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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 193602, 11 pages
http://dx.doi.org/10.1155/2013/193602
Research Article

Global Asymptotic Stability for Nonlinear Functional Integral Equation of Mixed Type

Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou, Zhejiang 310023, China

Received 29 May 2013; Revised 27 August 2013; Accepted 27 August 2013

Academic Editor: Naseer Shahzad

Copyright © 2013 Zhinan Xia. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The existence results of global asymptotic stability of the solution are proved for functional integral equation of mixed type. The measure of noncompactness and the fixed-point theorem of Darbo are the main tools in carrying out our proof. Furthermore, some examples are given to show the efficiency and usefulness of the main findings.

1. Introduction

It is well known that functional integral equation of various types creates an important subject of numerous mathematical investigations and constitutes a significant branch of nonlinear analysis. It has great applications in physics, engineering, economics, and biologyed modelling problems connect with real world. With the help of several tools of functional analysis, topology, and fixed-point theory, many authors have made important contributions to this theory [15].

Let denote the real line and . In this paper, we investigate the nonlinear functional integral equation of mixed type, namely, It is worthwhile mentioning that (1) contains numerous functions and integral and functional integral equations encountered in nonlinear analysis. For example, the classical Volterra integral equation, the famous Chandrasekhar integral equation, the Urysohn integral equation, and the Volterra-Stieltjes integral equation are the special cases of (1).

The goal of this paper is to investigate the existence and asymptotic behavior of solutions for (1). Using the technique associated with a suitable measure of noncompactness and the fixed-point theorem of Darbo, we show that (1) has at least one solution under rather general and convenient assumptions. We also obtain some asymptotic characterization of the solutions of (1). The results of the present paper generalize several ones obtained previously in the papers [1, 3, 68] and references therein.

2. Preliminaries

In this section, we introduce some notations, definitions, and preliminary facts which are used in this paper. Let be a real function defined on , and let denotes the variation of on ; if is finite, we say that is of bounded variation. Let be a function, and let indicate the variation of the function on , where is arbitrarily fixed in . Similarly, define . If and are two real functions defined on , define the Stieltjes integral and say that is Stieltjes integrable on with respect to .

Lemma 1 (see [9]). If is Stieltjes integrable on with respect to of bounded variation; then

Lemma 2 (see [9]). Let , be Stieltjes integrable functions on with respect to a nondecreasing function such that for , then

We also consider Stieltjes integrals , where and indicates the integration with respect to . The details concerning the integral of this type will be given later.

Let be a real Banach space with zero element . Denote by the closed ball centered at and with radius , stands for the ball . If is a nonempty subset of , we denote by and the closure and the closed convex hull of , respectively. The family of all nonempty and bounded subsets of is denoted by and its subfamily consisting of all relatively compact sets is denoted by .

Definition 3 (see [10]). A function is said to be a measure of noncompactness in if it satisfies the following conditions. (1)The family is nonempty and . (2). (3). (4) for . (5)If is a sequence of closed sets from such that and if , then the set is nonempty.
The family in (1) is called the kernel of . Note that from (5) is a member of .
Consider the Banach space of all real functions defined, bounded and continuous on , and equipped with the maximum norm . Fix a nonempty bounded subset of and , and for , , define Let Moreover, for a fixed , define Finally, define the function on by It can be shown that is a measure of noncompactness in [10].

Remark 4. contains nonempty and bounded sets such that functions from are locally equicontinuous on and tend to zero at infinity uniformly with respect to .

Assume is a nonempty subset of and is an operator defined on with values in . Consider the operator equation

Definition 5 (see [8]). The solution of (8) is said to be locally attractive if there exists a ball in such that for arbitrary solutions and of (8) belonging to such that

Definition 6 (see [3]). The solution of (8) is said to be globally attractive, if (9) holds for each solutions of (8). If (9) is satisfied uniformly with respect to , we say that solutions of (8) are globally asymptotically stable (or uniformly globally attractive).

Theorem 7 (Darbo fixed-point theorem [10]). Let be a nonempty bounded closed and convex subset of Banach space and let be a continuous mapping such that for any nonempty subset of of , where is a constant. Then has a fixed-point in .

From Theorem 7, it can be shown that the set fix of fixed-points of is a member of .

3. Main Results

In this section, we will investigate the existence and asymptotic behavior of solutions for (1) in . Assume that the following conditions are satisfied. are continuous functions. satisfies the following conditions. For all such that , the function is nondecreasing on . The function is nondecreasing on . The functions and are continuous on for each fixed or , respectively. is a continuous function and there exist a continuous function and a continuous, nondecreasing function such that is a continuous function, and there exist a continuous function and a continuous, nondecreasing functions such that and is a continuous function and there exist a constant and continuous functions such that for all , . Moreover, the function is bounded with . is a continuous function and there exists a constant such that Moreover, and ., . Consider There exists a positive solution of the inequality such that , where

Remark 8. There are many examples of function which verify , such as(i), (ii), (iii)(iv), where is a bounded and integrable function. One can see [2] for more details.

In what follows, we provide some properties of the function .

Lemma 9 (see [9]). Under assumptions and , the function is nondecreasing on for any fixed .

Lemma 10 (see [9]). Assume that the function satisfies the assumption ; then for arbitrarily fixed such that , the function is nondecreasing on .

Remark 11. Observe that if satisfies (15), then Thus, if one of the terms of , , , and dose not vanish, then is automatically satisfied.

For the sake of convenience and simplicity, we will use the following notations in the paper. For fixed , let

The main result of this section is the following theorem.

Theorem 12. Under assumptions , (1) has at least one solution . Moreover, solutions of (1) are globally asymptotically stable.

Proof. Consider the operator defined on by Taking into account , , and Lemma 9, we deduce that is well defined on .
First, we show that maps into itself. It is clear that is continuous on for each . By and Lemma 1, for , one has By , it is clear that .
Next, we show that is continuous on . Let be given, and since holds, there exists , such that and for . Consider the following two cases.
Case 1. For , with ,
Case 2. For , with , by Lemmas 9 and 10, one has where By (21) and (22), Note that , as , and is continuous on by (24).
Now, and let be a nonempty set of and , fix arbitrarily and , let such that and , where Moreover, where By Lemma 1, one has where By , one has where By (25), (27), (29), and (31), one has and then By the uniform continuity of the functions , , , , , and on compact set, one has so and then, by , On the other hand, take arbitrary and fix ; one has and hence By , one has By (37) and (40), one has where is the measure of noncompactness defined in (7). Hence, in view of Theorem 7, we conclude that has at least one fixed-point , which is a solution of (1). Moreover, taking into account the description of sets belonging to (see Remark 4), we deduce that the solutions of (1) are globally asymptotically stable. This completes the proof.

Remark 13. Equation (1) is quite general and includes some nonlinear integral equations studied earlier by various authors as special cases. For example, we have the following. (i) If , , , (1) reduces to the nonlinear functional integral equation which studied by [7], that is, (ii) If , , , (1) reduces to the nonlinear quadratic integral equation of Volterra-Stieltjes type which is studied by the author of [2], that is, (iii) If , , , , (1) reduces to the quadratic Urysohn integral equation which is studied by the author of [22], that is, For more detail, one can see Table 1.

tab1
Table 1: Some of papers studied earlier as special cases, where , , , , are suitable continuous functions.

4. Examples

In this section, we present some examples to illustrate our main result.

Example 14. Consider the nonlinear functional integral equation of Volterra-Stieltjes type
Equation (45) is a special case of (1), where , , . Since , is satisfied with , , and . By Theorem 12, one has the following.

Theorem 15. Assume that , , , and hold; then (45) has at least one solution in which is globally asymptotically stable.

Remark 16. When , (45) is investigated by [3], and if , , it is investigated by the author of [6], see Table 1 for more details. Theorem 15 improves, and generalizes the results of [3, 6].

Remark 17. If , the following quadratic nonlinear functional integral equation of Volterra-Stieltjes type: When in (46), it is investigated by the authors of [2]. For (46), by Theorem 12, one has a similar result. Here we omit it.

Example 18. Consider the quadratic Urysohn functional integral equation
Equation (47) is a special case of (1), where , , and . By Theorem 12, one has the following theorem.

Theorem 19. Assume that and hold, and (47) has at least one solution in which is globally asymptotically stable.

Finally, we provide an example of concrete nonlinear functional integral equation of (1) which satisfies all the assumptions of Theorem 12.

Example 20. Consider the functional integral equation
Observe that (48) is a special case of (1), where , , , , , , , and . Then so , , , and , and then hold.
By and , it is easy to see that and are satisfied with , , ,