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On stability of additive mappings. (English) Zbl 0739.39013
Let $$E_ 1$$, $$E_ 2$$ be real normed spaces with $$E_ 2$$ complete, and let $$p$$, $$\varepsilon$$ be real numbers with $$\varepsilon\geq 0$$. When $$f: E_ 1\to E_ 2$$ satisfies the inequality $$\| f(x+y)-f(x)- f(y)\|\leq\varepsilon(\| x\|^ p+\| y\|^ p)$$ for all $$x,y\in E$$, it was shown by T. M. Rassias [Proc. Amer. Math. Soc. 72, 299-300 (1978; Zbl 0398.47040)] that there exists a unique additive mapping $$T: E_ 1\to E_ 2$$ such that $$\| f(x)- T(x)\|\leq\delta\| x\|^ p$$ for all $$x\in E_ 1$$, providing that $$p<1$$, where $$\delta=2\varepsilon/(2-2^ p)$$.
The relationship between $$f$$ and $$T$$ was given by the formula $$T(x)=\lim_{n\to\infty}2^{-n}f(2^ nx)$$. Rassias also proved that if the mapping from $$\mathbb{R}$$ to $$E_ 2$$ given by $$t\to f(tx)$$ is continuous for each fixed $$x\in E$$, then $$T$$ is linear.
In the present paper the author extends these results to the case $$p>1$$, but now the additive mapping $$T$$ is given by $$T(x)=\lim_{n\to\infty}2^ nf(2^{-n}x)$$, and the corresponding value of $$\delta$$ is $$\delta=2\varepsilon/(2^ p-2)$$. The author also gives a counterexample to show that the theorem is false for the case $$p=1$$, and any choice of $$\delta>0$$ when $$\varepsilon>0$$.
Reviewer: Hyers, Donald H.

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