Open Access

Fixed Points and Stability in Nonlinear Equations with Variable Delays

Fixed Point Theory and Applications20102010:195916

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

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.

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.

In this paper, we consider the nonlinear delay differential equations
(1.1)
(1.2)

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

(A1) is differentiable,

(A2) the functions is strictly increasing,

(A3) as .

Many authors have investigated the special cases of (1.1) and (1.2). Since Burton [1] used fixed point theory to investigate the stability of the zero solution of the equation , many scholars continued his idea. For example, Zhang [2] has studied the equation
(1.3)
Becker and Burton [3] have studied the equation
(1.4)
Jin and Luo [4] have studied the equation
(1.5)

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

Theorem 1.1 (Burton [1]).

Suppose that , a constant, and there exists a constant such that
(1.6)

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]).

Suppose that is differentiable, the inverse function of exists, and there exists a constant such that for
  1. (i)
    (1.7)
     
  1. (ii)
    (1.8)
     
where . Then, the zero solution of (1.3) is asymptotically stable if and only if
  1. (iii)
    (1.9)
     

Theorem 1.3 (Burton [7]).

Suppose that , a constant. Let be odd, increasing on , and satisfies a Lipschitz condition, and let be nondecreasing on . Suppose also that for each , one has
(1.10)
and there exists such that
(1.11)

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

Theorem 1.4 (Becker and Burton [3]).

Suppose is odd, strictly increasing, and satisfies a Lipschitz condition on an interval and that is nondecreasing on . If
(1.12)
where is the unique solution of , and if a continuous function exists such that
(1.13)
on , then the zero solution of (1.5) is stable at . Furthermore, if is continuously differentiable on with and
(1.14)

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:

(i) , and there exists a constant so that if , then
(2.1)
(ii)there exists a constant and a continuous function such that
(2.2)
(iii) 
(2.3)

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

(iv) 
(2.4)

Proof.

First, suppose that (iv) holds. We set
(2.5)

Let , then is a Banach space.

Multiply both sides of (1.1) by , and then integrate from 0 to to obtain
(2.6)
By performing an integration by parts, we have
(2.7)
or
(2.8)
Let
(2.9)
Then, is a complete metric space with metric for . For all , define the mapping
(2.10)
By (i) and ,
(2.11)

Thus, when , .

We now show that as . Since and as , for each , there exists a such that implies . Thus, for ,
(2.12)
Hence, as . And
(2.13)
By (ii) and (iv), there exists such that implies
(2.14)

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.

Conversely, suppose (iv) fails. Then, by (iii), there exists a sequence , as such that for some . We may choose a positive constant satisfying
(2.15)
for all . To simplify the expression, we define
(2.16)
for all . By (ii), we have
(2.17)
This yields
(2.18)
The sequence is bounded, so there exists a convergent subsequence. For brevity of notation, we may assume that
(2.19)
for some and choose a positive integer so large that
(2.20)

for all , where satisfies .

By (iii), in (2.5) is well defined. We now consider the solution of (1.1) with and for . We may choose so that for and
(2.21)
It follows from (2.10) with that for ,
(2.22)
On the other hand, if the solution of (1.1) as , since as and (ii) holds, we have
(2.23)

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:

(i) , and there exists a constant so that if , then
(2.24)
  1. (ii)
    there exists a constant such that for all , one has
    (2.25)
     
  1. (iii)
    (2.26)
     

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

(iv) 
(2.27)

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:

(i)there exists a constant such that ,
  1. (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 ,

(iv)for each , one has

(2.28)

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

Proof.

By (iv), there exists such that
(2.29)
Let be the space of all continuous functions such that
(2.30)

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

The equation (1.2) can be transformed as
(2.31)
By the variation of parameters formula, we have
(2.32)
Let
(2.33)
then is a complete metric space with metric for . For all , define the mapping
(2.34)
By (i), (iii), and (2.29), we have
(2.35)

Thus, there exists such that and . Hence, .

We now show that is a contraction mapping in . For all ,
(2.36)
Since
(2.37)

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 ,

(iv)for each , one has

(2.38)
Then, the zero solution of the equation
(2.39)

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 ,

(iv)for each , one has

(2.40)
Then, the zero solution of
(2.41)

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

We use an example to illustrate our theory. Consider the following differential equation:
(3.1)

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

Choosing , we have
(3.2)

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

(1)
Department of Mathematics and System Science, College of Science, National University of Defence Technology
(2)
Air Force Radar Academy

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Copyright

© Liming Ding et al. 2010

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.