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Zbl 0957.39008
Lee, Yang-Hi; Jun, Kil-Woung
A generalization of the Hyers-Ulam-Rassias stability of the Pexider equation. (English)
[J] J. Math. Anal. Appl. 246, No.2, 627-638 (2000). ISSN 0022-247X

Let $V$ be a normed vector space and $X$ a Banach space, and let $f,g,h: V\to X$. The authors prove that the Pexider equation $$f(x+y)= g(x)+h(y)$$ is stable in the following sense: If there exists a real number $p\ne 1$, such that $$\bigl\|f(x+y)- g(x)-h(y) \bigr\|\le\|x \|^p+ \|y\|^p$$ for all $x,y\in V\setminus \{0\}$, then there exists exactly one additive map $T:V\to X$ such that $$\bigl\|f(x)- T(x)-f(0) \bigr\|\le C(p)\|x\|^p$$ for all $x\in V$. Here $C(p)$ is a certain specified constant.
[Henrik Stetkaer (Aarhus)]

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

Keywords: Hyers-Ulam stability; Banach space; Pexider equation

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