A criterion for the exponential stability of linear difference equations. (English) Zbl 1068.39019

Based on a result by B. Aulbach and 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.


39A11 Stability of difference equations (MSC2000)


Zbl 0880.39009
Full Text: DOI


[1] Aulbach, B.; Van Minh, N., The concept of spectral dichotomy for linear difference equations II, J. difference eq. appl., 2, 251-262, (1996) · Zbl 0880.39009
[2] Kučer, D.L., On some criteria for the boundedness of the solutions of a system of differential equations, Dokl. akad. nauk SSSR, 69, 603-606, (1949), (in Russian)
[3] Dalecki, Ju.L.; Kren, M.G., (), Transl. Math. Monographs
[4] Coppel, W.A., Stability and asymptotic behavior of differential equations, (1965), Heath Rhode Island · Zbl 0154.09301
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.