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Hyers-Ulam stability of the linear recurrence with constant coefficients. (English) Zbl 1095.39024
Extending the result of 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 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 \(r_1, r_2, \dots, r_p, | r_k| \neq 1\), \(1 \leq k \leq p\), \(\varepsilon > 0\), 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 \(\| x_{n+p}-a_1x_{n+p-1}-\cdots -a_{p-1}x_{n+1}-a_px_n-b_n\| \leq \varepsilon\) (\(n \geq 0\)). 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 \(\| x_n - y_n\| \leq \frac{\varepsilon}{| (| r_1| -1) \cdots (| r_p| -1)| }\) (\(n \geq 0\)).

39B82 Stability, separation, extension, and related topics for functional equations
39B52 Functional equations for functions with more general domains and/or ranges
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