On an efficient iterative scheme for a class of generalized nonexpansive operators in Banach spaces

Abstract

The main purpose of this work is to connect the recently introduced iterative approximation scheme known as [Formula: see text]-iterative scheme with the class of generalized [Formula: see text]-nonexpansive mappings. We consider the setting of a Banach space to prove our weak and strong convergence results. Once again, we prove that the class of [Formula: see text]-nonexpansive mapping includes properly the classes of Suzuki nonexpansive, generalized [Formula: see text]-nonexpansive and as well Reich–Suzuki type nonexpansive mappings by providing a numerical example. Eventually, we prove that [Formula: see text]-iterative scheme of this example is essentially more efficient than the other well-known iterative scheme of the literature. The presented work is new and extends some other well-known results in the iterations theory.