The Research Of Multiple Periodic Solutions For Second-order Neutral Functional Differential System With Variable Coefficients

In this thesis of master, we investigate the existence and multiplicity of periodicsolutions for second-order neutral functional di?erential system with variable coe?-cients, by employing Krasnoselskii fixed theorem and Leggett-Williams fixed theorem.The thesis is composed of three chapters listed as follows.In the first chapter, the historical background and recently research of functionaldi?erential equations are brie?y reviewed. Furthermore, we simply introduce the mainwork in this paper and give some preliminary knowledge for this thesis.In the second chapter, we research the existence and multiplicity of periodic so-lutions for a kind of second-order neutral functional di?erential systems with two pa-rameters and two variable coe?cients(?)(?) and ai,bi,gi,fi areω?periodic functions (i =1,2). Firstly, by transforming it into a equivalent periodic boundary value problem,we get Green function and the fixed operator. Using Krasnoselskii fixed theorem, wefind out two parallel lines:Γ1 andΓ2, which separates the parameter plane into threeparts: (?) such that the system have at least two periodic solutions when(?) and has no periodic solution when (?). Moreover, we can find acontinous curveΓ∈?3 such that the system has at least a periodic solution. Then,an example intended for demonstrating the e?ciency of our results is provided there.Lastly, we discuss how to extend those results to higher system.In the third chapter, we research a kind of second-order neutral functional differ-ential equation, which is the above system with only a independent variable(?)periodic solutions. As the secondchapter, by transforming it into a equivalent periodic boundary value problem, we geta fixed operator. Using Leggett-Williams fixed theorem, we obtain the conditions forthe existence of three periodic solutions.