Existence Of Positive Solutions For Several Kinds Of Boundary Value Problems Of Differential Equations On Time Scales

The theory of time scales was introduced by Hilger’s research about the con-sistency of difference and differential. It is applied widely in various aspects of mathematics and physics, especially in the aspects of computer and biochemistry. And, the boundary value problems of differential equations are closely related to the applied mathematics, theoretical physics, engineering controls and optimiza-tion theory, also have an important role in modern academic disciplines. In recent years, more and more people take an interest in the existence of positive solutions of boundary value problems on time scales.This paper mainly discusses the existence of solutions for several kinds of boundary value problems of nonlinear differential equations on time scales by using Leggett-Williams fixed point theorem, a fixed index theorem and Avery-Peterson fixed point theorem. There are three chapters in this paper.Chapter1investigates the existence of positive solutions of a three-point boundary value problem for second order dynamic equations: where T is a time scale. In [3], the author used Guo-Krasnosel’skii fixed point the-orem and Leggett-Williams fixed point theorem to discuss the existence of positive solutions of a second-order three-point boundary value problem on time scales. In [4], the author gave the existence results of a three-point boundary value problem. In [5], the author used twin fixed point theorem to study the existence of at least two positive solutions for a boundary value problem on time scales.However, the results of the above-mentioned literature [3]-[5] were the exis-tence of positive solutions with nonlinear terms not involving the derivative. We will face a lot of difficulties when discussing the existence of positive solutions with the nonlinear terms containing the derivative of the unknown function explicitly. The reason is that it is very difficult for us to control the nonlinear terms. Naturally, it is necessary for us to consider the existence of positive solutions to p-Laplacian boundary value problems when the nonlinear term is involved with the derivative explicitly. This chapter considers the problem and obtains at least three positive solutions by using Leggett-Williams fixed point theorem.Chapter2considers the following second-order boundary value problem with integral boundary conditions on time scales: Inspired by a three-order boundary value problem with integral boundary conditions in [16] and second-order three-point integral boundary value problem in [17], this chapter gets at least two positive solutions of the above second-order boundary value problem with integral boundary conditions by using fixed index theorem.Chapter3studies the existence of triple solutions for a second-order m point BVP of nonlinear p-Laplacian dynamic equations with derivative on time scales: We denote the p-Laplacian operator by<φp(u), i.e. φp(u)=|u|p-2. u,p>1.(φp)-1=φq,1/p+1/q=1,p>1. In [22], the author considered a m point B-VP with p-Laplacian and obtained at least one and two positive solutions by using the fixed point theory of cone. In [23], the author considered the existence of positive solutions to p-Laplacian boundary value problems when the nonlinear term was involved with the derivative explicitly and got at least three positive so- lutions by using Leggett-Williams fixed point theorem. This chapter considers the problem and obtains at least three positive solutions by using Avery-Peterson fixed point theorem.