# Fixed Points and Stability in Nonlinear Equations with Variable Delays

- Liming Ding
^{1, 2}Email author, - Xiang Li
^{1}and - Zhixiang Li
^{1}

**2010**:195916

https://doi.org/10.1155/2010/195916

© Liming Ding et al. 2010

**Received: **9 July 2010

**Accepted: **18 October 2010

**Published: **24 October 2010

## Abstract

We consider two nonlinear scalar delay differential equations with variable delays and give some new conditions for the boundedness and stability by means of the contraction mapping principle. We obtain the differences of the two equations about the stability of the zero solution. Previous results are improved and generalized. An example is given to illustrate our theory.

## Keywords

## 1. Introduction

Fixed point theory has been used to deal with stability problems for several years. It has conquered many difficulties which Liapunov method cannot. While Liapunov's direct method usually requires pointwise conditions, fixed point theory needs average conditions.

where , , , are continuous functions. We assume the following:

(A2) the functions is strictly increasing,

Burton [5] and Zhang [6] have also studied similar problems. Their main results are the following.

Theorem 1.1 (Burton [1]).

for all and . Then, for every continuous initial function , the solution of (1.3) is bounded and tends to zero as .

Theorem 1.2 (Zhang [2]).

- (i)

Theorem 1.3 (Burton [7]).

Then, the zero solution of (1.4) is stable.

Theorem 1.4 (Becker and Burton [3]).

then the zero solution of (1.4) is asymptotically stable.

In the present paper, we adopt the contraction mapping principle to study the boundedness and stability of (1.1) and (1.2). That means we investigate how the stability property will be when (1.3) and (1.4) are added to the perturbed term . We obtain their differences about the stability of the zero solution, and we also improve and generalize the special case . Finally, we give an example to illustrate our theory.

## 2. Main Results

From existence theory, we can conclude that for each continuous initial function there is a continuous solution on an interval for some and on . Let denote the set of all continuous functions and . Stability definitions can be found in [8].

Theorem 2.1.

Suppose that the following conditions are satisfied:

Then, the zero solution of (1.1) is asymptotically stable if and only if

Proof.

Apply (ii) to obtain . Thus, as . Similarly, we can show that the rest term in (2.10) approaches zero as . This yields as , and hence .

Also, by (ii), is a contraction mapping with contraction constant . By the contraction mapping principle, has a unique fixed point in which is a solution of (1.1) with on and as .

In order to prove stability at , let be given. Then, choose so that . Replacing with in , we see there is a such that implies that the unique continuous solution agreeing with on satisfies for all . This shows that the zero solution of (1.1) is asymptotically stable if (iv) holds.

which contradicts (2.22). Hence, condition (iv) is necessary for the asymptotically stability of the zero solution of (1.1). The proof is complete.

When , a constant, , we can get the following.

Corollary 2.2.

Suppose that the following conditions are satisfied:

Then, the zero solution of (1.1) is asymptotically stable if and only if

Remark 2.3.

We can also obtain the result that is bounded by on . Our results generalize Theorems 1.1 and 1.2.

Theorem 2.4.

Suppose that a continuous function exists such that and that the inverse function of exists. Suppose also that the following conditions are satisfied:

(iii) and are odd, increasing on . is nondecreasing on ,

Then, the zero solution of (1.2) is stable.

Proof.

where is a constant. Then, is a Banach space, which can be verified with Cauchy's criterion for uniform convergence.

Thus, there exists such that and . Hence, .

we have . That means . Hence, is a contraction mapping in with constant . By the contraction mapping principle, has a unique fixed point in , which is a solution of (1.2) with on and .

In order to prove stability at , let be given. Then, choose so that . Replacing with in , we see there is a such that implies that the unique continuous solution agreeing with on satisfies for all . This shows that the zero solution of (1.2) is stable. That completes the proof.

When , a constant, we have the following.

Corollary 2.5.

Suppose that the following conditions are satisfied:

(i)there exists a constant such that ,

(ii)there exists a constant such that satisfy a Lipschitz condition with constant on an interval ,

(iii) and are odd, increasing on . is nondecreasing on ,

is stable.

Corollary 2.6.

Suppose that the following conditions are satisfied:

(i)there exists a constant such that ,

(ii)there exists a constant such that , , satisfy a Lipschitz condition with constant on an interval ,

(iii) and are odd, increasing on . is nondecreasing on ,

is stable.

Remark 2.7.

The zero solution of (1.2) is not as asymptotically stable as that of (1.1). The key is that is not complete under the weighted metric when added the condition to that as .

Remark 2.8.

Theorem 2.4 makes use of the techniques of Theorems 1.3 and 1.4.

## 3. An Example

where , , , , and , . This equation comes from [4].

Let , when is sufficiently small, . Then, the condition (ii) of Theorem 2.1 is satisfied.

Let , then the condition (i) of Theorem 2.1 is satisfied.

And , then the condition (iii) and (iv) of Theorem 2.1 are satisfied.

According to Theorem 2.1, the zero solution of (3.1) is asymptotically stable.

## Authors’ Affiliations

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