Some results for a new three steps iteration scheme in Banach spaces

Abstract

In this paper, we introduce a three step iteration scheme and establish that this iterative method can be used to approximate fixed point of weak contraction mappings in the framework of Banach spaces. We also establish that our newly proposed iterative scheme is faster than some existing iterative processes in literature and the stability of this iterative scheme is established. Furthermore, we prove that this iterative method is equivalent to M iterative scheme introduced by Ullah et al. in [19], the iterative scheme introduced by Karakaya et al. in [11], and the Mann iterative iteration process. Finally, we established that the rate of convergence of our newly proposed iterative scheme is the same as that of the M iteration scheme introduced by Ullah et al. in [19], the iterative scheme introduced by Karakaya et al. in [11] and we present an analytic proof and also a numerical example to support our claim.