In this paper, we provide sufficient conditions for the existence of nonnegative solutions of a nonlocal boundary value problem for a second-order ordinary differential equation. By applying Krasnoselskii's fixed-point theorem in a cone, first we prove the existence of solutions of an auxiliary BVP formulated by truncating the response function. Then the Arzela-Ascoli Theorem is used to take C¹ limits of sequences of such solutions.