Functional equation arising in behavioral sciences: solvability and collocation scheme in H\"older spaces

TL;DR

This paper investigates a generalized nonlocal functional equation modeling animal learning processes, proving its solvability in H"older spaces and developing a convergent collocation numerical scheme.

Contribution

It establishes existence and uniqueness of solutions in H"older spaces and introduces a new collocation method with proven convergence for this class of equations.

Findings

Existence and uniqueness of solutions in H"older spaces.

Convergence of the collocation scheme with order matching H"older continuity.

Numerical simulations confirming theoretical convergence rates.

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 H\"older 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 H\"older space. Numerical simulations verify that the order of convergence of the method (measured in the supremum norm) is equal to the order of H\"older continuity.