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


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.


Banach Space Bounded Operator Linear Bounded Operator Bounded Sequence Recurrence Sequence 
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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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