On the boundedness of one recurrent sequence in a banach space - Vyatchaninov, O. V.; Gorodnii, M. F.; Вятчанінов, O. В.; Городній, М. Ф. (Institute of Mathematics, NAS of Ukraine, 2009)
We establish necessary and sufficient conditions under which a sequence $x_0 = y_0,\; x_{n+1} = Ax_n  + y_{n+1},\; n ≥ 0$, is bounded for each bounded sequence $\{y_n : n ⩾ 0\} ⊂ \left\{x ∈ ⋃^{∞}_{n=1} D(A_n)|\sup_{n ⩾ 0} ∥A^nx∥ < ∞\right\}$, where $A$ is a closed operator in a complex Banach space with domain of definition $D(A)$.