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Zbl 1101.39021
Tabor, Jacek
Stability of the Cauchy functional equation in quasi-Banach spaces.
(English)
[J] Ann. Pol. Math. 83, No. 3, 243-255 (2004). ISSN 0066-2216; ISSN 1730-6272/e

Summary: Let $X$ be a quasi-Banach space. We prove that there exists $K>0$ such that for every function $w:{\Bbb R} \to X$ satisfying $\|w(s+t)-w(s)-w(t)\| \leq \varepsilon (|s|+|t|) \text{ for } s,t \in \Bbb R,$ there exists a unique additive function $a:\Bbb R \to X$ such that $a(1)=0$ and $\|w(s)-a(s)-s \theta(\log_2|s|)\|\leq K\varepsilon |s|\text{ for }s \in \Bbb R,$ where $\theta :\Bbb R \to X$ is defined by $\theta(k):=w(2^k)/2^k$ for $k \in \Bbb Z$ and extended in a piecewise linear way over the rest of~$\Bbb R$.
MSC 2000:
*39B82 Stability, separation, extension, and related topics
39B52 Functional equations for functions with more general domains
46A16 Non-locally convex linear spaces

Keywords: stability; quasi-Banach space

Cited in: Zbl 1127.39055 Zbl 1123.39023

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