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

Suppose that XX is a sequentially complete Hausdorff locally convex space over a scalar field KK, VV is a bounded subset of XX, (an)n≥0(an)n≥0 is a sequence in K∖{0}K∖{0} with the property lim infn→∞|an|>1lim infn→∞|an|>1, and (bn)n≥0(bn)n≥0 is a sequence in XX. We show that for every sequence (xn)n≥0(xn)n≥0 in XX satisfying xn+1−anxn−bn∈V(n≥0) there exists a unique sequence (yn)n≥0(yn)n≥0 satisfying the recurrence yn+1=anyn+bn(n≥0), and for every qq with 1