Progressive Delay Differential Difference Equation And Neural Network Model For Qualitative Research

This Ph.D.Thesis is composed of five chapters. In the first chapter, we introduce the historical background of problems which will be investigated and the main works of this paper. In chapter 2, by using the method of comparison, we study the asymptotic behavior of a delay difference equation, and some sufficient conditions, by which every solution (bounded solution) of this equation asymptotically tends to a constant, are obtained. Chapter 3 mainly considers the asymptotic behavior of solutions for a class of nonautonomous neutral delay differential equations. Some results that solution of the equation convergent to constant are obtained by considering the corresponding ordinary differential equation. The corresponding discrete results are obtained too. Our results improve and generalize the some known results. The purpose of chapter 4 is to study the large-time dynamics of discrete-time neural networks with McCulloch-Pitts nonlinearity. We give the detailed analysis of the role of threshold in preventing delay-induced oscillations of the model. Finally in chapter 5, we consider the globally asymptotic stability of the dynamical threshold neuron model with delay. Necessary and sufficient conditions are obtained for the existence of a globally asymptotically stable equilibrium of the system. Our method is based on the Lyapunov functional and linearization system. The globally attractiveness of the corresponding discrete-time system is studied too.