#### Abstract

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.

#### 1. 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 and the operator appearing in (1.1) are given while 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

Using the monotone iterative technique, Constantin [13] got a sufficient condition which ensures the existence of positive solutions of the nonlinear integral equation

Roberts [21] examined the below nonlinear Volterra integral equation which arose from certain models of a diffusive medium that can experience explosive behavior; utilizing the Darbo fixed point theorem and the measure of noncompactness in [7], Banaś and Dhage [6], Banaś et al. [8], Banaś and Rzepka [9, 10], Hu and Yan [16] and Liu and Kang [19] investigated the existence and/or asymptotic stability and/or global asymptotic stability of solutions for the below class of integral equations of Volterra type: respectively. By means of the Schauder fixed point theorem and the measure of noncompactness in [7], Banaś and Rzepka [11] studied the existence of solutions for the below nonlinear quadratic Volterra integral equation:

Banaś and Chlebowicz [5] got the solvability of the following functional integral equation in the space of Lebesgue integrable functions on . El-Sayed [15] studied a differential equation of neutral type with deviated argument, which is equivalent to the functional-integral equation by the technique linking measures of noncompactness with the classical Schauder fixed point principle. Using an improvement of the Krasnosel’skii type fixed point theorem, Taoudi [22] discussed the existence of integrable solutions of a generalized functional-integral equation

Dhage [14] used the classical hybrid fixed point theorem to establish the uniform local asymptotic stability of solutions for the nonlinear quadratic functional integral equation of mixed type

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.

#### 2. Preliminaries

In this section, we give a collection of auxiliary facts which will be needed further on. Let and . Assume that is an infinite dimensional Banach space with zero element and stands for the closed ball centered at and with radius . Let denote the family of all nonempty bounded subsets of .

*Definition 2.1. *Let be a nonempty bounded closed convex subset of the space . A operator is said to be a *Darbo operator* if it is continuous and satisfies that for each nonempty subset of , where is a constant and is a measure of noncompactness on .

The Darbo fixed point theorem is as follows.

Lemma 2.2 (see [23]). *Let be a nonempty bounded closed convex subset of the space and let be a Darbo operator. Then has at least one fixed point in . *

Let denote the Banach space of all bounded and continuous functions equipped with the standard norm

For any nonempty bounded subset of and , define

It can be shown that the mapping is a measure of noncompactness in the space [4].

*Definition 2.3. *Solutions of an integral equation are said to be *asymptotically stable* if there exists a ball in the space such that for any , there exists with
for all solutions of the integral equation and any .

It is clear that the concept of asymptotic stability of solutions is equivalent to the concept of uniform local attractivity [9].

Lemma 2.4. *Let and be functions with
**
Then
*

*Proof. *Let . It follows that for each , there exists such that
Equation (2.4) means that there exists satisfying

Using (2.6) and (2.7), we infer that
that is, (2.5) holds. This completes the proof.

Lemma 2.5. *Let be a differential function. If for each , there exists a positive number satisfying
**
then
*

*Proof. *Let . It is clear that (2.9) yields that the function is nondecreasing in and for any , there exists satisfying
by the mean value theorem. Notice that (2.9) means that for each , which together with (2.11) gives that
which yields that (2.10) holds. This completes the proof.

Lemma 2.6. *Let be a function with and be a nonempty bounded subset of . Then
*

*Proof. *Since is a nonempty bounded subset of , it follows that . That is, for given , there exists satisfying

It follows from that there exists satisfying

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

#### 3. Main Results

Now we formulate the assumptions under which (1.1) will be investigated. is continuous with and ; satisfy that and have nonnegative and bounded derivative in the interval for each and are continuous and is nondecreasing and is continuous; there exist five positive constants , and and four continuous functions , such that is nondecreasing and satisfies that is a Darbo operator with respect to the measure of noncompactness of with a constant and

Theorem 3.1. *Under Assumptions , (1.1) has at least one solution . *

* Proof. *Let and define

It follows from (3.9) and Assumptions that is continuous on and that
which means that is bounded on and .

We now prove that

Let be a nonempty subset of . Using (3.2), (3.6), and (3.9), we conclude that
which yields that
which together with (3.3), Assumption and Lemma 2.4 ensures that
that is,

For each and , put

Let and with . It follows from that there exist and satisfying

In light of (3.2), (3.6), (3.9), (3.16), (3.17), and Lemma 2.5, we get that
which implies that

Notice that Assumptions imply that the functions and are uniformly continuous on the sets , and , respectively. It follows that

In terms of (3.19) and (3.20), we have

letting in the above inequality, by Assumption and Lemma 2.6, we infer that

By means of (3.15), (3.22), and Assumption , we conclude immediately that
that is, (3.11) holds.

Next we prove that is continuous on the ball . Let and with . It follows from (3.3) that for given , there exists a positive constant such that

Since is uniformly continuous in , it follows from (3.16) that there exists satisfying

By Assumption and , we know that there exists a positive integer such that

In view of (3.2), (3.6), (3.9), (3.24)–(3.26), and Assumption , we gain that for any and
which yields that
that is, is continuous at each point .

Thus Lemma 2.2 ensures that has at least one fixed point . Hence (1.1) has at least one solution . This completes the proof.

Now we discuss (1.2) under below hypotheses: is continuous with and ; are continuous and is nondecreasing; there exist two continuous functions and four positive constants and satisfying (3.3)–(3.5),

Theorem 3.2. *Under Assumptions and , (1.2) has at least one solution . Moreover, solutions of (1.2) are asymptotically stable.*

*Proof. *As in the proof of Theorem 3.1, we conclude that (1.2) possesses at least one solution in .

Now we claim that solutions of (1.2) are asymptotically stable. Note that , and are positive numbers and , it follows from (3.31) that . In terms of (3.3), we infer that for given , there exists such that

Let be two arbitrarily solutions of (1.2) in . According to (3.29)–(3.32), we deduce that
which means that
whenever are solutions of (1.2) in and . Hence solutions of (1.2) are asymptotically stable. This completes the proof.

*Remark 3.3. *Theorems 3.1 and 3.2 generalize Theorem 3.1 in [6], Theorem 2 in [9], Theorem 3 in [10], Theorem 1 in [15], Theorem 2 in [16], and Theorem 3.1 in [19]. Examples 4.1 and 4.2 in the fourth section show that Theorems 3.1 and 3.2 substantially extend the corresponding results in [6, 9, 10, 15, 16, 19].

#### 4. Examples

In this section, we construct two nontrivial examples to support our results.

*Example 4.1. *Consider the following nonlinear integral equation of Volterra type:
Put

Let

It is easy to verify that (3.6) and Assumptions and are satisfied. Notice that
that is, (3.2)–(3.5) and (3.7) hold. Hence all Assumptions of Theorem 3.1 are fulfilled. Consequently, Theorem 3.1 ensures that (4.1) has at least one solution . However Theorem 3.1 in [6], Theorem 2 in [9],Theorem 3 in [10], Theorem 1 in [15], Theorem 2 in [16], and Theorem 3.1 in [19] are unapplicable for (4.1).

*Example 4.2. *Consider the following nonlinear integral equation of Volterra type:

Put

It is easy to verify that (3.30), (3.31) and Assumptions and are satisfied. Notice that
which yield (3.3)–(3.5). That is, all Assumptions of Theorem 3.2 are fulfilled. Therefore, Theorem 3.2 guarantees that (4.5) has at least one solution . Moreover, solutions of (4.5) are asymptotically stable. But Theorem 2 in [9], Theorem 3 in [10], Theorem 1 in [15], Theorem 2 in [16], and Theorem 3.1 in [19] are invalid for (4.5).

#### Acknowledgments

The authors would like to thank the referees for useful comments and suggestions. This research was supported by the Science Research Foundation of Educational Department of Liaoning Province (L2012380).