Proximal normal structure and relatively nonexpansive mappings

A. Anthony Eldred, W. A. Kirk, and P. Veeramani. "Proximal normal structure and relatively nonexpansive mappings." Studia Mathematica 171.3 (2005): 283-293. <http://eudml.org/doc/284693>.

@article{A2005,
abstract = {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.},
author = {A. Anthony Eldred, W. A. Kirk, P. Veeramani},
journal = {Studia Mathematica},
keywords = {proximal normal structure; relatively nonexpansive mappings; proximal points; fixed points},
language = {eng},
number = {3},
pages = {283-293},
title = {Proximal normal structure and relatively nonexpansive mappings},
url = {http://eudml.org/doc/284693},
volume = {171},
year = {2005},
}

TY - JOUR
AU - A. Anthony Eldred
AU - W. A. Kirk
AU - P. Veeramani
TI - Proximal normal structure and relatively nonexpansive mappings
JO - Studia Mathematica
PY - 2005
VL - 171
IS - 3
SP - 283
EP - 293
AB - 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.
LA - eng
KW - proximal normal structure; relatively nonexpansive mappings; proximal points; fixed points
UR - http://eudml.org/doc/284693
ER -