Research On Some Problems Of Nonlinear Integro-Differential Equations

The present paper employs the cone theory,fixed point index theory,and Krasnoselskii fixed point theorem,the method of lower and upper solutions and so on,to investigate the existence of solutions,multiple solutions to several kinds of singular and semipositone boundary value problems(system) on finite intervals and on the half-line. Besides,we establish sufficient and necessary conditions for relative compactness of two abstract unbounded function groups.Finally we investigate the existence of solutions of initial value problem for nonlinear second order impulsive integro-differential equations of mixed type in Banach space.By deep study,we obtain some new interesting. results under weaker conditions.The thesis is divided into six chapters.In ChapterⅠ,we mainly introduce the background of nonlinear functional analysis and some concepts.In ChapterⅡ,we are concerned with the existence of solution,multiple solutions of one kind of singular boundary value problems on the half-line and establish two decision theorems of relative compactness for unbounded function groups in an infinite interval.In§2.2,the conditions for the existence of at least one solution and three solutions are established for a class of singular second-order three-point boundary value problem on the halfline, where the nonlinear term can take a negative value and includes the first order derivative.In§2.3,sufficient and necessary conditions are given for relative compactness of an abstract unbounded continuous function group and an abstract differentiable function group.Furthermore,the existence of unbounded solutions for a second-order boundary value problem in infinite interval are obtained.In ChapterⅢ,we discuss two kinds of singular three-point boundary value problems and a periodic boundary value problem by the first eigenvalue corresponding to the relevant linear operator. In§3.2,the existence criteria of positive solutions for a singular three-point boundary value problem is established.What we study in§3.3 is a three-point boundary value problem with a more general form which possesses the first derivative.§3.4 deals with the existence of nontrivial solutions to a periodic boundary value problem without any nonnegative assumption on nonlinearity,by using the topological degree.ChapterⅣfocuses on the study of two kinds of semipositone boundary value problems and one semipositone system.§4.2 establishes the existence of nontrivial solutions for a singular second-order m-point boundary value problem with a sign-changing nonlinear term which may be unbounded from below.§4.3 deals with the existence of nontrivial solutions for a singular higher-order boundary value problem with a sign-changing nonlinear term.In§4.4,the existence of positive solutions for a semipositone system of conjugate boundary value problems is established.ChapterⅤdeals with the existence of positive solutions to a class of singular systems.In ChapterⅥ,by employing a new comparison result and the M(o|¨)nch fixed point theorem,the existence of solutions of initial value problem for second order impulsive integro-differential equations of mixed type in Banach space is investigated.