Hyers-Ulam stability of a system of first order linear recurrences with constant coefficients
Summary: We study the Hyers-Ulam stability in a Banach space \(X\) of the system of first order linear difference equations of the form \(\mathbf{x}_{n + 1} = A \mathbf{x}_n + \mathbf{d}_n\) for \(n \in \mathbb{N}_0\) (nonnegative integers), where \(A\) is a given \(r \times r\) matrix with real or complex coefficients, respectively, and \((\mathbf{d}_n)_{n \in \mathbb{N}_0}\) is a fixed sequence in \(X^r\). That is, we investigate the sequences \((\mathbf{y}_n)_{n \in \mathbb{N}_0}\) in \(X^r\) such that \(\delta : = \sup_{n \in \mathbb{N}_0} \| \mathbf{y}_{n + 1} - A \mathbf{y}_n - \mathbf{d}_n\| < \infty\) (with the maximum norm in \(X^r\)) and show that, in the case where all the eigenvalues of \(A\) are not of modulus 1, there is a positive real constant \(c\) (dependent only on \(A\)) such that, for each such a sequence \((\mathbf{y}_n)_{n \in \mathbb{N}_0}\), there is a solution \((\mathbf{x}_n)_{n \in \mathbb{N}_0}\) of the system with \(\sup_{n \in \mathbb{N}_0} \| \mathbf{y}_n - \mathbf{x}_n\| \leq c \delta\).
scientific article; zbMATH DE number 7025286
scientific article