Asymptotic behavior of individual orbits of discrete systems
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- by Nguyen Van Minh
- Proc. Amer. Math. Soc. 137 (2009), 3025-3035
- DOI: https://doi.org/10.1090/S0002-9939-09-09871-2
- Published electronically: February 11, 2009
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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 $\sigma (B) \cap \{ |z|=1\} \subset \{ 1\}$, 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.References
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Bibliographic Information
- Nguyen Van Minh
- Affiliation: Department of Mathematics, University of West Georgia, Carrollton, Georgia 30118
- Email: vnguyen@westga.edu
- 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
- © Copyright 2009 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 137 (2009), 3025-3035
- MSC (2000): Primary 47D06; Secondary 47A35, 39A11
- DOI: https://doi.org/10.1090/S0002-9939-09-09871-2
- MathSciNet review: 2506461