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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.

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 [

In 2001, Burton and Furumochi [

Later, in 2005, by using contraction mappings, Burton [

Later, in 2011, Pi [

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 [

Raffoul [

In [

Zhang and Liu [

Ardjouni and Djoudi [

In 2010, Tunç [

In [

Further, Tunç [

More recently, by using Lyapunov’s function and functional approach, Tunç [

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:

We can write (

For each

It will cause no confusion even if we use

The zero solution of system (

We make the following basic assumptions on the delay functions

It is also worth mentioning that throughout the papers [

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 (

Let

Conversely, if the continuous function

Let

Applying the integration by parts formula for the last two terms, we have

Conversely, we assume that a continuous function

Let

Define the set

Since

It is also clear that

We suppose that there exists a constant

the metric space

(i) We change the supremum norm to an exponentially weighted norm

(ii) Let

For

Further for

Thus, we have

We suppose that the assumption

There exists a positive constant

There exist an

There exist constants

Choose

Thus, we can get

From (

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.

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

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.