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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
##### Keywords:
quasi-Banach space; $$p$$-norm; stability
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