Weak and strong convergence theorems for nonexpansive semigroups in Banach spaces
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We introduce an implicit iterative process for a nonexpansive semigroup and then we prove a weak convergence theorem for the nonexpansive semigroup in a uniformly convex Banach space which satisfies Opial’s condition. Further, we discuss the strong convergence of the implicit iterative process. 1. Introduction Let C be a closed convex subset of a Hilbert space and let T be a nonexpansive mapping from C into itself. For each t ∈ (0,1), the contraction mapping Tt of C into itself defined by Tt x = tu + (1 − t)Tx
(1.1)
for every x ∈ C, has a unique fixed point xt , where u is an element of C. Browder [4] proved that {xt } converges strongly to a fixed point of T as t → 0 in a Hilbert space. Motivated by Browder’s theorem [4], Takahahi and Ueda [20] proved the strong convergence of the following iterative process in a uniformly convex Banach space with a uniformly Gˆateaux differentiable norm (see also [14]):
1 1 xk = x + 1 − Txk k k
(1.2)
for every k = 1,2,3,..., where x ∈ C. On the other hand, Xu and Ori [21] studied the following implicit iterative process for finite nonexpansive mappings T1 ,T2 ,...,Tr in a Hilbert space: x0 = x ∈ C and
xn = αn xn−1 + 1 − αn Tn xn
(1.3)
for every n = 1,2,..., where {αn } is a sequence in (0,1) and Tn = Tn+r . And they proved a weak convergence of the iterative process defined by (1.3) in a Hilbert space. Sun et al. [17] studied the iterations defined by (1.3) and proved the strong convergence of the iterations in a uniformly convex Banach space, requiring one mapping Ti in the family to be semi compact. Copyright © 2005 Hindawi Publishing Corporation Fixed Point Theory and Applications 2005:3 (2005) 343–354 DOI: 10.1155/FPTA.2005.343
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Weak and strong convergence theorems
In this paper, using the idea of [17, 21], we introduce an implicit iterative process for a nonexpansive semigroup and then prove a weak convergence theorem for the nonexpansive semigroup in a uniformly convex Banach space which satisfies Opial’s condition. Further, we discuss the strong convergence of the implicit iterative process (see also [1, 2, 7, 12, 13]). 2. Preliminaries and notations Throughout this paper, we denote by N and Z+ the set of all positive integers and the set of all nonnegative integers, respectively. Let E be a real Banach space. We denote by Br the set {x ∈ E : x ≤ r }. A Banach space E is said to be strictly convex if x + y /2 < 1 for each x, y ∈ B1 with x = y, and it is said to be uniformly convex if for each ε > 0, there exists δ > 0 such that x + y /2 ≤ 1 − δ for each x, y ∈ B1 with x − y ≥ ε. It is wellknown that a uniformly convex Banach space is reflexive and strictly convex (see [19]). Let C be a closed subset of a Banach space and let T be a mapping from C into itself. We denote by F(T) and Fε (T) for ε > 0, the sets {x ∈ C : x = Tx} and {x ∈ C : x − Tx ≤ ε}, respectively. A mapping T of C into itself is said to be compact if T is continuous and maps bounded sets into relatively compact sets. A mapping T of C into itself is said to be demicompact
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