A stability of the generalized sine functional equations. (English) Zbl 1119.39024

It is well known that some functional equations (in particular of trigonometric type, i.e. sine, cosine etc.) show the superstability effect: posing the Hyers-Ulam stability problem either the function involved is bounded, or it fulfills the equation considered. It is shown in the paper that this is the case for the pexiderized version of the sine functional equation \[ g(x)h(y)=f\left(\frac{x+y}{2}\right)^2+f\left(\frac{x-y}{2}\right)^2, \quad x,y\in G\tag{1} \] considered for \(f,g,h:G\to{\mathbb C}\), where \((G,+)\) is a uniquely 2-divisible abelian group. The main result reads as follows.
{Theorem} Let \(f,g,h:G\to{\mathbb C}\) satisfy for all \(x,y\in G\) the inequality
\[ \left| g(x)h(y)-f\left(\frac{x+y}{2}\right)^2-f\left(\frac{x-y}{2}\right)^2 \right| \leq\varepsilon. \]
Then the following two statements hold.
{(i)} Either \(g\) is bounded or \(h\) satisfies the sine functional equation (i.e. (1) with all functions equal)
{(ii)} If \(g(0)=0\) or \(f(x)^2=f(-x)^2\) then either \(h\) is bounded or \(g\) satisfies the sine functional equation.
The related results for functions into the Banach algebra are also proved.


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


[1] Badora, R., On the stability of cosine functional equation, Rocznik Nauk.-Dydakt. Prace Mat., 15, 1-14 (1998) · Zbl 1159.39313
[2] Badora, R.; Ger, R., On some trigonometric functional inequalities, (Functional Equations—Results and Advances (2002), Kluwer Academic: Kluwer Academic Netherlands), 3-15 · Zbl 1010.39012
[3] Baker, J. A., The stability of the cosine equation, Proc. Amer. Math. Soc., 80, 411-416 (1980) · Zbl 0448.39003
[4] 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
[5] Cholewa, P. W., The stability of the sine equation, Proc. Amer. Math. Soc., 88, 631-634 (1983) · Zbl 0547.39003
[6] Friis, P. D.P., d’Alembert’s and Wilson’s equations on Lie groups, Aequationes Math., 67, 12-25 (2004) · Zbl 1060.39026
[7] Hyers, D. H., On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA, 27, 222-224 (1941) · Zbl 0061.26403
[8] Kannappan, Pl., The functional equation \(f(x y) + f(x y^{−1}) = 2 f(x) f(y)\) for groups, Proc. Amer. Math. Soc., 19, 69-74 (1968) · Zbl 0169.48102
[9] Kannappan, Pl.; Kim, G. H., On the stability of the generalized cosine functional equations, Ann. Acad. Paedagogicae Cracoviensis—Studia Math., 1, 49-58 (2001) · Zbl 1139.39316
[10] Kim, G. H.; Lee, S. H., Stability of the d’Alembert type functional equations, Nonlinear Funct. Anal. Appl., 9, 593-604 (2004) · Zbl 1067.39040
[11] Ulam, S. M., Problems in Modern Mathematics (1964), Wiley: Wiley New York, Chapter VI · Zbl 0137.24201
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