Convergence Of Several Classes Of Functional Differenitial Equations And Difference Equations

In this dissertation, we describe convergence of several classes of non-linear func-tional diferential equations and their deference models, which includes the convergence of bounded solutions, existence and exponential convergence of the anti-periodic solutions and the almost periodic solutions. It is consists of seven chapters.As the introductions, in first chapter, the background and history of these functional diferential equation models are briefly addressed, and the foundation that the non-linear dy-namic models are considered as dynamic system is expressed sketchly, and some notations and definitions are given in this chapter. And the research content and research methods of this present dissertation are given.In the second chapter, from Bernfeld-Haddock conjecture (1976), we propose a class of very wide range of delay differential equations model, and pointed out its practical meaning. Under certain conditions (weaker than the local Lipschitz conditions), we analysis of the order-preserving of the system, as well as the relationship between the?-Set limits with a constant function. Then, by using monotonou technique, we obtain the convergence of this system. Our results are new and complement of previously known results. We also give a simple, reasonable and strong proof of Bernf eld-Haddock conjecture(1976).In the third chapter, motived by conjecture Haddock (1987), we have put forward more than the previous range of high-dimensional system or neutral equations. First of all, we studied the order-preserving of this system, and analysised the relationship on order of?-limit and entire orbict with a focus on constant functions. With this results, by using a number of sub-analysis, we have established a convergence of this system. Our results correct and improve the existing ones and also confirms the Haddock's conjecture.In the fourth chapter, motived by three-dimensional Bernf eld-Haddock Conjecture, we analyze the convergence of discrete dynamical systems. Generally speaking, we can get a better order property of contiuous system, but due to the lack of continuity and connectivity of?-limit set, and this brings some difficulties on study of discrete dynamical systems, in turn, it also shows that there is the need to study them. In fact, under some better order conditions, using the provision methods of Chapter 2, we obtain the convergence on bounded solutions of such discrete system. It also reveals the the discrete form of three-dimensional Bernf eld-Haddock conjecture (1976) is still correct. In addition, Our results extend and improve some previous ones in the literature.In the fifth chapter, using differential inequality and its related mathematical analysis techniques, the convergence behavior of the delayed high-order Hopfield neural networks (HHNNs) with time-varying coefficients and recurrent neural networks (RNNs) are consid-ered. Without assuming Lipschitz condition, some sufficient conditions are established to ensure that all solutions of the networks converge to to an equilibrium, which are new and complement of previously known results.In sixth chapter and seventh chapter, we consider the third-order nonlinear differential equations with a deviating argument. We obtain some sufficient conditions for the existence and exponential stability of the anti-periodic solutions and almost solutions, which are new and complement previously known results.