eu EXISTENCE AND OSGOOD TYPE UNIQUENESS OF MILD SOLUTIONS OF NONLINEAR INTEGRODIFFERENTIAL EQUATION WITH NONLOCAL CONDITION

The aim of the present paper is to study existence, Osgood type uniqueness and qualitative properties of mild solutions of nonlinear integrodifferential equation with nonlocal condition. The main tools employed in the analysis are based on the applications of the Leray-Schauder alternative, rely on a priori bounds of solutions and the well known Bihari’s integral inequality. AMS Subject Classification: 34A12, 37C25, 34B10


Introduction
Using Tychonov's fixed point theorem, the method of successive approximations, Received: August 1, 2015 c 2015 Academic Publications, Ltd. url: www.acadpubl.eu§ Correspondence author and the comparison method, S. Sugiyama [18] studied the existence and uniqueness of solutions of the following problem: for 0 ≤ t ≤ t 1 , with the conditions x(0) = x 0 , where x and f represent n-dimensional vectors (see [18] for details) and Stokes [17] has discussed the same problems as above for nonlinear differential equations.
Recently, in the interesting paper [16], B. G. Pachpatte has studied the existence and uniqueness of solutions (1.1)-(1.3)by the Leray-Schauder alternative and well known Bihari's integral inequality.
From the above works, we can see a fact, although the integrodifferential problems have been investigated by some authors.However, to our knowledge, the integrodifferential equation with nonlocal conditions and an infinitesimal generator of operators has not been discussed extensively.So motivated by all the works above, the aim of this paper is to prove the existence and uniqueness of solutions of the integrodifferential of the form: x ′ (t) + Ax(t) = f t, x(t), t 0 k(t, s, x(s))ds, x(t − 1) , (1.4) for t ∈ J = [0, b], (b > 0) under the conditions x(t − 1) = φ(t) (0 ≤ t < 1), (1.5) where −A is an infinitesimal generator of a strongly continuous semigroup of bounded linear operators T (t) in X, f ∈ C(J × X × X, X), g ∈ C(C(J, X), X) and φ(t) is a continuous function for 0 ≤ t < 1, lim t→1−0 φ(t) exists, for which we denote by φ(1 − 0) = c 0 .If we consider the solutions of (1.4) for t ∈ J, we obtain a function x(t − 1) which is unable to define as solution for 0 ≤ t < 1.Hence, we have to impose some condition, for example the condition (1.5).We note that, if 0 ≤ t < 1, the problem is reduced to integrodifferential equation with initial condition x(0) + g(x) = x 0 .Here, it is essential to obtain the solutions of (1.4)-(1.6)for 0 ≤ t ≤ b, so that, we suppose in the sequel b is not less than 1.
Our main objective here is to investigate the global existence of solution to (1.4)-(1.6)by using the topological transversality theorem of Granas [ [7], p. 61], also known as Leray-Schauder alternative.Osgood type uniqueness result for the solutions of (1.4)-(1.6) is established by using the well known Bihari's integral inequality.Our general formulation of (1.4)- (1.6) is an attempt to generalize the results of [5,16].
The paper is organized as follows.In Section 2, we present the preliminaries and hypotheses.Section 3 deals with existence and Osgood type uniqueness of the solutions.Section 4 discuss the boundedness of solutions.Finally, in Section 5 we discuss result on continuous dependence of solutions on initial data.

Preliminaries and Hypotheses
Before proceeding to the statement of our main results, we shall set forth some preliminaries and hypotheses that will be used in our subsequent discussion.
Let X be the Banach space with norm • .Let B = C(J, X) be the space of all continuous functions from J into X endowed with the supremum norm Definition 2.1.Let −A is the infinitesimal generator of a C 0 −semigroup T (t), t ≥ 0, on a Banach space X.The function x ∈ B given by for 0 ≤ t < 1, and for 1 ≤ t ≤ b, is called the mild solution of the problem (1.4)- (1.6).
For completeness, we state here the fixed point result by Granas in ( [7], p. 61).
Lemma 2.2.(Leray-Schauder Alternative).Let S be a convex subset of a normed linear space E and assume 0 ∈ S. Let F : S → S be a completely continuous operator and let U (F ) = {x ∈ S : x = λF x} for some 0 < λ < 1.Then either U (F ) is unbounded or F has a fixed point.
We also need the following integral inequality, often referred to as Bihari's inequality [ [13], p. 107].
W −1 is the inverse function of W and t 1 ∈ R + be chosen so that , We list the following hypotheses: (H 1 ) −A is the infinitesimal generator of a semigroup of bounded linear operators T (t) in X, which is compact for t > 0, and there exist constant M ≥ 1 such that T (t) ≤ M, t ≥ 0.
(H 3 ) There exists a continuous function q : J → R + such that t 0 k(t, s, x(s))ds ≤ q(t) x for every t ≥ s ≥ 0 and x ∈ X.
(H 4 ) There exist constant G > 0 such that g(x) ≤ G for every x ∈ B.
(H 6 ) For each t, s ∈ J, the function k(t, s, •) : J × J × X → X is continuous and for each x ∈ X, the function k(•, •, x) : J × J × X → X is strongly measurable.
(H 7 ) For every positive integer m there exists α m ∈ L 1 (J) such that sup x ≤m, y ≤m, z ≤m f (t, x, y, z) ≤ α m (t) for t ∈ J a.e.
(H 9 ) The function k in (1.4) satisfies the condition for every x, x ∈ X, where q ∈ C(R + , R + ).

Existence and Uniqueness Results
The following theorem deals with the Wintner type global existence result for the solution of (1.4)-(1.6). where Proof.We define an operator for 0 ≤ t < 1, and ) In order to use 2.2, we establish the priori bounds on the solutions of the problem under the initial conditions (1.5)-(1.6)forλ ∈ (0, 1).Let x(t) be a solution of (3.5) with (1.5)-(1.6),then we consider the following two cases.
Case I: 0 ≤ t < 1.From the hypotheses, we have Let u(t) be defined by the right hand side of (3.6), then u(0 Integration of (3.7) from 0 to t (0 ≤ t < 1), the change of variable t → s = u(t), and the condition (3.1) gives From this inequality and the mean value theorem, we observe that, there is a constant γ 1 independent of λ ∈ (0, 1) such that u(t) ≤ γ 1 for 0 ≤ t < 1 and hence x(t) ≤ γ 1 .
Case II: 1 ≤ t ≤ b.From the hypotheses, we have where By making the change of variable, from (3.10), we obtain Using this inequality in (3.9), we obtain Let v(t) be defined by the right hand side of (3.12), then v(0 Integration of (3.13) from 0 to t, 1 ≤ t ≤ b, the change of variable, and the condition (3.1) give From (3.14) we conclude that there is a constant γ 2 independent of λ ∈ (0, 1) such that v(t) ≤ γ 2 and hence Obviously, x(t) ≤ γ for t ∈ J and consequently, x = sup{ x(t) : t ∈ J} ≤ γ.
Next we prove that F is completely continuous.Let B m = {x ∈ B : x(t) ≤ m, t ∈ J}, for some m ≥ 1.Then for each m ≥ 1, the set B m is clearly closed, convex and bounded subset of B. First we show that F B m is uniformly bounded.We have to consider the two cases.
Case I: 0 ≤ t < 1.From the definition of the operator F as in (3.3) , hypotheses and the fact that x ∈ B m , we obtain (3.15) Case II: 1 ≤ t ≤ b.From (3.4) , hypotheses and the fact that x ∈ B m , then looking at Case I immediately we have (3.16) From (3.15) and (3.16), it follows that {F B m } is uniformly bounded.We now show that F maps B m into an equicontinuous family.Let x ∈ B m .We must consider three cases.
Case I: t 1 and t 2 are contained in 0 ≤ t < 1.From (3.3), it follows that From the above equality and hypotheses, we have k(s, σ, x(σ))dσ, φ(s) ds Case II: t 1 and t 2 are contained in 1 ≤ t ≤ b.From (3.4), it follows that ) From the above equality and hypotheses, we have

.20)
Case III: t 1 and t 2 are respectively contained in [0, 1) and [1, b].From (3.3) and (3.4), it follows that (3.21) From the above equality and hypotheses, we have )dσ, φ(s) ds From (3.18), (3.20), (3.22), the right side of each one is independent of x ∈ B m and tends to zero as t 1 − t 2 → 0; since the compactness of T (t) for t > 0 implies the continuity in the uniform operator topology.Thus F maps B m into an equicontinuous family of functions.It is easy to see that the family B m is uniformly bounded.Next, we show that F B m is compact.Since we have shown that F B m is an equicontinuous and uniformly bounded collection, it is sufficient by Arzela-Ascoli theorem (see [ [4,9]]) to show that F maps B m into a precompact set in X.We have to consider the following two cases.
Case I: Let 0 < t < 1 be fixed and ǫ a real number satisfying 0 < ǫ < t.For x ∈ B m we define Therefore there are precompact sets arbitrary close to the set Case II: Let 1 < t < b be fixed and ǫ a real number satisfying 1 < ǫ < t.For x ∈ B m we define Therefore there are precompact sets arbitrary close to the set On combining these two cases we conclude that the set It remains to show that F : B → B is continuous.Let {v n } be a sequence of elements of B converging to v in B. Then there exists an integer q such that v n (t) ≤ q for all n and t ∈ J, so v n ∈ B q and u ∈ B q .We consider the following two cases.
Case I:For each t ∈ [0, 1) and by (H 4 ) − (H 7 ), we have Thus, since we have by dominated convergence Case II:For each t ∈ [1, b] and by (H 4 ) − (H 7 ), we have Thus, since we have by dominated convergence From (3.25) and (3.26), we conclude that the operator F is continuous.This completes the proof that F is completely continuous.Finally, the set U (F ) = {x ∈ B : x = λF x, λ ∈ (0, 1)} is bounded which was proved in the first part.Consequently, by Lemma 2.2, the operator F has a fixed point in B. This means that the problem (1.4)-(1.6)has a solution.This completes the proof of theorem.Remark 3.2.We note that the advantage of our approach here is that, it yields simutaneously the existence of solution of (1.4)-(1.6)and maximal interval of existence.In the special case, if we take p(t) = 1 in (3.1) and the integral on the right hand side in (3.1) is assumed to diverge, then the solution of (1.4)-(1.6)exists for every b < ∞; that is, on the entire interval R + .Our result in Theorem 3.1 yields existence of solution of (1. given in [22] The next theorem deals with the Osgood type uniqueness result for the solutions of (1.4)-(1.6).Proof.Let x(t), y(t) be two solutions of equation (1.4), under the initial conditions and let u(t) = x(t) − y(t) , t ∈ R + .We consider the following two cases.
Remark 3.4.We note that the hypothesis (H 8 ) corresponds to the Osgood type condition concerning the uniqueness of solutions in the theory of differential equations (see [ [4], p. 35]).

Boundedness of Solutions
In this section, we obtain estimates on the solutions of equations (1.4)-(1.6)under some suitable assumptions on the functions involved therein.
The following theorem concerning the estimate on the solution of equation (1.4).Theorem 4.1.Suppose that (H 1 ), (H 3 ), (H 6 ) hold.Let for 0 ≤ t < 1 and for 1 ≤ t < ∞, where In particular, if Proof.Let x(t) be a solution of the problem (1.4)-(1.4).We consider the following two cases.
Case I: 0 ≤ t < 1.Using the fact that the solution x(t) of the problem (1.4)-(1.6)and the hypotheses, we have Now, an application of Lemma 2.3 to (4.3) yields (4.1).

Continuous Dependence
In this section, we shall deals with the continuous dependence of solutions of (1.4) on the given initial data.