For a continuous function A:R→R the existence of at least one positive solution of the nonlocal differential equation −A∫01(g∘u)(s)dsu′′(t)=λf(t,u(t)),t∈(0,1)is considered in the specific case in which g is a concave, strictly increasing continuous function. As a model case we consider −A∫01|u(s)|qdsu′′(t)=λf(t,u(t)),t∈(0,1),where 0<q<1. The results rely on the use of a nonstandard order cone together with an application of topological fixed point theory. Due to this the coefficient function A is neither required to be strictly positive nor monotone nor satisfy any particular growth condition. As a consequence of the theory developed for the preceding problem we also provide results for the nonlocal equation −A((b∗(g∘u))(1))u′′(t)=λf(t,u(t)),t∈(0,1)where t↦(b∗(g∘u))(t) is a finite convolution.