Hyers-Ulam stability of an additive-quadratic functional equation

Abstract

In this paper, we introduce the following (a, b, c)-mixed type functional equation of the form g(ax1 + bx2 + cx3) − g(−ax1 + bx2 + cx3) + g(ax1 − bx2 + cx3) − g(ax1 + bx2 − cx3) + 2a2 [g(x1) + g(−x1)] + 2b2 [g(x2) + g(−x2)] + 2c2 [g(x3) + g(−x3)] + a[g(x1) − g(−x1)] + b[g(x2) − g(−x2)] + c[g(x3) − g(−x3)] = 4g(ax1 + cx3) + 2g(−bx2) + 2g(bx2) where a, b, c are positive integers with a > 1, and investigate the solution and the Hyers-Ulam stability of the above functional equation in Banach spaces by using two different methods.