Approximating fixed points of nonexpansive mappings in uniformly convex hyperbolic spaces

Abstract

This paper investigates the convergence of an iterative process to a fixed point of nonexpansive mappings in uniformly convex hyperbolic spaces. First, we analyze the iteration scheme introduced by Karakaya et al. for such mappings, establishing its key properties. Under specific conditions, we prove both ∆−convergence and strong convergence of the iteration to a fixed point. Additionally, we show that, if the iteration ∆−converges or strongly converges to a fixed point, then every subsequence exhibits the same behavior. These results extend the theory of iterative methods to uniformly convex hyperbolic spaces, broadening their applicability in nonlinear functional analysis.