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

Abstract

Let X and Y be Banach spaces and \(f\,{:}\,X \rightarrow Y\). Generalizing the result of Gilanyi (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.