Abstract
The present paper seeks to illustrate approximation theorems to the fixed point for generalized α -nonexpansive mapping with the Mann iteration process. Furthermore, the same results are established with the Ishikawa iteration process in the uniformly convex Banach space setting. The presented results expand and refine many of the recently reported results in the literature.
Highlights
Consider a Banach space (BS) X, together with its subset Dð ≠ φÞ
(2) quasi-nonexpansive provided that FixðTÞ ≠ φ, and for all u ∈ DðTÞ and v ∈ FixðTÞ, the following assertion holds: kTðuÞ − vk ≤ ku − vk Notably, there is a relationship between a nonexpansive mapping and a quasi-nonexpansive mapping
Consider a glz α-nonexpansive self-mapping T defined on a closed convex subset Dð≠ φÞ of a BS X
Summary
Consider a Banach space (BS) X, together with its subset Dð ≠ φÞ. Let us consider the following notations FixðTÞ, ⇀ , and ⟶ to represent the set of fixed points of T, weak convergence, and strong convergence, correspondingly. In 1974, Senter and Dotson [10] established a strong convergence fixed point theorem with regard to the Mann iteration of a nonexpansive mapping. In 1993, Xu and Tan [11] generalized the results of Reich [12] and Senter and Dotson [10] by using the Ishikawa iterative procedure instead of the Mann process. Piri et al [17] in 2019 have shown some interesting examples of the glz α-nonexpansive mapping and presented certain comparative convergence behaviors with regard to some powerful iteration procedures including the famous Mann and Ishikawa iterations among others. The present paper is aimed at establishing certain strong and weak convergence theorems of FP for the glz α-nonexpansive mapping via the application of the Mann iteration. These results happen to be an extension of the results presented in [1, 11]
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