Existence of Solutions of Nonlinear m-Point Boundary-Value Problems

We study the existence of positive solutions to the boundary-value problem u + a t f u = 0 t∈ 0 1 x 0 = m −2 i=1 bix ξi x 1 = m −2 i=1 aix ξi where ξi ∈ 0 1 with 0 < ξ1 < ξ2 < · · · < ξm−2 < 1 ai bi ∈ 0 ∞ with 0 < m−2 i=1 ai < 1, and m−2 i=1 bi < 1. We show the existence of at least one positive solution if f is either superlinear or sublinear by applying the fixed point theorem in cones.