In this paper, the author considers the following nonlocal differential equation \[ -A((a\ast u^q)(1))u''(t)=\lambda B((b\ast u^p)(1))f(t,u(t)),~~ t\in (0,1), \] where \(\lambda>0\) is a parameter, \(p, q \in (1, +\infty)\), \(\ast\) denotes the finite convolution and \(A\), \(B \) are continuous functions. A model case is the equation \[ -A\Big(\int_0^1(u(r))^q dr\Big)u''(t)=\lambda B\Big(\int_0^1(u(r))^p dr\Big)f(t,u(t)),~~ t\in (0,1). \]
By using a nonstandard order cone of the form \[ \mathcal{K}:=\Big\{u\in \mathcal{C}([0,1]): u\ge 0, (a\ast u)(1)\ge C_0\|u\|, \min_{t\in [c,d]}u(t)\ge \eta_0\|u\|\Big\} \] for the constants \(0\le c<d\le 1,\) \(C_0\in (0,1],\) \(\eta_0\in (0,1],\) together with topological fixed point theory, the existence of a positive solution is studied for boundary value problems consisting of the above equations equipped with a variety of boundary conditions. The importance of the use of a nonstandard order cone lies in fact that some assumptions which are used in the literature are eliminated or weakened, with the most important being that the results are achieved without assuming, as usually, that the coefficient functions are strictly positive, but may change sign and, in particular, may be equal to zero on a set of positive measure.