Abstract
In this paper, we provide and study the concept of multi-valued generalized (α,β)-nonexpansive mappings, which is the multi-valued version of the recently developed generalized (α,β)-nonexpansive mappings. We establish some elementary properties and fixed point existence results for these mappings. Moreover, a multi-valued version of the M-iterative scheme is proposed for approximating fixed points of these mappings in the weak and strong senses. Using an example, we also show that M-iterative scheme converges faster as compared to many other schemes for this class of mappings.
Highlights
The author proved in [25] that the Thauker-New iterative process is fast in terms of convergence when compared to Picard, Mann, Ishikawa, Noor, Agarwal, and Abbas iteration processes for Suzuki generalized nonexpansive mappings
Let B be a subset of a Banach space and J : B → S( B)
Let B be a nonempty subset of a Banach space X and let J : B → Scb ( B) be a generalized (α, β)-nonexpansive multi-valued mapping
Summary
We often denote the set of natural numbers by using the notation N. Gohde [5] (cf Kirk [6] and others) provided a well-known result, which shows that, in a uniformly convex Banach space (UCBS) setting, nonexpansive selfmaps admit fixed points. The purpose of the present work is to provide the multi-valued version of these maps and to study the related fixed point results. The author proved in [25] that the Thauker-New iterative process is fast in terms of convergence when compared to Picard, Mann, Ishikawa, Noor, Agarwal, and Abbas iteration processes for Suzuki generalized nonexpansive mappings. For multi-valued nonexpansive mappings, initially, Sastry and Babu [30] worked on the convergence of Mann and Ishikawa iterative processes in Hilbert spaces.
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