EXISTENCE OF SOLUTIONS FOR NEUTRAL FUNCTIONAL VOLTERRA-FREDHOLM INTEGRODIFFERENTIAL EQUATIONS

In this paper, we study the existence of mild solutions of nonlinear neutral functional VolterraFredholm integrodifferential equations with nonlocal conditions. The results are obtained by using fractional power of operators and Sadovskii’s fixed point theorem.


INTRODUCTION
Many phenomena in several branches have mathematical model in terms of differential equations. Differential equations are like a bridge which links mathematics and science with applications. It is a rightly considered as a language of sciences. Many branches of science have led to some kind of differential equations.
The study of integrodifferential equations has emerged in recent years as an independent branch of Balachandran and Chandrasekaran (1996), Lin and Liu (1996) and Murugesu and Suguna (2010).
In this paper, we extend this problem to neutral functional Volterra-Fredholm type integro differential equations with nonlocal conditions and discuss the existence of solutions for nonlinear neutral functional Volterra-Fredholm integro differential equations with nonlocal conditions of the form d x(t)  F (t, x(t), x(b (t)),, x(b (t))  Ax(t)  G(t, x(t), x(a (t)),, x(a (t))) dt 1 m 1 n  t a   modern research because of its connections to many applied fields such as elasticity, biology, epidemics and other branches of science and engineering. Neutral differential equations arises in many areas of applied mathematics and for this reasons this equations have received much attention in the last few decades.
The advantages of using nonlocal conditions is that measurable at more places can be incorporated to get better models. The nonlocal Cauchy problem for abstract evolution differential equation was first considered by Byszewski (Byszwski, 1991) Subsequently, several authors have investigated the problem for different types of nonlinear differential equations and integrodifferential equations including functional differential equations in Banach spaces (Balachandran, 1998;Byszwski and Acka, 1998;Balachandran and Park, 2001a, b;Fu and Ezzinbi, 2004).
In the past several years theorems about existence, uniqueness and stability of differential and functional differential abstract evolution Cauchy problem have been studied by Byszewski and Lakshmikantham (1990), Byszewski (1997Byszewski ( , 1998, where -A generates an analytic semigroup and F, G, K, k, h are given functions to specifed later. This paper has the following subsections. In section 2, we present some preliminary lemmas and definitions which will be used to prove our main results. In section 3, we present the existence of mild solution of the system (1) using Sadovskii's fixed point theorem (Sodovskii, 1967).

PRELIMINARIES
Throughout this work, let -A is the infinitesimal generator of a compact analytic semigroup of uniformly bounded linear operators T(t) defined in the Banach spaces X. Let 0 then define the fractional power A  , for 0    1, as a closed linear operator on its domain D(A  ) which is dense in X. Further, D(A) is a Banach space under the norm || x ||  = || A  x ||, xD(A  ) Which we denote by X . Then for each 0 <   1, X  X  for 0 <  <   1 and the imbedding is compact whenever the resolvent operator of A is For our convenience, let us take compact. We assume that Let M0 = || A - || and also assume the following b) For any a > 0, there exists a positive constant C  , (2) hypotheses: (H1) F : [0,a]  X m+1  X is a continuous functions and there exists a (0,1) and L, L1 >0 such that the function A  F satisfies the Lipschitz condition: (Pazy, 1983) Let X be a Banach space, a one parameter family T(t), 0 t<+, of bounded linear operators   (Sodovskii, 1967) sup ||x0 ||,,||xn ||n || G(t, x 0 , x 1 ,, x n ) || n (t ) 1 a Let  be a condensing operator on a Banach space X, that is  is continuous and takes bounded and lim  q n (s)ds   1  Next we will show that the operator Q has a fixed point on Br : Let us decompose Q as Q = Q1+Q2 where the operators Q1 and Q2 are defined on Br respectively by Where M0 Proof: For the sake of brevity, we write that x(a1(t)),…, x(am(t))) = (t, u(t)).
Define the operator Q on E by the formula for 0  t  a, and we will verify that Q1 is contraction and Q2 is a compact operator.

Claim : Q1 is a contraction
Let x , x  B . Then for each t  [0,a] and by  (5), we have then for each r, Br is clearly a bounded closed convex set in E. Since by (2) and (3) the following relation holds: (t  s) 1 1 Claim : Q2 is compact then from Bochners theorem (Marle, 1974) it follows that AT(t-s)F(s,v(s)) is integrable on [0,a], so Q is well defined on Br.

Claim : there exists a positive integer r such that QBr  Br :
If it is not true, then for each positive integer To prove this we have to prove that Q2 is continuous on Br.
Let {xn}Br with xnx in Br, then by (H2) (i), we have G(s, u n (s))  G(s, u(s)), n   r, there is a function x (.)B , but Qx (t)B , that is by the dominated convergence theorem, we have Dividing on bothsides by r and taking the limit as r This contradicts (6). Hence for some positive integer r, QBr  Br.
Next, we prove that {Q2x : xBr} is a family of equicontinuous functions. To see this we fix t1>0 and t2>t1 and >0 be enough small. Then Balachandran, K. and J.Y. Park, (2001b
We can prove that the function Q2x, xBr are equicontinuous at t=0. Hence Q2 maps Br into a family of equicontinuous function.