Journal of Function Spaces and Applications

Volume 2013 (2013), Article ID 159435, 7 pages

http://dx.doi.org/10.1155/2013/159435

Research Article

Stability for a Class of Differential Equations with Nonconstant Delay

Jin Liang,¹ Tzon-Tzer Lu,² and Yashan Xu³

¹Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China

²Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung 80424, Taiwan

³School of Mathematical Sciences, Fudan University, Shanghai 200433, China

Received 6 February 2013; Accepted 6 March 2013

Academic Editor: James H. Liu

Copyright © 2013 Jin Liang et al. 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.

Abstract.

Stability is investigated for the following differential equations with nonconstant delay where , , , and and with ( is a positive constant) are continuous functions. A criterion is given for the zero solution of this delay equation being uniformly stable and asymptotically stable.

1. Introduction

Delays are inherent in many physical and technological systems. In particular, pure delays are often used to ideally represent the effects of transmission, transportation, and inertia phenomena. Delay differential equations constitute basic mathematical models of real phenomena, for instance in biology, mechanics, and economics (cf., e.g., [1–17] and references therein). Stability analysis of delay differential equations is particularly relevant in control theory, where one cause of delay is the finite speed of communication. There have been a lot of results on the study of stability of delay differential equations. For example, we can see many earlier results on this issue from Burton's book [2]. Recently, in 2004, Butcher et al. [4] studied the stability properties of delay differential equations with time-periodic parameters. By employing a shifted Chebyshev polynomial approximation in each time interval with length equal to the delay and parametric excitation period, the system is reduced to a set of linear difference equations for the Chebyshev expansion coefficients of the state vector in the previous and current intervals. In 2005, Wahi and Chatterjee [16] used Galerkin-projection to reduce the infinite dimensional dynamics of a delay differential equation to one occurring on a finite number of modes. In 2009, Kalmár-Nagy [7] demonstrated that the method of steps for linear delay differential equation together with the inverse Laplace transform can be used to find a converging sequence of polynomial approximants to the transcendental function determining stability of the delay equation. Most recently, Berezansky and Braverman [3] gave some explicit conditions of asymptotic and exponential stability for the scalar nonautonomous linear delay differential equation with several delays and an arbitrary number of positive and negative coefficients.

This paper is concerned with the following differential equations with nonconstant delay:

where , , , and and with

( is a positive constant) are continuous functions. We aim at giving general criterion for the zero solution of this delay equation being uniformly stable and asymptotically stable.

2. Main Result

Denote by the Banach space of continuous functions from to with the sup-norm

We consider (1) for with the initial conditions (for any )

where .

For an initial function , we denote by the solution of (1) such that (4) holds.

Definition 1. The zero solution of (1) is said to be stable if for any and , there exists such that if

then

The zero solution of (1) is uniformly stable if the above is independent of .

Definition 2. The zero solution of (1) is said to be asymptotically stable if it is stable and if for any , there exists such that if

then

Theorem 3. Assume that

(1)the zero solution to (1) is unique;(2)if is nontrivial function and is nontrivial in any interval , then

for a constant ; (3); (4)if , then

where .

Then the zero solution of (1) is uniformly stable.

Proof. For each , we set

and when is a nontrivial function and is nontrivial in any interval (), we set

From (3) and (2), it follows that for every , there exists such that

and when is a nontrivial function and is nontrivial in any interval (), such that

We claim that for any and , if

then

which means that the zero solution of (1) is eventually uniformly stable. Actually, if this is not true, then there exist

and a solution

to (1) with and

such that there is a ,

Define

Then, together with (21) and (22), we obtain

and, for ,

and for arbitrary , there exists such that

Therefore,

This implies that

In fact, if

then by (23)–(25), we have

It is not hard to see that we can choose and above to make have constant sign in .

Case I. When or

where is a positive real number.

In this case, if , then

which contradicts with (28). Moreover, if

for a positive real number , then it is clear that we can require . Hence,

which contradicts with (28) too.

Consequently, in this case we have the following observation.

Case I-1. If , then we deduce by (23), (24), (1), and (11) that

This is clearly impossible.

Case I-2. If , then we deduce by (23), (24), (1), (11), and (14) that

This is clearly impossible too.

Therefore, in this case, the zero solution of (1) is eventually uniformly stable. This, together with assumption (1), implies that the zero solution of (1) is uniformly stable.

Case II. is a nontrivial function and is nontrivial in any interval ().

In this case, by virtue of (1), and assumption (2), (12), (13), and (16), we get

which contradicts with (28).

Consequently, in this case we have the following observation,

Case II-1. If , then we deduce by (23), (24), (1), (11), (12), (14), and (15) that

This is a contradiction.

Case II-2. If , then we deduce by (23), (24), (1), (11), (12), (14), and (15) that

This is a contradiction too.

Therefore, in this case, the zero solution of (1) is eventually uniformly stable. This, together with assumption (1), implies that the zero solution of (1) is uniformly stable.

Theorem 4. Assume that

(1)the zero solution to (1) is unique;(2)if or

for a positive real number , then

(3)if is nontrivial function and is nontrivial in any interval   , then

for a constant ;(4); (5)if , then

where . Then the zero solution of (1) is asymptotically stable.

Proof. It follows from Theorem 3 that the zero solution of (1) is uniformly stable; that is, for arbitrarily given and , there exists such that if

then

Next, we will prove that

First, we show that

Suppose that this is not true. Then

Hence, for the arbitrarily given

there exist and such that

or

Let us now consider

Case I. When or

for a positive real number , we obtain by assumption (2), (46), (50), and (53)

This implies that

which contradicts with (53).

Case II. When is a nontrivial function and is nontrivial in any interval (), we obtain by assumptions (3), (46), (50), and (53)

This, together with assumption (2), implies that

which contradicts with (53).

Moreover, in a similar way, we can prove that

is impossible.

Therefore, (48) is true.

Based on (48), we will show that

Actually, if this is not true, that is,

then by (48) we see that there are with

and two sequences and such that

and for ,

By the same reason as that in the proof of Theorem 3, we know that

Define , , , and as those in the proof of Theorem 3. Then when is large enough, we have

Case I. When or

where is a positive real number.

Case I-1. If , then we deduce that

This is impossible.

Case I-2. If , then we obtain

This is clearly impossible too.

Consequently, (60) is true in this case.

Case II. When is nontrivial function and is nontrivial in any interval ().

Case II-1. If , then we deduce that

This is a contradiction.

Case II-2. If , then we obtain

This is a contradiction too.

Therefore, (60) is true in this case. So, (60) holds truly. This means that the zero solution of (4) is asymptotically stable.

Remark 5. Our results are new comparing with the results in [2, 3] since could go to or a big number as and in this case also could be very large in our theorems. Moreover, for the case of , the condition on in our results is very weak.

Acknowledgment

This work was supported by the NSF of China (11171210).

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