A New Delay Vector Integral Inequality and Its Application
© Li Xiang et al. 2010
Received: 22 September 2010
Accepted: 14 December 2010
Published: 21 December 2010
A new delay integral inequality is established. Using this inequality and the properties of nonnegative matrix, the attracting sets for the nonlinear functional differential equations with distributed delays are obtained. Our results can extend and improve earlier publications.
The attracting set of dynamical systems have been extensively studied over the past few decades and various results are reported. It is well known that one of the most researching tools is inequality technique. With the establishment of various differential and integral inequalities [1–12], the sufficient conditions on the attracting sets for different differential systems are obtained [12–16]. However, the inequalities mentioned above are ineffective for studying the attracting sets of a class of nonlinear functional differential equations with distributed delays.
Motivated by the above discussions, in this paper, a new delay vector integral inequality is established. Applying this inequality and the properties of nonnegative matrix, some sufficient conditions ensuring the global attracting set for a class of nonlinear functional differential equations with distributed delays are obtained. The result in  is extended.
In this section, we introduce some notations and recall some basic definitions.
which includes all positive eigenvectors of provided that the nonnegative matrix has at least one positive eigenvector (see ).
Definition 2.1 (Xu ).
Lemma 2.2 (Lasalle ).
3. Delay Integral Inequality
This contradicts the equality in (3.9), and so (3.8) holds. The proof is complete.
Throughout this section, we suppose the following:
This work was supported by the National Natural Science Foundation of China under Grant no. 10971147. Thanks are due to the instruction of Professor Xu.
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