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

Authors: D. R. Sahu; Ngai-Ching Wong; Jen-Chih Yao
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

LetX be a real reflexive Banach space with a weakly continuous duality mapping Jφ. 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: yn=αnu+(1−αn)T(tn)yn, for all n∈ℕ, converges strongly to a common fixed point of the semigroup ℱ for some suitably chosen parameters {αn} and {tn}. Our results extend and improve corresponding ones of Suzuki (2002), Xu (2005), and Zegeye and Shahzad (2009).


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## Body

## 1. Introduction

LetC 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 {kn} in [1,+∞) with limn→∞kn=1 such that ∥Tnx-Tny∥≤kn∥x-y∥ for all x,y in C and n in ℕ.Setℝ+:=[0,+∞). We call a one-parameter family ℱ:={T(t):t∈ℝ+} of mappings from C into C astrongly continuous semigroup of Lipschitzian mappings if(1)
for eacht>0, there exists a bounded function k(·):(0,+∞)→(0,+∞) such that
(1)∥T(t)x-T(t)y∥≤k(t)∥x-y∥∀x,y∈C,(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 eachx in C, the mapping T(·)x from ℝ+ into C is continuous.Note thatliminft→0+k(t)≥1. If k(t)=L for all t>0 in (1), then ℱ is called astrongly continuous semigroup of uniformlyL-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 limt→+∞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 subsetB of C, we have
(2)limt→0+supx∈B∥T(t)x-x∥=0.We denote byF(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 subsetC 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 Gt by
(3)Gtx=tu+(1-t)Tx∀x∈C.
Let xt be the fixed point of Gt; that is,
(4)xt=tu+(1-t)Txt.
Browder [3] (Reich [4], resp.) proves that as t→0+, the point xt 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 {xt} for asymptotically nonexpansive self-mappings T in Banach spaces under the additional assumption xt-Txt→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:
(5)yn=αnu+(1-αn)T(tn)yn,∀n∈ℕ.
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.
LetC be a nonempty closed convex bounded subset of a real uniformly convex Banach space X with a weakly continuous duality mapping Jφ 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 {tn}⊂(0,+∞) and {αn}⊂(0,1) such that (1-αn)k(tn)≤1 for all n∈ℕ, limn→∞tn=limn→∞(αn/tn)=0, and {αn/(1-(1-αn)k(tn))} is bounded. Let u in C be fixed. (1)
There exists a sequence{yn} in C such that
(6)yn=αnu+(1-αn)T(tn)yn∀n∈ℕ,which converges strongly to an element ofF(ℱ).(2)
Every sequence{xn} defined iteratively with any x1 in C, and
(7)xn+1=αnu+(1-αn)T(tn)xn∀n∈ℕ,converges strongly to an element ofF(ℱ), provided that ∥xn+1-xn∥≤Dαn 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

LetC be a nonempty subset of a (real) Banach space X with dual space X*. We call a mapping T:C→C  weakly contractive if
(8)∥Tx-Ty∥≤∥x-y∥-ψ(∥x-y∥)∀x,y∈C,
where ψ:[0,+∞)→[0,+∞) is a continuous and nondecreasing function such that ψ(0)=0,  ψ(t)>0 for t>0, and limt→+∞ψ(t)=+∞.By agauge we mean a continuous strictly increasing function φ defined on ℝ+:=[0,+∞) such that φ(0)=0 and limr→+∞φ(r)=+∞. Associated with a gauge φ, the (generally multivalued) duality mappingJφ:X→X* is defined by
(9)Jφ(x):={x*∈X*:〈x,x*〉=∥x∥φ(∥x∥),∥x*∥=φ(∥x∥)}.
Clearly, the (normalized) duality mapping J corresponds to the gauge φ(t)=t. In general,
(10)Jφ(x)=φ(∥x∥)∥x∥J(x),x≠0.Recall thatX is said to have a weakly (resp, sequentially) continuous duality mapping if there exists a gauge φ such that the duality mapping Jφ is single valued and (resp, sequentially) continuous from X with the weak topology to X* with the weak* topology. Every ℓp(1<p<+∞) space has a weakly continuous duality mapping with the gauge φ(t)=tp-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 {xn} is a sequence weakly convergent to x in X, then there holds the inequality
(11)limsupn→∞∥xn-x∥<limsupn→∞∥xn-y∥,y∈Xwithy≠x.Browder [19] initiated the study of certain classes of nonlinear operators by means of a duality mapping Jφ. Define
(12)Φ(t):=∫0t‍φ(s)ds∀t≥0.
Then, it is known that Jφ(x) is the subdifferential of the convex function Φ(∥·∥) at x; that is,
(13)Jφ(x)=∂Φ(∥x∥),x∈X,
where ∂ denotes the subdifferential in the sense of convex analysis. We need the subdifferential inequality
(14)Φ(∥x+y∥)≤Φ(∥x∥)+〈y,j(x+y)〉∀x,y∈X,j(x+y)∈Jφ(x+y).
For a smooth X, we have
(15)Φ(∥x+y∥)≤Φ(∥x∥)+〈y,Jφ(x+y)〉∀x,y∈X,
or considering the normalized duality mapping J, we have
(16)∥x+y∥2≤∥x∥2+2〈y,J(x+y)〉∀x,y∈X.Assume that a sequence{xn} in X converges weakly to a point x in X. Then the following identity holds;
(17)limsupn→∞Φ(∥xn-y∥)=limsupn→∞Φ(∥xn-x∥)+Φ(∥x-y∥)∀y∈X.Remark 2.
For anyk with 0≤k≤1, we have
(18)φ(kt)≤φ(t)∀t>0,Φ(kt)=∫0kt‍φ(s)ds=k∫0t‍φ(ku)du≤k∫0t‍φ(u)du=kΦ(t)∀t>0.We need the following demiclosedness principle for asymptotically nonexpansive mappings in a Banach space.Lemma 3 (see [17, Corollary  5.6.4], [10]).
LetX be a Banach space with a weakly continuous duality mapping Jφ: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 {xn} is a sequence in C which converges weakly to x and if the sequence {xn-Txn} converges strongly to zero, then x-Tx=0.LetC 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]).
LetC 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{αn} and {γn} be two real sequences such that(i)
{αn}⊂[0,1] and ∑n=1∞‍αn=+∞,(ii)
limsupn→∞γn≤0.
Let{λn} be a sequence of nonnegative numbers which satisfies the inequality
(19)λn+1≤(1-αn)λn+αnγn,n∈ℕ.
Then, limn→∞λn=0.


## 3. Existence of Common Fixed Points

We begin with the following.Proposition 6.
LetC 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 {bn} be a sequence in (0,1) and {tn} a sequence in (0,+∞) with k(tn)-1<bn for all n in ℕ. Assume that u∈C and bnu+(1-bn)T(tn)x∈C for all x in C and n in ℕ. (a)
There exists a sequence{yn}inCdefined by(20)yn=bnu+(1-bn)T(tn)yn∀n∈ℕ.(b)
If the sequence{yn}described by (20) is bounded and  limn→∞tn=limn→∞(bn/tn)=0, then(21)yn-T(t)yn⟶0asn⟶∞,∀t>0.Proof.
(a) Setϱn:=(k(tn)-1)/bn. Since k(tn)-1<bn for all n in ℕ, it follows that ϱn<1≤k(tn), and hence 0<(1-bn)k(tn)=(1-bn)(1+ϱnbn)<1 for all n in ℕ. For each n in ℕ, the mapping Gn:C→C defined by
(22)Gny:=bnu+(1-bn)T(tn)y,y∈C
is a contraction with Lipschitz constant (1-bn)k(tn). Therefore, there exists a sequence {yn} in C described by (20).(b)
Suppose that the sequence{yn} in C described by (20) is bounded and limn→∞tn=limn→∞(bn/tn)=0. From (20), we have
(23)∥T(tn)yn∥=11-bn∥yn-bnu∥≤11-bn(∥yn∥+bn∥u∥).
Without loss of generality, we may assume that{bn} is bounded away from 1. Then, there exists a positive constant δ such that bn≤δ<1 for all n in ℕ. Since B:={yn:n∈ℕ} is bounded, it follows from (23) that {T(tn)yn} is bounded.
SetL:=sup{k(t):t∈ℝ+}. For t>0, we have
(24)∥yn-T(t)yn∥≤∑i=0[t/tn]-1∥T(itn)yn-T((i+1)tn)yn∥+∥T([ttn]tn)yn-T(t)yn∥≤[ttn]L∥yn-T(tn)yn∥+L∥T(t-[ttn]tn)yn-yn∥=Lbn[ttn]∥u-T(tn)yn∥+L∥T(t-[ttn]tn)yn-yn∥≤tL(bntn)∥u-T(tn)yn∥+Lsupy∈B∥T(sn)y-y∥,
for all n in ℕ, where sn=t-[t/tn]tn. Note that sn=t-[t/tn]tn≤tn→0 as n→∞. Since ℱ is uniformly continuous at 0, it follows that supy∈B∥T(sn)y-y∥→0 as n→∞. Therefore, yn-T(t)yn→0 as n→∞.Remark 7.
(a) IfC is star shaped with respect to u in C, then the assumption “bnu+(1-bn)T(tn)x∈C for all x in C and n in ℕ” in Proposition 6 is automatically satisfied.
(b) Ifℱ={T(t):t∈ℝ+} is a strongly continuous semigroup of nonexpansive mappings with F(ℱ)≠∅, the sequence {yn} in C described by (20) is bounded (see [15, Theorem 3.3]).Theorem 8.
LetX be a real reflexive Banach space with a weakly continuous duality mapping Jφ 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 {bn} be a sequence in (0,1) and {tn} a sequence in (0,+∞) with k(tn)-1<bn for all n in ℕ satisfying the condition
(25)limn→∞tn=limn→∞bntn=limn→∞k(tn)-1bn=0.
Assume that u∈C such that the sequence {yn} described by (20) is bounded in C. Then, (a)
F(ℱ)≠∅.(b)
{yn} converges strongly to an element y*∈F(ℱ) which holds the inequality
(26)〈y*-u,J(y*-v)〉≤0∀v∈F(ℱ).Proof.
By Proposition6 (b), we have yn-T(t)yn→0 as n→∞ for all t>0. Since {yn} is bounded, there exists a subsequence {yni} of {yn} such that yni⇀y*∈C.(a)
For eachr>0, we have
(27)∥T(r)mx-T(r)my∥=∥T(mr)x-T(mr)y∥≤k(mr)∥x-y∥∀x,y∈C,m∈ℕ,that is, each T(r) is asymptotically nonexpansive mapping. Since limn→∞∥yn-T(t)yn∥=0 for all t>0, it follows from Lemma 3 that T(t)y*=y* for all t>0. Hence, F(ℱ)≠∅.(b)
Since{yn} is bounded, there exists a constant M≥0 such that ∥yn-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
(28)Φ(∥yn-y*∥)=Φ(∥bnu+(1-bn)T(tn)yn-y*∥)≤Φ((1-bn)∥T(tn)yn-y*∥)+bn〈u-y*,Jφ(yn-y*)〉≤Φ((1-bn)k(tn)∥yn-y*∥)+bn〈u-y*,Jφ(yn-y*)〉≤(1-bn)k(tn)Φ(∥yn-y*∥)+bn〈u-y*,Jφ(yn-y*)〉.Since k(t)≥1 for all t>0, we have
(29)Φ(∥yn-y*∥)≤k(tn)-1bnΦ(∥yn-y*∥)+〈u-y*,Jφ(yn-y*)〉≤k(tn)-1bnΦ(M)+〈u-y*,Jφ(yn-y*)〉.
Observing that (k(tni)-1)/bni→0, yni⇀y*, and Jφ is weakly continuous, we conclude from (29) that yni→y*∈C as i→∞ because C is closed.
We prove that{yn} converges strongly to y*. Suppose, for contradiction, that {ynj} is another subsequence of {yn} such that ynj→z*∈C with z*≠y*. Since limn→∞(yn-T(t)yn)=0 for all t>0, we have z*∈F(ℱ). For any v in F(ℱ), we have
(30)〈yn-T(tn)yn,Jφ(yn-v)〉=〈yn-v+T(tn)v-T(tn)yn,Jφ(yn-v)〉=∥yn-v∥φ(∥yn-v∥)-〈T(tn)yn-T(tn)v,Jφ(yn-v)〉≥-(k(tn)-1)∥yn-v∥φ(∥yn-v∥)∀n∈ℕ.
From (20), we have
(31)〈yn-u,Jφ(yn-v)〉=(1-bn)〈T(tn)yn-u,Jφ(yn-v)〉=(1-bn)〈T(tn)yn-yn+yn-u,Jφ(yn-v)〉.
It follows that
(32)〈yn-u,Jφ(yn-v)〉≤1-bnbn〈T(tn)yn-yn,Jφ(yn-v)〉≤(1-bn)k(tn)-1bn∥yn-v∥φ(∥yn-v∥)≤k(tn)-1bnMφ(M)
for all v in F(ℱ) and n in ℕ. Since limn→∞(k(tn)-1)/bn=0, we obtain from (32) that
(33)〈y*-u,Jφ(y*-z*)〉≤0,〈z*-u,Jφ(z*-y*)〉≤0.
Addition of (33) yields
(34)〈y*-z*,Jφ(y*-z*)〉≤0.
Hence, y*=z*, a contradiction. Therefore, {yn} converges strongly to y*∈C.
Finally, from (32), we conclude that y* satisfies (26).Letℱ={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
(35)Eℱu(C)={λu+(1-λ)T(t)x:x∈C,λ∈[0,1],t>0}.Corollary 9.
LetX be a real reflexive Banach space with a weakly continuous duality mapping Jφ 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 {bn} be a sequence in (0,1) and {tn} a sequence in (0,+∞) with k(tn)-1<bn for all n in ℕ satisfying condition (25). Assume that u∈C such that C is star shaped with respect to u, and set Eℱu(C) defined by (35) is bounded. (a)
For eachn in ℕ, there is exactly one point yn in C described by (20).(b)
F(ℱ) is nonempty. Moreover, {yn} converges strongly to an element y* in F(ℱ) which holds (26).Corollary9 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 ofC is not required.(2)
The assumption “uniform convexity” of the underlying space is not required.(3)
For convergence of{yn}, condition “F(ℱ) is a sunny nonexpansive retract of C” is not assumed.Next, we show thatF(ℱ) is a nonempty sunny nonexpansive retract of C.Theorem 10.
LetX be a real reflexive Banach space with a weakly continuous duality mapping Jφ 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 Theorem8 (a), F(ℱ)≠∅. As in Theorem 8 (b), for each u in C the sequence {yn} converges strongly to an element y* in F(ℱ). Define a mapping Q:C→F(ℱ) by
(36)Qu=limn→∞yn.
By (32), we have
(37)〈yn-u,Jφ(yn-v)〉≤k(tn)-1bndiam(C)φ(diam(C))
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
(38)〈Qu-u,Jφ(Qu-v)〉≤0∀v∈F(ℱ).
Therefore, by Lemma 4, we conclude that Q is sunny nonexpansive.Remark 11.
In view of Remark7 (b), the nonemptهness of the common fixed point set of a uniformly continuous semigroup of nonexpansive mappings implies that the sequence {yn} 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.
LetX be a real reflexive Banach space with a weakly continuous duality mapping Jφ 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 {bn} be a sequence in (0,1) and {tn} a sequence in (0,+∞) such that limn→∞tn=limn→∞(bn/tn)=0. Assume that u∈C and that C is star shaped with respect to u. (a)
For eachn in ℕ, there is exactly one point yn in C described by (20).(b)
{yn} converges strongly to an element y* in F(ℱ) which holds (26).Corollary12 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.
LetX be a real reflexive Banach space with a weakly continuous duality mapping Jφ 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 {xn} be a sequence in C generated by (7), where {αn} is a sequence in (0,1) and {tn} is a decreasing sequence in (0,+∞) satisfying the following conditions.(C1)
limn→∞tn=limn→∞(αn/tn)=0 and ∑n=1∞‍αn=∞,(C2)
limn→∞(|αn-αn+1|/αn)=0 or ∑n=1∞‍|αn-αn+1|<+∞,(C3)
limn→∞(∥T(tn-tn+1)xn-xn∥/αn+1)=0,(C4)
(1-αn)k(tn)≤1foralln∈ℕ and limn→∞((k(tn)-1)/αn)=0.
Then,(a)
F(ℱ)≠∅.(b)
{xn} converges strongly to QF(ℱ)(u), where QF(ℱ) is a sunny nonexpansive retraction of C onto F(ℱ).Proof.
It follows from Theorem10 that F(ℱ)≠∅, and there is a sunny nonexpansive retraction QF(ℱ) of C onto F(ℱ). Set y*:=QF(ℱ)(u), B:={xn}, L:=sup{k(t):t>0}, and M:=sup{∥xn-y*∥:n∈ℕ}. Since C is bounded, there exists a constant K>0 such that
(39)∥xn+1-xn∥≤K,∥T(tn)xn-u∥≤K∀n∈ℕ.
Hence, for all n>1, we have
(40)∥xn+1-xn∥≤∥(αn-αn-1)(u-T(tn-1)xn-1)+(1-αn)(T(tn)xn-T(tn-1)xn-1)∥≤|αn-αn-1|K+(1-αn)∥T(tn)xn-T(tn-1)xn-1∥≤|αn-αn-1|K+(1-αn)(∥T(tn)xn-T(tn)xn-1∥+∥T(tn)xn-1-T(tn-1)xn-1∥)≤|αn-αn-1|K+(1-αn)k(tn)∥xn-xn-1∥+∥T(tn)xn-1-T(tn-1)xn-1∥≤(1-αn)k(tn)∥xn-xn-1∥+|αn-αn-1|K+k(tn)∥T(tn-1-tn)xn-1-xn-1∥≤(1-αn)∥xn-xn-1∥+|αn-αn-1|K+L∥T(tn-1-tn)xn-1-xn-1∥+(1-αn)(k(tn)-1)∥xn-xn-1∥≤(1-αn)∥xn-xn-1∥+|αn-αn-1|K+L∥T(tn-1-tn)xn-1-xn-1∥+(k(tn)-1)K.
Hence, by Lemma 5 and assumptions (C1)~(C4), we conclude that xn+1-xn→0 as n→∞. Fix t>0, we have
(41)∥xn-T(t)xn∥≤∑i=0[t/tn]-1∥T(itn)xn-T((i+1)tn)xn∥+∥T([ttn]tn)xn-T(t)xn∥≤[ttn]L∥xn-T(tn)xn∥+L∥T(t-[ttn]tn)xn-xn∥≤L[ttn](∥xn-xn+1∥+∥xn+1-T(tn)xn∥)+L∥T(t-[ttn]tn)xn-xn∥≤L(ttn)∥xn-xn+1∥+tL(αntn)∥u-T(tn)xn∥+Lmax{∥T(s)xn-xn∥:0≤s≤tn}
for all n in ℕ, which gives that xn-T(t)xn→0 as n→∞. We may assume that xni⇀p∈C as i→∞. In view of the assumption that the duality mapping Jφ is weakly sequentially continuous, it follows from Lemma 3 that p∈F(ℱ). Thus, by the weak continuity of Jφ and Lemma 4, we have
(42)limsupn→∞〈u-QF(ℱ)(u),Jφ(xn-QF(ℱ)(u))〉=limi→∞〈u-QF(ℱ)(u),Jφ(xni-QF(ℱ)(u))〉=〈u-QF(ℱ)(u),Jφ(p-QF(ℱ)(u))〉≤0.
Define γn:=max{0,〈u-y*,Jφ(xn+1-y*)〉}. From (7), we obtain
(43)Φ(∥xn+1-y*∥)=Φ(∥αn(u-y*)+(1-αn)(T(tn)xn-y*)∥)≤Φ((1-αn)∥Tnxn-y*∥)+αn〈u-y*,Jφ(xn+1-y*)〉≤Φ((1-αn)k(tn)∥xn-y*∥)+αnγn.
Since (1-αn)k(tn)≤1 for all n in ℕ, it follows from Remark 2 that
(44)Φ(∥xn+1-y*∥)≤(1-αn)k(tn)Φ(∥xn-y*∥)+αnγn≤(1-αn)Φ(∥xn-y*∥)+αnγn+(k(tn)-1)Φ(M).
Note that limn→∞γn=0 and limn→∞((k(tn)-1)/αn)=0. Using Lemma 5, we obtain that {xn} converges strongly to y*.One can carry over Theorem13 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.
LetX be a real reflexive Banach space with a weakly continuous duality mapping Jφ 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 {xn} be a sequence in C generated by
(45)xn+1=αnfxn+(1-αn)T(tn)xn∀n∈ℕ.
Here, {αn} is a sequence in (0,1), and {tn} is a decreasing sequence in (0,+∞) satisfying conditions (C1)~(C4). Then, (a)
F(ℱ)≠∅.(b)
{xn} converges strongly to x* in F(ℱ), where x*=QF(ℱ)(fx*) and QF(ℱ) is a sunny nonexpansive retraction of C onto F(ℱ).Proof.
It follows from Theorem10 that F(ℱ)≠∅, and there is a sunny nonexpansive retraction QF(ℱ) of C onto F(ℱ). Since QF(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*=QF(ℱ)(fx*). Such x* in C is an element of F(ℱ). Now, we define a sequence {zn} in C by
(46)zn+1=αnfx*+(1-αn)T(tn)zn∀n∈ℕ.
By Theorem13, we have that zn→x*=QF(ℱ)(fx*). By boundedness of {xn} and {zn}, there exists a constant M≥0 such that ∥xn-zn∥≤M for all n∈ℕ. Observe that
(47)∥xn+1-zn+1∥≤αn∥fxn-fx*∥+(1-αn)∥T(tn)xn-T(tn)zn∥≤αn(∥fxn-fzn∥+∥fzn-fx*∥)+(1-αn)∥T(tn)xn-T(tn)zn∥≤αn(∥xn-zn∥-ψ(∥xn-zn∥)+∥zn-x*∥)+(1-αn)k(tn)∥xn-zn∥≤∥xn-zn∥-αnψ(∥xn-zn∥)+αn∥zn-x*∥+(1-αn)(k(tn)-1)∥xn-zn∥≤∥xn-zn∥-αnψ(∥xn-zn∥)+αn[∥zn-x*∥+(k(tn)-1)αnM].
By [25, Lemma 3.2], we obtain ∥xn-zn∥→0. Therefore, xn→x*=QF(ℱ)(fx*).Theorem 15.
LetX be a real reflexive Banach space with a weakly continuous duality mapping Jφ 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 {xn} be a sequence in C generated by (45), where {αn} is a sequence in (0,1) and {tn} is a decreasing sequence in (0,+∞) satisfying conditions (C1)~(C3). Then, {xn} converges strongly to x*∈F(ℱ), where x*=QF(ℱ)(fx*) and QF(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 procedurezn+1:=(λ/kn)Tn(zn) 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 “∥xn+1-xn∥≤Dαn for some D>0” implies that ∥xn+1-xn∥→0 as αn→0. Thus, the assumption “∥xn+1-xn∥≤Dαn 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 {xn} 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.



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