Positive Solutions Of Second Order Non-local Boundary Value Problems

This Master Degree Thesis is mainly concerned with the following second ordernonlinear ordinary differential equationu'' + k2u +λb(t)f(u(t)) = 0.Under several difference nonlocal boundary conditions, we show the existence, no-existence and multiplicity of positive solutions of the differential equation by applyingfixed-point index theory in cones.In Chapter 1, we introduce the historical background and the recent developmentof nonlocal boundary value problems for ordinary differential equations. Also, themain results of this thesis are brie?y introduced.Chapter 2 is mainly concerned with the following nonlinear three-point boundaryvalue problemb(t) is allowed to have a singularity at t = 0 and/or t = 1. Applying fixed-pointindex theory in cones, we show the relationship between the asymptotic behaviors ofnonlinearity f (at zero and infinity) and the parameterλ, such that the problem has no,one and multiple positive solution(s), respectively. Also, a key condition of discussingthe existence of positive solutions for the three-point boundary value problem is putforward, which is stated as 0 <σsin kη< sin k.Chapter 3 mainly discuss the following nonlocal boundary problem with nonlin-ear boundary conditions By analyzing the growth of f and g(at zero and infinity)) and the open intervals ofthe parameterλ, we establish the existence, no-existence and multiplicity of positivesolutions for the problem.In further, Chapter 4 consider the following nonlocal boundary problem withnonlinear boundary conditionsBy a similar approach with the Chapter 3, we show the existence, nonexistence andmultiplicity of positive solutions for the problem.