$\alpha-$Cubic and $\beta-$Cubic Functional Equations

Abstract

In this paper, we established the general solution and generalized Ulam - Hyers stability of $\alpha-$cubic functional equation $2[\alpha f (w - \alpha z) + f (\alpha w + z)] = \alpha(\alpha^2 + 1)[ f (w + z) + f (w - z)] - 2(\alpha^4 - 1) f (z)$, where $\alpha \ne 0, \pm 1$ and $\beta-$cubic functional equation $\beta f (w + \beta z) - f(\beta w + z) - [ \beta f (w - \beta z) -f(\beta w - z) ] = 2 ( \beta ^ 4 -1 )f(z)$, where $\beta \ne 0, \pm 1$ in Banach Space using direct and fixed point methods.