A criterion for the exponential stability of linear difference equations.
Based on a result by \textit{B. Aulbach} and \textit{Nguyen Van Minh} [J. Difference Eq. Appl.~2, 251--262 (1996; Zbl 0880.39009)] the author proves the following stability criterion for the linear nonautonomous difference equation \[ x_{n+1}=A_nx_n, \] where each \(A_n\), \(n\in{\mathbb N}\), is a bounded linear operator on a real or complex Banach space, and one has \(\sup_{n\in{\mathbb N}}| A_n| <\infty\): The above linear equation is exponentially stable if and only if for every \(p\)-summable sequence \(f_n\), \(1<p<\infty\), the solution of the inhomogeneous initial value problem \(x_{n+1}=A_nx_n+f_n\), \(x_1=0\) is bounded.
scientific article; zbMATH DE number 2140742