Solutions To Several Classes Of Nonlinear Differential Equations And Integral Equations

In this paper,we discuss the existence and multiplicity of postive solutions or solutions for several classes of nonlinear differential and intergral equations in Banach space. The paper includes three chapters.In Chapter 1,two kinds of boundary value problems of second-order functional differential equations are considered:one is two-point boudary value problem for delay differential equations with a parameter,the other is three-point boundary value problem for functional differential equations with an advanced argument.In §1.2, some sufficient conditions for the existence and nonexistence of positive solutions of two-point boundary value problem for delay differential equations with a parameterwhere Τ > 0 and λ> 0, are obtained by using the Krasnosel'skii fixed-point theorem. In §1.3, we consider the three-point boundary value problem for second-order functional differential equations with an advanced argumentwhere η∈ (0, 1),α > 0, and 1 — αη > 0. By using the fixed-point index method,we established the existence of at least one or two positive solutions to problem(1.1.2).As an application,we also give an example to demonstrate our result.In Chapter 2, we mainly consider the existence of positive periodic solutions for multi-delay differential equationsandIt must be pointed out that problem of periodic solutions for differential equations and two-point boundary value problem for differential equations have the same ideas,which results that the methods of the two kinds of problems are samiliar. And in this chapter.we mainly employ the fixed point theorem of cone expansion and compression. In Chapter 3, we mainly consider a kind of nonlinear impulsive Fredholm integral equation in Banach spacex{t)=xo(t) + \ H(t,s,x(s))ds+ Y, ak(t)h(x(tk)), (3.1.1)Jt° to