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On the Hyers-Ulam stability of linear mappings. (English) Zbl 0789.46037
Let $$H$$ be a monotonically increasing symmetric homogeneous function of degree $$p$$, where $$p\in (0,\infty)\backslash\{1\}$$. Let $$f$$ be a mapping from a real normed space $$X$$ into a real Banach space $$Y$$. Assume that $\| f(x+ y)- f(x)- f(y)\|\leq H(\| x\| \| y\|)\quad \forall x,\;y\in X.$ The authors proved that $T(x)=\lim_{n\to\infty} 2^{-n} f(2^ n x)\quad(\text{resp. } T(x)=\lim_{n\to\infty} 2^ n f(2^{-n} x))\tag{i}$ exists for every $$x\in X$$ when $$p<1$$ (resp. $$p>1$$).
(ii) $$T$$ is the unique additive mapping satisfying $\| f(x)- T(x)\|\leq {H(1,1)\over | 2-2^ p|} \| x\|^ p\quad\forall x\in X.$ Furthermore, $$T$$ is linear if for every fixed $$x$$ in $$X$$ there exists a real number $$\delta_ x>0$$ such that the function $$t\to \| f(x)\|$$ is bounded on $$[0,\delta_ x]$$. A counterexample has been given for the cae $$p=1$$. These results partially answer a question of Ulam about the existence of additive mapping near an approximately additive mapping.

##### MSC:
 46G05 Derivatives of functions in infinite-dimensional spaces
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