Periodic Solutions Of Multi-delayed Neutral Functional Differential Equations

In this thesis, by applying the method of the nonlinear functional analysis, we respectively discuss first-order and second-order multi-delayed neutral functional differential equations, and obtain the conclusions of the existence and nonexistence of the positive w-periodic solutions. This paper is composed of third chapters. The main contents are as follows:In the first chapter, section I introduces the background knowledge of the func-tional equations, and mainly talked about the introduction of functional differential equations, its research significance, its main research methods and its present re-search trends. Sections II and III introduced the main work of this article and the related prior knowledge. The final section lists the related symbols.In the second chapter, because the functional differential equations exists the neutral project, by using the classic methods, we study the existence of the periodic solution of the functional differential equations, it has the certain difficulty. There-fore, we firstly construct an appropriate bounded linear operator A, and transform the existence of periodic solutions of the functional differential equations into the existence of the fixed point of the operator. Then, by applying the fixed-point theorem on the cone with integral operator, we prove the existence of one, two, or none periodic solutions for the first-order multi-delayed neutral functional differen-tial equations. Compared with the current available literature, our results are more general, and which can be directly applied to a number of biological models.In the third chapter, by applying the fixed-point theorem on the cone with polynomial operator, In the section IV, we discuss the second-order multi-delayed neutral functional differential equations, the conclusions about the existence of one, two, or none periodic solutions are obtained. Compared with the ordinary differential equation, the functional differential equations considers the historical factor, which affect the system, Therefore, it more accurately reflect the actual problem. As far as we know, the article about the periodic-solution existence of the multi-delayed neutral functional differential equation is still few, our results are relatively new.Finally, we point out that the research in this dissertation have some places, which must be perfect, for example, we can consider two kinds of the equations with three periodic solutions. Moreover, in practical problems, most of the models exists the form of the system, then whether the systems have the periodic solutions, etc.