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Approximate solutions of the Gołąb-Schinzel equation. (English) Zbl 1083.39025

The Gołąb-Schinzel equation for \(f:\mathbb{R}\to\mathbb{R}\) reads as \(f(x+f(x)y)=f(x)f(y)\), \(x,y\in\mathbb{R}\). For the stability version of this equation, the inequality \(| f(x+f(x)y)-f(x)f(y)|\leq\varepsilon\), \(x,y\in\mathbb{R}\), it is known that every unbounded continuous solution is already a solution of the equation. This phenomenon is called superstability. The corresponding phenomenon also appears for the functional equation of the exponential function. Modifying the inequality by measuring the distance of the quotient from \(1\) rather than the distance of the difference from \(0\) gives a “normal” stability behaviour. The author does the same for the Golab-Schinzel equation and shows that the phenomenon of superstability is still present.

MSC:

39B82 Stability, separation, extension, and related topics for functional equations
39B22 Functional equations for real functions
Full Text: DOI

References:

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