Volume 2009, Article ID 824374,13pages doi:10.1155/2009/824374
Research Article
Convergence Comparison of Several Iteration
Algorithms for the Common Fixed Point Problems
Yisheng Song and Xiao Liu
College of Mathematics and Information Science, Henan Normal University, 453007, China
Correspondence should be addressed to Yisheng Song,songyisheng123@yahoo.com.cn
Received 20 January 2009; Accepted 2 May 2009
Recommended by Naseer Shahzad
We discuss the following viscosity approximations with the weak contraction A for a non-expansive mapping sequence{Tn}, yn αnAyn 1−αnTnyn, xn1 αnAxn 1−αnTnxn. We prove that Browder’s and Halpern’s type convergence theorems imply Moudafi’s viscosity approximations with the weak contraction, and give the estimate of convergence rate between Halpern’s type iteration and Mouda’s viscosity approximations with the weak contraction.
Copyrightq2009 Y. Song and X. Liu. 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
The following famous theorem is referred to as the Banach Contraction Principle.
Theorem 1.1Banach1. LetE, dbe a complete metric space and letAbe a contraction onX, that is, there existsβ∈0,1such that
dAx, Ay≤βdx, y, ∀x, y∈E. 1.1
ThenAhas a unique fixed point.
In 2001, Rhoades 2 proved the following very interesting fixed point theorem
which is one of generalizations of Theorem 1.1 because the weakly contractions contains
contractions as the special casesϕt 1−βt.
Theorem 1.2Rhoades2, Theorem 2. LetE, dbe a complete metric space, and letAbe a weak contraction onE, that is,
for someϕ:0,∞ → 0,∞is a continuous and nondecreasing function such thatϕis positive on0,∞andϕ0 0. ThenAhas a unique fixed point.
The concept of the weak contraction is defined by Alber and Guerre-Delabriere3
in 1997. The natural generalization of the contraction as well as the weak contraction is
nonexpansive. Let K be a nonempty subset of Banach space E,T : K → K is said to be
nonexpansiveif
Tx−Ty≤x−y, ∀x, y∈K. 1.3
One classical way to study nonexpansive mappings is to use a contraction to approximate a
nonexpansive mapping. More precisely, taket∈0,1and define a contractionTt:K → K
byTtx tu 1−tTx, x∈ K,whereu ∈ Kis a fixed point. Banach Contraction Principle
guarantees thatTthas a unique fixed pointxtinK, that is,
xttu 1−tTxt. 1.4
Halpern4also firstly introduced the following explicit iteration scheme in Hilbert spaces:
foru, x0 ∈K,αn∈0,1,
xn1 αnu 1−αnTxn, n≥0. 1.5
In the case of T having a fixed point, Browder 5 resp. Halpern 4 proved that ifE is
a Hilbert space, then {xt} resp. {xn} converges strongly to the fixed point of T, that is,
nearest to u. Reich 6 extended Halpern’s and Browder’s result to the setting of Banach
spaces and proved that ifEis a uniformly smooth Banach space, then{xt}and{xn}converge
strongly to a same fixed point ofT, respectively, and the limit of {xt}defines theunique
sunny nonexpansive retraction fromKonto FixT. In 1984, Takahashi and Ueda7obtained
the same conclusion as Reich’s in uniformly convex Banach space with a uniformly Gˆateaux
differentiable norm. Recently, Xu8showed that the above result holds in a reflexive Banach
space which has a weakly continuous duality mappingJϕ. In 1992, Wittmann 9 studied
the iterative scheme 1.5 in Hilbert space, and obtained convergence of the iterations. In
particular, he proved a strong convergence result9, Theorem 2under the control conditions
C1lim
n→ ∞αn0, C2 ∞
n1
αn∞, C3 ∞
n1
|αn−αn1| <∞. 1.6
In 2002, Xu10,11extended wittmann’s result to a uniformly smooth Banach space, and
gained the strong convergence of{xn}under the control conditionsC1,C2,and
C4 lim
n→ ∞ αn1
αn
1. 1.7
Actually, Xu 10, 11 and Wittmann 9 proved the following approximate fixed points
Theorem 1.3. LetKbe a nonempty closed convex subset of a Banach spaceE. provided thatT :K → Kis nonexpansive withFixT/∅, and{xn}is given by1.5andαn ∈0,1satisfies the condition
C1,C2,andC3(orC4). Then{xn}is bounded andlimn→ ∞ xn−Txn 0.
In 2000, for a nonexpansive selfmapping T with FixT/∅ and a fixed contractive
selfmappingf, Moudafi14introduced the following viscosity approximation method for
T:
xn1αnfxn 1−αnTxn, 1.8
and proved that {xn} converges to a fixed pointp of T in a Hilbert space. They are very
important because they are applied to convex optimization, linear programming, monotone
inclusions, and elliptic differential equations. Xu 15 extended Moudafi’s results to a
uniformly smooth Banach space. Recently, Song and Chen12,13,16–18obtained a number
of strong convergence results about viscosity approximations 1.8. Very recently, Petrusel
and Yao 19, Wong, et al.20 also studied the convergence of viscosity approximations,
respectively.
In this paper, we naturally introduce viscosity approximations1.9and1.10with
the weak contractionAfor a nonexpansive mapping sequence{Tn},
ynαnAyn 1−αnTnyn, 1.9
xn1αnAxn 1−αnTnxn. 1.10
We will prove that Browder’s and Halpern’s type convergence theorems imply Moudafi’s viscosity approximations with the weak contraction, and give the estimate of convergence rate between Halpern’s type iteration and Moudafi’s viscosity approximations with the weak contraction.
2. Preliminaries and Basic Results
Throughout this paper, a Banach spaceEwill always be over the real scalar field. We denote
its norm by · and its dual space byE∗. The value ofx∗∈E∗aty∈Eis denoted byy, x∗
and thenormalized duality mappingJfromEinto 2E∗is defined by
Jx f ∈E∗:x, f x f, x f , ∀x∈E. 2.1
Let FixTdenote the set of all fixed points for a mappingT, that is, FixT {x∈E:Txx},
and letNdenote the set of all positive integers. We writexn xresp.xn x∗ to indicate
that the sequencexn weaklyresp. weak∗converges tox; as usualxn → xwill symbolize
strong convergence.
In the proof of our main results, we need the following definitions and results. Let
SE :{x ∈E; x 1}denote the unit sphere of a Banach spaceE.Eis said to haveia
Gˆateaux differentiable normwe also say thatEissmooth, if the limit
lim
t→0
xty− x
exists for eachx, y ∈ SE;iia uniformly Gˆateaux differentiable norm, if for eachy inSE,
the limit 2.2 is uniformly attained for x ∈ SE; iii a Fr´echet differentiable norm, if for
each x ∈ SE, the limit 2.2 is attained uniformly fory ∈ SE;iv a uniformly Fr´echet
differentiable norm we also say that E is uniformly smooth, if the limit 2.2 is attained
uniformly for x, y ∈ SE×SE. A Banach space E is said to be v strictly convex if
x y 1, x /yimplies xy/2 <1;viuniformly convexif for allε∈0,2,∃δε>0
such that x y 1 with x−y ≥ εimplies xy /2 < 1−δε.For more details on
geometry of Banach spaces, see21,22.
IfCis a nonempty convex subset of a Banach spaceE,andD is a nonempty subset
ofC, then a mappingP : C → D is called aretractionif P is continuous with FixP D.
A mappingP : C → D is calledsunnyifPP xtx−P x P x,for allx ∈ Cwhenever
P xtx−P x∈C,andt >0. A subsetDofCis said to be asunny nonexpansive retractofC
if there exists a sunny nonexpansive retraction ofContoD. We note that ifKis closed and
convex of a Hilbert spaceE, then the metric projection coincides with the sunny nonexpansive
retraction fromContoD. The following lemma is well known which is given in22,23.
Lemma 2.1see22, Lemma 5.1.6. LetCbe nonempty convex subset of a smooth Banach space
E,∅/D⊂C,J :E → E∗the normalized duality mapping ofE, andP :C → Da retraction. Then
Pis both sunny and nonexpansive if and only if there holds the inequality:
x−P x, Jy−P x≤0, ∀x∈C, y∈D. 2.3
Hence, there is at most one sunny nonexpansive retraction fromContoD.
In order to showing our main outcomes, we also need the following results. For completeness, we give a proof.
Proposition 2.2. LetKbe a convex subset of a smooth Banach spaceE. LetCbe a subset ofKand letP be the unique sunny nonexpansive retraction fromKontoC. SupposeAis a weak contraction with a functionϕonK,andTis a nonexpansive mapping. Then
ithe composite mappingTAis a weak contraction onK;
iiFor eacht∈0,1, a mappingTt 1−tTtAis a weak contraction onK. Moreover,
{xt}defined by2.4is well definition:
xttAxt 1−tTxt; 2.4
iiiz PAz if and only if z is a unique solution of the following variational
inequality:
Az−z, Jy−z≤0, ∀y∈C. 2.5
Proof. For anyx, y∈K, we have
TAx−T
So,TA is a weakly contractive mapping with a functionϕ. For each fixed t ∈ 0,1,and
ψs tϕs, we have
Ttx−TtytAx 1−tTx−tAy 1−tTy ≤1−tTx−TytAx−Ay
≤1−tx−ytx−y−tϕx−y
x−y−ψx−y.
2.7
Namely,Ttis a weakly contractive mapping with a functionψ. Thus,Theorem 1.2guarantees
thatTthas a unique fixed pointxtinK, that is,{xt}satisfying2.4is uniquely defined for
eacht∈0,1.iandiiare proved.
Subsequently, we showiii. Indeed, by Theorem 1.2, there exists a unique element
z∈K such thatz PAz. Such az∈Cfulfils2.5byLemma 2.1. Next we show that the
variational inequality2.5has a unique solutionz. In fact, supposep∈Cis another solution
of2.5. That is,
Ap−p, Jz−p≤0, Az−z, Jp−z≤0. 2.8
Adding up gets
ϕp−zp−z≤p−z2−Ap−Azp−z≤p−z−Ap−Az, Jp−z≤0.
2.9
Hencezpby the property ofϕ. This completes the proof.
Let {Tn} be a sequence of nonexpansive mappings with F
∞
n0FixTn/∅ on a
closed convex subsetKof a Banach spaceEand let{αn}be a sequence in0,1withC1.
E, K,{Tn},{αn}is said to haveBrowder’s propertyif for eachu∈K, a sequence{yn}defined
by
yn 1−αnTnynαnu, 2.10
for n ∈ N, converges strongly. Let {αn}be a sequence in 0,1 with C1and C2. Then
E, K,{Tn},{αn}is said to haveHalpern’s propertyif for eachu∈K, a sequence{yn}defined
by
yn1 1−αnTnynαnu, 2.11
forn∈N, converges strongly.
We know that if E is a uniformly smooth Banach space or a uniformly convex
Banach space with a uniformly Gˆateaux differentiable norm, K is bounded, {Tn} is a
constant sequence T, thenE, K,{Tn},{1/n} has both Browder’s and Halpern’s property
Lemma 2.3see24, Proposition 4. LetE, K,{Tn},{αn}have Browder’s property. For each∈ K, putP ulimn→ ∞ yn, where{yn}is a sequence inKdefined by2.10. ThenPis a nonexpansive
mapping onK.
Lemma 2.4 see24, Proposition 5. Let E, K,{Tn},{αn} have Halpern’s property. For each ∈ K, putP u limn→ ∞ yn, where{yn}is a sequence inKdefined by 2.11. Then the following
hold: (i)P udoes not depend on the initial pointy1. (ii)Pis a nonexpansive mapping onK.
Proposition 2.5. LetEbe a smooth Banach space, andE, K,{Tn},{αn}have Browder’s property.
Then F is a sunny nonexpansive retract of K, and moreover, P u limn→ ∞ yn define a sunny
nonexpansive retraction fromKtoF.
Proof. For eachp∈F, it is easy to see from2.10that
u−yn, J
p−yn
1−αn αn
yn−pTnp−Tnyn, J
p−yn
≤ 1−αn αn
Tnp−Tnynp−yn−p−yn2
≤0,
2.12
u−yn, J
p−yn
u−P u, Jp−yn
P u−yn, J
p−yn
. 2.13
This implies for anyp∈Fand someL≥ yn−p ,
u−P u, Jp−yn
≤yn−P u, J
p−yn
≤Lyn−P u → 0. 2.14
The smoothness ofEimplies the norm weak∗continuity ofJ22, Theorems 4.3.1, 4.3.2, so
lim
n→ ∞
u−P u, Jp−yn
u−P u, Jp−P u. 2.15
Thus
u−P u, Jp−P u≤0, ∀p∈F. 2.16
ByLemma 2.1,P ulimn→ ∞ynis a sunny nonexpansive retraction fromKtoF.
We will use the following facts concerning numerical recursive inequalitiessee25–
27.
Lemma 2.6. Let{λn},and{βn}be two sequences of nonnegative real numbers, and{αn}a sequence
of positive numbers satisfying the conditions∞n0γn∞,andlimn→ ∞ βn/αn0. Let the recursive
inequality
be given whereψλis a continuous and strict increasing function for allλ≥0withψ0 0. Then ( 1){λn}converges to zero, asn → ∞; ( 2) there exists a subsequence{λnk} ⊂ {λn}, k 1,2, . . . , such that
λnk ≤ψ
−1
1
nk
m0αm
βnk
αnk
,
λnk1≤ψ
−1
1
nk
m0αm
βnk
αnk
βnk,
λn≤λnk1−
n−1
mnk1
αm θm, nk
1< n < nk1, θm m
i0
αi,
λn1≤λ0− n
m0
αm
θm ≤λ0, 1≤n≤nk−1,
1≤nk≤smaxmax
s;
s
m0
αm θm ≤λ0
.
2.18
3. Main Results
We first discuss Browder’s type convergence.
Theorem 3.1. Let E, K,{Tn},{αn} have Browder’s property. For each u ∈ K, put P u
limn→ ∞yn, where{yn}is a sequence inKdefined by2.10. LetA:K → Kbe a weak contraction
with a functionϕ. Define a sequence{xn}inKby
xnαnAxn 1−αnTnxn, n∈N. 3.1
Then{xn}converges strongly to the unique pointz∈KsatisfyingPAz z.
Proof. We note that Proposition 2.2ii assures the existence and uniqueness of {xn}. It
follows fromProposition 2.2iandLemma 2.3thatP Ais a weak contraction onK, then by
Theorem 1.2, there exists the unique elementz ∈Ksuch thatPAz z. Define a sequence
{yn}inKby
ynαnAz 1−αnTnyn, for any n∈N. 3.2
Then by the assumption,{yn}converges strongly toPAz. For everyn, we have
xn−yn≤1−αnTnxn−Tnynαn Axn−Az
≤1−αnxn−ynαnAxn−AynαnAyn−Az ≤xn−yn−αnϕxn−ynαnyn−z−ϕ xn−z
,
then
ϕxn−yn≤yn−z. 3.4
Therefore,
lim
n→ ∞ϕxn−yn≤0, i.e., nlim→ ∞xn−yn0. 3.5
Hence,
lim
n→ ∞ xn−z ≤nlim→ ∞xn−ynyn−z0. 3.6
Consequently,{xn}converges strongly toz. This completes the proof.
We next discuss Halpern’s type convergence.
Theorem 3.2. LetE, K,{Tn},{αn}have Halpern’s property. For eachu∈K, putP ulimn→ ∞yn,
where{yn}is a sequence inKdefined by2.11. LetA:K → Kbe a weak contraction with a function ϕ. Define a sequence{xn}inKbyx1∈Kand
xn1αnAxn 1−αnTnxn, n∈N. 3.7
Then{xn}converges strongly to the unique pointz∈KsatisfyingPAz z. Moreover, there exist
a subsequence{xnk} ⊂ {xn}, k1,2, . . ., and∃{εn} ⊂0,∞withlimn→ ∞εn0such that
yn
k−xnk≤ϕ− 1
1
nk
m0αm
εnk
,
xn
k1−ynk1≤ϕ− 1
1
nk
m0αm
εnk
αnkεnk,
xn−yn≤xn
k1−ynk1−
n−1
mnk1
αm θm, nk
1< n < nk1, θm m
i0
αi,
yn1−xn1≤x0−y0−n m0
αm θm ≤
y0−x0, 1≤n≤nk−1,
1≤nk≤smaxmax
s;
s
m0
αm θm ≤
y0−x0.
3.8
Proof. It follows fromProposition 2.2iandLemma 2.4thatP Ais a weak contraction onK,
then byTheorem 1.2, there exists a unique elementz∈Ksuch thatzPAz. Thus we may
define a sequence{yn}inKby
Then by the assumption,yn → PAzasn → ∞. For everyn, we have
xn1−yn1≤αn Axn−Az 1−αnTnxn−Tnyn
≤αnAxn−AynAyn−Az 1−αnxn−yn ≤xn−yn−αnϕxn−ynαnyn−z−ϕyn−z.
3.10
Thus, we get forλn xn−yn the following recursive inequality:
λn1≤λn−αnϕλn βn, 3.11
whereβnαnεn, andεn yn−z . Thus byLemma 2.6,
lim
n→ ∞xn−yn0. 3.12
Hence,
lim
n→ ∞ xn−z ≤nlim→ ∞xn−ynyn−z0. 3.13
Consequently, we obtain the strong convergence of {xn} toz PAz, and the remainder
estimates now follow fromLemma 2.6.
Theorem 3.3. Let E be a Banach space E whose norm is uniformly Gˆateaux differentiable, and {αn} satisfies the condition (C2). Assume that E, K,{Tn},{αn} have Browder’s property and
limn→ ∞ yn −Tmyn 0 for every m ∈ N, where{yn} is a bounded sequence inK defined by
2.10. thenE, K,{Tn},{αn}have Halpern’s property.
Proof. Define a sequence{zm}inKbyu∈Kand
zmαmu 1−αmTmzm, m∈N. 3.14
It follows fromProposition 2.5and the assumption thatP ulimm→ ∞zmis the unique sunny
nonexpansive retraction fromKtoF.Subsequently, we approved that
∀ε >0, lim sup
n→ ∞
u−P u, Jyn−P u
In fact, sinceP u∈F, then we have
zm−yn2 1−
αm
Tmzm−yn, J
zm−yn
αm
u−yn, J
zm−yn
1−αm
Tmzm−Tmyn, J
zm−yn
Tmyn−yn, J
zm−yn
αm
u−P u, Jzm−yn
αm
P u−zm, J
zm−yn
αm
zm−yn, J
zm−yn
≤yn−zm2Tmyn−ynMαm
u−P u, Jzm−yn
αm zm−P u M,
3.16
then
u−P u, Jyn−zm
≤ yn−Tmyn
αm MM zm−P u ,
3.17
where Mis a constant such that M ≥ yn −zm by the boundedness of{yn}, and {zm}.
Therefore, using limn→ ∞ yn−Tmyn 0, andzm → P u, we get
lim sup
m→ ∞
lim sup
n→ ∞
u−P u, Jyn−zm
≤0. 3.18
On the other hand, since the duality mapJ is norm topology to weak∗ topology uniformly
continuous in a Banach spaceEwith uniformly Gˆateaux differentiable norm, we get that as
m → ∞,
u−P u, J
yn−P u
−Jyn−zm → 0, ∀n . 3.19
Therefore for anyε >0,∃N >0 such that for allm > Nand alln≥0, we have that
u−P u, Jyn−P u
<u−P u, Jyn−zm
ε. 3.20
Hence noting3.18, we get that
lim sup
n→ ∞
u−P u, Jyn−P u
≤lim sup
m→ ∞ lim supn→ ∞
u−P u, Jyn−zm
ε≤ε. 3.21
3.15is proved. From2.10andP u∈F, we have for alln≥0,
yn1−P u2
αn
u−P u, Jyn1−P u
1−αn
Tnyn−P u, J
yn1−P u
≤1−αn
Tnyn−P u2
Jyn1−P u2
2 αn
u−P u, Jyn1−P u
≤1−αn
yn−P u2
2
yn1−P u2
2 αn
u−P u, Jyn1−P u
.
Thus,
yn1−P u2≤
yn−P u2−αnyn−P u22αn
u−P u, Jyn1−P u
. 3.23
Consequently, we get forλn yn−P u 2the following recursive inequality:
λn1 ≤λn−αnψλn βn, 3.24
whereψt t,andβn2αnε.The strong convergence of{yn}toP ufollows fromLemma 2.6.
Namely,E, K,{Tn},{αn}have Halpern’s property.
4. Deduced Theorems
Using Theorems3.1,3.2, and3.3, we can obtain many convergence theorems. We state some
of them.
We now discuss convergence theorems for families of nonexpansive mappings. LetK
be a nonempty closed convex subset of a Banach spaceE. Aone parameternonexpansive
semigroups is a familyF{Tt:t >0}of selfmappings ofKsuch that
iT0xxforx∈K;
iiTtsxTtTsxfort, s >0,andx∈K;
iiilimt→0Ttxxforx∈K;
ivfor eacht >0, Ttis nonexpansive, that is,
Ttx−Tty≤x−y, ∀x, y∈K. 4.1
We will denote byFthe common fixed point set ofF, that is,
F:FixF {x∈K:Ttxx, t >0}
t>0
FixTt. 4.2
A continuous operator semigroupFis said to beuniformly asymptotically regularin
short, u.a.r. see28–31onKif for allh≥0 and any bounded subsetCofK,
lim
t→ ∞supx∈C ThTtx−Ttx 0. 4.3
Recently, Song and Xu31showed thatE, K,{Ttn},{αn}have both Browder’s
and Halpern’s property in a reflexive strictly convex Banach space with a uniformly
Gˆateaux differentiable norm whenevertn → ∞n → ∞. As a direct consequence
of Theorems3.1,3.2, and3.3, we obtain the following.
that limn→ ∞ tn ∞, and βn ∈ 0,1satisfies the condition C1, and αn ∈ 0,1 satisfies the
conditionsC1andC2. If{yn}and{xn}defined by
ynβnAyn
1−βn
Ttnyn, n∈N,
xn1αnAxn 1−αnTtnxn, n≥1.
4.4
Then as n → ∞, both {yn}, and{xn} strongly converge to z PAz, where P is a sunny
nonexpansive retraction fromKtoF.
Let{tn}a sequence of positive real numbers divergent to∞, and for each t > 0 and
x∈K,σtxis the average given by
σtx
1
t t
0
Tsxds. 4.5
Recently, Chen and Song 32 showed that E, K,{σtn},{αn} have both Browder’s and
Halpern’s property in a uniformly convex Banach space with a uniformly Gˆaeaux
differentiable norm whenevertn → ∞n → ∞. Then we also have the following.
Theorem 4.2. LetEbe a uniformly convex Banach space with uniformly Gˆateaux differentiable norm,
and letK, Abe as inTheorem 4.1. Suppose that{Tt}a nonexpansive semigroups fromKinto itself such thatF:FixF t>0 FixTt/∅,{yn},and{xn}defined by
ynβnAyn
1−βn
σtn
yn
, n∈N,
xn1αnAxn 1−αnσtnxn, n∈N,
4.6
wheretn → ∞, andβn ∈0,1satisfies the conditionC1, andαn ∈0,1satisfies the conditions
C1andC2. Then asn → ∞, both{yn},and{xn}strongly converge tozPAz, wherePis a
sunny nonexpansive retraction fromKtoF.
Acknowledgments
The authors would like to thank the editors and the anonymous referee for his or her valuable suggestions which helped to improve this manuscript.
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