Existence of Solutions for Neutral Functional Integrodifferential Evolution Equations with Non Local Conditions

We study the existence of mild and strong solutions for nonlinear neutral functional integrodifferential evolution equations with nonlocal conditions in Banach spaces. The results are obtained by using the fractional powers of operators and Sadovskii’s fixed point theorem.


Introduction
Neutral differential equations arise in many areas of applied mathematics and for this reason these equations have received much attention in the last few decades. The literature related to ordinary neutral functional differential equations is very extensive. The work in partial neutral functional differential equations with infinite delay was initiated by Hernandez and Henriquez. Firstorder partial neutral functional differential equations have been studied by different authors. The reader can consult Adimy 1 , Hale 13,14 and Wu 25 for systems with finite delay and Hernandez Henriquez 17,18 and Hernandez 16 for the unbounded delay case. Hernandez 15 established the existence results for partial neutral functional differential equations with nonlocal conditions modelled as in a separable Hilbert space, where t = max{t 1 , t 2 }, t 1 , t 2 > 0. An extensive theory for ordinary neutral functional differential equations which includes qualitative behavior of classes of such equations and applications to biological and engineering processes. Several authors have studied the existence of solutions of neutral functional differential equations in Banach space 2,3,4,6,11,12,13,15,17,18,23 .
The nonlocal Cauchy problem for semilinear evolution equations in Banach space was studied first by Byszekswi 7,8,9 where he established the existence and uniqueness of mild and classical solutions. The nonlocal conditions were motivated by physical problems and their importance is discussed in [7][8][9] . Balachandran et al [3][4][5]21 studied the nonlocal Cauchy problem for various type of nonlinear integrodifferntial equations. In addition, our result can also be regarded as an extension of the corresponding results on classical problem in 10,22 .
x a t x a t s ds

Preliminaries
Let −A be the infinitesimal generator of a compact analytic semigroup of uniformly bounded linear operators U(t, s) defined in the Banach space X. Let 0 ∈ ρ(A), then define the fractionl power A α , for 0 ≤ α ≤ 1, as a closed linear operator on its domain D(A α (t)) which is dense in X. Further D(A(t)) is a Banach space under the norm which we denote by X α . Then for each 0 ≤ α ≤ 1, X α → X β for 0 < β < α ≤ 1 and the imbedding is compact whenever the resolvent operator of A is compact. We assume that with n a positive integer, and there exists consants L, L 1 >0 such that the function A(t)F satisfies the Lipschitz condition: (H 2 ) The function G : [0, T] × X m+1 → X satisfies the following condition; (i) for each t ∈ [0, T], the function G(t, .) : X m+1 → X is continuous, and for each ( , and there exists a constant L 2 > 0 such that ||g(x)|| ≤ L 2 ||x|| for each x ∈ H.

Theorem 2.1 (Sadovskiiʹs fixed point theorem, cf. 24 ).
Let P be a condensing operator on a Banach space X, i.e.,

Existence of Mild Solutions
[0, t) and the following integral equation is verified:
We will show that the operator P has a fixed point on B k , which implies that equation (1.1) has a mild solution. To this end, we decompose P into P = P 1 + P 2 , where the operator P 1 , P 2 are defined on B k respectively by ∫ ∫ 0 ≤ t ≤ T, and will verify that P 1 is contraction which P 2 is compact operator.
To prove P 1 is a contraction, we take x 1 , x 2 ∈ B k , then for each t ∈ [0, T] and by condition (H 1 ) and (6), we have Noting that ||G(s, u(s))|| ≤ g k (s) and g k (s) ∈ Lʹ, we see that ||(P 2 x)(t 2 )−(P 2 x)(t 1 )|| tends to zero independently of x ∈ B k as t 2 − t 1 → 0 since the compactness of {U(t, s), t > s} implies the continuity of {U(t, s), t > s} in t in the uniform operator topology uniformly for s. Hence P 2 maps B k into a family of equicontinuous functions.
It remains to prove that Let 0 < t ≤ T be fixed, 0 < ∈ < t, for x ∈ B k , we define

U t x U t s G s u s ds
Then from the compactness of U(t, s)(t − s > 0), we obtain that is relatively compact in X for every, 0 < ∈ < t. Mortover, x ∈ B y , we have  Therefore, there are relatively compact sets arbitrarily close to the set V(t). Hence the set V(t) is also relatively compact in X.
Thus by Arzela-Ascoli theorem P 2 is compact operator. These arguments above enable us to conclude that P = P 1 + P 2 is condense mapping on B k , and by Theorem 2.1 there exists a fixed point z(.) for P on B k , therefore the which shows that P 1 is contraction.
To prove that P 2 is compact, firstly we prove that P 2 is continuous on B k , Let {x n } ⊆ B k with x n → x is B k , then by (H 2 )(i), we have

Existence of Strong Solutions
In this section, we provide conditions which allow the differentiation of the mild solutions obtained in section 3, i.e., these derivatives are shown to satisfy the differential equations of the form(1.1).

Definition 4.2
A