Volume 2008, Article ID 284613,8pages doi:10.1155/2008/284613
Research Article
Strong Convergence of Monotone Hybrid Algorithm for Hemi-Relatively Nonexpansive Mappings
Yongfu Su,1Dongxing Wang,1and Meijuan Shang1, 2
1Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China
2Department of Mathematics, Shijiazhuang University, Shijiazhuang 050035, China
Correspondence should be addressed to Yongfu Su,[email protected] Received 1 June 2007; Revised 5 September 2007; Accepted 16 October 2007 Recommended by Simeon Reich
The purpose of this article is to prove strong convergence theorems for fixed points of closed hemi- relatively nonexpansive mappings. In order to get these convergence theorems, the monotone hy- brid iteration method is presented and is used to approximate those fixed points. Note that the hybrid iteration method presented by S. Matsushita and W. Takahashi can be used for relatively nonexpansive mapping, but it cannot be used for hemi-relatively nonexpansive mapping. The re- sults of this paper modify and improve the results of S. Matsushita and W. Takahashi2005, and some others.
Copyrightq2008 Yongfu Su 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
In 2005, Shin-ya Matsushita and Wataru Takahashi1proposed the following hybrid iteration methodit is also called the CQ methodwith generalized projection for relatively nonexpan- sive mappingTin a Banach spaceE:
x0∈C chosen arbitrarily, ynJ−1
αnJxn 1−αn
JTxn
, Cn
z∈C:φ z, yn
≤φ z, xn
, Qn
z∈C:xn−z, Jx0−Jxn ≥0 , xn1 Π
Cn∩Qn
x0 .
1.1
They proved the following convergence theorem.
Theorem 1.1MT. LetEbe a uniformly convex and uniformly smooth real Banach space, letCbe a nonempty, closed, and convex subset ofE, letTbe a relatively nonexpansive mapping fromCinto itself, and let{αn}be a sequence of real numbers such that 0 ≤ αn < 1 and lim supn→∞αn < 1. Suppose that{xn}is given by1.1, whereJis the duality mapping onE. If the setFTof fixed points ofT is nonempty, then{xn}converges strongly toΠFTx0, whereΠFT·is the generalized projection from ContoFT.
The purpose of this article is to prove strong convergence theorems for fixed points of closed hemi-relatively nonexpansive mappings. In order to get these convergence theorems, the monotone hybrid iteration method is presented and is used to approximate those fixed points. Note that the hybrid iteration method presented by S.Matsushita and W. Takahashi can be used for relatively nonexpansive mapping, but it cannot be used for hemi-relatively non- expansive mapping. The results of this paper modify and improve the results of S.Matsushita and W. Takahashi1, and some others.
2. Preliminaries
LetEbe a real Banach space with dualE∗. We denote byJthe normalized duality mapping fromEto 2E∗defined by
Jx
f∈E∗:x, fx2f2
, 2.1
where·,·denotes the generalized duality pairing. It is well known that if E∗is uniformly convex, thenJis uniformly continuous on bounded subsets ofE. In this case,Jis singe valued and also one to one.
Recall that ifCis a nonempty, closed, and convex subset of a Hilbert spaceH andPC : H →Cis the metric projection ofHontoC, thenPCis nonexpansive. This is true only when H is a real Hilbert space. In this connection, Alber2has recently introduced a generalized projection operatorΠCin a Banach spaceEwhich is an analogue of the metric projection in Hilbert spaces.
Next, we assume thatEis a smooth Banach space. Consider the functional defined as 2,3by
φx, y x2−2x, Jyy2 forx, y∈E. 2.2 Observe that, in a Hilbert spaceH,2.2reduces toφx, y x−y2,x, y∈H.
The generalized projectionΠC:E → Cis a map that assigns to an arbitrary pointx∈E the minimum point of the functionalφy, x,that is,ΠCx x,wherexis the solution to the minimization problem
φx, x min
y∈Cφy, x, 2.3
existence and uniqueness of the operator ΠC follow from the properties of the functional φy, xand strict monotonicity of the mappingJsee, e.g.,2–4. In Hilbert space,ΠCPC.It is obvious from the definition of the functionφthat
y − x2≤φy, x≤
yx2 ∀x, y∈E. 2.4
Remark 2.1. IfEis a reflexive strict convex and smooth Banach space, then forx, y∈E,φx, y 0 if and only ifxy. It is sufficient to show that ifφx, y 0, thenxy. From2.4, we have xy. This impliesx, Jyx2Jy2.From the definition ofJ,we haveJxJy, that is,xy; see5for more details.
We refer the interested reader to the6, where additional information on the duality mapping may be found.
Let C be a closed convex subset of E, and Let T be a mapping from C into itself.
We denote by FT the set of fixed points ofT. T is called hemi-relatively nonexpansive if φp, Tx≤φp, xfor allx∈Candp∈FT.
A pointpinCis said to be an asymptotic fixed point ofT7ifCcontains a sequence{xn} which converges weakly topsuch that the strong limn→∞Txn−xn 0.The set of asymptotic fixed points ofTwill be denoted byFT . A hemi-relatively nonexpansive mappingT fromC into itself is called relatively nonexpansive1,7,8ifFT FT.
We need the following lemmas for the proof of our main results.
Lemma 2.2Kamimura and Takahashi4,1, Proposition 2.1. LetEbe a uniformly convex and smooth real Banach space and let{xn},{yn}be two sequences ofE. Ifφxn, yn → 0 and either{xn} or{yn}is bounded, thenxn−yn → 0.
Lemma 2.3Alber2,1, Proposition 2.2. LetCbe a nonempty closed convex subset of a smooth real Banach spaceEandx∈E. Then,x0 ΠCxif and only if
x0−y, Jx−Jx0
≥0 ∀y∈C. 2.5
Lemma 2.4Alber2,1, Proposition 2.3. LetE be a reflexive, strict convex, and smooth real Banach space, letCbe a nonempty closed convex subset ofEand letx∈E.Then
φ y,Π
cx φ Π
cx, x
≤φy, x ∀y∈C. 2.6
By using the similar method as1, Proposition 2.4, the following lemma is not hard to prove.
Lemma 2.5. LetEbe a strictly convex and smooth real Banach space, letCbe a closed convex subset ofE, and letTbe a hemi-relatively nonexpansive mapping fromCinto itself. ThenFTis closed and convex.
Recall that an operatorTin a Banach space is called closed, ifxn → x, Txn → y, then Txy.
3. Strong convergence for hemi-relatively nonexpansive mappings
Theorem 3.1. Theorem 3.1LetEbe a uniformly convex and uniformly smooth real Banach space, let Cbe a nonempty closed convex subset ofE, letT : C → Cbe a closed hemi-relatively nonexpansive mapping such thatFT/∅. Assume that{αn}is a sequence in0,1such that lim supn→∞αn <1.
Define a sequence{xn}inCby the following algorithm:
x0∈C chosen arbitrarily, ynJ−1
αnJxn 1−αn
JTxn , Cn
z∈Cn−1∩Qn−1:φ z, yn
≤φ z, xn
, C0
z∈C:φ z, y0
≤φ z, x0
, Qn
z∈Cn−1∩Qn−1:
xn−z, Jx0−Jxn
≥0 , Q0C,
xn1 Π
Cn∩Qn
x0
,
3.1
whereJ is the duality mapping onE. Then{xn} converges strongly toΠFTx0, whereΠFT is the generalized projection fromContoFT.
Proof. We first show thatCnandQnare closed and convex for eachn≥0. From the definition ofCnandQn, it is obvious thatCnis closed andQnis closed and convex for eachn≥ 0. We show thatCnis convex for anyn≥0. Since
φ z, yn
≤φ z, xn
3.2
is equivalent to
2
z, Jxn−Jyn
≤xn2−yn2, 3.3 it follows thatCnis convex.
Next, we show thatFT⊂Cnfor alln≥0. Indeed, we have for allp∈FTthat φ
p, yn
φ p, j−1
αnjxn 1−αn
jtxn
≤ p2−2
p, αnjxn 1−αn
jtxn
αnxn2
1−αntxn2 αnφ
p, xn
1−αn
φ p, txn
≤αnφ p, xn
1−αn
φ p, xn φ
p, xn
.
3.4
That is,p∈Cnfor alln≥0.
Next, we show thatFT ⊂Qnfor alln≥ 0, we prove this by induction. Forn0,we haveFT⊂CQ0.Assume thatFT⊂Qn.Sincexn1is the projection ofx0ontoCn∩Qn, by Lemma 2.3, we have
xn1−z, Jx0−Jxn1
≥0, ∀z∈Cn∩Qn. 3.5 AsFT⊂Cn∩Qnby the induction assumptions, the last inequality holds, in particular, for all z∈FT. This together with the definition ofQn1implies thatFT⊂Qn1.
Sincexn1 ΠCn∩Qnx0andCn∩Qn⊂Cn−1∩Qn−1for alln≥1, we have φ
xn, x0
≤φ xn1, x0
3.6 for alln≥0. Therefore,{φxn, x0}is nondecreasing. In addition, it follows from the definition ofQnandLemma 2.3thatxn ΠQnx0. Therefore, byLemma 2.4, we have
φ xn, x0
φ Π
Qnx0, x0
≤φ p, x0
−φ p, xn
≤φ p, x0
, 3.7
for eachp∈FT⊂Qnfor alln≥ 0.Therefore,φxn, x0is bounded, this together with3.6 implies that the limit of{φxn, x0}exists. Put
limn→∞φ xn, x0
d. 3.8
FromLemma 2.4, we have, for any positive integerm, that φ
xnm, xn
φ xnm,Π
Cn
x0
≤φ
xnm, x0
−φ Π
Cn
x0, x0
φ
xnm, x0
−φ xn, x0
, 3.9
for alln≥0.Therefore,
n→∞limφ
xnm, xn
0. 3.10 We claim that{xn}is a Cauchy sequence. If not, there exists a positive real numberε0>0 and subsequence{nk},{mk} ⊂ {n}such that
xnkmk−xnk≥ε0, 3.11 for allk≥1.
On the other hand, from3.8and3.9we have φ
xnkmk, xnk
≤φ
xnkmk, x0
−φ xnk, x0
≤φ
xnkmk, x0
−d|d−φ xnk, x0
| −→0, k−→ ∞. 3.12
Because from3.8we know thatφxn, x0is bounded, this and2.4imply that{xn}is also bounded, so byLemma 2.2we obtain
k→∞limxnkmk−xnk0. 3.13
This is a contradiction, so that{xn}is a Cauchy sequence, therefore there exists a pointp∈C such that{xn}converges strongly top.
Sincexn1 ΠCn∩Qnx0∈Cn, from the definition ofCn, we have φ
xn1, yn
≤φ
xn1, xn
. 3.14
It follows from3.10,3.14that
φ xn1, yn
−→0. 3.15
By usingLemma 2.2, we have
limn→∞xn1−ynlim
n→∞xn1−xn0. 3.16
SinceJis uniformly norm-to-norm continuous on bounded sets, we have limn→∞Jxn1−Jynlim
n→∞Jxn1−Jxn0. 3.17
Noticing that
Jxn1−JynJxn1−
αnJxn 1−αn
JTxn αn
Jxn1−Jxn
1−αn
Jxn1−JTxn 1−αn
Jxn1−Jtxn
−αn
Jxn−Jxn1
≥
1−αnJxn1−Jtxn−αnJxn−Jxn1,
3.18
which implies that
Jxn1−JTxn≤ 1 1−αn
Jxn1−JynαnJxn−Jxn1. 3.19
This together with3.17and lim supn→∞αn<1 implies that
limn→∞Jxn1−JTxn0. 3.20
SinceJ−1is also uniformly norm-to-norm continuous on any bounded sets, we have
limn→∞xn1−Txn0. 3.21
Observe that
xn−Txn≤xn−xn1xn1−Txn. 3.22
It follows from3.16and3.21that
n→∞limxn−Txn0. 3.23
SinceTis a closed operator andxn→p, thenpis a fixed point ofT.
Finally, we prove thatp ΠFTx0. FromLemma 2.4, we have φ p, Π
FTx0
φ Π
FTx0, x0
≤φ p, x0
. 3.24
On the other hand, sincexn1 ΠCn∩Qn andCn∩Qn ⊃FT, for alln, we get fromLemma 2.4 that
φ Π
FTx0, xn1
φ xn1, x0
≤φ Π
FTx0, x0
. 3.25
By the definition of φx, y, it follows that both φp, x0 ≤ φΠFTx0, x0 and φp, x0 ≥ φΠFTx0, x0, whenceφp, x0 φΠFTx0, x0. Therefore, it follows from the uniqueness of ΠFTx0thatp ΠFTx0. This completes the proof.
Theorem 3.2. Let E be a uniformly convex and uniformly smooth real Banach space, let C be a nonempty, closed, and convex subset of E, and let T : C → Cbe a closed relative nonexpansive mapping such thatFT/∅. Assume that{αn}is a sequences in0,1such that lim supn→∞αn<1.
Define a sequence{xn}inCby the following algorithm:
x0∈C chosen arbitrarily, ynJ−1
αnJxn 1−αn
JTxn
, Cn
z∈Cn−1∩Qn−1:φ z, yn
≤φ z, xn
, C0
z∈C:φ z, y0
≤φ z, x0
, Qn
z∈Cn−1∩Qn−1:
xn−z, Jx0−Jxn
≥0 , Q0C,
xn1 Π
Cn∩Qn
x0 ,
3.26
whereJ is the duality mapping onE. Then{xn} converges strongly toΠFTx0, whereΠFT is the generalized projection fromContoFT.
Proof. Since every relatively nonexpansive mapping is a hemi-relatively one, Theorem 3.2is implied byTheorem 3.1.
Remark 3.3. In recent years, the hybrid iteration methods for approximating fixed points of nonlinear mappings have been introduced and studied by various authors1,8–11. In fact, all hybrid iteration methods can be replacedor modifiedby monotone hybrid iteration methods, respectively. In addition, by using the monotone hybrid method we can easily show that the iteration sequence {xn} is a Cauchy sequence, without the use of the Kadec-Klee property, demiclosedness principle, and Opial’s condition or other methods which make use of the weak topology.
Acknowledgments
The authors would like to thank the referee for valuable suggestions which helped to improve this manuscript. This project is supported by the National Natural Science Foundation of China under Grant no. 10771050.
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