Research On Qualitative Properties Of Solutions For Several Differential Equations

As the focus of today's academic research, the related problems of stochastic differ-ential equations attract lots of scholars'attention and research. In recent decades, SDE plays an important role in physics,chemistry,biology,economic and finance,control theory,Aerospace engineering etc. So the research of SDE with stochastic disturbance has more important realistic significance. In this paper, we mainly considered the sta-bility of a class of special SDE-stochastic neural networks. In addition, in the field of chemistry,biology,meteorology,medical science and economic, the existence of positive periodic solutions for neutral functional differential equation and periodic boundary value problems of differential equations also have widely applications. Therefore the existence of positive periodic solutions and boundary value problems have important significance in both theoretical research and practical applications.This thesis is composed of four chapters:In chapter 1, we give background of the research and introduce the preliminary knowledge which is necessary in the thesis.In chapter 2, several criteria on theΨγstability in the mean square of neutral stochas-tic neural networks with multiple delays are obtained by using Lyapunov functional sta-bility theory and the linear matrix inequality (LMI) approach.In chapter 3, Under the suitable conditions and using Leggett-Williams multiple fixed point theorem, Green's Function and analysis techniques, we give sufficient conditions of the existence of multiple positive periodic solutions of one-order and second-order neutral functional differential equations with infinite delays.In chapter 4, by applying abstract fixed-point theorems in cone, one and multiple positive solutions of periodic boundary value problems for nonlinear second-order differ-ential systems are obtained.