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The fixed point method for fuzzy stability of the Jensen functional equation. (English) Zbl 1179.39039

The Jensen functional equation is

$2f\left(\frac{x+y}{2}\right)=f\left(x\right)+f\left(y\right)$

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 [A. K. Mirmostafaee, M. Mirzavaziri and 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.

##### MSC:
 39B82 Stability, separation, extension, and related topics 46S40 Fuzzy functional analysis 39B52 Functional equations for functions with more general domains and/or ranges 46S50 Functional analysis in probabilistic metric linear spaces