Singular Differential Equations Solutions Of Boundary Value Problems Existence

In this paper, we use the fixed point theorem of generalized cone expansion and compression, the generalized fixed point index theorem as well as Leray-Schauer nonlinear alternative to study the existence of the solutions for several kinds of boundary value problems for nonlinear singular differential equation.The thesis is divided into four chapters according to contents.In chapter 1, we chiefly narrate the background of the nonlinear analysis and innovation of this paper.In chapter 2, we discuss the following singular fourth-order Sturm-Liouville boundary value problem where f∈C([0,1]×[0,+∞), [0,+∞)), g∈C((0,1), [0,+∞)), and g(t)(?)0, g(t) is allowed to be singular at t= 0,1.αi,βi,γi,δi are nonnegative constants, and△i=αiγi+αiδi+βiγi＞0(i= 1,2). In this paper, we obtain the existence and nonexistence of positive solutions and the main results obtained improve and generalize the existing results.In chapter 3, we study the following singular fourth-order m-point boundary value problem where f∈C([0,1]×[0,+∞), [0,+∞)), g∈C((0,1), [0,+∞)), and g(t)(?)0, g(t) is allowed to be singular at t= 0,1.αj,βj,γj,δj(j= 1,2) are nonnegative constants,0＜ξ1＜ξ2＜…＜ξm-2＜1, a1i, a2i≥0, i= 1,2,…, m-2. We use the fixed point theorem of generalized cone expansion and compression to study the existence of positive solutions, whose method is different from some known results.In chapter 4, we use Leray-Schauer nonlinear alternative to consider the existence and uniqueness of nontrivial solution for the following singular fourth-order integral boundary value problem where f∈C([0,1] x (-∞,+∞) x (-∞,+∞), (-∞,+∞)), h∈C((0,1), [0,+∞)), and h(t)(?)0, h(t) is allowed to be singular at t= 0,1. g1, g2∈C([0,1], [0,+∞)),αi,βi,γi,δi(i= 1,2) are nonnegative constants,μis a positive parameter.