The notion of proximal normal structure is introduced and used to study mappings that are "relatively nonexpansive" in the sense that they are defined on the union of two subsets A and B of a Banach space X and satisfy ∥ Tx-Ty∥ ≤ ∥ x-y∥ for all x ∈ A, y ∈ B. It is shown that if A and B are weakly compact and convex, and if the pair (A,B) has proximal normal structure, then a relatively nonexpansive mapping T: A ∪ B → A ∪ B satisfying (i) T(A) ⊆ B and T(B) ⊆ A, has a proximal point in the sense that there exists x₀ ∈ A ∪ B such that ∥ x₀-Tx₀∥ = dist(A,B). If in addition the norm of X is strictly convex, and if (i) is replaced with (i)' T(A) ⊆ A and T(B) ⊆ B, then the conclusion is that there exist x₀ ∈ A and y₀ ∈ B such that x₀ and y₀ are fixed points of T and ∥ x₀ -y₀∥ = dist(A,B). Because every bounded closed convex pair in a uniformly convex Banach space has proximal normal structure, these results hold in all uniformly convex spaces. A Krasnosel'skiĭ type iteration method for approximating the fixed points of relatively nonexpansive mappings is also given, and some related Hilbert space results are discussed.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm171-3-5,
author = {A. Anthony Eldred and W. A. Kirk and P. Veeramani},
title = {Proximal normal structure and relatively nonexpansive mappings},
journal = {Studia Mathematica},
volume = {166},
year = {2005},
pages = {283-293},
zbl = {1078.47013},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm171-3-5}
}
A. Anthony Eldred; W. A. Kirk; P. Veeramani. Proximal normal structure and relatively nonexpansive mappings. Studia Mathematica, Tome 166 (2005) pp. 283-293. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm171-3-5/