Fuzzy Stability of a Quadratic-Additive Functional Equation

A classical question in the theory of functional equations is “when is it true that a mapping, which approximately satisfies a functional equation, must be somehow close to an exact solution of the equation?” Such a problem, called a stability problem of the functional equation, was formulated by Ulam 1 in 1940. In the next year, Hyers 2 gave a partial solution of Ulam’s problem for the case of approximate additive mappings. Subsequently, his result was generalized by Aoki 3 for additive mappings, and by Rassias 4 for linear mappings, to considering the stability problem with the unbounded Cauchy differences. During the last decades, the stability problems of functional equations have been extensively investigated by a number of mathematicians, see 5–16 . In 1984, Katsaras 17 defined a fuzzy norm on a linear space to construct a fuzzy structure on the space. Since then, some mathematicians have introduced several types of fuzzy norm in different points of view. In particular, Bag and Samanta 18 , following Cheng and Mordeson 19 , gave an idea of a fuzzy norm in such a manner that the corresponding fuzzymetric is of the Kramosil andMichálek type 20 . In 2008,Mirmostafaee andMoslehian 21 introduced for the first time the notion of fuzzy Hyers-Ulam-Rassias stability. They obtained a fuzzy version of stability for the Cauchy functional equation