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A fixed point approach to the stability of quadratic functional equation. (English) Zbl 1113.39031
Let $$(X, \| .\| _\beta)$$ be a $$\beta$$-normed space $$(0<\beta \leq 1)$$. Recall that the only substantial difference of the $$\beta$$-normed space from the normed space is that $$\| \lambda x\| _\beta = | \lambda| ^\beta \| x\| _\beta$$ for all scalars $$\lambda$$ and all vectors $$x$$. The authors use a fixed point method of L. Cădariu and V. Radu [Grazer Math. Ber. 346, 43–52 (2004; Zbl 1060.39028)] to prove the Hyers-Ulam-Rassias stability of the quadratic functional equation $$f(x+y)+f(x-y)=2f(x)+2f(y)$$, where f is a function from a vector space into a complete $$\beta$$-normed space. Another fixed point approach concerning the orthogonal stability of the Pexiderized quadratic functional equation can be found in M. Mirzavaziri and M. S. Moslehian [Bull. Braz. Math. Soc. (N.S.) 37, No. 3, 361–376 (2006; Zbl 1118.39015)].

##### MSC:
 39B82 Stability, separation, extension, and related topics for functional equations 39B52 Functional equations for functions with more general domains and/or ranges 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc.
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