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On stability of the translation equation in some classes of functions. (English) Zbl 1104.39024
The authors deal with the problem of the stability of the translation equation: \[ F(F(\alpha,k),l)=F(\alpha,k\cdot l)\tag{1} \] where \(F\colon X\times G\to X\) with \((G,\cdot)\) being a monoid and \((X,\varrho)\) being a metric space. Two definitions of the stability of the above equation are introduced. In the first one, it is required that if the distances \[ \varrho(F(F(\alpha,k),l),F(\alpha,k\cdot l))\tag{2} \] are bounded (for all \(\alpha\in X,k,l\in G\)) then there exists a true solution \(H\) of the equation (1) such that all the distances \[ \varrho(F(\alpha,k),H(\alpha,k))\tag{3} \] are also bounded. According to the second definition, the equation (1) is stable if for each \(\varepsilon>0\) there exists \(\delta>0\) such that if for \(F\) the distances (2) are always non greater than \(\delta\), then there exists an exact solution \(H\) of (1) such that (3) is always non greater than \(\varepsilon\). Some examples are given showing the lack of stability with respect to both definitions, in general. Then, the authors prove that the translation equation (1) is stable, in both senses, in two classes of functions: \[ {\mathbf{CB}}:=\{H\colon X\times G\to X:\;H(\alpha_0,\cdot)\;\text{is bijective for some } \alpha_0\in G\} \] and \[ {\mathbf{CI}}:=\{H\colon X\times G\to X:\;H(\alpha_0,\cdot)\;\text{is injective for some } \alpha_0\in G\;\text{and}\;H(\alpha_0, G)=H(X,1)\}. \] (In fact, the above result is only a corollary from a more general theorem.) It is remarked also that the exact solution \(H\) of (1) need not be unique.
For the translation equation itself and various definitions of the stability cf. the second author’s papers: Aequationes Math. 50, No. 1–2, 17–37 (1995; Zbl 0876.39007); ibid. 68, No. 3, 260–274 (2004; Zbl 1060.39031).

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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