2017
Том 69
№ 9

All Issues

On the Boundedness of a Recurrence Sequence in a Banach Space

Gomilko A. M., Gorodnii M. F., Lagoda O. A.

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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 zC, | z | ≤ 1.

English version (Springer): Ukrainian Mathematical Journal 55 (2003), no. 10, pp 1699-1708.

Citation Example: Gomilko A. M., Gorodnii M. F., Lagoda O. A. On the Boundedness of a Recurrence Sequence in a Banach Space // Ukr. Mat. Zh. - 2003. - 55, № 10. - pp. 1410-1418.

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