A Fixed Point Approach to Stability of the Quadratic Equation

Abstract

In this paper, by using the fixed point method in Banach spaces, we prove the Hyers–Ulam–Rassias stability for the quadratic functional equation $$\displaystyle{f\left (\sum _{i=1}^{m}x_{ i}\right ) =\sum _{ i=1}^{m}f(x_{ i}) + \frac{1} {2}\sum _{1\leq i<j\leq m}\{f(x_{i} + x_{j}) - f(x_{i} - x_{j})\}.}$$ The concept of the Hyers–Ulam–Rassias stability originated from Rassias’ stability theorem that appeared in his paper: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 72(2), 297–300 (1978).