Positive Solutions Of Boundary Value Problems For Some Nonlinear Differential Equations

Nonlinear functional analysis as a research subject,is with the development of society and the progress of science and technology,to social science and engineering technology and natural science in the field of nonlinear problems arising in the background.Its appearance not only has the profound theoretical significance,but also has the widespread application prospect.In theory,it sets up a series of general theories and methods to deal with nonlinear problems.The main contents include topological degree theory,fixed point theory of cone tension and cone compression,critical point theory,monotone operator theory and partial order theory Methods and so on.Naturally,the results obtained through these theories can be widely used in many nonlinear problems,such as mathematics,control theory,optimization theory,dynamical system and economic mathematics.Therefore,the nonlinear functional analysis in solving nonlineax Integral equation,ordinary differential equation and partial differential equation problem has an irreplaceable role.In the nonlinear functional analysis,the boundary value problem is the most active and the most research significance and value of the field.In the boundary value problem of differential equation,the most practical one is positive solution,so the existence of positive solution of differential equation becomes the focus and core of most scholars.In the course of the study,scholars usually transform the problem into the problem of whether the fixed point of the integral operator exists on the cone.In the nonlinear functional analysis,the fixed point theorem and the topological degree theory are usually used to solve the existence of fixed points,such as Schauder fixed point theorem,Krasnosel’skii fixed point theorem and Leggett-Williams Fixed point theorem.In the past years,although there are many scholars on the two-point or even multi-point boundary value problem,in particular,the existence of its positive solution to the issue of in-depth study,the results are quite rich,but because some commonly used fixed point theorem there are many Conditional constraints,making the application of these theorems have some limitations.Based on the research of other scholars,this paper broadens some limitations and makes the conditions more general,which improves and popularizes the results of previous studies.In this paper,we study the existence of positive solutions and multiple solutions of non-linear differential equation（group）boundary value problems and the uniqueness of positive solutions by using a priori estimation method and the fixed point index theory on cone.This paper is divided into three chapters:In Chapter One,We study the existence and uniqueness of positive solutions for the second-order integral boundary value problemwhere f ∈ C（R,R）;a ≥ 0;6 ≥ 0;α and β are right continuous on[0,1),left continuous at t = 1,and nondecreasing on[0,1],with α（0）= β（0）= 0;∫₀¹u（τ）dα（τ）and ∫₀¹u（τ）dβ（τ）denote the Riemann-Stielties integrals of u with respect to α and β,respectively.This chapter improves and generalizes the original results under a wide range of conditions.In Chapter Two,we study the existence of positive solutions for the system of p-Laplacian boundary value problems where p>0,q>0,f ∈ C（[0,1]×R⁺2,R⁺）,g ∈ C（[0,1]×R⁺2,R⁺）.In Chapter Three,we study the existence,multiplicity and uniqueness of positive solutions for the fourth order p-Laplacian boundary value problemwhere p>0,f ∈ C（[0,1]× R⁺,R⁺,R⁺）.we establish a priori estimates by use of the Jensen integral inequality for the positive solutions.On this basis,we using the properties of concave and fixed point index theorem proved the main result of this chapter.