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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.

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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