#### Abstract

We study a class of integrodifferential functional differential equations with variable delay. By using the fixed point theory, we establish necessary and sufficient conditions ensuring that the zero solution of this equation is asymptotically stable.

#### 1. Introduction

It is well known that differential equations have many applications in engineering, control theory, neural networks, biology, and so on. The stability of the solution of functional differential equations has drawn a lot of attention for many years. Liapunovâ€™s direct method has been widely applied to study the stability problems for a long time; see, for example, [1â€“3]. Nowadays, many authors have applied the fixed points theory to study the stability of solution of functional differential equations; some authors even use fixed point theory to study stability conditions for stochastic equations with delay; see, for example, [4â€“7] and the references therein.

Burton studied stability of a nonconvolution equation where was a positive constant. He obtained sufficient conditions ensuring that the zero solution was asymptotically stable via fixed point theorem; see [8].

Becker and Burton studied the following differential equation: with variable delay and its special case for , is a positive constant, , where function . They found sufficient conditions ensuring that the zero solution was asymptotically stable by changing the supremum metric to an exponentially weighted metric. Moreover, they hoped to reduce condition that the function was strictly increasing; see [9]. Jin and Luo reduced this condition in their paper. Moreover, they established necessary and sufficient conditions that could ensure that the zero solution of this equation is asymptotically stable; see, for example, [10].

Ardjouni et al. studied the stability of the following neutral integrodifferential equation: via fixed point theorem. Also . Their main result was as follows.

Theorem A. *Suppose that is twice differentiable and for all and there exist continuous functions for and a constant such that, for ,*(i)*,*(ii)*â€‰**where
**Then the zero solution is asymptotically stable if and only if
*

They improved the results of [9]; see [11]. Other results on fixed points and stability properties in equations with variable delays can be found in [5].

Recently, Zhao and Yuan investigate the stability of a generalized Volterra-Levin equation: They give new conditions on the stability of zero solution of this equation; their conditions are different from [5]; see, for example, [12].

The stability of second order equation has always been hot issue of research. Levin and Nohel [13] studied the global asymptotic stability of a class of nonlinear systems: They obtained asymptotic stability by constructing proper Lyapunov function.

Burton [14] considered the following equation with delay: where is a positive constant, and obtained some sufficient conditions under which each solution satisfies via the fixed point theorem.

We generalized the above equation to an equation with a variable delay [15] as follows:
and obtained some results on asymptotic stability of the zero solution. Before we introduce our new results we recall the main results in [15]. There are* basic assumptions* on the delay function which is strictly increasing and . is the inverse of . Moreover for some constant .

The main results in [15] can be stated as follows.

Theorem B. *Suppose and the following conditions hold.*(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, for all , , , . The function is continuous on , .*

If, for each , , then the zero solution of (11) is asymptotically stable.

In [16], we have studied the following equation: 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. Moreover, we establish necessary and sufficient conditions ensuring that the zero solution is asymptotically stable.

In this paper we consider for , where functions , , , and are all continuous, where . We assume that as . .

For each , define , . Let . Also is a function space endowed with function norm , where . We will also use to express the supremum on later. It is well known that, in [2], for a given continuous function , there exists a solution of (15) on an interval ; if the solution remains bounded, then . We denote by the solution . Denote by .

We can write (15) in an equivalent form as

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. 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.

#### 2. Statement of Main Results

First of all, we consider a special case of (15). We consider the following equation: Rewrite this equation as Our main results are as follows.

Theorem 1. *Assume that and the following conditions hold.*(i)*Consider that . There exist a constant and a function such that, for , and the inequality hold. There exists a constant such that .*(ii)*There exist and a continuous function such that for . For , , *(iii)*There exist constants and such that, for each , if , then
*â€‰*Then the zero solution of (17) is stable.*

In addition, we have the following.

Theorem 2. *Assume that , is differentiable, , and there exist some functions such that for the following conditions hold.*(i)*There exist a constant and some functions such that, for ,
*â€‰*There exists a constant such that , .*(ii)*Let . There exist a constant and a continuous function such that . For , , ,
*â€‰*where .*(iii)*There exist constants and such that, for each , if , then
*â€‰*Then the zero solution of (15) is asymptotically stable if and only if
*

#### 3. Proof of Theorem 1

In this section, we will prove Theorem 1 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 [12].

Lemma 3. *Let the function , , . Then
**
is equivalent to
*

Lemma 4. *Let be a given continuous function; if is the solution of (17) satisfying , , and , then is the solution of the following integral equation:
**
where .*

*Proof. *We apply the variation of parameters formula to the second equation of (18); we obtain
By Lemma 3, (28) can be written as
multiplying both sides of the above equation by , we see that

If we integrate the last term by parts, we have

Let be the Banach space of bounded continuous functions on with the supremum norm. For a given continuous initial function , define the set by and its subset where is a given initial function and is a positive constant. Let be a mapping defined on as follows: for , if , . If , Easy calculation shows that

We can easily obtain that is a contraction mapping by using condition (ii). We continue to prove Theorem 1. Choose and such that Condition (i) implies that , since satisfies the Lipschitz condition with respect to , such that exists and . By the expression of , condition (ii), and condition (iii) of Theorem 1, we have

According to condition (ii) and (36), we have Note that if , then . We obtain that , . Thus . Since we have proved that is a contraction mapping, hence has a unique fixed point and .

Recalling (28), we have then It follows that For any given ; we only need to replace by . Then we obtain the stability of zero solution. This completes the proof of Theorem 1.

#### 4. Proof of Theorem 2

In this section, we will prove Theorem 2. First of all, we will obtain a useful expression for the solution of (15) in the following proof.

We apply the variation of parameters formula to the second equation of (15); then where .

Lemma 5. *The equation
**
is equivalent to
**
where
*

*Proof. *Differentiating the integral term in (44), we have
It follows that if , , then (43) is equivalent to (44). Easy calculation shows that

By Lemma 5, (42) can be written as We define Multiplying both sides of the above equation by , easy calculation shows that where . Performing an integration by parts, then Let be a mapping defined on as follows: for , if , . If , we define

If , since as , , hence 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 The second term of , as .

Note that is bounded; we can check the eighth term of , as similarly.

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 . Also as .

In order to obtain the asymptotic stability, we still need to show that the zero solution is stable. Let be given, we choose and such that Then we have Easy calculation shows that Hence Therefore It follows that Therefore the zero solution is stable; since we have obtained that as , it follows that the zero solution is asymptotically stable.

A necessary condition is as follows. For each , we denote that . We will prove that by way of contradiction. If , since , being a sequence , as such that , where is a finite number. Choose such that holds, for all .

Denote that where .

By conditions of Theorem 2, 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 is satisfying

Now we consider the solution of (11) which satisfies We can obtain that by a similar argument with (60) if we replace by 1. Consider

By (27) and , if , we have This implies that Note that ; it follows that