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Zbl 1179.39039
Miheţ, Dorel
The fixed point method for fuzzy stability of the Jensen functional equation.
(English)
[J] Fuzzy Sets Syst. 160, No. 11, 1663-1667 (2009). ISSN 0165-0114

The Jensen functional equation is $$2f\left(\frac{x+y}{2}\right) = f(x) + f(y)$$ where the unknown $f$ is a mapping between linear spaces. In this paper, however, the unknown is considered as a mapping between fuzzy-normed linear spaces. The author comes up with an alternative proof and a slight improvement of a recently obtained generalized Hyers-Ulam-Rassias stability of such Jensen equation [{\it A. K. Mirmostafaee, M. Mirzavaziri} and {\it M. S. Moslehian}, Fuzzy stability of the Jensen functional equation", Fuzzy Sets Syst. 159, No. 6, 730--738 (2008; Zbl 1179.46060)]. The proof is based, besides several ideas of the original approach, on the fixed-point theory for the probabilistic metric spaces.
[Peter Sarkoci (Bratislava)]
MSC 2000:
*39B82 Stability, separation, extension, and related topics
46S40 Fuzzy functional analysis
39B52 Functional equations for functions with more general domains
46S50 Functional analysis in probabilistic metric spaces

Keywords: probabilistic metric space; fuzzy normed space; Jensen functional equation; additive mapping; Hyers-Ulam-Rassias stability; continuity; fixed point method

Citations: Zbl 1179.46060

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