Existence Of Positive Solutions For Boundary Value Problems With P-Laplacian

Boundary value problem is an active and fruitful field in the theory of nonlinear science, and it was widely used in physics, biology, medical science, astronomy, economics, and so on. This theory can be applied to practice only the properties of the solutions, especially existences of positive solutions are known. So, the existences of positive solutions for boundary value problems become new hot issues. With the development of the nonlinear analysis and practical problems, there are many new directions, such as singular boundary value problems, multi-point boundary value problems, boundary value problems with p-Laplacian, etc..In the real world, many models such as species model, Volterra-Lotka predator-prey model are discrete because of the restrictions for data acquisition, research contents and so on. The theory of dynamic equations on time scales provides a unified theory for continuous and discrete equations or systems. Moreover, it enriches the theory of dynamic systems due to the generality and complexity, it explains the discrepancies which occur in parallel statements in continuous and discrete cases and other more complicated cases, and helps with better understanding them.In this paper, we consider some class of boundary value problems of dynamic equations with p-Laplacian, and give some new existence theorems.This paper consists of four chapters and covers the existence of positive solutions for boundary value problems of dynamic equations with p-Laplacian, existence of positive solution for boundary value problem of fractional differential equations, eventually positive and bounded solutions of even-order neutral difference equations with continuous arguments, oscillation criteria for second-order nonlinear functional differential equations.In chapter 1, we introduce some background of problems and the main works of this paper. Also, we give some preliminary knowledge which is needed in this thesis.In chapter 2, we study a class of multi-point boundary value problems of dynamic equations with p-Laplacian on time scales. In the first section, by using Guo-Krasnosel'skii fixed point theorem, we establish some sufficient conditions for the existence of at least one positive solution of the above problems. In the second section, we use a new fixed point theorem due to Avery (Avery-Henderson fixed point theorem), and obtain some sufficient conditions for the existence of at least two positive solutions. In the third section, we give some existence of at least three positive solutions by using the Leggett-Williams fixed point theorem.In chapter 3, we consider Sturm-Liouville-like boundary value problems of dynamic equations with p-Laplacian on time scales. We derive some existence of positive solutions by using Guo-Krasnosel'skii's fixed point theorem, Avery-Henderson fixed point theorem, and Leggett-Williams fixed point theorem,respectively. Our results are new ones even for boundary valueproblems of difference equations.In the first section of chapter 4, by using the properties of Green function and Guo-Krasnosel'skii fixed point theorem, we discuss the eigenvalue intervals of boundary value problem of fractional differential equation and get some sufficient conditions for the existence of at least one positive solution for the boundary value problem. In the second section, we consider the even-order neutral difference equations with continuous arguments. We use Lebesgue'dominated convergence theorem and obtain a necessary and sufficient condition for the existence of eventually positive and bounded solutions. In the third section, we obtain some oscillation criteria for a second-order nonlinear functional differential equation.