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.
Citation: Hye Jeang Hwang, Gwang Hui Kim. Superstability of the $ p $-power-radical functional equation related to sine function equation[J]. Electronic Research Archive, 2023, 31(10): 6347-6362. doi: 10.3934/era.2023321
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.
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Hye Jeang Hwang, Gwang Hui Kim. Superstability of the $ p $-power-radical functional equation related to sine function equation[J]. Electronic Research Archive, 2023, 31(10): 6347-6362. doi: 10.3934/era.2023321
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