The Study Of Neutral Operators And Periodic Solutions For Second-Order Neutral Differential Equations

Neutral functional differential equations were widely used in Biology,Mechanics,Economics,Medical Science and many other fields.In recent years,there are many articles in studying neutral functional differential equations by using topological degree theory,fixed point theorem in cones,Krasnoselskii fixed point theorem combined with the properties of neutral operators(A1x)(t)=x(t)-cx(t-?),(A2x)(t)=x(t)-?,(A3x)(t)=x(t)-c(t)x(t-?),(A4x)(t)=x(t)-cx(t-?(t)).However,most of the literatures focus on the case that |c|<1 or |c(t)|<1 for all t ? R.In this thesis,we study the properties of the following two kinds of neutral operators with variable coefficients and variable delays(Ax)(t):=x(t)-c(t)x(t-?(t)),(Ax)(t):?and the existence of positive solutions for two kinds of second-order neutral functional differential equations with these operators in periodic function space.By employing Krasnoselskii fixed-point theorem,the properties of Green's function of second-order linear differential equations and the properties of the corresponding neutral operators we obtained in papers,we get the existence results of positive periodic solutions for the two kinds of functions(described in Chapter 2 and Chapter 3 respectively).Our results complement and improve previous results in the literature.The structure of this thesis is as follows:In Chapter 1,we mainly introduce the research background and progress of neutral functional differential equations.And the working contents of this paper are determined by the current research situation.Moreover,we introduce the function space and some lemmas which used in this paper.In Chapter 2,we first study the properties of the neutral operator(Ax)(t)=x(t)-c(t)x(t-?(t))and discuss the new sufficient conditions for the existence of positive periodic solutions for the second-order function with this neutral operator in the case that |c(t)|<1 and |c(t)|>1,for all t ? R,it fills in the gap that only the case that|c(t)|<1,for all t E R,was discussed in the previous literature.In Chapter 3,we improve the form of the neutral operator A and study the properties of the neutral operator with many variable coefficients and variable delays?,we give some sufficient conditions for existence for the second-order function with this neutral operator by application of the properties of operator A.In Chapter 4,we present a summary of our discussion.