Strong Convergence Theorems for Semigroups of Asymptotically Nonexpansive Mappings in Banach Spaces

Abstract

Let 𝑋 be a real reflexive Banach space with a weakly continuous duality mapping 𝐽_𝜑. Let 𝐶 be a nonempty weakly closed star-shaped (with respect to 𝑢) subset of 𝑋. Let ℱ = {𝑇(𝑡)∶𝑡∈[0,+∞)} be a uniformly continuous semigroup of asymptotically nonexpansive self-mappings of 𝐶, which is uniformly continuous at zero. We will show that the implicit iteration scheme: 𝑦_𝑛=𝛼_𝑛𝑢+(1−𝛼_𝑛)𝑇(𝑡_𝑛)𝑦_𝑛, for all 𝑛∈ℕ, converges strongly to a common fixed point of the semigroup ℱ for some suitably chosen parameters {𝛼_𝑛} and {𝑡_𝑛}. Our results extend and improve corresponding ones of Suzuki (2002), Xu (2005), and Zegeye and Shahzad (2009).

1. Introduction

Let 𝐶 be a nonempty subset of a (real) Banach space 𝑋 and 𝑇∶𝐶→𝐶 a mapping. The fixed point set of 𝑇 is defined by 𝐹(𝑇)={𝑥∈𝐶∶𝑇𝑥=𝑥}. We say that 𝑇 is nonexpansive if ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖ for all 𝑥,𝑦 in 𝐶 and asymptotically nonexpansive if there exists a sequence {𝑘_𝑛} in [1,+∞) with lim_𝑛→∞𝑘_𝑛=1 such that ‖𝑇^𝑛𝑥−𝑇^𝑛𝑦‖≤𝑘_𝑛‖𝑥−𝑦‖ for all 𝑥,𝑦 in 𝐶 and 𝑛 in ℕ.

Set ℝ⁺∶=[0,+∞). We call a one-parameter family ℱ∶={𝑇(𝑡)∶𝑡∈ℝ⁺} of mappings from 𝐶 into 𝐶 a strongly continuous semigroup of Lipschitzian mappings if(1) for each 𝑡>0, there exists a bounded function 𝑘(⋅)∶(0,+∞)→(0,+∞) such that 𝑇(𝑡)𝑥−𝑇(𝑡)𝑦≤𝑘(𝑡)𝑥−𝑦∀𝑥,𝑦∈𝐶, (1)(2) 𝑇(0)𝑥=𝑥 for all 𝑥 in 𝐶,(3) 𝑇(𝑠+𝑡)=𝑇(𝑠)𝑇(𝑡) for all 𝑠,𝑡 in ℝ⁺, (4) for each 𝑥 in 𝐶, the mapping 𝑇(⋅)𝑥 from ℝ⁺ into 𝐶 is continuous.

Note that liminf_𝑡→0⁺𝑘(𝑡)≥1. If 𝑘(𝑡)=𝐿 for all 𝑡>0 in (1), then ℱ is called a strongly continuous semigroup of uniformly 𝐿-Lipschitzian mappings. If 𝑘(𝑡)=1 for all 𝑡>0 in (1), then ℱ is called a strongly continuous semigroup of nonexpansive mappings. If 𝑘(𝑡)≥1 for all 𝑡>0 and lim_𝑡→+∞𝑘(𝑡)=1 in (1), then ℱ is called a strongly continuous semigroup of asymptotically nonexpansive mappings. Moreover, ℱ is said to be (right) uniformly continuous if it also holds:(5) For any bounded subset 𝐵 of 𝐶, we have lim_𝑡→0+sup_𝑥∈𝐵‖𝑇(𝑡)𝑥−𝑥‖=0. (2)

We denote by 𝐹(ℱ) the set of common fixed points of ℱ; that is, 𝐹(ℱ)∶=⋂_𝑡∈ℝ⁺𝐹(𝑇(𝑡)).

The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [1] in 1972. They proved that if 𝐶 is a nonempty closed convex bounded subset of a uniformly convex Banach space, then every asymptotically nonexpansive mapping has a fixed point. Several authors have studied the problem of the existence of fixed points of asymptotically nonexpansive mappings in Banach spaces having rich geometric structure; see [2] and the references therein.

Consider a nonempty closed convex subset 𝐶 of a (real) Banach space 𝑋. A classical method to study nonexpansive mappings is to approximate them by contractions. More precisely, for a fixed element 𝑢 in 𝐶, define for each 𝑡 in (0,1) a contraction 𝐺_𝑡 by 𝐺_𝑡𝑥=𝑡𝑢+(1−𝑡)𝑇𝑥∀𝑥∈𝐶. (3) Let 𝑥_𝑡 be the fixed point of 𝐺_𝑡; that is, 𝑥_𝑡=𝑡𝑢+(1−𝑡)𝑇𝑥_𝑡. (4) Browder [3] (Reich [4], resp.) proves that as 𝑡→0⁺, the point 𝑥_𝑡 converges strongly to a fixed point of 𝑇 if 𝑋 is a Hilbert space (uniformly smooth Banach space, resp.). Many authors (see, e.g., [5–13]) have studied strong convergence of approximates {𝑥_𝑡} for asymptotically nonexpansive self-mappings 𝑇 in Banach spaces under the additional assumption 𝑥_𝑡−𝑇𝑥_𝑡→0 as 𝑛→∞. This additional assumption can be removed when 𝑇 is uniformly asymptotically regular.

Suzuki [14] initiated the following implicit iteration process for a semigroup ℱ∶={𝑇(𝑡)∶𝑡∈ℝ⁺} of nonexpansive mappings in a Hilbert space: 𝑦_𝑛=𝛼_𝑛𝑢+1−𝛼_𝑛𝑇𝑡_𝑛𝑦_𝑛,∀𝑛∈ℕ. (5) Xu [15] extended Suzuki’s result to uniformly convex Banach spaces with weakly sequentially continuous duality mappings. Recently, Zegeye and Shahzad [16] extended results of Xu [15] and established the following strong convergence theorem.

Theorem ZS. Let 𝐶 be a nonempty closed convex bounded subset of a real uniformly convex Banach space 𝑋 with a weakly continuous duality mapping 𝐽_𝜑 with gauge 𝜑. Let ℱ∶={𝑇(𝑡)∶𝑡∈ℝ⁺} be a strongly continuous asymptotically nonexpansive semigroups with net {𝑘(𝑡)}⊂[1,+∞). Assume that 𝐹(ℱ) is a sunny nonexpansive retract of 𝐶 with 𝑃 as the sunny nonexpansive retraction. Assume that {𝑡_𝑛}⊂(0,+∞) and {𝛼_𝑛}⊂(0,1) such that (1−𝛼_𝑛)𝑘(𝑡_𝑛)≤1 for all 𝑛∈ℕ, lim_𝑛→∞𝑡_𝑛=lim_𝑛→∞(𝛼_𝑛/𝑡_𝑛)=0, and {𝛼_𝑛/(1−(1−𝛼_𝑛)𝑘(𝑡_𝑛))} is bounded. Let 𝑢 in 𝐶 be fixed. (1) There exists a sequence {𝑦_𝑛} in 𝐶 such that 𝑦_𝑛=𝛼_𝑛𝑢+1−𝛼_𝑛𝑇𝑡_𝑛𝑦_𝑛∀𝑛∈ℕ, (6) which converges strongly to an element of 𝐹(ℱ). (2) Every sequence {𝑥_𝑛} defined iteratively with any 𝑥₁ in 𝐶, and 𝑥_𝑛+1=𝛼_𝑛𝑢+1−𝛼_𝑛𝑇𝑡_𝑛𝑥_𝑛∀𝑛∈ℕ, (7) converges strongly to an element of 𝐹(ℱ), provided that ‖𝑥_𝑛+1−𝑥_𝑛‖≤𝐷𝛼_𝑛 for some 𝐷>0.

Problem 1. Is it possible to drop the uniform convexity assumption in Theorem ZS?

Motivated by Schu [13], the purpose of this paper is to further analyze strong convergence of (6) and (7) for strongly continuous semigroups of asymptotically nonexpansive mappings defined on a set which is not necessarily convex. It is important and actually quite surprising that we are able to do so for the class of Banach spaces which are not necessarily uniformly convex. It should be noted that, in this generality, Theorem ZS does not apply. Our results are definitive, settle Problem 1, and also improve results of Suzuki [14], Xu [15], and Zegeye and Shahzad [16].

2. Preliminaries

Let 𝐶 be a nonempty subset of a (real) Banach space 𝑋 with dual space 𝑋^∗. We call a mapping 𝑇∶𝐶→𝐶  weakly contractive if 𝑇𝑥−𝑇𝑦≤𝑥−𝑦−𝜓𝑥−𝑦∀𝑥,𝑦∈𝐶, (8) where 𝜓∶[0,+∞)→[0,+∞) is a continuous and nondecreasing function such that 𝜓(0)=0,  𝜓(𝑡)>0 for 𝑡>0, and lim_𝑡→+∞𝜓(𝑡)=+∞.

By a gauge we mean a continuous strictly increasing function 𝜑 defined on ℝ⁺∶=[0,+∞) such that 𝜑(0)=0 and lim_𝑟→+∞𝜑(𝑟)=+∞. Associated with a gauge 𝜑, the (generally multivalued) duality mapping 𝐽_𝜑∶𝑋→𝑋^∗ is defined by 𝐽_𝜑(𝑥)∶=𝑥^∗∈𝑋^∗∶𝑥,𝑥^∗=‖𝑥‖𝜑𝑥,𝑥(‖‖)_∗=𝜑(‖𝑥‖). (9) Clearly, the (normalized) duality mapping 𝐽 corresponds to the gauge 𝜑(𝑡)=𝑡. In general, 𝐽_𝜑(𝑥)=𝜑(‖𝑥‖)‖𝑥‖𝐽(𝑥),𝑥≠0. (10)

Recall that 𝑋 is said to have a weakly (resp, sequentially) continuous duality mapping if there exists a gauge 𝜑 such that the duality mapping 𝐽_𝜑 is single valued and (resp, sequentially) continuous from 𝑋 with the weak topology to 𝑋^∗ with the weak* topology. Every ℓ^𝑝(1<𝑝<+∞) space has a weakly continuous duality mapping with the gauge 𝜑(𝑡)=𝑡^𝑝−1 (for more details see [17, 18]). We know that if 𝑋 admits a weakly sequentially continuous duality mapping, then 𝑋 satisfies Opial’s condition; that is, if {𝑥_𝑛} is a sequence weakly convergent to 𝑥 in 𝑋, then there holds the inequality limsup_𝑛→∞𝑥_𝑛−𝑥<limsup_𝑛→∞𝑥_𝑛−𝑦,𝑦∈𝑋with𝑦≠𝑥. (11)

Browder [19] initiated the study of certain classes of nonlinear operators by means of a duality mapping 𝐽_𝜑. Define Φ(𝑡)∶=^𝑡₀𝜑(𝑠)𝑑𝑠∀𝑡≥0. (12) Then, it is known that 𝐽_𝜑(𝑥) is the subdifferential of the convex function Φ(‖⋅‖) at 𝑥; that is, 𝐽_𝜑(𝑥)=𝜕Φ(‖𝑥‖),𝑥∈𝑋, (13) where 𝜕 denotes the subdifferential in the sense of convex analysis. We need the subdifferential inequality Φ𝑥+𝑦≤Φ(‖𝑥‖)+𝑦,𝑗𝑥+𝑦∀𝑥,𝑦∈𝑋,𝑗𝑥+𝑦∈𝐽_𝜑𝑥+𝑦. (14) For a smooth 𝑋, we have Φ𝑥+𝑦≤Φ(‖𝑥‖)+𝑦,𝐽_𝜑𝑥+𝑦∀𝑥,𝑦∈𝑋, (15) or considering the normalized duality mapping 𝐽, we have 𝑥+𝑦²≤‖𝑥‖²+2𝑦,𝐽𝑥+𝑦∀𝑥,𝑦∈𝑋. (16)

Assume that a sequence {𝑥_𝑛} in 𝑋 converges weakly to a point 𝑥 in 𝑋. Then the following identity holds; limsup_𝑛→∞Φ𝑥_𝑛−𝑦=limsup_𝑛→∞Φ𝑥_𝑛−𝑥+Φ𝑥−𝑦∀𝑦∈𝑋. (17)

Remark 2. For any 𝑘 with 0≤𝑘≤1, we have 𝜑()(𝑡)Φ()=𝑘𝑡≤𝜑∀𝑡>0,𝑘𝑡₀^𝑘𝑡𝜑(𝑠)𝑑𝑠=𝑘^𝑡₀𝜑(𝑘𝑢)𝑑𝑢≤𝑘^𝑡₀𝜑(𝑢)𝑑𝑢=𝑘Φ(𝑡)∀𝑡>0. (18)

We need the following demiclosedness principle for asymptotically nonexpansive mappings in a Banach space.

Lemma 3 (see [17, Corollary  5.6.4], [10]). Let 𝑋 be a Banach space with a weakly continuous duality mapping 𝐽_𝜑∶𝑋→𝑋^∗ with gauge function 𝜑. Let 𝐶 be a nonempty closed convex subset of 𝑋 and 𝑇∶𝐶→𝐶 an asymptotically nonexpansive mapping. Then, 𝐼−𝑇 is demiclosed at zero; that is, if {𝑥_𝑛} is a sequence in 𝐶 which converges weakly to 𝑥 and if the sequence {𝑥_𝑛−𝑇𝑥_𝑛} converges strongly to zero, then 𝑥−𝑇𝑥=0.

Let 𝐶 be a convex subset of a Banach space 𝑋 and 𝐷 a nonempty subset of 𝐶. Then, a continuous mapping 𝑃 from 𝐶 onto 𝐷 is called a retraction if 𝑃𝑥=𝑥 for all 𝑥 in 𝐷; that is, 𝑃²=𝑃. A retraction 𝑃 is said to be sunny if 𝑃(𝑃𝑥+𝑡(𝑥−𝑃𝑥))=𝑃𝑥 for each 𝑥 in 𝐶 and 𝑡≥0 with 𝑃𝑥+𝑡(𝑥−𝑃𝑥) in 𝐶. If the sunny retraction 𝑃 is also nonexpansive, then 𝐷 is said to be a sunny nonexpansive retract of 𝐶. The sunny nonexpansive retraction 𝑄 from 𝐶 onto 𝐷 is unique if 𝑋 is smooth.

Lemma 4 (see Goebel and Reich [20, Lemma 13.1]). Let 𝐶 be a convex subset of a smooth Banach space 𝑋, 𝐷 a nonempty subset of 𝐶, and 𝑃 a retraction from 𝐶 onto 𝐷. Then, the following are equivalent. (a)𝑃  is sunny and nonexpansive. (b)⟨𝑥−𝑃𝑥,𝐽(𝑧−𝑃𝑥)⟩≤0  for all  𝑥∈𝐶,𝑧∈𝐷.(c)⟨𝑥−𝑦,𝐽(𝑃𝑥−𝑃𝑦)⟩≥‖𝑃𝑥−𝑃𝑦‖²  for all  𝑥,𝑦∈𝐶.

Lemma 5 (see [21]). Let {𝛼_𝑛} and {𝛾_𝑛} be two real sequences such that (i){𝛼_𝑛}⊂[0,1] and ∑^∞_𝑛=1𝛼_𝑛=+∞, (ii)limsup_𝑛→∞𝛾_𝑛≤0.
Let {𝜆_𝑛} be a sequence of nonnegative numbers which satisfies the inequality 𝜆_𝑛+1≤1−𝛼_𝑛𝜆_𝑛+𝛼_𝑛𝛾_𝑛,𝑛∈ℕ. (19) Then, lim_𝑛→∞𝜆_𝑛=0.

3. Existence of Common Fixed Points

We begin with the following.

Proposition 6. Let 𝐶 be a nonempty closed subset of a Banach space 𝑋. Let ℱ={𝑇(𝑡)∶𝑡∈ℝ⁺} be a uniformly continuous semigroup of asymptotically nonexpansive mappings from 𝐶 into itself with a net {𝑘(𝑡)∶𝑡∈(0,+∞)}. Let {𝑏_𝑛} be a sequence in (0,1) and {𝑡_𝑛} a sequence in (0,+∞) with 𝑘(𝑡_𝑛)−1<𝑏_𝑛 for all 𝑛 in ℕ. Assume that 𝑢∈𝐶 and 𝑏_𝑛𝑢+(1−𝑏_𝑛)𝑇(𝑡_𝑛)𝑥∈𝐶 for all 𝑥 in 𝐶 and 𝑛 in ℕ. (a) There exists a sequence {𝑦_𝑛} in 𝐶 defined by𝑦_𝑛=𝑏_𝑛𝑢+1−𝑏_𝑛𝑇𝑡_𝑛𝑦_𝑛∀𝑛∈ℕ. (20)(b) If the sequence {𝑦_𝑛} described by (20) is bounded and  lim_𝑛→∞𝑡_𝑛=lim_𝑛→∞(𝑏_𝑛/𝑡_𝑛)=0, then𝑦_𝑛−𝑇(𝑡)𝑦_𝑛⟶0𝑎𝑠𝑛⟶∞,∀𝑡>0. (21)

Proof. (a) Set 𝜚_𝑛∶=(𝑘(𝑡_𝑛)−1)/𝑏_𝑛. Since 𝑘(𝑡_𝑛)−1<𝑏_𝑛 for all 𝑛 in ℕ, it follows that 𝜚_𝑛<1≤𝑘(𝑡_𝑛), and hence 0<(1−𝑏_𝑛)𝑘(𝑡_𝑛)=(1−𝑏_𝑛)(1+𝜚_𝑛𝑏_𝑛)<1 for all 𝑛 in ℕ. For each 𝑛 in ℕ, the mapping 𝐺_𝑛∶𝐶→𝐶 defined by 𝐺_𝑛𝑦∶=𝑏_𝑛𝑢+1−𝑏_𝑛𝑇𝑡_𝑛𝑦,𝑦∈𝐶 (22) is a contraction with Lipschitz constant (1−𝑏_𝑛)𝑘(𝑡_𝑛). Therefore, there exists a sequence {𝑦_𝑛} in 𝐶 described by (20).(b) Suppose that the sequence {𝑦_𝑛} in 𝐶 described by (20) is bounded and lim_𝑛→∞𝑡_𝑛=lim_𝑛→∞(𝑏_𝑛/𝑡_𝑛)=0. From (20), we have 𝑇𝑡_𝑛𝑦_𝑛=11−𝑏_𝑛𝑦_𝑛−𝑏_𝑛𝑢≤11−𝑏_𝑛𝑦_𝑛+𝑏_𝑛‖𝑢‖. (23)
Without loss of generality, we may assume that {𝑏_𝑛} is bounded away from 1. Then, there exists a positive constant 𝛿 such that 𝑏_𝑛≤𝛿<1 for all 𝑛 in ℕ. Since 𝐵∶={𝑦_𝑛∶𝑛∈ℕ} is bounded, it follows from (23) that {𝑇(𝑡_𝑛)𝑦_𝑛} is bounded.
Set 𝐿∶=sup{𝑘(𝑡)∶𝑡∈ℝ⁺}. For 𝑡>0, we have 𝑦_𝑛−𝑇(𝑡)𝑦_𝑛≤^{[𝑡/𝑡𝑛]−1}_𝑖=0𝑇𝑖𝑡_𝑛𝑦_𝑛−𝑇(𝑖+1)𝑡_𝑛𝑦_𝑛+𝑇𝑡𝑡_𝑛𝑡_𝑛𝑦_𝑛−𝑇(𝑡)𝑦_𝑛≤𝑡𝑡_𝑛𝐿𝑦_𝑛−𝑇𝑡_𝑛𝑦_𝑛𝑇𝑡+𝐿𝑡−𝑡_𝑛𝑡_𝑛𝑦_𝑛−𝑦_𝑛=𝐿𝑏_𝑛𝑡𝑡_𝑛𝑢−𝑇𝑡_𝑛𝑦_𝑛𝑇𝑡+𝐿𝑡−𝑡_𝑛𝑡_𝑛𝑦_𝑛−𝑦_𝑛𝑏≤𝑡𝐿^𝑛𝑡_𝑛𝑢−𝑇𝑡_𝑛𝑦_𝑛+𝐿sup_𝑦∈𝐵𝑇𝑠_𝑛𝑦−𝑦, (24) for all 𝑛 in ℕ, where 𝑠_𝑛=𝑡−[𝑡/𝑡_𝑛]𝑡_𝑛. Note that 𝑠_𝑛=𝑡−[𝑡/𝑡_𝑛]𝑡_𝑛≤𝑡_𝑛→0 as 𝑛→∞. Since ℱ is uniformly continuous at 0, it follows that sup_𝑦∈𝐵‖𝑇(𝑠_𝑛)𝑦−𝑦‖→0 as 𝑛→∞. Therefore, 𝑦_𝑛−𝑇(𝑡)𝑦_𝑛→0 as 𝑛→∞.

Remark 7. (a) If 𝐶 is star shaped with respect to 𝑢 in 𝐶, then the assumption “𝑏_𝑛𝑢+(1−𝑏_𝑛)𝑇(𝑡_𝑛)𝑥∈𝐶 for all 𝑥 in 𝐶 and 𝑛 in ℕ” in Proposition 6 is automatically satisfied.
(b) If ℱ={𝑇(𝑡)∶𝑡∈ℝ⁺} is a strongly continuous semigroup of nonexpansive mappings with 𝐹(ℱ)≠∅, the sequence {𝑦_𝑛} in 𝐶 described by (20) is bounded (see [15, Theorem 3.3]).

Theorem 8. Let 𝑋 be a real reflexive Banach space with a weakly continuous duality mapping 𝐽_𝜑 with gauge function 𝜑, and 𝐶 a nonempty weakly closed subset of 𝑋. Let ℱ={𝑇(𝑡)∶𝑡∈ℝ⁺} be a uniformly continuous semigroup of asymptotically nonexpansive mappings from 𝐶 into itself with a net {𝑘(𝑡)∶𝑡∈(0,+∞)}. Let {𝑏_𝑛} be a sequence in (0,1) and {𝑡_𝑛} a sequence in (0,+∞) with 𝑘(𝑡_𝑛)−1<𝑏_𝑛 for all 𝑛 in ℕ satisfying the condition lim_𝑛→∞𝑡_𝑛=lim_𝑛→∞𝑏_𝑛𝑡_𝑛=lim_𝑛→∞𝑘𝑡_𝑛−1𝑏_𝑛=0. (25) Assume that 𝑢∈𝐶 such that the sequence {𝑦_𝑛} described by (20) is bounded in 𝐶. Then, (a) 𝐹(ℱ)≠∅. (b) {𝑦_𝑛} converges strongly to an element 𝑦^∗∈𝐹(ℱ) which holds the inequality 𝑦^∗−𝑢,𝐽𝑦^∗−𝑣≤0∀𝑣∈𝐹(ℱ). (26)

Proof. By Proposition 6 (b), we have 𝑦_𝑛−𝑇(𝑡)𝑦_𝑛→0 as 𝑛→∞ for all 𝑡>0. Since {𝑦_𝑛} is bounded, there exists a subsequence {𝑦_𝑛𝑖} of {𝑦_𝑛} such that 𝑦_𝑛𝑖⇀𝑦^∗∈𝐶.(a) For each 𝑟>0, we have 𝑇(𝑟)^𝑚𝑥−𝑇(𝑟)^𝑚𝑦=𝑇()()𝑦(𝑚𝑟𝑥−𝑇𝑚𝑟≤𝑘𝑚𝑟)𝑥−𝑦∀𝑥,𝑦∈𝐶,𝑚∈ℕ, (27)that is, each 𝑇(𝑟) is asymptotically nonexpansive mapping. Since lim_𝑛→∞‖𝑦_𝑛−𝑇(𝑡)𝑦_𝑛‖=0 for all 𝑡>0, it follows from Lemma 3 that 𝑇(𝑡)𝑦^∗=𝑦^∗ for all 𝑡>0. Hence, 𝐹(ℱ)≠∅.(b) Since {𝑦_𝑛} is bounded, there exists a constant 𝑀≥0 such that ‖𝑦_𝑛−𝑝‖≤𝑀 for all 𝑛 in ℕ and 𝑝 in 𝐹(ℱ). For any 𝑣 in 𝐹(ℱ), from (15) and Remark 2, we have Φ𝑦_𝑛−𝑦^∗=Φ𝑏_𝑛𝑢+1−𝑏_𝑛𝑇𝑡_𝑛𝑦_𝑛−𝑦^∗≤Φ1−𝑏_𝑛𝑇𝑡_𝑛𝑦_𝑛−𝑦^∗+𝑏_𝑛𝑢−𝑦^∗,𝐽_𝜑𝑦_𝑛−𝑦^∗≤Φ1−𝑏_𝑛𝑘𝑡_𝑛𝑦_𝑛−𝑦^∗+𝑏_𝑛𝑢−𝑦^∗,𝐽_𝜑𝑦_𝑛−𝑦^∗≤1−𝑏_𝑛𝑘𝑡_𝑛Φ𝑦_𝑛−𝑦^∗+𝑏_𝑛𝑢−𝑦^∗,𝐽_𝜑𝑦_𝑛−𝑦^∗. (28)Since 𝑘(𝑡)≥1 for all 𝑡>0, we have Φ𝑦_𝑛−𝑦^∗≤𝑘𝑡_𝑛−1𝑏_𝑛Φ𝑦_𝑛−𝑦^∗+𝑢−𝑦^∗,𝐽_𝜑𝑦_𝑛−𝑦^∗≤𝑘𝑡_𝑛−1𝑏_𝑛Φ(𝑀)+𝑢−𝑦^∗,𝐽_𝜑𝑦_𝑛−𝑦^∗. (29) Observing that (𝑘(𝑡_𝑛𝑖)−1)/𝑏_𝑛𝑖→0, 𝑦_𝑛𝑖⇀𝑦^∗, and 𝐽_𝜑 is weakly continuous, we conclude from (29) that 𝑦_𝑛𝑖→𝑦^∗∈𝐶 as 𝑖→∞ because 𝐶 is closed.
We prove that {𝑦_𝑛} converges strongly to 𝑦^∗. Suppose, for contradiction, that {𝑦_𝑛𝑗} is another subsequence of {𝑦_𝑛} such that 𝑦_𝑛𝑗→𝑧^∗∈𝐶 with 𝑧^∗≠𝑦^∗. Since lim_𝑛→∞(𝑦_𝑛−𝑇(𝑡)𝑦_𝑛)=0 for all 𝑡>0, we have 𝑧^∗∈𝐹(ℱ). For any 𝑣 in 𝐹(ℱ), we have 𝑦_𝑛−𝑇𝑡_𝑛𝑦_𝑛,𝐽_𝜑𝑦_𝑛=𝑦−𝑣_𝑛−𝑣+𝑇𝑡_𝑛𝑣−𝑇𝑡_𝑛𝑦_𝑛,𝐽_𝜑𝑦_𝑛=𝑦−𝑣_𝑛−𝑣𝜑𝑦_𝑛−𝑇𝑡−𝑣_𝑛𝑦_𝑛−𝑇𝑡_𝑛𝑣,𝐽_𝜑𝑦_𝑛𝑘𝑡−𝑣≥−_𝑛−1𝑦_𝑛−𝑣𝜑𝑦_𝑛−𝑣∀𝑛∈ℕ. (30) From (20), we have 𝑦_𝑛−𝑢,𝐽_𝜑𝑦_𝑛=−𝑣1−𝑏_𝑛𝑇𝑡_𝑛𝑦_𝑛−𝑢,𝐽_𝜑𝑦_𝑛=−𝑣1−𝑏_𝑛𝑇𝑡_𝑛𝑦_𝑛−𝑦_𝑛+𝑦_𝑛−𝑢,𝐽_𝜑𝑦_𝑛−𝑣. (31) It follows that 𝑦_𝑛−𝑢,𝐽_𝜑𝑦_𝑛≤−𝑣1−𝑏_𝑛𝑏_𝑛𝑇𝑡_𝑛𝑦_𝑛−𝑦_𝑛,𝐽_𝜑𝑦_𝑛≤−𝑣1−𝑏_𝑛𝑘𝑡_𝑛−1𝑏_𝑛𝑦_𝑛−𝑣𝜑𝑦_𝑛≤𝑘𝑡−𝑣_𝑛−1𝑏_𝑛𝑀𝜑(𝑀) (32) for all 𝑣 in 𝐹(ℱ) and 𝑛 in ℕ. Since lim_𝑛→∞(𝑘(𝑡_𝑛)−1)/𝑏_𝑛=0, we obtain from (32) that 𝑦^∗−𝑢,𝐽_𝜑𝑦^∗−𝑧^∗≤0,⟨𝑧^∗−𝑢,𝐽_𝜑𝑧^∗−𝑦^∗⟩≤0. (33) Addition of (33) yields 𝑦^∗−𝑧^∗,𝐽_𝜑𝑦^∗−𝑧^∗≤0. (34) Hence, 𝑦^∗=𝑧^∗, a contradiction. Therefore, {𝑦_𝑛} converges strongly to 𝑦^∗∈𝐶.
Finally, from (32), we conclude that 𝑦^∗ satisfies (26).

Let ℱ={𝑇(𝑡)∶𝑡∈ℝ⁺} be a strongly continuous semigroup of asymptotically nonexpansive mappings from 𝐶 into itself and 𝑢∈𝐶. Motivated by Morales and Jung [22, Theorem 1] and Morales [23, Theorem 2], we define 𝐸^𝑢_ℱ(𝐶)={𝜆𝑢+(1−𝜆)𝑇(𝑡)𝑥∶𝑥∈𝐶,𝜆∈[0,1],𝑡>0}. (35)

Corollary 9. Let 𝑋 be a real reflexive Banach space with a weakly continuous duality mapping 𝐽_𝜑 with gauge function 𝜑 and 𝐶 a nonempty weakly closed subset of 𝑋. Let ℱ={𝑇(𝑡)∶𝑡∈ℝ⁺} be a uniformly continuous semigroup of asymptotically nonexpansive mappings from 𝐶 into itself with a net {𝑘(𝑡)∶𝑡∈(0,+∞)}. Let {𝑏_𝑛} be a sequence in (0,1) and {𝑡_𝑛} a sequence in (0,+∞) with 𝑘(𝑡_𝑛)−1<𝑏_𝑛 for all 𝑛 in ℕ satisfying condition (25). Assume that 𝑢∈𝐶 such that 𝐶 is star shaped with respect to 𝑢, and set 𝐸^𝑢_ℱ(𝐶) defined by (35) is bounded. (a)For each 𝑛 in ℕ, there is exactly one point 𝑦_𝑛 in 𝐶 described by (20).(b)𝐹(ℱ) is nonempty. Moreover, {𝑦_𝑛} converges strongly to an element 𝑦^∗ in 𝐹(ℱ) which holds (26).

Corollary 9 improves and generalizes several recent results of this nature. Indeed, it extends [13, Theorem 1.7] from the class of asymptotically nonexpansive mappings to a uniformly continuous semigroup of asymptotically nonexpansive mappings without uniformly asymptotic regularity assumption. In particular, Corollary 9 improves Theorem ZS in the following ways. (1)Convexity of 𝐶 is not required. (2)The assumption “uniform convexity” of the underlying space is not required. (3)For convergence of {𝑦_𝑛}, condition “𝐹(ℱ) is a sunny nonexpansive retract of 𝐶” is not assumed.

Next, we show that 𝐹(ℱ) is a nonempty sunny nonexpansive retract of 𝐶.

Theorem 10. Let 𝑋 be a real reflexive Banach space with a weakly continuous duality mapping 𝐽_𝜑 with gauge function 𝜑. Let 𝐶 be a nonempty closed convex-bounded subset of 𝑋. Let ℱ={𝑇(𝑡)∶𝑡∈ℝ⁺} be a uniformly continuous semigroup of asymptotically nonexpansive mappings from 𝐶 into itself with net {𝑘(𝑡)∶𝑡∈(0,+∞)}. Then, 𝐹(ℱ) is a nonempty sunny nonexpansive retract of 𝐶.

Proof. By Theorem 8 (a), 𝐹(ℱ)≠∅. As in Theorem 8 (b), for each 𝑢 in 𝐶 the sequence {𝑦_𝑛} converges strongly to an element 𝑦^∗ in 𝐹(ℱ). Define a mapping 𝑄∶𝐶→𝐹(ℱ) by 𝑄𝑢=lim_𝑛→∞𝑦_𝑛. (36) By (32), we have 𝑦_𝑛−𝑢,𝐽_𝜑𝑦_𝑛≤𝑘𝑡−𝑣^𝑛−1𝑏_𝑛diam(𝐶)𝜑(diam(𝐶)) (37) for all 𝑢 in 𝐶,𝑣 in 𝐹(ℱ), and 𝑛 in ℕ, where diam(𝐵) is the diameter of set 𝐵. Letting 𝑛→∞, we obtain that 𝑄𝑢−𝑢,𝐽_𝜑(𝑄𝑢−𝑣)≤0∀𝑣∈𝐹(ℱ). (38) Therefore, by Lemma 4, we conclude that 𝑄 is sunny nonexpansive.

Remark 11. In view of Remark 7 (b), the nonemptهness of the common fixed point set of a uniformly continuous semigroup of nonexpansive mappings implies that the sequence {𝑦_𝑛} in 𝐶 described by (20) is bounded. So we can drop the boundedness assumption of domain 𝐶 in Theorem 10, provided that 𝐹(ℱ)≠∅.

Corollary 12. Let 𝑋 be a real reflexive Banach space with a weakly continuous duality mapping 𝐽_𝜑 with gauge function 𝜑 and 𝐶 a nonempty weakly closed subset of 𝑋. Let ℱ={𝑇(𝑡)∶𝑡∈ℝ⁺} be a uniformly continuous semigroup of nonexpansive mappings from 𝐶 into itself with 𝐹(ℱ)≠∅. Let {𝑏_𝑛} be a sequence in (0,1) and {𝑡_𝑛} a sequence in (0,+∞) such that lim_𝑛→∞𝑡_𝑛=lim_𝑛→∞(𝑏_𝑛/𝑡_𝑛)=0. Assume that 𝑢∈𝐶 and that 𝐶 is star shaped with respect to 𝑢. (a)For each 𝑛 in ℕ, there is exactly one point 𝑦_𝑛 in 𝐶 described by (20).(b){𝑦_𝑛} converges strongly to an element 𝑦^∗ in 𝐹(ℱ) which holds (26).

Corollary 12 is an improvement of [14, Theorem 3] and [15, Theorem 3.3], where strong convergence theorems were established in Hilbert and uniformly convex Banach spaces, respectively.

4. Approximation of Common Fixed Points

Theorem 13. Let 𝑋 be a real reflexive Banach space with a weakly continuous duality mapping 𝐽_𝜑 with gauge function 𝜑. Let 𝐶 be a nonempty closed convex bounded subset of 𝑋 and ℱ={𝑇(𝑡)∶𝑡∈ℝ⁺} a uniformly continuous semigroup of asymptotically nonexpansive mappings from 𝐶 into itself with a net {𝑘(𝑡)∶𝑡∈(0,+∞)}. For given 𝑢,𝑥₁ in 𝐶, let {𝑥_𝑛} be a sequence in 𝐶 generated by (7), where {𝛼_𝑛} is a sequence in (0,1) and {𝑡_𝑛} is a decreasing sequence in (0,+∞) satisfying the following conditions.(C1) lim_𝑛→∞𝑡_𝑛=lim_𝑛→∞(𝛼_𝑛/𝑡_𝑛)=0 and ∑^∞_𝑛=1𝛼_𝑛=∞,(C2) lim_𝑛→∞(|𝛼_𝑛−𝛼_𝑛+1|/𝛼_𝑛)=0 or ∑^∞_𝑛=1|𝛼_𝑛−𝛼_𝑛+1|<+∞,(C3) lim_𝑛→∞(‖𝑇(𝑡_𝑛−𝑡_𝑛+1)𝑥_𝑛−𝑥_𝑛‖/𝛼_𝑛+1)=0, (C4) (1−𝛼_𝑛)𝑘(𝑡_𝑛)≤1forall𝑛∈ℕ and lim_𝑛→∞((𝑘(𝑡_𝑛)−1)/𝛼_𝑛)=0.
Then,(a) 𝐹(ℱ)≠∅. (b) {𝑥_𝑛} converges strongly to 𝑄_𝐹(ℱ)(𝑢), where 𝑄_𝐹(ℱ) is a sunny nonexpansive retraction of 𝐶 onto 𝐹(ℱ).

Proof. It follows from Theorem 10 that 𝐹(ℱ)≠∅, and there is a sunny nonexpansive retraction 𝑄_𝐹(ℱ) of 𝐶 onto 𝐹(ℱ). Set 𝑦^∗∶=𝑄_𝐹(ℱ)(𝑢), 𝐵∶={𝑥_𝑛}, 𝐿∶=sup{𝑘(𝑡)∶𝑡>0}, and 𝑀∶=sup{‖𝑥_𝑛−𝑦^∗‖∶𝑛∈ℕ}. Since 𝐶 is bounded, there exists a constant 𝐾>0 such that 𝑥_𝑛+1−𝑥_𝑛≤𝐾,𝑇𝑡_𝑛𝑥_𝑛−𝑢≤𝐾∀𝑛∈ℕ. (39) Hence, for all 𝑛>1, we have 𝑥_𝑛+1−𝑥_𝑛≤𝛼_𝑛−𝛼_𝑛−1𝑢−𝑇𝑡_𝑛−1𝑥_𝑛−1+1−𝛼_𝑛𝑇𝑡_𝑛𝑥_𝑛−𝑇𝑡_𝑛−1𝑥_𝑛−1≤𝛼_𝑛−𝛼_𝑛−1𝐾+1−𝛼_𝑛𝑇𝑡_𝑛𝑥_𝑛−𝑇𝑡_𝑛−1𝑥_𝑛−1≤𝛼_𝑛−𝛼_𝑛−1𝐾+1−𝛼_𝑛𝑇𝑡_𝑛𝑥_𝑛−𝑇𝑡_𝑛𝑥_𝑛−1+𝑇𝑡_𝑛𝑥_𝑛−1−𝑇𝑡_𝑛−1𝑥_𝑛−1≤𝛼_𝑛−𝛼_𝑛−1𝐾+1−𝛼_𝑛𝑘𝑡_𝑛𝑥_𝑛−𝑥_𝑛−1+𝑇𝑡_𝑛𝑥_𝑛−1−𝑇𝑡_𝑛−1𝑥_𝑛−1≤1−𝛼_𝑛𝑘𝑡_𝑛𝑥_𝑛−𝑥_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝐾+𝑘𝑡_𝑛𝑇𝑡_𝑛−1−𝑡_𝑛𝑥_𝑛−1−𝑥_𝑛−1≤1−𝛼_𝑛𝑥_𝑛−𝑥_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝐾+𝐿𝑇𝑡_𝑛−1−𝑡_𝑛𝑥_𝑛−1−𝑥_𝑛−1+1−𝛼_𝑛𝑘𝑡_𝑛−1𝑥_𝑛−𝑥_𝑛−1≤1−𝛼_𝑛𝑥_𝑛−𝑥_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝐾+𝐿𝑇𝑡_𝑛−1−𝑡_𝑛𝑥_𝑛−1−𝑥_𝑛−1+𝑘𝑡_𝑛−1𝐾. (40) Hence, by Lemma 5 and assumptions (C1)~(C4), we conclude that 𝑥_𝑛+1−𝑥_𝑛→0 as 𝑛→∞. Fix 𝑡>0, we have 𝑥_𝑛−𝑇(𝑡)𝑥_𝑛≤^{[𝑡/𝑡𝑛]−1}_𝑖=0𝑇𝑖𝑡_𝑛𝑥_𝑛−𝑇(𝑖+1)𝑡_𝑛𝑥_𝑛+𝑇𝑡𝑡_𝑛𝑡_𝑛𝑥_𝑛−𝑇(𝑡)𝑥_𝑛≤𝑡𝑡_𝑛𝐿𝑥_𝑛−𝑇𝑡_𝑛𝑥_𝑛𝑇𝑡+𝐿𝑡−𝑡_𝑛𝑡_𝑛𝑥_𝑛−𝑥_𝑛𝑡≤𝐿𝑡_𝑛𝑥_𝑛−𝑥_𝑛+1+𝑥_𝑛+1−𝑇𝑡_𝑛𝑥_𝑛𝑇𝑡+𝐿𝑡−𝑡_𝑛𝑡_𝑛𝑥_𝑛−𝑥_𝑛𝑡≤𝐿𝑡_𝑛𝑥_𝑛−𝑥_𝑛+1𝛼+𝑡𝐿^𝑛𝑡_𝑛𝑢−𝑇𝑡_𝑛𝑥_𝑛+𝐿max𝑇(𝑠)𝑥_𝑛−𝑥_𝑛∶0≤𝑠≤𝑡_𝑛 (41) for all 𝑛 in ℕ, which gives that 𝑥_𝑛−𝑇(𝑡)𝑥_𝑛→0 as 𝑛→∞. We may assume that 𝑥_𝑛𝑖⇀𝑝∈𝐶 as 𝑖→∞. In view of the assumption that the duality mapping 𝐽_𝜑 is weakly sequentially continuous, it follows from Lemma 3 that 𝑝∈𝐹(ℱ). Thus, by the weak continuity of 𝐽_𝜑 and Lemma 4, we have limsup_𝑛→∞𝑢−𝑄_𝐹(ℱ)(𝑢),𝐽_𝜑𝑥_𝑛−𝑄_𝐹(ℱ)(𝑢)=lim_𝑖→∞𝑢−𝑄_𝐹(ℱ)(𝑢),𝐽_𝜑𝑥_𝑛𝑖−𝑄_𝐹(ℱ)(𝑢)=𝑢−𝑄_𝐹(ℱ)(𝑢),𝐽_𝜑𝑝−𝑄_𝐹(ℱ)(𝑢)≤0. (42) Define 𝛾_𝑛∶=max{0,⟨𝑢−𝑦^∗,𝐽_𝜑(𝑥_𝑛+1−𝑦^∗)⟩}. From (7), we obtain Φ𝑥_𝑛+1−𝑦^∗=Φ𝛼_𝑛𝑢−𝑦^∗+1−𝛼_𝑛𝑇𝑡_𝑛𝑥_𝑛−𝑦^∗≤Φ1−𝛼_𝑛𝑇^𝑛𝑥_𝑛−𝑦^∗+𝛼_𝑛𝑢−𝑦^∗,𝐽_𝜑𝑥_𝑛+1−𝑦^∗≤Φ1−𝛼_𝑛𝑘𝑡_𝑛𝑥_𝑛−𝑦^∗+𝛼_𝑛𝛾_𝑛. (43) Since (1−𝛼_𝑛)𝑘(𝑡_𝑛)≤1 for all 𝑛 in ℕ, it follows from Remark 2 that Φ𝑥_𝑛+1−𝑦^∗≤1−𝛼_𝑛𝑘𝑡_𝑛Φ𝑥_𝑛−𝑦^∗+𝛼_𝑛𝛾_𝑛≤1−𝛼_𝑛Φ𝑥_𝑛−𝑦^∗+𝛼_𝑛𝛾_𝑛+𝑘𝑡_𝑛−1Φ(𝑀). (44) Note that lim_𝑛→∞𝛾_𝑛=0 and lim_𝑛→∞((𝑘(𝑡_𝑛)−1)/𝛼_𝑛)=0. Using Lemma 5, we obtain that {𝑥_𝑛} converges strongly to 𝑦^∗.

One can carry over Theorem 13 to the so-called viscosity approximation technique (see Xu [21]). We derive a more general result in this direction which is an improvement upon several convergence results in the context of viscosity approximation technique.

Theorem 14. Let 𝑋 be a real reflexive Banach space with a weakly continuous duality mapping 𝐽_𝜑 with gauge function 𝜑. Let 𝐶 be a nonempty closed convex-bounded subset of 𝑋, 𝑓∶𝐶→𝐶 a weakly contraction with function 𝜓, and ℱ={𝑇(𝑡)∶𝑡∈ℝ⁺} a uniformly continuous semigroup of asymptotically nonexpansive mappings from 𝐶 into itself with a net {𝑘(𝑡)∶𝑡∈(0,+∞)}. For an arbitrary initial value 𝑥₁ in 𝐶, let {𝑥_𝑛} be a sequence in 𝐶 generated by 𝑥_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑇𝑡_𝑛𝑥_𝑛∀𝑛∈ℕ. (45) Here, {𝛼_𝑛} is a sequence in (0,1), and {𝑡_𝑛} is a decreasing sequence in (0,+∞) satisfying conditions (C1)~(C4). Then, (a)𝐹(ℱ)≠∅. (b){𝑥_𝑛} converges strongly to 𝑥^∗ in 𝐹(ℱ), where 𝑥^∗=𝑄_𝐹(ℱ)(𝑓𝑥^∗) and 𝑄_𝐹(ℱ) is a sunny nonexpansive retraction of 𝐶 onto 𝐹(ℱ).

Proof. It follows from Theorem 10 that 𝐹(ℱ)≠∅, and there is a sunny nonexpansive retraction 𝑄_𝐹(ℱ) of 𝐶 onto 𝐹(ℱ). Since 𝑄_𝐹(ℱ)𝑓 is a weakly contractive mapping from 𝐶 into itself, it follows from [24, Theorem 1] that there exists a unique element 𝑥^∗ in 𝐶 such that 𝑥^∗=𝑄_𝐹(ℱ)(𝑓𝑥^∗). Such 𝑥^∗ in 𝐶 is an element of 𝐹(ℱ). Now, we define a sequence {𝑧_𝑛} in 𝐶 by 𝑧_𝑛+1=𝛼_𝑛𝑓𝑥^∗+1−𝛼_𝑛𝑇𝑡_𝑛𝑧_𝑛∀𝑛∈ℕ. (46)
By Theorem 13, we have that 𝑧_𝑛→𝑥^∗=𝑄_𝐹(ℱ)(𝑓𝑥^∗). By boundedness of {𝑥_𝑛} and {𝑧_𝑛}, there exists a constant 𝑀≥0 such that ‖𝑥_𝑛−𝑧_𝑛‖≤𝑀 for all 𝑛∈ℕ. Observe that 𝑥_𝑛+1−𝑧_𝑛+1≤𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑥^∗+1−𝛼_𝑛𝑇𝑡_𝑛𝑥_𝑛−𝑇𝑡_𝑛𝑧_𝑛≤𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑧_𝑛+𝑓𝑧_𝑛−𝑓𝑥^∗+1−𝛼_𝑛𝑇𝑡_𝑛𝑥_𝑛−𝑇𝑡_𝑛𝑧_𝑛≤𝛼_𝑛𝑥_𝑛−𝑧_𝑛−𝜓𝑥_𝑛−𝑧_𝑛+𝑧_𝑛−𝑥^∗+1−𝛼_𝑛𝑘𝑡_𝑛𝑥_𝑛−𝑧_𝑛≤𝑥_𝑛−𝑧_𝑛−𝛼_𝑛𝜓𝑥_𝑛−𝑧_𝑛+𝛼_𝑛𝑧_𝑛−𝑥^∗+1−𝛼_𝑛𝑘𝑡_𝑛−1𝑥_𝑛−𝑧_𝑛≤𝑥_𝑛−𝑧_𝑛−𝛼_𝑛𝜓𝑥_𝑛−𝑧_𝑛+𝛼_𝑛𝑧_𝑛−𝑥^∗+𝑘𝑡_𝑛−1𝛼_𝑛𝑀. (47) By [25, Lemma 3.2], we obtain ‖𝑥_𝑛−𝑧_𝑛‖→0. Therefore, 𝑥_𝑛→𝑥^∗=𝑄_𝐹(ℱ)(𝑓𝑥^∗).

Theorem 15. Let 𝑋 be a real reflexive Banach space with a weakly continuous duality mapping 𝐽_𝜑 with gauge function 𝜑. Let 𝐶 be a nonempty closed convex subset of 𝑋 and 𝑓∶𝐶→𝐶 a weakly contraction. Let ℱ={𝑇(𝑡)∶𝑡∈ℝ⁺} be a uniformly continuous semigroup of nonexpansive mappings from 𝐶 into itself with 𝐹(ℱ)≠∅. For given 𝑥₁ in 𝐶, let {𝑥_𝑛} be a sequence in 𝐶 generated by (45), where {𝛼_𝑛} is a sequence in (0,1) and {𝑡_𝑛} is a decreasing sequence in (0,+∞) satisfying conditions (C1)~(C3). Then, {𝑥_𝑛} converges strongly to 𝑥^∗∈𝐹(ℱ), where 𝑥^∗=𝑄_𝐹(ℱ)(𝑓𝑥^∗) and 𝑄_𝐹(ℱ) is a sunny nonexpansive retraction of 𝐶 onto 𝐹(ℱ).

Remark 16. (a) For a related result concerning the strong convergence of the explicit iteration procedure 𝑧_𝑛+1∶=(𝜆/𝑘_𝑛)𝑇^𝑛(𝑧_𝑛) to some fixed point of an asymptotically nonexpansive mapping 𝑇 on star-shaped domain in a reflexive Banach space with a weakly continuous duality mapping, we refer the reader to Schu [13].
(b) We remark that condition “‖𝑥_𝑛+1−𝑥_𝑛‖≤𝐷𝛼_𝑛 for some 𝐷>0” implies that ‖𝑥_𝑛+1−𝑥_𝑛‖→0 as 𝛼_𝑛→0. Thus, the assumption “‖𝑥_𝑛+1−𝑥_𝑛‖≤𝐷𝛼_𝑛 for some 𝐷>0” imposed in Theorem ZS is very strong. In our results, such assumption is avoided. Under a mild assumption, Theorem 13 shows that the sequence {𝑥_𝑛} generated by (7) converges strongly to a common fixed point of a uniformly continuous semigroup of asymptotically nonexpansive mappings in a real Banach space without uniform convexity. Therefore, Theorem 13 is a significant improvement of a number of known results (e.g., Theorem ZS and [12, Theorem 4.7]) for semigroups of asymptotically nonexpansive mappings. Corollary 9 and Theorem 13 provide an affirmative answer to Problem 1.

Acknowledgments

This research is supported partially by Taiwan NSC Grants 99-2115-M-037-002-MY3, 98-2923-E-037-001-MY3, 99-2115-M-110-007-MY3. The authors would like to thank the referees for their careful reading and helpful comments.