On a Novel Iterative Algorithm in CAT(0) Spaces with Qualitative Analysis and Applications

This study presents a novel and efficient iterative scheme in the setting of CAT(0) spaces and investigates the convergence properties for a generalized class of mappings satisfying the Garcia–Falset property using the proposed iterative scheme. Strong and weak convergence results are established in CAT(0) spaces, generalizing many existing results in the literature. Furthermore, we discuss the stability and data dependence of the new iterative process. Numerical experiments include an analysis of error values, the number of iterations, and computational time, providing a comprehensive assessment of the method’s performance. Moreover, graphical comparisons demonstrate the efficiency and reliability of the approach. The obtained results are utilized in solving integral equations. Additionally, the paper concludes with a polynomiographic study of the newly introduced iterative process, in comparison with standard algorithms, such as Newton, Halley, or Kalantari’s $B_4$ iteration, emphasizing symmetry properties.