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Zbl 0933.39053
Lee, Yang-Hi; Jun, Kil-Woung
A generalization of the Hyers-Ulam-Rassias stability of Jensen's equation. (English)
[J] J. Math. Anal. Appl. 238, No.1, 305-315 (1999). ISSN 0022-247X

The following generalization of the stability of the Jensen's equation in the spirit of Hyers-Ulam-Rassias is proved: Let $V$ be a normed space, $X$ -- a Banach space, $p<1$ and $0\leq a<3$. If $f:V\rightarrow X$ satisfies $$\left\|2f\left({x+y\over 2}\right)-f(x)-f(y)\right\|\leq \|x\|^p+\|y\|^p$$ for all $x,y\in V$ with $\|x\|,\|y\|>a$, then there exists a unique additive mapping $T:V\rightarrow X$ such that $$\left\|f(x)-T(x)-f(0)\right\|\leq {3+3^p\over 3-3^p}\|x\|^p$$ for all $x\in V$ with $\|x\|>a$. For the case $p>1$ a corresponding result is obtained.
[Szymon Wasowicz (Bielsko-Biala)]

MSC 2000:: *39B82 Stability, separation, extension, and related topics
39B62 Systems of functional equations
39B52 Functional equations for functions with more general domains

Keywords: Hyers-Ulam-Rassias stability; Jensen functional equation; Banach space; additive mapping

Cited in: Zbl 1106.39028 Zbl 1035.39017 Zbl 0993.39024

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