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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 $$\Vert f(x+ y)- f(x)- f(y)\Vert\le H(\Vert x\Vert \Vert y\Vert)\quad \forall x,\ y\in X.$$ The authors proved that $$T(x)=\lim\sb{n\to\infty} 2\sp{-n} f(2\sp n x)\quad(\text{resp. } T(x)=\lim\sb{n\to\infty} 2\sp n f(2\sp{-n} x))\tag i$$ exists for every $x\in X$ when $p<1$ (resp. $p>1$). (ii) $T$ is the unique additive mapping satisfying $$\Vert f(x)- T(x)\Vert\le {H(1,1)\over \vert 2-2\sp p\vert} \Vert x\Vert\sp p\quad\forall x\in X.$$ Furthermore, $T$ is linear if for every fixed $x$ in $X$ there exists a real number $\delta\sb x>0$ such that the function $t\to \Vert f(x)\Vert$ is bounded on $[0,\delta\sb x]$. 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.

46G05Derivatives, etc. (functional analysis)
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