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IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-3008, p-ISSN:2319-7676. Volume 10, Issue 2 Ver. II (Mar-Apr. 2014), PP 129-138 www.iosrjournals.org

Existence of Solutions for Quasilinear Neutral Integrodifferential Equations in Banach Spaces R.Prabakaran 1, and V. Vinoba † Abstract: In this paper, we devoted to study the existence of mild solutions for quasilinear integrodifferential equation in Banach spaces. The results are established by using Hausdorff’s measure of noncompactness and the fixed point theorem. Application is provided to illustrate the theory. Keywords: Mild solution, neutral differential equation, Hausdorff measure of noncompactness, fixed point theorem. 2000 AMS Subject Classification: 34A37, 34K05, 47H10

I. Introduction In various fields of engineering and physics, many problems that are related to linear viscoelasticity, nonlinear elasticity have mathematical models and are described by the problems of differential or integral equations or integrodifferential equations. Our work centers on the problems described by the integrodifferential models. It is important to note that when we describe the systems which are functions of space and time by partial differential equations, in some situations, such a formulation may not accurately model the physical system because, while describing the system as a function at a given time, it may fail to take into account the effect of past history. Neutral differential equations arise in many areas of applied mathematics and for this reason these equations have received much attention during the last few decades [1, 2, 3]. A good guide to the literature for neutral functional differential equations is the book by Hale and Verduyn Lunel [4] and the references therein. The existence of solution to evolution equations with nonlocal conditions in Banach space was studied first by Byszewski [5, 6]. Byszewski and Lakshmikanthan [7] proved an existence and uniqueness of solutions of a nonlocal Cauchy problem in Banach spaces. Ntouyas and Tsamatos [8] studied the existence for semilinear evolution equations with nonlocal conditions. The problem of existence of solutions of evolution equations in Banach space has been studied by several authors [9, 10]. Measures of noncompactness are a very useful tool in many branches of mathematics. They are used in the fixed point theory, linear operators theory, theory of differential and integral equations and others [11]. There are two measures which are the most important ones. The Kuratowski measure of noncompactness  (X ) of a bounded set X in a metric space is defined as infimum of numbers r > 0 such that X can be covered with a finite number of sets of diameter smaller than r . The Hausdorff measure of noncompactness  (X ) defined as infimum of numbers r > 0 such that X can be covered with a finite number of balls of

r. There exist many formulae on  (X ) in various spaces [11, 14]. Let E be a Banach space and F be a subspace of E. Let E (X ) , F (X ) ,  E (X ) ,  F (X ) denote Hausdorff and Kuratowski measures in spaces E,F, respectively. Then, for any bounded X  F we radii smaller than

have

E ( X )  F ( X )   F ( X ) =  E ( X )  2E ( X ).

The notion of a measure of weak compactness was introduced by De Blasi [12] and was subsequently used in numerous branches of functional analysis and the theory of differential and integral equations. Several authors have studied the measures of noncompactness in Banach spaces [13, 14, 15]. Motivated by [9, 11, 16, 17], in this paper, we study the existence results for quasilinear equation represented by first-order neutral integrodifferential equations using the semigroup theory and the measure of noncompactness.

1

Department of Science and Humanities, Sri Krishna College of Technology, Coimbatore-42, India.

† Department of Mathematics, K.N. Government of Arts College for Woman, Thanjavur, India. 

Corresponding author-email: prabakaranr67@gmail.com

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Existence of Solutions for Quasilinear Neutral Impulsive Integrodifferential Equations in Banach Spaces

II. Preliminaries We consider the quasilinear integrodifferential equations with nonlocal condition of the form t d [ x(t )  e(t , x(t ),  k (t , s, x( s))ds)]  A(t , x(t )) x(t ) 0 dt t

= f (t , x(t ))   g (t , s, x( s))ds, t [0, b], t  tk ,

(1)

0

x(0)  h( x) = x0 ,

(2)

A : [0, b]  X  X are continuous functions in Banach space X , x0  X , f : [0, b]  X  X , g :   X  X , h : C([0, b], X )  X , e : [0, b]  X  X  X , k :   X  X . Here  = {(t , s) : 0  s  t  b}. Let X be a Banach space with norm ||  || and denoted L([0, b], X ) by the space of X -valued Bochner integrable functions on [0,b] with the form where

b

x L =  x(t ) dt. 0

 Y is defined by  (B) = inf {r > 0, B can be covered by finite number of balls with radii r}, for bounded set B in a Banach space Y . Lemma 2.1 [11]. Let Y be a real Banach space and B, E  Y be bounded, with the following The Hausdorff’s measure of noncompactness

properties: (i). B is precompact if and only if

 X (B) = 0.

Y (B) = Y (B ) = Y (conB), where B and con B mean the closure and convex hull of B respectively. (iii). Y (B)  Y ( E ), where B  E. (iv). Y (B  E )  Y (B)  Y ( E ), where B  E = {x  y : x  B, y  E}. (v). Y (B  E )  max{Y (B), Y ( E )}. (vi). Y (B) |  | Y ( B), for any   R. (vii). If the map F : D(F)  Y  Z is Lipschitz continuous with constant r , then  Z (FB)  rY (B), for any bounded subset B  D(F), where Z be a Banach space. (viii). Y ( B) = inf {dY (B, E ); E  Y is precompact} = inf {dY ( B, E ); E  Y is finite valued}, where dY (B, E ) means the non-symmetric (or symmetric) Hausdorff distance between B and E in Y.  (ix). If {Wn }n =1 is decreasing sequence of bounded closed nonempty subsets of Y and (ii).

lim n Y (Wn ) = 0, then



Wn is nonempty and compact in Y .

n =1

F : W  Y  Y is said to be a  Y -contraction if there exists a positive constant r < 1 such that Y (F(B))  rY (B) for any bounded closed subset B  W , where Y is a Banach space. The map

W  Y is bounded closed and convex, the continuous map F : W  W is a  Y -contraction, the map F has atleast one fixed point in W. We denote by  the Hausdorff’s measure of noncompactness of X and also denote  c by the Hausdorff’s measure of noncompactness of PC([0, b], X ) . Lemma 2.2 (Darbo-Sadovskii [11]). If

Before we prove the existence results, we need the following Lemmas. www.iosrjournals.org

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Existence of Solutions for Quasilinear Neutral Impulsive Integrodifferential Equations in Banach Spaces

W  PC([0, b], X ) is bounded, then  (W (t ))   c (W ), for all t [0,b], where W (t ) = {u(t ); u  W}  X . Furthermore if W is equicontinuous on [0,b] , then  ( W (t )) is continuous on [0,b] and  c (W ) = sup{ (W (t )), t [0, b]}. Lemma 2.3 If

Lemma 2.4 [18, 19]. If {un }n=1  L ([0, b], X ) is uniformly integrable, then the function

 ({un (t )}n=1 )

1

is measurable and

t

 (  un ( s)ds 0

t

n =1

)  2  ({un ( s)}n=1 )ds.

(3)

0

W  PC([0, b], X ) is bounded and equicontinuous, then  (W (t )) is continuous

Lemma 2.5 If and

t

t

0

0

 ( W (s)ds)    (W (s))ds, for all t [0, b], where

t

t

0

0

(4)

 W (s)ds = { u(s)ds : u  W}.

C0 semigroup U u (t , s) is said to be equicontinuous if (t , s)  {U u (t , s)u(s) : u  B} is equicontinuous for t > 0, for all bounded set B in X . The following lemma is obvious. The

Lemma 2.6 If the evolution family

{U u (t , s)}0st b is equicontinuous and   L([0, b], R ), then

t

the set

{ U u (t , s)u (s)ds, || u ( s) ||  ( s), for a.e s [0,b]}, is equicontinuous for t [0,b]. 0

u  PC([0, b], X ) there exist a unique continuous function U u : [0, b]  [0, b]  B( X ) defined on [0, b]  [0, b] such that We know that, for any fixed t

U u (t , s) = I   Au ( w)U u ( w, s)dw,

(5)

s

B(X ) denote the Banach space of bounded linear operators from X to X with the norm || F ||= sup {|| Fu ||:|| u ||= 1}, and I stands for the identity operator on X , Au (t ) = A(t , u(t )), we have U u (t , t ) = I , U u (t , s)U u (s, r ) = U u (t , r ), (t , s, r ) [0, b]  [0, b]  [0, b], where

U u (t , s) = Au (t )U u (t , s), for almost all t , s  [0, b]. t III. The Existence of Mild Solution Definition 3.1 A function satisfies the integral equation

x  PC([0, b], X ) is said to be a mild solution of (1)  (2) if it

t x(t ) = U x (t ,0) x0  U x (t ,0)h( x)  U x (t ,0)e(0, x(0),0)  e t , x(t ),  k (t , s, x( s))ds  0   t s   A( s, x( s))U x (t , s)e s, x( s),  k ( s, , x( ))d ds 0 0   t

s

0

0

  U x (t , s)[ f ( s, x( s))   g ( s, , x( ))d ]ds, 0  t  b. In this paper, we denote generality, we let

M0 = sup{ U x (t , s) : (t , s) [0, b]  [0, b]}, for all x  X . Without loss of

x0 = 0. www.iosrjournals.org

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Existence of Solutions for Quasilinear Neutral Impulsive Integrodifferential Equations in Banach Spaces Assume the following conditions: (H1). The evolution family {U x (t , s)}0 s t b generated by

A(t , x(t )) is equicontinuous and

|| U x (t , s) || M0 , for almost t , s [0, b]. (H2). (a). The function h :PC([0, b]  X )  X is continuous and compact. (b). There exists N0 > 0 such that || h( x) || N0 , for all u  PC([0, b]; X ) .

f : [0, b]  X  X satisfies the Caratheodory-type conditions; that is, f (, x) is measurable for all x  X and f (t ,) is continuous, for a.e t [a, b].  (ii). There exists a function   L([0, b], R ) such that for every x  X , we have f (t , x)   (t )(1  x ), a.e t [0, b].

(H3). (i). The nonlinear function

(iii). There exists a function

m f  L([0, b], R ) such that, for every bounded D  X , we

 ( f (t , K))  m f (t )  (K), a.e t [0, b]. The nonlinear function g : [0, b]  [0, b]  X  X satisfies the Caratheodory-type conditions; i.e., g (,, x) is measurable, for all x  X and g (t , s,) is continuous for a.e t  [a, b].   There exist two functions 1  L([0, b], R ) and  2  L([0, b], R ) such that for every x  X , we have g (t , s, x( s))  1 (t )  2 ( s)(1  x( s) ), a.e t [0, b]. have

(H4). (i).

(ii).

(iii). There exists a function we have

mg , ng  L([0, b], R ) such that, for every bounded D  X ,

 ( g (t , s, K))  mg (t )ng (s)  (K), a.e t [0, b]. t

Assume that the finite bound of

m 0

g

( s)ds is G0 .

e : [0, b]  X  X  X satisfies the Caratheodory-type conditions; that is, e(, x, x1 ) is measurable, for all x, x1  X and e(t ,,) is continuous, for a.e t [0,b].  (ii). There exists a function   L([0, b], R ) such that for every x, x1  X , we have e(t , x, x1 )   (t )(1  x )  x1 , a.e t [0, b]. (iii). The nonlinear function q : [0, b]  [0, b]  X  X satisfies the Caratheodory-type conditions; i.e. k (,, x) is measurable, for all x  X and k (t , s,) is continuous, for a.e t [0,b].

(H5). (i). The function

(iv). There exist two functions every

1  L([0, b], R )

and

2  L([0, b], R )

such that for

x  X , we have k (t , s, x(s))  1 (t )2 (s)(1  x(s) ), a.e t [0, b].

(v). There exists a function   L([0, b], R

) such that for every x, x1  X , we have

A(t , x(t ))e(t , x, x1 )   (t ) e(t , x, x1 ) , a.e t [0, b]. 

(vi). There exists a function me  L([0, b], R ) such that, for every bounded have

D  X , we

 (e(t , K, K1 ))  me (t )  (K)   (K1 ), a.e t [0, b]. www.iosrjournals.org

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Existence of Solutions for Quasilinear Neutral Impulsive Integrodifferential Equations in Banach Spaces t

 m (s)ds is G .

Assume that the finite bound of

e

0

1

(vii). There exists a function mk , nk  L([0, b], R ) such that for every bounded have

D  X , we

 (k (t , s, K))  mk (t )nk (s)  (K), a.e t [0, b]. t

Assume that the finite bound of (H6). For every

 m (s)ds is G . k

0

2

t [0,b] and there exist positive constants N1 and N2 , the scalar equation t

m(t )=M0N0   1 (1  m( s))  M0 0  M0C1 (t )(1  m(s))   (t ) (t )C1  1 (s)ds 0

t

t

 M0  [ ( s)(1  m( s))ds  C0  1 ( s)(1  m( s))ds]. 0

0

s

Where

C0    (t )dt. 0

Theorem: 3.1 Assumptions

( H1 )  ( H 7 ) holds, then the quasilinear neutral impulsive

problem (1)  (3) has at least one mild solution. Proof. Let

m(t ) be a solution of the scalar equation t

m(t )=M0N0   1 (1  m( s))  M0 0  M0C1 (t )(1  m(s))   (t ) (t )C1  1 (s)ds 0

t

t

0

0

 M0  [ ( s)(1  m( s))ds  C0  1 ( s)(1  m( s))ds]. t



Let us assume that the finite bound of

0

F : C([0, b], X )  C([0, b], X ) defined by

2

(6)

( s)ds is C0 , for t [0,b]. Consider the map

t (Fx)(t ) = U x (t ,0)h( x)  U x (t ,0)e(0, x(0),0)  e t , x(t ),  k (t , s, x( s))ds  0   t s   A( s, x( s))U x (t , s)e s, x( s),  k ( s, , x( ))d ds 0 0  

t

s

0

0

  U x (t , s)[ f ( s, x(s))   g ( s, , x( ))d ]ds, 0  t  b,

(8)

for all x  C([0, b], X ). Let us take W0 = {x C([0, b], X ), || x(t ) || m(t ), for all t [0, b]}.

W0  C([0, b]; X ) is bounded and convex. We define W1 = con K ( W0 ), where con means the closure of the convex hull in C([0, b], X ). As U x (t , s) is equicontinuous, h is compact and Then

W0  C([0, b], X ) is bounded, due to Lemma 2.6 and using the assumptions, W1  C([0, b], X ) is bounded closed convex nonempty and equicontinuous on [0,b]. For any u  F( W0 ), we know t x(t )  U x (t ,0)h(u )  U x (t ,0)e(0, x(0),0)  e t , x(t ),  k (t , s, x( s))ds  0   t s   A(t , x(t ))U x (t , s)e t , x(t ),  k ( s, , x( ))d  ds 0 0   t

s

0

0

  U x (t , s)[ f ( s, x( s))   g ( s, , x( s))d ] ds t

 M0N0  M0 0   1 (1  x )   k (t , s, x( s))ds 0

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Existence of Solutions for Quasilinear Neutral Impulsive Integrodifferential Equations in Banach Spaces t s  M0 (t )  e t , x(t ),  k ( s, , x( ))d  ds 0  0  t

t s

0

0 0

 M0 [  || f ( s, x( s)) || ds    || g ( s, , x( )) || dds] t

 M0N0  M0 0   1 (1  x )  1 (t ) 2 ( s)(1  x )ds 0

t

s

0

0

 M0 [ [ (s) (s)ds   1 ( s)2 ( )dsd ](1  x )] t

t s

0

0 0

M0   (s)(1  x(s) )ds  M0   1 (s) 2 ( )(1  x( ) )dds t

 M0N0   1 (1  m(s))  M0 0  M0C1 (t )(1  m( s))   (t ) (t )C1  1 (s)ds 0

t

t

0

0

 M0  [ ( s)(1  m( s))ds  C0  1 ( s)(1  m(s))ds], for t [0, b].

= m(t ). It follows that

W1  W0 . We define Wn1 = con F(Wn ), for n = 1,2,3,. From above we know that

 n n =1

{W } is a decreasing sequence of bounded, closed, convex, equicontinuous on [0,b] and nonempty subsets in C([0, b], X ). Now for n  1 and t [0, b], Wn (t ) and F( Wn (t )) are bounded subsets of X , hence, for any

 > 0, there is a sequence {xk }k =1  Wn ,  (Wn1 (t ))= (FWn (t ))

such that (see, e.g. [7], pp.125).

t

 2  (e(t ,{xk (s)}k =1 ,  k (t , s,{xk (t )}k =1 )ds)) 0

t

s

0

0

2M0 (t )  (e(s,{xk (s)}k =1 ,  k ( s, ,{xk ( )}k =1 )d ))ds t

t s

 2M0   ( f (s,{xk (s)}k =1 ))ds  4M0    ( g ( s, ,{uk ( )}k =1 ))dds   0

0 0

 2me (t ) {xk (t )}

 k =1

t

 2mk (t )  mk (s) {xk ( s)}k =1 ds 0

t

t s

 2M0 (t )[ me (s) {xk ( s)}k =1 ds  2  mk (s)mk ( ) {xk ( )}k =1 dds] 0

t

0 0 t s

 2M0  m f (s)  ({uk (s)} )ds4M0   mg ( s)ng ( )  ({uk ( )}k =1 )dds    k =1

0

0 0

 2[me (t )  mk (t )G2  M0 (t )G1 ] (Wn (t )) t

t

0

0

 2M0 [ {2G2 mk (s)  m f (s)} (Wn (s))ds  2G0  ng (s)  (Wn ( s))ds]  

Since

 > 0 is arbitrary, it follows that from the above inequality that  (Wn1 (t ))  2[me (t )  mk (t )G2  M0 (t )G1 ] (Wn (t ))

(8)

t

 2M0 [ [2G2 mk (s)  m f (s)  2G0 ng ( s)] ( Wn ( s))]ds, for all t [0, b]. 0

Because

Wn is decreasing for n, we have

 (t ) = lim  (Wn (t )), for all

t [0,b]. From (8), we have

n

 (t )  2[me (t )  mk (t )G2  M0 (t )G1 ] (t ) t

 2M0 [  [2G2 mk ( s)  m f ( s)  2G0 ng ( s)] ( s)ds], 0

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Existence of Solutions for Quasilinear Neutral Impulsive Integrodifferential Equations in Banach Spaces for

t [0,b], which implies that  (t ) = 0, for all ti [0,b]. By Lemma 2.3, we know that 

lim n  (Wn (t )) = 0. Using Lemma 2.1 we know that W = n=1Wn is convex compact and nonempty in PC([0, b], X ) and F(W )  W. By the Schauder fixed point theorem, there exist at least one mild solution u of the initial value problem (1)  (2) , where x  W is a fixed point of the continuous map F. Remark 3.2. If the functions f , g and I i are compact or Lipschitz continuous (see e.g [8, 10]), then

( H 3 )  ( H 6 ) is automatically satisfied.

In some of the early related results in references and above results, it is supposed that the map h is uniformly bounded. In fact, if h is compact, then it must be bounded on bounded set. Here we give an existence result under growth condition of assumptions

f , g and I i , when h is not uniformly bounded. Precisely, we replace the

( H 3 )  ( H 6 ) by

(H7). There exists a function that

p  L([0, b], R ) and a increasing function  : R  R such f (t , x)  L f (t ) ( x ),

for a.e

t [0,b], for all x  C([0, b], X ).

(H8). There exist two functions function

~ Lg  L([0, b], R ) and Lg  L([0, b], R  ) and a increasing

 : R  R such that ~ g (t , s, x)  Lg (t ) Lg ( s)( x ),

for a.e

t [0,b] and for all Lg  C([0, b], X ). Assume that the finite bound of

t

 L (s)ds is G . g

0

3

(H9). There exists a function Le  L([0, b], R ) and a increasing function that

 : R  R such

e(t , x, x1 )  Le (t )( x )  x1 for a.e

t [0,b] and for all Lg  C([0, b], X ). Assume that the finite bound of

t

 L (s)ds is G . 0

e

5

(H10). There exist two functions Lk  L([0, b], R ) and function

~ Lk  L([0, b], R  ) and a increasing

 : R  R such that ~ k (t , s, x)  Lk (t ) Lk ( s)( x ),

for a.e

t [0,b] and for all Lk  C([0, b], X ). Assume that the finite bound of

t

 L (s)ds is G . 0

k

4

Theorem: 3.2 Suppose that the assumptions equation

( H1 )  ( H 2 ) and ( H 7 )  ( H10 ) are satisfied, then the

(1)  (2) has at least one mild solution if 1 lim sup {M0 [ (r )  Le (t )]  Le (t )( x ) r  r www.iosrjournals.org

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Existence of Solutions for Quasilinear Neutral Impulsive Integrodifferential Equations in Banach Spaces t ~  G3 Lk (t )(r )   (t )M0 [G4 (r )  G3(r )  Lk ( s)ds] 0

t t ~  M0 [ (r )  L f ( s)ds  G2 (r )  Lg ( s)ds]} < 1, 0

where

(9)

0

 (r ) = sup{|| h( x) ||, || x || r}.

(10) implies that there exist a constant r > 0 such that M0 [ (r )  Le (t )]  Le (t )( x )  G3 Lk (t )(r )   (t )M0 [G4 (r )

Proof. The inequality

t ~ t t ~  G3(r )  Lk ( s)ds]  M0 [ (r )  p( s)ds  G2 (r )  Lg ( s)ds] < r , 0

As in the proof of Theorem 3.1, let

0

0

W0 = {x C([0, b], X ), || x(t ) || r} and

W1 = con FW0 . Then for any x  W1 , we have t x(t )  U x (t ,0)h( x)  U x (t ,0)e(0, x(0),0)  e t , x(t ),  k (t , s, x( s))ds  0   t s   A(t , x(t ))U x (t , s)e t , x(t ),  k ( s, , x( ))d  ds 0 0   t

s

0

0

  U x (t , s)[ f ( s, x( s))   g ( s, , x( s))d ] ds

~ t  M0 [ (r )  Le (t )]  Le (t )( x )   Lk (t ) Lk ( s)( x )ds 0

~ t s   (t )M0  [ p( s)( x )   Lk ( s) Lk ( )( x )d ]ds 0

0

~ t t s  M0 [  L f ( s) ( x( s) )ds   Lg ( s) Lg ( ) ( x( ) dds] 0

0 0

 M0 [ (r )  Le (t )]  Le (t )( x )  G3 Lk (t )( x ) t ~   (t )M0 [G4 ( x )  G3  Lk ( s)( x )ds] 0

t t ~  M0 [  p( s) ( x( s) )ds  G2  Lg (s)( x( s) )ds] 0

0

x(t )  M0 [ (r )  Le (t )]  Le (t )( x )  G3 Lk (t )(r ) t ~   (t )M0 [G4 (r )  G3(r )  Lk ( s)ds] 0

t t ~  M0 [ (r )  p( s)ds  G2 (r )  Lg ( s)ds] 0 0 < r, for t  [0,b]. It means that W1  W0 . So we can complete the proof similarly to Theorem 3.1.

IV. Application The notion of controllability is of great importance in mathematical control theory. Many fundamental problems of control theory such as pole-assignment, stabilizability and optimal control may be solved under the assumption that the system is controllable. It means that it is possible to steer any initial state of the system to any final state in some finite time using an admissible control. During the last few decades, several authors [21, 23] have discussed the existence, uniqueness, and asymptotic behavior of the solution of these systems. Apart www.iosrjournals.org

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Existence of Solutions for Quasilinear Neutral Impulsive Integrodifferential Equations in Banach Spaces from these, the study of controllability and observability properties of a system in control theory is certainly, at present, one of the most active interdisciplinary areas of research. Control theory arises in most modern applications. On the other hand, control theory has remained a discipline where many mathematical ideas and methods have fused to produce a new body of important mathematics. As an application of Theorem 3.1 we shall consider the system (1)  (2) with a control parameter such as t d [ x(t )  e(t , x(t ),  k (t , s, x( s))ds)] A(t , x(t )) x(t ) 0 dt

(10)

t

= f (t , x(t ))Cv(t ) g (t , s, x( s))ds, t [0, b], t  ti , 0

x(0)  h( x) = x0 , where A, f , g , Q, h and I k are as before and C is a bounded linear operator from a Banach space V into

X

and the control function

v  L2 ( J ,V ). The mild solution of (16)  (18) is given by

t x(t ) = U x (t ,0) x0  U x (t ,0)h( x)  U x (t ,0)e(0, x(0),0)  e t , x(t ),  k (t , s, x( s))ds  0   t s   A(t , x(t ))U x (t , s)e t , x(t ),  k ( s, , x( ))d ds 0 0  

t

s

0

0

  U x (t , s)[ f (s, x(s))  Cv(s)   g ( s, , x( ))d ]ds, 0  t  b.

(10)  (11) is said to be controllable on the interval J , if for every x0 , x1  X , there exists a control v  L2 ( J ,V ) such that the mild solution u () of (10)  (11) satisfies x(0) = x0 and x(b) = x1. Definition 5.1 ([22, 23]) System

To study the controllability result we need the following additional condition: 1. The linear operator

W : L2 ( J ,V )  X , defined by b

Wv =  U x (b, s)Cv( s)ds, 0

induces an inverse operator

W 1 defined an L2 ( J ,V )/kerW and there exists a positive constant

M1 > 0 such that CW 1  M1. Theorem: 5.1 If the assumptions controllable on J . Proof. Using the assumption

( H1 )  ( H11) are satisfied, then the system (10)  (11) is

( H13 ), for an arbitrary function u (), define the control

1

v(t ) = W [u1  U x (b,0) x0  U x (b,0)h( x)  U x (b,0)e(0, x(0),0) b  e b, x(b),  k (b, s, x( s))ds  0   b s   A(b, x(b))U x (b, s)e b, x(b),  k ( s, , x( ))d ds 0 0  

b

s

0

0

  U x (b, s)[ f ( s, x(s))   g (s, , x( ))d ]ds

We shall show that when using this control, the operator H : Z  Z defined by t (Hv)(t ) = U x (t ,0) x0  U x (t ,0)h( x)  U x (t ,0)e(0, x(0),0)  e t , x(t ),  k (t , s, x( s))ds  0   t s   A(t , x(t ))U x (t , s)e t , x(t ),  k ( s, , x( ))d ds 0 0  

t

  U x (t , s)[ f ( s, x(s))  CW 1[u1  U x (b,0) x0  U x (b,0)h( x) 0

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(11)


Existence of Solutions for Quasilinear Neutral Impulsive Integrodifferential Equations in Banach Spaces b  U x (b,0)e(0, x(0),0)  e b, x(b),  k (b, s, x( s))ds  0   b s   A(b, x(b))U x (b, s)e b, x(b),  k ( s, , x( ))d ds 0 0  

b

s

0

0

  U x (b, s)[ f ( s, x(s))   g ( s, , x( ))d ]ds 

U 0<tk <t

s

x

(t , tk ) I k ( x(tk ))](s)   g ( s, , x( ))d ]ds 0

(10)  (11) . Clearly, (Hv)(b) = x(b) = x1 , which means that the control v steers the system (10)  (11) from the initial state x0 to the final state x1 at time b, provided we can obtain a fixed point of the nonlinear operator H. The remaining part of the proof is similar has a fixed point. This fixed point is, then a solution of

to Theorem 3.1 and hence, it is omitted.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]

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