Fixed Point Theory and Applications Volume 2010, Article ID 150539,9pages doi:10.1155/2010/150539
Research Article
Strong Convergence Theorems for Strict Pseudocontractions in Uniformly Convex Banach Spaces
Liang-Gen Hu,
1Wei-Wei Lin,
2and Jin-Ping Wang
11Department of Mathematics, Ningbo University, Zhejiang 315211, China
2School of Computer Science and Engineering, South China University of Technology, Guangzhou 510640, China
Correspondence should be addressed to Liang-Gen Hu,[email protected] Received 20 April 2010; Accepted 26 August 2010
Academic Editor: W. Takahashi
Copyrightq2010 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.
The viscosity approximation methods are employed to establish strong convergence theorems of the modified Mann iteration scheme toλ-strict pseudocontractions inp-uniformly convex Banach spaces with a uniformly Gˆateaux differentiable norm. The main result improves and extends many nice results existing in the current literature.
1. Introduction
LetEbe a real Banach space, and letCbe a nonempty closed convex subsetE. We denote by Jthe normalized duality map fromEto 2E∗defined by
Jx
x∗∈E∗: x, x∗x2x∗2,∀x∈E
. 1.1
A mappingT :C → Cis said to be aλ-strictly pseudocontractive mappingsee, e.g., 1if there exists a constant 0≤λ <1 such that
Tx−Ty2≤x−y2λI−Tx−I−Ty2, 1.2
for allx, y∈C. We note that the class ofλ-strict pseudocontractions strictly includes the class of nonexpansive mappings which are mappingTonCsuch that
Tx−Ty≤x−y, 1.3 for allx, y∈C. Obviously,Tis nonexpansive if and only ifT is a 0-strict pseudocontraction.
A mappingT : C → Cis said to be aλ-strictly pseudocontractive mapping with respect to pif, for allx, y∈C, there exists a constant 0≤λ <1 such that
Tx−Typ≤x−ypλI−Tx−I−Typ. 1.4 A mappingf:C → Cis calledk-contraction if there exists a constantk∈0,1such that
fx−f
y≤kx−y, ∀x, y∈C. 1.5 We denote by FixTthe set of fixed point ofT, that is, FixT {x∈C: Txx}.
Recall the definition of Mann’s iteration; letCbe a nonempty convex subsetE,and letT be a self-mapping of C. For anyx1∈C, the sequence{xn}is defined by
xn1 1−αnxnαnTxn, n≥1, 1.6 where{αn}is a real sequence in0,1.
In the last ten years or so, there have been many nice papers in the literature dealing with the iteration approximating fixed points of Lipschitz strongly pseudocontractive mappings by utilizing the Mann iteration process. Results which had been known only for Hilbert spaces and Lipschitz mappings have been extended to more general Banach spaces and more general class of mappings; see, for example,1–6and the references therein for more information about this problem.
In 2007, Marino and Xu 2 showed that the Mann iterative sequence converges weakly to a fixed point of λ-strict pseudocontractions in Hilbert spaces. Meanwhile, they have proposed an open question; that is, is the result of 2, Theorem 3.1true in uniformly convex Banach spaces with Fr´echet differentiable norm? In other words, can Reich’s theorem7, Theorem 2, with respect to nonexpansive mappings, be extended to λ-strict pseudocontractions in uniformly convex Banach spaces?
In 2008, using the Mann iteration and the modified Ishikawa iteration, Zhou 3 obtained some weak and strong convergence theorems for λ-strict pseudocontractions in Hilbert spaces which extend the corresponding results in2.
Recently, Hu and Wang 4 obtained that the Mann iterative sequence converges weakly to a fixed point ofλ-strict pseudocontractions with respect topinp-uniformly convex Banach spaces.
In this paper, we first introduce the modified Mann iterative sequence. Let C be a nonempty closed convex subset of E,and letf : C → Cbe ak-contraction. For anyx1 ∈ C, the sequence{xn}is defined by
xn1αnxn 1−αnTn
βnfxn 1−βn
xn
, n≥1, 1.7
whereTnx: 1−μnxμnTx, for allx∈C,{αn},{βn}, and{μn}in0,1. The iterative sequence 1.7is a natural generalization of the Mann iterative sequences1.6. If we takeβn ≡ 0, for alln≥1, in1.7, then1.7is reduced to the Mann iteration.
The purpose in this paper is to show strong convergence theorems of the modified Mann iteration scheme for λ-strict pseudocontractions with respect to p in p-uniformly convex Banach spaces with uniformly Gateaux differentiable norm by using viscosity approximation methods. Our theorems improve and extend the comparable results in the following four aspects: 1 in contrast to weak convergence results in 2–4, strong convergence theorems of the modified Mann iterative sequence are obtained inp-uniformly convex Banach spaces;2in contrast to the results in 7,8, these results with respect to nonexpansive mappings are extended to λ-strict pseudocontractions with respect to p;3 the restrictions∞
n1|αn1−αn|<∞and∞
n1|βn1−βn|<∞in8, Theorem 3.1are removed;
4our results partially answer the open question.
2. Preliminaries
The modulus of convexity ofEis the functionδE: 0,2 → 0,1defined by δE inf 1−
xy 2
:x1,y1,x−y≥
, 0≤≤2. 2.1
E is uniformly convex if and only if, for all 0 < ≤ 2 such thatδE > 0.Eis said to be p-uniformly convex if there exists a constanta >0 such thatδE≥ap. Hilbert spaces,Lpor lpspaces1< p <∞and Sobolev spacesWmp1< p <∞arep-uniformly convex. Hilbert spaces are 2-uniformly convex, while
Lp, lp, Wmp are
⎧⎨
⎩
2-uniformly convex if 1< p≤2,
p-uniformly convex ifp≥2. 2.2
A Banach spaceEis said to have Gateaux differentiable norm if the limit
limt→0
xty− x
t 2.3
exists for eachx, y∈U, whereU{x∈E: x1}. The norm ofEis a uniformlyGateaux differentiable if for eachy∈U, the limit is attained uniformly forx∈U. It is well known that ifEis a uniformly Gateaux differentiable norm, then the duality mappingJis single valued and norm-to-weak∗uniformly continuous on each bounded subset ofE.
Lemma 2.1see4. LetEbe a real p-uniformly convex Banach space, and letCbe a nonempty closed convex subset ofE. LetT : C → Cbe aλ-strict pseudocontraction with respect top, and let {ξn}be a real sequence in0,1. IfTn: C → Cis defined byTnx: 1−ξnxξnTx, for allx∈C, then for allx, y∈C, the inequality holds
Tnx−Tnyp≤x−yp−
wpξncp−ξnkI−Tx−I−Typ, 2.4
wherecpis a constant in [9, Theorem 1]. In addition, if 0≤λ <min{1,2−p−2cp},ξ1−λ·2p−2/cp, andξn∈0, ξ, thenTnx−Tny ≤ x−y, for allx, y∈C.
Lemma 2.2see10. Let{xn}and{yn}be bounded sequences in a Banach spaceEsuch that
xn1αnxn 1−αnyn, n≥0, 2.5
where{αn}is a sequence in0,1such that 0<lim infn→ ∞αn≤lim supn→ ∞αn<1. Assuming lim sup
n→ ∞
yn1−yn− xn1−xn
≤0, 2.6
then limn→ ∞xn−yn0.
Lemma 2.3. LetEbe a real Banach space. Then, for allx, y∈Eandjxy∈Jxy, the following inequality holds:
xy2≤ x22 y, j
xy
. 2.7
Lemma 2.4see11. Let{an}be a sequence of nonnegative real number such that
an1≤1−δnanδnηn, ∀n≥0, 2.8
where{δn}is a sequence in0,1and{ηn}is a sequence inRsatisfying the following conditions:
i ∞
n1δn ∞;iilim supn→ ∞ηn ≤0 or∞
n1δn|ηn|<∞. Then, limn→ ∞an0.
3. Main Results
Theorem 3.1. Let E be a real p-uniformly convex Banach space with a uniformly Gateaux differentiable norm, and letC be a nonempty closed convex subset of E which has the fixed point property for nonexpansive mappings. LetT :C → Cbe aλ-strict pseudocontraction with respect to p,λ ∈ 0,min{1,2−p−2cp}and FixT/∅. Letf : C → Cbe ak-contraction withk ∈ 0,1.
Assume that real sequences{αn},{βn}, and{ξn}in0,1satisfy the following conditions:
i0<lim infn→ ∞αn≤lim supn→ ∞αn<1, iilimn→ ∞βn0 and∞
n1βn ∞,
iii0<infnξn≤ξand limn→ ∞|ξn1−ξn|0, whereξ1−λ·2p−2/cp. For anyx1∈C, the sequence{xn}is generated by
xn1αnxn 1−αnTn
βnfxn 1−βn
xn
, n≥1, 3.1
whereTnx: 1−ξnxξnTx, for allx∈C. Then, the sequence{xn}converges strongly to a fixed point ofT.
Proof. Equation3.1can be expressed as follows:
xn1αnxn 1−αnTnyn, 3.2 where
ynβnfxn 1−βn
xn, ∀n≥1. 3.3
Takingp∈FixT, we obtain fromLemma 2.1 xn1−p≤αnxn−p 1−αnTnyn−p
≤αnxn−p 1−αn
βnfxn−p
1−βnxn−p
≤αnxn−p 1−αn
βnkxn−pβnf p
−p
1−βnxn−p
1−1−αnβn1−kxn−p 1−αnβn1−k 1 1−kf
p
−p
≤max x1−p, 1 1−kf
p
−p .
3.4
Therefore, the sequence{xn}is bounded, and so are the sequences{fxn},{Tnyn}, and{yn}.
SinceTnyn 1−ξnynξnTynand the conditioniii,we know that{Tyn}is bounded. We estimate from3.3that
yn1−yn≤βn1fxn1−fxn
1−βn1
xn1−xn βn1−βnfxn−xn
≤
1−βn11−k
xn1−xnβn1−βnfxn−xn.
3.5
SinceTn: 1−ξnIξnT, whereIis the identity mapping, we have
Tn1yn1−Tnyn≤1−ξn1yn1ξn1Tyn1−1−ξn1yn−ξn1Tyn |ξn1−ξn|yn−Tyn
≤yn1−yn|ξn1−ξn|yn−Tyn.
3.6
limn→ ∞βn0 and limn→ ∞|ξn1−ξn|0 imply from3.5and3.6that lim sup
n→ ∞
Tn1yn1−Tnyn− xn1−xn
≤0. 3.7
Hence, byLemma 2.2, we obtain
nlim→ ∞Tnyn−xn0. 3.8
From3.3, we get
nlim→ ∞yn−xn lim
n→ ∞βnfxn−xn0, 3.9
and so it follows from3.8and3.9that limn→ ∞yn−Tnyn0. Sinceyn−Tnynξnyn−Tyn and infnξn>0, we have
nlim→ ∞yn−Tyn lim
n→ ∞
yn−Tnyn
ξn 0. 3.10
For anyδ∈0, ξ, definingTδ: 1−δIδT, we have
nlim→ ∞yn−Tδyn lim
n→ ∞δyn−Tyn0. 3.11
Since Tδ is a nonexpansive mapping, we have from 12, Theorem 4.1 that the net {xt} generated byxt tfxt 1−tTδxtconverges strongly toq∈FixTδ FixT, ast → 0.
Clearly,
xt−yn 1−t
Tδxt−yn t
fxt−yn
. 3.12
In view ofLemma 2.3, we find
xt−yn2≤1−t2Tδxt−yn22t
fxt−yn, J
xt−yn
≤
1−2tt2xt−ynTδyn−yn22t
fxt−xt, J
xt−yn 2txt−yn2,
3.13
and hence fxt−xt, J
yn−xt
≤ t
2xt−yn2
1t2yn−Tδyn 2t
2xt−ynyn−Tδyn. 3.14 Since the sequences{yn},{xt}, and{Tδyn}are bounded and limn→ ∞yn−Tδyn/2t0, we obtain
lim sup
n→ ∞
fxt−xt, J
yn−xt
≤ t
2M, 3.15
whereMsupn≥1,t∈0,1{xt−yn2}. We also know that f
q
−q, J
yn−q
fxt−xt, J
yn−xt
f q
−fxt xt−q, J
yn−xt
f q
−q, J yn−q
−J
yn−xt
. 3.16
From the facts thatxt → q∈FixT, ast → 0,{yn}is bounded, and the duality mappingJis norm-to-weak∗uniformly continuous on bounded subset ofE, it follows that
f q
−q, J yn−q
−J
yn−xt
−→0, ast−→0, f
q
−fxt xt−q, J
yn−xt
−→0, ast−→0. 3.17
Combining3.15,3.16, and the two results mentioned above, we get lim sup
n→ ∞
f q
−q, J
yn−q
≤0. 3.18
From3.9and the fact that the duality mappingJ is norm-to-weak∗uniformly continuous on bounded subset ofE, it follows that
nlim→ ∞fxn−f q
, J yn−q
−J
xn−q0. 3.19
Writing
xn1−qαn xn−q
1−αnTn yn−q
, 3.20
and fromLemma 2.3, we find
xn1−q2≤αnxn−q2 1−αnβn
fxn−q
1−βn
xn−q2
≤αnxn−q2 1−αn
1−βn2xn−q2 21−αnβn
fxn−q, J
yn−q
≤αnxn−q2 1−αn
1−βn2xn−q221−αnβnkxn−q2 21−αnβn
f q
−q, J
yn−q 21−αnβn
fxn−f q
, J yn−q
−J
xn−q
≤
1−21−αn1−kβnxn−q221−αnβn
×
βnxn−qfxn−f q
, J yn−q
−J
xn−q f
q
−q, J
yn−q 1−1−kδnxn−q2δnηn,
3.21
where
δn21−αnβn,
ηnβnxn−qfxn−f q
, J yn−q
−J
xn−qf q
−q, J yn−q
. 3.22
From 3.18, 3.19, and the conditions i,ii, it follows that ∞
n1δn ∞ and
lim supn→ ∞ηn ≤ 0. Consequently, applying Lemma 2.4 to 3.21, we conclude that limn→ ∞xn−q0.
Corollary 3.2. Let E,C,T, {αn},{βn}, and {ξn} be as in Theorem 3.1. For any u, x1 ∈ C, the sequence{xn}is generated by
xn1αnxn 1−αnTn βnu
1−βn xn
, n≥1, 3.23
whereTnx: 1−ξnxξnTx, for allx∈C. Then the sequence{xn}converges strongly to a fixed point ofT.
Remark 3.3. Theorem 3.1andCorollary 3.2improve and extend the corresponding results in 2–4,7,8essentially since the following facts hold.
1 Theorem 3.1 and Corollary 3.2 give strong convergence results in p-uniformly convex Banach spaces for the modification of Mann iteration scheme in contrast to the weak convergence result in2, Theorem 3.1,3, Theorem 3.1 and Corollary 3.3, and4, Theorems 3.2 and 3.3.
2 In contrast to the results in7, Theorem 2, and8, Theorem 3.1, these results with respect to nonexpansive mappings are extended to λ-strict pseudocontraction in p-uniformly convex Banach spaces.
3In contrast to the results in8, Theorem 3.1, the restrictions∞
n1|αn1−αn| < ∞ and∞
n1|βn1−βn|<∞are removed.
Acknowledgments
The authors would like to thank the referees for the helpful suggestions. Liang-Gen Hu was supported partly by Ningbo Natural Science Foundation 2010A610100, the NNSFC 60872095, the K. C. Wong Magna Fund of Ningbo University and the Scientific Research Fund of Zhejiang Provincial Education Department Y200906210. Wei-Wei Lin was supported partly by the Fundamental Research Funds for the Central Universities, SCUT20092M0103. Jin-Ping Wang were supported partly by the NNSFC60872095 and Ningbo Natural Science Foundation2008A610018.
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