In the present paper, we introduce a new three-step fixed point iterative scheme for approximating the fixed points of non-expansive mappings. Here, we prove that the rate of convergence of our proposed iterative scheme is faster than that of Picard, Mann, Ishikawa, Agarwal et al., Abbas et al. and Thakur et al. iterative schemes. To support the analytic proof and the proper comparison of our proposed iterative scheme, we discuss a numerical example of Thakur et al. in which we approximate the fixed point by using a Matlab program. Finally, we establish some convergence theorems for the non-expansive mappings by using our proposed iterative scheme.

fixed point, non-expansive mapping, new three-step iterative scheme, rate of convergence of iterative scheme, strong and weak convergence theorems.