Fixed Points, Inner Product Spaces, and Functional Equations

Rassias introduced the following equality ∑i,j=1n∥xi-xj∥2=2n∑i=1n∥xi∥2, ∑i=1nxi=0, for a fixed integer n≥3. Let V,W be real vector spaces. It is shown that, if a mapping f:V→W satisfies the following functional equation ∑i,j=1nf(xi-xj)=2n∑i=1nf(xi) for all x1,…,xn∈V with ∑i=1nxi=0, which is defined by the above equality, then the mapping f:V→W is realized as the sum of an additive mapping and a quadratic mapping. Using the fixed point method, we prove the generalized Hyers-Ulam stability of the above functional equation in real Banach spaces