Periodic Solutions And Permanence For Some Classes Of Periodic Systems

Affected by the natural or contrived factors, large numbers of complex periodic models exist in many fields, which are generally described by periodic functional differential equations, periodic impulsive differential equations, periodic difference equations and so on. This dissertation consists of six chapters and some classes of periodic systems are investigated, which are extensively adapted to many fields such as economics and ecology. The existence of periodic solutions and permanence for these systems are obtained and our results substantially extend or improve some important existing ones.In the first chapter, we introduce researching situations concerning existence of periodic solutions for some functional differential equations, impulsive differential equations, difference equations and biological models. Our main work in this dissertation and the background is also generally presented.In Chapter 2, an n-dimension periodic functional differential system with a parameter is concerned. By using the eigenvalue problems of completely continuous operators and theory of α-concave or -α-convex operators, we prove the existence of the positive periodic solution for this system. Moreover, the results on the uniqueness of positive periodic solutions of its special case are obtained. As an application, sufficient conditions for the existence of positive periodic solutions of several population models are established. Our conclusions extend some existing ones.In the third chapter, by employing a fixed-point theorem on cones, we establish some criteria for existence of positive periodic solutions of a class of high-dimensional periodic functional differential equations with impulses, which improve and extend some existing results. Finally, we apply main results obtained to several biomathematical models and new results which have great potential in application are obtained.A coupled delay difference system depending on two parameters is considered in the fourth chapter. By using Krasnoselskii's fixed point index theroem, we investigate the nonexistence, existence and multiplicity of positive periodic solutions of the system. Some satisfactory results are obtained.In Chapter 5, by using Mawhin's continuation theorem, we study the existence of positive periodic solutions for a difference model, which can be seen as discrete analogue of a continuous periodic stage-structure biological model with diffusion...