Nonuniform Nonresonant Singular Dirichlet Boundary Value Problemsfor The One-Dimension P-Laplacian With Sign Changing Nonlinearity

Singular boundary value problems for nonlinear ordinary differential equations, which arise in a variety of areas such as mechanics, boundary layer theory, diffusion and reaction equations, biology, ect, become an impotant topic in ordinary equations fields.In this paper , Existence results are presented for the nonuniform nonresonant singular boundary value problemwhere φ(s) = |s|^p-2s,p > 1. The singularity may appear at u = 0, t = 0 or t = 1, and the function f may change sign. The existence of solutions is obtained via an upper and lower solutions method.The present work is a direct extension of some results in [ 12, 22] for the singular problem, i.e., f is singular at u = 0 and nonuniform nonresonance at the first eigenvalue. Our technique relies essentially on a method of upper and lower solutions in [18, 20] which we believe is well adapted to this type of problem, when / is allowed to change sign, and f(t, u, v) may be singular at u = 0, t =0 and t = 1, [3] obtained existence results under nonuniform nonresonance, when p = 2. We extented the results of p = 2 to p ≠ 2 using the upper and lower solutions method established in [3].This thesis is composed of two parts. In the first part, we introduce the historical background of problems which will be investigated and the main promlem studied in this paper,namely the Singular One-Dimension p-Laplacian Boundary Value Problems. There is a brief summarize of results of this problem in other literatures, and introduction of some basic knowledge and propositions that needed in the proof of the theorem.The second part is the main text, we establish the existence result for the singular nonlinear boundary value problem and prove it.The contents and methods from [3] and [12] are needed in this process, but the main basis is still the upper and lower solutions approach.