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Volume 2019

# On new hyperstability results for the generalized $p$-radical functional equation in quasi-Banach spaces with the illustrative example-by Laddawan Aiemsomboon, Wutiphol Sintunavarat

Abstract: The goal of this paper is to investigate the general solution and the hyperstability results of the generalized $p$-radical functional equation
\begin{equation*}
f\left(\sqrt[p]{\sum_{i=1}^kx_i^p}\right)=\sum_{i=1}^kf(x_i)
\end{equation*}
for all $x_1,\dots,x_k \in \mathbb{R}$,
where $f: \mathbb{R} \to Y$ is a mapping, $p$ is a natural number, $k$ is a natural number such that $k \geq 2$, and $Y$ is a quasi-Banach space. Furthermore, the hyperstablity result for the inhomogeneous generalized $p$-radical functional equations is derived from the main results. Finally, we give the illustrative example showing the validity of the obtained results while the main results of Almahalebi et al. is not applicable.

Keywords: Fixed point; hyperstability; quasi-Banach space; $p$-radical functional equation.

DOI: 10.30697/rfpta-2019-012

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How to cite this article: Laddawan Aiemsomboon, Wutiphol Sintunavarat, On new hyperstability results for the generalized $p$-radical functional equation in quasi-Banach spaces with the illustrative example, Results in Fixed Point Theory and Applications, Vol. 2019, Article ID 2019012, 14 pages, 2019.  DOI: https://doi.org/10.30697/rfpta-2019-012

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Copyright © 2018 SUMA Publishing Group., This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.