This paper introduces new fixed point theorems for non-expansive mappings specifically within complete CAT(0) spaces, which are vital instances of geodesic metric spaces characterized by non-positive curvature. We specifically demonstrate the efficiency of the established SNIA iterative process in approximating fixed points. This is achieved using Δ-convergence, in CAT(0) Space. The results presented here enhance and integrate various known convergence theorems, establishing a more comprehensive context for analyzing iterative schemes in CAT(0) spaces.