\(G\)-ultrametric dynamics and some fixed point theorems for single valued mappings in \(G\)-ultrametric spaces (Q1703748)

In the present paper, the main theorem states that every strongly contractive mapping \(T:X\to X\) defined on a spherically complete generalized ultrametric space \(X\) has a unique fixed point. It is shown by some examples that the strong contraction or spherical completeness hypothesis in the theorem cannot be relaxed. This approach lies in the fact that every generalized ultrametric space is topologically equivalent to an ultrametric space. In fact, the underlying generalized metric space was due to \textit{Z. Mustafa} and \textit{B. Sims} [J. Nonlinear Convex Anal. 7, No. 2, 289--297 (2006; Zbl 1111.54025)].