Positive Solutions For A Kind Of Second Order Boundary Value Problems Of Ordinary Differential Equations(Systems)

The existence of solutions of boundary value problem for nonlinear ordinary differential equation, especially the existence of positive solutions, is an interesting and critical question in application and theory, playing an important role in the research field of ordinary differential equations. Particularly, the boundary value problem for second order ordinary differential equations is always an important subject in the research field of differential equations. It has extensive applied backgrounds and significant theory value in many research fields, such as physics, astronomy, biology and sociology, etc.In the past decades, with nonlinear functional analysis arising, mathematicians using the upper and lower solution method, topological degree method, fixed point theorem on cone to solve the boundary value problem could always get good results and have made some great developments and success. More and more mathematicians get many significant achievements(see [4-19]), also including some results about two-point boundary value problems for nonlinear ordinary differential equations(see [4-6,8-16]). Among them, by using the fixed point index theory, Zhanbing Bai, Weigao Ge generalized the Leggett-Williams fixed-point theorem in reference [6], and applied it into a class of nonlinear boundary value problem. Be inspired by [6], we study a kind of Robin boundary value problem for second order nonlinear ordinary differential equations in this paper.Nonlinear boundary value problem for systems of differential equations originates from fluid mechanics, the theory of boundary layer, nonlinear optics, etc, which is at present one of the most active fields in analysis mathematics. Since the structures they present have profound significance of physical backgrounds and practical mathematical models that coincide with the natural phenomenon, and there are lots of models in applied mathematics and engineering could be come down to the existence of positive solutions of boundary value problems for systems of differential equations. The investigation of positive solutions of boundary value problems for systems of differential equations has profound intrinsic value.There are abundant results for the study of boundary value problems for second order nonlinear ordinary differential equations. By contrast, it's much harder to study the boundary value problems for systems of second order nonlinear ordinary differential equations. There are fewer people to investigate, and the relevant references are slim pickings(see [20-32]). And in these references, there isn't first-order derivative explicitly in the nonlinear term.This paper is divided into three chapters, employing the theory and method of nonlinear functional analysis and fixed point theory on cones to investigate the existence of positive solutions of a class of second order two-point ordinary differential equations and ordinary differential systems.Chapter 1 introduces the aim and significance of the research for nonlinear boundary value problems, the general situation of domestic and foreign research, as well as some relevant preparatory knowledge including some definitions and basic theorems.Chapter 2 mainly discusses the existence of positive solutions of Robin boundary value problems of nonlinear second order ordinary differential equation. Under certain conditions, we gets the existence theorem of at least three positive solutions of the following boundary value problem.where f:[0,1]×[0,∞)×R→[0,∞) is continuous.In chapter 3, we construct a existence theorem of positive solutions of the follow-ing second-order ordinary differential systems with Robin boundary value problem by generalizing the theorem of chapter 2.where f,g:[0,1]×[0,∞)×R→[0,∞) is continuous.At present, the references about second order ordinary differential systems with two- point boundary value problems(see [20-29]) mostly consider the following forms:where, f∈C([0,1]×R⁺,R⁺),g∈C-([0,1]×R⁺,R⁺),f(x,0)≡0,g(x,0)≡0,wheref₁,f₂∈C(I×R⁺×R⁺,R⁺),I=[0,1],R⁺=[0,+∞).whereλ>0 is a parameter, f₁,f₂:R⁺×R⁺→R is continuous, R⁺ = [0,∞) Moreover, as we learn there are very few references studying the Robin boundary value problem with first-order derivative explicitly in the nonlinear term. In this chapter, we get a existence theorem of at least three positive solutions for BVP(B).