Felix e. Browder, Nonlinear Accretive Operators in Banach Spaces

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N O N L I N E A R A C C R E T I V E O P E R A T O R S I N B A N A C H S P A C E S

BY FELI X E . BROW DER

Communicated December 8, 1966

Introduction. Le t I be a rea l B ana ch space , X* i t s con juga te

space wi th the pa i r ing be tween w in X* a n d x in X denoted by (# , w).

T h e d u a l i t y m a p p i n g J of X i n to 2X i s g iven by

(1) ƒ(*) = {w\ w G X*; [Ml = ||*||; (*, «0 = ||*|| . | H | } ,

for each * in X. For any Banach space X and any e lement * o f X t

J(x) i s a nonempty c losed convex subset of the sphere of radius | |x | |

about ze ro in X*. I f X * is s t r i c t ly con vex , J is a s inglevalued m ap pin g

of X i n to X* an d i s con t inuo us f rom the s t rong topology of X to the

weak* topology of X*. J i s cont inuous in the s t rong topologies i f and

only i f the norm in X is C l on the complement o f the o r ig in .

D E F I N I T I O N . If T is a (possibly) nonlinear mapp ing with domain

D(T) in X and w ith range R(T) in X, then T is said to be accretive if for

each pair x and y in D(T),(2 ) (T(x) - T(y), J(x - y)) £ 0,

i.e. (T(x) — T(y), w) ^ 0 for all w in J(x—y).

If X* i s s t r i c t ly convex , the nonl inear accre t ive opera tors f rom X

to X co inc ide w i th t he / -mono tone ope ra to r s s t ud i ed i n Brow de r [3 ] ,

[4 ] an d B row der -de F igue i redo [8] . Fo r l inear T9 T is accretive if and

only if (-T) i s d iss ipat ive in the sense of Lumer-Phi l l ips [ lO] . F u n c

t iona l equa t ions fo r nonl inear accre t ive opera tors have a l so been con

s idered by Vainberg [15] and re la ted c lasses o f opera tors and probl em s h a v e b ee n s t u d i e d b y H a r t m a n [9 ] , M a m e d o v [ l l ] , M u r a k a m i

[13 ] , a n d P e t r y s h y n [ 1 4 ] .

I t i s our ob jec t in the p rese n t pap er to p resen t a nu m be r of genera l

ex is tence theorems for so lu t ions o f nonl inear func t iona l equa t ions

invo lv ing non l inea r a cc r e t i ve ope ra to r s w h ich d r a s t i c a l l y improve

ear l ie r resu l t s in th i s d i rec t ion as g iven in the papers ment ioned

above . The de ta i led proofs o f these genera l ex i s tence theorems a re

g iven in an o t he r pape r [7 ] . In the second sec tion of the p rese n t pap er ,

w e g ive unde r somew ha t sha rpe r hypo these s a p rocedu re o f p ro j ec -t iona l o r Gale rk in type fo r comput ing such so lu t ions . The con

vergence proof which is given in full is of special interest because i t

depends in an essen t ia l way upon the fac t tha t the ex is tence theorem

for so lu t ions has been g iven an independent proof. (We should a lso

remark that the proofs of the exis tence theorems of [7] are a lso

cons t ruc t ive bu t in a more compl ica ted fash ion . )

470



NONLINEAR ACCRETIVE OPERATORS IN BANACH SPACES 4 7 1

W h e n X is a H ub er t space , J = / , and th e acc re t ive ope ra to r s r educeto the class of monotone operators . The special interest of the class ofnon l inea r acc re t ive ope ra to r s T with domain and range in a Banachspace X i s tha t under the hypotheses of the Theorems be low, theo p e r a t o r s (~-T) coincide with the inf ini tesimal generators of one-pa ra m et er Co semig roups of non l inear cont ra c t ion o pera tors in X,

(i.e. of nonl inear opera tors { U(t), t^O} such that for al l x a n d y in X,

|| U{t)x— U(i)y\\ ^\\x—y\\. W e d i s t ingu i sh con t rac t ions o r nonexpan-s ive opera tors in the te rminology of Browder [3] f rom s t r ic t cont ract ions F for which th ere exis t s a con s ta nt k< 1 such th a t | | V(x) — V(y)\\

^k\\x—y\\9 (x, yÇzX)). Nonl inea r acc re t ive ope ra to r s ex tend th i sp ro pe r ty of m o no ton e ope ra to r s in H ub er t space , r a the r tha n thecharac ter i s t ic proper t ies of gradients of funct ionals on X, as for thec lass of monotone opera tors T from X to X*. (For a survey of thel i te r a tu re on the la t te r c lass , cf. [ l ] , [2] , [5 ] , [6 ] . )

1 . The present sec t ion i s devoted to the s ta tement of the genera l

exis tence theorems, in a s imple opera tor - theore t ica l form.

T H E O R E M 1. Let X be a Banach space with a strictly co nvex conjugate

space X* and with the duality m apping J uniformly continuous from

the unit sphere in the strong topology of X to the strong topology of X * .

Let T be an operator with domain and range in X of the form T= — L

+ To, where L is a closed densely defined linear operator in X which is

the infinitesimal genera tor of a C 0 contraction semigroup, while TQ is a

continuous nonlinear accretive mapping of X into X which maps

bounded sets into bounded sets. Suppose that the inverse image under T of

each bounded set is bounded.

Then R (T), the range of T, is dense in X.

T H E O R E M 2. Suppose that in addition to the hypotheses of Theorem 1,

we assume that for each bounded subset B of X, there exists a compact

mapping C of B into X and a strictly increasing continuous function

\from i ? + = {*| * è 0 } to R+ w ith$0) = 0, such that

(3) | | ( r + C)x - ( r + C>y|| è A(| |* - y\$; (x, y G B H Z > ( ï U

Then the range of T is all of the space X.

T H E O R E M 3 . In Theorems 1 and 2, the hypothesis on the boundedness

of inverse images under T of bounded sets in X can be replaced by the

stronger hypothesis of J-coerciveness of T, namely :

(4) (r(«), 7(«))/||«|| - • + « (|M| -* + » , u e D(T)).

A corol lary of the preceding resul ts is the fol lowing general izat ion

of a theorem of M in ty in Huber t space [12] :

http://from/

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472 F. E. BROWDER [May

T H E O R E M 4 . Let X be a Banach space with strictly convex X* and J

uniformly continuous on the unit sphere. Let T== — L + To, with L the

linear generator of a contraction semigroup, T0 a nonlinear accretive

operator which is continuous from X to X and maps bounded sets into

bounded sets.

Then T+kl, for any k>0 , maps D(T) onto X and has a Lipschitzian

inverse mapping X into X.

T H E O R E M 5 . In Theorems 1, 2, 3, and 4 , if the Banach space X is

reflexive, the continu ity condition on TQ can be replaced by demicontinu-ity, i.e. T0 is continuous from the strong topology of X to the weak topol

ogy of X.

T H E O R E M 6. Suppose that under the hypotheses of Theorem 1, we

assume that there exists a mapp ing J\ of X into X * of the form Ji(x)

=fi(\\x\\)J(x) with ft(r)>O for r>0 such that J\ is continuous from the

weak topology of X to the weak topology of X*. Suppose that X is reflexive

and To merely demicontinuous.

Then the range of T is all of X.

Th eor em 6 con ta ins as a specia l case fo r L = 0, the resul t of Browder

and de F igue i redo [8] ob ta ined under the add i t iona l condi t ion tha t

X i s separable and sat isf ies condi t ion (T)I of §2 below. A very specia l

case of Th eo rem 2 w as es tabl ished b y V ainb erg [15] for L = 0 , To

sa t i s fy ing a L ipsch i tz condi t ion on each bounded se t , and T=To

sa t i s fy ing the inequa l i ty

(5 ) (T(x) - T(y), J(x - y)) g: m(r)\\x - y\\\ (\\x\\, \\y\\ â r) ,

w i t h m(t) a posi t ive decreasing funct ion on R+ s u c h t h a t tm(t)-++ <*>

as t—*+ oo.

The proofs of our resul ts are based upon a general ized method of

s teep es t desce n t ob ta in in g th e convergence of so lu t ions of d i ffe ren t ia l

equa t ions o f the fo rm du/dt= — T(u)+R(t, u) fo r su i tab le per tu rba

t i on t e rms R(t, u) w i th R(t, u)~-»0 as /—»+ oo. T h e proof of con ve r

genc e is base d up on new a pr ior i in equ al i t ies for solu t ions of su ch

equa t ions . (We r emark t ha t a d i s c r e t e ana logue o f t he o rd ina ry

m eth od of s teep es t desc en t was app l ied by Vainberg in [ IS] . )

2. We reca l l t ha t a Banach space X is said to satisfy co nd ition

(w)c for C ^ 1, if X i s separable and i f there exis ts a project ional sys

t e m w i t h c o n s t a n t C in X f i . e . a sequence {Xn} of finite dimensional

subspaces o f X, inc reas ing wi th n and wi th the i r un ion dense in X,

and for each n f a pro jec t ion P n of X on Xn (i.e. Pl=P n a n d R(P n)

= X„ ) fo r which | | P n | | â C . E v e r y B a n a c h s p a c e w i t h a S c h a u d e r



1967] NONLINEAR ACCRETIVE OPERATORS IN BANACH SPACES 4 7 3

bas i s has a p ro jec t iona l sys te m for som e co ns tan t C, an d a space wi th

an abso lu te Schauder bas i s sa t i s f ies condi t ion (w)i.

D E F I N I T I O N . Let Xbea Banach space with a projectional system^ T a

mapp ing of X into X tfa given element of X, Then the equation T(x) = ƒ

is said to be projectiona lly solvable in the strong (wea k) sense if the fol

lowing two assertions hold:

(a) For each n*zl, the equation

Tn(xn) = PnT(x n) = P n(f)

has an unique solution x n in Xn, and exactly one.

(b) As n— »+<*>, these solutions xn converge strongly (weakly) in

X to a solution x of the equation T(x) = ƒ .

Us ing the tech niqu es of [8 ] , Pe t ry sh yn [14] has show n th a t if a

re f lex ive Banach space X sat isf ies condi t ion (w)% and has a weakly

c o n t i n u o u s d u a l i t y m a p p i n g J\ i n to X*, then for every con t inuous

m a p p i n g T of X i n to X w hich maps bounded s e t s i n to bounded s e t s

and sat isf ies an inequal i ty of the form

(T(«) - T(y), J(x - y)) 2: A(| |* - y | | ) | | * - y\\

for a fun ction X from R+ to R+ w hich is con t inuo us w i th X (0 )= 0 ,

X(r)—>+ oo as r— >+ oo, the equ a t io n T(x) = ƒ is p ro jec t iona l ly so lvab le

in th e s t r on g sense for each ƒ in X.

T h e a rg um en t s of B row de r -de F igue i r edo [8 ] and Pe t ry sh yn [14]

do no t app ly w i thou t a s sumpt ions l i ke t ha t o f t he w eak con t inu i ty

of the mapping J i . However , the wr i te r has observed the fo l lowing

genera l fac t which should be very usefu l in fu r ther inves t iga t ions :

If we know already that the equation T(x) = ƒ has a solution, then rela

tively weak assumptions ensure that the same equation is projectionally

solvable in the strong sense.

T H E O R E M 7. Let X be a separable Banach space with a projectional

system ({Xn}, {P n}) with constant C ^ l , Ta continuous mapping of

X into X, and let Tn be the mapp ing of X into X given by Tn=PnTPn.

Suppose that there exists a function X from R+ to R+

with X continuous

and strictly increasing , X(0) = 0 , such that f or all n:

(6) || Tn(x) - r . (y ) | | à X( | |* - y | | ) , (* , yEX n).

Let f be a given element of X. Then the equation T(x) = ƒ has a solution

x in X if and only if the same equa tion is projectionally solvable in the

strong sense in X.

P R O O F O F T H E O R E M 7. Since project ional solvabi l i ty implies solva

bi l i ty , i t suff ices to prove the converse . The inequal i ty (6) implies



474 F. E. BROWDER [May

t h a t Tn as a mapping of Xn i n to X n i s in ject ive and has a c losed range.

S ince Xn i s of f in ite dime nsio n, th e rang e is open b y th e Bro uw er

theo rem o f i nva r i ance o f doma in . Hence P (P W ) = X W , and the equa

t ion Tn(xn) =Pnf has exac t ly one so lu t ion xn in Xn. It suffices to show

t h a t Xn' ^ X y w here x is a given solution of T(x) =ƒ, whose ex is tence

w e k n o w b y h y p o t h e s i s . W e k n o w t h a t

TmPnx = PnTx + Pm(T Pnx - Tx) = Pnf + Pn(T Pnx - Tx).

Since Tn(xn) =Pnf a n d b o t h xn a n d P nf lie in Xnt w e may app ly t heinequ a l i t y (6 ) and ob t a in :

\ ( \ \ p n x - x n \ \ ) ^ \ \ T n p n x - Tnx n \\ = | | p w ( r p n * - r * ) | |

^ c||rpn*- r*||.

How eve r , g iven 5 > 0, there exis ts yn in Xn fo r w sufficiently large such

t h a t | | # — y n | | < $ . F o r s u c h n, w e have Pn%—Pnyn+Pn(x-~yn)

= yn+Pn(x--yn), and there fore

| |* - Pnx \\ ^ \\x - yn\\ + \\P n(x - y n )| | g « + C*.

I t fol lows that P n # converges s t rongly to x as n — » o o . By the con t inu

i ty of r , TPnx—>Tx. T h u s :

| | * - * n | | ^ | | * - P « * | | + | | P n * - * n | | ^ | | * - P n * | |

+ <7(c||rpn*-r*||)

w i th c r the inverse func t ion of X which i s con t inuo us a t ze ro . H enc e

\\x—xn\\— >0, a n d xn converges s t rongly to * in X . q.e.d.R E M A R K . Pe t ry shyn [14 ] ha s show n unde r t he hypo these s o f The

orem 7 th a t if X if reflexive and X(r)—>+ °° as r—±+ oo, then the fol

lowing condi t ion is suff ic ient in order that the equat ion T(x) = ƒ b e

solva ble in th e s t ro ng sense for every ƒ in X: (I ) If xn converges weakly

to x with XnÇzXn, and Tnxn converges strongly to g, then T(x) = g . F o r

nonl inear accre t ive opera tors , however , a d i rec t ve r i f ica t ion of condi

t ion ( I ) seems to demand a hypothesis of the exis tence of a weakly

c o n t i n u o u s d u a l i t y m a p p i n g .

W e now app ly The ore m 7 to the p ro jec t iona l so lvab i l i ty of T(x) = ƒ ,w i t h T nonl inear and sa t i s fy ing accre t iveness condi t ions in a genera l

B a n a c h s p a c e .

T H E O R E M 8. Let X be a Banach space with a strictly convex conjugate

space X* and a duality mapping J uniformly continuous from the

unit sphere in X to X*. Suppo se that X has a projectional system



1967] NONLINEAR ACCRETIVE OPERATORS IN BANACH SPACES 475

({X n } , {Pn}) with constant C= 1. Let The a continuous mapping of X

into X, and suppose that there exists a continuous strictly increasing

mapping X of R+

onto R+, X(0) =0, such that for all x and y of X,

(7) (T(x) - T(y), J(x - y)) § X(||* - y||)||* - y||-

Then for every ƒ in X, the equation T(x) = ƒ is projectionally solvable

in the strong sense f or the projectional system ({Xn}, {Pn}).

P R O O F OF T H E O R E M 8. It follows immediately from the inequality

(7) that T satisfies the conditions of Theorem 2, and therefore tha t

the equation T(x) =ƒ has exactly one solution x for each ƒ in X. We

shall therefore derive the conclusion of projectional solvability from

Theorem 7, and must verify the hypotheses of that theorem. It suf

fices to verify the inequali ty (6) with the same function X as in the

hypothesis of Theorem 8.

By the argument of [8], P%Jw = Jw for w in Xn. Hence for x and y

in Xn, (Tnx-Tny, J(x~y)) = (Tx-Ty, J(x-y))*$\\x-y\$\\x-y\\9

and

\\Tnx - Tny\\ ^ A(||* - y| |), (x, y <E Xn). q.e.d.

B I B L I O G R A P H Y

1. F. E. Browder, Existence and uniqueness theorems for solutions of nonl inearboundary value problems, Proc. Sympos. Appl . Math. , Vol. 17, Amer. Math. Soc,

Providence, R. I., 1965, pp. 24-29.2. 1 Problèmes nonlinêaires, Univ. of Montreal Press, Montreal, 1966,

129 pp.

3# 1 Fixed poin t theorems for n on lin ear semicon tractive mappings in Banachspaces, Arch. R ational Mech. A nal. 21 (1966), 259-269.

4. , Con vergence of approximants to fixed points of non expan sive mappings in

Banach spaces, Arch Rational Mech. Anal 24 (1967), 82-90 .5. , On the un ification of the calculus of variations an d the theory of mon otone

nonlinear operators in Banach spaces, Proc. N at. Acad. Sci. U.S.A. 56 (1966), 419-425 .5# 1 Existence andapproximation of solutions of n on lin ear variation al in-

equalities' Proc. Nat. Acad. Sci. U.S.A. 56 (1966), 1080-1086.7# f Nonlinear equations of evolution and the method of steepest descent in

Ban ach spaces, (toappear).8. F. E. Browder and D. G. de Figueiredo, J-monotone nonlinear mappings in

Banach spaces, Kon. Nederl. Akad. Wetesch. 69 (1966), 412-42 0.9. P. Hartman, Gen eralized Lyapounov functions and n onlinear fun ction al equa

tions, Ann . M at. Pu ra Appl. 69 (1965), 303-320.10 G. Lumer and R. S. Phillips, Dissipative operators in Ban ach spaces, Pacific

J . Math . 11 (1961), 679-69 8.11. Ja. D. Mamedov, One-sided estimates in the conditions f or asym ptotic stability

of solutions of differential equation s with un boun ded operators, Dokl. Akad. NaukSSSR, 166 (1966) , 533-535= Soviet M ath. Dokl . 7 (1966), 110.



4 7 6 F. E. BROWDER

12 . G. J. Minty, Mon otone (n on lin ear) operators in Hilbert space, Duke Math. J,29 (1962), 341-346.

13 . H. Murakami, On n onlinear ordinary and evolution equations, Univ. of KansasTechnical Report No. 9, June 1966.

14. W. V. Petryshyn, Projection methods in n on lin ear n umerical fun ction al anal"ysis, (to appear).

15. M. M. Vainberg, On the con vergence of the process of steepest descent for non lin ear equation s, Sibirsk, Ma t. J. 2 (1961), 201-22 0.

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Felix e. Browder, Nonlinear Accretive Operators in Banach Spaces