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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.

MSC:

39A11 Stability of difference equations (MSC2000)

Citations:

Zbl 0880.39009
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References:

[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) · Zbl 0034.19802
[3] Dalecki, Ju. L.; Kren, M. G., (Stability of Solutions of Differential Equations in Banach Space, Volume 43 (1974), Amer. Math. Soc.), Transl. Math. Monographs · Zbl 0286.34094
[4] Coppel, W. A., Stability and Asymptotic Behavior of Differential Equations (1965), Heath: Heath Rhode Island · Zbl 0154.09301
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