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 |y n} and |α n } are sequences bounded in B, and A k, 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 |x n} is bounded for arbitrary bounded sequences |y n} 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 z ∈ C, | z | ≤ 1.
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Gomilko, A.M., Gorodnii, M.F. & Lagoda, O.A. On the Boundedness of a Recurrence Sequence in a Banach Space. Ukrainian Mathematical Journal 55, 1699–1708 (2003). https://doi.org/10.1023/B:UKMA.0000022074.96704.7e
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DOI: https://doi.org/10.1023/B:UKMA.0000022074.96704.7e