Volume 2008, Article ID 384629,10pages doi:10.1155/2008/384629
Research Article
Iterative Algorithms for Nonexpansive Mappings
Yeong-Cheng Liou1and Yonghong Yao2
1Department of Information Management, Cheng Shiu University, Kaohsiung 833, Taiwan
2Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China
Correspondence should be addressed to Yeong-Cheng Liou,simplex [email protected]
Received 13 September 2007; Accepted 25 November 2007
Recommended by Massimo Furi
We suggest and analyze two new iterative algorithms for a nonexpansive mappingT in Banach spaces. We prove that the proposed iterative algorithms converge strongly to some fixed point ofT.
Copyrightq2008 Y.-C. Liou and Y. Yao. 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
Let E be a real Banach space, C a nonempty closed convex subset of E, and T : C → C a nonexpansive mapping; namely,
Tx−Ty ≤ x−y, 1.1
for allx, y ∈C. We useFTto denote the set of fixed points ofT, that is,FT {x∈C:Tx
x}. Throughout the paper, we assume thatFT/∅.
Construction of fixed points of nonlinear mappings is an important and active research area. In particular, iterative methods for finding fixed points of nonexpansive mappings have received a vast investigation; see1–28.
It is well known that the Picard iterationxn1 Txn · · · Tn1xof the mappingT at
a pointx ∈ Cmay, in general, not behave well. This means that it may not converge even in the weak topology. One way to overcome this difficulty is to use Mann’s iteration method that produces a sequence{xn}via the recursive manner:
xn1αnxn
1−αnTxn, n≥0, 1.2
that∞n1αn1−αn ∞, then the sequence{xn}defined by1.2converges weakly to a fixed
point ofT. However, this scheme has only a weak convergence even in a Hilbert space.
Some attempts to construct iteration method so that strong convergence is guaranteed have recently been made. For a sequence{αn}of real numbers in0,1and an arbitraryu∈C,
let the sequence{xn}inCbe iteratively defined byx0∈C,
xn1 αnu1−αnTxn. 1.3
The iterative method1.3is now referred to as the Halpern iterative method. The interest and importance of Halpern iterative method lie in the fact that strong convergence of the sequence
{xn}is achieved under certain mild conditions on parameter{αn}in a general Banach space.
Recently, Su and Li 21 introduced the following two new iterative algorithms for a nonexpansive mappingT: for fixedu∈C, let the sequences{xn}and{yn}be generated by
xn1αnu1−αnTβnu1−βnxn, 1.4
yn1αnβnu1−βnTyn1−αnn11 n
i0
Tiyn, 1.5
respectively. Su and Li proved that the sequences {xn} and{yn}converge strongly to some
fixed point of T under some assumptions. Subsequently, Liu and Li 22 extended and im-proved the corresponding results of Su and Li 21. But we note that all of the above results have imposed some additional assumptions on parameters. Please see21,22for more details. Motivated and inspired by the above works, in this paper we construct two new itera-tive algorithms for approximating fixed points of a nonexpansive mappingT. We prove that the proposed iterative algorithms converge strongly to a fixed point of T under some mild conditions.
2. Preliminaries
LetEbe a real Banach space with its dualE∗. LetS{x∈E:x1}denote the unit sphere ofE. The norm onEis said to be Gˆateaux differentiable if the limit
lim
t→0
xty − x
t 2.1
exists for eachx, y ∈ Sand in this caseEis said to be smooth.Eis said to have a uniformly Frechet differentiable norm if the limit2.1is attained uniformly forx, y ∈Sand in this caseE is said to be uniformly smooth. It is well known that ifEis uniformly smooth, then the duality map is norm-to-norm uniformly continuous on bounded subsets ofE.
LetC ⊂ Ebe closed convex andP a mapping ofEontoC. Then,P is said to be sunny if PPxtx−Px Px, for allx ∈ Eand t ≥ 0. A mapping P ofE intoE is said to be a retraction ifP2P. If a mappingP is a retraction, thenPzzfor everyz∈RP, whereRP is the range ofP. A subsetCofEis said to be a sunny nonexpansive retract ofEif there exists a sunny nonexpansive retraction ofEontoCand it is said to be a nonexpansive retract ofEif there exists a nonexpansive retraction ofEontoC. IfEH, the metric projectionPCis a sunny
We need the following lemmas for proving our main results.
Lemma 2.1see16. LetEbe a real Banach space andJthe normalized duality map onE. Then, for any givenx, y∈E, the following inequality holds:
xy2≤ x22y, jxy, ∀jxy∈Jxy. 2.2
Lemma 2.2see20. Let {xn}and{zn}be bounded sequences in a Banach space E and let{αn}
be a sequence in0,1which satisfies the condition0<lim infn→∞αn≤lim supn→∞αn <1. Suppose
xn1 αnxn1−αnzn, n≥0, andlim supn→∞zn1−zn− xn1−xn≤0. Then,limn→∞zn−
xn0.
Lemma 2.3 see 17. Assume {an} is a sequence of nonnegative real numbers such that an1 ≤
1−bnancn, n≥0, where{bn}is a sequence in0,1and{cn}is a sequence inRsuch that
i∞n0bn∞,
iilim supn→∞cn/bn≤0or∞n0|cn|<∞.
Then,limn→∞an0.
Lemma 2.4see17. LetCbe a nonempty closed convex subset of a uniformly smooth Banach space
E. LetT : C → Cbe a nonexpansive mapping with FT/∅. Then, there exists a continuous path
t → zt, 0 < t < 1, satisfying zt tu 1−tTzt, for arbitrary but fixedu ∈ C, which converges
strongly to a fixed point ofT. Further, ifPulimt→0zt, for eachu∈C, thenPis a sunny nonexpansive retraction ofContoFT.
Lemma 2.5see17. LetCbe a nonempty closed convex subset of a uniformly smooth Banach space
E. LetT : C → Cbe a nonexpansive mapping withFT/∅. Let{xn} ⊂ Cbe a sequence. Suppose
that{xn}is bounded andlimn→∞xn−Txn0. Then,u−Pu, jxn−Pu ≤0.
3. Main results
LetCbe a nonempty closed convex subset of a real uniformly smooth Banach space E. First, for fixed two different anchorsu, v∈Cwe definext tu 1−tTxtandyttv 1−tTyt. It
follows fromLemma 2.4thatxtandytconverge strongly toPuandPv, respectively. Now we
assume that anchorsuandvsatisfy conditionP:PuPv,whereP is a sunny nonexpansive retraction fromContoFT.
Now we introduce the following iterative algorithm: for fixedu, v∈Cand givenx0∈C arbitrarily, find the approximate solution{xn}by
xn1αnuβnxnγnT
δnv1−δnxn, 3.1
where{αn},{βn},{γn}, and{δn}are real sequences in0,1.
Remark 3.1. Our iterative algorithm3.1can be reviewed as an extension of the iterative algo-rithm1.4.
Theorem 3.2. LetCbe a nonempty closed convex subset of a real uniformly smooth Banach spaceE. LetT : C → Cbe a nonexpansive mapping withFT/∅. Suppose the sequences{αn}, {βn},{γn},
and{δn}in0,1satisfyαnβnγn1, n≥0. Suppose the following conditions are satisfied: ilimn→∞αn0and∞n0αn∞;
ii0<lim infn→∞βn≤lim supn→∞βn<1;
iiilimn→∞δn 0andδn/αn≤Mfor some constantM≥0.
For fixed anchors u and v satisfying condition (P) and arbitrary given x0 ∈ C, the sequence {xn}
defined by3.1converges strongly toPu∈FT.
Proof. Settingynδnv 1−δnxnandσnαnγnδn. Forp∈FT, we have
xn1−p≤αnu−pβnxn−pγn δnv−p
1−δnxn−p αnu−pγnδnv−p βnγn1−δnxn−p ≤σnmaxv−p,u−p1−σnxn−p
≤maxu−p,v−p,x0−p,
3.2
which implies that{xn}is bounded.
From3.1, we writexn1βnxn 1−βnzn. It is easily seen that
zn1−zn αn1uγn1Tyn1
1−βn1 −
αnuγnTyn
1−βn
αn1 1−βn1 −
αn
1−βn
u γn1
1−βn1
Tyn1−Tyn
γn1 1−βn1 −
γn
1−βn
Tyn. 3.3
It follows that
zn1−zn≤
1−αnβn11−
αn
1−βn
uTyn1−γnβ1 n1
Tyn1−Tyn
≤ αn1 1−βn1−
αn
1−βn
uTyn1−γnβ1 n1
yn1−yn. 3.4
At the same time, we note that
yn1−yn≤δn1−δnvxn
1−δn1xn1−xn. 3.5
It follows from3.4and3.5that
zn1−zn≤
1α−nβn11 −
αn
1−βn
uTyn γn1
1−βn1
×δn1−δnvxn1−γnβ1 n1
Then, we have
zn1−zn≤ αn1 1−βn1 −
αn
1−βn |
uTyn1−γnβ1 n1
δn1−δn
×vxn αn1
1−βn1
xn1−xnxn1−xn, 3.7
which implies that
lim sup
n→∞
zn1−zn−xn1−xn≤0. 3.8
FromLemma 2.2and3.8, we obtain limn→∞zn−xn0. Consequently, limn→∞xn1−xn limn→∞1−βnzn−xn0 .
On the other hand, we observe that
xn−Txn≤xn1−xnxn1−Txn
≤xn1−xnαnu−Txnβnxn−TxnγnTyn−Txn ≤xn1−xnαnu−Txnβnxn−Txnγnδnv−xn,
3.9
that is,
xn−Txn≤ 1
1−βn
xn1−xnαnu−Txnγnδnv−xn−→0. 3.10
ByLemma 2.5and3.10, we can obtain
lim sup
n→∞
u−Pu, jxn−Pu≤0, lim sup n→∞
v−Pv, jxn−Pv≤0. 3.11
Sinceyn−xnδnv−xn →0 andPuPv, we have
lim sup
n→∞
u−Pu, jxn−Pu≤0, lim sup n→∞
v−Pu, jyn−Pu≤0. 3.12
Finally, we show thatxn→Pu. As a matter of fact, we have
xn1−Pu2≤β
nxn−PuγnTyn−Pu22αnu−Pu, jxn1−Pu
≤βnxn−Pu2γnyn−Pu22αnu−Pu, jxn1−Pu
≤βnxn−Pu2γn1−δnxn−Pu22δnv−Pu, jyn−Pu 2αnu−Pu, jxn1−Pu
≤1−αnxn−Pu22δnv−Pu, jyn−Pu2αnu−Pu, jxn1−Pu
1−αnxn−Pu2αnλn,
3.13
Corollary 3.3. LetCbe a nonempty closed convex subset of a real uniformly smooth Banach spaceE. Let T : C → C be a nonexpansive mapping withFT/∅. Suppose that the sequences {αn},{βn}, {γn}, and{δn}in0,1satisfyαnβnγn1, n≥0. Suppose the following conditions are satisfied:
ilimn→∞αn0and∞n0αn∞;
ii0<lim infn→∞βn≤lim supn→∞βn<1;
iiilimn→∞δn 0andδn/αn≤Mfor some constantM≥0.
For fixed anchoruand arbitraryx0∈C, define the sequence{xn}as
xn1αnuβnxnγnTδnu1−δnxn. 3.14
Then, the sequence{xn}converges strongly toPu∈FT.
Finally, we introduce another iterative algorithm: for fixed anchoru∈Cand givenx0 ∈
Carbitrarily, find the approximate solution{xn}by the iterative algorithm:
xn1βnxn1−βnAnyn,
ynαnu1−αnTxn, n≥0,
3.15
whereAn 1/n1ni0Ti.
Now we prove the strong convergence of the iterative algorithm3.15.
Theorem 3.4. LetCbe a nonempty closed convex subset of a real uniformly smooth Banach spaceE. LetT :C→Cbe a nonexpansive mapping withFT/∅. Let{αn}and{βn}be the sequences in0,1.
Suppose the following conditions are satisfied:
ilimn→∞αn0; ii∞n0αn∞;
iii0<lim infn→∞βn≤lim supn→∞βn<1.
For arbitraryx0∈C, the sequence{xn}defined by3.15converges strongly toPu∈FT,whereP is
a sunny nonexpansive retraction fromContoFT.
Proof. First, we note thatAn:C→Cis nonexpansive. In fact, for anyx, y ∈C, we have that
Anx−Any n11
n
i0
Tix− 1
n1
n
i0
Tiy
≤ n11
n
i0
x−yx−y. 3.16
At the same time, for anyx∈C,
An1x−Anx n12
n1
i0
Tix− 1
n1
n
i0
Tix
≤ n11n2
n
i0
Tix 1
n2T
n1x.
3.17
Now we show that{xn}is bounded. Indeed, forp∈FT, we have
yn−p ≤αnu−p
Hence, we have
xn1−p≤βnxn−p
1−βnAnyn−p≤βnxn−p1−βnyn−p ≤ 1−αn1−βnxn−p1−βnαnu−p ≤maxxn−p,u−p.
3.19
Now, an induction yields
xn−p≤maxx0−p,u−p
. 3.20
Therefore,{xn}is bounded, so is{Txn}.
Settingxn1βnxn 1−βnzn, whereznAnynfor alln≥0, then we have that zn1−zn≤An1yn1−Anyn1Anyn1−Anyn
≤An1yn1−Anyn1yn1−yn.
3.21
Observe that
yn1−ynαn1−αn
u1−αn1
Txn1−Txn
αn−αn1
Txn ≤αn1−αnuTxn
1−αn1xn1−xn.
3.22
So, we get
zn1−zn−xn1−xn≤An1yn1−Anyn1αn1−αnuTxn. 3.23
This together with3.17implies that
lim sup
n→∞
zn1−zn−xn1−xn≤0. 3.24
Consequently, fromLemma 2.2, we obtain
lim
n→∞zn−xn0, lim
n→∞xn1−xnnlim→∞
1−βnxn−zn0.
3.25
Next, we show that
lim
n→∞xn−Txn0. 3.26 We note from23that
zn−Tzn−→0. 3.27
Therefore, from3.25, and3.27, we have xn−Txn≤xn1−xnxn1−Txn
≤xn1−xnxn1−znzn−Txn
≤xn1−xnxn1−znzn−TznTzn−Txn ≤xn1−xnxn1−znzn−Tznzn−xn−→0.
From3.28andLemma 2.5, we have
lim sup
n→∞
u−Pu, jxn−Pu≤0. 3.29
Noting that
yn−xn≤αnu−xn
1−αnxn−Txn−→0, 3.30
hence,
lim sup
n→∞
u−Pu, jyn−Pu≤0. 3.31
Finally, applyingLemma 2.1to3.15, we obtain xn1−Pu2≤β
nxn−Pu21−βnzn−Pu2 ≤βnxn−Pu21−βnyn−Pu2
βnxn−Pu21−βnαnu−Pu 1−αnTxn−Pu2
≤βnxn−Pu21−βn 1−αnTxn−Pu22αnu−Pu, jyn−Pu
≤βnxn−Pu21−βn 1−αnxn−Pu22αnu−Pu, jyn−Pu
1−1−βnαnxn−Pu221−βnαnu−Pu, jyn−Pu.
3.32
ByLemma 2.3and3.32, we conclude thatxn→Pu. This completes the proof.
Remark 3.5. We drop the imposed assumptions in21:∞n1|αn−αn−1| < ∞and
∞
n1δn < ∞ on parameters{αn}and{δn}, respectively.
Acknowledgments
The authors are extremely grateful to the anonymous referees for their useful comments and suggestions. The first author was partially supported by Grant NSC 96-2221-E-230-003. The second author was partially supported by National Natural Science Foundation of China Grant 10771050.
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