Hyers-Ulam stability of linear recurrence with constant coefficients over the quaternion skew yield
Let \(\mathbb{H}\) denote the skew-field of the quaternions. The main aim of the authors is to investigate the Hyers-Ulam stability of the linear recursion \[ y_{n+p}= a_{1}y_{n+p-1}+ \cdots + a_{p-1}y_{n+1}+a_{p}y_{n}+b_{n}, \] where \(a_{t}\in \mathbb{H}\), \(t=1, \ldots, p\) and \(y_{n}, b_{n}\in \mathbb{H}\), \(n\in \mathbb{N}\cup \left\{ 0\right\}\). In Theorem 5.1 they show that if the constants \(a_{1}, \ldots, a_{p}\) fulfill some assumptions, then the above equation is stable in the sense of Hyers and Ulam. Among other results it is shown that if \(\varepsilon > 0\) and the \(\mathbb{H}\)-valued sequence satisfies \[ \left| x_{n+1}- 2\mathbf{k}x_n - (-1)^{n}\mathbf{i}\right| \leq \varepsilon \] for all \(n\in \mathbb{N}\cup \left\{ 0\right\}\), where \(\mathbf{i}, \mathbf{k}\) are quaternion imaginary units, then there exists an \(\mathbb{H}\)-valued sequence \((y_{n})_{n\in \mathbb{N}\cup \left\{ 0\right\}}\) such that \[ y_{n+1} = 2\mathbf{k}y_{n} + (-1)^{n}\mathbf{i} \qquad \left(n\in \mathbb{N}\cup \left\{ 0\right\} \right) \] and \[ \sup_{n\in \mathbb{N}\cup \left\{0\right\}} |x_n - y_n| \leq 4\varepsilon \] hold. They close the paper with some examples.
scientific article; zbMATH DE number 7636957
scientific article