Journal of Function Spaces and Applications

Volume 2013, Article ID 159435, 7 pages

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

## Stability for a Class of Differential Equations with Nonconstant Delay

^{1}Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China^{2}Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung 80424, Taiwan^{3}School 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).

#### References

- R. P. Agarwal, C. Cuevas, and M. V. S. Frasson, “Semilinear functional difference equations with infinite delay,”
*Mathematical and Computer Modelling*, vol. 55, no. 3-4, pp. 1083–1105, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. A. Burton,
*Stability and Periodic Solutions of Ordinary and Functional-Differential Equations*, Mathematics in Science and Engineering, Academic Press, Orlando, Fla, USA, 1985. View at MathSciNet - L. Berezansky and E. Braverman, “New stability conditions for linear differential equations with several delays,”
*Abstract and Applied Analysis*, vol. 2011, Article ID 178568, 19 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. A. Butcher, H. Ma, E. Bueler, V. Averina, and Z. Szabo, “Stability of linear time-periodic delay-differential equations via Chebyshev polynomials,”
*International Journal for Numerical Methods in Engineering*, vol. 59, no. 7, pp. 895–922, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Diagana, “Existence of almost automorphic solutions to some neutral functional differential equations with infinite delay,”
*Electronic Journal of Differential Equations*, no. 129, p. 14, 2008. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. K. Hale and S. M. Verduyn Lunel,
*Introduction to Functional-Differential Equations*, vol. 99 of*Applied Mathematical Sciences*, Springer, New York, NY, USA, 1993. View at MathSciNet - T. Kalmár-Nagy, “Stability analysis of delay-differential equations by the method of steps and inverse Laplace transform,”
*Differential Equations and Dynamical Systems*, vol. 17, no. 1-2, pp. 185–200, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Li and G. M. N'Guérékata, “An existence result for neutral delay integrodifferential equations with fractional order and nonlocal conditions,”
*Abstract and Applied Analysis*, vol. 2011, Article ID 952782, 20 pages, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - J. Liang, J. H. Liu, and T.-J. Xiao, “Periodic solutions of delay impulsive differential equations,”
*Nonlinear Analysis. Theory, Methods & Applications A*, vol. 74, no. 17, pp. 6835–6842, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Liang, H.-Y. Wang, and T.-J. Xiao, “On a comparison principle for delay coupled systems with nonlocal and nonlinear boundary conditions,”
*Nonlinear Analysis. Theory, Methods & Applications A*, vol. 71, no. 12, pp. e359–e365, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Liang and T.-J. Xiao, “Solutions to nonautonomous abstract functional equations with infinite delay,”
*Taiwanese Journal of Mathematics*, vol. 10, no. 1, pp. 163–172, 2006. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Liang and T.-J. Xiao, “Solutions to abstract integral equations and infinite delay evolution equations,”
*Bulletin of the Belgian Mathematical Society*, vol. 18, no. 5, pp. 793–804, 2011. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. H. Liu, G. M. N'Guérékata, and N. V. Minh,
*Topics on Stability and Periodicity in Abstract Differential Equations*, vol. 6 of*Series on Concrete and Applicable Mathematics*, World Scientific Publishing, Hackensack, NJ, USA, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - Y. Liu, W. Jiang, and F. Huang, “Robustness with respect to small delays for exponential stability of Pritchard-Salamon systems with admissible state feedback,”
*International Journal of Evolution Equations*, vol. 1, no. 3, pp. 225–263, 2005. View at Google Scholar · View at MathSciNet - P. H. A. Ngoc, N. van Minh, and T. Naito, “Stability radii of positive linear functional differential systems in Banach spaces,”
*International Journal of Evolution Equations*, vol. 2, no. 1, pp. 75–97, 2007. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - P. Wahi and A. Chatterjee, “Galerkin projections for delay differential equations,”
*Journal of Dynamic Systems, Measurement, and Control*, vol. 127, pp. 80–89, 2005. View at Google Scholar - T.-J. Xiao and J. Liang, “Blow-up and global existence of solutions to integral equations with infinite delay in Banach spaces,”
*Nonlinear Analysis. Theory, Methods & Applications A*, vol. 71, no. 12, pp. e1442–e1447, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet