On the stability of the Pexiderized cubic functional equation in multi-normed spaces

In this paper, we investigate the Hyers-Ulam stability of the orthogonally cubic equation and Pexiderized cubic equation: f (kx + y) + f (kx − y) = g(x + y) + g(x − y) + 2/k g(kx) − 2g(x), in multi-normed spaces by the direct method and the fixed point method. Moreover, we prove the Hyers-Ulam stability of the 2-variables cubic equation: f (2x + y, 2z + t) + f (2x − y, 2z − t) = 2f (x + y, z + t) + 2f (x − y, z − t) + 12f (x, z), and orthogonally cubic type and k-cubic equation in multi-normed spaces. A counter example for non-stability of the cubic equation is also discussed.