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Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras associated to the Pexiderized Cauchy functional equation. (English) Zbl 1123.39023
Let \(X\) be a real linear space. A quasi-norm is a real-valued function \(\| \cdot\| \) on \(X\) satisfying the following:
(i) \(\| x\| \geq 0\) for all \(x\in X\), and \(\| x\| =0\) if and only if \(x=0\);
(ii) \(\| \lambda x\| =| \lambda| \| x\| \) for all \(\lambda \in {\mathbb R} \) and all \(x\in X\);
(iii) there is a constant \(K\geq 1\) such that \(\| x+y\| \leq K(\| x\| +\| y\| )\) for all \(x, y\in X\).
A quasi-Banach space is a complete quasi-normed space. A quasi-norm \(\| \cdot\| \) is called a \(p\)-norm \((0 < p \leq 1)\) if \(\| x+y\| ^p\leq \| x\| ^p+\| y\| ^p\) for all \(x, y\in X\). By the Aoki–Rolewicz theorem [cf. S. Rolewicz, Metric linear spaces. PWN-Polish Scientific Publishers. Warszawa (1985; Zbl 0573.46001)] each qusi-norm is equivalent to some \(p\)-norm. The first result on the stability of the Cauchy functional equation in quasi-Banach spaces was given by Jacek Tabor [Ann. Pol. Math. 83, No. 3, 243–255 (2004; Zbl 1101.39021)]. In this paper the authors restrict their attention to \(p\)-norms and prove the stability of homomorphisms in quasi-Banach algebras associated to the Pexiderized Cauchy functional equation \(f(x+y)=g(x)+h(y)\) involving a product of powers of norms.

MSC:
39B82 Stability, separation, extension, and related topics for functional equations
39B52 Functional equations for functions with more general domains and/or ranges
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