# The General Iterative Methods for Asymptotically Nonexpansive Semigroups in Banach Spaces

Authors: Rabian Wangkeeree; Pakkapon Preechasilp
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We introduce the general iterative methods for finding a common fixed point of asymptotically nonexpansive semigroups which is a unique solution of some variational inequalities. We prove the strong convergence theorems of such iterative scheme in a reflexive Banach space which admits a weakly continuous duality mapping. The main result extends various results existing in the current literature.


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## Body

## 1. Introduction

LetE be a normed linear space. Let T be a self-mapping on E. Then T is said to be asymptotically nonexpansive if there exists a sequence {kn}⊂[1,∞) with limn→∞kn=1 such that for each x,y∈E,
(1.1)∥Tnx-Tny∥≤kn∥x-y∥,∀n≥1.
The class of asymptotically nonexpansive maps was introduced by Goebel and Kirk [1] as an important generalization of the class of nonexpansive maps (i.e., mappings T:E→E such that ∥Tx-Ty∥≤∥x-y∥, for all x,y∈E). We use F(T) to denote the set of fixed points of T, that is, F(T)={x∈E:Tx=x}. A self-mapping f:E→E is a contraction on E if there exists a constant α∈(0,1) such that
(1.2)∥f(x)-f(y)∥≤α∥x-y∥,∀x,y∈E.
We use ΠE to denote the collection of all contractions on E. That is, ΠE={f:f  is a contraction on  E}.A family𝒮={T(s):0≤s<∞} of mappings of E into itself is called a strongly continuous semigroup of Lipschitzian mappings on E if it satisfies the following conditions:  (i)
T(0)x=x for all x∈E;(ii)
T(s+t)=T(s)T(t) for all s,t≥0;(iii)
for eacht>0, there exists a bounded measurable function Lt:(0,∞)→[0,∞) such that ∥T(t)x-T(t)y∥≤Lt∥x-y∥, for all  x,y∈E;(iv)
for allx∈E, the mapping t↦T(t)x is continuous.A strongly continuous semigroup of Lipchitszian mappings𝒮 is called  strongly continuous semigroup of nonexpansive mappings if Lt=1 for all t>0 and  strongly continuous semigroup of asymptotically nonexpansive  if limsupt→∞Lt≤1. Note that for asymptotically nonexpansive semigroup 𝒮, we can always assume that the Lipchitszian constant {Lt}t>0 is such that Lt≥1 for each t>0,  Lt is nonincreasing in t, and limn→∞Lt=1; otherwise we replace Lt, for each t>0, with Lt:=max{sups≥tLs,1}. We denote by F(𝒮) the set of all common fixed points of 𝒮, that is,
(1.3)F(𝒮):={x∈E:T(t)x=x,0≤t<∞}=⋂t≥0F(T(t)).𝒮 is called uniformly asymptotically regular on C [2, 3] if for all h≥0 and any bounded subset B of C,
(1.4)limt→∞supx∈B∥T(h)T(t)x-T(t)x∥=0,
and  almost uniformly asymtotically regular  on C [4] if
(1.5)limt→∞supx∈B∥T(h)1t∫0t‍T(s)xds-1t∫0t‍T(s)xds∥=0.Letu∈C. Then, for each t∈(0,1) and for a nonexpansive map T, there exists a unique point xt∈C satisfying the following condition:
(1.6)xt=(1-t)Txt+tu,
since the mapping Gt(x)=(1-t)Tx+tu is a contraction. When H is a Hilbert space and T is a self-map, Browder [5] showed that {xt} converges strongly to an element of F(T) which is nearest to u as t→0+. This result was extended to more various general Banach space by Morales and Jung [6], Takahashi and Ueda [7], Reich [8], and a host of other authors.Many authors (see, e.g., [9, 10]) have also shown the convergence of the path xn=(1-αn)Tnxn+αnu, in Banach spaces for asymptotically nonexpansive mapping self-map T under some conditions on αn.It is an interesting problem to extend the above results to a strongly continuous semigroup of nonexpansive mappings and a strongly continuous semigroup of asymptotically nonexpansive mappings.Let𝒮 be a strongly continuous semigroup of nonexpansive self-mappings. In 1998 Shioji and Takahashi [11] introduced, in Hilbert space, the implicit iteration
(1.7)un=(1-αn)1tn∫0tn‍T(s)unds+αnu,u∈C,n≥0,
where {αn} is a sequence in (0,1), {tn} is a sequence of positive real numbers divergent to ∞. Under certain restrictions to the sequence {αn}, Shioji and Takahashi proved strong convergence of (1.7) to a member of F(𝒮). Recently, Zegeye et al. [4] introduced the implicit (1.7) and the following explicit iteration process for a semigroup of asymptotically nonexpansive mappings:
(1.8)un+1=(1-αn)1tn∫0tn‍T(s)unds+αnu,u∈C,n≥0,
where tn∈ℝ+ and αn∈(0,1) in a reflexive strictly convex Banach space with a uniformly Gâteaux differentiable norm. Suppose, in addition, that 𝒮 is almost uniformly asymptotically regular. Then the implicit sequence (1.7) and explicit sequence (1.8) converge strongly to a point of F(𝒮).On the other hand, by a gauge functionφ we mean a continuous strictly increasing function φ:[0,∞)→[0,∞) such that φ(0)=0 and φ(t)→∞ as t→∞. Let E* be the dual space of E. The duality mapping Jφ:E→2E* associated to a gauge function φ is defined by
(1.9)Jφ(x)={f*∈E*:〈x,f*〉=∥x∥φ(∥x∥),∥f*∥=φ(∥x∥)},∀x∈E.
In particular, the duality mapping with the gauge function φ(t)=t, denoted by J, is referred to as the normalized duality mapping. Clearly, there holds the relation Jφ(x)=(φ(∥x∥)/∥x∥)J(x) for all x≠0 (see [12]). Browder [12] initiated the study of certain classes of nonlinear operators by means of the duality mapping Jφ. Following Browder [12], we say that a Banach space E has a weakly continuous duality mapping if there exists a gauge φ for which the duality mapping Jφ(x) is single valued and continuous from the weak topology to the weak* topology; that is, for any {xn} with xn⇀x, the sequence {Jφ(xn)} converges weakly* to Jφ(x). It is known that lp has a weakly continuous duality mapping with a gauge function φ(t)=tp-1 for all 1<p<∞. Set
(1.10)Φ(t)=∫0t‍φ(τ)dτ,∀t≥0,
then
(1.11)Jφ(x)=∂Φ(∥x∥),∀x∈E,
where ∂ denotes the subdifferential in the sense of convex analysis.In a Banach spaceE having a weakly continuous duality mapping Jφ with a gauge function φ, an operator A is said to be strongly positive[13] if there exists a constant γ->0 with the property
(1.12)〈Ax,Jφ(x)〉≥γ-∥x∥φ(∥x∥),(1.13)∥αI-βA∥=sup∥x∥≤1|〈(αI-βA)x,Jφ(x)〉|,α∈[0,1],β∈[-1,1],
where I is the identity mapping. If E:=H is a real Hilbert space, then the inequality (1.12) reduces to
(1.14)〈Ax,x〉≥γ-∥x∥2∀x∈H.
A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert space H:
(1.15)minx∈C12〈Ax,x〉-〈x,b〉,
where C is the fixed point set of a nonexpansive mapping T on H and b is a given point in H. In 2009, motivated and inspired by Marino and Xu [14], Li et al. [15] introduced the following general iterative procedures for the approximation of common fixed points of a one-parameter nonexpansive semigroup {T(s):s≥0} on a nonempty closed convex subset C in a Hilbert space:
(1.16)yn=αnγf(yn)+(I-αnA)1tn∫0tn‍T(s)ynds,n≥0,xn+1=αnγf(xn)+(I-αnA)1tn∫0tn‍T(s)xnds,n≥0,
where {αn} and {tn} are sequences in [0,1] and (0,∞), respectively, A is a strongly positive bounded linear operator on H, and f is a contraction on H. And their convergence theorems can be proved under some appropriate control conditions on parameter {αn} and {tn}. Furthermore, by using these results, they obtained two mean ergodic theorems for nonexpansive mappings in a Hilbert space.All of the above brings us to the following conjectures.Question 1.
Could we obtain strong convergence theorems for the general class of strongly continuous semigroup of asymptotically nonexpansive mappings in more general Banach spaces? such as a reflexive Banach space which admits a weakly continuous duality mappingJφ, where φ:[0,∞)→[0,∞) is a gauge function.In this paper, inspired and motivated by Shioji and Takahashi [11], Zegeye et al. [4], Marino and Xu [14], Li et al. [15], and Wangkeeree et al. [13], we prove the strong convergence theorems of the iterative approximation methods (1.16) for the general class of the strongly continuous semigroup of asymptotically nonexpansive mappings in a reflexive Banach space which admits a weakly continuous duality mapping Jφ, where φ:[0,∞)→[0,∞) is a gauge function and A is a strongly positive bounded linear operator on a Banach space E. The results in this paper generalize and improve some well-known results in Shioji and Takahashi [11], Li et al. [15], and many others.


## 2. Preliminaries

Throughout this paper, letE be a real Banach space and E* be its dual space. We write xn⇀x (resp., xn⇀*x) to indicate that the sequence {xn} weakly (resp., weak*) converges to x; as usual xn→x will symbolize strong convergence. Let U={x∈E:∥x∥=1}. A Banach space E is said to uniformly convex if, for any ϵ∈(0,2], there exists δ>0 such that, for any x,y∈U, ∥x-y∥≥ϵ implies ∥(x+y)/2∥≤1-δ. It is known that a uniformly convex Banach space is reflexive and strictly convex (see also [16]). A Banach space E is said to be smooth if the limit limt→0((∥x+ty∥-∥x∥)/t) exists for all x,y∈U. It is also said to be uniformly smooth if the limit is attained uniformly for x,y∈U.Now we collect some useful lemmas for proving the convergence result of this paper.The first part of the next lemma is an immediate consequence of the subdifferential inequality and the proof of the second part can be found in [10].Lemma 2.1 (see [10]).
Assume that a Banach spaceE has a weakly continuous duality mapping Jφ with gauge φ.(i)
For allx,y∈E, the following inequality holds:
(2.1)Φ(∥x+y∥)≤Φ(∥x∥)+〈y,Jφ(x+y)〉.
In particular, for all x,y∈E,
(2.2)∥x+y∥2≤∥x∥2+2〈y,J(x+y)〉.(ii)
Assume that a sequence{xn} in E converges weakly to a point x∈E.
Then the following identity holds:(2.3)limsupn→∞Φ(∥xn-y∥)=limsupn→∞Φ(∥xn-x∥)+Φ(∥y-x∥),∀x,y∈E.The next valuable lemma is proved for applying our main results.Lemma 2.2 (see [13, Lemma 3.1]).
Assume that a Banach spaceE has a weakly continuous duality mapping Jφ with gauge φ. Let A be a strong positive linear bounded operator on E with coefficient γ->0 and 0<ρ≤φ(1)∥A∥-1. Then ∥I-ρA∥≤φ(1)(1-ργ-).In the following, we also need the following lemma that can be found in the existing literature [17, 18].Lemma 2.3 (see [18, Lemma 2.1]).
Let{an} be a sequence of a nonnegative real number satisfying the property
(2.4)an+1≤(1-γn)an+γnβn,n≥0,
where {γn}⊆(0,1) and {βn}⊆ℝ such that ∑n=0∞‍γn=∞ and limsupn→∞βn≤0. Then {an} converges to zero, as n→∞.


## 3. Main Theorem

Theorem 3.1.
LetE be a reflexive Banach space which admits a weakly continuous duality mapping Jφ with gauge φ such that φ is invariant on [0,1]. Let 𝒮={T(s):s≥0} be a strongly continuous semigroup of asymptotically nonexpansive mappings on E with a sequence {Lt}⊂[1,∞) and F(𝒮)≠∅. Let f∈ΠE with coefficient α∈(0,1), let A be a strongly positive bounded linear operator with coefficient γ->0 and 0<γ<φ(1)γ-/α, and let {αn} and {tn} be sequences of real numbers such that 0<αn<1,  tn>0. Then the following holds.(i)
If((1/tn)∫0tn‍Lsds-1)/αn<φ(1)γ--γα, for all  n≥0, then there exists a sequence {yn}⊂E defined by
(3.1)yn=αnγf(yn)+(I-αnA)1tn∫0tn‍T(s)ynds,n≥0.(ii)
Suppose, in addition, that𝒮 is almost uniformly asymptotically regular and the real sequences {αn} and {tn} satisfy the following conditions:
(B1)
limn→∞tn=∞;(B2)
limn→∞αn=0;(B3)
limn→∞((1/tn)∫0tn‍Lsds-1)/αn=0.
Then{yn} converges strongly as n→∞ to a common fixed point x~ in F(𝒮) which solves the variational inequality:
(3.2)〈(A-γf)x~,Jφ(x~-z)〉≤0,z∈F(𝒮).Proof.
We first show the uniqueness of a solution of the variational inequality (3.2). Suppose both x~∈F(𝒮) and x*∈F(𝒮) are solutions to (3.2), then
(3.3)〈(A-γf)x~,Jφ(x~-x*)〉≤0,〈(A-γf)x*,Jφ(x*-x~)〉≤0.
Adding (3.3), we obtain
(3.4)〈(A-γf)x~-(A-γf)x*,Jφ(x~-x*)〉≤0.
Noticing that for any x,y∈E,
(3.5)〈(A-γf)x-(A-γf)y,Jφ(x-y)〉=〈A(x-y),Jφ(x-y)〉-γ〈f(x)-f(y),Jφ(x-y)〉≥γ-∥x-y∥φ(∥x-y∥)-γ∥f(x)-f(y)∥∥Jφ(x-y)∥≥γ-Φ(∥x-y∥)-γαΦ(∥x-y∥)=(γ--γα)Φ(∥x-y∥)≥(γ-φ(1)-γα)Φ(∥x-y∥)≥0.
Therefore x~=x* and the uniqueness is proved. Below, we use x~ to denote the unique solution of (3.2).
Sincelimn→∞αn=0, we may assume, without the loss of generality, that αn<φ(1)∥A∥-1.
For each integern≥0, define a mapping Gn:E→E by
(3.6)Gn(y)=αnγf(y)+(I-αnA)1tn∫0tn‍T(s)yds,∀y∈E.
We show that Gn is a contraction mapping. For any x,y∈E,
(3.7)∥Gn(x)-Gn(y)∥=∥αnγf(x)+(I-αnA)1tn∫0tn‍T(s)xds-αnγf(y)-(I-αnA)1tn∫0tn‍T(s)yds∥≤∥αnγ(f(x)-f(y))∥+∥(I-αnA)(1tn∫0tn‍T(s)xds-1tn∫0tn‍T(s)yds)∥≤αnγα∥x-y∥+φ(1)(1-αnγ-)(1tn∫0tn‍Lsds)∥x-y∥=(αnγα+φ(1)1tn∫0tn‍Lsds-φ(1)αnγ-1tn∫0tn‍Lsds)∥x-y∥≤(1tn∫0tn‍Lsds-αn(φ(1)γ-1tn∫0tn‍Lsds-γα))∥x-y∥.
Since 0<((1/tn)∫0tn‍Lsds-1)/αn<φ(1)γ--γα, we have
(3.8)0<(1/tn)∫0tn‍Lsds-1αn<φ(1)γ--γα≤φ(1)γ-1tn∫0tn‍Lsds-γα.
It then follows that 0<((1/tn)∫0tn‍Lsds-αn(φ(1)γ-(1/tn)∫0tn‍Lsds-γα))<1. We have Gn is a contraction map with coefficient ((1/tn)∫0tn‍Lsds-αn(φ(1)γ-(1/tn)∫0tn‍Lsds-γα)). Then, for each n≥0, there exists a unique yn∈E such that Gnyn=yn, that is,
(3.9)yn=αnγf(yn)+(I-αnA)1tn∫0tn‍T(s)ynds,n≥0.
Hence (i) is proved.
(ii) We first show that{yn} is bounded. Letting p∈F(𝒮) and using Lemma 2.2, we can calculate the following:
(3.10)∥yn-p∥=∥αnγf(yn)+(I-αnA)1tn∫0tn‍T(s)ynds-p∥=∥1tn∫0tn‍αnγf(yn)-αnγf(p)+αnγf(p)+(I-αnA)1tn∫0tn‍T(s)ynds-(I-αnA)p-(αnA)p∥≤αnγ∥f(yn)-f(p)∥+αn∥γf(p)-A(p)∥+φ(1)(1-αnγ-)∥1tn∫0tn‍T(s)ynds-1tn∫0tn‍T(s)pds∥≤αnγα∥yn-p∥+αn∥γf(p)-A(p)∥+φ(1)(1-αnγ-)1tn∫0tn‍Lsds∥yn-p∥≤αnγα∥yn-p∥+αn∥γf(p)-A(p)∥+φ(1)1tn∫0tn‍Lsds∥yn-p∥-φ(1)αnγ-1tn∫0tn‍Lsds∥yn-p∥≤αnγα∥yn-p∥+αn∥γf(p)-A(p)∥+1tn∫0tn‍Lsds∥yn-p∥-φ(1)αnγ-1tn∫0tn‍Lsds∥yn-p∥.
Thus, we get that
(3.11)∥yn-p∥≤αn∥γf(p)-A(p)∥1-αnγα-(1/tn)∫0tn‍Ls+φ(1)αnγ-(1/tn)∫0tn‍Lsds.
Calculating the right-hand side of the above inequality, we have
(3.12)αn∥γf(p)-A(p)∥1-αnγα-(1/tn)∫0tn‍Ls+φ(1)αnγ-(1/tn)∫0tn‍Lsds+φ(1)αnγ-=αn∥γf(p)-A(p)∥1-αnγα-(1/tn)∫0tn‍Ls+φ(1)αnγ-(1/tn)∫0tn‍Lsds+φ(1)αnγ--φ(1)αnγ-=αn∥γf(p)-A(p)∥φ(1)αnγ--αnαγ+1-(1/tn)∫0tn‍Lsds+φ(1)αnγ-(1/tn)∫0tn‍Lsds-φ(1)αnγ-=αn∥γf(p)-A(p)∥αn(φ(1)γ--αγ)-((1/tn)∫0tn‍Lsds-1)+φ(1)αnγ-((1/tn)∫0tn‍Lsds-1)=αn∥γf(p)-A(p)∥αn(φ(1)γ--αγ)-(1-φ(1)αnγ-)((1/tn)∫0tn‍Lsds-1).
Thus, we get that
(3.13)∥yn-p∥≤∥γf(p)-A(p)∥(φ(1)γ--αγ)-(1-αnγ-)dn,
where dn=((1/tn)∫0tn‍Lsds-1)/αn. Thus, there exists N>0 such that ∥yn-p∥≤∥γf(p)-A(p)∥/(φ(1)γ--γα), for all n≥N. Therefore, {yn} is bounded and hence {f(yn)} and {(1/tn)∫0tn‍T(s)ynds} are also bounded.
Letδtn(yn)=(1/tn)∫0tn‍T(s)ynds. Then, from (3.1), we get
(3.14)∥yn-δtn(yn)∥=αn∥γf(yn)-Aδtn(yn)∥→0asn→∞.
Since 𝒮 is almost uniformly asymptotically regular and (3.14), we have
(3.15)∥yn-T(h)yn∥≤∥yn-δtn(yn)∥+∥δtn(yn)-T(h)δtn(yn)∥+∥T(h)δtn(yn)-T(h)yn∥≤∥yn-δtn(yn)∥+∥δtn(yn)-T(h)δtn(yn)∥+Ls∥δtn(yn)-yn∥→0.
It follows from the reflexivity of E and the boundedness of sequence {yn} that there exists {ynj} which is a subsequence of {yn} converging weakly to w∈E as j→∞. Since Jφ is weakly sequentially continuous, we have by Lemma 2.1 that
(3.16)limsupj→∞Φ(∥ynj-y∥)=limsupj→∞Φ(∥ynj-w∥)+Φ(∥y-w∥),∀x∈E.
Let
(3.17)H(x)=limsupj→∞Φ(∥ynj-y∥),∀y∈E.
It follows that
(3.18)H(y)=H(w)+Φ(∥y-w∥),∀y∈E.
For h≥0, from (3.15) we obtain
(3.19)H(T(h)w)=limsupj→∞Φ(∥ynj-T(h)w∥)=limsupj→∞Φ(∥T(h)ynj-T(h)w∥)≤limsupj→∞Φ(∥ynj-w∥)=H(w).
On the other hand, however,
(3.20)H(T(h)w)=H(w)+Φ(∥T(h)w-w∥).
It follows from (3.19) and (3.20) that
(3.21)Φ(∥T(h)w-w∥)=H(T(h)w)-H(w)≤0.
This implies that T(h)w=w for all h≥0, and so w∈F(𝒮). Next, we show that ynj→w as j→∞. In fact, since Φ(t)=∫0t‍φ(τ)dτ, for all t≥0, and φ:[0,∞)→[0,∞) is a gauge function, then for 1≥k≥0,  φ(kx)≤φ(x) and
(3.22)Φ(kt)=∫0kt‍φ(τ)dτ=k∫0t‍φ(kx)dx≤k∫0t‍φ(x)dx=kΦ(t).
Following Lemma 2.1, we have
(3.23)Φ(∥yn-w∥)=Φ(∥(I-αnA)1tn∫0tn‍T(s)ynds-(I-αnA)w+αn(γf(yn)-γf(w)+γf(w)-A(w))1tn∫0tn‍∥)≤Φ(∥(I-αnA)1tn∫0tn‍T(s)ynds-(I-αnA)w+αnγ(f(yn)-f(w))∥)+αn〈γf(w)-A(w),Jφ(yn-w)〉≤Φ(φ(1)(1-αnγ-)∥1tn∫0tn‍T(s)ynds-1tn∫0tn‍T(s)wds∥+αnγα∥yn-w∥)+αnγ〈f(w)-f(w),Jφ(yn-w)〉≤Φ(φ(1)(1-αnγ-)(1tn∫0tn‍Lsds)∥yn-w∥+αnγα∥yn-w∥)+αn〈γf(w)-A(w),Jφ(yn-w)〉≤Φ([φ(1)(1-αnγ-)(1tn∫0tn‍Lsds)+αnγα]∥yn-w∥)+αn〈γf(w)-A(w),Jφ(yn-w)〉≤[φ(1)(1-αnγ-)(1tn∫0tn‍Lsds)+αnγα]Φ(∥yn-w∥)+αn〈γf(w)-A(w),Jφ(yn-w)〉.
This implies that
(3.24)Φ(∥yn-w∥)≤11-φ(1)(1-αnγ-)((1/tn)∫0tn‍Lsds)+αnγααn〈γf(w)-A(w),Jφ(yn-w)〉,
also
(3.25)Φ(∥yn-w∥)≤1(φ(1)γ--αγ)-(1-αnγ-)dn〈γf(w)-A(w),Jφ(yn-w)〉,
where dn=((1/tn)∫0tn‍Lsds-1)/αn. Now observing that ynj⇀w implies Jφ(ynj-w)⇀*0, we conclude from the above inequality that
(3.26)Φ(∥ynj-w∥)→0asj→∞.
Hence ynj→w as j→∞. Next, we prove that w solves the variational inequality (3.2). For any z∈F(𝒮), we observe that
(3.27)〈(yn-1tn∫0tn‍T(s)ynds)-(z-1tn∫0tn‍T(s)zds),Jφ(yn-z)〉=〈yn-z,Jφ(yn-z)〉-〈1tn∫0tn‍T(s)ynds-1tn∫0tn‍T(s)zds,Jφ(yn-z)〉≥Φ(∥yn-z∥)-∥1tn∫0tn‍T(s)ynds-1tn∫0tn‍T(s)zds∥∥Jφ(yn-z)∥≥Φ(∥yn-z∥)-1tn∫0tn‍Lsds∥yn-z∥∥Jφ(yn-z)∥=Φ(∥yn-z∥)-1tn∫0tn‍LsdsΦ(∥yn-z∥)=(1-1tn∫0tn‍Lsds)Φ(∥yn-z∥).
Since
(3.28)yn=αnγf(yn)+(I-αnA)1tn∫0tn‍T(s)ynds,
we can derive that
(3.29)(A-γf)(yn)=-1αn(yn-1tn∫0tn‍T(s)ynds)+(A(yn)-A(1tn∫0tn‍T(s)ynds)).
Since Φ is strictly increasing and ∥yn-p∥≤M for some M>0, we have Φ(∥yn-p∥)≤Φ(M). Thus
(3.30)〈(A-γf)(yn),Jφ(yn-z)〉=-1αn〈(yn-1tn∫0tn‍T(s)ynds)-(z-1tn∫0tn‍T(s)zds),Jφ(yn-z)〉+〈A(yn)-A(1tn∫0tn‍T(s)ynds),Jφ(yn-z)〉≤((1/tn)∫0tn‍Lsds-1)αnΦ(∥yn-z∥)+〈A(yn)-A(1tn∫0tn‍T(s)ynds),Jφ(yn-z)〉≤((1/tn)∫0tn‍Lsds-1)αnΦ(M)+〈A(yn-1tn∫0tn‍T(s)ynds),Jφ(yn-z)〉.
Notice that
(3.31)ynj-1tnj∫0tnj‍T(s)ynjds→w-1tnj∫0tnj‍T(s)wds=w-w=0.
Now using (B3) and replacing n with nj in (3.30) and letting j→∞, we have
(3.32)〈(A-γf)w,Jφ(w-z)〉≤0.
So, w∈F(𝒮) is a solution of the variational inequality (3.2), and hence w=x~ by the uniqueness. In a summary, we have shown that each cluster point of {yn} (at n→∞) equals x~. Therefore, yn→x~ as n→∞. This completes the proof.IfA≡I, the identity mapping on E, and γ=1, then Theorem 3.1 reduces to the following corollary.Corollary 3.2.
LetE be a reflexive Banach space which admits a weakly continuous duality mapping Jφ with gauge φ such that φ is invariant on [0,1]. Let 𝒮={T(s):s≥0} be a strongly continuous semigroup of asymptotically nonexpansive mappings on E with a sequence {Lt}⊂[1,∞) and F(𝒮)≠∅. Let f∈ΠE with coefficient α∈(0,1) and let {αn} and {tn} be sequences of real numbers such that 0<αn<1 and tn>0. Then the following holds.(i)
If(1/tn)∫0tn‍Lsds-1<αn(1-α),  for all  n∈ℕ, then there exists a sequence {yn} defined by
(3.33)yn=αnf(yn)+(1-αn)1tn∫0tn‍T(s)ynds,n≥0.(ii)
Suppose, in addition, that𝒮 is almost uniformly asymptotically regular and the real sequences {αn} and {tn} satisfy the following:
(B1)
limn→∞tn=∞;(B2)
limn→∞αn=0;(B3)
limn→∞((1/tn)∫0tn‍Lsds-1)/αn=0.
Then{yn} converges strongly as n→∞ to a common fixed point x~ in F(𝒮) which solves the variational inequality:
(3.34)〈(I-f)x~,Jφ(x~-z)〉≤0,z∈F(𝒮).IfE:=H is a Hilbert space and 𝒮={T(s):s≥0} is a strongly continuous semigroup of nonexpansive mappings on H, then we have Lt≡1 and Theorem 3.1 reduces to the following corollary.Corollary 3.3 (see [15, Theorem 3.1]).
LetH be a real Hilbert space. Suppose that f:H→H is a contraction with coefficient α∈(0,1) and 𝒮={T(s):s≥0} a strongly continuous semigroup of nonexpansive mappings on H such that F(𝒮)≠∅. Let A be a strongly positive bounded linear operator with coefficient γ->0 and let {αn} and {tn} be sequences of real numbers such that 0<αn<1,  tn>0 such that limn→∞tn=∞ and limn→∞αn=0, then for any 0<γ<γ-/α, there is a unique {yn} in H such that
(3.35)yn=αnγf(yn)+(I-αnA)1tn∫0tn‍T(s)ynds,n≥0
and the iterative sequence {yn} converges strongly as n→∞ to a common fixed point x~ in F(𝒮) which solves the variational inequality:
(3.36)〈(A-γf)x~,x~-z〉≤0,z∈F(𝒮).Theorem 3.4.
LetE be a reflexive strictly convex Banach space which admits a weakly continuous duality mapping Jφ with gauge φ such that φ is invariant on [0,1]. Let 𝒮={T(s):s≥0} be a strongly continuous semigroup of asymptotically nonexpansive mappings on E with a sequence {Lt}⊂[1,∞) and F(𝒮)≠∅. Let f∈ΠE with coefficient α∈(0,1); let A be a strongly positive bounded linear operator with coefficient γ->0 and 0<γ<φ(1)γ-/α. For any x0∈C, let the sequence {xn} be defined by
(3.37)xn+1=αnγf(xn)+(I-αnA)1tn∫0tn‍T(s)xnds,n≥0.
Suppose, in addition, that 𝒮 is almost uniformly asymptotically regular. Let {αn} and {tn} be sequences of real numbers such that 0<αn<1,  tn>0,  (C1)
limn→∞tn=∞;(C2)
limn→∞αn=0,  ∑n=0∞‍αn=∞;(C3)
limn→∞((1/tn)∫0tn‍Lsds-1)/αn=0.
Then{xn} converges strongly as n→∞ to a common fixed point x~ in F(𝒮) which solves the variational inequality (3.2).Proof.
First we show that{xn} is bounded. By condition (C3) and given 0<ε<φ(1)γ--αγ there exists N>0 such that ((1/tn)∫0tn‍Lsds-1)/αn<ε for all n≥N. Thus
(3.38)(1-φ(1)γ-αn)(1tn∫0tn‍Lsds-1)≤1tn∫0tn‍Lsds-1<εαn,
for all n≥N. Since limn→∞αn=0, we may assume, without the loss of generality, that αn<φ(1)∥A∥-1.
Claim that∥xn-p∥≤M,  n≥0, where M:=max{∥x0-p∥,…,∥xN-p∥,∥f(p)-p∥/(φ(1)γ--αγ-ε)}.
Letp∈F(𝒮). Then from (3.56) we get that
(3.39)∥xn+1-p∥=∥αnγf(xn)+(I-αnA)1tn∫0tn‍T(s)xnds-p∥≤∥αnγf(xn)+(I-αnA)1tn∫0tn‍T(s)xnds-αnA(p)-(I-αnA)p∥≤∥αnγf(xn)-αnγf(p)+αnγf(p)-αnA(p)∥+∥(I-αnA)(1tn∫0tn‍T(s)xnds-1tn∫0tn‍T(s)pds)∥≤∥αnγf(xn)-αnγf(p)+αnγf(p)-αnA(p)∥+∥I-αnA∥∥1tn∫0tn‍T(s)xnds-1tn∫0tn‍T(s)pds∥≤αn∥γf(p)-Ap∥+αnαγ∥xn-p∥+φ(1)(1-αnγ-)(1tn∫0tn‍Lsds)∥xn-p∥=αn∥γf(p)-Ap∥+(αnαγ+φ(1)(1-αnγ-)(1tn∫0tn‍Lsds))∥xn-p∥≤αn∥γf(p)-Ap∥+(αnαγ+1tn∫0tn‍Lsds-φ(1)αnγ-1tn∫0tn‍Lsds)∥xn-p∥≤αn∥γf(p)-Ap∥+(1+(1-φ(1)αnγ-)(1tn∫0tn‍Lsds-1)-αn(φ(1)γ--αγ))∥xn-p∥≤αn∥f(p)-p∥+(1-αn(φ(1)γ--αγ)+εαn)∥xn-p∥=αn∥f(p)-p∥+(1-αn(φ(1)γ--αγ-ε))∥xn-p∥≤max{∥f(p)-p∥(φ(1)γ--αγ-ε),∥xn-p∥}.
By induction,
(3.40)∥xn-p∥≤max{∥f(p)-p∥(φ(1)γ--αγ-ε),∥xN-p∥},∀n≥N,
and hence {xn} is bounded, so are {f(xn)} and {(1/tn)∫0tn‍T(s)xnds}. Let δtn(xn):=(1/tn)∫0tn‍T(s)xnds. Then, since αn→0 as n→∞, we obtain that
(3.41)∥xn+1-δtn(xn)∥=αn∥γf(xn)-Aδtn(xn)∥→0asn→∞.
For any h>0, we have
(3.42)∥T(h)xn+1-xn+1∥≤∥T(h)xn+1-T(h)δtn(xn)∥+∥T(h)δtn(xn)-δtn(xn)∥+∥δtn(xn)-xn+1∥≤Lh∥xn+1-δtn(xn)∥+∥T(h)δtn(xn)-δtn(xn)∥+∥δtn(xn)-xn+1∥,
it follows from (3.41) and 𝒮 is almost uniformly asymptotically regular that
(3.43)∥T(h)xn+1-xn+1∥→0asn→∞.
Next, we prove that
(3.44)limsupn→∞〈γf(x~)-Ax~,Jφ(xn-x~)〉≤0.
Let {xnk} be a subsequence of {xn} such that
(3.45)limk→∞〈γf(x~)-Ax~,Jφ(xnk-x~)〉=limsupn→∞〈γf(x~)-Ax~,Jφ(xn-x~)〉.
It follows from the reflexivity of E and the boundedness of sequence {xnk} that there exists {xnki} which is a subsequence of {xnk} converging weakly to w∈E as i→∞. Since Jφ is weakly continuous, we have by Lemma 2.1 that
(3.46)limsupi→∞Φ(∥xnki-x∥)=limsupi→∞Φ(∥xnki-w∥)+Φ(∥x-w∥),∀x∈E.
Let
(3.47)H(x)=limsupi→∞Φ(∥xnki-x∥),∀x∈E.
It follows that
(3.48)H(x)=H(w)+Φ(∥x-w∥),∀x∈E.
From (3.43), for each h>0, we obtain
(3.49)H(T(h)w)=limsupi→∞Φ(∥xnki-T(h)w∥)=limsupi→∞Φ(∥T(h)xnki-T(h)w∥)≤limsupi→∞Φ(∥xnki-w∥)=H(w).
On the other hand, however,
(3.50)H(T(h)w)=H(w)+Φ(∥T(h)w-w∥).
It follows from (3.49) and (3.50) that
(3.51)Φ(∥T(h)w-w∥)=H(T(h)w)-H(w)≤0.
This implies that T(h)w=w for all h>0, and so w∈F(𝒮). Since the duality map Jφ is single valued and weakly continuous, we get that
(3.52)limsupn→∞〈γf(x~)-Ax~,Jφ(xn-x~)〉=limk→∞〈γf(x~)-Ax~,Jφ(xnk-x~)〉=limi→∞〈γf(x~)-Ax~,Jφ(xnki-x~)〉=〈(A-γf)x~,Jφ(x~-w)〉≤0
as required. Finally, we show that xn→x~ as n→∞. It follows from Lemma 2.1(i) that
(3.53)Φ(∥xn+1-x~∥)=Φ(∥(I-αnA)1tn∫0tn‍T(s)xnds-(I-αnA)x~+αn(γf(xn)-γf(x~)+γf(x~)-A(x~))(I-αnA)1tn∫0tn‍T(s)xnds-(I-αnA)x~∥)≤Φ(∥(I-αnA)1tn∫0tn‍T(s)xnds-(I-αnA)x~+αnγ(f(xn)-f(x~))∥)+αn〈γf(x~)-A(x~),Jφ(yn-x~)〉≤Φ(φ(1)(1-αnγ-)∥1tn∫0tn‍T(s)xnds-1tn∫0tn‍T(s)x~ds∥+αnγα∥xn-x~∥)+αnγ〈f(x~)-f(x~),Jφ(yn-x~)〉≤Φ(φ(1)(1-αnγ-)(1tn∫0tn‍Lsds)∥xn-x~∥+αnγα∥xn-x~∥)+αn〈γf(x~)-A(x~),Jφ(xn-x~)〉≤Φ([φ(1)(1-αnγ-)(1tn∫0tn‍Lsds)+αnγα]∥xn-x~∥)+αn〈γf(x~)-A(x~),Jφ(xn-x~)〉≤[φ(1)(1-αnγ-)(1tn∫0tn‍Lsds)+αnγα]Φ(∥xn-x~∥)+αn〈γf(x~)-A(x~),Jφ(xn-x~)〉≤[(1-φ(1)αnγ-)(1tn∫0tn‍Lsds-1)+1-αn(φ(1)γ--γα)]Φ(∥xn-x~∥)+αn〈γf(x~)-A(x~),Jφ(xn-x~)〉≤(1-αn(φ(1)γ--γα))Φ(∥xnX-x~∥)+(1-φ(1)αnγ-)(1tn∫0tn‍Lsds-1)M+αn〈γf(x~)-A(x~),Jφ(xn-x~)〉,
where M>0 such that Φ(∥xn-x~∥)≤M. Put
(3.54)sn=αn(φ(1)γ--γα),tn=(1-φ(1)αnγ-φ(1)γ--γα)((1/tn)∫0tn‍Lsds-1αn)M+1(φ(1)γ--γα)〈γf(x~)-A(x~),Jφ(xn-x~)〉.
Then (3.53) is reduced to
(3.55)Φ(∥xn+1-x~∥)≤(1-sn)Φ(∥xn-x~∥)+sntn.
Applying Lemma 2.3 to (3.55), we conclude that Φ(∥xn+1-x~∥)→0 as n→∞; that is, xn→x~ as n→∞. This completes the proof.IfA≡I, the identity mapping on E, and γ=1, then Theorem 3.4 reduces to the following corollary.Corollary 3.5.
LetE be a reflexive strictly convex Banach space which admits a weakly continuous duality mapping Jφ with gauge φ such that φ is invariant on [0,1]. Let 𝒮={T(s):s≥0} be a strongly continuous semigroup of asymptotically nonexpansive mappings from C into C with a sequence {Lt}⊂[1,∞), F(𝒮)≠∅. Let f∈ΠE with coefficient α∈(0,1) and the sequence {xn} be defined by x0∈C,
(3.56)xn+1=αnf(xn)+(1-αn)1tn∫0tn‍T(s)xnds,n≥0.
Suppose, in addition, that 𝒮 is almost uniformly asymptotically regular. Let {αn} and {tn} be sequences of real numbers such that 0<αn<1,  tn>0,  (C1)
limn→∞tn=∞;(C2)
limn→∞αn=0,  ∑n=0∞‍αn=∞;(C3)
limn→∞((1/tn)∫0tn‍Lsds-1)/αn=0.
Then{xn} converges strongly as n→∞ to a common fixed point x~ in F(𝒮) which solves the variational inequality:
(3.57)〈(I-f)x~,Jφ(x~-z)〉≤0,z∈F(𝒮).IfE:=H is a Hilbert space and 𝒮={T(s):s≥0} is a strongly continuous semigroup of nonexpansive mappings on H, then we have Lt≡1 and Theorem 3.4 reduces to the following corollary.Corollary 3.6 (see [15, Theorem 3.2]).
LetH be a real Hilbert space. Suppose that f:H→H is a contraction with coefficient α∈(0,1) and 𝒮={T(s):s≥0} a strongly continuous semigroup of nonexpansive mappings on H such that F(𝒮)≠∅. Let A be a strongly positive bounded linear operator with coefficient γ->0 and 0<γ<γ-/α and let the sequence {xn} be defined by x0∈C,
(3.58)xn+1=αnγf(xn)+(I-αnA)1tn∫0tn‍T(s)xnds,n≥0.
Let {αn} and {tn} be sequences of real numbers such that 0<αn<1,  tn>0,  (C1)
limn→∞tn=∞;(C2)
limn→∞αn=0,  ∑n=0∞‍αn=∞.
Then{xn} converges strongly as n→∞ to a common fixed point x~ in F(𝒮) which solves the variational inequality:
(3.59)〈(A-γf)x~,x~-z〉≤0,z∈F(𝒮).



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*Source: 695183-2012-12-30.xml*