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Ukrainian Mathematical Journal

, Volume 55, Issue 10, pp 1699–1708 | Cite as

On the Boundedness of a Recurrence Sequence in a Banach Space

  • A. M. Gomilko
  • M. F. Gorodnii
  • O. A. Lagoda
Article
  • 22 Downloads

Abstract

We investigate the problem of the boundedness of the following recurrence sequence in a Banach space B: \(x_n = \sum\limits_{k = 1}^\infty {A_k x_{n - k} + y_n } ,{ }n \geqslant 1,{ }x_n = {\alpha}_n ,{ }n \leqslant 0,\) where |yn} and |α n } are sequences bounded in B, and Ak, k ≥ 1, are linear bounded operators. We prove that if, for any ε > 0, the condition \(\sum\limits_{k = 1}^\infty {k^{1 + {\varepsilon}} \left\| {A_k } \right\| < \infty } \) is satisfied, then the sequence |xn} is bounded for arbitrary bounded sequences |yn} and |α n } if and only if the operator \(I - \sum {_{k = 1}^\infty {\text{ }}z^k A_k } \) has the continuous inverse for every zC, | z | ≤ 1.

Keywords

Banach Space Bounded Operator Linear Bounded Operator Bounded Sequence Recurrence Sequence 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Plenum Publishing Corporation 2003

Authors and Affiliations

  • A. M. Gomilko
    • 1
  • M. F. Gorodnii
    • 2
  • O. A. Lagoda
    • 3
  1. 1.Institute of HydromechanicsUkrainian Academy of SciencesKiev
  2. 2.Shevchenko Kiev National UniversityKiev
  3. 3.Kiev National University of Technologies and DesignKiev

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