Positive solutions for nonhomogeneous m-point boundary value problems

Let 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, where εi ε (0, 1) and αi ε (0, ∞) (for i ε {1, … , m − 2}) are given constants satisfying . 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*.