%0 Journal Article
%T Fuzzy Stability of a Quadratic-Additive Functional Equation
%A Sun Sook Jin
%A Yang Hi Lee
%J International Journal of Mathematics and Mathematical Sciences
%D 2011
%I Hindawi Publishing Corporation
%R 10.1155/2011/504802
%X We investigate a fuzzy version of stability for the functional equation . 1. 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, 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¨C16]. 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 fuzzy metric is of the Kramosil and Mich¨¢lek type [20]. In 2008, Mirmostafaee and Moslehian [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 whose solution is called an additive mapping. In the same year, they [22] proved a fuzzy version of stability for the quadratic functional equation whose solution is called a quadratic mapping. Now we consider the quadratic-additive functional equation whose solution is called a quadratic-additive mapping. In [23], Chang et al. obtained a stability of the quadratic-additive functional equation by taking and composing an additive mapping and a quadratic mapping to prove the existence of a quadratic-additive mapping which is close to the given mapping . In their processing, is approximate to the odd part of and is close to the even part of it, respectively. In this paper, we get a general stability result of the quadratic-additive functional equation in the fuzzy normed linear space. To do it, we introduce a Cauchy sequence starting from a given mapping , which converges to the desired mapping in the fuzzy sense. As we mentioned before, in previous studies of stability problem of (1.3), Chang et al. attempted to get stability theorems by handling the odd and even part of , respectively. According to our
%U http://www.hindawi.com/journals/ijmms/2011/504802/