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Hyers-Ulam stability of the linear recurrence with constant coefficients. (English) Zbl 1095.39024
Extending the result of {\it Z. Páles} [Publ. Math. 58, No. 4, 651--666 (2001; Zbl 0980.39022)], the author solves the Hyers-Ulam stability problem for an $n$-order linear recurrence with constant coefficients in $X$ as the following theorem [see also {\it D. Popa}, PU.M.A., Pure Math. Appl. 15, No. 1--2, 285--293 (2005; Zbl 1112.39024)]: Suppose that $X$ is a Banach space, $a_1, a_2, \dots, a_p$ are scalars such that the equation $r^p-a_1r^{p-1}-\cdots -a_{p-1}r-a_p=0$ admits the roots $r_1, r_2, \dots, r_p, \vert r_k\vert \neq 1$, $1 \leq k \leq p$, $\varepsilon > 0$, and $(b_n)_{n\geq 0}$ is a sequence in $X$. Suppose that $(x_n)_{n\geq 0}$ is a sequence in $X$ with the property $\Vert x_{n+p}-a_1x_{n+p-1}-\cdots -a_{p-1}x_{n+1}-a_px_n-b_n\Vert \leq \varepsilon$ ($n \geq 0$). Then there exists a sequence $(y_n)_{n\geq 0}$ in $X$ given by the recurrence $y_{n+p}=a_1y_{n+p-1}+\cdots +a_{p-1}y_{n+1}+a_py_n+b_n$ ($n \geq 0$) such that $\Vert x_n - y_n\Vert \leq \frac{\varepsilon}{\vert (\vert r_1\vert -1) \cdots (\vert r_p\vert -1)\vert }$ ($n \geq 0$).

39B82Stability, separation, extension, and related topics
39B52Functional equations for functions with more general domains and/or ranges
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