Positive solutions for nonhomogeneous m-point boundary value problems

AbstractLet a ϵ C[0,1], b ϵ C([0,1], (−∞,0]). Let d ϵ R and d > 0. Let φ1(t) be the unique solution of the linear boundary value problem u″(t)+a(t)u′(t)+b(t)u(t)=0, tϵ(0,1),u(0)=0, u(1)=1.We study the existence of positive solutions for the m-point boundary value problem u″+a(t)u′(t)+b(t)u+h(t)f(u)=0,u(0)=0, u(1)−∑i=1m−2αiu(Ei)=d, where ϵi ϵ (0, 1) and αi ϵ (0, ∞) (for i ϵ {1, … , m − 2}) are given constants satisfying ∑i=1m−2αix δ1(ϵ)<1. Under suitable conditions, we show that there exists a positive number d∗ such that the problem has at least one solution for 0 < d < d∗ and no solution for d > d∗