Abstract and Applied Analysis

Volume 2014 (2014), Article ID 371639, 11 pages

http://dx.doi.org/10.1155/2014/371639

Research Article

Stability Conditions of Second Order Integrodifferential Equations with Variable Delay

Dingheng Pi

School of Mathematical Sciences, Huaqiao University, Quanzhou, Fujian 362021, China

Received 7 March 2014; Accepted 23 April 2014; Published 7 May 2014

Academic Editor: Yonghui Xia

Copyright © 2014 Dingheng Pi. 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.

We investigate integrodifferential functional differential equations with variable delay. By using the fixed point theory, we obtain conditions which ensure that the zero solution of this equation is stable under an exponentially weighted metric. Then we establish necessary and sufficient conditions ensuring that the zero solution is asymptotically stable. We will give an example to apply our results.

1. Introduction

Functional differential equations have many applications in control theory, biology, and so on. The stability of the solution of functional differential equations has been a hot issue for researchers for many years. It is well known that Lyapunov’s direct method has been widely applied to study the stability problems for a long time; see, for example, [1, 2]. Recently, many authors have applied the fixed points theory to study the stability of solution of integral equations and several functional differential equations with variable delays; see, for example, [3–5] and the references therein.

In [6], Levin and Nohel investigated the behavior of solution of a nonlinear equation

where is a constant. This equation was equivalent to

Burton studied stability of a nonconvolution equation

where was a positive constant. He gave conditions on functions and to ensure that the zero solution was asymptotically stable by applying fixed point theorem; see [7].

Becker and Burton studied the following differential equation:

and equation

for , where , , , are continuous. In addition, they assumed that

is differentiable;the function is strictly increasing; as .

They obtained sufficient conditions ensuring that the zero solution was asymptotically stable by changing the supremum metric to an exponentially weighted metric. Moreover, they hoped to relax condition ; see, for example, [8].

Jin and Luo succeeded in eliminating condition in their work; they did not need the condition that was invertible. Moreover, they established necessary and sufficient conditions that could ensure that the zero solution of this equation was asymptotically stable; see, for example, [9]. Dung [10] studied linear case of this equation and gave new stability results by using a new expression of the solution. Other results on fixed points and stability properties in equations with variable delays can be found in [3, 11] and the references therein.

Levin and Nohel [12] studied the global asymptotic stability of a class of nonlinear systems

Burton [13] used the fixed points theory to study the stability problems of some second order functional differential equations. He considered the equation

where is a positive constant. He obtained sufficient conditions under which each solution satisfied via the fixed point theorem.

We generalized the above equation to an equation with a variable delay [11]

and obtained some results on asymptotic stability of the zero solution. Before we introduce our new results we recall the main results in [11]. There are basic assumptions on the delay function .

is strictly increasing and . The inverse of exists and denotes it by . Moreover, for some constant .

The main results in [11] can be stated as follows.

Theorem 1. Suppose and the following conditions.

(i)There exists a constant such that satisfies the Lipschitz condition on . The function is odd and is strictly increasing on , and is nondecreasing on .(ii)There exist an and a continuous function such that for , , , and

(iii)There exist constants and such that, for each , if , then

(iv)There exist continuous functions and such that , , , . The function is continuous on , .

The following statements hold.

(a)If, for each ,

then the zero solution of (8) is asymptotically stable.(b)If the zero solution of (8) is asymptotically stable, then

This theorem failed to offer a necessary and sufficient condition which ensures that the zero solution was asymptotically stable. In this paper, we will establish a necessary and sufficient condition which ensures that the zero solution of related equation is asymptotically stable.

In this paper we consider equation

for , where . , , , are all continuous, where . We assume that as .

For each , define . Set with the continuous function norm , where . It will cause no confusion even though we use to express the supremum on later. It is well known that in [2], for a given continuous function , there exists a solution of (13) on an interval ; if the solution remains bounded, then . We denote by the solution .

We will give a necessary and sufficient condition ensuring that the zero solution of this equation is asymptotically stable. To our knowledge, there are few results about its stability. From the solution , we denote . We can write (13) as

For each , let with the continuous function norm , where .

This paper is organized as follows. In the next section we will state our main results. Their proofs will be given in Sections 3 and 4. We will give an example to apply our results in Section 5.

2. Statement of Main Results

We make the following basic assumptions on the delay function of (13).

. is the inverse of . and . There exists a constant such that . The following are our main results.

Theorem 2. Assume that holds and the following conditions hold.

(i)There exists a constant such that satisfies Lipschitz condition on . is odd and it is strictly increasing on , and is nondecreasing on .(ii)There exist an and a continuous function such that for . For , is increasing with respect to , and is bounded and for and for , , ,

(iii)There exist constants and such that, for each , if , then

Then the zero solution of (13) is stable.

In addition, we have the following.

Theorem 3. Assume that and there exists a function such that for the following conditions hold.

(i)There exists constant such that satisfies Lipschitz condition on . is the Lipschitz constant. is odd and it is strictly increasing on , and is nondecreasing on .(ii)There exist a constant and a continuous function such that . For , is bounded and for , , ,

(iii)There exist constants and such that, for each , if , then

Then the zero solution of (13) is asymptotically stable if and only if(iv)

Remark 4. We give some new notations:

3. Proof of Theorem 2

In this section, we will prove Theorem 2 by applying the fixed point theory. We will give the expression of the solution of the related equation. The following result can be found in [8].

Lemma 5. Let the function denote the inverse of . Then

is equivalent to

Lemma 6. Let be a given continuous function; if is the solution of (13) on satisfying , , and , then is the solution of the following integral equation:

Proof. We apply the variation of parameters formula to the second equation of (14); then we obtain

Equation (24) can be written as

Therefore,

Since , we have

This implies that the integral is convergent. Hence, we have

Then we have

For , by the variation of parameters formula, we obtain that

If we integrate the last several terms by parts, we have (23). This ends the proof of this lemma.

Let be the Banach space of bounded continuous functions on with the supremum norm. For a given continuous initial function , define the set by

where is a given initial function and is a positive constant. We will also use to denote the supremum norm of an initial function. Let be a mapping defined on as follows: for , if , . If ,

Note that may not be a contraction mapping. We solve this problem in Lemma 7 by introducing an exponentially weighted metric.

Lemma 7. Suppose that there exists a constant such that satisfies Lipschitz condition on . Then there exists a metric on such that

(i)the metric space is complete;(ii) is a contraction mapping on if maps into itself.

Proof. (i) We change the supremum norm to an exponentially weighted norm , which is defined on . Let be the space of all continuous functions such that

where , is a constant and , and is the common Lipschitz constant for and . Then is a Banach space. Thus, is a complete metric space, where denotes the induced metric: , where . Under this metric, the space is a closed subset of . Therefore, the metric space is complete.

(ii) Suppose that . For , since and , then

For , since , we have

For , since , we have

For ,

Since , then

For ,

Easy calculation shows that

For , . Hence, . Note that ; thus is a contraction mapping on .

We continue to prove Theorem 2. Choose and satisfying such that

Since (i) implies that , thus . Since satisfies the Lipschitz condition on , thus is continuous on , so such a exists and .

By the expression of , and condition (ii), we have

By condition (iii), some easy computation shows that

Hence,

Observe that if , then . We obtain that , . Thus, . Since we have proved that is a contraction mapping, hence has a unique fixed point and .

Recall (24); we have

Since is bounded, a constant such that , then

It follows that

To show the stability of zero solution, let be given; we only need to replace by . This completes the proof of Theorem 2.

4. Proof of Theorem 3

In this section, we will prove Theorem 3. First of all, we will obtain a new expression of the solution of (13). We multiply by both sides of (30); then

We have

Performing an integration by parts, then we have

We define

Let be a mapping defined on as follows: for , if , . If , we define

If , since as , the first term and fourth term of , as .

Note that

Since , then

For a given , there exists such that . For and , we have

For , we have

Then this term of , as . Analogously, we can prove other terms of , as . Then we can easily check that is a contraction mapping on by using condition (ii). By the contraction mapping principle, has a unique fixed point in . Thus, as .

Remark  8. By using the new expression (53), we do not need to change the supremum metric to an exponentially weighted metric. We can easily check that is a contraction mapping.

In order to obtain the asymptotic stability, we now need to show that the zero solution is stable. Let be given; we choose and satisfying such that

By (51), we have

We have obtained that

Hence,

Recall (24); we have

It follows that

Therefore, the zero solution is stable; since we have obtained that as , it follows that the zero solution is asymptotically stable.

The necessary condition is as follows: for each , we denote

We will prove that

by way of contradiction. If

since , a sequence , as such that , for a certain finite number . Choose such that holds, for .

Denote

By conditions of Theorem 3, we have

Then

Thus, the sequence is bounded; there exists a convergent subsequence; we assume that , . We can choose a positive integer large enough such that

for , where satisfying

Now we consider the solution of (8) which satisfies

We can obtain that by a similar argument with (61) if we replace by 1. Then

By (23) and , if , we have

This implies that

Note that ; it follows that

If the zero solution is asymptotically stable, then as . By the mean value theorem, easy computation shows that

So this term tends to 0 as .

Easy computation shows that

tends to 0 as tends to . This is a contradiction to (76).

Hence,

This completes the proof of Theorem 3.

5. An Example

In this section, we will give an example to apply our results. Let

where is a very small positive constant. Consider

This implies that is increasing with respect to for . Consider

Set ; we have for , . Consider

Since , then

Thus, we have

Easy computation shows that ; thus, is bounded. Consider

Since , thus is bounded. Hence, conditions (ii) and (iii) in Theorem 2 hold. Choose . Let ; satisfies the Lipschitz condition on . , on . Thus, is nondecreasing on . Then by Theorem 2 the zero solution of (13) is stable.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author is grateful to the reviewer for his or her useful suggestions. This work is partially supported by NNSF of China Grant nos. 11226145 and 11271046 and a research foundation of Huaqiao University (12BS112).

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