Abstract
We investigate first the existence of periodic solution in general Cohen-Grossberg BAM neural networks with multiple time-varying delays by means of using degree theory. Then using the existence result of periodic solution and constructing a Lyapunov functional, we discuss global exponential stability of periodic solution for the above neural networks. Our result on global exponential stability of periodic solution is different from the existing results. In our result, the hypothesis for monotonicity ineqiality conditions in the works of Xia (2010) Chen and Cao (2007) on the behaved functions is removed and the assumption for boundedness in the works of Zhang et al. (2011) and Li et al. (2009) is also removed. We just require that the behaved functions satisfy sign conditions and activation functions are globally Lipschitz continuous.
Highlights
In 1983, Cohen and Grossberg 1 constructed a kind of simplified neural networks that are called Cohen-Grossberg neural networks CGNNs ; they have received increasing interesting due to their promising potential applications in many fields such as pattern recognition, parallel computing, associative memory, and combinatorial optimization
The hypotheses for monotonicity inequalities in 27, 28 on behaved functions are replaced with sign conditions and the assumption for boundedness in 29, 30 on activation functions is removed
We investigate first the existence of the periodic solution in general Cohen-Grossberg BAM neural networks with multiple time-varying delays by means of using degree theory
Summary
In 1983, Cohen and Grossberg 1 constructed a kind of simplified neural networks that are called Cohen-Grossberg neural networks CGNNs ; they have received increasing interesting due to their promising potential applications in many fields such as pattern recognition, parallel computing, associative memory, and combinatorial optimization. A few authors 17, 21–26 discussed global stability of Cohen-Grossberg BAM neural networks. A few authors discussed the existence and stability of periodic solution to Cohen-Grossberg BAM neural networks with delays 27–31. In 28 , the authors discussed the following Cohen-Grossberg-type BAM neural networks with time-varying delays: dxi t dt dyj t dt. By using the analysis method and inequality technique, some sufficient conditions were obtained to ensure the existence, uniqueness, global attractivity, and exponential stability of the periodic solution to this neural networks. When time scale T becomes R, the existence and global exponential stability of periodic solution are obtained in 29, 30 under the assumptions that activation functions satisfy global Lipschitz conditions and boundedness conditions and behaved functions satisfy some inequality conditions. For monotonicity inequalities in 27, 28 on behaved functions are replaced with sign conditions and the assumption for boundedness in 29, 30 on activation functions is removed
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