Existence Of Positive Solutions For A Singular Nonlocal Boundary Value Problem

This paper mainly studies the existence, nonexistence of positive solutions of nonlinear boundary value problems subject to nonlocal boundary conditions of Riemann-Stieltjes type by utilizing the theory of topological degrees of nonlinear function analysis. There are three chapters.In chapter 1 we introduce the research background and give some important definitions and basic lemmas.In chapter 2 we consider the nonlinear boundary value problem with nonlocal boundary conditionswhere ?[u]=?01u(t)dA(t),?[u]=?01u(t)dB(t);A and B are functions of bounded variation; a> 0, b> 0; the nonlinearity/:[0,1]×[0,+?)?R is continuous with derivative dependence and is allowed to change sign.We obtain at least two positive solutions by using a fixed point theorem in double cones. And by using Leggett-Williams' theorem, we prove that there exist at least three positive solutions.In chapter 3 we concern the following boundary value problemwhere ?[u]=?01u(t)dA(t),?[u]=?01u(t)dB(t);A and B are functions of bounded variation; a> 0, b> 0; the nonlinearity f:[0,1]×(0,+?)×(-?,+?)?R is continuous with derivative dependence and is allowed to change sign.In the first section, we consider the singular semi-positone boundary value problem, that is the case f(t,u,y)=F(t,u,y)-?(t) where F(t,u,y)?C([0,1]×(0,+?)×(-?,+?),(0,+?)).Using fixed point index theory, we present multiplicity results for the singular semi-positone nonlocal boundary value problems with derivative dependence.In section two, we discuss the singular boundary value problem with sign-changing nonlinearity.Now the nonlinearity f with derivative dependence may be singular in its sec-ond variable and it is allowed to change sign. We establish the results for the nonex-istence and the existence of positive solutions for second order singular nonlocal boundary value problems when the nonlinear term f(t,u, y) satisfies the conditions above as well as our assumptions in the text.