Hyers-Ulam stability of an n-variable quartic functional equation

Abstract

In this note we investigate the general solution for the quartic functional equation of the form

(3n + 4) f(Sigma(n)(i=1) x(i)) + Sigma(n)(j=1)f(-nx(j) + Sigma(n)(i=1,i not equal j) x(i)) = (n(2) + 2n + 1) Sigma(n)(i=1,i not equal j not equal k) f(x(i) + x(j) + x(k))

-1/2(3n(3) - 2n(2) - 13n - 8) Sigma(n)(i=1,i not equal j) f(x(i) + x(j))

+1/2(n(3) + 2n(2) + n) Sigma(n)(i=1,i not equal j) f(x(i) - x(j))

+ 1/2(3n(4) - 5n(3) - 7n(2) +13n + 12) Sigma(n)(i=1) f(x(i))

(n epsilon N, n ˃ 4) and also investigate the Hyers-Ulam stability of the quartic functional equation in random normed spaces using the direct approach and the fixed point approach.