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

Abstract

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: $ \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*} $ from an approximation of the $ p $-power-radical functional equation: $ \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*} $ 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.

Keywords

• stability
• superstability
• sine functional equation
• $ p $-radical functional equation
• $ p $-power-radical functional equation