A Hybrid Iterative Algorithm for Fixed Point Approximation in CAT (0) Spaces

Fixed-point approximation for nonlinear mappings in geodesic metric spaces has attracted considerable attention because of its importance in nonlinear analysis, optimization, and related areas. This paper proposes a hybrid iterative algorithm for approximating fixed points of total asymptotically nonexpansive mappings in CAT (0) spaces. The proposed algorithm is developed for a nonempty closed convex subset of a CAT (0) space and incorporates suitable control parameters to generate successive approximations to a fixed point of the underlying mapping. The convergence analysis is established by exploiting the fundamental geometric properties of CAT (0) spaces, particularly the CAT (0) convexity inequality, together with the defining properties of total asymptotically nonexpansive mappings. Under appropriate conditions on the control parameters and asymptotic error sequences, it is proved that the sequence generated by the proposed hybrid algorithm converges strongly to a fixed point of the mapping. The obtained result demonstrates the effectiveness of the hybrid approach within the framework of nonlinear metric spaces and contributes to the existing theory of iterative fixed-point approximation. The study also discusses the limitations of the proposed scheme and identifies possible directions for further investigations, including extensions to other classes of nonlinear mappings and metric spaces, as well as applications to optimization, variational inequalities, and equilibrium problems.