By principal motivation from the results of the new iterative scheme that produces faster results than K-iteration. In this article, we study generalized results by a new iteration scheme to approximate fixed points of generalized contraction and Suzuki non-expansive mappings. We establish strong convergence results of generalized contraction mappings of closed convex Banach space and also deduce data dependent results. Furthermore, we prove some weak and strong convergence theorems in the sense of generalized Suzuki non-expansive mapping by applying condition (C).

Some real life mathematical problems can be converted in the form of nonlinear equations. Solving such problems by analytical approaches is dificult in many situations. Hence numerical solution is the best way in this case. In this paper, a twelth-order iterative scheme for solvingnonlinear equations is presented and analyzed in terms of eciency. The new scheme is derived from the well-known King's method with order of convergence eight. We extend eighth-order King's method to an iterative method with memory of order 12:16 by using famous Newton'sinterpolating polynomial of degree 6 to avoid the derivative used in King's method. The new derived method is a three-step and is totally derivative free with twelth order of convergence. The method requires four functional evaluations at each iteration introducing high efficiency index of (12:16) 1/4 = 1:8673: Convergence order of new method is also studied. It is achieved by using matrix method of Herzburger. Numerical results are also provided to support theoretical analysis. Comparison of the derived scheme with previously well-known iterative schemes of the same order is also presented. As diffierent schemes of same order has efficiency index of (12)1/6 = 1:5131 because they requires six functional evaluations at each iteration, hence the proposed scheme is better than other schemes.