Solutions To Boundary Value Problems For Diferential Equations On The Half-line

Two order boundary value problem on the half-line originate in the study ofradially symmetric solutions of nonlinear elliptic diferential equations and porousgas pressure model on the half-line. In recent years, boundary value problems fordiferential equations on the half-line often appear in a variety of theoretical andapplied problems, for example, there is a wide range of applications on unsteadyflow of gas through a semi-infinite porous media problem, the theory of drain flows,determining the electrical potential in an isolated neutral atom. Now people paymore and more attention to the existence of positive solution for diferential equa-tions on the half-line, and made many outstanding achievements(see[3-16,18-24]).Along with the problem study thoroughly, cone theory, fixed point theorem, themethod of upper and lower solutions, partial ordering method and the variationalmethod have become the tools to study the existence of positive solution for bound-ary value problems on the half-line.This paper deeply discusses the existence of solutions for several kinds of bound-ary value problems on the half-line mainly by using the cone theory, fixed pointtheorem, fixed point index theorem and the method of upper and lower solutions.The paper mainly contains four chapters:In chapter 1, we investigate the two-order boundary value problem on thehalf-line wheref∈C(J×J, J ), 0≤α< 1,η∈(0, +∞). We use fixed point theorem andfixed point index theorem to get one or two positive solutions of the boundary valueproblem.In chapter 2, we consider the boundary value problem with integral boundaryvalue condition which f contain derivative term0g(s)x(s)ds,wheref : 0sg(s)ds < 1. Here thenonlinear termf dependent on derivative term x′. It brings some difculties to ourproblems. We mainly use nonlinear alternative an a kind of three-solution theoremto get one and three positive solutions to the boundary value problem.In chapter 3, we consider the existence of positive solutions for the two-orderboundary value problem with derivative term in the followingwheref, g : R+×R+×Râ†'R+are continuous functions, m, We mainly use fixed point theorem and fixed point index theorem to get two positivesolutions of the boundary value problem.In chapter 4, we consider the existence of positive solutions for the two-ordermulti-point boundary value problem in the followingwheref :â†'Ris continuous function, By using the method of upper and loweri=1solutions and Leray-Schauder degree theory, we get the existence of one and three nontrivial solutions for the boundary value problem.