A link between the Perron-Frobenius theorem and Perron's theorem for difference equations
The paper deals with the growth of the solutions of the difference equation \[ x_{n+1}=A_nx_n, \;\;n \in \mathbb{N}, \] where the coefficients \(A_n\) are complex matrices, of size \(k \times k\), asymptotically constant, that is, \[ \lim_{n \rightarrow \infty} A_n = A, \] for some \(k \times k\) complex matrix \(A\). In this work, the author is interested in those solutions \(x=(x_n)_{n \in \mathbb{N}}\) of the difference equation which are nonnegative, that is, \(x_n \geq 0\) for all \(n \in \mathbb{N}\). The main result of this paper establishes that if \(x=(x_n)_{n \in \mathbb{N}}\) is a nonnegative solution of the difference equation, then either {\parindent6.5mm \begin{itemize}\item[(i)] \(x_n=0\) for all large \(n\), or \item[(ii)] the limit \(\rho(x)=\lim_{n \rightarrow \infty} \sqrt[n]{\|x_n \|}\) is an eigenvalue of \(A\) with a nonnegative eigenvector. \end{itemize}} This result provides not only new information about the nonnegative solutions from the Perron type theorem, but, moreover, it contains the weak form of the Perron-Frobenius theorem (the spectral radius of a nonnegative matrix \(A\) is an eigenvalue of \(A\) with a nonnegative eigenvector) as a special case.
scientific article; zbMATH DE number 5833887