The purpose of this article is to study iterative algorithm in hyperbolic space. We prove the weak -stability, data dependence and convergence results of the proposed iterative algorithm for contractive-like mappings in hyperbolic spaces. Furthermore, we study several strong and -convergence analysis for fixed points of generalized Reich–Suzuki nonexpansive-type mappings. Some new numerical examples are provided to compare the efficiency and applicability of the proposed iterative algorithm over existing iterative algorithms. As an application, we use the proposed iterative method to approximate the solution of a delay nonlinear Volterra integral equation in hyperbolic spaces. We also furnished an example which validate the mild conditions in the application results. Our results are new and improve several results in the current literature.