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

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

2010-06-10

arxiv.org/abs/1006.1940v1

Nonlinear Anal. (TMA) 73 (2010), no. 9, 2792-2799

Mathematics

Functional Analysis

13 pages, to appear in Nonlinear Analysis: Theory, Methods & Applications

Scientific paper

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

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