Singular One-dimension P-Laplace Boundary Value Problems For The Second-order Difference Systems

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 theory for single and multiple positive solutions are presented to singular discrete boundary value problem whereφ(s)=|s|^p-2s, p>1. The singularity may appear at (x,y)=(0,0). The existence result is obtained via fixed point theorem in cones and Leray-Schauder alternative.The present work is a direct extension of some results in [15,16] for the singular problem, namely, f_k(i, x, y) is singular at (x, y)=(0, 0). Our technique relies essentially on Leray-Schauder alternative theorem in [14] and fixed point theorem in cones in [5,9], which we believe is well adapted to this type of problem, when p=2 has been proved. We extented the results of p=2 to p≠2 using Leray-Schauder alternative theorem in [14] and fixed point theorem in cones in[5,9].This thesis is composed of three parts. In the first part, we introduces the historical background of problems which will be investigated and the main promlem studied in this paper,namely the singular boundary value problems of the second-order difference systems onedimension p-Laplace. There is a brief summary of results of this problem in other literatures, and introduction of some basic knowledge and propositions which will be needed in the proof of the theorem.In the second part, we establish existence principles for nonsingular discrete Dirichlet boundary value problem to the one-dimension p-Laplacian. In this part, we will using contents and methods from [5,9] and [14].The last part is about the main findings of the present study, we establish and prove the existence theorem for single and multiple positive solutions to singular Dirichlet boundary value problems for the second-order difference systems one-dimension p-Laplacian. The proof of these theoriems adopt the results of the part two.