Fixed Points, Inner Product Spaces, and Functional Equations

Abstract

Rassias introduced the following equality Sigma(n)(i,j)=1 parallel to x(i) - x(j)parallel to(2) = 2n Sigma(n)(i=1) parallel to x(i)parallel to(2), Sigma(n)(i=1) x(i) = 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 Sigma(n)(i,j-1) f(x(i) -x(j)) = 2n Sigma(n)(i-1) f(x(i)) for all x(1), ... , x(n) is an element of V with Sigma(n)(i-1) x(i) = 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.