Fuzzy stability of a mixed type functional equation

Introduction 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, for considering the stability problem with unbounded Cauchy differences. During the last decades, the stability problems of functional equations have been extensively investigated by a number of mathematicians, see [5-17]. In 1984, Katsaras [18] 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 [19], following Cheng and Mordeson [20], gave an idea of a fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosil and Michalek type [21]. In 2008, Mirmostafaee and Moslehian [22] obtained a fuzzy version of stability for the Cauchy functional equation: f (x + y) − f (x) − f (y) = 0. (1:1)