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.
Keywords : Hyperstability, $(2,\beta)$-normed space, two-variable radical functional equation, fixed point theorem
MSC numbers : Primary 39B82, 39B62; Secondary 65Q20, 47H10
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