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

TL;DR

This paper investigates the stability of first-order linear recurrence relations in topological vector spaces, establishing conditions under which approximate solutions are close to exact solutions in a general locally convex setting.

Contribution

It extends stability results of linear recurrences to sequentially complete Hausdorff locally convex spaces with new bounds and convergence criteria.

Findings

Existence and uniqueness of exact solutions close to approximate solutions.

Quantitative bounds on the deviation between approximate and exact solutions.

Conditions involving the limit inferior of the coefficients' magnitudes.

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*}