Existence And Global Structure Of Positive Solutions To Nonlocal Boundary Value Problems

This thesis mainly investigates the existence of positive solutions to multi-point boundary value problems and the eigenvalue problems on time scales for singular dif-ferential equations,and the global structure of positive solutions for nonlocal boundary value problems by utilizing the theory of topological degrees of nonlinear function analysis.There are six chapters.Chapter 1 presents the research background and main results of this thesis.Chapter 2 is about preliminaries of this thesis.Basic definitions,formulas and theorems on time scales are stated.Chapter 3 is concerned with the following three-point boundary value problems for singular 2nth-order differential equations whereη∈(a,b),βi≥0,1<γi 0; The nonlinearity w∈(7) by using Leray-Schauder theorem and analysis technique. Furthermore, we discuss the case of nonexistence of positive solutions.Chapter 6 investigates the singular second-order differential equation with integral boundary conditionwhere∫01u(s)dA(s) is a Stieltjes integral, A is nondecreasing,λis a positive parame-ter, g∈C((0,1), (0,+∞)),f∈C([0,+∞), (0,+∞)), and g may be singular at t= 0 and/or t=1, f(u)> 0 for u> 0, and f∞=+∞. We obtain criteria of the existence, multiplicity and nonexistence of positive solutions to the equation (8) by utilizing the Sturm-Liouville eigenvalue theory, Leray-Schauder global continuation theorem, and fixed point index theory. Meanwhile, we give the asymptotic of the solutions and the corresponding interval of the parameterλ.