Nonlocal Cauchy problem for analytic resolvent integrodifferential equations in Banach spaces

Applied Mathematics and Computation 204 (2008) 352–362

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Nonlocal Cauchy problem for analytic resolvent integrodifferential equations in Banach spaces R. Ravi Kumar Coimbatore Institute of Engineering and Information Technology, Coimbatore 641 109, India

a r t i c l e

i n f o

Keywords: Existence theorem Integrodifferential equations Fixed point theorem Nonlocal condition

a b s t r a c t In this paper, we prove the existence of semi classical and mild solutions of nonlinear integrodifferential equations with nonlocal conditions in Banach spaces. The results are obtained by using the theory of analytic resolvents and Schaefer’s fixed point theorem. Ó 2008 Elsevier Inc. All rights reserved.

1. Introduction Several papers have appeared on the problem of existence of solutions of semilinear differential equations and integrodifferential equations in Banach spaces [1–3,11,14–17]. Using the method of semigroup, existence and uniqueness of mild, strong and classical solutions of semilinear evolution equations have discussed by Pazy [18]. Hernandez and Henriquez [12] studied the existence problem for neutral functional differential equations in Banach space and also [10] have discussed about the Existence results for a class of semilinear evolution equations. Fu and Ezzinbi [7] also studied the neutral functional differential evolution equations by using Sadovskiis fixed point theorem. Balachandran and Sakthivel [4], and Dauer and Balachandran [5] investigated the existence of solutions of nonlinear neutral integrodifferential equations in Banach spaces. Grimmer [8] obtained the representation of solutions of integrodifferential equations by using resolvent operators in Banach space and also Grimmer and Pritchard [9] have discussed about the analytic resolvent for integral equations in a Banach space. Lin and Liu [13] studied the nonlocal Cauchy problem for semilinear integrodifferential equations by using resolvent operators. Liu [14] discussed the Cauchy problem for integrodifferential evolution equations in abstract spaces. The purpose of this paper is to prove the existence of semiclassical and mild solutions for neutral integrodifferential equations with the help of contraction mapping principle and Scheafers fixed point theorem. 2. Preliminaries Consider the nonlinear integrodifferential equation of the form

    Z t Z t d ½xðtÞ þ gðt; xðtÞÞ ¼ A xðtÞ þ Fðt  sÞxðsÞds þ f t; xðtÞ; aðt; s; xðsÞÞds ; dt 0 0

ð1Þ

xð0Þ ¼ x0  qðt1 ; t 2 ; . . . ; tp ; xðÞÞ;

ð2Þ

E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2008.06.050

R. Ravi Kumar / Applied Mathematics and Computation 204 (2008) 352–362

353

where A is the infinitesimal generator of a compact, analytic resolvent operator RðtÞ; t > 0 in a Banach space X. f : J  X  X ! X, g : J  X ! X, a : J  J  X ! X and q : ½0; Tp  X ! X are given functions. FðtÞ : Y ! Y and for xðÞ is continuous in Y. AFðÞxðÞ 2 L1 ð½0; T : XÞFðtÞ 2 BðXÞ, t 2 J and for x 2 X. F 0 ðtÞx is continuous in t 2 J, where BðXÞ is the space all bounded linear operator on X and Y is the Banach space formed from DðAÞ, the domain of A, endowed with the graph norm. Here J ¼ ½0; T. It is well known that [8] there exists a constant M P 1 and a real number w such that

kRðtÞk 6 Mewt ;

t P 0:

We assume that kRðtÞk is uniformly bounded by M and analytic resolvent such that 0 2 qðAÞ. In this case it is possible to define the fractional power ðAÞa , for 0 < a < 1, as a closed linear operator with domain DððAÞa Þ. Furthermore, the subspace DððAÞa Þ is dense in X and the expression

kxka ¼ kðAÞa xk; defines a norm on DððAÞa Þ which will be denoted by X a . The following properties are well known [9]. Lemma 2.1. Under the above conditions we have (i) ðAÞa : X a ! X, then X a is a Banach space for 0 6 a 6 1. (ii) If the resolvent operator of A is compact then X a ! X b is continuous and compact for 0 < b 6 a. (iii) For every a > 0 there exists a positive constant C a such that

kðAÞa k 6

Ca 0 < t 6 T; ta

kðAÞb k 6 M 0

and kFðt  sÞk 6 M2 .

Definition 2.1. A family of bounded linear operator RðtÞ 2 BðXÞ for t 2 J is called a resolvent operator for [6,12]

  Z t dx ¼ A xðtÞ þ Fðt  sÞxðsÞds dt 0 if (i) Rð0Þ ¼ I (the identity operator on X), (ii) for all x 2 X; RðtÞx is continuous for t 2 J, (iii) RðtÞ 2 BðYÞ; t 2 J. For y 2 Y, RðtÞy 2 C 1 ð½0; T; XÞ \ Cð½0; T; YÞ,

  Z t Z t d RðtÞy ¼ A RðtÞy þ Fðt  sÞRðsÞy ds ¼ RðtÞAy þ Rðt  sÞAFðsÞy ds; dt 0 0

t 2 J:

Definition 2.2. A function u 2 Cð½0; rÞ : XÞ is called a semiclassical solution of the abstract Cauchy problem (1) and (2) if d xð0Þ ¼ x0 , dt ðxðtÞ þ gðt; xðtÞÞ is continuous on ð0; rÞ, xðtÞ 2 DðAÞ for all t 2 ð0; rÞ, and xðÞ satisfies (1) and (2) on ð0; rÞ. Our results are based on the properties of analytic resolvent of linear operators and the ideas used in [10].

3. Semiclassical solution The existence of semiclassical solution requires some additional assumptions on the functions f , g and a. In particular, we assume the following assumption: (H) There exist 0 < a < b < 1 and an open set Xa  X a such that the functions f ; a and ðAÞb g are continuous on ½0; t 2 J, and there exist L; L1 ; M 1 ; N 1 > 0 and 0 < c1 < 1 such that for every ðt 1 ; x1 Þ; ðt 2 ; x2 Þ 2 J  Xa , ðt; x1 ; y1 Þ; ðt; x2 ; y2 Þ 2 J  Xa  Xa and ðt; s; y1 Þ; ðt; s; y2 Þ 2 J  J  Xa we have

kðAÞb gðt; xÞk 6 L; kðAÞb gðt 1 ; x1 Þ  ðAÞb gðt 2 ; x2 Þk 6 L1 fjt 1  t 2 jc1 þ kx1  x2 ka g; kf ðt; x1 ; y1 Þ  f ðt; x2 ; y2 Þk 6 M 1 fkx1  x2 ka þ ky1  y2 ka g; kaðt; s; x1 Þ  aðt; s; x2 Þk 6 N1 fkx1  x2 ka g; L1 kðAÞab k < 1:

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R. Ravi Kumar / Applied Mathematics and Computation 204 (2008) 352–362

Theorem 3.1. Let xð0Þ 2 Xa and assume that f ; g and a are satisfies the hypothesis ðHÞ, that g is DðAÞ-valued continuous and that 1  b < minfb  a; c1 g. Then there exists a unique semiclassical solution xð; x0 Þ 2 Cð½0; r : XÞ for some 0 < r < T. Proof. Let 0 < r 1 < T and d > 0 such that

V ¼ fðt; xÞ 2 ½0; r 1   X a : kðAÞa x  ðAÞa xð0Þk < dg  ½0; TÞ  Xa : Assuming that the function f ; a and ðAÞb g are bounded on V by C 1 > 0, we choose 0 < r < r 1 such that

ð1  lÞd ð1  lÞd ; kðRðtÞ  IÞðAÞa gð0; xð0ÞÞk 6 ; 6 6 ba 1þba 1a r r r ð1  lÞd ; þ M 2 C 1bþa C 1 þ CaC1 < kðAÞab kL1 r c1 þ C 1bþa C 1 6 ba ba 1a rab r 1a < ð1  lÞ; ð1 þ M 2 TÞL1 C 1bþa þ NC a ab 1a

kðRðtÞ  IÞðAÞa xð0Þk 6

where l ¼ L1 kðAÞab k, C a and C 1bþa are the constants in Lemma 2.1. Define the set

S ¼ fy 2 Cð½0; r : XÞ : yð0Þ ¼ ðAÞa xð0Þ; kyðtÞ  ðAÞa xð0Þk 6 d; t 2 ½0; rg and define the operator W : S ! S by

WðyÞðtÞ ¼ RðtÞðAÞa ðxð0Þ þ gð0; xð0ÞÞ  ðAÞa gðt; ðAÞa yðtÞÞ þ

Z

t

ðAÞ1bþa Rðt  sÞðAÞb gðs; ðAÞa yðsÞÞds Z t Z s Z t þ ðAÞ1bþa Rðt  sÞ Fðs  sÞðAÞb gðs; ðAÞa yðsÞÞdsds þ ðAÞa 0 0 0   Z s  Rðt  sÞf s; ðAÞa yðsÞ; aðs; s; ðAÞa yðsÞÞds ds: 0

0

For the mapping W we consider the decomposition W ¼ W1 þ W2 , where

W1 ðyÞðtÞ ¼ RðtÞðAÞa ðxð0Þ þ gð0; xð0ÞÞ  ðAÞa gðt; ðAÞa yðtÞÞ þ þ

Z

t

ðAÞ1bþa Rðt  sÞ

W2 ðyÞðtÞ ¼

Z

t

t

ðAÞ1bþa Rðt  sÞðAÞb gðs; ðAÞa yðsÞÞds

0

Z

0

Z

s

Fðs  sÞðAÞb gðs; ðAÞa yðsÞÞdsds;

0

  Z s ðAÞa Rðt  sÞf s; ðAÞa yðsÞ; aðs; s; ðAÞa yðsÞÞds ds:

0

0

Next, we prove that W1 and W2 are well defined, that W satisfies a Lipschitz condition and that the ranges of W is contained in S. Since RðtÞ is analytic the function s ! ARðt  sÞ is continuous in the uniform operator topology on ½0; TÞ, consequently the function ARðt  sÞgðs; ðAÞa yðsÞÞ is continuous on ½0; tÞ. Moreover, from Lemma 2.1, we have

kðAÞ1bþa Rðt  sÞðAÞb gðs; ðAÞa yðsÞÞk 6

C 1bþa ðt  sÞ1bþa

C1;

s 2 ½0; tÞ;

which implies that kðAÞ1bþa Rðt  sÞðAÞb gðs; ðAÞa yðsÞÞk is integrable on ½0; tÞ. We thus conclude that W2 is well defined with values in Cð½0; r : XÞ. It is clear from the previous remark that W1 is also well defined with values in Cð½0; r : XÞ. It remains to show that the operator W is a contraction on S. Let y be a function in S. Then for t 2 ½0; r we get

kWðyÞðtÞ  ðAÞa xð0Þk 6 kðRðtÞ  IÞðAÞa xð0Þk þ kðRðtÞ  IÞðAÞa gð0; xð0ÞÞk þ kðAÞa gð0; xð0ÞÞ  ðAÞa gðt; ðAÞa yðtÞÞk Z t Z t C 1bþa C 1bþa kðAÞb gðs; ðAÞa yðsÞÞkds þ þ 1bþa 1bþa 0 ðt  sÞ 0 ðt  sÞ Z s Z t Ca  a   kFðs  sÞðAÞb gðs; ðAÞa yðsÞÞkdsds þ a f ðs; ðAÞ yðsÞ; 0 0 ðt  sÞ  Z s  2ð1  lÞd þ kðAÞab kL1 fr c1 þ kðAÞa xð0Þ  yðtÞkg aðs; s; ðAÞa yðsÞÞdsÞ  ds 6 6 0 ! Z t C 1bþa C 1bþa Ca 2ð1  lÞd þ C 1 þ M2 r C1 þ C 1 ds 6 þ kðAÞab kL1 fr c1 þ dg 6 ðt  sÞa ðt  sÞ1bþa ðt  sÞ1bþa 0 þ C 1bþa C 1

r ba r 1þba r 1a þ M 2 C 1bþa C 1 þ CaC1 6 d: ba ba 1a

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R. Ravi Kumar / Applied Mathematics and Computation 204 (2008) 352–362

From the choice of r we conclude that kWðyÞ  ðAÞa xð0Þk 6 d, so that WðyÞ 2 S. On the other hand, for y1 ðÞ; y2 ðÞ 2 S and t 2 ½0; r

kWðy1 ÞðtÞ  Wðy2 ÞðtÞk 6 kðAÞa gðt; ðAÞa y1 ðtÞÞ  ðAÞa gðt; ðAÞa y2 ðtÞÞk þ  gðs; ðAÞa y2 ðsÞÞkds þ

Z

t

Z

C 1bþa

Z 0

t

C 1bþa ðt  sÞ1bþa

kðAÞb ½gðs; ðAÞa y1 ðsÞÞ

s

kFðs  sÞðAÞb ½gðs; ðAÞa y1 ðsÞÞ ðt  sÞ 0  Z t Z s Ca   f ðs; ðAÞa y1 ðsÞ; aðs; s; ðAÞa y1 ðsÞÞdsÞ  gðs; ðAÞa y2 ðsÞÞkdsds þ a 0 ðt  sÞ 0  Z s  ab aðs; s; ðAÞa y2 ðsÞÞdsÞ   ðs; ðAÞa y2 ðsÞ; ds 6 L1 kðAÞ kky1  y2 k 0 Z t( C 1bþa C 1bþa þ L1 ky1 ðsÞ  y2 ðsÞk þ M 2 TL1 ky1 ðsÞ  y2 ðsÞk 1bþa ðt  sÞ ðt  sÞ1bþa 0   Z s Ca a a þ ds M ky ðsÞ  y ðsÞk þ kaðs; s ; ðAÞ y ð s ÞÞ  aðs; s ; ðAÞ y ð s ÞÞkd s 1 1 2 1 2 ðt  sÞa 0 ( Z t C 1bþa 6 L1 kðAÞab kky1  y2 k þ ð1 þ M 2 TÞL1 ky1 ðsÞ  y2 ðsÞk ðt  sÞ1bþa 0 (  Ca þ M ½ky ðsÞ  y ðsÞk þ N Tky ð s Þ  y ð s Þk ds 6 L1 kðAÞab k 1 1 1 2 1 2 ðt  sÞa ) r ba r 1a þ Ca N ky1  y2 kr ; where N ¼ M1 þ M 1 N1 T: þ 1 þ M 2 TÞL1 C 1bþa ba 1a 1bþa

0

The last estimate and the choice of r imply that W is a contraction mapping on S. So there exists a unique fixed point yðÞ of the operator W in S. We prove that yðÞ is a locally Holder continuous on ð0; rÞ. If c2 ¼ minf1; c1 g and 0 6 t 6 r, then

kyðt þ hÞ  yðtÞk 6 kRðt þ hÞ  RðtÞ½ðAÞa ðxð0Þ þ gð0; xð0ÞÞk þ kðAÞa ½gðt þ h; ðAÞa yðt þ hÞÞ  gðt; ðAÞa yðtÞÞk Z tþh Z t kðAÞ1bþa ½Rðt þ h  sÞ  Rðt  sÞðAÞb gðs; ðAÞa yðsÞÞkds þ kðAÞ1bþa Rðt þ h  sÞ þ 0

t b

a

 ðAÞ gðs; ðAÞ yðsÞÞkds  Z t Z s   ðAÞ1bþa ½Rðt þ h  sÞ  Rðt  sÞ þ Fðs  sÞðAÞb gðs; ðAÞa yðsÞÞds  ds 0

þ

Z

0

tþh

t

  Z s   b a ðAÞ1bþa Rðt þ h  sÞ ds Fðs  s ÞðAÞ gð s ; ðAÞ yð s ÞÞd s   0

 Z s Z t   a ðAÞa ½Rðt þ h  sÞ  Rðt  sÞf ðs; ðAÞa yðsÞ; ds aðs; s ; ðAÞ yð s ÞÞd s Þ þ   0

þ

Z

0

tþh

t

6 kðAÞ

  Z s   ðAÞa Rðt þ h  sÞf ðs; ðAÞa yðsÞ; aðs; s; ðAÞa yðsÞÞdsÞ  ds

1b

b

RðtÞkkðAÞ kðkxð0ÞkÞh þ

 ðAÞb gð0; xð0ÞÞkh þ

Z

Z

0

t

kðAÞ1b RðtÞkkFðsÞkkðAÞb kðkxð0ÞkÞdsh þ kðAÞ1b RðtÞ

0 t

kðAÞ1b RðtÞkkFðsÞkkðAÞb gð0; xð0ÞÞkdsh þ kðAÞab kkðAÞb fgðt

0

þ h; ðAÞa yðt þ hÞÞ  gðt; ðAÞa yðtÞÞgk þ

Z

t

kðAÞ2bþa Rðt  sÞðAÞb gðs; ðAÞa yðsÞÞkdsh

0

þ

 Z Z s Z t   ðAÞ2bþa Rðt  sÞ Fðs  sÞðAÞb gðs; ðAÞa yðsÞÞds  dsh þ 0

0

 ðAÞb gðs; ðAÞa yðsÞÞkds þ

tþh

kðAÞ1bþa Rðt þ h  sÞ

t

 Z t Z s   b a ðAÞ2bþa Rðt  sÞ dsh Fðs  s ÞðAÞ gð s ; ðAÞ yð s ÞÞd s   0

0

 Z t Z s Z s   b a ðAÞ2bþa Rðt  sÞ dsh þ Fðs  s Þ Fð s  hÞðAÞ gðh; ðAÞ yðhÞÞdhd s   0

þ

Z

t

0

tþh

0

  Z s   ðAÞ1bþa Rðt þ h  sÞ Fðs  sÞðAÞb gðs; ðAÞa yðsÞÞds  ds 0

 Z t Z s   a ðAÞ1þa Rðt  sÞf ðs; ðAÞa yðsÞ; dsh þ aðs; s ; ðAÞ yð s ÞÞd s Þ   0

0

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R. Ravi Kumar / Applied Mathematics and Computation 204 (2008) 352–362

þ

 Z t Z s Z s   a a dsh ðAÞ1þa Rðt  sÞ Fðs  s Þf ð s ; ðAÞ yð s Þ; að s ; h; ðAÞ yðhÞÞdhÞd s   0

þ

Z

0

tþh

t

0

  Z s   a ðAÞa Rðt þ h  sÞf ðs; ðAÞa yðsÞ; ds 6 M0 C 1b T b1 ðkxð0ÞkÞh aðs; s ; ðAÞ yð s ÞÞd s Þ   0

c

þ M 0 M 2 C 1b T b ðkxð0ÞkÞh þ C 1b T b1 C 1 h þ M2 C 1b T b C 1 h þ kðAÞab kL1 ½h 1 þ kyðt þ hÞ  yðtÞk ba

þ C 2bþa C 1

rba1 r ba h r baþ1 þ 2M2 C 2bþa C 1 h þ 2M 2 C 2bþa C 1 h þ C 1bþa C 1 h ba1 ba1 ba ba1 ba

þ M 2 TC 1bþa C 1

1a

h ra h c þ C 1þa C 1 h þ C a C 1 6 C 1 ðkxð0ÞkÞh þ C 2 h 2 þ L1 kðAÞab kkyðt þ hÞ ba a 1a

 yðtÞk þ C 3 h þ C 4 h

ba

þ C5h

1a

;

 ¼ minf1; c2 ; b  a; 1  ag, the last inequality can be written in the form where the constant C i are independent of t. If q

kyðt þ hÞ  yðtÞk 6

Cða; b; 1; t; xð0ÞÞ q h : 1l

 -Holder continuous on ð0; rÞ, moreover, we can to assume Since l ¼ L1 kðAÞab k < 1. Therefore, the function yðÞ is locally q R  þ b > 1. Now it is easy to show that s ! ðAÞb gðs; ðAÞa yðsÞÞ and s ! f ðs; ðAÞa yðsÞ; 0s aðs; s; ðAÞa yðsÞdsÞ are that q q-Holder continuous on ð0; rÞ, where q ¼ minfq ; c2 g and q þ b > 1. From this remark in Pazy [18, Theorem 2.4.1] and Lemma 2 of [10] we infer that the function

Z t Z t xðtÞ ¼ RðtÞðxð0Þ þ gð0; xð0ÞÞ  gðt; ðAÞa yðtÞÞ þ ðAÞ1b Rðt  sÞðAÞb gðs; ðAÞa yðsÞÞds þ ðAÞ1b Rðt  sÞ 0 0 Z t Z s Z s Fðs  sÞðAÞb gðs; ðAÞa yðsÞÞdsds þ Rðt  sÞf ðs; ðAÞa yðsÞÞ; aðs; s; ðAÞa yðsÞÞdsÞds  0

0

0 1

is X a -valued that the integral terms of above function in C ð½0; r : XÞ and that xðtÞ 2 DðAÞ for all t 2 ð0; rÞ. Operating on xðÞ with ðAÞa , we conclude that ðAÞa y ¼ x and hence that xðtÞ þ gðt; xðtÞÞ is a C 1 function on ð0; TÞ. 4. Existence of mild solutions Definition 4.1. A continuous function xðÞ : ½0; T ! X is called a mild solution of Eqs. (1) and (2) if the function ARðt  sÞgðs; xðsÞÞ; s 2 ½0; T, is integrable and the following integral equation

Z t Z t xðtÞ ¼ RðtÞ½x0  qðt 1 ; . . . ; t p ; xðÞÞ þ gð0; xð0ÞÞ  gðt; xðtÞÞ  ARðt  sÞgðs; xðsÞÞds  ARðt  sÞ 0 0 Z t Z s Z s Fðs  sÞgðs; xðsÞÞdsds þ Rðt  sÞf ðs; xðsÞ; aðs; s; xðsÞÞdsÞds  0

0

0

is satisfied. We need the following fixed point theorem due to Schaefer [19]. Schaefer’s Theorem: Let E be a normed linear space. Let W : E ! E be a completely continuous operator, that is, it is continuous and the image of any bounded set is contained in a compact set and let

fðWÞ ¼ fx 2 E : x ¼ kWx for some 0 < k < 1g: Then either fðWÞ is unbounded or W has a fixed point. Assume the following conditions hold: (H1) For each ðt; sÞ 2 J  J, the function aðt; s; Þ : X ! X is continuous and for each g 2 X, then the function að; ; gÞ : J  J ! X is strongly measurable. (H2) For each t 2 J, the function f ðt; ; Þ : X  X ! X is continuous and for each g; / 2 X then the function f ð; g; /Þ : J ! X is strongly measurable. (H3) For each positive integer k there exists lk 2 L1 ð½0; TÞ such that

sup kf ðt; /; gÞk 6 lk ðtÞ for t 2 J:

k/k;jgj6k

(H4) The function g : J  X ! X is completely continuous and for any bounded set Q in CðJ : XÞ the set ft ! gðt; xðtÞÞ : x 2 Q g is equicontinuous in CðJ : XÞ. (H5) There exists b 2 ð0; 1Þ and constants b1 ; b2 > 0 such that

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R. Ravi Kumar / Applied Mathematics and Computation 204 (2008) 352–362

kðAÞb gðt; /Þk 6 b1 ; x 2 X

and

kðAÞb gðt; /Þ  ðAÞb gðs; gÞk 6 b2 ½jt  sj þ k/  gk; for t; s 2 J; /; g 2 X: (H6) There exists an integrable function m1 : J  J ! ½0; 1Þ such that

kaðt; s; /Þk 6 m1 ðt; sÞX0 ðk/kÞ;

0 6 s 6 t 6 T; / 2 X;

where X0 : ½0; 1Þ ! ½0; 1Þ is a continuous nondecreasing function. (H7) There exists an integrable function m2 : J ! ð0; 1Þ such that

kf ðt; /; gÞk 6 m2 ðsÞX1 ðk/k þ jgjÞ;

t 2 J; /; g 2 X;

where X1 : ½0; 1Þ ! ½0; 1Þ is a continuous nondecreasing function. (H8) q : CðJ p : XÞ ! X is continuous, compact and there exists a constant H > 0 such that t 2 J

kqðt 1 ; . . . ; t p ; xðÞÞk 6 H for x 2 X:

n Rt (H9) The function m ðtÞ ¼ max Mm2 ðtÞ; m1 ðt; tÞ; 0

Z

T

m ðsÞds <

Z

0

1

c

o

om1 ðt;sÞ ds ot

satisfies

ds ; 2X0 ðsÞ þ X1 ðsÞ

where

c ¼ M½kx0 k þ H þ M0 b1  þ M 0 b1 þ C 1b b1

Tb T 1þb þ M 2 C 1b b1 : b b

Theorem 4.1. If the assumption ðH1 Þ—ðH9 Þ are satisfied then the problem (1) and (2) has a mild solution on J. Proof. To prove the existence of mild solution of (1) and (2) we have to apply Schaefers theorem for the following operator equation

xðtÞ ¼ kWxðtÞ 0 < k < 1;

ð3Þ

where W : Z ! Z is defined as

Z t ðWxÞðtÞ ¼ RðtÞ½x0  qðt 1    tp ; xðÞÞ þ gð0; xð0ÞÞ  gðt; xðtÞÞ  ARðt  sÞgðs; xðsÞÞds 0 Z s Z t Z s Z t ARðt  sÞ Fðs  sÞgðs; xðsÞÞdsds þ Rðt  sÞf ðs; xðsÞ; aðs; s; xðsÞÞdsÞds:  0

0

0

ð4Þ

0

Then from (3) and (4) we have

Z t kxðtÞk 6 kRðtÞ½x0  qðt1    t p ; xðÞÞ þ gð0; xð0ÞÞk þ kAb ½Ab gðt; xðtÞÞk þ kðAÞ1b Rðt  sÞðAÞb gðs; xðsÞÞkds 0   Z t Z s Z s Z t     b Rðt  sÞf ðs; xðsÞ; ds ðAÞ1b Rðt  sÞ ds þ Fðs  s ÞðAÞ gð s ; xð s ÞÞd s aðs; s ; xð s ÞÞd s Þ þ     0

0

6 M½kx0 k þ H þ M 0 b1  þ M 0 b1 þ Z s m1 ðs; sÞX0 ðkxðsÞkÞdsÞds: þ

Z

0

t

C 1b ðt  sÞb1 b1 ds þ M 2

0

Z

0

t

C 1b ðt  sÞb1 b1 Tds þ M

Z

0

t

m2 ðsÞX1 ðkxðsÞk 0

0

Let us take the right hand side of the above inequality as vðtÞ. Then we have xð0Þ ¼ vð0Þ ¼ c, where

c ¼ M½kx0 k þ H þ M0 b1  þ M 0 b1 þ C 1b

v0 ðtÞ ¼ Mm2 ðtÞX1 ðkxðtÞk þ

Z

t

Tb T 1þb b1 þ M2 C 1b b1 b b

and kxðtÞk 6 vðtÞ;

  Z t m1 ðt; sÞX0 ðkxðsÞkÞdsÞ 6 Mm2 ðtÞX1 vðtÞ þ m1 ðt; sÞX0 ðvðsÞÞds ;

0

0

since v is obviously increasing and let

wðtÞ ¼ vðtÞ þ

Z

t

m1 ðt; sÞX0 ðvðsÞÞds: Then wð0Þ ¼ vð0Þ ¼ c; and vðtÞ 6 wðtÞ;

0

Z

t

om1 ðt; sÞ X0 ðvðsÞÞds ot Z t om1 ðt; sÞ 6 Mm2 ðtÞX1 ðwðtÞÞ þ m1 ðt; tÞX0 ðwðtÞÞ þ X0 ðwðsÞÞds 6 m ðtÞf2X0 ðwðtÞÞ þ X1 ðwðtÞÞg: ot 0

w0 ðtÞ ¼ v0 ðtÞ þ m1 ðt; tÞX0 ðvðtÞÞ þ

0

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R. Ravi Kumar / Applied Mathematics and Computation 204 (2008) 352–362

This implies

Z

wðtÞ

wð0Þ

ds 6 2X0 ðsÞ þ X1 ðsÞ

Z

T

m ðsÞds <

Z

0

c

1

ds : 2X0 ðsÞ þ X1 ðsÞ

This inequality implies that there exists a constant K such that wðtÞ 6 K, t 2 J and hence xðtÞ 6 K where K depends only on T and on the functions m1 ; m2 ; X0 and X1 . We shall prove that the operator W : Z ! Z is a completely continuous operator. Let

Bk ¼ fx 2 Z : kxðtÞk 6 k; for some k P 1g; we first show that W maps Bk into an equicontinuous family. Let y 2 Bk and let t 1 ; t 2 2 J. Then if 0 < t1 < t2 < T.

kðWxÞðt 1 Þ  ðWxÞðt2 Þk ¼ kRðt 1 Þ  Rðt 2 Þ½xð0Þ þ gð0; xð0ÞÞ  ½gðt 1 ; xðt 1 ÞÞ  gðt 2 ; xðt 2 ÞÞ Z t2 Z t1 ðAÞ½Rðt1  sÞ  Rðt2  sÞgðs; xðsÞÞds þ ðAÞRðt 2  sÞgðs; xðsÞÞds þ 0

þ

Z

t1

ðAÞ½Rðt1  sÞ  Rðt2  sÞ

0

Z

t1 s

Fðs  sÞgðs; xðsÞÞdsds þ

0

Z

t2

ðAÞRðt2  sÞ

t1

  Z s s t1 Fðs  sÞgðs; xðsÞÞdsds þ ½Rðt 1  sÞ  Rðt 2  sÞf s; xðsÞ; aðs; s; xðsÞÞds ds  0 0 0    Z s Z t2  1b b Rðt 2  sÞf s; xðsÞ; aðs; s; xðsÞÞds ds þ  6 kðAÞ Rðt 1 ÞðAÞ ðxð0Þ þ gð0; xð0ÞÞkjðt 1  t 2 Þj Z

Z

0

t

 Z1 t   1b b b b  þ  ðAÞ Rðt 1 ÞkFðsÞðAÞ ðxð0Þ þ gð0; xð0ÞÞkdsjðt1  t 2 Þj þ kðAÞ kk½ðAÞ ½gðt1 ; xðt1 ÞÞ 0 Z t 1  Z t1     ðAÞ2b Rðt 1  sÞðAÞb gðs; xðsÞÞds ðAÞ2b Rðt 1  sÞ  gðt 2 ; xðt 2 ÞÞk þ   jðt 1  t2 Þj þ  0 0  Z t2  Z s      Fðs  sÞðAÞb gðs; xðsÞÞdsds ðAÞ1b Rðt2  sÞðAÞb gðs; xðsÞÞds jðt 1  t 2 Þj þ   0

t

1 Z t1  Z t 1 Z s    2b b   jðt þ ðAÞ Rðt1  sÞ Fðs  sÞðAÞ gðs; xðsÞÞdsds  t Þj þ ðAÞ2b Rðt 1  sÞ 1 2   0 0 0  Z t2 Z s Z s    jðt  Fðs  sÞ Fðs  hÞðAÞb gðh; xðhÞÞdhdsds  t Þj þ ðAÞ1b Rðt2  sÞ 1 2  

0

0

t

1  Z t1   1b b  Fðs  sÞðAÞ gðs; xðsÞÞdsds ðAÞ Rðt 1  sÞðAÞ f þ 0 0 Z t1    Z s Z s   1b b  jðt  s; xðsÞ; aðs; s; xðsÞÞds ds  t Þj þ ðAÞ Rðt  sÞðAÞ Fðs  sÞf ðs; xðsÞ; 2 1  1  0 0 0      Z s Z s  Z t2  b b  jðt  ds aðs; h; xðhÞÞdhÞdsds  t Þj þ ðAÞ Rðt  sÞðAÞ f s; xðsÞ; aðs; s ; xð s ÞÞd s 2 2    1

Z

s

b

0

t1

0

6 M 0 C 1b T b1 ðkxð0ÞkÞjt 1  t 2 j þ M0 M2 C 1b T b ðkxð0ÞkÞjt1  t2 j þ C 1b T b1 b1 jt1  t2 j C 2b b1 T b1 jt 1  t2 j þ M 2 C 1b T b b1 jt1  t 2 j þ M 0 b2 ½jt 1  t2 j þ kxðt 1 Þ  xðt 2 Þk þ b1 C 2b b C 1b C 2b bþ1 C 1b þ 2M 2 T b1 jt1  t 2 j þ b1 jt 1  t2 jb þ 2M2 T b1 jt1  t 2 j þ M 2 T b1 jt 1  t 2 jb b1 b b b C 1b b C 1b 1þb C 1b þ M0 l ðsÞjt1  t2 jb : T lk ðsÞjt 1  t2 j þ M 0 M 2 T lk ðsÞjt1  t 2 j þ M 0 b b b k The right hand side is independent of x 2 Bk and tends to zero as t1  t2 ! 0, since f ; g; a are completely continuous and the compactness of RðtÞ for t > 0 implies continuity in the uniform operator topology. Thus, W maps Bk into an equicontinuous family of functions. Next, we show that WBk is compact. Since we have shown WBk is equicontinuous collection, by the Arzela–Ascoli theorem it suffices to show that W maps Bk into precompact set in X. Let 0 6 t 6 T be fixed and let  be a real number satisfying 0 <  < t. For x 2 Bk , we define

Z t ðW xÞðtÞ ¼ RðtÞ½xð0Þ þ gð0; xð0ÞÞ  gðt; xðtÞÞ þ ðAÞRðt  sÞgðs; xðsÞÞds 0 Z t Z s Z t Z s þ ðAÞRðt  sÞ Fðs  sÞgðs; xðsÞÞdsds þ Rðt  sÞf ðs; xðsÞ; aðs; s; xðsÞÞdsÞds: 0

0

0

0

R. Ravi Kumar / Applied Mathematics and Computation 204 (2008) 352–362

359

Since RðtÞ is compact, the set fW xðtÞ : x 2 Bk g is precompact in X for every , 0 <  < t. Moreover for every x 2 Bk , we have

kðWxÞðtÞ  ðW xÞðtÞk 6

Z

t

kðAÞRðt  sÞgðs; xðsÞÞkds þ

t

þ

Z

Z

t

kðAÞRðt  sÞ

t

Z

s

Fðs  sÞgðs; xðsÞÞdskds

0

t

t

kRðt  sÞklk ðsÞds

Therefore, there are precompact sets arbitrary close to the set fWxðtÞ : x 2 Bk g. Hence the set fWxðtÞ : x 2 Bk g precompact in X. It remains to show that W : Z ! Z is continuous. Let fxn g1 0  Z with xn ! x in Z. Then there is an integer l such that kxn ðtÞk 6 l for all n and t 2 J, so xn 2 Bk and x 2 Bk . By ðH3 Þ

    Z t Z t f t; xn ðtÞ; aðt; s; xn ðsÞÞds ! f t; xðtÞ; aðt; s; xðsÞÞds 0

0

for each t 2 J and since

  Z t Z t   f ðt; xn ðtÞ;  6 2l ðtÞ; aðt; s; x ðsÞÞdsÞ  f ðt; xðtÞ; aðt; s; xðsÞÞdsÞ n l   0

0

we have by the dominated convergence theorem

 Z t    kðWxn Þ  ðWxÞk 6 kgðt; xn ðtÞÞ  gðt; xðtÞÞk þ   ðAÞRðt  sÞ½gðs; xn ðsÞÞ  gðs; xðsÞÞds 0

Z t  Z s    þ ðAÞRðt  sÞ Fðs  s Þ½gð s ; x s ÞÞ  gð s ; xð s ÞÞd s ds n   0 0 Z t       Z s Z s   þ  f s; xðsÞ; ds Rðt  sÞ f s; x ðsÞ; aðs; s ; x ð s ÞÞd s aðs; s ; xð s ÞÞd s n n   0

0

0

! 0 as n ! 1: Thus, W is continuous. This completes the proof that W is completely continuous. Finally, the set fðWÞ ¼ fx 2 Z : x ¼ kWx; k 2 ð0; 1Þg is bounded. Consequently by Schaefers theorem the operator W has a fixed point in Z. This means that any fixed point of W is a mild solution of (1) and (2) on J satisfying ðWxÞðtÞ ¼ xðtÞ. 5. Example Consider the following partial functional integrodifferential equation

    Z t Z p Z t o o2 z2 ðt; xÞ zðt; xÞ  aðs; y; xÞzðs; yÞdyds ¼ 2 zðt; xÞ þ bðt  sÞzðs; yÞds þ ot ox ð1 þ tÞð1 þ t2 Þ 0 0 0 Z t zðs; xÞ þ ds; 2 2 2 0 ð1 þ tÞð1 þ t Þ ð1 þ sÞ 0 6 t 6 T; 0 6 x 6 p;

ð5Þ

zðt; 0Þ ¼ zðt; pÞ ¼ 0; zð0; xÞ þ

i¼p Z p X i¼0

kðx; yÞzðti ; yÞdy ¼ z0 ðxÞ;

0 6 x 6 p;

ð6Þ

0

where T 6 p; p is a positive integer, 0 < t0 < t1 <    < t p < T, z0 ðxÞ 2 X ¼ L2 ð½0; pÞ and kðx; yÞ 2 L2 ð½0; p  ½0; pÞ. In order to write Eqs. (5) and (6) in the abstract form of (1) and (2), take X ¼ L2 ð½0; pÞ and let A be defined by

Aw ¼ w00 with domain

DðAÞ ¼ fw 2 X : w0 w00 2 X and wð0Þ ¼ wðpÞ ¼ 0g: Then A generates a strongly continuous semigroup RðtÞ which is compact,analytic and self-adjoint. Furthermore, A has a dis 1 crete spectrum, the eigenvalues are n2 , n 2 N with the corresponding normalized eigenvectors zn ðxÞ ¼ p2 2 sinðnxÞ [7,11] and bðt  sÞ is continuous then there exist a constant k1 > 0 such that jbðt  sÞj 6 k1 . Then the following properties hold: (i) If w 2 DðAÞ, then

Aw ¼

1 X n¼1

n2 < w;

zn > zn :

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R. Ravi Kumar / Applied Mathematics and Computation 204 (2008) 352–362

P 1=2 (ii) For each w 2 X, A1=2 w ¼ 1 k ¼ 1. n¼1 1=n < w; zn > zn . In particular, kA 1=2 (iii) The operator A is given by

A1=2 w ¼

1 X

n < w;

zn > zn :

n¼1

on the space DðA1=2 Þ ¼ fwðÞ 2 X;

P1

n¼1 n

< w; zn > zn 2 Xg:

Define f : J  X  X ! X g : J  X ! X and k : E ! X by

gðt; wÞ ¼

Z

t

Z p

aðs; y; xÞzðyÞdy ds; Z t z ðt; wÞ zðs; wÞ f ðt; w; Þ ¼ ds; þ 2 2 2 ð1 þ tÞð1 þ t2 Þ 0 ð1 þ tÞð1 þ t Þ ð1 þ sÞ p X Kwðt i Þ: and qðwÞ ¼ 0

0 2

i¼0

Since the analytic resolvent RðtÞ is compact, there exist constants k2 ; k3 > 0 such that kRðtÞk 6 k2 and kðAÞa Rðt  sÞk 6 k3 ðt  sÞa for each t 2 J and 0 < a < 1. Further, the functions a : J  J  ½0; p ! ½0; p is completely continuous and uniformly bounded that is there exist constants n1 ; n2 > 0 such that

kðAÞb aðs; y; xÞk 6 n1 ; kðAÞb ½aðs; y1 ; x1 Þ  aðs; y2 ; x2 Þk 6 n2 ½js  sj þ ky1  y2 k þ kx1  x2 k: Also the function f : J  ½0; p  ½0; p ! ½0; p are measurable and there integrable functions l1 : J  J ! ½0; 1Þ and l2 : J ! ½0; 1Þ such that

    zðs; wÞ    6 l1 ðs; sÞX0 ðkvkÞ  ð1 þ tÞð1 þ t 2 Þ2 ð1 þ sÞ2 

and

    z2 ðt; wÞ zðs; wÞ   þ  6 l2 ðtÞX1 ðkvk þ kwkÞ;  ð1 þ tÞð1 þ t 2 Þ ð1 þ tÞð1 þ t 2 Þ2 ð1 þ sÞ2  where X0 ; X1 : ½0; 1Þ ! ð0; 1Þ are continuous nondecreasing and

Z

T

n ðsÞds <

0

Z c

1

ds

X0 ðsÞ þ X1 ðsÞ

;

n Rt b 1þb where c ¼ k2 ½kx0 k þ H þ M 0 n1  þ M 0 n1 þ k3 Tb n1 þ k1 k3 T b n1 and n ðsÞ ¼ max k2 l2 ðtÞ; l1 ðt; tÞ; 0 ditions of the Theorem 4.1 is satisfies Eqs. (5) and (6) has a mild solution on ½0; T.

o l ðt; sÞds ot 1

o . Since all the con-

6. Application As an application of Theorem 4.1 we shall consider the system (1) and (2) with a control parameter such as

    Z t Z t d ½xðtÞ þ gðt; xðtÞÞ ¼ A xðtÞ þ Fðt  sÞxðsÞds þ BuðtÞ þ f t; xðtÞ; aðt; s; xðsÞÞds ; dt 0 0

ð7Þ

xð0Þ ¼ x0  qðt1 ; t 2 ; . . . ; tp ; xðÞÞ;

ð8Þ 2

where B is a bounded linear operator from U, a Banach space, to X and u 2 L ðJ; UÞ. The mild solution is given by

Z t xðtÞ ¼ RðtÞ½x0  qðt 1 ; . . . ; t p ; xðÞÞ þ gð0; xð0ÞÞ  gðt; xðtÞÞ  ARðt  sÞgðs; xðsÞÞds 0   Z s Z t Z s Z t ARðt  sÞ Fðs  sÞgðs; xðsÞÞdsds þ Rðt  sÞ BuðsÞ þ f ðs; xðsÞ; aðs; s; xðsÞÞdsÞ ds:  0

0

0

0

Definition 6.1. System (7) is said to be controllable with nonlocal condition (8) on the interval J if for every x0 ; xT 2 X, there exists a control u 2 L2 ðJ; UÞ such that the mild solution xðÞ of (7) and (8) satisfies

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R. Ravi Kumar / Applied Mathematics and Computation 204 (2008) 352–362

xð0Þ ¼ x0  qðt1 ; t 2 ; . . . ; t p ; xðÞÞ;

xðTÞ ¼ xT :

To establish the result, we need the following additional conditions ðH10 Þ The linear operator W : L2 ðJ; UÞ ! X, defined by

Wu ¼

Z

T

RðT  sÞBuðsÞds;

0

~ ~ induces an inverse operator W 1 defined on L2 ðJ; UÞ=kerW and there exists a constant M 3 > 0 such that kBW 1 k 6 M 3 . R t om1 ðt;sÞ  ðH11 Þ The function m ðtÞ ¼ maxfMm2 ðtÞ; m1 ðt; tÞ; 0 dt dsg satisfies

Z

T

m ðsÞds <

Z

0

c

1

ds 2X0 ðsÞ þ X1 ðsÞ 1þb

b

where c ¼ M½kx0 k þ H þ M 0 b1  þ M 0 b1 þ C 1b b1 Tb þ M 2 C 1b b1 T b þ MNT and

Z N ¼ M 3 kxT k þ M½kx0 k þ H þ M 0 b1  þ M 0 b1 þ þM

T

C 1b ðt  sÞ1b b1 ds þ M 2

Z

0

Z

T

C 1b ðt  sÞ1b b1 ds

0

   Z s m2 ðsÞX1 kuðsÞk þ mðs; sÞX0 ðkuðsÞkÞds ds

0

T

0

Theorem 6.1. If the hypothesis ðH1 Þ—ðH8 Þ and ðH10 Þ—ðH11 Þ are satisfied then the system (7) and (8) are controllable on J. Proof. Using the hypothesis ðH10 Þ, for an arbitrary function xðÞ define the control

~ uðtÞ ¼ W 1 ½xT  RðTÞ½x0  qðt1 ; t 2 ; . . . ; tp ; xðÞÞ  gð0; xð0ÞÞ  gðT; xðTÞÞ þ 

Z

s

Fðs  sÞgðs; xðsÞÞdsds 

0

Z

T

ARðT  sÞgðs; xðsÞÞds þ

Z

0

Z

T

  Z s RðT  sÞf s; xðsÞ; aðs; s; xðsÞÞds dsðtÞ:

0

T

ARðT  sÞ

0

0

We shall show that when using this control the operator U : Z ! Z defined by

Z t Z t ðUxÞðtÞ ¼ RðtÞ½x0  qðt1 ; . . . ; tp ; xðÞÞ þ gð0; xð0ÞÞ  gðt; xðtÞÞ  ARðt  sÞgðs; xðsÞÞds  ARðt  sÞ 0 0    Z t Z s Z s Fðs  sÞgðs; xðsÞÞdsds þ Rðt  sÞ BuðsÞ þ f s; xðsÞ; aðs; s; xðsÞÞds ds  0

0

0

has a fixed point. This fixed point is then a solution of (7) and (8). Clearly,ðUxÞðTÞ ¼ xT , which means that the control u steers the system (7)–(8) from the initial state x0 to xT in time T provided we can obtain a fixed point of the nonlinear operator U. The remaining part of the proof is similar to Theorem 4.1 and hence it is omitted. Acknowledgements The author wishes to thank Prof. K. Balachandran, Department of Mathematics, Bharathiar University, Coimbatore, for fruitful discussions regarding this problem and to thank the Management and Principal of Coimbatore Institute of Engineering and Information Technology, for providing the facilities and to carryout this work. References [1] O. Arino, R. Benkhali, K. Ezzinbi, Existence results for initial value problem neutral functional differential equations, Journal of Differential Equations 138 (1997) 188–193. [2] K. Balachandran, J.P. Dauer, Existence of solutions of a nonlinear mixed neutral equations, Applied Mathematics Letters 11 (1998) 23–28. [3] K. Balachandran, J.Y. Park, Existence of solutions and controllability of nonlinear integrodifferential systems in Banach spaces, Mathematical Problems in Engineering 2003 2 (2003) 67–79. [4] K. Balachandran, R. Sakthivel, Existence of solutions of neutral functional integrodifferential equations in Banach spaces, Proceedings of the Indian Academic Sciences and Mathematical Sciences 109 (1999) 325–332. [5] J.P. Dauer, K. Balachandran, Existence of solutions of nonlinear neutral integrodifferential equations in Banach spaces, Journal of Mathematical Analysis and Applications 251 (2000) 93–105. [6] W. Desch, R. Grimmer, W. Schappacher, Some considerations for linear integrodifferential equations, Journal of Mathematical Analysis and Applications 104 (1984) 219–234. [7] X.L. Fu, K. Ezzinbi, Existence of solutions for neutral functional differential evolution equations with nonlocal conditions, Nonlinear Analysis 54 (2003) 215–227. [8] R. Grimmer, Resolvent operators for integral equations in a Banach space, Transactions of the American Mathematical Society 273 (1982) 333–349. [9] R. Grimmer, A.J. Pritchard, Analytic resolvent operators for integral equations in a Banach space, Journal of Differential Equations 50 (1983) 234–259.

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