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The stability of the sine equation. (English) Zbl 0547.39003

The author considers the stability problem for the sine equation. Let G be an abelian group in which division by 2 is uniquely performed. It is shown that every unbounded complex-valued function f on G is a solution of the sine functional equation \(f(x+y)f(x-y)=f(x)^ 2-f(y)^ 2\) for all x,\(y\in G\) if f satisfies the inequality \(| f(x+y)f(x-y)-f(x)^ 2+f(y)^ 2|\leq \delta\) for all x,\(y\in G\) where \(\delta\) is a positive real constant.
Reviewer: G.Dial

MSC:

39B52 Functional equations for functions with more general domains and/or ranges
39B99 Functional equations and inequalities
39B72 Systems of functional equations and inequalities
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References:

[1] John A. Baker, The stability of the cosine equation, Proc. Amer. Math. Soc. 80 (1980), no. 3, 411 – 416. · Zbl 0448.39003
[2] John Baker, J. Lawrence, and F. Zorzitto, The stability of the equation \?(\?+\?)=\?(\?)\?(\?), Proc. Amer. Math. Soc. 74 (1979), no. 2, 242 – 246. · Zbl 0397.39010
[3] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U. S. A. 27 (1941), 222 – 224. · Zbl 0061.26403
[4] S. M. Ulam, A collection of mathematical problems, Interscience Tracts in Pure and Applied Mathematics, no. 8, Interscience Publishers, New York-London, 1960. · Zbl 0086.24101
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