Functional equation arising in behavioral sciences: solvability and collocation scheme in Hölder spaces

Abstract

We consider a generalization of a functional equation that models the learning process in various animal species. The equation can be considered nonlocal, as it is built with a convex combination of the unknown function evaluated at mixed arguments. This makes the equation contain two terms with vanishing delays. We prove the existence and uniqueness of the solution in the Hölder space which is a natural function space to consider. In the second part of the paper, we devise an efficient numerical collocation method used to find an approximation to the main problem. We prove the convergence of the scheme and, in passing, several properties of the linear interpolation operator acting on the Hölder space. Numerical simulations verify that the order of convergence of the method (measured in the supremum norm) is equal to the order of Hölder continuity.