The focus of this paper is to present results on stability and data dependency for a new four steps iteration scheme when dealing with contractive-like mappings. We also proffer some weak and strong convergence results for generalized nonexpansive mappings endowed with the property (E) in the setting of uniformly convex Banach spaces. We provide a numerical example supported by graphs and tables to validate our proofs. We used this new iteration method to find the solution of a Volterra-Fredholm integral equation.