Abstract
The main intent of this article is to innovate a new iterative method to approximate fixed points of contraction and nonexpansive mappings. We prove that the new iterative method is stable for contraction and has a better rate of convergence than some distinctive iterative methods. Furthermore, some convergence results are proved for nonexpansive mappings. Finally, the solution of a nonlinear fractional difference equation is approximated via the proposed iterative method. Some numerical examples are constructed to support the analytical results and to illustrate the efficiency of the proposed iterative method.