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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.

46G05 Derivatives of functions in infinite-dimensional spaces
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