模糊赋范空间中一类二次函数方程的稳定性

Stability of a Quadratic Functional Equation in a Fuzzy Banach Space

本文在模糊赋范空间中讨论了二次函数方程f(2x+y)+f(2x-y)=2f(x+y)+2f(x-y)+4f(x)-2f(y)的Hyers-Ulam-Rassias稳定性,得到了如下主要结论:在满足适当条件下,按照模糊范数逼近的上述方程一定存在唯一的二次映射逼近;另外还得到了该二次映射的模糊连续性;最后,给出了一个具体例子来说明上述结论。所得结果可以看作是赋范空间框架下相应函数方程稳定性结论的推广。

In this paper,the Hyers-Ulam-Rassias stability of the following quadratic functional equation f(2x+y)+f(2x-y)=2f(x+y)+2f(x-y)+4f(x)-2f(y)is discussed in afuzzy Banach space.The following result is obtained:under some suitable conditions,an approximately quadratic function can be approximated by an unique quadratic mapping in a fuzzy sense.Moreover,the fuzzy continuity of this quadratic mapping is proved under certain conditions.Finally,a concrete example is given to illustrate the above conclusions.The results presented in this paper can be regarded as a generalization of the stability of the corresponding functional equations in the framework of normed spaces.