A one-dimensional Kirchhoff equation with generalized convolution coefficients

For $$q\ge 1$$ we consider the one-dimensional Kirchhoff-type problem $$\begin{aligned} -A((a*(u')^q)(1))u''(t)=\lambda f(t,u(t)),\quad t\in (0,1), \end{aligned}$$ where $$a*(u')^q$$ represents a finite convolution, subject to right-focal boundary conditions. Because the nonlocal coefficient is phrased in terms of convolution the results of this paper can accommodate all manner of nonlocal coefficients, such as a fractional derivative coefficient of Caputo type. A nonstandard order cone together with a specially tailored open set is used to deduce existence of at least one positive solution for this problem via topological fixed point theory.