, Volume 59, Issue 2, pp 235-250
Date: 13 Jul 2013

The generalized cubic functional equation and the stability of cubic Jordan \(*\) -derivations

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Abstract

In the current work, we obtain the general solution of the following generalized cubic functional equation

$$\begin{aligned}&f(x+my)+f(x-my)\\&\quad =2\left( 2\cos \left( \frac{m\pi }{2}\right) +m^2-1\right) f(x)-\frac{1}{2}\left( \cos \left( \frac{m\pi }{2}\right) +m^2-1\right) f(2x)\\&\qquad +m^2\{f(x+y)+f(x-y)\} \end{aligned}$$
for an integer \(m \ge 1\). We prove the Hyers–Ulam stability and the superstability for this cubic functional equation by the directed method and a fixed point approach. We also employ the mentioned functional equation to establish the stability of cubic Jordan \(*\)-derivations on \(C^*\)-algebras and \(JC^*\)-algebras.