Existence And Multiplicity Of Positive Solutions For Fourth-order Two-point Boundary Value Problems

With the advancement of science and technology, nonlinear problems gradually attractedpeople’s attention. Whether in nuclear physics, astronomy, fluid mechanics, or in terms ofaerospace technology, bio-technology, nonlinear differential equations with boundary valueproblems are closely linked with them. Therefore, the boundary value problems for nonlineardifferential equations have a great value to research. This paper investigates the existence andmultiplicity of positive solutions for several kinds of fourth-order two-point boundary valueproblems.Full paper is divided into four chapters, the main contents are as follows:In chapter one, we introduce the symbols, definitions and theorems which are referenced inthis paper. Then reviewed the development and research on nonlinear four-order boundary valueproblems. Finally, the arrangements of the main contents of the paper are given.In chapter two, the existence and multiplicity of positive solutions for a class of fourth-ordertwo-point boundary value problems is investigated. Using the lower and upper solutions methodand the topological degree theory, under a weaker conditions on the nonlinear term, we obtainthe existence of a positive solution and multiple positive solutions for this class of problems.In chapter three, we discuss the existence of solutions for another class of fourth-order andtwo-point boundary value problem. By constructing a function, under a satisfy certain conditionson the nonlinear term, using the fixed point theorem, we obtain the sufficient condition for theexistence of solutions for the problem. Under the parameters satisfy certain conditions, based onthe above, we obtain a sufficient condition for the existence of non-negative solutions of theproblem.Chapter four is the conclusion and prospect. We summarize the full paper, and give somesuggestions for the following research.