Existence And Multiplicity Of Positive Solutions Of Boundary Value Problems For Nonlinear Differential Equations

With the development of science and technology, an overgrowing interest in various nonlinearproblems has been aroused. Therefore, nonlinear analysis, aimed at studying nonlinear problemshas become an important area which is of profound theoretical significance and wide applica-tionin modern mathematics.Nonlinear differential equations with boundary value problems in the natural science and en-gineering technology have a wide range of applications, for example, fourth-order boundary valueproblems that is well-known characterizations of the mathematical model of the elastic beam state.This paper investigates the existence and multiplicity of solutions for several kinds of fourth-orderboundary value problems.Full paper is divided into four chapters, the main contents as follows:In chapter one, we introduce the symbols, definitions and lemmas which are used in this paper,and then introduce the development and research on the boundary value problems for nonlineardifferential equations. Finally, the arrangements of the main contents of the paper are given.In chapter two, the existence of solutions and positive solutions for a class of fourth-orderthree-point boundary value problems is investigated in this chapter. Employing the Green functionof a third-order three-point boundary value problem, the equivalent integral equation system isestablished, and some sufficient conditions of existence of solutions and positive solutions areobtained for this class of fourth-order three point boundary value problems.In chapter three, we discuss the existence and multiplicity of positive solutions for a class offourth-order boundary value problems. By constructing a convex cone and employing the fixedpoint theorem of cone expansion or compression type, sufficient conditions of the existence ofpositive solutions are obtained for this class of fourth-order boundary value problems, and then themultiplicity of the positive solutions is studied.In chapter four, conclusion and prospect as the end of this paper summarize the full text.