On the stability of the generalized sine functional equations (Q1034215)

The superstability bounded by a constant for the sine functional equation \[ f(x)f(y)=f\big(\frac{x+y}{2}\big)^2-f\big(\frac{x-y}{2}\big)^2 \] was improved by \textit{G. H. Kim} [J. Math. Anal. Appl. 331, No. 2, 886--894 (2007; Zbl 1119.39024)] and \textit{R. Badora} and \textit{R. Ger} [Adv. Math., Dordr. 3, 3--15 (2002; Zbl 1010.39012)]. In this paper the author investigates the superstability for the generalized sine functional equations \[ g(x)f(y)=f\big(\frac{x+y}{2}\big)^2-f\big(\frac{x-y}{2}\big)^2 \] \[ f(x)g(y)=f\big(\frac{x+y}{2}\big)^2-f\big(\frac{x-y}{2}\big)^2 \] \[ g(x)g(y)=f\big(\frac{x+y}{2}\big)^2-f\big(\frac{x-y}{2}\big)^2 \] where \((G,+)\) is a uniquely 2-divisible Abelian group and \(f, g:G \to \mathbb C\) are nonzero functions. The author also obtains the superstability bounded by a constant for each equation and extends his results when \(G\) is a Banach algebra.