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 φ of a d-dimensional (d ≥ 2) Euclidean space is introduced that enables us to express the system in the form ft+1:=φ( ft), t = 0, 1, 2,. …. 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 ≥ 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.

For a given generalized Nevanlinna function Q(z) 2 N� (H), we study decompo- sitions that satisfy: Q = Q1 +Q2; Qi2 Ni (H), 0 � �i, i = 1, 2, and �1 +�2 = �, which we call desirable decompositions. In this paper, necessary and sufficient con- ditions for such decompositions of Q are obtained. One of the main results is a new, operator representation of ˆ

In this paper the analytic characterization of generalized poles of operator valued generalized Nevanlinna functions (including the length of Jordan chains of the representing relation) is completed. In particular, given a Jordan chain of the representing relation of length l, we show that there exists a pole cancellation function of order at least l, and, moreover, the construction shows that it is of surprisingly simple form.