Stability Results in Intuitionistic Fuzzy Normed Spaces for a Cubic Functional Equation

Sometime in modeling applied problems there may be a degree of uncertainty in the parameters used in the model or some measurements may be imprecise. Due to such features, we are tempted to consider the study of functional equations in the fuzzy setting. The notion of fuzzy sets was first introduced by Zadeh [31] in 1965 which is a powerful hand set for modeling uncertainty and vagueness in various problems arising in the field of science and engineering. For the last four decades, fuzzy theory has become very active area of research and a lot of developments have been made in the theory of fuzzy sets to find the fuzzy analogues of the classical set theory. The notion of intuitionistic fuzzy norm (see [13,16–19,22,29]) is also useful one to deal with the inexactness and vagueness arising in modeling. There are many situations where the norm of a vector is not possible to find and the concept of intuitionistic fuzzy norm seems to be more suitable in such cases, that is, we can deal with such situations by modeling the inexactness through the intuitionistic fuzzy norm. In 1940, S.M. Ulam [30] raised the following question. Under what conditions does there exist an additive mapping near an approximately addition mapping? The case of approximately additive functions was solved by D.H. Hyers [3] under certain assumption. In 1978, a generalized version of the theorem of Hyers for approximately linear mapping was given by Th.M. Rassias [26]. A number of mathematicians were attracted by the result of Th.M. Rassias. The stability concept that was introduced and investigated by Rassias is called the Hyers-Ulam-Rassias stability. During the last decades, the stability problems of several functional equations have been extensively investigated by a number of authors (c.f. [1, 4–12, 14, 15, 21, 23– 25, 27, 28]) and references therein. Recently, Bae, Lee and Park [2] established some stability results for the functional equation