- Open Access
Global attracting set for a class of nonautonomous neutral functional differential equations
© Teng et al.; licensee Springer 2012
- Received: 6 June 2011
- Accepted: 4 June 2012
- Published: 22 June 2012
In this paper, a class of nonlinear and nonautonomous neutral functional differential equations is considered. By developing a new integral inequality, we obtain sufficient conditions for the existence of a global attracting set of neutral functional differential equations with time-varying delays. The results have extended and improved the related reports in the literature.
MSC:26D10, 26D15, 34K40.
- integral inequality
- attracting set
- neutral functional differential equations
The asymptotic properties of neutral functional differential equations have attracted considerable attention in the past few decades, and many significant results have been obtained [1–6]. One of the most popular ways to analyze the stability property and asymptotic behavior is the method of Lyapunov functionals [1–3]. However, to construct a suitable Lyapunov functional is not easy for certain equations. The characteristic equation is another important researching tool. It seems to work well for constant delays in neutral equations [4, 5]. Meanwhile, the known approach to the study of differential inequalities for nonautonomous neutral functional differential equations was presented in Azbelev’s book . These inequality methods are based on the representation formula of a solution and the analysis of Cauchy, Green’s and fundamental matrices. In this way, various assertions about the estimates of solutions, maximum principles and stability on neutral differential equations were obtained. Important results in this direction can be found in [7–10] and in the monograph . Recently, by using differential and integral inequalities, Xu et al. have studied attracting and invariant sets of functional differential systems [12, 13] and impulsive functional differential equations . However, the inequalities mentioned above are ineffective in studying the attracting sets of a class of nonlinear and nonautonomous neutral functional differential equations.
Motivated by the above discussions, in this paper, we will improve the inequality established in  so that it is effective for neutral functional differential equations. Combining with the properties of nonnegative matrices, we obtained some sufficient conditions ensuring the global attracting set for a class of nonlinear and nonautonomous neutral differential equations with time-varying coefficients and unbounded delays. The results extend the earlier publications.
In this section, we introduce some notations and recall some basic definitions.
E denotes the identity matrix, is the set of real numbers and . () means that each pair of corresponding elements of A and B satisfies the inequality “≤ (<).” Especially, A is called a nonnegative matrix if , where 0 denotes the zero matrix.
denotes the space of continuous mappings from the topological space X to the topological space Y. Especially, let denote the family of all bounded continuous -valued functions ϕ on , here .
For , , , , we define , , , , .
where , , , , . We always assume that for any , the system (1) has at least one solution through denoted by or simply if no confusion should occur.
Definition 2.1 (Xu )
Especially, if and .
where dist and is a norm of .
which includes all positive eigenvectors of A provided that the nonnegative matrix A has at least one positive eigenvector (see Refs. ).
Lemma 2.1 (Lasalle )
where, , , , , . Assume that the following conditions are satisfied: () = as, and there exists a constant matrixsuch that
Proof By the condition and the properties of nonnegative matrices, there exists a positive vector z such that . Together with and Lemma 2.1, this implies that exists and .
where , , , .
where denotes the i th component of vector .
This contradicts the equality in (11), and so (10) holds. The proof is complete. □
where is the fundamental matrix of the linear equation .
For (1), we suppose the following: (): = , , where , , , , .; (): = , , . For , there exist a constant matrix and a vector such that
Theorem 3.2 Assume that ()-() hold. Thenis a global attracting set of (1).
where , , , .
Letting , we have , that is , and the proof is completed. □
Corollary 3.1 Suppose that the conditions of Theorem 3.2 hold and. Ifis an equilibrium point of System (1), then the equilibrium pointis globally asymptotically stable.
where , b are constants, and .
If , we can get is the global attracting set for (23).
This work was supported by the National Natural Science Foundation of China under Grant No.10971147.
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