Hyperstability results for generalized p-radical functional equations in non-Archimedean Banach spaces with the secret key in the client-server environment

Abstract

Let $p$ be a natural number, $k$ be a natural number such that $k \geq 2$,
$a_i, b_i\in \mathbb{R}\setminus\{0\}$
for all $i=1, \ldots, k$ and $Y$ is a non-Archimedean Banach space.
The purpose of this work is to introduce the following generalized $p$-radical functional equation of the form
\begin{equation}\label{A}
f\left(\sqrt[p]{\sum_{i=1}^ka_ix_i^p}\right)=\sum_{i=1}^kb_if(x_i) \tag{A}
\end{equation}
for all $x_1,\dots,x_k \in \mathbb{R}$, where $f: \mathbb{R} \to Y$ is an unknown function. The hyperstability results for the generalized $p$-radical functional equation \eqref{A} are proved by using the fixed point method in non-Archimedean Banach spaces. Also, we apply these results for obtaining the hyperstability results for the generalized inhomogeneous $p$-radical functional equation. Finally, the generating secret keys in the client-server environment is discussed from the functional equation \eqref{A}, and its solutions.