Fixed point properties of saturated and unsaturated contractive mappings in CAT(0) spaces

This paper analyzes seven substantial distinct classes of contractive-type mappings using the technique of geodesic average perturbation within the framework of CAT(0) spaces. These classes of mappings are shown to be either saturated, in the sense that the geodesic average perturbation technique does not yield any significant new fixed point results, or unsaturated, in the sense that the technique provides genuine new fixed point results. The results establish that the class of strictly pseudocontractive self-mappings and the class of demicontractive self-mappings are saturated. Furthermore, the unsaturated category includes the class of Banach contractions, the class of Kannan contractions, the class of Bianchini contractions, the class of nonexpansive mappings, and the class of Ćirić–Reich–Rus contraction mappings. Our findings extend results from Hilbert spaces to convex (in the geodesic sense) metric spaces. This work provides an avenue for investigating fixed point results for several other important classes of contractive mappings using the geodesic average perturbation technique within the framework of geodesic spaces such as Hadamard manifolds, Hilbert balls, hyperbolic spaces, and CAT(k) spaces for some \(k \in {\mathbb {R}}\) $k\in \mathbb{R}$ .