The Existence Of Positive Solutions Of Nonlinear Differential Equations

Nonlinear functional analysis is an important branch of mathematics, andit can explain several kinds of natural phenomena, so more and more mathe-maticians are devoting their time to it. The boundary value problems (BVP)for nonlinear differential equations arise in a variety of areas of applied mathe-matics, physics and variational problems of control theory, it is at present oneof the most active fields in analysis mathematics. Among them, muliti-pointBVP come from a lot of branches of applied mathematics and physics, and it isvery meaningful in both practical and theoretical aspects. The present paperemploys the cone theory, fixed point theory, topological degree theory and soon, to investigate the existence of positive solutions to BVP of some kinds ofnonlinear differential equations, and have obtained some new results.The thesis is divided into three chapters according to contents.In chapter 1, we use the Leggett-Williams fixed point theorem, we showthe existence of at least three solutions for a class of second order boundaryvalue problems on the half-line as followingIn Chapter 2, by employing the well-known fixed point index theorem, weshow the existence of symmetric positive solution for a singular second-orderthree-point boundary value problem with sign-changing nonlinear terms-u″{t)= a{t)f{u{t)) + b{t)q(t), 0 < t < 1,u(0)-u(1)=0, u′(0)-u′(1) = u(1/2),where a : (0,1)→[0,∞) is continuous, and a(t) is symmetric on (0,1) andmay be singular at t = 0 and t = 1; f : [0,∞)→[0,∞) is continuous; q : (0,1)→(-∞, +∞) is symmetric and continuous, q(t) may be singular att = 0 and t = 1; b(t) is symmetric and continuous.In Chapter 3, we are concerned with the two-point boundary value prob-lems for systems of second order ordinary differential equations of the form:where f(t,x,y)∈C((0,∞)×[0,∞)×[0,∞), [0,∞)) and g(t,x, y)∈C((0,∞)×[0,∞)×[0,∞),[0,∞)); k > 0;λ> 0. Under some conditions, we show theexistence of positive solutions of the above problem.