Solvability of Nonlinear Integral Equations of Volterra Type

This paper deals with the existence of continuous bounded solutions for a rather general nonlinear integral equation of Volterra type and discusses also the existence and asymptotic stability of continuous bounded solutions for another nonlinear integral equation of Volterra type. The main tools used in the proofs are some techniques in analysis and the Darbo fixed point theorem via measures of noncompactness. The results obtained in this paper extend and improve essentially some known results in the recent literature. Two nontrivial examples that explain the generalizations and applications of our results are also included.


Introduction
It is well known that the theory of nonlinear integral equations and inclusions has become important in some mathematical models of real processes and phenomena studied in mathematical physics, elasticity, engineering, biology, queuing theory economics, and so on see, 1-3 . In the last decade, the existence, asymptotical stability, and global asymptotical stability of solutions for various Volterra integral equations have received much attention, see, for instance, 1, 4-22 and the references therein.
In this paper, we are interested in the following nonlinear integral equations of Volterra type: where the functions f, h, u, a, b, c, α and the operator H appearing in 1.1 are given while x x t is an unknown function.
To the best of our knowledge, the papers dealing with 1.1 and 1.2 are few. But some special cases of 1.1 and 1.2 have been investigated by a lot of authors. For example, Arias et al. 4 studied the existence, uniqueness, and attractive behaviour of solutions for the nonlinear Volterra integral equation with nonconvolution kernels x t t 0 k t, s g x s ds, ∀t ∈ R . 1.3 Using the monotone iterative technique, Constantin  The purpose of this paper is to prove the existence of continuous bounded solutions for 1.1 and to discuss the existence and asymptotic stability of continuous bounded solutions for 1.2 . The main tool used in our considerations is the technique of measures of noncompactness 7 and the famous fixed point theorem of Darbo 23 . The results presented in this paper extend proper the corresponding results in 6, 9, 10, 15, 16, 19 . Two nontrivial examples which show the importance and the applicability of our results are also included. This paper is organized as follows. In the second section, we recall some definitions and preliminary results and prove a few lemmas, which will be used in our investigations. In the third section, we state and prove our main results involving the existence and asymptotic stability of solutions for 1.1 and 1.2 . In the final section, we construct two nontrivial examples for explaining our results, from which one can see that the results obtained in this paper extend proper several ones obtained earlier in a lot of papers.

Preliminaries
In this section, we give a collection of auxiliary facts which will be needed further on. Let R −∞, ∞ and R 0, ∞ . Assume that E, · is an infinite dimensional Banach space with zero element θ and B r stands for the closed ball centered at θ and with radius r. Let B E denote the family of all nonempty bounded subsets of E. Let BC R denote the Banach space of all bounded and continuous functions x : For any nonempty bounded subset X of BC R , x ∈ X, t ∈ R , T > 0 and ε ≥ 0, define

2.2
It can be shown that the mapping μ is a measure of noncompactness in the space BC R 4 .

Definition 2.3.
Solutions of an integral equation are said to be asymptotically stable if there exists a ball B r in the space BC R such that for any ε > 0, there exists T > 0 with for all solutions x t , y t ∈ B r of the integral equation and any t ≥ T . It is clear that the concept of asymptotic stability of solutions is equivalent to the concept of uniform local attractivity 9 .

2.10
Proof. Let T > 0. It is clear that 2.9 yields that the function a is nondecreasing in 0, T and for any t, s ∈ 0, T , there exists ξ ∈ 0, T satisfying |a t − a s | a ξ |t − s| ≤ a T |t − s| 2.11 by the mean value theorem. Notice that 2.9 means that a t ∈ a 0 , a T ⊆ 0, a T for each t ∈ 0, T , which together with 2.11 gives that which yields that 2.10 holds. This completes the proof.
Lemma 2.6. Let ϕ : R → R be a function with lim t → ∞ ϕ t ∞ and X be a nonempty bounded subset of BC R . Then Proof. Since X is a nonempty bounded subset of BC R , it follows that ω 0 X lim T → ∞ ω T 0 X . That is, for given ε > 0, there exists M > 0 satisfying It follows from lim t → ∞ ϕ t ∞ that there exists L > 0 satisfying ϕ T > M, ∀T > L.

2.15
By means of 2.14 and 2.15 , we get that which yields 2.13 . This completes the proof.

Main Results
Now we formulate the assumptions under which 1.1 will be investigated.
H5 H : BC R → BC R satisfies that H : B r → BC R is a Darbo operator with respect to the measure of noncompactness of μ with a constant Q and It follows from 3.9 and Assumptions H1 -H5 that Fx is continuous on R and that which means that Fx is bounded on R and F B r ⊆ B r . We now prove that Let X be a nonempty subset of B r . Using 3.2 , 3.6 , and 3.9 , we conclude that 3.14 that is, For each T > 0 and ε > 0, put v ∈ −r, r , w ∈ −g r , g r , z ∈ −M, M .

3.16
Abstract and Applied Analysis 9 Let T > 0, ε > 0, x ∈ X and t, s ∈ 0, T with |t − s| ≤ ε. It follows from H2 that there exist a T and b T satisfying In light of 3.2 , 3.6 , 3.9 , 3.16 , 3.17 , and Lemma 2.5, we get that which implies that

Abstract and Applied Analysis
Notice that Assumptions H1 -H3 imply that the functions α α t , f f t, p, q, v and u u t, y, z are uniformly continuous on the sets 0, T , 0, T × −r, r × −g r , g r × −M, M and 0, T × 0, α T × −r, r , respectively. It follows that 3.20 In terms of 3.19 and 3.20 , we have letting T → ∞ in the above inequality, by Assumption H2 and Lemma 2.6, we infer that By means of 3.15 , 3.22 , and Assumption H5 , we conclude immediately that that is, 3.11 holds. Next we prove that F is continuous on the ball B r . Let x ∈ B r and {x n } n≥1 ⊂ B r with lim n → ∞ x n x. It follows from 3.3 that for given ε > 0, there exists a positive constant T such that sup m 3 t α t 0 |u t, s, v c s − u t, s, w c s |ds : v, w ∈ B r < ε 3 , ∀t > T.

3.24
Since u u t, s, v is uniformly continuous in 0, T × 0, α T × −r, r , it follows from 3.16 that there exists δ 0 > 0 satisfying By Assumption H5 and lim n → ∞ x n x, we know that there exists a positive integer N such that 1 M 1 x n − x M 2 Hx n − Hx < 1 2 min{ε, δ 0 }, ∀n > N.

3.26
Abstract and Applied Analysis

11
In view of 3.2 , 3.6 , 3.9 , 3.24 -3.26 , and Assumption H2 , we gain that for any n > N and t ∈ R

3.32
Let z z t , y y t be two arbitrarily solutions of 1.2 in B r . According to 3.29 -3.32 , we deduce that