EXISTENCE OF SOLUTIONS TO SOBOLEV-TYPE PARTIAL NEUTRAL DIFFERENTIAL EQUATIONS

where τ =max{τ1,τ2}, τi > 0, i = 1,2, T <∞, φ ∈ 0 := C([−τ,0],X), and B and A are linear operators with the domains contained in X and the ranges contained in Y . f : [0,T]×X → Y , g : [0,T]×X ×X → Y are two given functions, and the map h is defined from 0 into 0. Here t := C([−τ, t];X) for t ∈ [0,T] is a Banach space of all continuous functions from [−τ, t] into X endowed with the norm ‖ψ‖t := sup ∥ψ(θ) ∥ ∥ X : θ ∈ [−τ, t] } . (1.2)


Introduction
Let X and Y be two real Banach spaces.We consider the following neutral differential equation of Sobolev type with a nonlocal history condition: where τ = max{τ 1 ,τ 2 }, τ i > 0, i = 1,2, T < ∞, φ ∈ Ꮿ 0 := C([−τ,0],X), and B and A are linear operators with the domains contained in X and the ranges contained in Y .f : [0,T] × X → Y , g : [0,T] × X × X → Y are two given functions, and the map h is defined from Ꮿ 0 into Ꮿ 0 .Here Ꮿ t := C([−τ,t];X) for t ∈ [0,T] is a Banach space of all continuous functions from [−τ,t] into X endowed with the norm 2) The study of differential equations with nonlocal conditions is of significance due to its applications in problems in physics and other areas of applied mathematics.Byszewski [6] proved the existence of mild, strong, and classical solutions for the nonlocal Cauchy problem.Some more results on the existence, uniqueness, and stability of solutions are 2 Sobolev-type neutral partial differential equations given by Byszewski and Akca [7], Balachandran and Chandrasekaran [2], and Byszewski and Lakshmikantham [8].Recently, Bahuguna [1] obtained existence and regularity results for functional differential equation with nonlocal condition using semigroup theory.Brill [5] and Showalter [14,15] established the existence of solutions of semilinear evolution equations of Sobolev type in Banach spaces.On the other hand, Balachandran and Uchiyama [4] considered an integrodifferential equation of Sobolev type with a nonlocal condition and proved the existence of mild and strong solutions.
The theory of neutral differential equations has been extensively studied in the literature.Hernández [10] established the existence results for partial neutral functional differential equations with nonlocal conditions modeled as where A is the infinitesimal generator of an analytic semigroup T(t) on a Banach space.He made use of fixed point theorems and the results mentioned in Pazy's [13].For results on neutral partial differential equations with nonlocal and classical conditions, we refer to the papers of Hernández and Henríquez [11], Balachandran and Sakthivel [3], Fu and Ezzinbi [9], and references therein.Our aim in this paper is to study the existence of a solution of partial neutral differential equation of form (1.1) by using Schauder's fixed point theorem.For this purpose, we first prove the existence of a solution of (1.1) on [−τ, T] for some 0 < T ≤ T and then prove that this solution can be extended to a solution of (1.1) either on [−τ,T] or on the maximal interval of existence [−τ,t max ], 0 < t max ≤ T, and in the latter case we show that lim t→tmax− u(t) X = ∞.Finally, we present an example to show an application of the existence theorem.

Preliminaries
In this section, we introduce notations, definitions, and preliminary results which we require to establish the existence of a solution of (1.1).
Henceforth, (X, • X ) and (Y , • Y ) will denote two real Banach spaces.The space of continuous linear mapping from X into Y will be denoted by L(X,Y ).
To prove our main theorem, we consider the following assumptions on the operators From the above fact and the closed graph theorem, we get the boundedness of the linear operator AB −1 : Y → Y .Consequently, AB −1 generates a uniformly continuous semigroup e −tAB −1 , t ≥ 0.
Under the above assumptions, we prove the existence of a solution u of (1.1) in the sense that there exists a continuous function u Note that Bu(t) itself may not be differentiable on the interval of existence.Now we mention a few results needed to establish our main result. (2.3) respectively, where Then, the following hold for 0 ≤ s ≤ t ≤ T: (I) 4 Sobolev-type neutral partial differential equations (II) (III) (2.9) Proof.Let Y * be the dual space of Y and let the duality map : Y → 2 Y * be given by (2.12) It implies which gives then the estimate (II) is followed by (I).
(III) By using the fact Ax Y ≤ w Bx Y for every x ∈ D(B) in (II), we obtain (2.16) Corollary 2.2.Let F : [0,T] → Y be a continuous function and let u be a solution of (2.4).Also, suppose that the assumptions (i)-(v) are satisfied.Then, there exist constants α = α(T) and (2.17) Proof.Let us take b as the norm of B −1 ∈ L(Y ,X).Using Proposition 2.1(II) and (III), we obtain (2.18) Set α = be wT and β = ce wT .Thus, 6 Sobolev-type neutral partial differential equations

Existence of solutions
We start by establishing a result of existence of a solution to the following abstract Cauchy problem: Proof.Since AB −1 : Y → Y is a bonded linear operator, it follows that the problem where has a solution w : [−τ, T] → Y , given by Consequently, a function u(t) = B −1 (w(t)− f (t,χ(t − τ 1 ))) is a solution of (3.1).Uniqueness of a solution is obtained by using Proposition 2.1(I).
Our main existence theorem is the following.
Theorem 3.2.Let the assumptions (i)-(v) be satisfied and suppose that g is a continuous function from [0,T] × X × X into Y .Then, for any given χ(0) ∈ D(B), there exists a solution of the abstract Cauchy problem (2.2) on the subinterval Proof.Let 0 < T ≤ T and R > 0 be real numbers and define the set It is easy to see that S 0 is a bounded, closed, and convex subset of C([0, T];Y ).Since all the hypotheses of Proposition 3.1 are verified, there exists a solution u F , F ∈ S 0 , of the problem (3.1).
We define a mapping Z : S 0 → C([0, T];Y ) by S. Agarwal and D. Bahuguna 7 The proof will be given in three steps.
Suppose that B −1 Y = b, c > 0 is a constant, and 0 In view of these conditions and Proposition 2.1(II), we can easily show that Z maps S 0 into S 0 .Now, we will show that Z is continuous.To this end, we introduce and using Proposition 2.1(II), we observe that (3.9) The right-hand side tends to zero as t 2 − t 1 → 0, since from Corollary 2.2 we obtain and g is a uniformly continuous function.Thus, Z maps S 0 into an equicontinuous family of functions.
As a consequence of Step 2 and Step 3 together with the Ascoli-Arzela theorem, we infer that Z(S 0 ) is relatively compact in Y .Hence, by Schauder's fixed point theorem, we deduce that the operator Z has a fixed point.This means that the problem (2.2) has a solution.
Next we will prove the following global existence result.
Theorem 3.3.Suppose that all the hypotheses of Theorem 3.2 are satisfied.Then, (2.2) has a solution either on [−τ,T] or on the maximal interval of existence [−τ,t max ), 0 < t max ≤ T, and in the latter case, lim t→tmax− u(t) X = ∞.
Proof.Since all the assumptions of Theorem 3.2 are satisfied, there exists a solution u of (1.1) on [−τ, T].
8 Sobolev-type neutral partial differential equations Suppose T < T and consider the problem where Since χ(0) = u( T) ∈ D(B), f satisfies the assumption (v), and g is a continuous function, we may proceed as before and prove the existence of a solution w : [−τ,T 1 ] → X, 0 < T 1 ≤ T − T, of the considered problem (3.10).
Then, the function ū : [−τ, T + T 1 ] → X, given by is a solution of (2.2) on [−τ, T + T 1 ].Continuing this way, we may prove the existence either on the whole interval [−τ,T] or on the maximal interval of existence [−τ,t max ), 0 < t max ≤ T. In the latter case, if lim t→tmax− u(t) X < ∞, then as u(t) ∈ D(B) for t ∈ [0,t max ), we have that lim t→tmax− u(t) is in the closure of D(B) in X, and if it is in D(B), then proceeding as before, we may extend u(t) beyond t max , but this will contradict the definition of the maximal interval of existence.Therefore, lim t→tmax− u(t) X = ∞.

1 .
If assumptions (i)-(v) are satisfied and F : [0, T] → Y is a continuous function, then the Cauchy problem (3.1) has a unique solution.
.3) Then, A and B are closed, linear operators.Furthermore, A is an accerative operator and D(A) is compactly imbedded in Y .Therefore, A and B verify the assumptions (i)-(iii).We assume the following conditions.(i) k 1 (t,x, y) and k 2 (t,x, y) are real-valued Lipschitz continuous and continuous functions, respectively, on [0, T] × Ω × Ω satisfying = 1, i = 1,2.