%0 Journal Article
%T Stability and Existence of Periodic Solutions for Cellular Neural Networks with State Dependent Delays on Time Scales
%A Yaping Ren
%A Yongkun Li
%J Discrete Dynamics in Nature and Society
%D 2012
%I Hindawi Publishing Corporation
%R 10.1155/2012/386706
%X We study delayed cellular neural networks on time scales. Without assuming the boundedness of the activation functions, we establish the exponential stability and existence of periodic solutions. The results in this paper are completely new even in case of the time scale or and improve some of the previously known results. 1. Introduction Consider the following cellular neural networks with state-dependent delays on time scales: where , is an -periodic time scale which has the subspace topology inherited from the standard topology on , , will be defined in the next section, , corresponds to the number of units in the neural network, corresponds to the state of the th unit at time , denotes the output of the th unit on th unit at time , denotes the strength of the th unit on the th unit at time , denotes the external bias on the th unit at time , corresponds to the transmission delay along the axon of the th unit, represents the rate with which the th unit will reset its potential to the resting state in isolation when disconnected from the network and external inputs. It is well known that the cellular neural networks have been successfully applied to signal processing, pattern recognition, optimization, and associative memories, especially in image processing and solving nonlinear algebraic equations. They have been widely studied both in theory and applications [1¨C3]. Many results for the existence of their periodic solutions and the exponential convergence properties for cellular neural networks have been reported in the literatures. See, for instance, [4¨C17] and references cited therein. In fact, continuous and discrete systems are very important in implementation and applications. It is well known that the theory of time scales has received a lot of attention which was introduced by Stefan Hilger in order to unify continuous and discrete analysis. Therefore, it is meaningful to study dynamic systems on time scales which can unify differential and difference systems see [18¨C28]. When , , (1.1) reduces to where . By using Mawhin¡¯s continuation theorem and Liapunov functions, the authors [6, 14] obtained the existence and stability of periodic solutions of (1.2), respectively. Furthermore, (1.1) also covers discrete system (for when ; see [15]) where , . In [15], the author firstly obtained the discrete-time analogue of (1.3) by the semidiscretization technique [29, 30], and then some sufficient conditions for the existence and global asymptotical stability of periodic solutions of (1.3) were established by using Mawhin¡¯s continuation theorem and
%U http://www.hindawi.com/journals/ddns/2012/386706/