A new hyperstability result for the Apollonius equation on a restricted domain and some applications

Let \((G,+)\) be an abelian group equipped with a complete ultrametric d that is invariant (i.e., \(d(x + z, y + z)= d(x, y\)) for \(x, y, z \in G\)), X be a normed space and \(U\subset X\setminus \{0\} \) be a nonempty subset. Under some weak natural assumptions on U and on the function \(\chi :U^3\rightarrow 0,\infty )\), we study new hyperstability results when \(f:U\rightarrow G\) satisfy the following Apollonius inequality

$$\begin{aligned}&d\Big (4f\Big (z-\frac{x+y}{2}\Big )+f(x-y),2f(x-z)+2f(y-z)\Big )\leqslant \chi (x,y,z),\\ {}&\quad x, y, z\in U,x-z,y-z,x-y,z-\frac{x+y}{2}\in U. \end{aligned}$$

Moreover, we derive some consequences from our main results.