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Stability of the Cauchy functional equation in quasi-Banach spaces. (English) Zbl 1101.39021
Summary: Let \(X\) be a quasi-Banach space. We prove that there exists \(K>0\) such that for every function \(w:{\mathbb R} \to X\) satisfying \(\|w(s+t)-w(s)-w(t)\| \leq \varepsilon (|s|+|t|) \text{ for } s,t \in \mathbb R, \) there exists a unique additive function \(a:\mathbb R \to X\) such that \(a(1)=0\) and \(\|w(s)-a(s)-s \theta(\log_2|s|)\|\leq K\varepsilon |s|\text{ for }s \in \mathbb R, \) where \(\theta :\mathbb R \to X\) is defined by \(\theta(k):=w(2^k)/2^k\) for \(k \in \mathbb Z\) and extended in a piecewise linear way over the rest of \(\mathbb R\).

MSC:
39B82 Stability, separation, extension, and related topics for functional equations
39B52 Functional equations for functions with more general domains and/or ranges
46A16 Not locally convex spaces (metrizable topological linear spaces, locally bounded spaces, quasi-Banach spaces, etc.)
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