A Generalized Halanay Inequality for Stability of Nonlinear Neutral Functional Differential Equations
© Wansheng Wang. 2010
Received: 22 March 2010
Accepted: 18 July 2010
Published: 1 August 2010
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.
Halanay used the inequality as follows.
Lemma 1.1 (Halanay's inequality, see ).
Baker and Tang  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 ).
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.
if and only if for any fixed characteristic equation (2.4) has at least one nonnegative root .
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).
If (2.2) holds, then
(i)for any fixed , characteristic equation (2.4) does not have any nonnegative root but has a negative root ;
We consider the following two cases successively.
which implies that is a strictly monotone increasing function. Therefore, for any fixed the characteristic equation (2.4) has a negative root .
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 2.6 (see ).
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.  and Bellen and Zennaro . 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.
To prove our main results in this section, we need the following lemma.
Lemma 3.1 (cf. Li ).
Thus, the application of Theorem 2.1 and Corollary 2.5 to (3.13) and (3.14) leads to Theorem 3.2.
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.
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).
one has (3.9) and (3.11).
the last assertion follows.
3.3. Comparison with the Existing Results
- (i)In 2004, Wang and Li  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:
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.
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).
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