The Existence Of Positive Solutions For Fourth-order Nonlinear Differential Equations With Boundary Value Problems

Nonlinear differential equations with boundary value problems in physics, applied mathematics, control theory, aerospace, biology and other fields have a wide range of applications. Therefore, the research on the existence of solutions for them is meaningful both in theory and in practice.Currently, the research on the existence of fixed points for integral operator, nonlinear functional analysis theory is commonly used by the cone theory and fixed point index theory. In many papers a variety of nonlinear problems and the existence of solution and multiple solutions of nonlinear second, third-order ordinary differential equations with boundary value have been widely studied and a lot of satisfactory results are obtained. However, it is more difficult to study the fourth-order nonlinear boundary value problems more than the second, third-order ones. Although many papers have been with many results of the fourth-order nonlinear boundary value problems, but a lot of results and proof of them will be further enriched. On this basis, this paper further study fourth-order boundary value problems, to enrich the theory and corresponding results of them.Full paper is divided into three chapters, the main contents are as follows:The first chapter gives a class of fourth-order boundary value problems of nonlinear differential equations: The Green function is given in two forms. Under the nonlinearity f satisfying the appropriate conditions, by using Krasnosellskii fixed point theorem, the conclusions of the existence of positive solution are obtained and proved. This section has been published in the "Heilongjiang August First Land Reclamation University".In the second chapter, by defining the reasonable and completely continuous operator for a special class of boundary value problems: the results about the existence of positive solutions of the boundary value problems are obtained by using the fixed point index theorem.In the third chapter, given the boundary value problems with variable coefficients: Under the nonlinearity f and variable coefficients K (t) satisfying the appropriate conditions, using respectively the fixed point index theorem and Krasnosellskii fixed point theorem, we prove the existence of positive solutions.Finally, we summarize the full paper.