This paper presents an implicit midpoint scheme for approximating fixed points of enriched nonexpansive mappings. The scheme is developed within the framework of unique geodesic spaces, with its core convergence properties established specifically in CAT(0) spaces. The sequence generated by the scheme is shown to be an approximate fixed point sequence that \(\Delta \) -converges to a fixed point of the underlying mapping. The convergence analysis is conducted using geometric inequalities, which allow the transformation of non-convex problems into geodesically convex ones, enabling the application of optimization techniques. Furthermore, the decay rate of the residual associated with the generated sequence is examined, and the convergence rate of the iterates is established under additional assumptions. Empirical investigations on two distinct examples in non-Hilbert CAT(0) spaces are conducted to evaluate the behavior of the proposed iterative scheme under varying parameter settings, and its performance is compared with that of the Krasnosel’skiĭ–Mann iteration.
