Existence and uniqueness of solutions in the Lipschitz space of a functional equation and its application to the behavior of the paradise fish

TL;DR

This paper proves the existence and uniqueness of solutions for a functional equation in Lipschitz space and applies it to model the learning behavior of paradise fish, offering practical numerical approximation methods.

Contribution

It establishes theoretical results on solvability in Lipschitz space and demonstrates their application to a biological learning model with efficient approximation techniques.

Findings

Proved existence and uniqueness of solutions in Lipschitz space.

Applied results to a fish learning behavior model.

Developed efficient numerical approximation methods.

Abstract

In this paper, we examine the solvability of a functional equation in a Lipschitz space. As an application, we use our result to determine the existence and uniqueness of solutions to an equation describing a specific type of choice behavior model for the learning process of the paradise fish. Finally, we present some concrete examples where, using numerical techniques, we obtain approximations to the solution of the functional equation. As the straightforward Picard's iteration can be very expensive, we show that an analytical suboptimal least-squares approximation can be chosen in practice, resulting in very good accuracy.