Existence Of The Positive Solutions For A Class Of Third-order Three Point Differential Equations

The third-order ordinary differential equations boundary value problems often encountered in our real life, it applies in mathematics, physics, chemistry and many other scientific fields. It has been widely studied in recent years. The main research methods include:upper and lower solution methods, degree theory, Guo-Krasnoselskii, Leggett-Williams and other fixed-point theorems.This paper studies the following third-order three-point boundary value problems for the existence of its positive solutions. with a＞0,0＜η＜1,γ≥0,δ＞0.The main method is constructing the Green function of the related linear boundary value problems, analysising some important properties, and finally get one or three positive solution's existence by different fixed-point theorems.Chapter one describes the contents of the boundary value problem and related background, developments, and describes the main question studied in this article.The second chapter focuses on the relevant Green function, solving, proof of certification and its main character.By using the methods in functional analysis, we change the existence of positive solutions of PVBs into the existence of fixed points of some operators, and three get the existence of one positive solution by Guo-Krasnoselskii fixed-point theorem with certain restrictions in Chapter three.Chapter four get the existence of three positive solutions by Leggett-Williams fixed-point theorem with certain restrictions.It shows the overall framework and possible extension of this thesis in the Summary part.The main idea of this thesis is converting the existence of solutions for BVPs to the existence of fixed points of some abstract operators by using functional analysis methods. The key is constructing of the abstract operator and the difficulty is the problem becoming more complicated for the participa-tion of the second order derivative. It makes the analysis and the proof more diffcult.