#### 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 [120] 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 delaysand 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 equationsIn [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 aswhere is a positive constant to be determined. Using condition (), we haveand .
Suppose that is a solution of (6). ThenBy conditions () and () and , we haveThen,Choosing , we haveTherefore, 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. ButThus, choosing , we haveTherefore, 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 aswhere and are two positive constants to be determined. Using condition (), we haveand then .
Let be a solution of (17). Then,Applying conditions () and () and , we haveBy condition (), we haveThus, we choose and . Using conditions () and (), we haveTherefore, 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. ButThus, choosing and , we haveThen, 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).

#### 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).