Solutions For Several Kinds Of Multi-point Boundary Value Problems Of Differential Equations

Multi-point boundary value problems originated in many different areas of ap-plied mathematics. In recent decades, a variety of nonlinear problems emerged in mathematics, physics, engineering and control theory, biology, economics and many other fields of science. When solving these nonlinear problems, there appeared many important analytical methods and theories, including:semi-order method, the up-per and lower solution method, fixed point theory, topological degree methods, the cone theory and bifurcation theory, which become the effective tools to solve lots of nonlinear problems emerging in science and technology.This paper mainly applied the fixed point theory, the cone theory and topolog-ical degree theory to solve some nonlinear differential equations. Since1980’s, the existence theory of differential equations and multi-point boundary value problems have been extensively studied (eg,[3-7],[9-48]). Based on which, this paper have further studied several classes of multi-point boundary value problems.Chapter1investigates the following multi-point boundary value problem with p-Laplace operator and impulseThe authors considered the existence of one positive solution for this problem by using Krasnosel’skii fixed point theorem in [7]. But as far as we know, there are few papers to investigate the existence of multiple solutions for this problem. This chapter considers this problem. Using the completely continuous operator and fixed point index theorem, we get the existence of multiple positive solutions for this multi-point boundary value problem.Using Banach fixed point theorem, the first part of Chapter2studies the fol-lowing first-order multi-point boundary value problemIn the second part of this chapter, we use the fixed point theorem of mixed monotone operators to study the existence and uniqueness of the following nonlinear first-order multi-point boundary value problem:The general form of the Green’s function for the above boundary value problem has been derived in [13]. Moreover, the author has given the sign of the Green’s function and the solution of the above linear problem. It is noted that the author concluded that the results obtained in the article is very helpful to solve the nonlinear problems. Thus, this chapter discusses the existence and uniqueness of solutions for the above nonlinear problem.Chapter3considers the following fractional differential equation D0+α(t)+f(t,u(t))=0,t∈(0,1), subject to the following two boundary conditions, respectively, u(0)=0, D0+βu(ζ), and u(0)=0, u(1)-m2u(ζ).Literatures [38,39] obtained some positive characters of the Green’s functions for above systems, respectively. Using fixed point theorem and mixed monotone method, they obtained some conditions for the the existence of positive solutions. However, there are few results for the existence of three or more positive solutions for the above problems. Therefore, this chapter uses Leggett-Williams fixed point theorem to study the existence of three non-negative solutions for the above two kinds of boundary value problems.