Periodic Solutions Of Several Types Of Functional Equations

This dissertation studies a type of predator-prey systems, one-order and second-order nonlinear differential systems, one-order and second-order neutral functional differential equations respectively. The results on the existence of one periodic solution or several periodic solutions are established. The dissertation is divided into six chapters. Main contents are as follows:In chapter one, we introduce a survey to the development of periodic solutions for functional differential equations. Main results in this dissertation are summarized.Chapter two is about a class of delayed predator-prey systems with stocking and Michaelis-Menten functional response. Employing Mawhin's continuation theorem of coincidence degree theory and some analysis techniques, we obtain the sufficient conditions for the existence of one positive periodic solution. The system we study in this chapter is genenralized, containing several kinds of specialized predator-prey systems. Our results are applied to the existence of positive periodic solutions for many classes of specialized systems.In chapter three, we firstly introduce the development of the Schaefer fixed-point theorem, including three steps: Schaefer fixed-point theorem(1955), Schaefer fixed-point theorem improved by Burton and Kirk(1998) and Schaefer fixed-point theorem generalized by Liu and Li(2006). Then, by the generalized Schaefer fixed-point theorems, we study one-order neutral functional differential equations (or systems) and establish the results of the exstence of one periodic solution. To our knowledge, this is the first work for the existence of periodic solutions for neutral systems by generalized Schaefer fixed point theorem of seperate contraction mapping.In chapter four, two classes of functional differential systems depending on parameters are discussed. Applying Deimling fixed point theorem and some analysis techniques, we prove that the number of positive periodic solutions depends on the value of the parameters and the asymptotical behavior of the nonlinear terms. First, one class of nonlinear functional differential systems with feedback contral is studied. The existence of two positive periodic solutions and one positive periodic solution is obtained. Then, one class of second-order semilinear differential systems with two parameters is considered. We show that there exists a continuous curve such that the system has at least one positive periodic solution for any...