Abel-Gontscharoff boundary value problems

Mathl. Cornput. Modelling Vol. 17, No. 7, pp. 37-55, 1993 Printed in Great Britain

0895-7177193 $6.00 + 0.00 Pergamon Press Ltd

ABEL-GONTSCHAROFF BOUNDARY VALUE PROBLEMS

RAVI P. AGARWAL AND QIN SHENG

Department of Mathematics, National University of Singapore

Kent Ridge, Singapore 0511

PATRICIA J. Y. WONG

Division of Mathematics, Nanyang Technological University

Bukit Timah Road, Singapore 1025

(Received May 1992; accepted June 1992)

Abstract-In this paper, we shall provide necessary and sufficient conditions for the existence and uniqueness of solutions of general nth order nonlinear differential equations satisfying Abel- Gontscharoff boundary conditions. Sufficient conditions which guarantee the convergence of a general class of iterative methods are provided. Computational aspects of these iterative methods are also discussed. An example which dwells upon the importance of the obtained results is also included.

1. INTRODUCTION

In this paper, we shall consider the differential equation

y(n) = f(z, y, y’, . . . ,y+q U-1)

together with the Abel-Gontscharoff boundary conditions [1,2]

~%i+d = -%+I, O<i<n-1, (1.2)

where -co < a 5 al I a2 I .. . 5 a, 5 b < 00. Throughout, it will be assumed that the

function f in (1.1) is continuous at least in the interior of the domain of its definition. Boundary conditions (1.2), in particular, include the

(i) (kl,... , k,) right focal point conditions

yci)(aj) = Ai,j Sj-1 I i 5 Sj - 1, SO = 0, sj = 2 ki,

a 5 al < a2 < . . . < a, 5 b

(ii) two-point right focal conditions

yti)(al) = Ai, Olil(Y,

yli)(a2) = Ai, cr+l<i<n-1,

a 2 al < a2 5 b.

(1.3)

(1.4)

HCM 17:7-0 37

38 R.P. AGARWAL et al.

The qualitative theory of these boundary value problems is in a process of continuous develop- ment, as it is apparent from the large number of research papers dedicated to it [3-181. Most of these investigations provide necessary and sufficient conditions which ensure the existence and/or uniqueness of the solutions. While the quantitative results for the two point right focal boundary value problems with ai = a, u2 = b are available in [19-221, the general problem (l.l), (1.2) remains untouched. In this paper, we shall employ recently obtained best possible inequalities [23) to obtain easily verifiable necessary and sufficient conditions which ensure the existence and/or uniqueness of the solutions of (l.l), (1.2). These inequalities are further used to obtain sufficient conditions which guarantee the convergence of a general class of iterative methods for the problem (l.l), (1.2). The linear convergence of the Picard method and the quadratic convergence of the Newton method (Quasilinearization) are deduced from the general result. A priori necessary and sufficient conditions for the convergence of approximate iterative methods to the unique solution of (l.l), (1.2) are also provided. Finally, an example which dwells upon the importance of the obtained results is illustrated.

2. PRELIMINARIES

Throughout, in what follows, we shall assume that the function y(z) E C(n)[a, 61, although this restriction is not necessary.

LEMMA 2.1. The Abel-Gontscharoff interpolating polynomial P,_ 1 (x) of the function y(z), i.e., satisfying

P:‘,(ui+i) = Y%i+i), O<i<n--1, (2.1)

can be expressed as n-1

P,_I(z) = c Z(z) y%i+d, (2.2) i=o

where TO(Z) = 1, and Ti(z), 1 I i I n - 1 is the unique polynomial of degree i satisfying

T,!~)(u~+~) = 0, o<j<i-1,

T,(i’(ui) = 1,

and it can be written as

Ti(Z) = lr2,1 ., . . ...2.

/= fXl

1 a1 u; i-l . . . a1 (42 0 1 2uz . . . (i - l)u;-2 iu;-l

0 0 0 ::: (2.4)

i!u, (i-1)l Xi2 zi-l

= Ja, J,, . . .~~ld:‘,,,:, . . .&cl, (20 = z). (2.5)

(2.3)

PROOF. It is clear that Ti(zc) is a polynomial of degree i. Further, Ti(Z) defined in (2.4) satisfies (2.3) and follows from the usual properties of the determinants. Similarly, Ti(z) defined in (2.5) satisfies (2.3) and follows by the successive differentiation.

In particular, we have

COROLLARY 2.2. [4]

y(z), i.e., satisfying

To(x) = 1, Tl(z) = 2 - al,

T2(z) = ; [z2 - 2uzx + ui(2uz - al)] . I

The two point right focal interpolating polynomial Qn_ i (x) of the function

Q::&I> = Y%), Olila,

Q:i,(a2> = ~%2), (2.6)

cy+lliln-1,

Boundary value problems 39

can be written as

&n-l(z) = c a (x ;!al)* y(Q1) + n-g2 y(a+1+j)(,2). (2.7) i=O j=O

PROOF. Since al = a2 = . - - = a,+1 from (2.5), it is clear that Ti(x) = (x-a~)~/i!, 0 I i I cy+l. Further, since (a2 =)ua+2 = ua+3 = . . . = a, once again from (2.5), we have

gl;...l; (~“+j;a2)jdxctl...dxl

=~yy...~:_ (~(~~+~-~~~~~~)~~~~~-~)dx.,l...dxl

2 (x - al) a+l+yal _ a2)j-i =

((Y + 1 + i)! (j - i)! ’ l<j<n-a-2.

i=. I

LEMMA 2.3. In terms of repeated integrals the error function e(x) = y(x) - P,_l(x) can be written as

e(x) =~:~~...~:_-‘y(~)(x~)dx~...d~~. (2.8)

PROOF. It suffices to note that

Oljln-1. (2.9)

LEMMA 2.4. The error function e(x) = y(x) - P,+I(x) can be written as

e(x) = J b

!dx, t> y’“‘(t) dt, (2.10) la

where g(x, t) is the Green’s function of the boundary value problem

,(n) = 0,

Z(i)(Odj+l) = 0, OSiSn-1,

and appears as

dx,t) = ak 5 t 5 xv

-c;:; &[ (&+I - ty-1, x 5 t 5 ak+l,

k=O,l,..., n (a0 = a, a,+1 = b).

PROOF. It is clear that e(“)(x) = YCn)(x),

e(‘)(ai+l) = 0, OSiSn-1,

and, therefore, n-1

e(x) = c ~i(x>ai+l+ & Jz(x - t)+l y(“)(t) dt, i=o * a

(2.11)

(2.12)

40 R.P. ACARWAL et al.

where

oi+r = -

Thus, it follows that

n-1 e(z) = - C Ti(s) /ai+l(fp+l _ t)n-i-ly(n)(t)&

i=. (n-i - l)! 0

+(A)! 4x - J (x - t)“-ly(yt) dt

(2.13) =

However, from (2.8) it is obvious that

(x - t)n-1 n-l (n _ 1), = c 2xX) ‘“;‘nl_-i~;)i;’

i=o (2.14)

and hence, (2.13) is the same as (2.10) which follows from the definition of g(x, t) in (2.12). fl

COROLLARY 2.5. [4] Corresponding to the two point right focal conditions (1.4) the Green’s function h(z, t) of the boundary value problem

,(n) = 0 ,

“qzl) = 0, O<i<Cr,

Ji)(az) = 0, a+lli<n-1

(2.15)

is given by

(x - a# (a1 - tp-i-1, a<tIx,

(2.16) (z - al)” (a1 - ty-i-l, xltlb.

tither, for al 5 t, x 5 a2 the following inequalities hold

O<il(Y, (2.17)

(-1Y &i - , aw&t) > o

o+l<i<n-1. (2.18)

COROLLARY 2.6. Corresponding to the two point right focal conditions (1.4) the error function E(x) = y(x) - &-l(x) in the interval (al, 4 can be written as

where < E (ul,uz), and

a(x) = zl(4YW) , (2.19)

T,(4=-& n

.c(> 1 (x - a# (a1 - az)? r=a+1

(2.20)


PROOF. In view of (2.17), (-l)“-“-lh(x, t) 2 0, al < t, x 5 ~2; also since

J aah(x,t)dt= f ,g (y) (x - U#(Ul - u2y, a1 t=a+l

equality (2.19) follows on applying the mean value theorem in

J aa

a(x) = h(x, t)y’“‘(t) di!. 01

LEMMA 2.7. [4,24,25] For each 0 5 j 5 n - 1, the following holds

I I e(j)(x) I - (n ~ j)! ( r(nn_~‘_i;,2] > (b - u)n--3 M’

(2.21)

(2.22)

I

(2.23)

where A4 = max [Al. a<x<b

REMARK 2.1. For each 0 5 j 2 n - 1, the inequality (2.23) is the best possible, in the sense that equality holds if and only if y cn) = M. However, if al = a and a, = b, then (2.23) can be improved.

LEMMA 2.8. [23] Let (;Y and /3 be nonnegative integers such that

a=al = . . . = a,+1 < ua+2 5 f * * I u,_p < an-o+1 = . *. = a, = b.

Then, for each 0 I j I n - 1, the following holds

’ ifOLjln-P-1 (2.24) ifn-p<j<n-1

where T* = max{cr - j, p, [(n - j - 1)/2]}.

Inequalities (2.24) are the best possible.

COROLLARY 2.9. [4] Corresponding to the two point right focal conditions (1.4) let al = a and u2 = b. Then, for each 0 < j 5 n - 1 the following holds

if05 j<(r (b - u)“+M. (2.25)

ifa+llj<n-1

In view of Lemmas 2.7 and 2.8, hereafter, we shall assume that Cn,j are the best ‘available’ constants in the inequalities

I@(x)1 I Cn,j(b - a)“-j ,FJb Iy(“)(X)I , 0 5 j 5 n - 1. --

(2.26)

3. EXISTENCE AND UNIQUENESS

THEOREM 3.1. Suppose that

(i) K3 > 0, 0 5 j 5 n - 1 are given real numbers and let Q be the maximum of If(x, ‘yo,

Yl,..., Yn_l)l on the compact set [a, b] x DO, where Do = {(yo,yl,...,yn_l) :

j~jlS2Kj, OSj<n--1)

(ii) (b - u) 5 (&) 1’(n-j) , 0 5 j 5 n - 1

(iii) lAj+ll + Cyii-j $ ( [(i Zf,2])

(b -~~)~(Ai+j+ll = Sj 5 Kj, 0 5 j 5 n - 1.

Then, the boundary value problem (l.l), (1.2) has a solution in Do.


PROOF. The set

B[a,b] = Y(Z) E c (

(+I) [a, b] : Ily(j) 11 <2Kj, O<jln-1 >

,

where /[Y(j)]] = ,msnsb [y(j)(z)] is a closed convex subset of the Banach space C(+‘) [a, b]. Consider

an operator T : E(ntl) [a, b] + Cl”) [a, b] a~ follows

(Ty)(z) = I’,-I(Z) + /bg(~,t)f(t,~(t),...,~‘“-l’(t))dt. Ja

Obviously, any fixed point of (3.1) is a solution of (l.l), (1.2).

We note that (Ty)(x)-P,-I(Z) E C(“)[a,b], (Ty)‘j’(aj+l)-P~j_)l(aj+l) = 0, 0 5 j 5 n-l and

(Ty)tn)(z) - Ppl(z) = (TY)(~)(z) = f(z, y(z), . . . ,y(+‘)(z)), and hence, for all y(z) E B[a,b], it follows that

I((Ty)‘j’ - P~J,(I 5 Cn,j(b - a)“-jQ, 0 5 j < n - 1,

which in view of (ii) gives

I((Ty)‘j’I(~(l~~1/I+Kj, Olj<n-1. (3.2)

However, since n-1

I’:&) = c T,!j’(z) Ai+l, O<jln-1, i=j

from Lemma 2.7, we find

Thus, the hypothesis (iii) in (3.2) leads to

Il(~y)(j)ll <2Kj, Osj<n-1. (3.4)

Thus, T maps B[u,b] into itself. Further, the inequalities (3.4) imply that the sets {(Ty)(j)(z) : y(z) E B[u,b]}, 0 5 j 5 n - 1 are uniformly bounded and equicontinuous on [a, b]. Hence, m[u, b] is compact follows from the Ascoli-Arzela theorem. Therefore, the Schauder fixed point theorem is applicable and a fixed point of (3.1) in Ds exists. I

COROLLARY 3.2. Suppose that the function f(z,yO,yl,. . . , y,,_l) on [u, b] x Rn satisfies the following condition

n-1

If(z, YO, Yl, . . .7 Yn-1)l 5 L + C Li IYila’7 (3.5) i=o

where L, Li, 0 5 i 6 n - 1 are nonnegative constants, and 0 < Cri < 1, 0 5 i 5 n - 1. Then, the boundary value problem (3.1), (3.2) has a solution.

PROOF. For y(t) E B[u, b] the condition (3.5) implies that

n-l

1% Y(Z), . . . ) Y(~-‘)(cE))~ 5 L + C Li(2Ki)Q* = &I, i=O

say. Now Corollary 3.2 follows immediately by observing that the hypotheses of Theorem 3.1 are satisfied with Q replaced by Qr provided Kj, 0 5 j 5 n - 1 are sufficiently large. a


COROLLARY 3.3. Suppose that the conditions of Theorem 3.1 are satisfied. Then, for any E > 0

there is a solution Y(s) of the boundary value problem (3.1), (3.2) such that lY(j)(z) - Ptj-‘,(z)l < E, 0 5 j 5 n - 1 provided (b - a) is sufficiently small.

COROLLARY 3.4. Suppose that the conditions (i), (ii) of Theorem 3.1 are satisfied. Then, for

any dxc> E C (“-‘)[a, b] the differential equation (1.1) together with

y@(ai+l) = g(i)(ai+l) 7 O<i<n-1 (3.6)

has a solution provided

where Mj = JIIF~ [g(j)(x)1 , 0 5 j 5 n - 1. --

THEOREM 3.5. Suppose that the function f(z, 2/o, ~1,. . . ,yn_l) on [a, b] x D1 satisfies the following condition

n-1

If(&YO,Yl,... ,Yn-1)l 5 L + C J% IYil, (3.7) i-0

where

Dl = { (YO,Yl,...

L+c ,Yn-1): lyjl 5 Sj +Cn,j(b-a)n-j- 1_o, OLjln-1

I

and n-1

c = c LiSi i=o n-l

8 = c c*,i Li(b - ay < 1. i=o

Then, the boundary value problem (l.l), (1.2) has a solution in D1.

PROOF. It is clear that the boundary value problem (l.l), (1.2) is equivalent to the problem

W’“‘(Z) = f(z,?.u(z) + P,_l(Z), . . . ,w+l)(z) + P~;Q(z)), (3.8)

&)(ai+l) = 0, Oliln-1, (3.9)

where UJ(X) = y(z) - P,_l(x). We define M as the set of functions n times continuously differentiable on [a, b] satisfying the boundary conditions (3.9). If we introduce in M the norm llwll = ,~~x~ 1 d”)(z)l, then M becomes a Banach space. We shall show that the mapping

-- T : M --) M defined by

(Tw)(z) = fg(qt) f&w(t) + P,_l(t), . . . ,w@-l)(t) + P;;‘)(t))& (3.10)

maps the ball S = {W(X) L M : llwll 5 s} ’ into itself. For this, let ~(5) E S then from (2.26),

we have

I I

L+c w(j)(,) 5 Cn,j(b - q-j 1_, O<j<n-1

and hence, in view of (3.3), it follows that

5 sj + cn,) (b - a)+j 1-8, Oljln-1,

which implies that (

u)(z) + P,_~(x), . . . , d”-‘)(z) + F’~:;‘)(z)) E D1.


Further, from (3.10) we have

Thus, it follows from Schauder’s fixed point theorem that T has a fixed point in 5’. This fixed point w(z) is a solution of (3.8), (3.9) and hence the boundary value problem (l.l), (1.2) has a solution y(z) = W(X) + P,_r(s). I

THEOREM 3.6. Suppose that the differential equation (1.1) together with the boundary conditions

Y(i$&+r) = 0, Oli<n-1 (3.11)

has a nontrivial solution y(x) and the condition (3.7) with L = 0 is satisfied for all

(z,Yo,Y1,. * . ,Y,+~) E (a,b] x Dz, where

Dz = {(Yo,YI,-. ,yn_l):Iy~ljlIC*,j(b-u)n-jm, Olj<n-1)

and m = ,r_-nZyb 1 Y(~)(X) I. Then, it is necessary that B 2 1. - -

PROOF. Since y(z) is a nontrivial solution of (l.l), (3.11) it is necessary that m # 0. Thus, in view of (2.26), it follows that (z, y(z), . . . , y(“-‘l(z)) E [a, b] x D2. Thus, we have

=

C Li C,,i (b - u)~-~ m i=O

8m

and hence 8 2 1. Conditions of Theorem 3.6 ensure that in (3.7) at least one of the Li, 0 5 i 5 71 - 1 will not

be zero, otherwise on [a, b] the solution y(r) will be a polynomial of degree at most (n - 1) and will not be a nontrivial solution of (l.l), (3.11). Further, y(z) E 0 is obviously a solution of (l.l), (3.11) and, if 8 < 1, then it is also unique. I

n-1

THEOREM 3.7. Suppose that for all (z,ys, ~1,. . . ,Y,+~), (2, go, yl,. . . ,j~.~) E [u,b] x D1 the function f satisfies the Lipschitz condition

n-1

If (2, YO? Yl? . . ..Yn-l)-f(2.~O,~l,...,&n-1)1 I ~LilYi-~il~ , (3.12) i=o

where L = ~ng~$f(x,O,O,. . . , O)l. Then, the boundary value problem (l.l), (1.2) has a unique

solution in rS;.-

Boundary Mlue problems 45

PROOF. Lipschitz condition (3.12) implies (3.7) and hence the existence of a solution in D1 follows from Theorem 3.5. To show the uniqueness, we let ‘~1 (x) and We be two solu-

tions of (3.8), (3.9) in Dl. Then, (z, WI(Z) + P,_l(z), . . . , we-‘) + P:_<‘)(z)), (5, w2(z) +

P,_l(Z), . . . ,wp (z)+Py(z)) E [a$] XD 1, and hence, in view of (3.8) and (3.12) it follows that

Since 6 < 1, we find that W?‘(Z) = W?‘(X) for all 2 E [a, b]. Thus, yl(z) = yz(z) follows from the boundary conditions (3.9). I

4. CONVERGENCE OF ITERATIVE METHODS

DEFINITION 4.1. A function g(z) E C(“)(a, b] called an approximate solution of (1.1), (1.2) if there exist nonnegative constants 6 and e such that

max a<x<b

g(“)(z) - f(z, Q(Z), . . . , (4.1)

and

IAj+l - g(j)(q+~)( + n$;J; ( I(ii-i;i21) (b - ali (Ai+j+~ - gci+j) (ai+j+~)(

5 c Cn,j(b - uy--j, 0 5 j < TI - 1. (4.2)

The inequality (4.1) means that there exists a continuous function v(x) such that

jji’“‘(Z) = f@,&(X), . . . ,jj’“-“(z)) + q(2)

and

aFZ!b 177(x)1 I 6. --

(4.3)

(4.4)

Thus, the approximate solution y(z) can be written as

?-j(X) = &1(X) + I” [

s(z, t) f(& g(t), . . > 5’“-w + v(t)] dt,

where p,_l(z) is the (n - l)th degree polynomial expressed as

n-l

(4.5)

Pn_1(5) = c T&)g(qui+l). (4.6) i=o

In what follows, we shall consider the Banach space dn-‘)[u, 61 and for y(z) E C(n-‘)ju, b]

(4.7)

THEOREM 4.1. With respect to the boundary value problem (1.1), (1.2), we ILssume that there exists an approximate solution 5 (x) and

(i) the function f(z, ye,, yl, . . . , y,-1) is continuously differentiable with respect to all yi, 0 < i 5 n - 1 on [a, b] x &, where

D3 = 1 (YO,Yl,... rYn-I) : IYj - Q'%jj IN. Cnoy+ OIj<?z-1 ,

>

46

(ii)

(iii)

(iv) Then,

(1)

(2)

(3)

R.P. AGARWAL et al.

there exist nonnegative constants Li, 0 5 i 5 n - 1 such that for al‘1 (CC, yo, ~1,. . . ,y,.,-1) E

[a, bl x D3

&~9Yo7Y17...7 YA)i I&, O<i<n-1

the function P(z) iicontinuous on [a,b], P = ma~b~~(z)~, and 80 = (1 + 2/3)0 c 1, --

ivr = (1 - @)-‘(e + S) c,,s (b - a>n I N.

the sequence {ym(z)} generated by the process

Y$l(4 = m, Ym(Z), * * * 1 Y2%)) n-l

+ P(x) c (Yz+l i=o (4.8)

y$_l(ai+l) = Ai+13 Oli<n-1, m=O,l,..., (4.9)

with ys(z) = y(x) remains in S(Z, NI), the sequence {ym(z)} converges to the unique solution y*(x) of the boundary value prob lem (l.l), (1.2), a bound on the error is given by

IIYm-Y*Il 5 (‘:‘;~)“(~)-‘IIY1-Yoll

5 (‘:‘;r)” (1 - e,>-’ (e + a> c,,o (b - a)n.

(4.10)

(4.11)

PROOF. First, we shall show that {y,(x)} C S(y, Nr). For this, we define an implicit operator T as follows

(TY)(x) = J’,-I(Z) + Jbs(x, t) [f(t, y(t), . . . , y’“-‘)(t)) a

n-1

+ P(t) c ((TY)?~) - y’“‘(t)) i=O

& f(t,y(r), * f * ,r’“-l’(t))]&

(4.12)

whose form is patterned on the integral equation representatio-n of (4.8), (4.9). Since y(x) E S(y, Nr), it is sufficient to show that if y(x) E S(g, Nr), then (Ty)(x) E s(g, NI).

Let y(x) E S(&, Nr), then from the definition of norm (4.7), we have

which implies that

and hence, (y(z), . . . ,~J(~-~)(z)) E Ds. Thus, from (4.12) and (4.5), we have

(Ty)(x) - y(x) = P,-l(X) - K-l(X) + JbsW) [f(CY(t). . . .9Y+l)(t)) 0

n-1

+ P(t) c ((Td(“)(t) - d”‘(t))

i=O

& fk y(t), * * * 9 y’“-“(t))

- f(C x7(t), . f . 7 Y -‘“-l’(t)) -q(t) I dt.

Boundary value problems

Therefore, in view of (2.26), (4.2) and (4.4), we obtain

I(Ty)‘qz) - &(j)(z)1 I ECn,j(b - a)“+ + cn,j(b - q-j

x max a<x<b [ 1

- f(? Y(Z), * * * 7 Y’“--l)(d) - f(G G(z), . . .

+ an& py)(“)(s) - yo(z)( + 61, O<jln-1,

i=o

which gives

G,o (b - a)j Gj

((TY)‘-” - LP’(s)( 5 (E + 6)C*,o (b - ay n-1

+ c G&i@ - 4qY - &II i=O

n-1

+ PC Cn,iJ%(b - a>n-i{II(TY> - &II + IIY - LVIII7 O<j<n-I i=O

and hence,

jl(Ty) - gjl 5 (E + b)Cn,o (b - ajn + (1 + PMIY - all + Pell(TY) - ?7lI

Ol-

II - 011 I (1 - PW’[(c + v&,0 (b - ay + (1 + Lw~ll.

Thus, II - jj)) 5 IV1 follows from the definition of NI. Next, we shall show the convergence of the sequence {ym(z)}. From (4.8), (4.9), we have

Ym+lb) - Y?n(~) = f(h YvL($. . * ,Yp?)W - f(k Ym-l(t), . *. 7 Y$L%))

+ Pdg{ ( y$\&) - y;)(t)) --$---f(t,Ym@). . . .7YiF(tN (4.13) i==o m

- (y::‘(t) - y;L,(t)) ay’“;,(t) f(t,Ym-l(t),. . . ,Yk,‘) w }] f&k m

Thus, from (2.26) and the fact that {ym(z)} C s(g, Nl), we get

and hence,

CT&,0 @ - 4j ( ‘)

Gj IY;+&g - Ygi’(r)l L (1 + PP IlYm - Ym-111 + P4IYm+l - Ymll7 Oljln-1,

which provides

IlYmfl - Ymll I s IlYm - Ym-111.


The above inequality by an easy induction gives

IlYm+l -Ymll L: (‘:‘;~)“llY1-Yoll. (4.14)

Since (1 + a/3)0 < 1, inequality (4.14) implies that {ym(z)} is a Cauchy sequence and hence converges to some y*(z) E S(Y,IVr). This y*(z) is the unique solution of (l.l), (1.2) and can be easily verified.

The error bound (4.10) follows from (4.14) and the triangle inequality

IlYm+* - Ymll I lIYm+p - Ym+p-ill + ... + IIYm+l - Ymll

< P+PP m+p-l+**_ K ) + ((:i_;f)‘“] llYl -Yell

z (r;;;;+ @j-l llyI -yell

and now taking p -+ 00. Next, from (4.5), (4.8), (4.9), we have

y1(5) -Ye(Z) = P,-l(2) - K-l(~) + J

6L+,t) a

[

n-1 x p(t) 5 (yp’(t) - Yp(t)) ---&-p Yo(t)>.~. 7 Yr”w - v(t) 1 dt

and hence, as earlier, we find

which is the same as

llY1 - Yoll I (E + 6) cl,0 (b - aJn + PqYl - yoJ(,

IlYl - Yell < (1 - @WE + 6) cn,o (b - ay. (4.15)

Using (4.15) in (4.10), we obtain the required inequality (4.11). I

COROLLARY 4.2. With respect to the boundary value problem (1.1), (1.2), we assume that there exists an approximate solution Y(x) and

(i) the function f(x, yc, yr,. . . (ii) 8 < 1,

, y,_r) satisfies the Lipschitz condition (3.12) on [a, b] x Da,

(iii) IV2 = (1 - e)-lcE + 6) C,,O (b - a)n < N.

Then,

(1) the sequence {ym(x)} generated by the process

I 6

Ym+l(X) = pn-l(X) + dx, t>f(t, ym(t), . . ., Y!,?(Q) dt, m=O,l,..., (4.16) a

with ys(x) = Y(z) remains in S(Z, Nz), (2) the sequence {y,(x)} converges to the unique solution y*(x) of the boundary value prob-

lem (1.11, (1.2),

(3) a bound on the error is given by

11~~ - Y*H 5 ml - WilYl - ~~11

5 eyi - e)-l(c + 6) Cn,o (b - a)“.

(4.17)

(4.18)


THEOREM 4.3. Let in Theorem 4.1 the function p(z) E 1. Further, let f(z, 90, yl,. . . , Y,-~) be continuously twice differentiable with respect to all yi, 0 5 i 5 n - 1 on [a, b] x D3 and

&~(~vYo,YIY-..TY~-I) I LiLjK, 0 5 i,j 5 n - 1. s 3

Then,

(4.19)

(4.20)

where cy = K02/(2(1-e)C,,o (b-a)n). Th us, the convergence is quadratic if f K(E + 6) x (A) 2

< 1.

PROOF. From {y,(x)} C s(Z, Nl) with /? = 1, it follows that for all m, (ym(z), . . . , y?-“(z)) E D3. Further, since f is twice continuously differentiable, we have

f(s, y&z), . . . , y$y’(z)) = f(G Ym-l(Z), . . * 7 Yi?z’(~))

(4.21)

where pi(z) lies between Y;_~(E) and y$)(z), 0 I i 5 n - 1. Using (4.21) in (4 13) ke’, * ,

Ym+l(~) - Yn&) = ~~(r.i)(~(Y~~~(t)-Y~)(t))~~(t,Ym(t)....,Y~-Li(t))

+ ; [ ng (YW - Yi)4W) & a=0 1

2 f(C Po(4, . . . ,Pn-l(t)) &.

1

Thus, (2.26) provides

IY$$l(5) - Y$$‘(z)l = Cn,j (b - a)“-j i

n-1 c. C Lic(b - ~)-illYm+l - Ynll i=O

and hence,

IlYm+l - Ymll 5 ~llYm+l - hll + 2cn o;;2_ a>n IlYm - Ym-1 II23

which is the same as the first part of the inequality (4.20). The second part of (4.20) follows by an easy induction. Finally, the last part is an application of (4.15) with p = 1. I

In Theorem 4.1, the conclusion (3) ensures that the sequence {y,(z)}, generated from (4.8), (4.9), converges linearly to the unique solution of the boundary value problem (l.l), (1.2). For /3(z) E 1, Theorem 4.3 provides its quadratic convergence. However, in practical evaluation this


sequence is approximated by the computed sequence, say, {z~(x)} which satisfies the recurrence relation

&&i+d = -%+I, O<i<n-1, m=O,l,...,

with .zo(x) = ye(x) = y(x). With respect to fm, we shall assume the following condition.

CONDITION C1.

(4.23)

(i) The function fm(z, yo,y~, . . . , y,_l) is continuously differentiable with respect to all Yi,

O<i<n-lon[a,b]xDsand

$f&,Yo,Y1,. . . ,Yn-1) 5 Li, O<i<n-1. 2

(ii) For z,(z), . . . , z2-l) (x) obtained from (4.22), (4.23), the following inequality is satisfied

where A > 0 is a constant. The above inequality corresponds to the relative error in approximating f by fm for the

(m + l)th iteration.

THEOREM 4.4. With respect to the boundary value problem (l.l), (1.2), we assume that there exists an approximate solution g(z) and Condition Cl is satisfied. Further, we assume

(9 (ii)

(iii)

Then,

(1)

;:;

(4)

IV*

Conditions (i) and (ii) of Theorem 4.1, +,A = (1 + 2/3 + A)0 < 1, Ns = (1 - 8g,A)-‘(c + 6 + AF)C,,o(b - a)n 5 N, where F = ,ma& If(xcg(x), . . . ,

-- g’“-l’(x))\.

all the conclusions (l)-(3) of Theorem 4.1 are valid, the sequence {zm(z)} obtained from (4.22), (4.23) remains in s(&, N3), the sequence {Ed} converges to y*(x), the solution of (1.11, (1.2), if and only if lim a, = 0, where

Tn’M

b a m= II GTz+1(~) - PtI-1(x> - I g(x, qf(c k($. . .Y

a &y’(t)) q,

a bound on the error is given by

- Z,+~II 5 (1 - erl [(I + mwm+l - 41

+ AGo@ - a)” ,‘=yb If(x, z&x>, . . . , z:-“(x))/]. (4.24) - -


PROOF. Since 00,~ < 1 implies that 0~ < 1, and obviously Nr 5 Na, the conditions of Theo- rem 4.1 are satisfied and part (1) follows.

To prove (2), we note that jj E s(g,Ns) and, from (4.5), (4.22), (4.23), we have

q(z) -g(z) = P,_l(Z) - L(z) + JLS(L-J) [fo(t.lo(t), . . . ,$%) a

n-l

+ /3(t) 5 (zf’(t) - g(t)) ---&Joct, .zo(t>, . * * 9 4+“(t))

-f(C s(t), . . . 7 Y -‘“-l’(t)) - 7)(t) fit 1

and hence, from (2.26), we obtain

11~1 - 811 I (c + 6 + WG,o @ - a>” + Wlla - -zolI

and llzl - gl( I (1 - e/3)-‘(e + 6 + AF)G,,o@ - aIn

I N3. (4.25)

Thus, zl(z) E s(g, Na). Next, we assume that zm(z) E s(g, Ns) and show that z~+I(z) E @, N3). Once again from (4.5), (4.22), (4.23), we have

z,+1(z) - g(z) = Pn-l(2) - Pn_l(Z) + JLS(Zd) [f&Gn(% . *. 9 4rY”(w a

n-1

+ B(t) c (&(t) - 4w) &-p,lm(t), . * * 7 4pw i=o

-f(C s(t), . * * 2 Y -‘“-l’(t)) - q(t) 1

tit

and hence, as earlier we get

+ (1 + A) If(z,z,(z), . . . &-l)(z)) - f(z,z,,(z), . . . ,$-“(+))I

+ A f(z, zo(z), . . . , $--l) z ( ))I]

,

which provides

II~m+l - dl 5 (e + 6 + 4 Go (b - aY + Pellz,+l - tmll + (1 + ~MlG?l - zoll

5 (e + 6 + AF) Cn,o (b - aY + (I + /3 + A)e ll.zm - z,,ll

+ peiizm+l - ~~11.

F’rom the last inequality, we find that

IlGn+1 - &ll I (1 - pe)-’ [(c + 5 + AF)C,,O (b - ay + (1 + p + A)eNs]

= N3.

This completes the proof of part (2).


Next, from the definitions of ym+r(z) and zm+r(z), we have

I

b

Ym+l(Z) - G&+1(~) = pn-lb> + g(&t)f(t, Gn(q, * * * , &+(q)dt - %.+1(~) a

b + J [ g(a,t) f(k ym(t), *. .7 Y2-"W) - f(h &b(t), . . .Y &-“(w a

n-1

+ P(t) c (Y:!&) - Yim) &-pYm@)3.. . ) Ym] dt i=O

and hence, as earlier, we find

IIYm+l - Gn+111 I Gn + ~IlYm - Gnll + PqIYm+l - Ymll.

Using (4.14) in (4.26), we get

(4.26)

IIYm+l - Gn+1 II 2 hn + qynz - z,ll + pe (‘:‘;~)mllyl-Yoll.

Since ye(z) = Q(Z) = y(z), the above inequality provides

IIYm+l - z,+~ ll 2 2 e”-i i=. [~i+U~((:~~~)illyl-yOll].

Now from (4.27) and the triangle inequality, we obtain

IIz~+~ - Y* ii I IIY~+~ - Y*II + 2 em-i [ui + pe (e)’ IIY~ - yoke]. i=O

(4.27)

(4.28)

In (4.28), Theorem 4.1 ensures that ,liliW llyn+i - y*II = 0. Thus, from the Toeplitz lemma

“for any 0 5 CY < 1, let s,,, = CEO C-P-~&, m = 0, 1, . . . , then ,li*“, s, = 0 if and only if

lim d, = 0”, lim Ilz,+i - y*ll = 0 if and only if lim m--W0 m--rm m~oo [em + 00 (w)” llyl - y. tl] = 0.

However, since ,llr (w) m = 0, lim IIz,+i - y*ll = 0 if and only if lim a, = 0. m-C0 m+cc

To prove (4), we note that

and hence,

ll~* - ~~+~ll I oily* - hll + mlh+l - hll

+ AC,,, (b - “,“a~z~b If@, G(Z), . . . , d,?(d)/ , - -

which on using the triangle inequality lly* - z,II 5 lly* - z,+lll + IIzm+i - zmll gives (4.24). I

COROLLARY 4.5. With respect to the boundary value problem (1.1), (1.2), we assume that there

exists an approximate solution g(z), and for zm(z), . . . , z~-~)(z) obtained from b

Zm+1(~) = pn-l(Z) + J

g(t, Wn(t, &n(t), . . . , z$-“(Q) c-it m=O,l,..., (4.29) a

with ZO(Z) = ye(z) = g(z), the inequality (4.23) is satisfied. Further, we assume

(i) Condition (i) of Corollary 4.2, (ii) eA = (1 + A)6 < 1,

(iii) N4 = (I - e&‘(e + 6 + AF)C,,e (b - a)* 5 N.


Then,

(1) all the conclusions (l)-(3) of Corollary 4.2 are valid, (2) the sequence {z,(x)} obtained from (4.29) remains in s(p, Na), (3) the sequence {zm(x)} converges to y*(x), the solution of (1.1), (1.2) if and only if

flm a, = 0,

(4) a bound on the error is given by

5. AN EXAMPLE

Consider the boundary value problem

y(v) = ._A____ 10 + x2

ey + cos x2 + e-”

Y(O) = 1, y’ ($) =O, y”(;) =f, y$) =O, ~(‘~)(1)=0.

(5.1)

(5.2)

For this problem, we note that

(5.3)

and cs,o = &. Further, for the differential eqnation (5.1), it is sufficient to consider the set

Do = (90 : lyol I ‘.%}. Thus, it follows that Q = i e2Ko + 2. Hence, the conditions of

Theorem 3.1 are satisfied provided

l< ( L$Y+2)1’5 (5.4)

and

(5.5)

Inequality (5.4) implies that 0.1062 I Ko I 3.2177. Therefore, both the inequalities (5.4), (5.5) are satisfied provided 1.25 I KO I 3.2177. In conclusion, the problem (5.1), (5.2) has a solution y(x) such that [y(z)1 I 1.25.

Next, to apply Theorem 4.1 we assume that a(x) = Pd(x) so that the boundary conditions (5.2) are exactly satisfied, and hence E = 0. Further, the inequality (4.1) reduces to

1 max O--e P4(z) - cosx2 - e-” eg/’ + 2 = 6.

O<z<l 10 + 5

-- x2 $ (5.6)

Also, since for the problem (5.1), (5.2), the set D3 = {yo : Iyo - P4(x)I < IV}, 8 can be taken as

0 = -& x $ x eN+‘/’ < 1 (5.7)

and hence, N < 4.173317. (5.8)

Thus, the condition of Theorem 4.1 with p = 0 (or of Corollary 4.2) are satisfied provided

(1 - e)-’ ($eg/8+2).$<N, (5.9)


i.e., 0.117435 5 N < 4.145082. (5.10)

Clearly, (5.7) as well as (5.9) holds if (5.10) holds. Thus, in conclusion:

(i) the problem (5.1), (5.2) h as a solusion y*(z) in 0s = {ye :I yc - P4(2)1 5 0.117435}, (ii) this solution y*(z) is unique in Ds = {ye :) ys - Pd(z)[ 5 4.145082},

(iii) the sequence {y,(z)} generated by

y$$ (x) = & eymlx) + cos x2 + e-”

Ym+l(O) = 17 Yk+1 ; 0 =o, y;+1 ; =;, 0 Yii+1 a 0 =o, ?/g,“!(l)=0 m=0,1,...,

(5.11)

with ys(z) = Pd(z), converges to y*(s), moreover, if we choose N = 0.117435 then 8 = 0.01732019, and the following error estimate holds:

Iy,&) - y*(z)1 5 (0.01732019)m (0.117435082). (5.12)

The iterates ym+i(z) from (5.11) can be computed by using simple quadrature formulae. First eight iterates by employing the Trapezoidal rule with h = l/80 are presented in the following:

number of iterations

Table 1

computational error

3.26635247 x 1O-3

4.89861821 x lo-’

9.96934757 x lo-”

1.80966353 x lo-l4

2.22044605 x lo-l6

0.0

0.0

0.0

estimated error bound

1.17435082 x 10-l

2.03399790 x 10-s

3.52292293 x 10-S

6.10176933 x 1O-7

1.05683802 x 10-s

1.83046348 x lo-”

3.17039747 x 10-12

5.49118854 x lo-l4

Finally, we shall apply Theorem 4.3 to our problem (5.1), (5.2). For this, it is necessary that

81 = 38 < 1 (5.13)

and

(1 - 38)-i ($egjs+2) .h 2 N. (5.14)

Inequality (5.13) holds if and only if

N < 3.074705078 (5.15)

and (5.14) implies that 0.1540108 2 N 5 3.0359523. (5.16)

Obviously, both inequalities (5.15) and (5.16) hold if N satisfies (5.16). Thus, if we choose N = 0.1540108, then B = 0.0179655, LO = 0.35931 and it follows that K = 2.7831121. Therefore,

= 1.098779 x 1O-3 < 1.

Boundary value problems

In conclusion: the sequence generated by

55

y(V)&) = m+ & (ym+l(2) - ym(x)) eymcs) + cosz2 + e-” (5.17)

Ym+lP) = 1, Yin+1 ; 0 =o, y;+l ; =;, 0 YX+1 ; 0 =o, y;;!(l)=0 m=O,l,...,

with ye(z) = Pi converges to y*(z) quadratically. Double precision variables are used throughout the numerical computations.

1. 2. 3.

4.

5

6

7.

8.

9.

10.

11. 12. 13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24. 25.

26.

27.

REFERENCES

P. J. Davis, Interpolation and Approximation, Blaisdell Publishing Co., Boston, (1961). V.L. Gontscharoff, Theory of Interpolation and Approximation of finctions, Gostek-hizdat, Moscow, (1954). R.P. Agarwal, On the right focal point boundary value problems for linear ordinary differential equations, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 79, 172-177 (1985). R.P. Agarwal, Boundary Value Problems for Higher Order Differential Equations, World Scientific, Sings

pore, (1986). J. Ehme and D. Hankerson, Existence of solutions for right focal boundary value problems, Nonlinear Analysis 18, 191-197 (1992). U. Eliss, Focal points for a linear differential equation whose coefficients are of constant sign, Tmns. Amer. Math. Sot. 249, 187-202 (1979)). U. Elias, Green’s function for a non-disconjugate differential operator, J. Diflerential Equations 37, 318-350

(1980). P.W. Eloe, Sign properties of Green’s functions for two classes of boundary value problems, Canad. Math.

Bull. 30, 28-35 (1987). J. Henderson, Uniqueness of solutions of right focal point boundary value problems for ordinary differential equations, J. Dierential Equations 41, 218-227 (1981). J. Henderson, Existence of solutions of right focal point boundary value problems for ordinary differential equations, Nonlinear Analysis 5, 989-1002 (1981). J. Muldowney, A necessary and sufficient condition for disfocality, Proc. Amer. Math. Sot. 74, 49-55 (1979). Z. Nehari, Disconjugate linear differential operators, Trans. Amer. Math. Sot. 129, 500-516 (1967). A.C. Perterson, Green’s function for focal type boundary value problems, Rocky Mountain J. Math. 9, 721-732 (1979). A.C. Perterson, Existence-uniqueness for focal-point boundary value problems, SIAM J. Math. Anal. 12, 173-185 (1981). S. Umamaheswaram and M. Venkata Rama, Green’s functions for k-point focal boundary value problems, J. Math. Anal. Appl. 148, 35&359 (1990). S. Umamaheswaram and M. Venkata Rama, Focal subfunctions and second order differential inequalities, Rocky Mountain J. Math. 21, 1127-1142 (1991). S. Umamaheswaram and M. Venkata Rama, Existence theorems for focal boundary value problems, Difler- ential and Integral Equations 4, 883-889 (1991). S. Umamaheswaram and M. Venkata Rama, Existence theorems for some special type of boundary value problems, Nonlinear Analysis 6, 663-668 (1991). R.P. Agarwal and R.A. Usmani, Iterative methods for solving right focal point boundary value problems, J. Comput. Appl. Math. 14, 371-390 (1986). R.P. Agarwal and R.A. Usmani, On the right focal point boundary value problems for integredifferential equations, J. Math. Anal. Appl. 126, 51-69 (1987). R.P. Agarwal and R.A. Usmani, Monotone convergence of iterative methods for right focal point boundary value problems, J. Math. Anal. Appl. 130, 451-459 (1988). R.P. Agarwal, Existence-uniqueness and iterative methods for right focal point boundary value problems for differential equations with deviating arguments, Annales Polonici Mathematici 52, 211-230 (1991). P.J.Y. Wong and R.P. Agarwal, Abel-Gontscharoff interpolation error bounds for derivatives, Proc. Royal Sot. Edinburgh 119A, 367-372 (1991). W.A. Coppel, Disconjugacy, Lecture Notes in Mathematics 220, Springer-Verlag, New York, (1971). A.Ju. Levin, A bound for a function with monotonely distributed zeros of successive derivatives, Mat. Sb. 64 (106), 396-409 (1964). Donal O’Regan, Fourth (and higher) order singular boundary value problems, Nonlineur Analysis 14, 1001-1038 (1990). Donal O’Regan, Second and higher order systems of boundary value problems, J. Math. Anal. Appl. 156, 120-149 (1991).

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Abel-Gontscharoff boundary value problems