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Asymptotic behavior of individual orbits of discrete systems

Author: Nguyen Van Minh
Journal: Proc. Amer. Math. Soc. 137 (2009), 3025-3035
MSC (2000): Primary 47D06; Secondary 47A35, 39A11
Published electronically: February 11, 2009
MathSciNet review: 2506461
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Abstract: We consider the asymptotic behavior of bounded solutions of the difference equations of the form $ x(n+1)=Bx(n) + y(n)$ in a Banach space $ \mathbb{X}$, where $ n=1,2,...$, $ B$ is a linear continuous operator in $ \mathbb{X}$, and $ (y(n))$ is a sequence in $ \mathbb{X}$ converging to 0 as $ n\to\infty$. An obtained result with an elementary proof says that if $ \sigma (B) \cap \{ \vert z\vert=1\} \subset \{ 1\}$, then every bounded solution $ x(n)$ has the property that $ \lim_{n\to\infty} (x(n+1)-x(n)) =0$. This result extends a theorem due to Katznelson-Tzafriri. Moreover, the techniques of the proof are furthered to study the individual stability of solutions of the discrete system. A discussion on further extensions is also given.

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Additional Information

Nguyen Van Minh
Affiliation: Department of Mathematics, University of West Georgia, Carrollton, Georgia 30118

Keywords: Katznelson-Tzafriri Theorem, discrete system, individual orbit, stability, asymptotically almost periodic.
Received by editor(s): November 3, 2008
Published electronically: February 11, 2009
Additional Notes: The author is grateful to the anonymous referee for carefully reading the manuscript and for pointing out several inaccuracies and for making suggestions to improve the presentation of this paper.
Communicated by: Nigel J. Kalton
Article copyright: © Copyright 2009 American Mathematical Society

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