  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.
