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A note on stability of additive mappings. (English) Zbl 0844.39012
Rassias, Themistocles M. (ed.) et al., Stability of mappings of Hyers-Ulam type. Palm Harbor, FL: Hadronic Press. 19-22 (1994).
It is known that the following theorem is valid:
Let \(E_1\), \(E_2\) be two real normed linear spaces and assume that \(E_2\) is complete. Let \(f : E_1 \to E_2\) be a mapping for which there are \(c \in [0, + \infty)\) and \(p \in \mathbb{R} \backslash \{1\}\) such that \[ \bigl |f(x + y) - f(x) - f(y) \bigr |\leq c \bigl (|x |^p + |y |^p \bigr) \tag{1} \] for \(x,y \in E_1\). Then there exists a unique additive mapping \(T : E_1 \to E_2\) with \[ \bigl |f(x) - T(x) \bigr |\leq c |2^{p - 1} - 1 |^{-1} |x |^p \tag{2} \] for every \(x \in E_1\).
It is also known that the theorem is not true for \(p = 1\). We show that (2) gives the best possible estimate of the difference \(|f(x) - T(x) |\) for every \(p \in (0, + \infty)\), \(p \neq 1\).
For the entire collection see [Zbl 0835.00001].

39B72 Systems of functional equations and inequalities
39B52 Functional equations for functions with more general domains and/or ranges