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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.)
##### Keywords:
stability; quasi-Banach space
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