Strong convergence to common fixed points of nonexpansive mappings without commutativity assumption
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We introduce an iteration scheme for nonexpansive mappings in a Hilbert space and prove that the iteration converges strongly to common fixed points of the mappings without commutativity assumption. Copyright © 2006 Yonghong Yao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Let H be a real Hilbert space, and let C be a nonempty closed convex subset of H. A mapping T of C into itself is said to be nonexpansive if Tx − T y ≤ x − y ,
(1.1)
for each x, y ∈ C. For a mapping T of C into itself, we denote by F(T) the set of fixed points of T. We also denote by N and R+ the set of positive integers and nonnegative real numbers, respectively. Baillon [1] proved the first nonlinear ergodic theorem. Let C be a nonempty bounded convex closed subset of a Hilbert space H and letT be a nonexpansive mapping of C into itself. Then, for an arbitrary x ∈ C, {(1/(n + 1)) ni=0 T i x}∞ n=0 converges weakly to a fixed point of T. Wittmann [9] studied the following iteration scheme, which has first been considered by Halpern [3]: x0 = x ∈ C,
xn+1 = αn+1 x + 1 − αn+1 Txn ,
(1.2)
n ≥ 0,
∞ where a sequence {αn } in [0,1] is chosen so that limn→∞ αn = 0, ∞ n=1 αn = ∞, and n =1 |αn+1 − αn | < ∞; see also Reich [7]. Wittmann proved that for any x ∈ C, the sequence
Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2006, Article ID 89470, Pages 1–8 DOI 10.1155/FPTA/2006/89470
2
Nonexpansive mappings without commutativity assumption
{xn } defined by (1.2) converges strongly to the unique element Px ∈ F(T), where P is the metric projection of H onto F(T). Recall that two mappings S and T of H into itself are called commutative if
ST = TS,
(1.3)
for all x, y ∈ H. Recently, Shimizu and Takahashi [8] have first considered an iteration scheme for two commutative nonexpansive mappings S and T and proved that the iterations converge strongly to a common fixed point of S and T. They obtained the following result. Theorem 1.1 (see [8]). Let H be a Hilbert space, and let C be a nonempty closed convex subsetof H. Let S and T be nonexpansive mappings of C into itself such that ST = TS and F(S) F(T) is nonempty. Suppose that {αn }∞ n=0 ⊆ [0,1] satisfies (i) limn→∞ αn = 0, and (ii) ∞ n=0 αn = ∞. Then, for an arbitrary x ∈ C, the sequence {xn }∞ n=0 generated by x0 = x and
xn+1 = αn x + 1 − αn
n
2 Si T j xn , (n + 1)(n + 2) k=0 i+ j =k
n ≥ 0,
(1.4)
converges strongly to a common fixed point Px of S and T, where P is the metric projection of H onto F(S) F(T). Remark 1.2. At this point, we note that the authors have imposed the commutativity on the mappings S and T. But there are many mappings, that do not satisfy ST = TS. For example, if X = [−1/2,1/2], and S and T of X into itself are defined by S = x2 ,
T = sinx,
(1.5)
then ST = sin2 x, whereas TS = sinx2 . In this paper, we deal with the strong c
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