Solution of a 3-D cubic functional equation and its stability

Abstract

In this paper, we define and find the general solution of the following 3-D cubic functional equation

f (2x(1) + x(2) + x(3)) = 3f (x(1)+ x(2) + x(3)) + f (-x(1) + x(2) + x(3)) + 2f (x(1) + x(2)) + 2f (x(1)+ x(3)) -6f (x(1 )- x(2)) - 6f (x(1 )- x(3)) - 3f (x(2) + x(3)) + 2f (2x(1) - x(2)) + 2f (2x(1) - x(3)) - 18f (x(1)) - 6f (x(2)) - 6f (x(3))

We also prove the Hyers-Ulam stability of this functional equation in fuzzy normed spaces by using the direct method and the fixed point method.