Two Applications of Brouwer's Fixed Point Theorem: in Insurance and in Biology Models

TL;DR

This paper applies Brouwer's Fixed Point Theorem to establish existence and uniqueness of solutions in two different systems: a new insurance model based on difference equations and a biological Leslie-Gower model, highlighting their mathematical connection.

Contribution

It introduces a new difference equation system for insurance re-rating and proves solution existence and uniqueness using Brouwer's theorem, also analyzing a Leslie-Gower biological model.

Findings

Existence of solutions in the insurance model is proven.

Uniqueness of solutions is established under certain conditions.

The Leslie-Gower system's equilibrium is shown to be unique under general conditions.

Abstract

In the first part of the article, a new interesting system of difference equations is introduced. It is developed for re-rating purposes in general insurance. A nonlinear transformation $\varphi $ of a d-dimensional $(d \ge 2)$ Euclidean space is introduced that enables us to express the system in the form $f^{t+1}:=\varphi (f^t),\, t=0,\, 1,\, 2,\, \ldots $. Under typical actuarial assumptions, existence of solutions of that system is proven by means of Brouwer's fixed point theorem in normed spaces. In addition, conditions that guarantee uniqueness of a solution are given. The second, smaller part of the article is about Leslie-Gower's system of $d \ge 2$ difference equations. We focus on the system that satisfies conditions consistent with weak inter-specific competition. We prove existence and uniqueness of the equilibrium of the model under surprisingly simple and very general conditions. Even though the two parts of this article have applications in two different sciences, they are connected with similar mathematics, in particular by our use of Brouwer's Fixed point Theorem.