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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.

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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