Bounded and periodic solutions of a difference equation and its stochastic analogue in Banach space (Q1176944)

Conditions for the existence of bounded or periodic solutions \(\{x_ n;\;n\in\mathbb{Z}\}\) of the equation \(Ax_ n=\sum_{k\in\mathbb{Z}}A_ k x_{n+k}+y_ n, \qquad n\in\mathbb{Z},\) in a Banach space \(B\) are presented. Here \(A\) is a closed operator, \(\{A_ n:\;n\in\mathbb{Z}\}\) are bounded operators, and \(\{y_ n:\;n\in\mathbb{Z}\}\) is a sequence of elements of \(B\) or a \(B\)-valued random process.