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On the Boundedness of a Recurrence Sequence in a Banach Space

Authors:	A. M. Gomilko M. F. Gorodnii O. A. Lagoda

Affiliation:	(1) Institute of Hydromechanics, Ukrainian Academy of Sciences, Kiev;(2) Shevchenko Kiev National University, Kiev;(3) Kiev National University of Technologies and Design, Kiev

Abstract:	We investigate the problem of the boundedness of the following recurrence sequence in a Banach space B: $x_n = sumlimits_{k = 1}^infty {A_k x_{n - k} + y_n } ,{ }n geqslant 1,{ }x_n = {alpha}_n ,{ }n leqslant 0,$ where \|y_n} and \|_n} are sequences bounded in B, and A_k, k 1, are linear bounded operators. We prove that if, for any > 0, the condition $sumlimits_{k = 1}^infty {k^{1 + {varepsilon}} left\| {A_k } right\| < infty }$ is satisfied, then the sequence \|x_n} is bounded for arbitrary bounded sequences \|y_n} and \|_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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