Hyers-Ulam stability of an additive-quadratic functional equation

In this paper, we introduce the following \((a,b,c)\)-mixed type functional equation of the form \(g(ax_1+bx_2+cx_3 )-g(-ax_1+bx_2+cx_3 ) + g(ax_1-bx_2+cx_3 )\)\(-g(ax_1+bx_2-cx_3 ) + 2a^2 [g(x_1 ) + g(-x_1)] + 2b^2 [g(x_2 ) + g(-x_2)] + \)\(2c^2 [g(x_3 ) + g(-x_3)]+a[g(x_1 ) - g(-x_1)]+ b[g(x_2 )-g(-x_2)] + \) \(c[g(x_3 )-g(-x_3)]=4g(ax_1+cx_3 )+2g(-bx_2)+\) \(2g(bx_2)\) 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.