On the stability of the first-order linear recurrence in topological vector spaces

Suppose that X is a sequentially complete Hausdorff locally convex space over a scalar field K, V is a bounded subset of X, (a(n))(n >= 0) is a sequence in K \ {0} with the property lim inf(n ->infinity) vertical bar a(n)vertical bar > 1, and (bn)(n >= 0) is a sequence in X. We show that for every sequence (x(n))(n >= 0) in X satisfying x(n+1) - a(n)x(n) - b(n) is an element of V (n >= 0) there exists a unique sequence (y(n))(n >= 0) satisfying the recurrence y(n+1) = a(n)y(n) + b(n) (n >= 0), and for every q with 1 < q < lim inf(n ->infinity) vertical bar a(n)vertical bar there exists n(0) is an element of N such that x(n) - y(n) is an element of 1/q -1 (n >= n(0)). (C) 2010 Elsevier Ltd. All rights reserved.