Abstract
In this article, we prove some weak and strong convergence theorems for mappings satisfying condition (E) using the AK iterative scheme in the setting of Banach spaces. We offer a new example of mapping with condition (E) in support of our main result. Our results extend and improve many well-known corresponding results of the current literature.
Highlights
Let X be a Banach space, ∅ ≠ C ⊆ X and J: C ⟶ C
In 2008, Suzuki [4] introduced a new class of nonlinear mappings, which is the generalization of the class of nonexpansive mappings
In [18], Ullah and Arshad with the help of a numerical example proved that the iterative scheme (10) is better than the leading iterative scheme (9) for Suzuki mappings in Banach spaces
Summary
Let X be a Banach space, ∅ ≠ C ⊆ X and J: C ⟶ C. Garcia-Falset et al [5] proved that every Suzuki mapping satisfies condition (E) with μ 3. E Banach contraction principle suggests a Picard iterative scheme for finding the unique fixed point of a given contraction mapping. The Picard iterative scheme does not always converge to a fixed point of a nonexpansive mapping. In [17], akur et al with the help of a numerical example proved that the iterative scheme (9) converges faster than all of the iterative schemes (3)–(8) for the general setting of Suzuki mappings. In [18], Ullah and Arshad with the help of a numerical example proved that the iterative scheme (10) is better than the leading iterative scheme (9) for Suzuki mappings in Banach spaces. We give a new example of the GarciaFalset mapping and show that its AK iteration process is more efficient than all of the above schemes
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