Hyperstability criterion for a new type of 2-variable radical functional equations

Commun. Korean Math. Soc. 2021 Vol. 36, No. 2, 343-359 https://doi.org/10.4134/CKMS.c200238 Published online September 14, 2020 Printed April 30, 2021

Iz-iddine EL-Fassi Sidi Mohamed ben abdellah University

Abstract : The aim of this paper is to obtain the general solution of the 2-variable radical functional equations $$f\left(\sqrt[k]{x^{k}+z^{k}} ,\sqrt[\ell]{y^{\ell}+w^{\ell}}\right)=f(x,y)+f(z,w),\;\;\;x,y,z,w\in \mathbb{R}, $$for $f$ a mapping from the set of all real numbers $\mathbb{R}$ into a vector space, where $k$ and $\ell$ are fixed positive integers. Also using the fixed point result of Brzd\k{e}k and Ciepli\'{n}ski \cite[Theorem 1]{krs} in $(2,\beta)$-Banach spaces, we prove the generalized hyperstability results of the 2-variable radical functional equations. In the end of this paper we derive some consequences from our main results.