Positive Solution Of 2m Order Nonlinear Boundary Value And Eigenvalue Problems

Non-linear boundary value problems come from applied mathematics, physics, control theory and other applied sciences, and it is applied in the fields of nonlinear diffusion, gas dynamics, fluid dynamics and other subjects. Therefore, the research on existence and multiple solutions of nonlinear ordinary differential equation with boundary value is of significance both in theory and in practice.At present, the study of boundary value problems for ordinary differential equations are focused on problems on low-order ordinary differential equations with the two or multi-point boundary value, in recent years, due to the significance of higher order nonlinear differential problems with boundary value in practice and theory, existence of solution and multiple solutions of nonlinear higher order ordinary differential equation with boundary value have been widely studied in many papers and obtained a lot of satisfactory results. However, most authors study the problem under the conjugate boundary conditions or more simple boundary conditions, and papers constructed the research on high-order ordinary differential equations with more complex boundary conditions are not common, especially, the study on the boundary problems with the right-hand side boundary condition being higher-order derivative is more rare. This paper studied a class of 2m order differential equation boundary value problem with right-hand side boundary condition being higher order derivative, under different assumptions, we prove the existence of positive solutions under the nonsingular and singular conditions respectively, furthermore, we prove uniqueness of positive solutions . At the same time, the multiplicity of positive solutions and positive solutions of eigenvalues problems are also given, so we obtain the new research results concerning the existence of positive solutions for 2m-order boundary value problems.Full-paper is divided into four chapters, the main contents are as follows: In the first chapter, take literature [3] method as reference, the Green function method for a class of nonlinear differential equation with 2m order boundary is introduced and popularized,the estimate of upper and lower boundary and derivative of the Green function are established. Under the nonlinearity f satisfying the appropriate conditions, the conclusion that existence of positive solutions of the boundary problem is obtained, by using Krasnosellskii fixed-point theorem, which enriched further the literature results. This part has been published in the "Journal of Daqing Petroleum Institute."In chapter II, under the nonlinear terms satisfying the singuler conditions, by defining a reasonable completely continuous operator, using truncation method , existence of positive solutions of boundary value problem (1.1) is given by using Schauder fixed point theorem, and then the uniqueness of positive solutions is proved using reduction to absurdum method.In chapter III, as the nonlinearity f satisfies the appropriate conditions, multiple solu- tions of the boundary value problem in the first and second chapter are discussed by using Krasnosellskii fixed point theorem.The fourth chapter focused on the eigenvalue problem of even-order differential equation, through the cone fixed point theorem, we prove that the equation has at least one or two positive solution. Finally, we summarize the full paper.