Stability to a Kind of Functional Differential Equations of Second Order with Multiple Delays by Fixed Points

Abstract

We discuss the stability of solutions to a kind of scalar Liénard type equations with multiple variable delays by means of the fixed point technique under an exponentially weighted metric. By this work, we improve some related results from one delay to multiple variable delays.

1. Introduction

For more than one hundred years, Lyapunov’s direct (second) method has been very effectively used to investigate the stability problems in ordinary and functional differential equations. This method is one of the highly effective methods to determine the stability properties of solutions of ordinary and functional differential equations of higher order in the literature. However, till now, constructing or defining Lyapunov functions or functionals which give a meaningful discussion remains a general problem in the literature. In recent years, many researchers discussed that the fixed point theory has an important advantage over Lyapunov’s direct method. While Lyapunov’s direct method usually requires pointwise conditions, fixed point theory needs average conditions; see Burton [1].

In 2001, Burton and Furumochi [2] observed some difficulties that occur in studying the stability theory of ordinary and functional differential equations by Lyapunov's second (direct) method. Rather than inventing new modifications of the standard Lyapunov function(al) method to overcome the difficulties, the authors demonstrate by various examples that the contraction mapping principle can do the magic in many circumstances.

Later, in 2005, by using contraction mappings, Burton [3] investigated the scalar Liénard type equation with constant delay, : Burton [3] obtained conditions for each solution to satisfy as .

Later, in 2011, Pi [4] studied stability properties of solutions to a scalar functional Liénard type equation with variable delay, : By using fixed point theory under an exponentially weighted metric, Pi [4] obtained some interesting sufficient conditions ensuring that the zero solution of this equation is stable and asymptotically stable.

On the other hand, some recent relative results proceeded on the qualitative behaviors of delay differential equations, neutral differential equations, neutral Volterra integrodifferential equations, and certain nonlinear differential equations of second order with and without delay can be summarized as follows.

In [5], Fan et al. studied delay differential equations of the form and the authors established sufficient and necessary criteria for the asymptotic stability by using two different approaches, the contraction mapping principle and Schauder's fixed point theorem.

Raffoul [6] dealt with the stability of the zero solution of a scalar neutral differential equation. The author established sufficient conditions for the stability of the zero solution on the base of the contraction mapping principle.

In [7], Jin and Luo aimed to study the asymptotic stability for some scalar differential equations of retarded and neutral type by using a fixed point approach. The authors did not use Lyapunov’s method; they got interesting results for the stability even when the delay is unbounded. The authors also obtained necessary and sufficient conditions for the asymptotic stability.

Zhang and Liu [8] considered a nonlinear neutral differential equation. By using fixed point theory, they gave some conditions to ensure that the zero solution to the equation is asymptotically stable. Hence, some existing results were improved and generalized by this work.

Ardjouni and Djoudi [9] used the contraction mapping theorem to obtain an asymptotic stability result of the zero solution of a nonlinear neutral Volterra integrodifferential equation with variable delays. The asymptotic stability theorem with a necessary and sufficient condition was proved, which improves and extends the results in the literature.

In 2010, Tunç [10] considered the following Liénard type equation with multiple variable deviating arguments, : The author studied the problems of stability and boundedness of the solutions of this equation by using the Lyapunov second method and made a comparison with some earlier works in the literature.

In [11], the author considered the nonlinear differential equation of second order with a constant delay, : The author discussed the stability of the zero solution of this equation, when , and established two new results on the boundedness and uniform-boundedness of the solutions of the same equation, when . By this work, Tunç [11] improved the existing results on the stability and boundedness of the solutions of the differential equations of second order without a delay by imposing a few new criteria to the second order nonlinear and nonautonomous delay differential equations of the above form.

Further, Tunç [12] took into consideration the vector Liénard equation with the multiple constant deviating arguments, Based on the Lyapunov-Krasovskii functional approach, the asymptotic stability of the zero solution and the boundedness of all solutions of this equation, when and , respectively, are discussed.

More recently, by using Lyapunov’s function and functional approach, Tunç [13, 14] and Tunç and Yazgan [15] discussed some problems on stability, the boundedness, and the existence of periodic solutions of a certain second order vector and scalar nonlinear differential equations without and with delay. In [16], Tunç also gave certain sufficient conditions for the existence of periodic solutions to a Rayleigh-type equation with state-dependent delay.

By the mentioned papers, the authors contributed to the subject for a class of ordinary and functional differential equations.

In this paper, instead of the mentioned equations, we consider the scalar Liénard type equation with multiple variable delays: where , , are bounded and continuous functions, ,  , , and are all continuous functions such that .

We can write (7) as follows:

For each , we define and with the continuous function norm , where

It will cause no confusion even if we use as the supremum on . It can be seen from [9] that, for a given continuous function and a number , there exists a solution of system (8) on an interval ; if the solution remains bounded, then . Let denote the solution .

Definition 1. The zero solution of system (8) is stable if for each there exists such that implies that for .

We make the following basic assumptions on the delay functions . Let be strictly increasing and The inverses of exist, denoted by and , . Let . Hence, .

It is also worth mentioning that throughout the papers [10–15] the authors discussed the qualitative behavior of solutions of certain scalar and vector ordinary and functional differential equations of second order by means of the Lyapunov function or functional approach. In this paper, instead of the mentioned methods, we use the fixed point technique under an exponentially weighted metric to discuss stability of solutions to a kind of scalar Liénard type equations with multiple variable delays. This approach has a contribution to the topic in the literature, and it may be useful for researchers to work on the qualitative behaviors of solutions.

2. Main Result

In this section, sufficient conditions for stability are presented by the fixed point theory. We give some results on stability of the zero solution of (7). Before giving our main result, we introduce some auxiliary results.

Lemma 2. Let be a given continuous function. If is the solution of system (8) on satisfying , , and , then is the solution of the following integral equation:
Conversely, if the continuous function , is the solution of (10) on , then is the solution of system (8) on .

Proof. Let . Then, (8) can be written as the following system: so that Multiplying both sides of (12) by and then integrating from to , we obtain and hence If we choose , then it follows that Let Then, (15) can be written in the form of Hence so that Thus, it can be written that Let . Then, Multiplying both sides of (21) by and then integrating from to , then so that
Applying the integration by parts formula for the last two terms, we have
Conversely, we assume that a continuous function for and satisfies the integral equation on . Then, it is differentiable on . Hence, it is only needed to differentiate the integral equation. When we differentiate the integral equation, we can conclude the desired result.
Let be the Banach space of bounded continuous functions on with the supremum norm for . Let denote the supremum metric and , where . Next, let be a given continuous initial function.
Define the set by and its subset where is a given initial function and is a positive constant. Define the mapping by and if , then
Since satisfy the Lipschitz condition, let denote the common Lipschitz constants for and .
It is also clear that But since are nonlinear, then may not be small enough. Hence, may not be a contracting mapping. We can solve this problem by giving an exponentially weight metric via the next lemma.

Lemma 3. We suppose that there exists a constant such that satisfy the 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 are the common Lipschitz constants for and . Then is a Banach space. Thus is a complete metric space with , where . Under this metric, the space is a closed subset of . Thus the metric space is complete.
(ii) Let . It is clear that and . Then, for , we can get
For , since , we have
Further for , it follows that Since , then we have Hence Therefore,
Thus, we have For , . Thus, Therefore, is contraction mapping on .

Theorem 4. We suppose that the assumption holds. Moreover, we assume the following.(i)There exists a positive constant such that satisfy the Lipschitz condition on and are odd and they are strictly increasing on , and are nondecreasing on .(ii)There exist an and a continuous function such that for , , , (iii)There exist constants and such that, for each , if , then Then there exists such that, for each initial function and satisfying , there is a unique continuous function satisfying , which is a solution of (7) on . Moreover, the zero solution of (7) is stable.

Proof. Choose and such that In view of the assumptions and , it follows that . Since satisfies Lipschitz condition on , thus is continuous function on . Then, there exists a constant such that .
Thus, we can get It also follows that From assumption (ii), we have Hence Using condition (iii) of the theorem, we get Thus, and so It is obvious that if , then . Moreover, for , we get Therefore, . Since is a contraction mapping, then has unique fixed point such that .
From (14), we have Since, for , , then Hence If we replace by , then we show that the zero solution of (7) is stable. This result completes the proof of the theorem.

3. Conclusion

A kind of scalar Liénard type equations with multiple variable delays is considered. The stability of the zero solution of this equation is investigated. In proving our main result, we use the fixed points theory by giving an exponentially weight metric. Our result extends and improves some recent results in the literature.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors of this paper would like to expresses their sincere appreciation to the anonymous referees for their valuable comments and suggestions which have led to an improvement in the presentation of the paper.