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

Pituk, M.

Appl. Math. Lett. 17, No. 7, 779-783 (2004).

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.

Reviewer: Christian Poetzsche (Minneapolis)

MSC:

39A11

Stability of difference equations (MSC2000)

Keywords:

difference equations in Banach space; linear equations; \(l_p\)-spaces; exponential; stability; bounded linear operator; initial value problem

Citations:

Zbl 0880.39009

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)
[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