Approximation of a bounded solution of a difference equation by solutions of corresponding boundary problems in Banach space (Q1311116)

The author considers a complex separable Banach space \(\mathcal B\), a linear bounded operator \(A\) from \(\mathcal B\) into \({\mathcal B}\) and the difference equation \(x_{n + 1} + x_{n - 1} = A x_ n + y_ n\), \(n \in\mathbb{Z}\), where \(x_ n\) is a given bounded sequence in \(\mathcal B\). To this equation he assigns the difference boundary problem \(u_{n + 1} - u_{n - 1} = Au_ n + y_ n\), \(-p + 1 \leq n \leq q -1\), \(u_{-p} = a\), \(u_ q = b\) where \(a\), \(b\) are fixed elements of the space \({\mathcal B}\); and \(p\), \(q\), are natural numbers. He investigates the problem of the existence and uniqueness of a solution of the difference boundary problem and also the problem of approximation of a bounded solution of the difference equation by solutions of a family of boundary problems for \(p, q \rightarrow +\infty\).