Volume 2010, Article ID 948529,8pages doi:10.1155/2010/948529
Research Article
Iterative Algorithms with Variable Coefficients for
Asymptotically Strict Pseudocontractions
Ci-Shui Ge,
1Jin Liang,
2and Ti-Jun Xiao
31Department of Mathematics and Physics, Anhui University of Architecture, Hefei, Anhui 230022, China
2Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China
3School of Mathematical Sciences, Fudan University, Shanghai 200433, China
Correspondence should be addressed to Jin Liang,[email protected]
Received 8 October 2009; Revised 29 November 2009; Accepted 22 January 2010
Academic Editor: Anthony To Ming Lau
Copyrightq2010 Ci-Shui Ge 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.
We introduce and study some new CQ-type iterative algorithms with variable coefficients for asymptotically strict pseudocontractions in real Hilbert spaces. General results for asymptotically strict pseudocontractions are established. The main result extends the previous results.
1. Introduction
LetHbe a real Hilbert space,Ca nonempty closed convex subset ofH,T :C → Ca self-mapping ofCand FixT:{x∈C:Txx}.
Recall that a mappingT :C → Cis called to be nonexpansive if
Tx−Ty≤x−y, ∀x, y∈C. 1.1
T is called to be asymptotically nonexpansive1if there exists a sequence{kn}withkn ≥1
and limn→ ∞kn1 such that
Tnx−Tny≤k
nx−y, ∀x, y∈C, and all integers n≥1. 1.2
T is called to be an asymptoticallyκ-strict pseudocontraction, if there exist 0≤κ <1 and 0≤
γn → 0n → ∞such that Tnx−Tny2≤
1γnx−y2κI−Tnx−I−Tny2 1.3
Asκ0, asymptoticallyκ-strict pseudocontractionT is asymptotically nonexpansive. In2, Nakajo and Takahashi studied the iterative approximation of fixed points of nonexpansive mappings and proved the following strong convergence theorem.
Theorem A. Let C be a nonempty closed convex subset of a Hilbert space H and let T be a nonexpansive mapping ofCinto itself such thatFixT/∅. Suppose{xn}is given by
x0∈Cchosen arbitrarily, ynαnxn 1−αnTxn,
Cnz∈C:yn−z≤ xn−z ,
Qn{z∈C:xn−z, x0−xn ≥0},
xn1PCn∩Qnx0, n∈N,
1.4
wherePCn∩Qn is the metric projection from C ontoCn∩Qnandαnis chosen so that0≤αn ≤a <1. Then,{xn}converges strongly toPFixTx0, wherePFixTis the metric projection from C ontoFixT.
Such algorithm in1.4is referred to be theCQalgorithm in3, due to the fact that each iteratexn1is obtained by projectingx0onto the intersection of the suitably constructed
closed convex setsCnandQn.It is known that theCQalgorithm in1.4is of independent interest, and theCQalgorithm has been extended to various mappings by many authors
cf., e.g.,3–11.
Very recently, by extending theCQalgorithm, Takahashi et al.9studied a family of nonexpansive mappings and gave some good strong convergence theorems. Kim and Xu
5 extended the CQ algorithm to study asymptotically κ-strict pseudocontractions and established the following interesting result with the help of some boundedness conditions.
Theorem B. Let C be a closed convex subset of a Hilbert space H and let T : C → C be an asymptoticallyκ-strict pseudocontractions for some0≤κ <1.Assume that the fixed point setFixT
of T is nonempty and bounded. Let{xn}∞n0be the sequence generated by the following (CQ) algorithm:
x0∈C, chosen arbitrarily, ynαnxn 1−αnTnxn,
Cn
z∈C:yn−z2≤ xn−z 2 κ−αn1−αn xn−Txn 2θn
,
Qn{z∈C:xn−z, x0−xn ≥0}, xn1PCn∩Qnx0,
1.5
where
θn Δ2n1−αnγn−→0 n−→ ∞, Δnsup{ xn−z :z∈FixT}<∞. 1.6
Assume that control sequence{αn}∞n0is chosen so thatlim supn→ ∞αn<1−κ.Then{xn}converges
It is our purpose in this paper to try to obtain some new fixed point theorems for asymptotically strict pseudocontractions without the boundedness conditions as in Theorem B. Motivated by Nakajo and Takahashi2, Takahashi et al. 9, and Kim and Xu 5, we introduce and study certain new CQ-type iterative algorithms with variable coefficients for asymptotically strict pseudocontractions in real Hilbert spaces. Our results improve essentially the corresponding results of5.
2. Results and Proofs
Throughout this paper,
ixn xmeans that{xn}converges weakly tox.
iixn → xmeans that{xn}converges strongly tox.
iiiωwxn:{x:∃xnj x}, that is, the weakω-limit set of{xn}.
ivBrx0:{x∈H: x−x0 ≤r}.
vNis the set of nonnegative integers.
The following lemmas are basiccf., e.g.,6forLemma 2.1, and5for Lemmas 2.2-2.3.
Lemma 2.1. LetKbe a closed convex subset of a real Hilbert spaceH. Givenx∈ H, z∈K. Then zPKxif and only if
x−z, y−z ≤0, ∀y∈K, 2.1
wherePKxis the unique point inKwith the property
x−PKx ≤x−y, ∀y∈K. 2.2
Lemma 2.2. Let K be a closed convex subset of a real Hilbert space H,{xn} ⊂ H, u ∈ H, and qPKu. Suppose that{xn}satisfies
xn−u ≤u−q, ∀n∈N, 2.3
andωwxn⊂K. Thenxn → q.
Lemma 2.3. LetCbe a closed convex subset of a Hilbert spaceHandT:C → Can asymptotically κ-strict pseudocontraction. Then
Ifor eachn≥1,Tnsatisfies the Lipschitz condition:
Tnx−Tny≤L
nx−y, ∀x, y∈C, 2.4
where
IIif{xn}is a sequence inCsuch thatxnxand
lim sup
m→ ∞ lim supn→ ∞ xn−T
mx
n 0, 2.6
then
I−Txn−→0⇒I−Tx0. 2.7
In particular,
xnx, I−Txn−→0⇒I−Tx0. 2.8
IIIFixTis closed and convex so that the projectionPFixTis well defined.
Theorem 2.4. LetCbe a closed convex subset of a Hilbert spaceH,T :C → Can asymptotically κ-strict pseudocontraction for some0≤κ <1, andFixT/∅.Let{xn}be the sequence generated by
the following CQ-type algorithm with variable coefficients:
x0∈Cchosen arbitrarily,
yn
1−βn
xnβnTnxn,
Cn
z∈C:yn−z2 ≤ xn−z 2βn
κβn−1
xn−Tnxn 2θn
,
Qn{z∈C:xn−z, x0−xn ≥0},
xn1PCn∩Qnx0, n∈N,
2.9
where
βn βn
1 xn−x0 2
, βn∈
1 2,1
, θn2
1r02βnγn, 2.10
the sequence{βn}is chosen so thatβn → 1n → ∞, the positive real numberr0 is chosen so that Br0x0∩FixT/∅, and{γn}is as in1.3. Then{xn}converges strongly toPFixTx0.
Proof. We divide the proof into five steps.
Step 1. We prove thatCn∩Qnis nonempty, convex and closed.
Clearly, bothQnandCn are convex and closed, so isCn∩Qn. SinceT :C → Cis an
asymptoticallyκ-strict pseudocontraction, we have by1.3,
Tnx−p2≤
1γnx−p2κI−Tnx−I−Tnp2
≤1γnx−p2κ x−Tnx 2,
2.11
By2.9and2.11, we deduce that for eachp∈Br0x0∩FixT, n∈N,
yn−p2
1−βnxn−pβnTnxn−p 2
1−βnxn−p2βnTnxn−p2−βn
1−βn
xn−Tnxn 2
1−βnxn−p2βn1γnxn−p2κ xn−Tnxn 2
−βn
1−βn xn−Tnxn 2
≤xn−p2βn
κβn−1
xn−Tnxn 2βnγn
2 xn−x0 2x0−p2
1 xn−x0 2
≤xn−p2βn
κβn−1
xn−Tnxn 22
1r02βnγn
xn−p2βn
κβn−1
xn−Tnxn 2θn.
2.12
Therefore,
Brx0∩FixT⊂Cn, ∀n∈N. 2.13
Next, we prove by induction that
Br0x0∩FixT⊂Qn, ∀n∈N. 2.14
Obviously,Br0x0∩FixT⊂ C Q0, that is,2.14holds forn 0. Assume thatBr0x0∩ FixT ⊂ Qn for somen ∈ N.Then, 2.13implies thatBr0x0∩FixT ⊂ Cn∩Qn/∅and
xn1PCn∩Qnx0is well defined.
ByLemma 2.1, we getxn1−z, x0−xn1 ≥ 0,∀ z∈ Cn∩Qn.In particular, for each
z ∈ Br0x0∩FixT,we havexn1−z, x0−xn1 ≥ 0.This together with the definition of
Qn1, the inequality2.14holds forn1. So2.14is true.
Step 2. We prove that limn→ ∞ xn1−xn 0.
By the definition ofQnandLemma 2.1, we getxn PQnx0.Hence,
xn−x0 ≤p−x0, ∀p∈Br0x0∩FixT. 2.15
DenotingM: x0 p−x0 , we have xn ≤M,for alln∈N,and
whereq PFixTx0 ⊂ Br0x0∩FixT.The definition ofxn1 shows thatxn1 ∈ Qn, that is, xn1−xn, xn−x0 ≥0.This implies that
xn1−xn 2 xn1−x0 2− xn−x0 2−2xn1−xn, xn−x0
≤ xn1−x0 2− xn−x0 2.
2.17
Thus{ xn−x0 }is increasing. Since{xn}is bounded, limn→ ∞ xn−x0 exists and
lim
n→ ∞ xn1−xn 0. 2.18
Step 3. We prove that limn→ ∞ xn−Tnxn 0.
The definition ofxn1shows thatxn1∈Cn, that is,
yn−xn12
≤ xn−xn1 2βn
κβn−1
xn−Tnxn 2θn. 2.19
By2.19and the definition ofynin2.9, we deduce that
β2
n xn−Tnxn 2yn−xn2
≤yn−xn12 xn1−xn 22yn−xn1· xn1−xn
≤βn
κβn−1
xn−Tnxn 2θn2 xn1−xn 22yn−xn1· xn1−xn .
2.20
Further, we have
1−κβn xn−Tnxn 2≤2 xn1−xn 22 xn1−xn ·yn−xn1θn. 2.21
Thus,2.19and2.21imply that
1−κβn xn−Tnxn 2≤4 xn1−xn 22 xn1−xn · xn−Tnxn
βnκβn−1
2 xn1−xn θnθn.
2.22
Noticing xn ≤M, βn∈1/2,1, we get
βn βn
1 xn 2
≥ 1
21M2>0. 2.23
From limn→ ∞ xn1−xn 0, limn→ ∞θn0,and2.22, it follows that
lim
n→ ∞ xn−T
nx
Step 4. We prove that
lim
n→ ∞ xn−Txn 0. 2.25
ByLemma 2.3and the definition ofT, we obtain
xn−Txn ≤ xn−xn1 xn1−Tn1xn1Tn1xn1−Tn1xnTn1xn−Txn
≤1Ln1 xn1−xn xn1−Tn1xn1L1 xn−Tnxn ,
2.26
where
Ln κ 11−γκn1−κ, γnis as in1.3. 2.27
By2.18,2.24, and2.26, we know that2.25holds.
Step 5. Finally, byLemma 2.3and2.25, we haveωwxn⊂ FixT. Furthermore, it follows
from2.16andLemma 2.2that the sequence{xn}converges strongly toqPFixTx0.
Remark 2.5. Theorem 2.4 improves 5, Theorem 4.1 since the condition that θn → 0 is
satisfied and the boundedness of FixTis dropped off.
Theorem 2.6. LetCbe a closed convex subset of a Hilbert spaceH,T :C → Can asymptotically κ-strict pseudocontraction for some0 ≤κ < 1, andFixTbe nonempty and bounded. Let{xn}the
sequence generated by the following CQ-type algorithm with variable coefficients:
x0∈Cchosen arbitrarily,
yn
1−βn
xnβnTnxn,
Cn
z∈C:yn−z2≤ xn−z 2βnκβn−1 xn−Tnxn 2θn,
Qn{z∈C:xn−z, x0−xn ≥0},
xn1PCn∩Qnx0, n∈N,
2.28
where
βn βn
1 xn−x0 2
, βn∈
1 2,1
, θn
sup
z∈FixT xn−z
2
βnγn, 2.29
the sequence{βn}is chosen so thatβn → 1n → ∞,and{γn}is as in1.3. Then{xn}converges
Proof. It is easy to see that θn → 0 in Theorem 2.6. Following the reasoning in the proof
of Theorem 2.4and using FixT instead ofBr0x0∩FixT, we deduce the conclusion of
Theorem 2.6.
Acknowledgments
The authors are very grateful to the referee for his/her valuable suggestions and comments. The work was supported partly by the NSF of China 10771202, the Research Fund for Shanghai Key Laboratory for Contemporary Applied Mathematics 08DZ2271900, and the Specialized Research Fund for the Doctoral Program of Higher Education of China
2007035805. This work is dedicated to W. Takahashi.
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