Relatively Nonexpansive Mappings in k-Uniformly Convex Banach Spaces. (18th November 2021)

• Record Type: Journal Article
• Authors: Normohamadi, Z.; Moosaei, M.; Gabeleh, M.
• Is Part Of: Numerical functional analysis and optimization. Volume 42:Number 15(2021)
• Page Start: 1785
• Page End: 1801
• Publication Date: 2021-11-18
• Subjects: Best proximity pair -- k-uniformly convex Banach space -- proximal normal structure -- noncyclic relatively nonexpansive mapping -- T-regular pair; 47H10 -- 47H09 -- 34A12; Functional analysis -- Periodicals; Numerical analysis -- Periodicals; Mathematical optimization -- Periodicals; Numerical Analysis, Computer-Assisted; 515.705
• DOI: 10.1080/01630563.2021.2001751
• Languages: English
• ISSNs: 0163-0563

Abstract

Abstract: Let A and B be two nonempty subsets of a normed linear space X . A mapping T : A ∪ B → A ∪ B is said to be noncyclic if T ( A ) ⊆ A and T ( B ) ⊆ B . In the current paper, we consider the problem of finding the best proximity pair for the noncyclic mapping T, that is, two fixed points of T which achieve the minimum distance between the sets A and B . We do it from some different approaches. The common condition on these results is relatively nonexpansivity of the mapping T . At the first conclusion, we obtain the existence of best proximity pairs in the setting of uniformly convex in every direction Banach spaces where the pair ( A, B ) is nonconvex. Then we conclude a similar result by replacing the geometric property of Opial's property of the Banach space and adding another assumption on the mapping T, called condition ( C ) . We also show that the same result is true when X is a 2-uniformly convex Banach space. In the setting of k -uniformly convex Banach spaces, we prove that every nonempty, and convex pair of subsets has a geometric notion of proximal normal structure and then, we deduce the existence of best proximity pairs for relatively nonexpansive mappings in such spaces.