# A Generalized Halanay Inequality for Stability of Nonlinear Neutral Functional Differential Equations

- Wansheng Wang
^{1}Email author

**2010**:475019

https://doi.org/10.1155/2010/475019

© Wansheng Wang. 2010

**Received: **22 March 2010

**Accepted: **18 July 2010

**Published: **1 August 2010

## Abstract

This paper is devoted to generalize Halanay's inequality which plays an important rule in study of stability of differential equations. By applying the generalized Halanay inequality, the stability results of nonlinear neutral functional differential equations (NFDEs) and nonlinear neutral delay integrodifferential equations (NDIDEs) are obtained.

## Keywords

## 1. Introduction

Halanay used the inequality as follows.

Lemma 1.1 (Halanay's inequality, see [1]).

Baker and Tang [2] give a generalization of Halanay inequality as Lemma 1.2 which can be used for discussing the stability of solutions of some general Volterra functional differential equations.

Lemma 1.2 (see [2]).

In recent years, the Halanay inequality has been extended to more general type and used for investigating the stability and dissipativity of various functional differential equations by several researchers (see, e.g., [3–7]). In this paper, we consider a more general inequality and use this inequality to discuss the stability of nonlinear neutral functional differential equations (NFDEs) and a class of nonlinear neutral delay integrodifferential equations (NDIDEs).

## 2. Generalized Halanay Inequality

In this section, we first give a generalization of Lemma 1.1.

Theorem 2.1 (generalized Halanay inequality).

To prove the theorem, we need the following lemmas.

Lemma 2.2.

if and only if for any fixed characteristic equation (2.4) has at least one nonnegative root .

Proof.

If systems (2.6) have nontrivial solution , , then is obviously a nonnegative root of the characteristic equation (2.4). Conversely, if characteristic equation (2.4) has nonnegative root for any fixed , then and , , are obviously a nontrivial solution of (2.6).

Lemma 2.3.

If (2.2) holds, then

(i)for any fixed , characteristic equation (2.4) does not have any nonnegative root but has a negative root ;

Proof.

We consider the following two cases successively.

which is a contradiction, and therefore .

After simply calculating, we have which contradicts the assumption. Thus, (2.4) does not have any nonnegative root.

which implies that is a strictly monotone increasing function. Therefore, for any fixed the characteristic equation (2.4) has a negative root .

which is a contradiction, and therefore .

Lemma 2.4.

If (2.6) has a solution with exponential form , , , , then for any , any nontrivial solution , of (2.1) satisfies (2.3).

Proof.

a contradiction proving the lemma.

Proof of Theorem 2.1.

By Lemma 2.3, we can find that for any fixed , characteristic equation (2.4) only has negative root and . Thus from Lemma 2.2 we know that systems (2.6) have not nontrivial solution with the form , , , . However, it is easily verified that systems (2.6) have nontrivial solution , , , . The result now follows from Lemma 2.4.

Corollary 2.5.

Proof.

Corollary 2.6 (see [3]).

## 3. Applications of the Halanay Inequality

In this section, we consider several simple applications of Theorem 2.1 to the study of stability for nonlinear neutral functional differential equations (NFDEs) and nonlinear neutral delay-integrodifferential equations (NDIDEs).

### 3.1. Stability of Nonlinear NFDEs

Neutral functional differential equations (NFDEs) are frequently encountered in many fields of science and engineering, including communication network, manufacturing systems, biology, electrodynamics, number theory, and other areas (see, e.g., [8–11]). During the last two decades, the problem of stability of various neutral systems has been the subject of considerable research efforts. Many significant results have been reported in the literature. For the recent progress, the reader is referred to the work of Gu et al. [12] and Bellen and Zennaro [13]. However, these studies were devoted to the stability of linear systems and nonlinear systems with special form, and there exist few results available in the literature for general nonlinear NFDEs. Therefore, deriving some sufficient conditions for the stability of nonlinear NFDEs motivates the present study.

Let be a real or complex Banach space with norm . For any given closed interval , let the symbol denote a Banach space consisting of all continuous mappings , on which the norm is defined by .

and throughout this paper, , , and , for all , denote continuous functions. The existence of a unique solution on the interval of (3.1) will be assumed.

where we assume the initial function is also a given continuously differentiable mapping, but it may be different from in problem (3.1).

To prove our main results in this section, we need the following lemma.

Lemma 3.1 (cf. Li [14]).

Theorem 3.2.

Proof.

Thus, the application of Theorem 2.1 and Corollary 2.5 to (3.13) and (3.14) leads to Theorem 3.2.

Remark 3.3.

where denotes the identity matrix, and denotes the logarithmic norm induced by .

Remark 3.4.

From (3.9), we know that and have an exponential asymptotic decay when the conditions of Theorem 3.2 are satisfied.

Example 3.5.

Since it can be equivalently written in the pattern of IVP (3.1) in NFDEs, on the basis of Theorem 3.2, we can assert that the system is exponentially stable if the assumptions of Theorem 3.2 are satisfied.

Example 3.6.

where there exists a constant such that . It is easy to verify that , , , and . Then, according to Theorem 3.2 presented in this paper, we can assert that the system (3.19) is exponentially stable.

### 3.2. Asymptotic Stability of Nonlinear NDIDEs

Since (3.20) is a special case of (3.1), we can directly obtain a sufficient condition for stability of (3.20).

Theorem 3.7.

one has (3.9) and (3.11).

Theorem 3.8.

Proof.

the last assertion follows.

### 3.3. Comparison with the Existing Results

- (i)In 2004, Wang and Li [16] were among the first who studied IVP in nonlinear NDDEs with a single delay in a finite dimensional space , that is,

They obtained the asymptotic stability result (3.31) for the cases of (3.25), (3.26) and (3.25), and (3.30) under the following assumptions:

() is a strictly increasing function on the interval ;

- (ii)

Note that in this case, , and condition (3.26) is equivalent to condition (3.30). Since Theorem 3.7 or Theorem 3.8 of the present paper can be applied to (3.41) with a variable delay , , and (3.9), (3.11) can be obtained under condition (3.26), the results of these two theorems are more general and deeper than these obtained by Zhang and Vandewalle mentioned above.

## Declarations

### Acknowledgments

This work was partially supported by the National Natural Science Foundation of China (Grant no. 10871164) and the China Postdoctoral Science Foundation Funded Project (Grant nos. 20080440946 and 200902437).

## Authors’ Affiliations

## References

- Halanay A:
*Differential Equations: Stability, Oscillations, Time Lags*. Academic Press, New York, NY, USA; 1966:xii+528.MATHGoogle Scholar - Baker CTH, Tang A: Generalized Halanay inequalities for Volterra functional differential equations and discretized versions. In
*Volterra Equations and Applications (Arlington, TX, 1996), Stability and Control: Theory and Applications*.*Volume 10*. Edited by: Corduneanu C, Sandberg IW. Gordon and Breach, Amsterdam, The Netherlands; 2000:39–55.Google Scholar - Liz E, Trofimchuk S: Existence and stability of almost periodic solutions for quasilinear delay systems and the Halanay inequality.
*Journal of Mathematical Analysis and Applications*2000, 248(2):625–644. 10.1006/jmaa.2000.6947MathSciNetView ArticleMATHGoogle Scholar - Tian H: Numerical and analytic dissipativity of the -method for delay differential equations with a bounded variable lag.
*International Journal of Bifurcation and Chaos in Applied Sciences and Engineering*2004, 14(5):1839–1845. 10.1142/S0218127404010096MathSciNetView ArticleMATHGoogle Scholar - Gan SQ: Dissipativity of -methods for nonlinear Volterra delay-integro-differential equations.
*Journal of Computational and Applied Mathematics*2007, 206(2):898–907. 10.1016/j.cam.2006.08.030MathSciNetView ArticleMATHGoogle Scholar - Wen LP, Yu YX, Wang WS: Generalized Halanay inequalities for dissipativity of Volterra functional differential equations.
*Journal of Mathematical Analysis and Applications*2008, 347(1):169–178. 10.1016/j.jmaa.2008.05.007MathSciNetView ArticleMATHGoogle Scholar - Wen LP, Wang WS, Yu YX: Dissipativity and asymptotic stability of nonlinear neutral delay integro-differential equations.
*Nonlinear Analysis: Theory, Methods & Applications*2010, 72(3–4):1746–1754. 10.1016/j.na.2009.09.016MathSciNetView ArticleMATHGoogle Scholar - Hale JK, Lunel SM:
*Introduction to Functional-Differential Equations, Applied Mathematical Sciences*.*Volume 99*. Springer, New York, NY, USA; 1993:x+447.View ArticleMATHGoogle Scholar - Gil' MI:
*Stability of Finite- and Infinite-Dimensional Systems, The Kluwer International Series in Engineering and Computer Science no. 455*. Kluwer Academic Publishers, Boston, Mass, USA; 1998:xviii+358.View ArticleGoogle Scholar - Kolmanovskii V, Myshkis A:
*Introduction to the Theory and Applications of Functional-Differential Equations, Mathematics and Its Applications*.*Volume 463*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1999:xvi+648.View ArticleMATHGoogle Scholar - Richard J-P: Time-delay systems: an overview of some recent advances and open problems.
*Automatica*2003, 39(10):1667–1694. 10.1016/S0005-1098(03)00167-5MathSciNetView ArticleMATHGoogle Scholar - Gu K, Kharitonov VL, Chen J:
*Stability of Time-Delay Systems*. Birkhäuser, Boston, Mass, USA; 2003.View ArticleMATHGoogle Scholar - Bellen A, Zennaro M:
*Numerical Methods for Delay Differential Equations, Numerical Mathematics and Scientific Computation*. The Clarendon Press, New York, NY, USA; 2003:xiv+395.View ArticleMATHGoogle Scholar - Li SF:
*Theory of Computational Methods for Stiff Differential Equations*. Hunan Science and Technology, Changsha, China; 1997.Google Scholar - Bartoszewski Z, Kwapisz M: Delay dependent estimates for waveform relaxation methods for neutral differential-functional systems.
*Computers & Mathematics with Applications*2004, 48(12):1877–1892. 10.1016/j.camwa.2004.05.011MathSciNetView ArticleMATHGoogle Scholar - Wang WS, Li SF: Stability analysis of nonlinear delay differential equations of neutral type.
*Mathematica Numerica Sinica*2004, 26(3):303–314.MathSciNetGoogle Scholar - Zhang C, Vandewalle S: Stability analysis of Volterra delay-integro-differential equations and their backward differentiation time discretization.
*Journal of Computational and Applied Mathematics*2004, 164–165: 797–814.MathSciNetView ArticleMATHGoogle Scholar - Zhang C, Vandewalle S: Stability analysis of Runge-Kutta methods for nonlinear Volterra delay-integro-differential equations.
*IMA Journal of Numerical Analysis*2004, 24(2):193–214. 10.1093/imanum/24.2.193MathSciNetView ArticleMATHGoogle Scholar

## Copyright

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.