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On a Hyers-Ulam-Aoki-Rassias type stability and a fixed point theorem. (English) Zbl 1209.39006
M. Mirzavaziri and M. S. Moslehian [Bull. Braz. Math. Soc. (N.S.) 37, No. 3, 361–376 (2006; Zbl 1118.39015)] applied a fixed point approach to a perturbation problem. In this paper the authors prove the Hyers-Ulam-Aoki-Rassias stability for the commutative diagram below, where \(X\) is a set with a binary operation \(\circ\) and \((Y,d)\) is a complete metric space with a binary operation \(\diamond\) and \(f, g:X \to Y\) are two mappings.
\[ \begin{tikzcd} X\times X \rar["\circ"]\dar["g\times g" '] & X\dar["f"]\\ Y \times Y \rar["\diamond" '] &Y \end{tikzcd} \]
They also apply their results to the stability of following diagram,
\[ \begin{tikzcd} X \rar["\sigma"]\dar["g" '] & X\dar["f"]\\Y \rar["\tau" '] &Y \end{tikzcd} \]
where \(\sigma\) and \(\tau\) are selfmaps of \(X\) and \(Y\), respectively.

MSC:
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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