A COUPLED SYSTEM OF FUNCTIONAL DIFFERENTIAL EQUATIONS IN REFLEXIVE BANACH SPACES

We present an existence theorem for at least one weak solution for the coupled system of functional differential equations x ′(t) = f1(t, y ′(t)), t ∈ (0, T ], y ′(t) = f2(t, x ′(t)), t ∈ (0, T ] in reflexive Banach spaces. AMS Subject Classification: 35D30, 34Gxx


Introduction and Preliminaries
Let E be a reflexive Banach space with norm .and dual E * , and L 1 (I) be the space of Lebesgue integrable functions on the interval I = [0, T ].Denote by C[I, E] the Banach space of strongly continuous functions x : I → E with sup-norm .0 .Such systems appear in many problems of applied nature (see [1], [3], [4], [9]- [11] and [13]).Su [18] discussed a two-point boundary value problem for a coupled system of fractional differential equations.Gafiychuk et al. [11] analyzed the solutions of coupled nonlinear fractional reaction-diffusion equations.The solvability of the coupled systems of integral equations in reflexive Banach space was proved in (see [5]- [7]).
In [15], O'Regan studied, in the reflexive Banach space, the existence of weak solutions of the initial value problem Recently, the authors studied the existence of weak solutions of the initial value problem in reflexive and nonreflexive Banach spaces (see [8]).
In this paper we study the existence of weak solutions for the coupled system of the functional differential equations in the reflexive Banach space E. For this aim we study, firstly, the existence of weak solutions for the coupled system of the functional equations in the reflexive Banach space E. Now, we shall present some auxiliary results that will be needed in this work.
(3) A function h : E → E is said to be weakly sequentially continuous if h maps weakly convergent sequences in E to weakly convergent sequences in E.
It is clear that (1) implies ( 2) and ( 2) implies (3).If h linear, then (2) and ( 3) are equivalent.The relation between weak and weak sequentially continuous of mapping is studied in details in [2].Now, we have the following theorem due to Rubin (see [16]) and some propositions which will be used in the sequel (see [7] and [17]).
Theorem 1.If E is metrizable (i.e., the topology is induced by a metric).Then the weakly sequentially continuous functions are weakly continuous.Proposition 1.A subset of a reflexive Banach space is weakly compact if and only if it is closed in the weak topology and bounded in the norm topology.
The following result follows directly from the Hahn-Banach theorem.
Also, we have the following fixed point theorem, due to O'Regan, in reflexive Banach space (see [14]).
Theorem 2. (O'Regan fixed point theorem) Let E be a Banach space and let Q be a nonempty, bounded, closed and convex subset of the space E and let F : Q → Q be a weakly sequentially continuous and assume that F Q(t) is relatively weakly compact in E for each t ∈ I .Then, F has a fixed point in the set Q.

Coupled System of Functional Equations
Here, we discuss the existence of weak solutions for the coupled system (3)-( 4) in the reflexive Banach space E. The coupled system (3)-(4) will be investigated under the following assumptions : (ii) For each t ∈ I, f i (t, .)are weakly Lipschitz with Lipschitz constants K i , where a i = sup{φ(f i (t, 0)) : t ∈ I}.Now, let X be the class of all ordered pairs (u, v), u, v ∈ C[I, E] with the norm Definition 1.By a solution to the coupled system (3)-( 4), we mean the pair of functions (u, v) ∈ X, u, v ∈ C[I, E] which satisfies (3)-( 4) weakly.This is equivalent to finding Now, we can prove the following existence theorem.Theorem 3.Under the assumptions (i)-(ii), the coupled system (3)-( 4) has at least one weak solution

Proof. Define the operator A by
where The reminder of the proof will be given in four steps.
Step 1: The operator A maps X into itself.For this, let by proposition 2 we have Similarity, we can show that Therefore, Which prove that A : X → X.
Step 3: AΩ(t) is relatively weakly compact in E. Note that Ω is nonempty, closed, convex and bounded subset of X.According to proposition 1, AΩ is bounded in X and closed in weak topology, hence AΩ is relatively weakly compact in X implies AΩ(t) is relatively weakly compact in E for each t ∈ I.