Solvability To Boundary Value Problemsof Nonlinear Fourth-order Ordinary Differential Equations

Nonlinear differential equations with boundary value problems can be applied in a wide variety of fields, such as physics, mechanics, biology, aerospace technology and others. Therefore, the research of the problems has important theoretical and practical significance, which draws researchers’ wide attention. This paper investigates the solvability to two kinds of boundary value problems of nonlinear fourth-order ordinary differential equations. The full paper is composed of four chapters.Chapter one is the article’s preface. Firstly, the relevant symbols,definitions and theorems of this article are introduced. Moreover, we review the research background and present progress of boundary value problems of nonlinear fourth-order ordinary differential equations. Finally, the basic idea and structure of the paper are presented.In chapter two, the existence of positive solutions to integral boundary value problems of nonlinear fourth-order ordinary differential equations is investigated. By using the Krasnoselskii fixed point theorem in cones, under the nonlocal boundary value conditions, some sufficient conditions for the existence of one and two positive solutions are obtained for this class of problems under the superlinear and sublinear cases.In chapter three, we mainly discuss the existence of the solutions for another class of fourth-order mixed boundary value problem with the parameter in the nonlinear terms. When discussing parameter r>0 and r<0, some sufficient conditions for the existence of solutions are obtained for this problem. Compared with the study of fourth-order boundary value problems in the paper[31], the boundary value conditions studied in this chapter is more general.Chapter four is the conclusion and prospect. We summarize our research work and briefly propose further research in the future.