Positive Solutions For Two Classes Of Singular Boundary Value Problems

Over the last 30 years, singular boundary value problems of ordinary differential equations have been paid to close attention for their extensive applications in science and technology, and a large number of results have arisen in this field. The existence of positive solutions and multiple solutions is an important subject in the research of singular boundary value problems. Up to now, only a few papers consider problems with sign changing nonlinearities. In this paper, We use the method of upper and lower solutions, Schauder fixed point theorem and approximation theory to study two classes of singular second-order boundary value problems with sign changing nonlinearities and provide new existence results of positive solutions.This paper is composed of three chapters.In the first chapter, we narrated the applied background of singular boundary value problems which motivate our study.In the second chapter, a new existence theorem is established for the singular second-order mixed boundary value problemwhere the nonlinearities may change sign. Our main result is the following theorem.Theorem 2.1 Suppose the following conditions are satisfied:(H1) p ∈ C [0,1] ∩ C~1 (0,1) with p > 0 on [0,1].(H2) f : [0,1) × (0, ∞) × R→ R is continuous and f is singular at u = 0 and t = 1.(H3) (?)L > 0 such that for any compact set l (?) [0,1) there exists ε_l > 0 withf (t, u,v) > L for all (t, u, v) ∈ l × (0, ε_l] × R. (H4) for any δ > 0 there are two functions qδ and ψδ such that...