Some Questions On The Application Of Differential And Difference Equations' Theory

Differential and difference equations are mathematical models from practical problems. So it is important to analyze the property of the solution to an equation by the equation itself. Stability is the important part of a dynamical system; the question about periodic solutions and almost periodic solutions is the most remarkable one.In this paper, we discuss respectively the existence of positive periodic solution and almost periodic solution, and the stability of solution and the estimation of upper bound of delays to several kinds of equations. This paper consists of five main parts as follows:In chapter 1, we simply introduce the development of our investigation and the motivation of this paper.In chapter 2, we devote ourself to the study of almost periodic solution of shunting inhibitory cellular neural networks. We obtain a sufficient condition about the existence of almost periodic solution and all its solutions converge to the almost periodic solution. In the last, an example is given to illustrate that the conditions of our results are feasible.In chapter 3, we obtain some stability criteria for non-linear systems of ordinary differential equations by the methods involving the Lozinskii measures of matrices and derive a necessary and sufficient condition on the stability of the non-linear system.In chapter 4, by using Mawhin's Coincidence degree theory, we deal with the existence of a positive periodic solution to a genotype selection model and obtain a new sufficient condition for the existence of positive periodic solution of the difference model.In chapter 5, by constructing suitable Lyapunov function and some analysis techniques, we study a bi-directional associative memory networks with delays and provide an upper bound of delays for global asymptotic stability of the equilibrium. Such bound has important significance in both theory and application for the design of the globally stable networks.