Remarks on Minimal Sets for Cyclic Mappings in Uniformly Convex Banach Spaces. (4th March 2017)

• Record Type: Journal Article
• Authors: Gabeleh, Moosa
• Is Part Of: Numerical functional analysis and optimization. Volume 38:Number 3(2017)
• Page Start: 360
• Page End: 375
• Publication Date: 2017-03-04
• Subjects: Best proximity point -- cyclic relatively nonexpansive mapping -- proximal normal structure -- uniformly convex Banach space; 47H10 -- 47H09 -- 46B20; Functional analysis -- Periodicals; Numerical analysis -- Periodicals; Mathematical optimization -- Periodicals; Numerical Analysis, Computer-Assisted; 515.705
• DOI: 10.1080/01630563.2016.1276074
• Languages: English
• ISSNs: 0163-0563

Abstract

ABSTRACT: In this article, we prove that every nonempty and convex pair of subsets of uniformly convex in every direction Banach spaces has the proximal normal structure and then we present a best proximity point theorem for cyclic relatively nonexpansive mappings in such spaces. We also study the structure of minimal sets of cyclic relatively nonexpansive mappings and obtain the existence results of best proximity points for cyclic mappings using some new geometric notions on minimal sets. Finally, we prove a best proximity point theorem for a new class of cyclic contraction-type mappings in the setting of uniformly convex Banach spaces and so, we improve the main conclusions of Eldred and Veeramani.