We develop a practical method for constructing canonical sets of root and pole cancellation functions for a matrix-valued function $Q(z)$ that is meromorphic on the extended complex plane $\bar{\mathbb{C}}:=\mathbb{C} \cup \left\{ \infty \right\}$. This method is applied to solve a nonlinear system of $n\in \mathbb{N}$ differential equations of order $l\in \mathbb{N}$ with $n $ unknown functions $u_{i}\left( t \right)$, where $i=1,\, \mathellipsis ,\,n $. In addition, for a function $Q\in \mathcal{N}_{\kappa}(\mathcal{H}) ,\, \kappa \in \mathbb{N} \cup \lbrace 0 \rbrace$, with a pole at infinity of order $m \in \mathbb{N}$, we establish the following factorization \[ Q(z)=(z-\beta)^{m}\tilde{Q}(z), \, z\in \mathcal{D}(Q), \] where $\beta \in \mathbb{R}$ is a regular point of $Q$, and $\tilde{Q}\in \mathcal{N}_{\kappa'}(\mathcal{H})$ is holomotphic at $\infty$. Unlike the Krein-Langer representation of $Q$, which involves a linear relation $A$, the factorization utilizes a bounded operator $\tilde{A}$ in the Krein-Langer representation of $\tilde{Q}$. The operator $\tilde{A}$ and the relation $A$ have identical spectra, except at two points: $\beta$ and $\infty$. The main advantage of this approach is that both the operator $\tilde{A}$ and the entire representation can be constructed explicitly and in a practically applicable manner for certain meromorphic functions $Q\in \mathcal{N}_{\kappa}^{n \times n}$ on $\bar{\mathbb{C}}$. We illustrate all main results through examples.

<jats:p>First, we present a method for obtaining a canonical set of root functions and Jordan chains of the invertible matrix polynomial <jats:italic>L</jats:italic>(<jats:italic>z</jats:italic>) through elementary transformations of the matrix <jats:italic>L</jats:italic>(<jats:italic>z</jats:italic>) alone. This method provides a new and simple approach to deriving a general solution of the system of ordinary linear differential equations <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$L\left( \frac{d}{dt}\right) u=0$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>L</mml:mi> <mml:mfenced> <mml:mfrac> <mml:mi>d</mml:mi> <mml:mrow> <mml:mi>dt</mml:mi> </mml:mrow> </mml:mfrac> </mml:mfenced> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula>, where <jats:italic>u</jats:italic>(<jats:italic>t</jats:italic>) is <jats:italic>n</jats:italic>-dimensional unknown function. We illustrate the effectiveness of this method by applying it to solve a high-order linear system of ODEs. Second, given a matrix generalized Nevanlinna function <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$Q\in N_{\kappa }^{n \times n}$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Q</mml:mi> <mml:mo>∈</mml:mo> <mml:msubsup> <mml:mi>N</mml:mi> <mml:mrow> <mml:mi>κ</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>×</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msubsup> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula>, that satisfies certain conditions at <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$\infty $$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>∞</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula>, and a canonical set of root functions of <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$\hat{Q}(z):= -Q(z)^{-1}$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mover> <mml:mi>Q</mml:mi> <mml:mo>^</mml:mo> </mml:mover> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>z</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>:</mml:mo> <mml:mo>=</mml:mo> <mml:mo>-</mml:mo> <mml:mi>Q</mml:mi> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>z</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula>, we construct the corresponding Pontryagin space <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$(\mathcal {K}, [.,.])$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>K</mml:mi> <mml:mo>,</mml:mo> <mml:mo>[</mml:mo> <mml:mo>.</mml:mo> <mml:mo>,</mml:mo> <mml:mo>.</mml:mo> <mml:mo>]</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula>, a self-adjoint operator <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$A:\mathcal {K}\rightarrow \mathcal {K}$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>:</mml:mo> <mml:mi>K</mml:mi> <mml:mo>→</mml:mo> <mml:mi>K</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula>, and an operator <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$\Gamma : \mathbb {C}^{n}\rightarrow \mathcal {K}$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Γ</mml:mi> <mml:mo>:</mml:mo> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>→</mml:mo> <mml:mi>K</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula>, that represent the function <jats:italic>Q</jats:italic>(<jats:italic>z</jats:italic>) in a Krein–Langer type representation. We illustrate the application of main results with examples involving concrete matrix polynomials <jats:italic>L</jats:italic>(<jats:italic>z</jats:italic>) and their inverses, defined as <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$Q(z):=\hat{L}(z):= -L(z)^{-1}$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Q</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>z</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>:</mml:mo> <mml:mo>=</mml:mo> <mml:mover> <mml:mi>L</mml:mi> <mml:mo>^</mml:mo> </mml:mover> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>z</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>:</mml:mo> <mml:mo>=</mml:mo> <mml:mo>-</mml:mo> <mml:mi>L</mml:mi> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>z</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula>.</jats:p>

Given Krein and Hilbert spaces $\left( \mathcal{K},[.,.] \right)$ and $\left( \mathcal{H}, \left( .,. \right) \right)$, respectively, the concept of the boundary triple $\Pi =(\mathcal{H}, \Gamma _{0}, \Gamma_{1})$ is generalized through the abstract Green's identity for the isometric relation $\Gamma$ between Krein spaces $\left( \mathcal{K}^{2}, \left[ .,.\right]_{\mathcal{K}^{2}} \right) $ and $\left(\mathcal{H}^{2}, \left[ .,.\right]_{\mathcal{H}^{2}} \right) $ without any conditions on $\dom\, \Gamma$ and $\ran\, \Gamma$. This also means that we do not assume the existence of a closed symmetric linear relation $S$ such that $\dom\, \Gamma=S^{+}$, which is a standard assumptions in all previous research of boundary triples. The main properties of such a general Green's boundary model are proven. In the process, some useful properties of the isometric relation $V$ between two Krein spaces $X$ and $Y$ are proven. Additionally, surprising properties of the unitary relation $\Gamma : \mathcal{K}^{2} \rightarrow\mathcal{H}^{2}$ and the self-adjoint main transformation $\tilde{A}$ of $\Gamma$ are discovered. Then, two statements about generalized Nevanlinna families are generalized using this Green's boundary model. Furthermore, several previously known boundary triples involving a Hilbert space $\mathcal{K}$ and reduction operator $\Gamma : \mathcal{K}^{2} \rightarrow\mathcal{H}^{2}$, such as AB-generalized, B-generalized, ordinary, isometric, unitary, quasi-boundary, and S-generalized boundary triples, have been extended to a Krein space $\mathcal{K}$ and linear relation $\Gamma$ using the Green's boundary model approach.

Let S be a symmetric linear relation in the Pontyagin space (K, [., .]) and let Π = (H,Γ0,Γ1) be the corresponding boundary triple. We prove that the corresponding Weyl function Q satisfies Q ∈ Nκ(H). Conversely, for regular Q ∈ Nκ(H), we find linear relation S ( A, where A is representing self-adjoint linear relation of Q, and we prove that Q is the Weyl function of the relation S. We also prove  = kerΓ1, where  is the representing relation of the Q̂ := −Q−1. In addition, if we assume that the derivative at infinity Q ′ (∞) := lim z→∞ zQ(z) is a boundedly invertible operator then we are able to decompose A,  and S in terms of S, i.e. we express relation matrices of A,  and S in terms of S, which is a bounded operator in this case.

We provide the necessary and sufficient conditions for a generalized Nevanlinna function $Q$ ($Q\in N_{\kappa }\left( \mathcal{H} \right)$) to be a Weyl function (also known as a Weyl-Titchmarch function). We also investigate an important subclass of $N_{\kappa }(\mathcal{H})$, the functions that have a boundedly invertible derivative at infinity $Q'\left( \infty \right):=\lim \limits_{z \to \infty}{zQ(z)}$. These functions are regular and have the operator representation $Q\left( z \right)=\tilde{\Gamma}^{+}\left( A-z \right)^{-1}\tilde{\Gamma},z\in \rho \left( A \right)$, where $A$ is a bounded self-adjoint operator in a Pontryagin space $\mathcal{K}$. We prove that every such strict function $Q$ is a Weyl function associated with the symmetric operator $S:=A_{\vert (I-P)\mathcal{K}}$, where $P$ is the orthogonal projection, $P:=\tilde{\Gamma} \left( \tilde{\Gamma}^{+} \tilde{\Gamma} \right)^{-1} \tilde{\Gamma}^{+} $. Additionally, we provide the relation matrices of the adjoint relation $S^{+}$ of $S$, and of $\hat{A}$, where $\hat{A}$ is the representing relation of $\hat{Q}:=-Q^{-1}$. We illustrate our results through examples, wherein we begin with a given function $Q\in N_{\kappa }\left( \mathcal{H} \right)$ and proceed to determine the closed symmetric linear relation $S$ and the boundary triple $\Pi$ so that $Q$ becomes the Weyl function associated with $\Pi$. 2020 Mathematics Subject Classification. 34B20, 47B50, 47A06, 47A56

UDC 517.9We present necessary and sufficient conditions for the reducibility of a self-adjoint linear relation in a Krein space. Then a generalized Nevanlinna function Q represented by a self-adjoint linear relation A in a Pontryagin space is decomposed by means of the reducing subspaces of A . The sum of two functions Q i ∈ N κ i ( ℋ ) , i = 1,2 , minimally represented by the triplets ( 𝒦 i , A i , Γ i ) is also studied. For this purpose, we create a model ( 𝒦 ˜ , A ˜ , Γ ˜ ) to represent Q : = Q 1 + Q 2 in terms of ( 𝒦 i , A i , Γ i ) . By using this model, necessary and sufficient conditions for κ = κ 1 + κ 2 are proved in the analytic form. Finally, we explain how degenerate Jordan chains of the representing relation A affect the reducing subspaces of A and the decomposition of the corresponding function Q .

Let $\left(\mathcal{H},\left(.,.\right)\right)$ be a Hilbert space and let $\mathcal{L}\left(\mathcal{H}\right)$ be the linear space of bounded operators in $\mathcal{H}$. In this paper, we deal with $\mathcal{L}(\mathcal{H})$-valued function $Q$ that belongs to the generalized Nevanlinna class $\mathcal{N}_{\kappa} (\mathcal{H})$, where $\kappa$ is a non-negative integer. It is the class of functions meromorphic on $C \backslash R$, such that $Q(z)^{*}=Q(\bar{z})$ and the kernel $\mathcal{N}_{Q}\left( z,w \right):=\frac{Q\left( z \right)-{Q\left( w \right)}^{\ast }}{z-\bar{w}}$ has $\kappa$ negative squares. A focus is on the functions $Q \in \mathcal{N}_{\kappa} (\mathcal{H})$ which are holomorphic at $ \infty$. A new operator representation of the inverse function $\hat{Q}\left( z \right):=-{Q\left( z \right)}^{-1}$ is obtained under the condition that the derivative at infinity $Q^{'}\left( \infty\right):=\lim\limits_{z\to \infty}{zQ(z)}$ is boundedly invertible operator. It turns out that $\hat{Q}$ is the sum $\hat{Q}=\hat{Q}_{1}+\hat{Q}_{2},\, \, \hat{Q}_{i}\in \mathcal{N}_{\kappa_{i}}\left( \mathcal{H} \right)$ that satisfies $\kappa_{1}+\kappa_{2}=\kappa $. That decomposition enables us to study properties of both functions, $Q$ and $\hat{Q}$, by studying the simple components $\hat{Q}_{1}$ and $\hat{Q}_{2}$.

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