We establish necessary and sufficient conditions under which a sequence x 0 = y 0 , x n+1 = Ax n + y n+1 , n ≥ 0, is bounded for each bounded sequence \(\left\{ {y_n :n \geqslant 0} \right\} \subset \left\{ {\left. {x \in \bigcup\nolimits_{n = 1}^\infty {D\left( {A^n } \right)} } \right|\sup _{n \geqslant 0} \left\| {A^n x} \right\| < \infty } \right\}\), where A is a closed operator in a complex Banach space with domain of definition D(A) .
References
N. Dunford and J. T. Schwartz, Linear Operators. Part I: General Theory, Interscience, New York (1962).
Yu. V. Tomilov, “Asymptotic behavior of one recurrent sequence in the Banach space,” in: Asymptotic Integration of Nonlinear Equations [in Ukrainian], Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv (1992), pp. 146–153.
Author information
Authors and Affiliations
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, No. 9, pp. 1293–1296, September, 2009.
Rights and permissions
About this article
Cite this article
Horodnii, M.F., Vyatchaninov, O.V. On the boundedness of one recurrent sequence in a banach space. Ukr Math J 61, 1529–1532 (2009). https://doi.org/10.1007/s11253-010-0294-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-010-0294-x