Qualities Of Periodic Solutions For Differential Equations On Time Scales

In this paper, we study qualities of periodic solutions for three types differential equations on time scales which includes:a ratio-dependent predator-prey system of two species with variant time delays; cellular neural networks with time delays:nonlinear impulsive partial differential equations with time delays. By using the continuation theorem of coincidence degree theory we study respectively the existence of periodic solutions for a ratio-dependent predator-prey system of two species with variant time delays and cellular neural networks with time delays on time scales. Constructing Lyapunov functional, we study the exponential stability of periodic solution for cellular neural networks with time delays on time scales. Meanwhile, through upper and lower solutions,we investigate the existence of periodic solutions for nonlinear impulsive partial differential equations with time delays on the real number.This paper is divided into four chapters. In the first chapter, we introduce the concept of time scale, and the background of predator-prey system, cellular neural networks and nonlinear impulsive partial differential equations. We also give some preliminary including some definitions and lemmas. In the next sections the existences of periodic solutions for a ratio-dependent predator-prey system of two species with variant time delays and cellular neural networks with time delays on time scales have been studied respectively by the continuation theorem of coincidence degree theory proposed by Gains and Mawhin; and sufficient conditions for the exponential stability of periodic solution for cellular neural networks with time delays on time scales has been obtained by the method of Lyapunov functional; then an example is given to illustrate theories are effective. Also, we get the existence of periodic solutions for nonlinear impulsive partial differential equations with time delays on the real number through upper and lower solutions.At the end of the paper, we summarize the innovations, propose the direction of future research work and list the related literatures.