Functional equation arising in behavioral sciences: solvability and collocation scheme in Hölder spaces
Josefa Caballero, Hanna Okrasińska-Płociniczak, Łukasz Płociniczak, Kishin Sadarangani
TL;DR
This paper addresses the solvability of a generalized nonlocal functional equation arising in learning dynamics, f(t)=\varphi(t)f(\varphi_1(t))+(1-\varphi(t))f(\varphi_2(t)), with boundary conditions f(0)=0 and f(1)=1, by working in Hölder spaces $H^\gamma[0,1]$ and establishing a contraction framework. It proves existence and uniqueness of the solution via a fixed-point argument under a precise smallness condition on the coefficient maps, and develops a collapse-free collocation scheme on a uniform grid that converges with order tied to the Hölder exponent. The analysis provides sharp bounds for the linear projection $P_h$ and shows the collocation method achieves an error bound $\|f-f_h\|_\infty = O(h^\gamma)$ (and $O(h^{1+\gamma})$ for smoother $f$ in $H_0^{1,\gamma}$). Numerically, the method is validated on Hölder test functions, with observed convergence rates matching theory, offering a robust tool for nonlocal learning models in behavioral sciences.
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.