Periodic Solution And Almost Periodic Solution Of Several Kinds Of Functional Differential Equations

In this dissertation, the existences and uniqueness of periodic solution and almost periodic solutions for several kinds of functional differential equations are investigated. Some sufficient condition to the existence and uniqueness of almost periodic solutions of neutral functional differential equations are obtained by utilizing exponential dichotomy, matrix measure and fix point theorem, which generalized several results in corresponding reference. Meantime, it provides some new condition to certify the existence of periodic solutions of discrete-time Lotka-Volterra and bi-directional associative memory neural networks models.This dissertation is organized by five parts: In Chapter 1, we simply introduce the development of periodic solution and almost periodic solutions and the motivation of this paper; In Chapter2, by using space, matrix measure and fixed theorem, the almost periodic solution of neutral functional differential equation with infinite delay are obtained. Especially, we derive the sufficient conditions to the existence of the unique and uniformly stable almost periodic solution, which generalize several results in reference [15]; In Chapter3, the similar results about that one with finite delay by using exponential dichotomy and fixed theorem are obtained; In Chapter4, we investigate the positive periodic solution of the discrete-time Predator-Prey system with diffusion and delay under some identical sufficient conditions for this discrete model. We derived the existence positive periodic solution by using the continuation theorem of coincidence degree theory. This shows that the discrete analogue replicates some of the dynamical characteristics of the continuous one. In Chapter5, We formulate discrete-time analogues of bi-directional associative memory models with delays. The discrete-time system can replicate many dynamical characteristics of their continuous counterparts under no restriction on the discretization step-size. Such as the existence and exponential stability to the unique equilibrium, the existence and exponential stability of the periodic oscillatory solution. All these results are obtained by constructing suitable Lyapunov-type function and employing some analysis.