On an abstract nonlinear second order integrodifferential equation

The aim of the present paper is to study the global existence of solutions of nonlinear second order integrodifferential equation in Banach space. Our analysis is based on an application of the Leray-Schauder alternative and rely on a priori bounds of solutions..


Introduction
Let X be a Banach space with norm .Let B = C([0, T ], X) be the Banach space of all continuous functions from [0, T ] into X endowed with supremum norm x B = sup { x(t) : t ∈ [0, T ]} .The purpose of this paper is to prove the global existence of solutions of the following initial value problem for second order integrodifferentioal equations of the form (r(t)x (t)) = f t, x(t), t 0 k(t, s)g(s, x(s))ds , t ∈ [0, T ] (1. 1) x(0) = x 0 , x (0) = 0; (1.2) where r(t) is real-valued positive and sufficiently smooth function defined on [0, T ], f : [0, T ] × X × X → X, g : [0, T ] × X → X, k : [0, T ] × [0, T ] → R are continuous functions and x 0 is a given element of X.
Let x ∈ L 1 (0, T ; X).By a solution of the initial value problem (1.1) − (1.2), we mean the function x ∈ C([0, T ]; X) given by The main purpose of this paper is to study the global existence of solutions of initial value problem (1.1)-(1.2) by using an application of the topological transversality theorem known as Leray-Schauder alternative.The interesting and useful aspect of the method employed here is that it yields simultaneously the existence of a solution and the maximal interval of existence.Our work is motivated by interesting results obtained by W.F. Trench [4,5] and influenced by the work of the authors [1] and D.B. Pachpatte [3].

Statement of result
Our Theorem is based on the following theorem, which is a version of the topological transversality theorem given in J. Dugundji and A. Granas [2, p. 61] Theorem G. Let Y be a convex subset of a normed linear space E and assume 0 ∈ Y. Let F : Y → Y be a completely continuous operator and let U (F ) = {x ∈ Y : x = λF x f or some 0 < λ < 1} .Then either U (F ) is unbounded or F has a fixed point.
For convenience, we list the following hypotheses used in our subsequent discussion.
(H 2 ) There exists a continuous function p : for every t ∈ [0, T ], and x, y ∈ X.

Proof of result
In order to prove the Theorem we apply Theorem G. First we establish the priori bounds for a solution to the initial value problem (1.1) λ − (1.2), λ ∈ (0, 1) where Let x(t) be a solution of (1.1) λ − (1.2).Then it satisfies the equivalent integral equation Using (3.1),(3.2),hypotheses (H 1 ),(H 2 ) and the fact that λ ∈ (0, 1), we obtain Define a function u(t) by the right hand side of (3.3).By using the fact that H is continuous non-decreasing function, we get x(t) ≤ u(t), u(0) = x 0 = c and It is clear that v(t) is also an increasing function.Differentiating, we have Integrating (3.6) from 0 to t and by making use of the change of variables t → s = v(t) and the condition (2.1), we get (3.7) From (3.7) we conclude by the mean value theorem that there is a constant Now, we rewrite the problem (1.1)-(1.2) as follows: If y ∈ B and x(t) defined by x(t) = x 0 + y(t), t ∈ [0, T ], then it is easy to observe that y(t) satisfies an integral equation 2) or their equivalent integral equation Now, we prove that F : B 0 → B 0 is continuous.Let {y n } be a sequence of elements of B 0 converging to y in B 0 .Then by using (H 3 ) and (H 4 ), we get . By using hypotheses (H 3 ) and (H 4 ) and the dominated convergence theorem, we have Now, we prove that F maps a bounded set of B 0 into a precompact set in B 0 .Let B m = {y ∈ B 0 : y B ≤ m} for m ≥ 1.We first show that F maps B m into an equicontinuous family of functions with values in X.Let y ∈ B m , t 1 , t 2 ∈ [0, T ] and 0 ≤ t 1 ≤ t 2 ≤ T. Using (3.2),hypotheses (H 1 ), (H 2 ) and condition (2.1), we have The right hand side of (3.10) tends to zero as t 2 − t 1 → 0. Thus F Bm is an equicontinuous family of functions with values in X.
We next show that F Bm is uniformly bounded.From the equation (3.8) and using (3.2), hypotheses (H 1 ), (H 2 ), the condition (2.1) and the fact that y B ≤ m, we have This yields that the set {(F y)(t): y B ≤ m, 0 ≤ t ≤ T } is bounded in X and hence, {F Bm } is uniformly bounded.
We have already shown that F Bm is an equicontinuous and uniformly bounded collection.To prove the set F Bm is precompact in B, it is sufficient, by Arzela-Ascoli's argument, to show that the set {(F y)(t): y ∈ B m } is precompact in X for each t ∈ [0, T ].Since (F y)(0) = 0 for y ∈ B m , it suffices to show this for 0 < t ≤ T. Let 0 < t ≤ T be fixed and a real number satisfying 0 < < t.For y ∈ B m , we define By making use of (3.2), hypotheses (H 1 ), (H 2 ), the condition (2.1) and the fact that y(s) ≤ m, we have

5 )
For every positive integer m there exists α m ∈ L 1 (0, T ) such that f (t, x, y) ≤ α m (t) for x,y satisfying x ≤ m, y ≤ m and for a.e.t ∈ [0, T ].We next establish the following Theorem which deals with the global existence of solution of initial value problem (1.1)-(1.2).

s 0 p 0 L 1 L 1
(τ )[ y(τ ) + x 0 + τ LH( y(η) + x 0 )dη]dτ ds ≤ LH(m + c)dη]dτ ds ≤ [m + c + M * H(m + c)T ] t t− N (s)dsThis implies that there exist precompact sets arbitrarily close to the set{(F y)(t): y ∈ B m }.Hence the set {(F y)(t): y ∈ B m } is precompact in X.Thus we have shown that F is completely continuous operator.Moreover, the setU (F ) = {y ∈ B 0 : y = λF y, λ ∈ (0,1)}, is bounded in B, since for every y in U (F ), the function x(t) = y(t) + x 0 is a solution of (1.1) λ − (1.2) for which we have proved x B ≤ Q and hence y B ≤ Q + c.Now, by virtue of Theorem G, the operator F has a fixed point in B 0 .Therefore, the initial value problem (1.1) − (1.2) has a solution.Thus the proof of the Theorem is complete.