On the Boundedness of a Recurrence Sequence in a Banach Space

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

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.
Published
25.10.2003
How to Cite
GomilkoA. M., GorodniiM. F., and LagodaO. A. “On the Boundedness of a Recurrence Sequence in a Banach Space”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 55, no. 10, Oct. 2003, pp. 1410-8, https://umj.imath.kiev.ua/index.php/umj/article/view/4009.
Section
Short communications