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

Abstract: Suppose that $\mathcal{X}$ is a sequentially complete Hausdorff locally convex space over a scalar field $\mathbb{K}$, $V$ is a bounded subset of $\mathcal{X}$, $(a_n){n\ge 0}$ is a sequence in $\mathbb{K}\setminus{0}$ with the property\ $\ds\liminf{n\to\infty} |a_n|>1$ and $(b_n){n\ge 0}$ is a sequence in $\mathcal{X}$. We show that for every sequence $(x_n){n\ge 0}$ in $\mathcal{X}$ satisfying \begin{eqnarray*} x_{n+1}-a_nx_n-b_n\in V\q(n\geq 0) \end{eqnarray*} there exists a unique sequence $(y_n){n\ge 0}$ satisfying the recurrence $y{n+1}=a_ny_n+b_n\,\,(n\geq 0)$ and for every $q$ with $1<q<\ds\liminf_{n\to\infty} |a_n|$, there exists $n_0\in \mathbb{N}$ such that \begin{eqnarray*} x_n-y_n\in \ds\f{1}{q-1}\ov{conv(V^b)}\q (n\geq n_0). \end{eqnarray*}