In this paper, we investigate fixed point approximation of mappings satisfying the Reich– Chatterjea–Suzuki-type condition ((RCSC)-condition) in uniformly convex hyperbolic spaces. By using the K-iteration process, we establish ∆-convergence and strong convergence results under suitable assumptions. An example illustrating the applicability of the obtained results is presented. Furthermore, an application to a nonlinear Fredholm integral equation is provided. The obtained results extend and complement several existing convergence theorems for generalized nonexpansive mappings.