Existence Of Positive Solutions For A Nonlinear Higher Order Three-Point Boundary Value Problems

This thesis is mainly concerned with the existence of positive solutions for a nonlinearhigher order three-point boundary value problems and a nonlinear higher order three-pointboundary value problem with one parameter. Some results are obtained by virtue of thefixed point index theory and the cone theory.It consists of three chapters.In Chapter 1, the historical background and development of multi-point boundary valueproblems of nonlinear higher order differential equations are addressed.The current researchand the main work of this paper are given as well.Chapter 2 deals with the existence of positive solutions for a nonlinear higher orderthree-point boundary value problems, which is discussed via turning into another two lowerorder boundary value problems. At the same time, the properties of the correspondingGreen's function is established by using the relationship between them. By applying thefixed point theorem on cone expansion-compression type, we obtain some sufficient con-ditions guaranteeing the existence of at least one positive solution and multiple positivesolutions of this problem followed by no existence of positive solutions in the consideringcone. The results obtained not only weaken the restrictions of the nonlinearity, but alsogeneralize and improve the corresponding results.In Chapter 3, we discuss the existence of positive solutions of a nonlinear higher orderdifferential equation three-point boundary value problem with one parameter. This chapteris inspired by the example of the previous chapter. One important lemma is given first, andthen the original problem can be turned into another equivalent problem. Meanwhile, a newoperator is defined and the Scha¨uder fixed point theorem is employed to establish sufficientconditions for the existence of positive solutions of the new boundary value problems. Fur-thermore, the existence of positive solutions of the boundary value problems of the equationwith one parameter is preliminary solved.At the end of both Chapter 2 and Chapter 3, some examples are presented to illustratethe validity of our results.