Boundary Value Problems Of Nonlinear Differential Equations

Along with science's and technology's development, various non-linear problem has aroused people's widespread interest day by day, and so the nonlinear analysis has become one important research directions in modern mathematics. The nonlinear functional analysis is an important branch in nonlinear analysis, because it can explain well various the natural phenomenon. The boundary value problem of nonlinear differential equation stems from the applied mathematics, the physics, the cybernetics and each kind of application discipline. It is one of most active domains of functional analysis studiesin at present. In this paper, we use the topological degree theory as well as the Leray-Schauder theory, to study several kinds of boundary value problems for nonlinear singular differential equation .The thesis is divided into three chapters according to contents.In Chapter 1, we use the topological degree theory to investigate the following singular fourth-order boundary value problem with a sign-changing nonlinear term in Banach spaces.where h(t) is allowed to be singular at t = 0 and/or t = 1.Moreover, f(t, x, y) : [0,1]×R~2â†'R is a sign-changing continuous function,we obtain the existenc(?) of nontrivial solution for singular boundary value problem (1.1.1).In Chapter 2, we study the existence of positive solutions of singular boundary value problems for second-order and fractional-order differential equation systems by Leray-Schauder Theorem. where 1 <α< 2,v > 0,α- v > 1,α_i∈C((0,1),R~+) (i=1,2),α_i(t) is allowed to be singular at t = 0 and/or t = 1,f : [0,1]×R×R~+â†'R~+ and g : [0,1]×R~+â†'R~+ are continuous,b : (0,1)â†'R is Lebesgue integrable and may have finitely many singularities in [0,1].In Chapter 3, This paper considered the existence of nontrivial solutions for a class of fourth-order integral boundary value problems with parameters under some suitable conditions concerning the first eigenvalue of the corresponding linear operator. The main tool of this paper is topological degree. The results presented here improve and generalize some known results.