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

Iz-iddine EL-Fassi
대한수학회 (The Korean Mathematical Society)|2021.04|대한수학회논문집 (Communications of the KMS)|제36권 제2호|pp. 343-359|DOI 10.4134/CKMS.c200238

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키워드

fixed point theorem$(2Hyperstability\beta)$-normed spacetwo-variable radical functional equation
주제분류수학

초록

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.

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.

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