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

Fixed-point approximation in geodesic metric spaces is an important problem in nonlinear analysis and optimization. This study develops a hybrid iterative algorithm for approximating fixed points of total asymptotically nonexpansive mappings in CAT(0) spaces and establishes conditions for its strong convergence. The algorithm is formulated on a nonempty closed convex subset of a CAT(0) space and uses control parameters to generate successive approximations. The convergence analysis combines the geometric properties of CAT(0) spaces, particularly the CAT(0) convexity inequality, with the defining properties of total asymptotically nonexpansive mappings. Under suitable conditions on the control parameters and asymptotic error sequences, the generated sequence is proved to converge strongly to a fixed point of the underlying mapping. This result establishes a theoretical convergence guarantee for the proposed scheme and contributes to iterative fixed-point approximation in nonlinear metric spaces. The study also discusses the scheme’s limitations and outlines directions for further research, including extensions to other classes of nonlinear mappings and metric spaces and potential applications to optimization, variational inequalities, and equilibrium problems.