Superstability of the $p$-radical trigonometric functional equation
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Abstract
In this paper, we solve and investigate the superstability of the $p$-radical functional equations
$$ \begin{align*} f\left(\sqrt[p]{x^{p}+y^{p}}\right) &-f\left(\sqrt[p]{x^{p}-y^{p}}\right)=\lambda f(x) g(y),\\ f\left(\sqrt[p]{x^{p}+y^{p}}\right) &-f\left(\sqrt[p]{x^{p}-y^{p}}\right)=\lambda g(x) f(y), \end{align*} $$
which is related to the trigonometric(Kim's type) functional equations, where $p$ is an odd positive integer and $f$ is a complex valued function. Furthermore, the results are extended to Banach algebras.
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References
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