Stabilities of Norm-Additive Functional Equations Over Noncommutative Group Through \((\delta , \theta , \epsilon )\)-isometry

In this research article, we focus on analyzing the generalized stability of norm-additive functional equations (FEs) \(\begin{aligned} \Vert \xi (gh)\Vert =\Vert \xi (g)+\xi (h)\Vert ,\end{aligned}\) $$\begin{array}{c}\hfill \Vert \xi (gh)\Vert =\Vert \xi (g)+\xi (h)\Vert ,\end{array}$$ and \(\begin{aligned} \Vert \xi (gh^{-1})\Vert =\Vert \xi (g)-\xi (h)\Vert \end{aligned}\) $$\begin{array}{c}\hfill \Vert \xi (g{h}^{-1})\Vert =\Vert \xi (g)-\xi (h)\Vert \end{array}$$ for every \(g, h \in G\) $g,h\in G$ , where \((G, \cdot )\) $(G,·)$ represents an arbitrary (noncommutative) group and B is a p-uniformly convex space, assuming that the mapping \(\xi \) $\xi$ is \((\delta , \theta )\) $(\delta ,\theta )$ -surjective. This proposed work generalizes and extends the findings of Y. Dong (2015), L. Sun (2023) and Y. Sun (2023) by applying \((\delta , \theta , \epsilon )\) $(\delta ,\theta ,ϵ)$ -isometry.