With The ¦Ñ-laplacian Operator Of Multi-point Boundary Value Problems

The study of boundary value problems for nonlinear ordinary differential equations has practical significance and has interested people for a long time and the research in the field is still very active. The study of multipiont boundary value problems for linear second-order ordinary differential equation was initiated by Il'in and Moiseev. Recently the existence and multiplicity solutions of multipoint boundary value problems for nonlinear ordinary differential equations attracts close attention, in which the nonlinear function is endowed with the conditions of some different kinds in much literature. Among them, the nonlinear multipoint boundary value problems with one-dimensional p-Laplacian is now one of the most active research field. Which arise in the study of non-Newtonian fluid theory and the turbulent flow of a gas in a porous medium, subsequently, which has wide application in a variety of different areas of applied mathematics and physics.The present paper mainly investigates existence of solution, multiplicity for some multipoint boundary value problems of nonlinear ordinary differential equations with one-dimensional p-Laplacian by using topological degree, cone theory and monotone iterative technique. It's made up of three chapters and the main contents are as follows:Chapter 2 investigate a multipoint boundary value problem of differential equation with a p-Laplacian operatorwhereΦ_p(s)=|s|^p-2s,p＞1,Φ_q=(Φ_p)^-1,1/p+-/q=1,1≤k≤s≤m-2, 0＜ξ₁＜ξ₂＜…＜ξ_m-2＜1. We obtain a sufficient condition for the existence of three positive solutions, by applying Avery-Peterson fixed point theorm.Chapter 3 by applying Krasnosel'skii fixed point theorem and fixed-point index theorem, we study one-dimensional multipoint p-Laplacian boundary value problem whereΦ_p(s)=|s|^P-2s,p＞1,Φ_q=(Φ_p)^-1,1/p+1/q=1,1≤k≤m-20＜ξ₁＜ξ₂＜…＜ξ_m-2＜1,a_i,b_i,a,f satisfying(H₂)f(t,u,v)∈C([0,1]×[0,∞)×R→[0,∞)),a(t): (0,1)→[0,∞) is easurable, and a(t) is not identically zero on any compact subinterval of (0,1).Furthermore a(t) satisfies 0＜(?)＜+∞.We obtain the existence theorem of positive solutions for the problem, meanwhile, by applying a monotone iterative method , obtain not only the existence of positive solutions for the problem, but also establish iterative schemes for approximating the solutions.In Chapter 4, we study the following multipoint boundary value problem with one-dimensional p -Laplacian at resonanceWhenΦ_p(s):|s|^p-2s,p＞1,e(t)∈L¹[0,1].α_i∈R,η_i∈(0,1),α_i＞0,(?)=1i=1,2,…,m-2,0＜η₁＜η₂＜…η_m-2＜1,f:[0,1]×R²â†’R, and satisfying the Caratheodory conditions. Some sufficient conditions for the existence of at least one solution to the problem are obtained, by using an extension of Mawhin's continuation theorem due to Ge.