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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].

##### MSC:
 39B72 Systems of functional equations and inequalities 39B52 Functional equations for functions with more general domains and/or ranges
##### Keywords:
stability; normed linear spaces; additive mapping