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On the stability of metric semigroup homomorphisms. (English) Zbl 1338.39044

For a given nonempty set \(G\), an operation \(\circ: G \times G \to G\) is called square symmetric if \[ (x \circ y) \circ (x \circ y) = (x \circ x) \circ (y \circ y) \;\;(x,y \in G). \] Let a nonempty set \(G\) be closed under a square-symmetric operation \(\circ\). For any function \(f: G \times G \to [0, \infty)\) we define \[ U(f,\circ) := \inf \{\Theta \geq 0: f(x \circ x,y \circ y) \leq \Theta f(x,y) \;\text{for \;\textrm all} \;x,y \in G\} \] and \[ L(f,\circ) := \sup \{\Theta \geq 0: f(x \circ x,y \circ y) \geq \Theta f(x,y) \;\text{for \;all} \;x,y \in G\}. \]
One of the results of this paper is the following theorem.
{ Theorem 3.} Assume that \(G_1\) and \(G_2\) are nonempty sets and \(\circ_1: G_1 \times G_1 \to G_1\) and \(\circ_2: G_2 \times G_2 \to G_2\) are square-symmetric operations on \(G_1\) and \(G_2\), respectively. Let \((G_2, d)\) be a complete metric space such that the operation \(\circ_2\) is continuous and the mapping \(x \mapsto x \circ_2 x\) is surjective on \(G_2\). Moreover, assume that \(\beta: G_1 \times G_1 \to [0, \infty)\) is a function such that \[ - \infty < U(\beta, \circ_1) < L(d, \circ_2) < + \infty. \] If a mapping \(f: G_1 \to G_2\) fulfils the inequality \[ d \left(f(x \circ_1 y), f(x) \circ_2 f(y) \right) \leq \beta(x,y) \;\;\text{for} \;\;x, y \in G_1, \] then there exists a unique solution \(A: G_1 \to G_2\) of the functional equation \[ A(x \circ_1 y) = A(x) \circ_2 A(y) \] such that \[ d \left(f(x), A(x) \right) \leq \frac{\beta(x,x)}{L(d, \circ_2) - U(\beta, \circ_1)} \;\;(x \in G_1). \] This theorem is a common generalization of some previous stability theorems (e.g. the well-known Hyers and Rassias theorems).

MSC:

39B82 Stability, separation, extension, and related topics for functional equations
39B52 Functional equations for functions with more general domains and/or ranges
54H25 Fixed-point and coincidence theorems (topological aspects)
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