Volume 2007, Article ID 64874,10pages doi:10.1155/2007/64874
Research Article
A New Iterative Algorithm for Approximating Common Fixed
Points for Asymptotically Nonexpansive Mappings
H. Y. Zhou, Y. J. Cho, and S. M. Kang
Received 28 February 2007; Accepted 13 April 2007
Recommended by Nan-Jing Huang
Suppose that K is a nonempty closed convex subset of a real uniformly convex and smooth Banach space E withP as a sunny nonexpansive retraction. Let T1,T2:K→
Ebe two weakly inward and asymptotically nonexpansive mappings with respect toP with sequences{Kn},{ln} ⊂[1,∞), limn→∞kn=1, limn→∞ln=1,F(T1)∩F(T2)= {x∈
K:T1x=T2x=x} =∅, respectively. Suppose that{xn}is a sequence inKgenerated it-eratively byx1∈K,xn+1=αnxn+βn(PT1)nxn+γn(PT2)nxn, for alln≥1, where{αn}, {βn}, and{γn}are three real sequences in [, 1−] for some>0 which satisfy condi-tionαn+βn+γn=1. Then, we have the following. (1) If one ofT1andT2is completely continuous or demicompact and∞n=1(kn−1)<∞,
∞
n=1(ln−1)<∞, then the strong convergence of{xn}to someq∈F(T1)∩F(T2) is established. (2) IfEis a real uniformly convex Banach space satisfying Opial’s condition or whose norm is Fr´echet differentiable, then the weak convergence of{xn}to someq∈F(T1)∩F(T2) is proved.
Copyright © 2007 H. Y. Zhou 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
LetK be a nonempty closed convex subset of a real uniformly convex Banach spaceE. A self-mappingT:K→K is said to be nonexpansive if T(x)−T(y) ≤ x−y for all
x,y∈K. A self-mappingT:K→K is called asymptotically nonexpansive if there exist sequences{kn} ⊂[1,∞),kn→1 asn→ ∞such that
Tn(x)−Tn(y)≤k
A self-mappingT:K→Kis said to be uniformlyL-Lipschitzian if there exists constant
L >0 such that
Tn(x)−Tn(y)≤L x−y , ∀x,y∈K,n≥1. (1.2)
A self-mappingT:K→Kis called asymptotically quasi-nonexpansive ifF(T)= ∅and there exist sequences{kn} ⊂[1,∞) withkn→1 asn→ ∞such that
Tn(x)−p≤k
n x−p , ∀x∈K, p∈F(T), n≥1. (1.3)
It is clear that, ifTis an asymptotically nonexpansive mapping fromKinto itself with a fixed point inK, thenTis asymptotically quasi-nonexpansive, but the converse may be not true.
As a generalization of the class of nonexpansive maps, the class of asymptotically non-expansive mappings was introduced by Goebel and Kirk [1] in 1972, who proved that if
Kis a nonempty bounded closed convex subset of a real uniformly convex Banach space andTis an asymptotically nonexpansive self-mapping ofK, thenThas a fixed point.
In 1978, Bose [2] first proved that ifK is a nonempty bounded closed convex subset of a real uniformly convex Banach spaceEsatisfying Opial’s condition andT:K→K is an asymptotically nonexpansive mapping, then the sequence{Tnx}converges weakly to a fixed point ofT, provided thatTis asymptotically regular atx∈K, that is,
lim n→∞T
nx−Tn+1x=0. (1.4)
In 1982, Passty [3] proved that Bose’s weak convergence theorem still holds if Opial’s condition is replaced by the condition thatEhas a Fr´echet differentiable norm.
Furthermore, Tan and Xu [4,5] later proved that the asymptotic regularity ofT atx can be weakened to the weakly asymptotic regularity ofTatx, that is,
ω−nlim
→∞
Tnx−Tn+1x=0. (1.5)
In all the above results (xn=Tnx), the asymptotic regularity ofT atx∈K is equiv-alent toxn−Txn→0 asn→ ∞. We wish that the later is a conclusion rather than an assumption.
In 1991, Schu [6,7] introduced a modified Mann iterative algorithm to approximate fixed points of asymptotically nonexpansive maps without assuming the asymptotic reg-ularity ofT atx∈K. Schu established the conclusion thatxn−Txn→0 asn→ ∞by choosing properly iterative parameters{αn}.
Schu’s iterative algorithm was defined as follows:
x1∈K,
xn+1=
1−αnxn+αnTnxn, ∀n≥1. (1.6)
Furthermore, Osilike and Aniagbosor [9] improved the main results of Schu [6]. Schu [7] and Rhoades [8], without assuming the boundedness condition, imposed onK. Re-cently, Chang et al. [10] established a more general demiclosed principle and improved the corresponding results of Bose [2], G ´ornicki [11], Passty [3], Reich [12], Schu [6,7], and Tan and Xu [4,5].
Some iterative algorithms for approximating fixed points of nonself nonexpansive mappings have been studied by various authors (see [13–18]). However, iterative algo-rithms for approximating fixed points of nonself asymptotically nonexpansive mappings have not been paid too much attention. The main reason is the fact that whenTis not a self-mapping, the mappingTnis nonsensical. Recently, in order to establish the conver-gence theorems for non-self-asymptotically nonexpansive mappings, Chidume et al. [19] introduced the following definition.
Definition 1.1. LetKbe a nonempty subset of real-normed linear spaceE. LetP:E→K
be the nonexpansive retraction ofEontoK.
(1) A non-self-mappingT:K→Eis calledasymptotically nonexpansiveif there exists a sequence{kn} ⊂[1,∞) withkn→1 asn→ ∞such that
T(PT)n−1(x)−T(PT)n−1(y)≤k
n x−y , ∀x,y∈K,n≥1. (1.7)
(2)Tis said to beuniformlyL-Lipschitzianif there exists a constantL >0 such that T(PT)n−1(x)−T(PT)n−1(y)≤L x−y , ∀x,y∈K,n≥1. (1.8)
By using the following iterative algorithm:
x1∈K,
xn+1=P
1−αnxn+αnT(PT)n−1xn, ∀n≥1, (1.9)
Chidume et al. [19] established the following demiclosed principle, strong and weak convergence theorems for non-self-asymptotically nonexpansive mappings in uniformly convex Banach spaces.
Theorem1.2 [19]. LetEbe a uniformly convex Banach space,Ka nonempty closed convex
subset of E. LetT:K→Ebe an asymptotically nonexpansive mapping with a sequence
{kn} ⊂[1,∞)andkn→1asn→ ∞. ThenI−Tis demiclosed at zero.
Theorem1.3 [19]. Let Ebe a uniformly convex Banach space and letK be a nonempty
closed convex subset ofE. LetT:K→Ebe completely continuous and asymptotically nonex-pansive mapping with a sequence{kn}⊂[1,∞)such that∞n=1, (kn2−1)<∞, andF(T)=∅.
Let{αn} ⊂(0, 1)be a sequence such that≤1−αn≤1−for alln≥1and some>0. For an arbitrary pointx1∈K, define the sequence{xn}by (1.9). Then,{xn}converges strongly
to some fixed point ofT.
Theorem1.4 [19]. LetEbe a uniformly convex Banach space which has a Fr´echet diff
eren-tiable norm and letKbe a nonempty closed convex subset ofE. LetT:K→Ebe an asymp-totically nonexpansive mapping with a sequence{kn} ⊂[1,∞)such thatn∞=1(k2n−1)<∞
and some>0. For an arbitrary pointx1∈K, let{xn}be the sequence defined by (1.9).
Then{xn}converges weakly to some fixed point ofT.
We now introduce the following definition.
Definition 1.5. LetKbe a nonempty subset of real normed linear spaceE. LetP:E→K
be a nonexpansive retraction ofEontoK.
(1) A non-self-mappingT:K→Eis calledasymptotically nonexpansivewith respect toPif there exists a sequence{kn} ⊂[1,∞) withkn→1 asn→ ∞such that
(PT)nx−(PT)ny≤k
n x−y , ∀x,y∈K,n≥1. (1.10)
(2)Tis said to beuniformlyL-Lipschitzianwith respect toPif there exists a constant
L >0 such that
(PT)nx−(PT)ny≤L x−y , ∀x,y∈K,n≥1. (1.11)
Remark 1.6. IfT is self-mapping, thenP becomes the identity mapping, so that (1.7), (1.8), and (1.9) reduce to (1.1), (1.2), and (1.6), respectively.
We remark in the passing that if T:K→Eis asymptotically nonexpansive in light of (1.7) andP:E→K is a nonexpansive retraction, thenPT:K→K is asymptotically nonexpansive in light of (1.1). Indeed, by definition (1.7), we have
(PT)nx−(PT)ny
=PT(PT)n−1x−PT(PT)n−1y
≤T(PT)n−1x−T(PT)n−1y
≤kn x−y , ∀x,y∈K,n≥1.
(1.12)
Conversely, it may not be true.
It is our purpose in this paper to introduce a new iterative algorithm (see (2.6)) for approximating common fixed points of two non-self-asymptotically nonexpansive map-pings with respect toP and to prove some strong and weak convergence theorems for such mappings in uniformly convex Banach spaces. As a consequence, the main results of Chidume et al. [19] are deduced.
2. Preliminaries
In this section, we will introduce a new iterative algorithm and prove a new demiclosed-ness principle for a non-self-asymptotically nonexpansive mapping in the sense of (1.10). Let Ebe a Banach space with dimensionE≥2. The modulus of Eis the function
δE: (0, 2]→[0, 1] defined by
δE()=inf
1−1
2(x+y)
: x =1, y =1,= x−y . (2.1)
A subsetK ofEis said to be retract if there exists a continuous mappingP:E→K
such thatPx=xfor allx∈K. Every closed convex subset of a uniformly convex Banach space is a retraction. A mappingP:E→Eis said to be a retraction ifP2=P. Note that if a mappingPis a retraction, thenPz=zfor allz∈R(P), the range ofP.
LetEbe a Banach space and letC,Dbe subsets ofE. Then, a mappingP:C→Dis said to be sunny if
PPx+t(x−Px)=Px, (2.2)
wheneverPx+t(x−Px)∈Cfor allx∈Candt≥0.
LetKbe a subset of a Banach spaceE. For allx∈K, define a setIK(x) by
IK(x)=x+λ(y−x) :λ >0, y∈K . (2.3)
A non-self-mappingT:K→Eis said to be inward ifTx∈Ik(x) for allx∈K andT is said to be weakly inward ifTx∈IK(x) for allx∈K.
The following facts are well known (see [20,18]).
Lemma2.1. LetCbe a nonempty convex subset of a smooth Banach spaceE,C0⊂C, let
J:E→E∗be the normalized duality mapping ofE, and letP:C→C0be a retraction. Then,
the following statements are equivalent:
(1)x−Px,J(y−Px) ≤0for allx∈Candy∈C0; (2)Pis both sunny and nonexpansive.
Lemma2.2. LetEbe a real smooth Banach space, letKbe a nonempty closed convex subset
ofEwithPas a sunny nonexpansive retraction, and letT:K→Ebe a mapping satisfying weakly inward condition. ThenF(PT)=F(T).
A Banach spaceEis said to satisfy Opial’s condition if for any sequence {xn}inE, xnximplies that
lim sup n→∞
xn−x<lim sup
n→∞
xn−y (2.4)
for ally∈Ewithy=x, wherexnxdenotes that{xn}converges weakly tox. It is well known that Hilbert space andlp (1< p <∞) admit Opial’s property, whileLp does not unlessp=2.
LetEbe a Banach space andS(E)= {x∈E: x =1}. The spaceEis said to be smooth if
lim t→0
x+ty − x
t (2.5)
exists for allx,y∈S(E). For any x,y∈E(x=0), we denote this limit by (x,y). The norm · ofEis said to be Fr´echet differentiable if for allx∈S(E), the limit (x,y) exists uniformly for ally∈S(E).
Let Ebe a real normed linear space, let K be a nonempty closed convex subset of
Ewhich is also a nonexpansive retraction ofEwith a retractionP. LetT1:K→Eand
T2:K→Ebe two non-self-asymptotically nonexpansive mappings with respect toP. For approximating the common fixed points of two non-self-asymptotically nonexpansive mappings, we introduce the following iterative algorithm:
x1∈K,
xn+1=αnxn+βnPT1 nx
n+γnPT2 nx
n, ∀n≥1, (2.6)
where{αn},{βn}, and{γn}are three real sequences in (0, 1) satisfyingαn+βn+γn=1.
Lemma2.3 [21]. Let{αn}and{tn}be two nonnegative real sequences satisfying
αn+1≤αn+tn, ∀n≥1. (2.7)
If∞n=1tn<∞, thenlimn→∞αnexists.
The following lemma can be found in Zhou et al. [22].
Lemma2.4 [22]. LetEbe a real uniformly convex Banach space and letBr(0)be the closed
ball ofEwith centre at the origin and radiusr >0. Then, there exists a continuous strictly increasing convex functiong: [0,∞)→[0,∞)withg(0)=0such that
λx+μy+γz 2≤λ x 2+μ y 2+γ z 2−λμg x−y (2.8)
for allx,y,z∈Br(0)andλ,μ,γ∈[0, 1]withλ+μ+γ=1.
The following demiclosedness principle for non-self-mapping follows from [10, The-orem 1].
Lemma2.5. LetEbe a real smooth and uniformly convex Banach space andKa nonempty
closed convex subset ofEwithP as a sunny nonexpansive retraction. Let T:K→Ebe a weakly inward and asymptotically nonexpansive mapping with respect toPwith a sequence
{kn} ⊂[1,∞)such that{kn} →1asn→ ∞. ThenI−Tis demiclosed at zero, that is,xnx
andxn−Txn→0imply thatTx=x.
Proof. Suppose that{xn} ⊂K converges weakly tox∗∈K andxn−Txn→0 asn→ ∞. We will prove thatTx∗=x∗. Indeed, since{xn} ⊂K, by the property ofP, we havePxn= xn for alln≥1 and soxn−PTxn→0 asn→ ∞. By Chang et al. [10, Theorem 1], we conclude thatx∗=PTx∗. SinceF(PT)=F(T) byLemma 2.2, we haveTx∗=x∗. This
completes the proof.
Remark 2.6. Lemma 2.5extends Chang et al. [10, Theorem 1] to non-self-mapping case.
Using the proof lines of Reich [12, Proposition], then we can prove the following lemma.
Lemma2.7. LetK be a closed convex subset of a uniformly convex Banach spaceEwith
a Fr´echet differentiable norm and let {Tn: 1≤n≤ ∞} be a family of Lipschitzian
sequence{Ln} such that ∞n=1(Ln−1)<∞. Ifx1∈K and xn+1=Tnxn for n≥1, then limn→∞(f1−f2,xn)exists for allf1=f2∈F.
Remark 2.8. Lemma 2.7is an extension of a proposition due to Reich [12].
3. Main results
In this section, we present some several strong and weak convergence theorems for two non-self-asymptotically nonexpansive mappings with respect toP.
Lemma 3.1. Let K be a nonempty closed convex subset of a normed linear space E. Let
T1,T2:K→Ebe two non-self-asymptotically nonexpansive mappings with respect toPwith
sequences {kn},{ln} ⊂[1,∞),∞n=1(kn−1)<∞, ∞
n=1(ln−1)<∞, respectively. Suppose
that{xn}is the sequence defined by (2.6). IfF(T1)∩F(T2)= ∅, thenlimn→∞ xn−q and limn→∞ yn−q exist for anyq∈F(T1)∩F(T2).
Proof. For anyq∈F(T1)∩F(T2), using the fact thatPis nonexpansive and (2.6), then we have
xn+1−q=αnxn+βnPT
1nxn+γnPT2nxn−Pq ≤αnxn−q+βnknxn−q+γnlnxn−q ≤mnxn−q,
(3.1)
wheremn=max{kn,ln}for alln≥1. It is clear that∞n=1(mn−1)<∞by the assumptions on{kn}and{ln}. It follows fromLemma 2.3that limn→∞ xn−q exists. This completes
the proof.
Lemma3.2. LetK be a nonempty closed convex subset of a real uniformly convex Banach
spaceE. LetT1,T2:K→Ebe two non-self-asymptotically nonexpansive mappings with
re-spect toPwith sequences{kn},{ln} ⊂[1,∞),∞n=1(kn−1)<∞, ∞
n=1(ln−1)<∞,
respec-tively. Suppose that{xn}is the sequence defined by (2.6), where{αn},{βn}, and{γn}are three sequences in[, 1−]for some>0. IfF(T1)∩F(T2)= ∅, then
lim n→∞xn−
PT1
xn=nlim
→∞xn−
PT2
xn=0. (3.2)
Proof. From (2.6), by the property ofP, andLemma 2.4, we have xn+1−q2
≤αnxn+βnPT1 nx
n+γnPT2 nx
n−q2
=αnxn−q+βnPT1 n
xn−q+γnPT2 n
xn−q2 ≤αnxn−q2+βnPT1nxn−q2+γnPT2nxn−q2
−αnβngxn−PT1nxn ≤mn2xn−q2−2gxn−PT1
nx n,
(3.3)
n→ ∞. Consequently,xn−(PT1)xn→0 as n→ ∞. Similarly, we can prove thatxn− (PT2)xn→0 asn→ ∞. This completes the proof.
Theorem3.3. LetKbe a nonempty closed convex subset of a real smooth uniformly
con-vex Banach spaceEwithP as a sunny nonexpansive retraction. LetT1,T2:K→Ebe two
weakly inward and asymptotically nonexpansive mappings with respect toPwith sequences
{kn},{ln} ⊂[1,∞),∞n=1(kn−1)<∞, ∞
n=1(ln−1)<∞, respectively. Let{xn} ⊂K be the
sequence defined by (2.6), where{αn},{βn}, and{γn}are three sequences in[, 1−)for some>0. If one ofT1andT2is completely continuous andF(T1)∩F(T2)= ∅, then{xn}
converges strongly to a common fixed point ofT1andT2.
Proof. ByLemma 3.1, limn→∞ xn−q exists for anyq∈F. It is sufficient to show that
{xn}has a subsequence which converges strongly to a common fixed point of T1 and
T2. ByLemma 3.2, limn→∞ xn−PT1xn =limn→∞ xn−PT2xn =0. Suppose thatT1 is completely continuous. Noting thatP is nonexpansive, we conclude that there exists subsequence{PT1xnj}of{PT1xn}such thatPT1xnj→q, and hencexnj→qas j→ ∞. By the continuity ofP,T1, andT2, we haveq=PT1q=PT2q, and soq∈F(T1)∩F(T2) by Lemma 2.2. Thus,{xn}converges strongly to a common fixed pointqofT1andT2. This
completes the proof.
Theorem3.4. LetK be a nonempty closed convex subset of a real smooth and uniformly
convex Banach spaceEwithP as a sunny nonexpansive retraction. LetT1,T2:K→Ebe
two weakly inward asymptotically nonexpansive mappings with respect toPwith sequences
{kn},{ln} ⊂[1,∞),∞n=1(kn−1)<∞, ∞
n=1(ln−1)<∞, respectively. Let{xn} ⊂K be the
sequence defined by (2.6), where{αn},{βn}, and{γn}are three sequences in[, 1−)for
some>0. If one ofT1andT2is demicompact andF(T1)∩F(T2)= ∅, then{xn}converges
strongly to a common fixed point ofT1andT2.
Proof. Since one ofT1andT2is demicompact, so is one ofPT1andPT2. Suppose thatPT1 is demicompact. Noting that{xn}is bounded, we assert that there exists a subsequence {PT1xnj}of{PT1xn}such thatPT1xnj converges strongly toq. ByLemma 3.2, we have
xnj→qas j→ ∞. SinceP,T1, andT2are all continuous, we haveq=PT1q=PT2qand
q∈F(T1)∩F(T2) byLemma 2.2. By Lemma 3.1, we know that limn→∞ xn−q exists. Therefore,{xn}converges strongly toqasn→ ∞. This completes the proof.
Theorem3.5. LetK be a nonempty closed convex subset of a real smooth and uniformly
convex Banach spaceEsatisfying Opial’s condition or whose norm is Fr´echet differentiable. LetT1,T2:K→Ebe two weakly inward and asymptotically nonexpansive mappings with
respect toP with sequences{kn},{ln} ⊂[1,∞),∞n=1(kn−1)<∞, ∞
n=1(ln−1)<∞,
re-spectively. Let{xn} ⊂K be the sequence defined by (2.6), where{αn},{βn}, and{γn}are
three sequences in [, 1−)for some>0. If F(T1)∩F(T2)= ∅, then{xn} converges
weakly to a common fixed point ofT1andT2.
a common fixed point ofT1andT2. Secondly, Opial’s condition andLemma 2.7 guaran-tee that the weakly subsequential limit of{xn}is unique. Consequently,{xn}converges weakly to a common fixed point ofT1andT2.This completes the proof.
Remark 3.6. The main results of this paper can be extended to a finite family of non-self-asymptotically nonexpansive mappings{Ti: 1≤i≤m}, wheremis a fixed positive integer, by introducing the following iterative algorithm:
x1∈K,
xn+1=αn1xn+αn2
PT1 nx
n+αn3
PT2 nx
n+···+αn(m+1)
PTmnxn, (3.4)
where{αn1},{αn2},..., and {αn(m+1)}arem+ 1 real sequences in (0, 1) satisfyingαn1+
αn2+···+αn(m+1)=1.
We close this section with the following open question.
How to devise an iterative algorithm for approximating common fixed points of an infinite family of non-self-asymptotically nonexpansive mappings?
Acknowledgment
The first author was supported by National Natural Science Foundation of China Grant no. 10471033.
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H. Y. Zhou: Department of Applied Mathematics, North China Electric Power University, Baoding 071003, China
Email address:[email protected]
Y. J. Cho: Department of Mathematics Education and RINS, College of Natural Sciences, Gyeongsang National University, Chinju 660-701, South Korea
Email address:[email protected]
S. M. Kang: Department of Mathematics Education and RINS, College of Natural Sciences, Gyeongsang National University, Chinju 660-701, South Korea
