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Zbl 0739.39013
Gajda, Zbigniew
(English)
[J] Int. J. Math. Math. Sci. 14, No.3, 431-434 (1991). ISSN 0161-1712; ISSN 1687-0425/e

Let $E\sb 1$, $E\sb 2$ be real normed spaces with $E\sb 2$ complete, and let $p$, $\varepsilon$ be real numbers with $\varepsilon\ge 0$. When $f: E\sb 1\to E\sb 2$ satisfies the inequality $\Vert f(x+y)-f(x)- f(y)\Vert\le\varepsilon(\Vert x\Vert\sp p+\Vert y\Vert\sp p)$ for all $x,y\in E$, it was shown by {\it T. M. Rassias} [Proc. Amer. Math. Soc. 72, 299-300 (1978; Zbl 0398.47040)] that there exists a unique additive mapping $T: E\sb 1\to E\sb 2$ such that $\Vert f(x)- T(x)\Vert\le\delta\Vert x\Vert\sp p$ for all $x\in E\sb 1$, providing that $p<1$, where $\delta=2\varepsilon/(2-2\sp p)$.\par The relationship between $f$ and $T$ was given by the formula $T(x)=\lim\sb{n\to\infty}2\sp{-n}f(2\sp nx)$. Rassias also proved that if the mapping from $\bbfR$ to $E\sb 2$ given by $t\to f(tx)$ is continuous for each fixed $x\in E$, then $T$ is linear.\par In the present paper the author extends these results to the case $p>1$, but now the additive mapping $T$ is given by $T(x)=\lim\sb{n\to\infty}2\sp nf(2\sp{-n}x)$, and the corresponding value of $\delta$ is $\delta=2\varepsilon/(2\sp p-2)$. 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 $\varepsilon>0$.
[Prof.D.H.Hyers]
MSC 2000:
*39B72 Functional inequalities involving unknown functions
39B52 Functional equations for functions with more general domains

Keywords: additive mappings; linear mappings; stability; normed spaces

Citations: Zbl 0398.47040

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