Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

Remote Access
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 47D06, 47A35, 39A11

Retrieve articles in all journals with MSC (2000): 47D06, 47A35, 39A11

Additional Information

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

PII: S 0002-9939(09)09871-2
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

Comments: Email Webmaster

© Copyright , American Mathematical Society
Contact Us · Sitemap · Privacy Statement

Connect with us Facebook Twitter Google+ LinkedIn Instagram RSS feeds Blogs YouTube Podcasts Wikipedia