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A note on Picard iterates of nonexpansive mappings

100%
Kim E., Kirk W.
Annales Polonici Mathematici
|
2001
|
tom 76
|
nr 3
189-196
EN
Let X be a Banach space, C a closed subset of X, and T:C → C a nonexpansive mapping. It has recently been shown that if X is reflexive and locally uniformly convex and if the fixed point set F(T) of T has nonempty interior then the Picard iterates of the mapping T always converge to a point of F(T). In this paper it is shown that if T is assumed to be asymptotically regular, this condition can be weakened much further. Finally, some observations are made about the geometric conditions imposed.
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Proximal normal structure and relatively nonexpansive mappings

81%
Eldred A., Kirk W., Veeramani P.
Studia Mathematica
|
2005
|
tom 171
|
nr 3
283-293
EN
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.
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