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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)].

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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