Abstract
In this paper, we establish weak and strong convergence theorems for mean nonexpansive maps in Banach spaces under the Picard–Mann hybrid iteration process. We also construct an example of mean nonexpansive mappings and show that it exceeds the class of nonexpansive mappings. To show the numerical accuracy of our main outcome, we show that Picard–Mann hybrid iteration process of this example is more effective than all of the Picard, Mann, and Ishikawa iterative processes.
Highlights
Introduction and PreliminariesSuppose Y is a Banach space and ∅≠ W ⊆ Y
If an element e0 ∈ W exists such that e0 = Se0, we say that e0 is a fixed point for S
In 1965, Browder [2] and Gohde [3] proved a fixed point theorem for a nonexpansive map S : W ⟶ W under the restriction that Y is a uniformly convex Banach space (UCBS) and ∅≠ W ⊆ Y is bounded as well as closed and convex
Summary
Suppose Y is a Banach space and ∅≠ W ⊆ Y. In 1965, Browder [2] and Gohde [3] proved a fixed point theorem for a nonexpansive map S : W ⟶ W under the restriction that Y is a uniformly convex Banach space (UCBS) and ∅≠ W ⊆ Y is bounded as well as closed and convex. Zhang [4] provided an existence of fixed point result for mean nonexpansive mappings in Banach space setting under the normal structure assumption. Khan [9] provided the weak and strong convergence of the scheme (7) for the class of nonexpansive operators He proved that the Picard–Mann hybrid iteration process is more effective than the Picard (4), Mann (5) and Ishikawa (6) iteration processes in the setting of nonexpansive maps.
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