Stability and Bifurcation Analysis of Differential Equations and its ApplicationsView this Special Issue
Research Article | Open Access
Stability Conditions of Second Order Integrodifferential Equations with Variable Delay
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.
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 , 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 .
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, .
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, . Dung  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  studied the global asymptotic stability of a class of nonlinear systems Burton  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  and obtained some results on asymptotic stability of the zero solution. Before we introduce our new results we recall the main results in . 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  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 , 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 .
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
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
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