Exploring the reciprocal functional equations: Approximations in diverse spaces

Abstract

In this study, we explore the generalized Hyers-Ulam-Rassias stability of a specific reciprocal-type functional equation. The equation is given by
\begin{equation*}\Omega(2u+v)+\Omega(2u-v)=\frac{2\Omega(u)\Omega(v)\displaystyle{\sum_{\substack{k=0\\ \text{$k$ is even}}}^{l}2^{l-k}\binom{l}{k}\Omega(u)^{\frac{k}{l}}\Omega(v)^{\frac{l-k}{l}}}}{\left(4\Omega(v)^{\frac{2}{l}}-\Omega(u)^{\frac{2}{l}}\right)^l}\end{equation*}
and we consider its behavior in both non-zero real and non-Archimedean spaces. Additionally, an appropriate counter-example is provided to demonstrate the failure of the stability result in the singular case.