Positive solutions for nonlinear m-point boundary value problems of dirichlet type via fixed-point index theory

AbstractLet a ϵ C[0,1], b ϵ C([0,1], (-∞, 0)). Let φ1(t) be the unique solution of the linear boundary value problem u″(t)+s(t)u′(t)+b(t)u(t)=0, tϵ(0,1),u(0)=0, u(1)=1. We study the multiplicity of positive solutions for the m-point boundary value problems of Dirichlet type u″+a(t)u′+b(t)u+g(t)f(u)=0,u(0)=0, u(1)−∑i=1m−2αiu(ξi)=0, where ξi ϵ (0,1) and αi ϵ (0, ∞), i ϵ {… , m−2), are given constants satisfying Σi=1m−1 αiφ1(ξi) < 1. The methods employed are fixed-point index theory

Nonexistence of positive solutions of nonlinear boundary value problems

We discuss the nonexistence of positive solutions for nonlinear boundary value problems. In particular, we discuss necessary restrictions on parameters in nonlocal problems in order that (strictly) positive solutions exist. We consider cases that can be written in an equivalent integral equation form which covers a wide range of problems. In contrast to previous work, we do not use concavity arguments, instead we use positivity properties of an associated linear operator which uses ideas related to the $u_0$-positive operators of Krasnosel'skii