Volume 2008, Article ID 824607,9pages doi:10.1155/2008/824607
Research Article
Generalized Mann Iterations for Approximating
Fixed Points of a Family of Hemicontractions
Liang-Gen Hu,1Ti-Jun Xiao,2and Jin Liang3
1Department of Mathematics, University of Science and Technology of China, Hefei 230026, China 2School of Mathematical Sciences, Fudan University, Shanghai 200433, China
3Department of Mathematics, Shanghai Jiaotong University, Shanghai 200240, China
Correspondence should be addressed to Jin Liang,[email protected]
Received 10 January 2008; Accepted 15 May 2008
Recommended by Hichem Ben-El-Mechaiekh
This paper concerns common fixed points for a finite family of hemicontractions or a finite family of strict pseudocontractions on uniformly convex Banach spaces. By introducing a new iteration process with error term, we obtain sufficient and necessary conditions, as well as sufficient conditions, for the existence of a fixed point. As one will see, we derive these strong convergence theorems in uniformly convex Banach spaces and without any requirement of the compactness on the domain of the mapping. The results given in this paper extend some previous theorems.
Copyrightq2008 Liang-Gen Hu 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
LetX be a real Banach space andKa nonempty closed subset ofX. A mappingT : K→Kis said to be pseudocontractivesee, e.g.,1if
Tx−Ty2≤ x−y2I−Tx−I−Ty2 1.1
holds for allx, y ∈K.T is said to be strictly pseudocontractive if, for allx, y ∈K, there exists a constantk∈0,1such that
Tx−Ty2≤ x−y2kI−Tx−I−Ty2. 1.2
Denote by FixT {x ∈ K : Tx x}the set of fixed points ofT. A mapT : K→Kis called hemicontractive if FixT/∅and for allx∈K,x∗∈FixT, the following inequality holds:
It is easy to see that the class of pseudocontractive mappings with fixed points is a subset of the class of hemicontractions.
There are many papers in the literature dealing with the approximation of fixed points for several classes of nonlinear mappingssee, e.g.,1–11, and the reference therein. In these works, there are two iterative methods to be used to find a point in FixT. One is explicit and
one is implicit.
The explicit one is the following well-known Mann iteration.
LetK be a nonempty closed convex subset ofX. For anyx0 ∈ K, the sequence{xn}is
defined by
xn1
1−αn
xnαnTxn, ∀n≥0, 1.4
where{αn}is a real sequence in0,1satisfying some assumptions.
It has been applied to many classes of nonlinear mappings to find a fixed point. However, for hemicontractive mappings and strictly pseudocontractive mappings, the iteration process of convergence is in general not strong see a counterexample given by Chidume and Mutangadura3. Most recently, Marino and Xu6proved that the Mann iterative sequence {xn} converges weakly to a fixed point for strictly pseudocontractive mappings in a Hilbert
space, while the real sequence{αn}satisfyingik < αn<1 andii
∞
n0αn−k1−αn ∞.
In order to get strong convergence for fixed points of hemicontractive mappings and strictly pseudocontractive mappings, the following Mann-type implicit iteration scheme is introduced.
LetKbe a nonempty closed convex subset of XwithKK⊆K.For anyx0∈K,the sequence {xn}is generated by
xnαnxn−11−αn
Txn, ∀n≥1, I
where{αn}is a real sequence in0,1satisfying suitable conditions.
Recently, in the setting of a Hilbert space, Rafiq12proved that the Mann-type implicit iterative sequence {xn} converges strongly to a fixed point for hemicontractive mappings,
under the assumption that the domainKof T is a compact convex subset of a Hilbert space, and{αn} ⊂δ,1−δfor someδ ∈0,1.
In this paper, we will study the strong convergence of the generalized Mann-type iteration schemeseeDefinition 2.1for hemicontractive and, respectively, pseudocontractive mappings. As we will see, our theorems extend the corresponding results in 12 in four aspects.1The space setting is a more general one: uniformly convex Banach space, which could not be a Hilbert space. 2The requirement of the compactness on the domain of the mapping is dropped. 3 A single mapping is replaced by a family of mappings. 4 The Mann-type implicit iteration is replaced by the generalized Mann iteration. Moreover, we give answers to a question asked in13.
2. Preliminaries and lemmas
Definition 2.1generalized Mann iteration. LetN ≥1 be a fixed integer,Λ:{1,2, . . . , N}, and Ka nonempty closed convex subset ofXsatisfying the conditionKK⊆K. Let{Ti:i∈Λ}:
K→Kbe a family of mappings. For eachx0 ∈K, the sequence{xn}is defined by
whereTnTnmodN,{an},{bn},and{cn}are three sequences in0,1withanbncn1 and
{un} ⊂Kis bounded.
The modulus of convexity ofXis the functionδX :0,2→0,1defined by
δXε inf
1−1
2xy
:xy1, x−y ≥ε
, 0≤ε≤2. 2.1
X is called uniformly convex if and only if, for all 0 < ε ≤ 2 such thatδXε > 0.X is called
p-uniformly convex if there exists a constanta >0, such thatδXε≥aεp. It is well knownsee
10that
Lp, lp, W1,p is
2-uniformly convex, if 1< p≤2, p-uniformly convex, if p≥2.
LetXbe a Banach space,Y ⊂X,andx∈X. Then, we denotedx, Y:infy∈Yx−y.
Definition 2.2see4. Letf:0,∞→0,∞be a nondecreasing function withf0 0 and
fr>0, for allr∈0,∞.
iA mappingT :K→Kwith FixT/∅is said to satisfyconditionAonKif there is a functionf such that for allx∈K,x−Tx ≥fdx,FixT.
iiA finite family of mappings{Ti : i ∈ Λ} : K→K with F : Ni1FixTi/∅are said
to satisfy condition Bif there exists a function f, such that max1≤i≤N{x−Tix} ≥
fdx, Fholds for allx∈K.
Lemma 2.3see8. LetXbe a real uniformly convex Banach space with the modulus of convexity of power typep≥2. Then, for allx, yinXandλ∈0,1, there exists a constantdp >0such that
λx 1−λyp ≤
λxp 1−λyp−wpλdpx−yp, 2.2
wherewpλ λp1−λ λ1−λp.
Remark 2.4. Ifp2 in the previous lemma, then we denoted2:d.
Lemma 2.5. LetXbe a real Banach space andJ : X→2X∗ the normalized duality mapping. Then for
anyx, yinXandjxy∈Jxy, such that
xy2≤ x22y, jxy. 2.3
Lemma 2.6see7. Let{αn},{βn},and{γn}be three nonnegative real sequences, satisfying
αn1≤
1βn
αnγn, ∀n≥1, 2.4
with∞n1βn<∞and
∞
n1γn<∞. Then,limn→∞αnexists. In addition, if{αn}has a subsequence
converging to zero, thenlimn→∞αn0.
Proposition 2.7. IfT is a strict pseudocontraction, thenT satisfies the Lipschitz condition
Tx−Ty ≤ 1
√
k
Proof. By the definition of the strict pseudocontraction, we have
Tx−Ty2≤ x−y2kI−Tx−I−Ty2≤x−ykI−Tx−I−Ty2.
2.6
A simple computation shows the conclusion.
3. Main results
Lemma 3.1. LetXbe a uniformly convex Banach space with the convex modulus of power typep≥2,K
a nonempty closed convex subset ofXsatisfyingKK⊆K, and{Ti:i∈Λ}:K→Khemicontractive
mappings with Ni1FixTi/∅. Let{an},{bn},{cn},{un},and{xn}be the sequences inIIand
i∞
n1
cn<∞,
ii
⎧ ⎪ ⎨ ⎪ ⎩
ε≤bn≤1−ε, for someε∈0,1, if d≥1,
bn1−bn≥ε, bn>1−dε, ε∈
0,d
2
, if d <1, ∀n≥1,
3.1
wheredis the constant inRemark 2.4. Then,
1limn→∞xn−qexists for allq∈F : Ni1FixTi,
2limn→∞dxn, Fexists,
3ifTii∈Λis continuous, thenlimn→∞xn−Tixn0, for alli∈Λ.
Proof. 1 Let q ∈ F Ni1FixTi. By the boundedness assumption on {un}, there exists a
constantM >0, for anyn≥1, such thatun−q ≤M. From the definition of hemicontractive
mappings, we have
Tixn−q2≤
xn−q2xn−Tixn2, ∀i∈Λ. 3.2
Using Lemmas2.3,2.5, and3.2, we obtain
xn−q2
1−bn
xn−1−q
bn
Tnxn−q
cn
un−xn−12
≤1−bn
xn−1−q
bn
Tnxn−q22cn
un−xn−1, j
xn−q
≤1−bnxn−1−q2bnTnxn−q2−bn
1−bn
dxn−1−Tnxn2
2cnun−qxn−1−qxn−q
≤1−bnxn−1−q2bnxn−q2bnxn−Tnxn2
−bn
1−bn
dxn−1−Tnxn22cnM
2cnMxn−q2cnxn−1−q2cnxn−q2.
Hence,
an−2cnMxn−q2 ≤
an2cnxn−1−q2bnxn−Tnxn2
−bn
1−bn
dxn−1−Tnxn22cnM.
3.4
It follows fromIIandLemma 2.5that
xn−Tnxn2
ancn
xn−1−Tnxn
cn
un−xn−12 ≤1−bn
2
xn−1−Tnxn22cn
un−xn−1, jxn−Tnxn
≤1−bn
2
xn−1−Tnxn22cnM22cnxn−1−q2cnxn−Tnxn2.
3.5
By the condition∞n1cn <∞, we may assume that
1 1−cn ≤
12cn, ∀n≥1. 3.6
Therefore,
xn−Tnxn2≤
1−bn 2
1−cn
xn−1−Tnxn2
2M2cn
12cn
2cn
12cnxn−1−q2.
3.7
Substituting3.7into3.4, we get
an−2cnMxn−q2≤
an2cn2bncn
12cnxn−1−q2 bn1−bn
2 1−cn
xn−1−Tnxn2
−bn
1−bn
dxn−1−Tnxn22cnM2cnbn
12cn
M2
an2cn2bncn
12cnxn−1−q2−bn
1−bn
d−1−bn
1−cn
×xn−1−Tnxn22cnM2cnbn
12cn
M2.
3.8
Assumptionsiandiiimply that there exists a positive integerN1such that for everyn >
N1,
an−2cnM ≥η >0, d−
1−bn
1−cn ≥ζ >0. 3.9
Hence, for alln > N1,
xn−q2≤ 1 2
M1bn
12cn
cn
an−2cnM
xn−1−q2
− bn
1−bn
an−2cnM
d−1−bn
1−cn
xn−1−Tnxn2
2Mbn
12cn
M1cn
an−2cnM
1λnxn−1−q2−σnxn−1−Tnxn2δn,
where
λn2
M1bn
12cn
cnη−1,
σn
bn
1−bn
an−2cnM
d−1−bn
1−cn
,
δn2M
bn
12cn
M1cnη−1.
3.11
From3.9and conditionsiandii, it follows that
∞
n1
λn<∞,
∞
n1
δn<∞, σn≥σ >0. 3.12
ByLemma 2.6, we see that limn→∞xn−qexists and the sequence{xn−q}is bounded.
2It is easy to verify that limn→∞dxn, Fexists.
3By the boundedness of{xn−q}, there exists a constantM1>0 such thatxn−q ≤
M1, for alln≥1. From3.10, we get, forn > N1,
σxn−1−Tnxn2 ≤xn−1−q2−xn−q2λnM1δn, 3.13
which implies
σ
∞
nN1
xn−1−Tnxn2≤ ∞
nN1
xn−1−q2−
xn−q2
∞
nN1
λnM1δn
<∞. 3.14
Thus,
∞
n1
xn−1−Tnxn2
<∞. 3.15
It implies that
lim
n→∞xn−1−Tnxn0. 3.16
Therefore, by3.7, we have
lim
n→∞xn−Tnxn0. 3.17
UsingII, we obtain
xn−xn−1≤ bn
an
xn−1−Tnxn cn
an
un−xn−1−→0, n−→ ∞,
xni−xn−→0, n−→ ∞, i∈Λ. 3.18
By a combination with the continuity ofTii∈Λ, we get
xn−Tnixn ≤xn−xnixni−TnixniTnixni−Tnixn−→0 n−→ ∞.
It is clear that for eachl∈Λ, there existsi∈Λsuch thatl nimodN. Consequently,
lim
n→∞xn−Tlxnnlim→∞xn−Tnixn0. 3.20
This completes the proof.
Theorem 3.2. Let the assumptions ofLemma 3.1hold, and letTii ∈ Λbe continuous. Then,{xn}
converges strongly to a common fixed point of{Ti:i∈Λ}if and only iflim infn→∞dxn, F 0.
Proof. The necessity is obvious.
Now, we prove the sufficiency. Since lim infn→∞dxn, F 0, it follows fromLemma 3.1
that limn→∞dxn, F 0.
For anyq∈F, we have
xn−xm≤xn−qxm−q. 3.21
Hence, we get
xn−xm≤inf q∈F
xn−qxm−qd
xn, F
dxm, F
−→0, n−→ ∞, m−→ ∞.
3.22
So, {xn}is a Cauchy sequence in K. By the closedness of K, we get that the sequence {xn}
converges strongly tox∗ ∈K. Let a sequence{qn} ∈ FixTi, for somei ∈Λ, be such that{qn}
converges strongly toq. By the continuity ofTii∈Λ, we obtain
q−Tiq≤q−qnqn−Tiqq−qnTiqn−Tiq−→0, n−→ ∞. 3.23
Therefore,q ∈FTi. This implies thatFTiis closed. Therefore,F : Ni1FixTiis closed. By
limn→∞dxn, F 0, we getx∗∈F. This completes the proof.
Theorem 3.3. Let the assumptions ofLemma 3.1hold. LetTii ∈Λbe continuous and{Ti: i ∈Λ}
satisfy conditionB. Then,{xn}converges strongly to a common fixed point of{Ti:i∈Λ}.
Proof. Since{Ti : i ∈ Λ}satisfies condition B, and limn→∞xn−Tixn 0 for each i ∈ Λ, it
follows from the existence of limn→∞dxn, Fthat limn→∞dxn, F 0. Applying the similar
arguments as in the proof of Theorem 3.2, we conclude that {xn} converges strongly to a
common fixed point of{Ti:i∈Λ}. This completes the proof.
As a direct consequence ofTheorem 3.3, we get the following result.
Corollary 3.4 see 12, Theorem 3. Let H be a real Hilbert space, K a nonempty closed convex subset ofHsatisfyingKK⊆K, andT :K→Kcontinuous hemicontractive mapping which satisfies condition A. Let{αn}be a real sequence in0,1with∞n11−αn2 ∞. For anyx0 ∈ K, the sequence{xn}is defined by
xnαnxn−11−αn
Txn, n≥1. 3.24
Proof. Employing the similar proof method ofLemma 3.1, we obtain by3.10
xn−q≤xn−1−q2− 1−αn
2
xn−1−Txn2. 3.25
This implies
∞
n1
1−αn 2
xn−1−Txn2≤x0−q2<∞. 3.26
By ∞n11−αn2 ∞, we have lim infn→∞xn−1 −Txn 0. Equation 3.7 implies that
lim infn→∞xn−Txn 0. SinceT satisfies condition Aand the limit limn→∞dxn, Fexists,
we get limn→∞dxn, F 0. The rest of the proof follows now directly fromTheorem 3.2. This
completes the proof.
Remark 3.5. Theorems3.2and3.3extend12, Theorem 3essentially since the following hold.
iHilbert spaces are extended to uniformly convex Banach spaces.
iiThe requirement of compactness on domainDTon12, Theorem 3is dropped. iiiA single mapping is replaced by a family of mappings.
ivThe Mann-type implicit iteration is replaced by the generalized Mann iteration. So the restrictions of {αn} with {αn} ⊂ δ,1 −δ for some δ ∈ 0,1 are relaxed to
∞
n11−αn2 ∞. The error term is also considered in the iterationII.
Moreover, ifKK⊆K, then{xn}is well defined byII. Hence, Theorems3.2and3.3are also
answers to the question proposed by Qing13.
Theorem 3.6. LetXandKbe as the assumptions ofLemma 3.1. Let{Ti :i ∈Λ}: K→Kbe strictly
pseudocontractive mappings with Ni1FixTibeing nonempty. Let{an},{bn},{cn},{un}, and{xn}
be the sequences inIIand
i∞
n1
cn<∞,
ii
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
ε≤bn≤1−ε, for someε∈0,1, ifd≥k,
bn−b2n≥ε, bn>1− d
k ε, for someε∈
0,
1− d
k d k −1
, ifk /0, d < k,
3.27
wheredis the constant inRemark 2.4. Then,
1{xn} converges strongly to a common fixed point of {Ti : i ∈ Λ} if and only if
lim infn→∞dxn, F 0.
2If{Ti:i∈Λ}satisfies condition (B) , then{xn}converges strongly to a common fixed point
Remark 3.7. Theorem 3.6extends the corresponding result6, Theorem 3.1.
Acknowledgments
The authors would like to thank the referees very much for helpful comments and suggestions. The work was supported partly by the National Natural Science Foundation of China, the Specialized Research Fund for the Doctoral Program of Higher Education of China, the NCET-04-0572 and Research Fund for the Key Program of the Chinese Academy of Sciences.
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