Mathematical Problems in Engineering
Volume 2018 (2018), Article ID 9519020, 6 pages
https://doi.org/10.1155/2018/9519020

Research Article

Stability Analysis of Solutions for a Kind of Integro-Differential Equations with a Delay

Jing Zhao1,2 and Fanwei Meng¹

¹School of Mathematical Science, Qufu Normal University, Qufu, 273165, China
²Science and Information College, Qingdao Agricultural University, Qingdao 266109, China

Correspondence should be addressed to Fanwei Meng; fwmeng@qfnu.edu.cn

Received 8 May 2018; Accepted 24 June 2018; Published 10 July 2018

Academic Editor: Weihai Zhang

Copyright © 2018 Jing Zhao and Fanwei Meng. 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.

Stability of zero solution for second-order integro-differential equations with a delay is analyzed and some new results are presented. Through constructing Lyapunov functional, we give the corresponding sufficient conditions on stability of zero solution for two integro-differential equations. Moreover, an illustrative example is considered to support our new results.

1. Introduction

Stability and the existence of solutions for nonlinear differential equations have been studied by many scholars [1–20] due to their many applications to problems in information theory, control theory, mechanics, chemistry, physics, and so on. In [5], the authors considered a second-order functional integro-differential equation with multiple delays

and they gave some new conditions on the continuability and boundedness of solutions. Li [6] studied -stability of the trivial solutions of the following three nonlinear systems with time delay,

and two Volterra integro-differential equations

In [1], Afuwape studied the asymptotic stability and the uniformly ultimate boundedness of the solutions for a kind of third-order delay differential equations and gave some sufficient conditions.

This paper investigates second-order integro-differential equations with a delay and obtains some new results on the stability of zero solution. By constructing Lyapunov functional, the corresponding sufficient conditions are present on stability of zero solution for two integro-differential equations. Moreover, an illustrative example is considered to show that our results are effective.

The rest of this paper is organized as follows. Section 2 presents the main results of this paper. In Section 3, an illustrative example is given to support our new results.

2. Main Results

Consider the following integro-differential equation with a delay:

where , is a constant, , and are continuous with , and with and .

We can rewrite (5) as the system

Theorem 1. Consider system (6). There exist nonnegative constants , , , , and , a positive number , and functions , such that the following conditions hold:

() when , .() , when .() , , .() .

Then the zero solution of system (6) is stable.

Proof. Define a Lyapunov functional as

where is a positive constant to be determined. Using condition (), we have

and .
Suppose that is a solution of (6). Then

By conditions () and () and , we have

Then,

Choosing , we have

Therefore, the zero solution of system (6) is stable and the proof is completed.

Theorem 1 can be generalized to the form with a variable delay . It only needs a new condition. Then, the following result is obtained.

Corollary 2. Consider system (6) with a variable delay . Conditions (1)-(3) of Theorem 1 are satisfied. Moreover,

() ,() there are and , such that and .

Then the zero solution of system (6) with a variable delay is stable.

Proof. The proof is similar to that of Theorem 1. But

Thus, choosing , we have

Therefore, the zero solution of system (6) with a variable delay is stable and the proof is completed.

Next, we consider another second-order integro-differential system with time delay

System (16) can be rewritten as follows:

Theorem 3. Consider system (17). There are nonnegative constants , , , , and , positive numbers , , and , and functions and , such that the conditions hold:

() when , .() , , , when .() , , , .() .() .

Then the zero solution of system (17) is stable.

Proof. Define a Lyapunov functional as

where and are two positive constants to be determined. Using condition (), we have

and then .
Let be a solution of (17). Then,

Applying conditions () and () and , we have

By condition (), we have

Thus, we choose and . Using conditions () and (), we have

Therefore, the zero solution of system (17) is stable and the proof is completed.

Corollary 4. Consider system (17) with a variable delay . Conditions (1)-(3) of Theorem 3 are satisfied. Moreover,

() ,() there are and , such that and .

Then the zero solution of system (17) with a variable delay is stable.

Proof. The proof is similar to that of Theorem 3. But

Thus, choosing and , we have

Then, the zero solution of system (6) with a variable delay is stable and the proof is completed.

3. An Illustrative Example

In this section, we give an illustrative example to the effectiveness of results obtained in this paper. Moreover, we list a graph of solutions of an integral-differential equation with a delay to verify the correctness of the conclusion.

Example 1. Consider the following example:

In the example, , , , , , and a delay . Obvious, this system is the same form as (5). Moreover,

() , ;() , ;() , , ;() = , .

Thus, all conditions of Theorem 1 are satisfied and the zero solution of system (27) is stable by the obtained result. To show the effectiveness of the result, we carry out a simulation result with the following choices. Initial Condition: for . The simulation result is shown in Figure 1, which is the stability of solutions for system (27).

Figure 1: Stability of solutions for system (27).

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

The first author is the main completer. The second author is the corresponding author and provides the thought of this paper and advice in the process of writing.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (G11671227, G11701310, and G61403223).

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