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Stability of a Monomial Functional Equation on Restricted Domains of Lebesgue Measure Zero

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Abstract

Let X and Y be Banach spaces and \(f\,{:}\,X \rightarrow Y\). Generalizing the result of Gilányi (Proc Natl Acad Sci USA 96:10588–10590, 1999) we study the Hyers–Ulam stability problem of the monomial functional equation

$$\begin{aligned} \Delta ^n_yf(x)-n!f(y)=0 \end{aligned}$$

in restricted domains of form \(e^{i\theta }{{\mathscr {H}}}^2\), where \({\mathscr {H}}\) is a subset of X such that \({{\mathscr {H}}}^c\) is of first category and \(e^{i\theta }{{\mathscr {H}}}^2\) denotes the rotation of \({{\mathscr {H}}}^2\) by the angle \(\theta \). As a consequence we solve a measure zero stability problem of the equation and obtain a hyperstability theorem on a set of Lebesgue measure zero.

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  1. Department of Mathematics and Liberal Education Institute, Kunsan National University, Gunsan, 54150, Republic of Korea

    Chang-Kwon Choi

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Choi, CK. Stability of a Monomial Functional Equation on Restricted Domains of Lebesgue Measure Zero. Results Math 72, 2067–2077 (2017). https://doi.org/10.1007/s00025-017-0742-0

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