Types Of Singular Second Order Boundary Value Problem Is The Existence Of Solutions

Nonlinear functional analysis is an important branch of mathematics. It is recognized by the mathematical world and the natural science world, as it can explain various natural phenomenon well. Now people have obtained some new results through the researching about nonlinear functional analysis.Singular second-order boundary value problem is the important part of the nonlinear functional analysis. Singular boundary value problem has a profound practical background. It appeared when people investigated atmosphere convection,globe evolvemcnt and several hydrodynamics issues.In recent years, people begin investigating more popular singular boundary value problem, and many authors use Lcray-Schauder continuity rule, Leray-Schauder non-linear choice, Krasnoselskii's fixed point theorem to study the second-order differential equation boundary value problem. Adding the singular condition, combining some boundary value term such as Sturm-Liouville condition,symmetrical boundary value condition, we can get some results but not perfect. Basing on this situation, by the arouse of documents [1-31], the present paper adds singular condition on second-order differential equation, cither changingequation structure or boundary value condition, we obtain the existence and the multiplicity of positive solutions or symmetic positive solutions. All the conclusionis based on fixed point theorem of cone expansion and compression theory and Krasnoselskii's fixed point theorem in cone.The thesis is divided into three chapters according to contents:In Chapter 1, by using fixed point theorem of cone expansion and compressiontheory, the existence of positive solutions is established for singular secondorderboundary value problemwhereα,γ> 0;β,δ≥0,△=γβ+ (αγ+αδ, the non-linear item f(t, u), g(t, u)∈C[(0,1)×(0,∞), R~+] all has singularity at t = 1, t = 0, u = 0.In Chapter 2, we concern with the existence and the multiplicity of symmetric In Chapter 2, we concern with the existence and the multiplicity of symmetric positive solutions for second-order three-point boundary value problemwhere a, b : (0,1)�'[0,∞) is symmetric on (0,1), and they arc likely to be singular at t = 0, t＝1. f, g : [0,1]×[0,∞)�'[0,∞) is continuous, for all u∈[0,∞),f(t,u) and g(t,u) is symmetric on [0,1]. If both f(t,u) and g(t, u) is suplinear (sublinear), we obtain at least one symmetic positive solution: If f(t,u.) and g(t, u) has the different character of suplinear and sublinear. we obtain at, least two symmetic positive solutions.Basing on Chapter 2, by the arouse of documents [3,4,6,14], we gained the following Chapter 3.In Chapter 3, we mainly concern with the existence and the multiplicity of positive solutions for second-order two-point boundary value problemwhere a∈C((0,1), [0,∞)) is singular at t = 0, t＝1. f : [0,1]×[0.∞)�'[0,∞) is continuous, andα,γ> 0;β,δ≥0,△=γβ+αγ+αδ. We give it some judge rule and obtain new results.