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On the stability of the generalized sine functional equations. (English) Zbl 1181.39021

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 G. H. Kim [J. Math. Anal. Appl. 331, No. 2, 886–894 (2007; Zbl 1119.39024)] and R. Badora and 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.

MSC:

39B82 Stability, separation, extension, and related topics for functional equations
39B52 Functional equations for functions with more general domains and/or ranges
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