Superstability of the $ p $-power-radical functional equation related to sine function equation

Abstract

<abstract><p>In this paper, we find solutions and investigate the superstability bounded by a function (Gǎvruta sense) for the $ p $-power-radical functional equation related to sine function equation:</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} f\left(\sqrt[p]{\frac{x^{p}+y^{p}}{2}}\right)^{2} -f\left(\sqrt[p]{\frac{x^{p}-y^{p}}{2}}\right)^{2} = f(x)f(y) \end{equation*} $\end{document} </tex-math></disp-formula></p> <p>from an approximation of the $ p $-power-radical functional equation:</p> <p><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \begin{align*} f\left(\sqrt[p]{\frac{x^{p}+y^{p}}{2}}\right)^{2} -f\left(\sqrt[p]{\frac{x^{p}-y^{p}}{2}}\right)^{2} = g(x)h(y), \end{align*} $\end{document} </tex-math></disp-formula></p> <p>where $ p $ is a positive odd integer, and $ f, g $ and $ h $ are complex valued functions on $ \mathbb{R} $. Furthermore, the obtained results are extended to Banach algebras.</p></abstract>