Positive Solutions For The Green's Function With A Variable Number Of Third-order Three-point Boundary Value Problem

Third-order differential equations arise from a variety of different areas of ap-plied mathematics and physics, e.g., in the deflection of a curved beam having a constant or varying cross section, a three-layer beam, electromagnetic waves or gravity driven flows and so on. Recently, the existence of positive solutions to some third-order three-point boundary value problems (BVPs for short) has received much attention from many authors. However, most of the work are achieved when the corresponding Green’s functions are nonnegative. There have been very few researches on the existence of positive solutions to boundary value problems when the Green’s functions are sign-changing. Therefore, no matter in theory or in real life, the study of this kind of problem has very important significance.In chapter2, by using the Guo-Krasnoselskii fixed-point theorem, some ex-istence criteria of at least two positive solutions are established for a three-point boundary value problem of third-order ordinary differential equation under the condition that the corresponding Green’s function is sign-changing.In chapter3, we discuss the existence of positive solution of third-order three-point BVP with sign-changing Green’s function Our main tool is the Guo-Krasnoselskii fixed point theorem.In chapter4, by using the Leggett-Williams fixed point theorem, we continue to study the BVP in chapter3and obtain the existence of three positive solutions to the problem, where Furthermore, for arbitrary positive integer m, existence results for at least2m-1positive solutions are obtained.In chapter5, by using monotone iteration method, we obtain the existence of monotone positive solution to the BVP studied by the third and fourth chapter, where η€[0, min{α+6/18-α,α+6/4(α+3)}]. Moreover, the initial value of the iterative sequence starts off with zero function.