Fixed Points and Fuzzy Stability of Functional Equations Related to Inner Product

Abstract

In , Th.M. Rassias introduced the following equality \sum_{i,j=1}^m \|x_i - x_j \|^2 = 2m \sum_{i=1}^m\|x_i\|^2, \qquad \sum_{i=1}^m x_i =0 for a fixed integer $m \ge 3$. Let $V, W$ be real vector spaces. It is shown that if a mapping $f : V \rightarrow W$ satisfies \sum_{i,j=1}^m f(x_i - x_j ) = 2m \sum_{i=1}^m f(x_i) for all $x_1, \ldots, x_{m} \in V$ with $\sum_{i=1}^m x_i =0$, then the mapping $f : V \rightarrow W$ is realized as the sum of an additive mapping and a quadratic mapping. From the above equality we can define the functional equation f(x-y) +f(2x+y) + f(x+2y)= 3f(x)+ 3f(y) + 3f(x+y) , which is called a {\it quadratic functional equation}. Every solution of the quadratic functional equation is said to be a {\it quadratic mapping}. Using fixed point theorem we prove the Hyers-Ulam stability of the functional equation () in fuzzy Banach spaces.