Fixed Point Theory and Applications Volume 2007, Article ID 29091,10pages doi:10.1155/2007/29091
Research Article
Iteration Scheme with Perturbed Mapping for Common Fixed
Points of a Finite Family of Nonexpansive Mappings
Yeong-Cheng Liou, Yonghong Yao, and Rudong Chen
Received 17 December 2006; Revised 6 February 2007; Accepted 6 February 2007
Recommended by H´el`ene Frankowska
We propose an iteration scheme with perturbed mapping for approximation of common fixed points of a finite family of nonexpansive mappings{Ti}Ni=1. We show that the pro-posed iteration scheme converges to the common fixed pointx∗∈N
i=1Fix(Ti) which solves some variational inequality.
Copyright © 2007 Yeong-Cheng Liou et al. This is an open access article distributed un-der the Creative Commons Attribution License, which permits unrestricted use, distri-bution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction and preliminaries
LetH be a real Hilbert space with inner product·,·and norm · , respectively. A mappingTwith domainD(T) and rangeR(T) inHis called nonexpansive if
Tx−T y ≤ x−y, ∀x,y∈D(T). (1.1)
Let{Ti}Ni=1be a finite family of nonexpansive self-maps ofH. Denote the common fixed points set of{Ti}Ni=1 by
N
i=1Fix (Ti). LetF:H→H be a mapping such that for some constants k,η >0, F isk-Lipschitzian and η-strongly monotone. Let{αn}∞n=1⊂(0, 1), {λn}∞n=1⊂[0, 1) and take a fixed numberμ∈(0, 2η/k2). The iterative schemes concern-ing nonlinear operators have been studied extensively by many authors, you may refer to [1–12]. Especially, in [13], Zeng and Yao introduced the following implicit iteration process with perturbed mappingF.
For an arbitrary initial pointx0∈H, the sequence{xn}∞n=1is generated as follows:
xn=αnxn−1+
1−αn
Tnxn−λnμF
Tnxn
, n≥1, (1.2)
Using this iteration process, they proved the following weak and strong convergence theorems for nonexpansive mappings in Hilbert spaces.
Theorem1.1 (see [13]). LetH be a real Hilbert space and letF:H→H be a mapping such that for some constantsk,η >0,F isk-Lipschitzain vcommentandη-strongly mono-tone. Let{Ti}Ni=1 beN nonexpansive self-mappings ofH such that
N
i=1Fix (Ti)= ∅. Let
μ∈(0, 2η/k2)andx
0∈H. Let{λn}∞n=1⊂[0, 1)and{αn}∞n=1⊂(0, 1)satisfying the
condi-tions∞n=1λn<∞andα≤αn≤β,n≥1, for someα,β∈(0, 1). Then the sequence{xn}∞n=1
defined by (1.2) converges weakly to a common fixed point of the mappings{Ti}Ni=1.
Theorem1.2 (see [13]). LetHbe a real Hilbert space and letF:H→Hbe a mapping such that for some constantsk,η >0,Fisk-Lipschitzain andη-strongly monotone. Let{Ti}Ni=1be N nonexpansive self-mappings ofH such that Ni=1Fix (Ti)= ∅. Letμ∈(0, 2η/k2)and
x0∈H. Let{λn}∞n=1⊂[0, 1) and{αn}∞n=1⊂(0, 1) satisfying the conditions
∞
n=1λn<∞
andα≤αn≤β,n≥1, for someα,β∈(0, 1). Then the sequence{xn}∞n=1 defined by (1.2)
converges strongly to a common fixed point of the mappings{Ti}Ni=1if and only if
lim inf n→∞ d
xn, N
i=1 FixTi
=0. (1.3)
Very recently, Wang [14] considered an explicit iterative scheme with perturbed map-pingFand obtained the following result.
Theorem1.3. LetH be a Hilbert space, letT:H→H be a nonexpansive mapping with
F(T)=∅,and letF:H→H be anη-strongly monotone andk-Lipschitzian mapping. For any givenx0∈H,{xn}is defined by
xn+1=αnxn+1−αnTλn+1xn, n≥0, (1.4)
where Tλn+1x
n=Txn−λn+1μF(Txn),{αn} and{λn} ⊂[0, 1) satisfy the following
condi-tions:
(1)α≤αn≤βfor someα,β∈(0, 1); (2)∞n=1λn<∞;
(3) 0< μ <2η/k2.
Then
(1){xn}converges weakly to a fixed point ofT,
(2){xn}converges strongly to a fixed point ofTif and only if
lim inf n→∞ d
xn,F(T)
=0. (1.5)
This naturally brings us the following questions.
Questions 1.4. LetTi:H→H(i=1, 2,...,N) be a finite family of nonexpansive mappings andFisk-Lipschitzain andη-strongly monotone.
(i) Could we construct an explicit iterative algorithm to approximate the common fixed points of the mappings{Ti}Ni=1?
Motivated and inspired by the above research work of Zeng and Yao [13] and Wang [14], in this paper, we will propose a new explicit iteration scheme with perturbed map-ping for approximation of common fixed points of a finite family of nonexpansive self-mappings ofH. We will establish strong convergence theorem for this explicit iteration scheme. To be more specific, letαn1,αn2,...,αnN ∈(0, 1], n∈N. Given the mappings
T1,T2,...,TN, following [15], one can define, for eachn, mappingsUn1,Un2,...,UnN by
Un1=αn1T1+
1−αn1
I,
Un2=αn2T2Un1+
1−αn2
I, ..
.
Un,N−1=αn,N−1TN−1Un,N−2+
1−αn,N−1
I,
Wn:=UnN=αnNTNUn,N−1+
1−αnN
I.
(1.6)
Such a mappingWnis called theW-mapping generated byT1,...,TNandαn1,...,αnN. First we introduce the following explicit iteration scheme with perturbed mappingF. For an arbitrary initial pointx0∈H, the sequence{xn}∞n=1is generated iteratively by
xn+1=βxn+ (1−β)Wnxn−λnμFWnxn, n≥0, (1.7)
where{λn} is a sequence in (0, 1),βis a constant in (0, 1),F isk-Lipschitzian andη -strongly monotone, andWnis theW-mapping defined by (1.6).
We have the following crucial conclusion concerningWn.
Proposition1.5 (see [15]). LetCbe a nonempty closed convex subset of a Banach space
E. LetT1,T2,...,TN be nonexpansive mappings of C into itself such that
N
i=1Fix (Ti)is
nonempty, and letαn1,αn2,...,αnN be real numbers such that0< αni≤b <1for anyi∈N.
For anyn∈N, letWn be theW-mapping of Cinto itself generated by TN,TN−1,...,T1
and αnN,αn,N−1,...,αn1. Then Wn is nonexpansive. Further, if E is strictly convex, then Fix (Wn)=Ni=1Fix (Ti).
Now we recall some basic notations. Let T:H→H be nonexpansive mapping and
F:H→H be a mapping such that for some constantsk,η >0,F isk-Lipschitzian and
η-strongly monotone; that is,Fsatisfies the following conditions:
Fx−F y ≤kx−y, ∀x,y∈H,
Fx−F y,x−y ≥ηx−y2, ∀x,y∈H, (1.8)
respectively. We may assume, without loss of generality, thatη∈(0, 1) and k∈[1,∞). Under these conditions, it is well known that the variational inequality problem—find
x∗∈N
i=1Fix (Ti) such that
VI
F, N
i=1 FixTi
:Fx∗,x−x∗≥0, ∀x∈ N i=1
FixTi
has a unique solution x∗∈N
i=1Fix (Ti). [Note: the unique existence of the solution
x∗∈N
i=1Fix (Ti) is guaranteed automatically becauseFisk-Lipschitzian andη-strongly monotone overNi=1Fix (Ti).]
For any given numbersλ∈[0, 1) andμ∈(0, 2η/k2), we define the mappingTλ:H→
Hby
Tλx:=Tx−λμF(Tx), ∀x∈H. (1.10)
Concerning the corresponding result ofTλx, you can find it in [16].
Lemma1.6 (see [16]). If0≤λ <1and0< μ <2η/k2, then there holds forTλ:H→H,
Tλx−Tλy≤(1−λτ)x−y, ∀x,y∈H, (1.11)
whereτ=1−1−μ(2η−μk2)∈(0, 1).
Next, let us state four preliminary results which will be needed in the sequel.Lemma 1.7is very interesting and important, you may find it in [17], the original prove can be found in [18]. Lemmas1.8and1.9well-known demiclosedness principle and subdiff er-ential inequality, respectively.Lemma 1.10is basic and important result, please consult it in [19].
Lemma1.7 (see [17]). Let{xn}and{yn}be bounded sequences in a Banach spaceXand
let{βn}be a sequence in[0, 1]with
0<lim inf
n→∞ βn≤lim supn→∞ βn<1. (1.12)
Suppose
xn+1=
1−βn
yn+βnxn, (1.13)
for all integersn≥0and
lim sup n→∞
yn+1−yn−xn+1−xn≤0. (1.14)
Then,limn→∞yn−xn =0.
Lemma1.8 (see [20]). Assume thatT is a nonexpansive self-mapping of a closed convex subsetCof a Hilbert spaceH. IfThas a fixed point, thenI−Tis demiclosed. That is, when-ever{xn}is a sequence inCweakly converging to somex∈Cand the sequence{(I−T)xn}
strongly converges to some y, it follows that(I−T)x=y. Here,I is the identity operator ofH.
Lemma1.9 (see [21]). x+y2≤ x2+ 2y,x+yfor allx,y∈H.
Lemma1.10 (see [19]). Assume that{an}is a sequence of nonnegative real numbers such
that
an+1≤
1−γn
where{γn}is a sequence in(0, 1)and{δn}is a sequence such that (1)∞n=1γn= ∞,
(2) lim supn→∞δn/γn≤0or
∞
n=1|δn|<∞.
Thenlimn→∞an=0.
2. Main result
Now we state and prove our main result.
Theorem2.1. Let H be a real Hilbert space and letF:H→H be a k-Lipschitzian and
η-strongly monotone mapping. Let{Ti}Ni=1be a finite family of nonexpansive self-mappings
ofH such thatNi=1Fix (Ti)= ∅. Letμ∈(0, 2η/k2). Suppose the sequences{αn,i}Ni=1
sat-isfylimn→∞(αn,i−αn−1,i)=0, for alli=1, 2,...,N. If{λn}∞n=1⊂[0, 1)satisfy the following
conditions:
(i) limn→∞λn=0; (ii)∞n=0λn= ∞,
then the sequence{xn}∞n=1defined by (1.7) converges strongly to a common fixed pointx∗∈
N
i=1Fix (Ti)which solves the variational inequality (1.9).
Proof. Letx∗be an arbitrary element ofNi=1Fix (Ti). Observe that
x
n+1−x∗=βxn+ (1−β)Wnλnxn−x∗
≤βxn−x∗+ (1−β)Wnλnxn−x∗,
(2.1)
whereWλn
n x:=Wnx−λnμF(Wnx). Note that
Wλn
n x∗=x∗−λnμFx∗. (2.2)
UtilizingLemma 1.6, we have
Wλn
n xn−x∗=Wnλnxn−Wnλnx∗+Wnλnx∗−x∗ ≤Wλn
n xn−Wnλnx∗+Wnλnx∗−x∗ ≤1−λnτxn−x∗+λnμF
x∗.
(2.3)
From (2.1) and (2.3), we have
xn+1−x∗≤β+ (1−β)1−λ
nτxn−x∗+ (1−β)λnμF
x∗
=1−(1−β)λnτxn−x∗+ (1−β)λnμF
x∗
≤maxx0−x∗,
μ
τ
Fx∗.
(2.4)
Hence,{xn}is bounded. We also can obtain that{Wnxn},{TiUn,jxn}(i=1,...,N; j= 1,...,N), and{F(Wnxn)}are all bounded.
We note that
Wλn+1
n+1xn+1−Wnλnxn
=Wn+1xn+1−Wnxn−λn+1μF
Wn+1xn+1
+λnμF
Wnxn
≤Wn+1xn+1−Wnxn+λn+1μF
Wn+1xn+1+λnμF
Wnxn
≤Wn+1xn+1−Wn+1xn+Wn+1xn−Wnxn+
λn+1+λn
M
≤xn+1−xn+Wn+1xn−Wnxn+
λn+1+λn
M.
(2.5)
From (1.6), sinceTNandUn,Nare nonexpansive,
Wn+1xn−Wnxn
=αn+1,NTNUn+1,N−1xn+
1−αn+1,N
xn−αn,NTNUn,N−1xn−
1−αn,N
xn
≤αn+1,NTNUn+1,N−1xn−αn,NTNUn,N−1xn+αn+1,N−αn,Nxn
≤αn+1,N
TNUn+1,N−1xn−TNUn,N−1xn+αn+1,N−αn,NTNUn,N−1xn
+αn+1,N−αn,Nxn
≤αn+1,NUn+1,N−1xn−Un,N−1xn+ 2Mαn+1,N−αn,N.
(2.6)
Again, from (1.6), we have
Un+1,N−1xn−Un,N−1xn
=αn+1,N−1TN−1Un+1,N−2xn+
1−αn+1,N−1
xn
−αn,N−1TN−1Un,N−2xn−1−αn,N−1
xn
≤αn+1,N−1TN−1Un+1,N−2xn−αn,N−1TN−1Un,N−2xn
+αn+1,N−1−αn,N−1xn
≤αn+1,N−1−αn,N−1xn+αn+1,N−1−αn,N−1M
+αn+1,N−1TN−1Un+1,N−2xn−TN−1Un,N−2xn
≤2Mαn+1,N−1−αn,N−1+αn+1,N−1Un+1,N−2xn−Un,N−2xn
≤2Mαn+1,N−1−αn,N−1+Un+1,N−2xn−Un,N−2xn.
Therefore, we have
Un+1,N−1xn−Un,N−1xn
≤2Mαn+1,N−1−αn,N−1+ 2Mαn+1,N−2−αn,N−2 +Un+1,N−3xn−Un,N−3xn
≤2M
N−1
i=2
αn+1,i−αn,i+Un+1,1xn−Un,1xn
=αn+1,1T1xn+
1−αn+1,1
xn−αn,1T1xn−
1−αn,1
xn
+ 2M
N−1
i=2
αn+1,i−αn,i,
(2.8)
then
Un+1,N−1xn−Un,N−1xn
≤αn+1,1−αn,1xn+αn+1,1T1xn−αn,1T1xn
+ 2M
N−1
i=2
αn+1,i−αn,i≤2MN−1 i=1
αn+1,i−αn,i. (2.9)
Substituting (2.9) into (2.6), we have
Wn+1xn−Wnxn≤2Mαn+1,N−αn,N+ 2αn+1,NMN−1 i=1
αn+1,i−αn,i
≤2M
N
i=1
α
n+1,i−αn,i.
(2.10)
Substituting (2.10) into (2.5), we have
Wλn+1
n+1xn+1−Wnλnxn≤xn+1−xn+ 2M N
i=1
αn+1,i−αn,i+
λn+1+λn
M, (2.11)
which implies that
lim sup n→∞
Wλn+1
n+1xn+1−Wnλnxn−xn+1−xn≤0. (2.12)
We note thatxn+1=βxn+ (1−β)Wnλnxnand 0< β <1, then fromLemma 1.7and (2.12), we have limn→∞Wnλnxn−xn =0. It follows that
lim
n→∞xn+1−xn=nlim→∞(1−β)Wλ
n
n xn−xn=0. (2.13)
On the other hand,
x
n−Wnxn≤xn+1−xn+xn+1−Wnxn
≤xn+1−xn+βxn−Wnxn+ (1−β)λnμF
Wnxn,
that is,
xn−Wnxn≤ 1
1−βxn+1−xn+λnμF
Wnxn, (2.15)
this together with (i) and (2.13) imply
lim
n→∞xn−Wnxn=0. (2.16)
We next show that
lim sup n→∞
−Fx∗,xn−x∗
≤0. (2.17)
To prove this, we pick a subsequence{xni}of{xn}such that
lim sup n→∞
−Fx∗,xn−x∗=lim i→∞
−Fx∗,xni−x∗
. (2.18)
Without loss of generality, we may further assume thatxni→zweakly for somez∈H.
ByLemma 1.8and (2.16), we have
z∈FixWn
, (2.19)
this together withProposition 1.5imply that
z∈ N i=1
FixTi. (2.20)
Sincex∗solves the variational inequality (1.9), then we obtain
lim sup n→∞
−Fx∗,xn−x∗
=−Fx∗,z−x∗≤0. (2.21)
Finally, we show thatxn→x∗. Indeed, fromLemma 1.9, we have
x
n+1−x∗ 2
=βxn−x∗+ (1−β)Wnλnxn−Wnλnx∗
+ (1−β)Wλn
n x∗−x∗ 2
≤βxn−x∗
+ (1−β)Wλn
n xn−Wnλnx∗ 2
+ 2(1−β)Wλn
n x∗−x∗,xn+1−x∗
≤βxn−x∗+ (1−β)Wnλnxn−Wnλnx∗ 2
+ 2(1−β)λnμ
−Fx∗,xn+1−x∗
≤βxn−x∗+ (1−β)
1−λnτxn−x∗ 2
+ 2(1−β)λnμ
−Fx∗,xn+1−x∗
≤1−(1−β)τλnxn−x∗ 2
+ (1−β)τλn
2μ
τ
−Fx∗,xn+1−x∗
.
(2.22)
Now applyingLemma 1.10and (2.21) to (2.22) concludes thatxn→x∗(n→ ∞). This
Acknowledgments
The authors thank the referees for their suggestions and comments which led to the present version. The research was partially supposed by Grant NSC 95-2221-E-230-017.
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Yeong-Cheng Liou: Department of Information Management, Cheng Shiu University, Kaohsiung 833, Taiwan
Email address:simplex [email protected]
Yonghong Yao: Department of Mathematics, Tianjin Polytechnic University, Tianji 300160, China Email address:[email protected]
