that p(1) increases for I between 0 and 1. Also ~(~)

has a positive maximum for I between -2 and -3 of approximately 0.23. Thus ~0 is

the smallest element of oh(T) and ch(T) is discrete. It is easy to see that C can

be chosen to fit the hypotheses of Theorem 4.

~FE~N~S

i. M.A. K~asnosel'skii, Topological methods in the theory of nonlinear integral equations, Pergamon, London and New York 1964.

2. W.V. Petryshyn, "On the approximation-solvability of equations involving A-proper and pseudo A-proper mappings", Bull. Amer. Math. Soc. 81 (1975), 223-312.

3. P.H. Rabinowitz, "Some global results for nonlinear eigenvalue problems", J. Funct. Anal. 7 (1971), 487-513.

4. C.A. Stuart, "Some bifurcation theory for k-set contractions", Proc. London Math. Soc. 27 (1973), 531-550.

5. A.E. Taylor, Introduction to Functional Anal~tsis, Wiley & Sons, New York and London, 1958.

6. J.F. Toland, "Topological methods for nonlinear eigenvalue problems", Battelle Mathematics report no. 77, (1973).

7. J.R.L. Webb, "Existence theorems for sums of k-ball contractions and accretive operators via A-proper mappings", Nonlinear Analysis TMA, 5 (1981), 891-896.

ON THE SINGULARITIES AND ASYMPTOTIC EXPANSIONS OF SINGULAR STURM-LIOUVILLE EXPANSIONS

A. I. Zayed

Let ¢(x,s) be the solution of the following singular Sturm-Liouville

problem :

y " - q ( s ) y = - t y ( ] . 1 )

l=s 2 and x>O with the boundary conditions

y(O,X):sina , y'(O,l)=-cos~

and

ly(0,x) I<~

where q ( x ) i s assumed to be

i ) a n a l y t i c i n the r i g h t h a l f - p l a n e Rez>0 and r e a l f o r z r e a l .

i i ) i n t e g r a b l e on [ 0 , ~ ) .

C o n d i t i o n i ) i m p l i e s t h a t ~ ( x , s ) i s a n a l y t i c i n t he h a l f - p l a n e Rez>0

and c o n d i t i o n i i ) i m p l i e s t h a t t he s p e c t r u m a s s o c i a t e d w i t h t he

p r o b l e m i s d i s c r e t e and b o u n d e d f rom be lo w f o r X~<0.

I n two r e c e n t p a p e r s [71 and [ 5 1 , we have d e v i s e d a t e c h n i q u e to

l o c a t e t h e s i n g u l a r p o i n t s o f t h e f u n c t i o n f ( x ) g i v e n by

o o

f(x)= fF(s) , (x,s)d0(s) (1.2) - c o

where F ( s ) i s t he " g e n e r a l i z e d F o u r i e r c o e f f i c i e n t " o f f ( x ) , i . e . ,

N F ( s ) = l . i . m . f f ( x ) , ( x , s ) d x ( 1 . 3 )

N - ~ 0

and dO(s) i s t he s p e c t r a l m e a s u r e . In t h i s t a l k we s h a l l assume t h a t

do has s u p p o r t i n [ 0 , ~ ) .

The t e c h n i q u e we u s e d i s b a s e d on r e l a t i n g t he s i n g u l a r i t i e s o f f ( x )

to t h o s e o f t h e f u n c t i o n g ( z ) g i v e n by

g(z) = I-/- ] F(s)eiSZds, Imz>O (I .4) 2/2~0

351

In 17] we assumed that F(s)=0(e -s) as s ~, however in [5] we

extended the results tIO the case where F(s)=O(s n) as s ~ where n is

a constant. It should be pointed out that the difference between the

case where F(s)=O(e -s) and the case where F(s)=O(s n) is by no means

trivial, since in the latter both f(x) and g(z) may not exist in the

ordinary sense. Nevertheless, we showed that both f and g exist as

generalized functions and in addition to that, we showed that the

singularities of the analytic representation of f can also be located

by comparing them with the singularities of the analytic representa-

tion of g.

Our investigation of the singularity problem has also led us to

consider the problem of finding the asymptotic expansion of f(x) as

x~ assuming that we know the asymptotic expansion of F(s) as s~0 +

and vise-versa. More precisely, we want to determine the asymptotic

expansion of f(x) as x~ assuming that F(s) has an asymptotic expan-

sion of the form

k as s~0 + (1.5) F ( s ) ~ Z aks k = o

As Watson's lemma may suggest, one may try intuitively to substitute

the series (1.5) into (1.2) and integrate term by term.

Unfortunately, this technique will not work because one will encoun-

ter integrals of type

fsn¢(x,s)dp(s) (1.6) 0

which a r e , i n g e n e r a l , d i v e r g e n t .

I f t he f u n c t i o n ~ ( x , s ) i s o f t he form ~ ( x s ) , t h e r e i s a number o f

me thods t h a t c an be u s e d to a t t a c k t h i s p r o b l e m and a s s i g n v a l u e s to

t h e s e d i v e r g e n t i n t e g r a l s ( s e e [6] f o r r e f e r e n c e s ) . Some o f t h e s e

me thods a r e t h e Abe l s u m m a b i l i t y and t he M e l l i n t r a n s f o r m t e c h n i q u e s .

U n f o r t u n a t e l y , t h e f u n c t i o n ~ ( x , s ) i s n o t i n g e n e r a l o f t y p e ~ ( xs )

and, in addition, the presence of the factor dp(s) adds complexity

to the problem. Furthermore, to evaluate the integrals (1.5) by

using either the Abel summability or the Mellin transform technique,

one has to know ¢(x,s) and dp(s) explicitly. In this talk we shall

discuss a new technique which a?oids some of these difficulties and

will enable us to assign values to the (generally divergent) integrals

(1.6). This technique is based on the theory of generalized functions

and the fact that the kernel of the integral transform (1.2) is a

solution to a singular Sturm-Liouville problem.

352

2. Preliminaries :

Let C ~ be the space of all infinitely differentiable functions on

(-=,~) provided with its standard topology and E be its dual space,

i.e. E is the space of all generalized functions with compact support.

Furthermore, let S be the space all rapidly decreasing C~-functions

on (-~,~) provided with its standard topology as described in [I].

Let S* denote the dual space of S. S is usually called the space of

tempered distributions.

Let feE, then the Fourier transform If of f and the analytic repre-

sentation f(z) of f are defined by

If(x)= I <f(t), e itx> (2.1)

and

f (z )= 1 <f( t ) , t l_~>, Imz~O. (2.2) 2~i

respectively.

It is known that f(z) is analytic at z as long as z does not belong

to the support of f. Moreover, f can be recovered from f(z) via

the formula

oo

lira ~ [ f ( x + i E ) - f ( x - i c ) ] ¢ ( x ) d x = < f , ¢ > e-~O -co

(2.3)

for any ~eE. •

These definitions can also be extended to the space S , see [I]. ,

We shall say that a generalized function feS is even (odd) if

<f,~>=O

for all odd (even) functions ~eS. For example the Dirac B-function

is even since 4(0)=0 for all odd functions ~eS. A tempered distribution

feS is said to vanish in an open set Uc(-~,~) if and only if

<f,~>=O for any element ~ of S whose support is contained in U. Two

tempered distributions f and g ~re said to be equal in U if and only if

<f,~>=<g,~> for any ~eS which has support contained in U. The support

of a tempered distribution f is the complement of the largest open set

on which f vanishes. We shall say that a generalized function f is con-

centrated at the point a if and only if the support of f is the set {a}.

353

oc

F i n a l l y , t he Abel l i m i t o f t he i n t e g r a l [ f ( t ) d t i s d e f i n e d to be 0

c o

lim ~f(t)e-Ctdt (2.4) e~0 0

If f is absolutely integrable on [0,~), then it is easy to see that

the Abel limit of the integral is the integral itself, i.e.,

co oo

lim f f(t)e-etdt=ff(t)dt (2.5) c~O 0 0

In f a c t , t h e e q u a l i t y i n ( 2 . 5 ) h o l d s as l o n g as the i n t e g r a l on the

right exists as an improper Riemann integral. However, the Abel limit

may exist when the integral on the right does not exist [2].

3. The main result :

Theorem 1. Let ¢ ( x , s ) and d p ( s ) be g i v e n as b e f o r e . Then

c o

i) The integral H(x)=f ¢(x,s)dp(s) is the sum of two generalized func- 0

tions ; one is concentrated at the origin and the other has support in

( -~ ,0) . i i ) I f ~ ( x , s ) i s an e v e n (odd) f u n c t i o n of x , t h e n so i s H ( x ) . F u r t h e r -

more , i n t h i s c a s e H(x) i s c o n c e n t r a t e d o n l y a t t h e o r i g i n .

co

2k iii) The integra] T(x)=f s ~(x,s)dp(s) is also the sum of two genera-

0

lized functions; one is concentrated at the origin and the other has

support in (-co,0), where k is a non-negative integer.

Proof : Let ?(x)eS such that the support of ~ is contained in (0,~)

and let ~(s) be its generalized Fourier transform as given by (1.3).

Then, it can be easily shown that H(x)eS and

c o 0 o o

<H,,>= ft~(x)H(x)dx= j '0(x)H(x)dx+]k~(x)H(x)dx _oo _co 0

oo eo co

=f * ( x ) d x f , ( x , s ) d p ( s ) = f ~ ( s ) d p ( s ) 0 0 0

But since supp?(x)c(0,@, it is not hard to see that [5]

(3.1)

354

ae ,(x)=g ~(s) ¢(x,s)dp(s)

and hence if s ina¢0, we have

0 ~ N ? - ( - ~ = f ~ ( s ~ d p : l . i . m ~ ~ ( s ) d p slna 6 * " a N+~

N

=l.i.m.N_,o~ of ~ ( x ) d x g ¢ ( x , s ) d p

From (3.1) and (3.3) we obtain that

< H , ¢ > = ) ( o ) _ 1 - ~ < a ?>

s i n a S l n a '

If sin~=0, we differentiate both sides of (3.2) to obtain

oo ~ ' ( O ) = - c o s c t f " ~ ( s ) d p ( s )

0

then repeat the same argument as above and we finally obtain

<H,~> = m r ~ ±, , _ < ~ ' %,>

C O S ~ ) -COS~

(3.2)

( 3 . 3 )

ii) Let ¢(x,s) be an even function of x. Then, it is readily seen that

co co <H,qa>=f * ( x ) f ¢ ( x , s ) d p ( s ) d x = O

- m 0

if ¢(x) is odd and

2 • <6,~> for sina¢0 ]_~ sln~

<H,~J> = -2 <6',¢> for sina=0 COSa

(3 .4 )

if ~ is even.

The proof for the case where ¢(x,s) is an odd function of x is

similar. Now assume that ¢(x,s) is an even function of x so that H(x)

is an even tempered distribution. Let ¢(x)~S but otherwise arbitrary.

Since ¢(x)=¢even(X)+~odd(X), we immediately obtain

<H.~b>=<H ', >+<H.~ . . > = < H . ~ > • " "even "-ocm " - e v e n

355

which implies that H is indeed concentrated at the origin in virtue

of (3 .4 ) and the f a c t t h a t ~ (0 )=~even(0 ) '" 0 ,Oeven( )=0.

i i i ) To prove ( i i i ) i t s u f f i c e s to show t h a t T(x) i s a comb ina t i on

o f H(x) and i t s d e r i v a t i v e s . Let k= l , then

co co

T (x) =f sZ~ ( x , s ) dp (s) =f (q (x) 0 ( x , s ) - 4 ~ " ( x , s ) ) dp (s) 0 0

=q (x)H (x) -H" (x)

which is well defined since q(x) is analytic. The general case is

proved by induction on k.

co

Theorem 2. Let H(x)= I ¢(x,s)dp(s) be a regular tempered distribution

except possibly at the origin. Let

-fe -~s H (x)- O(x,s)dp(s) , E>0 0

Then Hc(x)~H(x ) in S as ~0.

Furthermore , if Hc(x)~h(x ) (pointwise) as c~0 except possibly at

x=0, i.e. , if the Abel limit of H(x) exists, then

h(x)=H(x)=I-1(lim [If(x+iy)-If(x-iy)]), x~-0 (3.5) y-~O

where I f i s the a n a l y t i c r e p r e s e n t a t i o n o f the F o u r i e r t r a n s f o r m of f .

Proof : We only outline the proof since a more detailed version will

be submitted somewhere else . Let OcS, then

co o9 co

<H ,¢>= f~(x)fe-CS¢(x,s)dp(s)dx=fe-CS0(s)dp(s ) C

-~ 0 0

which converges to <H,~>=f~(s)dp as 0

Since the Fourier transform is a continuous linear onerator onS we

have that

IH ~IH a s E~O E

Assuming that h(x) also defines a regular tempered distribution with

a possible singularity at the origin, one can show that

356

IH ~Ih as c~-O E

and w i t h some e a s y c a l c u l a t i o n s one can a l s o show t h a t ( 3 . 5 ) h o ] d s .

Theorem 3. Let H be concentrated at the origin. Let F(s)~S and

assume that co

F(s)~ Z ak sk as s~0 ( 3 . 6 )

k=0

where t he a s y m p t o t i c e x p a n s i o n ( 3 . 6 ) i s i n f i n i t e l y d i f f e r e n t i a b l e .

Then

co ~ oo

f ( x ) = f F ( s ) ¢ ( x , s ) d p ( s ) ~ Z a2k+l f s 2 k + l ¢ ( x , s ) d p ( s ) 0 k=0 0

oo

as x ~ . I n p a r t i c u l a r , i f f s ~ ( x , s ) d 0 ( s ) i s e q u a l t o the g e n e r a l i z e d 0

f u n c t i o n ,,1,, t h e n f ( x ) has the a s y m p t o t i c e x p a n s i o n X

a2k+l (2k+1) ! f ( x ) - z a s x ~ .

k=0 x~k+~

Proof : Again we outline the proof. ~:e write

n - 1

F ( s ) = ~ aksk+Rn(S ) ( 3 . 7 ) k=0

where g n ( S ) = O ( ~ n) as s~0 , s u b s t i t u t e ( 3 . 7 ) i n t o ( 1 . 2 ) and i n t e g r a t e

t e rm by t e r m . S i n c e by t h e o r e m 1 a l l t h e i n t e g r a l s o f t he fo rm

f s 2 k * ( x , s ) d 0 ( s ) 0

a r e a l s o c o n c e n t r a t e d a t t he o r i g i n , i t f o l l o w s f rom t h e o r y 2 t h a t

a l l t h e even te rms a r e z e r o s and we f i n a l l y o b t a i n

n-i co oo

f ( x ) = Z a2k+t / s 2 k + l ¢ ( x , s ) d p ( s ) + ~ R n ( S ) ¢ ( x , s ) d p ( s ) k=o 0

To complete the proof we must show that the remainder term

satisfies

rn (X) =f Rn(S) ¢ ( x , s ) dp (s ) 0

r n ( X ) = 0 ( J ~ 2 n + l ¢ ( x , s ) d p ( s ) ) as x ~ 0

But this requires

more work and the proof will be omitted.

Corollary : Let F(s) satisfy the hypothise of theorem 3. Then

357

~ a2k+1(2k+1) ! f(x)=f F(s)cos ,~s~ x

0 k=0 X 2k+J

as x ~ .

This formula was obtained earlier by M.J. Lighthill ([4], pp.56)

by using different technique.

We close this talk'by giving an interesting example which is a by-

product of our work•

Example : Consider the singular Sturm-Liouville problem

y"=-ly with y(0)=sin~ , y' (0)=-cos~; ~<~

It is known that ¢(x,s)=sinacos~-cos~ sin(SX)an d S

s2ds dp(s)=-- cos2~+sin2a s ~

Thus by theorems I and 2 we obtain that

oo

lira f s2k(sin~cos~_cos~ sin SX)e-aS s2ds g~0 0 s cos2a+sin2~ s ~

for k=0,],2,.•.

Re fe rence s

I) H. Bremermann, "Distributions, Complex Variables and Fourier transforms", Addison-Wesley, New York 1965.

2) G Hardy, "Divergent Series", Oxford University Press (Clarendon) L o n d o n 1 9 4 9

3) B Levitan and I. Sargsjan, "Introduction to spectral theory", Math. Monos., Voi.39, Amer.Math. Soc. , Providence, R.I.1975

4) M J. Lighthill, "Introduction to Fourier Analysis and Generalized Functions", Cambridge University Press, Cambridge 1962

5) G Walter and A. Zayed, "On the real singularities of Sturm- Liouville expansions", submitted.

6) R Wong, "Error bound for asymptotic expansions of integrals", SIAM Review, Voi.22 N°4 ]980

7) A Zayed and G. Walter, "On the singularities of singular Sturm-Liouville expansions and an associated class of elliptic P.D.E's, to appear in SIAM J. of Math.Analysis.