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In this paper we shall study a neutral differential equation with deviated argument in an arbitrary Banach space X. With the help of the analytic semigroups theory and fixed point method we establish the existence and uniqueness of solutions of the given problem. Finally, we give examples to illustrate the applications of the abstract results.


Introduction
We consider the following neutral differential equation with deviated argument in a Banach space X: u(a(t)))] + A[u(t) + g(t, u(a(t)))] = f (t, u(t), u[h(u(t), t)]), 0 < t ≤ T < ∞, u(0) = u 0 , where −A is the infinitesimal generators of an analytic semigroup. f, g, h and a are suitably defined functions satisfying certain conditions to be specified later.
Initial results related to the differential equations with deviated arguments can be found in some research papers of the last decade but still a complete theory seems to be missing . For the initial works on existence, uniqueness and stability of various types of solutions of different kind of differential equations, we refer to [1]- [10] and the references cited in these papers.
Adimy et al [1] have studies the existence and stability of solutions of the following general class of nonlinear partial neutral functional differential equations: (u(t) − g(t, u t )) = A(u(t) − g(t, u t )) + f (t, u t ), t ≥ 0, where the operator A is the Hille-Yosida operator not necessarily densely defined on the Banach space B. The functions g and f are continuous from [0, ∞) × C 0 into B.
In this paper, we use the Banach fixed point theorem and analytic semigroup theory to prove existence and uniqueness of different kind of solutions to the given problem (1.1). The plan of the paper is as follows. In Section 3, we prove the existence and uniqueness of local solutions and in Section 4, the existence of global solution for the problem (1.1) is given. In the last section, we have given an example.
The results presented in this paper easily can be apply to the same problem (1.1) with nonlocal condition under some modified assumptions on the function f and operator A.

Preliminaries and Assumptions
We note that if −A is the infinitesimal generator of an analytic semigroup then for c > 0 large enough, −(A + cI) is invertible and generates a bounded analytic semigroup. This allows us to reduce the general case in which −A is the infinitesimal generator of an analytic semigroup to the case in which the semigroup is bounded and the generator is invertible. Hence without loss of generality we suppose that where ρ(−A) is the resolvent set of −A. It follows that for 0 ≤ α ≤ 1, A α can be defined as a closed linear invertible operator with domain D(A α ) being dense in X. We have X κ → X α for 0 < α < κ and the embedding is continuous. For more details on the fractional powers of closed linear operators we refer to Pazy [11]. It can be proved easily that X α := D(A α ) is a Banach space with norm x α = A α x and it is equivalent to the graph norm of A α . Also, for each α > 0, we define It can be seen easily that We set, where L is a suitable positive constant to be specified later and 0 ≤ α < 1. We assume the following conditions: (A1): 0 ∈ ρ(−A) and −A is the infinitesimal generator of an analytic semigroup {S(t) : t ≥ 0}.
The nonlinear map f : R + ×X α ×X α−1 → X satisfies the following condition, be an open subset of X α × R + and for each (x, t) ∈ U 2 there is a neighborhood V 2 ⊂ U 2 of (x, t). The map h : X α × R + → R + satisfies the following condition, there is a neighborhood V 3 ⊂ U 3 of (x, t). The function g : (ii) The function a is Lipschitz continuous; that is, there exist a positive constant L a such that and satisfies the initial condition u(0) = u 0 .
Definition 2.2 By a solution of the problem (1.1), we mean a function u : [0, T ] → X α satisfying the following four conditions:

Existence of Local Solutions
We can prove that assumptions Theorem 3.1 Let us assume that the assumptions (A1)-(A5) are hold and u 0 ∈ D(A α ) for 0 ≤ α < 1. Then, the differential equation (1.1) has a unique local mild solution if and We set We define a map F : W → W given by In order to proved this theorem first we need to show that Fu ∈ C α−1 We have, Also, we can see that We observe that, Now we use the inequality (3.8) to get the inequality given below, where We use the inequalities (3.6) (3.7) (3.9) and (3.10) in inequality (3.5) and get the following inequality, + A α g(s, u(a(s))) − A α g(0, u(a(0))) Hence, from inequalities (3.2) and (3.3), we get We have the following inequalities, (3.14) We use the inequalities (3.13) and (3.14) in the inequality (3.12) and get Hence from inequality (3.1), we get the following inequality given below Therefore, the map F has a unique fixed point u ∈ W which is given by, u(a(t))) Proof. In order to prove this theorem, we first need to prove that the mild solution u is Hölder continuous on (0, T 0 ]. From Theorem 2.6.13 in Pazy [11], it follows that for every 0 < β < 1 − α, t > s > 0 and every 0 < h < 1, we have where C = C β C α+β . For 0 < t < t + h ≤ T 0 , we have We calculate the first term of the above inequality (3.18) as follows; where M 1 = Ct −(α+β) { u 0 + g(0, u 0 ) } depends on t and blows up as t decreases to zero. Second term of the above inequality (3.18) we calculate as follows, Third and the fourth term of the inequality (3.18) can be calculated as follows: where M 3 and M 4 can be chosen to be independent of t. Therefore, where C is a positive constant. Thus, u is locally Hölder continuous on (0, T 0 ]. Hence, Hence, the map t → f (t, u(t), u[h(u(t), t)]) is locally Hölder continuous. Therefore, where 0 < β < min{θ 1 , β, θ 2 }. Similarly, we can prove that u(.) + g(., u(a(.))) is also Hölder continuous on (0, T 0 ]. Therefore from Theorem 3.1 pp. 110 and Corollary 3.3, pp. 113, Pazy [11], the function u(.) + g(., u(a(.))) ∈ C α−1

Existence of Global Solutions
Theorem 4.1 Suppose that 0 ∈ ρ(−A) and the operator −A generates the analytic semigroup S(t) with S(t) ≤ M , for t ≥ 0, the conditions (A1)-(A4) are satisfied and u 0 ∈ D(A α ). If there are continuous nondecreasing real valued function k 1 (t), k 2 (t) and k 3 (t) such that then the initial value problem (1.1) has a unique solution which exists for all t ∈ [0, T ].
Proof: By theorem (3.1) we can continue the solution of equation (1.1) as long as u(t) α stays bounded. It is therefore sufficient to show that if u exists on [0, T [ then u(t) α is bounded as t ↑ T.
We have the following inequality, Hence, where and . Hence by applying the Gronwall's lemma to the above inequality (4.5), we get the required results. This completes the proof of the theorem. Let X = L 2 (0, 1). We consider the following partial differential equations with deviated argument, The function f 3 : We define an operator A, as follows, Here clearly the operator A is self-adjoint with compact resolvent and is the infinitesimal generator of an analytic semigroup S(t). Now we take α = 1/2, D(A 1/2 ) = H 1 0 (0, 1) is the Banach space endowed with the norm, and we denote this space by X 1/2 . Also, for t ∈ [0, T ], we denote We observe some properties of the operators A and A 1/2 defined by (5.2). For u ∈ D(A) and λ ∈ R, with Au = −u = λu, we have Au, u = λu, u ; that is, and the conditions u(0) = u(1) = 0 imply that C = 0 and λ = λ n = n 2 π 2 , n ∈ N. Thus, for each n ∈ N, the corresponding solution is given by u n (x) = D sin( λ n x).
For the function a we can take (i) a(t) = kt, where t ∈ [0, T ] and 0 < k ≤ 1.