An explicit construction of sunny nonexpansive retractions in Banach spaces
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An explicit algorithmic scheme for constructing the unique sunny nonexpansive retraction onto the common fixed point set of a nonlinear semigroup of nonexpansive mappings in a Banach space is analyzed and a proof of convergence is given. 1. Introduction Throughout this paper all vector spaces are real and we denote by N and R+ the set of nonnegative integers and nonnegative real numbers, respectively. Let (X, · ) be a Banach space and let X ∗ be its dual. The value of y ∈ X ∗ at x ∈ X will be denoted by ∗ x, y . We also denote by J : X → 2X the normalized duality map from X into the family of nonempty (by the Hahn-Banach theorem) weak-star compact convex subsets of X ∗ , which is defined by Jx = {x∗ ∈ X ∗ : x,x∗ = x2 = x∗ 2 } for all x ∈ X. The Banach space X is said to be smooth or to have a Gˆateaux differentiable norm if the limit lim t →0
x + t y − x
t
(1.1)
exists for each x, y ∈ X with x = y = 1. The space X is said to have a uniformly Gˆateaux differentiable norm if, for each y ∈ X with y = 1, the limit (1.1) is attained uniformly in x ∈ X with x = 1. It is known [12, Lemma 2.2] that if the norm of X is uniformly Gˆateaux differentiable, then the duality map is single-valued and norm to weak star uniformly continuous on each bounded subset of X. Let C be a nonempty, closed and convex subset of X and E be a nonempty subset of C. A mapping Q : C → X is nonexpansive if Qx − Qy ≤ x − y for all x, y ∈ C. A mapping Q : C → E is called a retraction from C onto E if Qx = x for all x ∈ E. A retraction Q from C onto E is called sunny if Q has the following property: Q(Qx + t(x − Qx)) = Qx for all x ∈ C and t ≥ 0 with Qx + t(x − Qx) ∈ C. It is known [6, Lemma 13.1] that in a smooth Banach space X, a retraction Q from C onto E is both sunny and nonexpansive if and only if
x − Qx,J(y − Qx) ≤ 0
(1.2)
for all x ∈ C and y ∈ E. Hence, there is at most one sunny nonexpansive retraction from C onto E. For example, if E is a nonempty, closed and convex subset of a Hilbert space Copyright © 2005 Hindawi Publishing Corporation Fixed Point Theory and Applications 2005:3 (2005) 295–305 DOI: 10.1155/FPTA.2005.295
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Sunny nonexpansive retractions
H, then the nearest point projection PE from H onto E is the unique sunny nonexpansive retraction of H onto E. This is not true for all Banach spaces, since outside Hilbert space, nearest point projections, although sunny, are no longer nonexpansive. On the other hand, sunny nonexpansive retractions do sometimes play a similar role in Banach spaces to that of nearest point projections in a Hilbert space. So an interesting problem arises: for which subsets of a Banach space does a sunny nonexpansive retraction exist? If it does exist, how can one find it? It is known [6, Theorem 13.2] that if C is a closed convex subset of a uniformly smooth Banach space and T : C → C is nonexpansive, then the fixed point set of T is a sunny nonexpansive retract of C. More generally, Bruck [3, Theorem 2] proves that if C is a closed convex subset of a reflex
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