Large-time Behavior Of Solutions For Two Classes Of Systems Of Delay Difference Equations

Difference equation is one of the important constitute parts of mathematical study and is of great theoretical significance and practical value. Recently, with the rapid development of computer, the theory of difference equation has been extensively applied to many important problems such as numerical analysis, modern control, social economic, and so on. The investigation of difference equation is to discuss its large-time behavior of the solutions, including periodicity, oscillation, attractivity and stability, etc. In this thesis, we propose two classes of systems of delay difference equations as two new neural networks models, which are based on the well-known general Hopfield neural network model. This thesis is mainly concerned with the large-time behavior of these difference systems.In the first chapter, the historical background, the present searching situation of the research problems and the main works of this paper are described.In the second chapter, this thesis proposes a discrete-time difference neural network model of m neurons with delayed feedback, and introduces a class of non-linear discontinuous signal function which display obvious meaning. By introducing an auxiliary system and employing semicycle analysis as a powerful tool, it is shown that every solution of such a system is either truncated periodic or unbounded.In the third chapter, we study a nonlinear system of delay difference equa-tions, which describes the dynamics of a network of 2m neurons with no internal decays and excitatory-inhibitory delayed feedback. It is shown that every solution of such a system is oscillatory. Moreover, by employing semicycle analysis as a tool, it is shown that every solution of such a system is either truncated periodic or unbounded.