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.


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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[1] Ulam, S. M. ”Problems in Modern Mathematics” Chap. VI, Science Editions, Wiley, New York, 1964 · Zbl 0137.24201
[2] Hyers, D. H.: On the stability of the linear functional equation. Proc. Nat. Acad. Sci. U.S.A., 27, 222–224 (1941) · Zbl 0061.26403
[3] Baker, J., Lawrence, J., Zorzitto, F.: The stability of the equation f(x+y) = f(x)f(y). Proc. Amer. Math. Soc., 74, 242–246 (1979) · Zbl 0397.39010
[4] Baker, J. A.: The stability of the cosine equation. Proc. Amer. Math. Soc., 80, 411–416 (1980) · Zbl 0448.39003
[5] Gǎvruta, P.: On the stability of some functional equations, in: ”Stability of mappings of Hyers-Ulam type” (Eds. Th. M. Rassias and J. Tabor), Hadronic Press, 93–98, 1994
[6] Badora, R.: On the stability of cosine functional equation. Rocznik Naukowo-Dydak., Prace Mat., 15, 1–14 (1998)
[7] Badora, R., Ger, R.: On some trigonometric functional inequalities. Functional Equations Results and Advances, 3–15 (2002) · Zbl 1010.39012
[8] Elqorachi, E., Akkouchi, M.: On Hyers-Ulam stability of the generalized Cauchy and Wilson equations. Publ. Math. Debrecen, 66, 283–301 (2005) · Zbl 1084.39024
[9] Friis, P., de P.: d’Alembert’s and Wilson’s equations on Lie groups. Aequationes Math., 67, 12–25 (2004) · Zbl 1060.39026
[10] Kannappan, Pl., Kim, G. H.: On the stability of the generalized cosine functional equations. Annales Acadedmiae Paedagogicae Cracoviensis – Studia Mathematica, 1, 49–58 (2001) · Zbl 1139.39316
[11] Kim, G. H.: The Stability of the d’Alembert and Jensen type functional equations. Jour. Math. Anal & Appl., 325, 237–248 (2007) · Zbl 1106.39026
[12] Kim, G. H., Dragomir, Sever S.: On the the Stability of generalized d’Alembert and Jensen functional equation. Intern. Jour. Math. & Math. Sci., Vol 2006, Article ID 43185, (2006). · Zbl 1131.39027
[13] Sinopoulos, P.: Generalized sine equations. III., Aequationes Math., 51, 311–327 (1996) · Zbl 0853.39012
[14] Cholewa, P. W.: The stability of the sine equation. Proc. Amer. Math. Soc., 88, 631–634 (1983) · Zbl 0547.39003
[15] Kim, G. H.: A Stability of the generalized sine functional equations. Jour. Math. Anal & Appl., 331, 886–894 (2007) · Zbl 1119.39024
[16] Kannappan, Pl.: The functional equation f(xy) + f(xy ) = 2f(x)f(y) for groups. Proc. Amer. Math. Soc., 19, 69–74 (1968) · Zbl 0169.48102
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