Best Proximity Point Theorems for Asymptotically Relatively Nonexpansive Mappings. (2nd January 2016)

• Record Type: Journal Article
• Authors: Rajesh, S.; Veeramani, P.
• Is Part Of: Numerical functional analysis and optimization. Volume 37:Number 1(2016)
• Page Start: 80
• Page End: 91
• Publication Date: 2016-01-02
• Subjects: Asymptotically nonexpansive maps -- best proximity points -- property UC -- proximal pairs -- relatively nonexpansive maps; 47H09 -- 47H10; Functional analysis -- Periodicals; Numerical analysis -- Periodicals; Mathematical optimization -- Periodicals; Numerical Analysis, Computer-Assisted; 515.705
• DOI: 10.1080/01630563.2015.1079533
• Languages: English
• ISSNs: 0163-0563

Abstract

ABSTRACT: Let ( A, B ) be a nonempty bounded closed convex proximal parallel pair in a nearly uniformly convex Banach space and T : A ∪ B → A ∪ B be a continuous and asymptotically relatively nonexpansive map. We prove that there exists x ∈ A ∪ B such that ‖ x − Tx ‖ = dist( A, B ) whenever T ( A ) ⊆ B, T ( B ) ⊆ A . Also, we establish that if T ( A ) ⊆ A and T ( B ) ⊆ B, then there exist x ∈ A and y ∈ B such that Tx = x, Ty = y and ‖ x − y ‖ = dist( A, B ). We prove the aforementioned results when the pair ( A, B ) has the rectangle property and property UC . In the case of A = B, we obtain, as a particular case of our results, the basic fixed point theorem for asymptotically nonexpansive maps by Goebel and Kirk.