Volume 2007, Article ID 28619,8pages doi:10.1155/2007/28619
Research Article
An Iteration Method for Nonexpansive Mappings
in Hilbert Spaces
Lin Wang
Received 22 August 2006; Revised 2 November 2006; Accepted 2 November 2006
Recommended by Nan-Jing Huang
In real Hilbert spaceH, from an arbitrary initial pointx0∈H, an explicit iteration scheme is defined as follows: xn+1 =αnxn+ (1−αn)Tλn+1xn,n≥0, where Tλn+1xn=Txn− λn+1μF(Txn),T:H→His a nonexpansive mapping such thatF(T)= {x∈K:Tx=x} is nonempty,F:H→H is aη-strongly monotone andk-Lipschitzian mapping,{αn} ⊂ (0, 1), and{λn} ⊂[0, 1). Under some suitable conditions, the sequence{xn}is shown to converge strongly to a fixed point ofT and the necessary and sufficient conditions that {xn}converges strongly to a fixed point ofTare obtained.
Copyright © 2007 Lin Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and repro-duction in any medium, provided the original work is properly cited.
1. Introduction
LetHbe a Hilbert space with inner product·,·and norm · . A mappingT:H→His said to be nonexpansive ifTx−T y ≤ x−yfor anyx,y∈H. A mappingF:H→His said to beη-strongly monotone if there exists constantη >0 such thatFx−F y,x−y ≥
ηx−y2for anyx,y∈H.F:H→His said to bek-Lipschitzian if there exists constant k >0 such thatFx−F y ≤kx−yfor anyx,y∈H.
The interest and importance of construction of fixed points of nonexpansive map-pings stem mainly from the fact that it may be applied in many areas, such as imagine recovery and signal processing (see, e.g., [1–3]). Iterative techniques for approximat-ing fixed points of nonexpansive mappapproximat-ings have been studied by various authors (see, e.g., [1,4–10], etc.), using famous Mann iteration method, Ishikawa iteration method, and many other iteration methods such as, viscosity approximation method [6] and CQ method [7].
Let F:H→H be a nonlinear mapping andK nonempty closed convex subset of
that
VI(F,K)Fu∗,v−u∗≥0, ∀v∈K. (1.1)
The variational inequalities were initially studied by Kinderlehrer and Stampacchia [11], and ever since have been widely studied. It is well known that the VI(F,K) is equivalent to the fixed point equation
u∗=PK
u∗−μFu∗, (1.2)
where PK is the projection from H onto K and μ is an arbitrarily fixed constant. In fact, whenFis anη-strongly monotone and Lipschitzian mapping onKandμ >0 small enough, then the mapping defined by the right-hand side of (1.2) is a contraction.
For reducing the complexity of computation caused by the projectionPK, Yamada [12] proposed an iteration method to solve the variational inequalities VI(F,K). For arbitrary
u0∈H,
un+1=Tun−λn+1μF
Tun
, n≥0, (1.3)
whereT is a nonexpansive mapping fromH into itself,K is the fixed point set ofT,F
is anη-strongly monotone andk-Lipschitzian mapping onK,{λn}is a real sequence in [0, 1), and 0< μ <2η/k2. Then Yamada [12] proved that{u
n}converges strongly to the unique solution of the VI(F,K) as{λn}satisfies the following conditions:
(1) limn→∞λn=0, (2)∞n=0λn= ∞,
(3) limn→∞(λn−λn+1)/λ2n+1=0.
Motivated by the above work, we propose a new explicit iteration scheme with map-pingFto approximate the fixed point of nonexpansive mappingTin Hilbert space. The strong and weak convergence theorems to a fixed point ofTare obtained. The necessary and sufficient conditions for strong convergence of this iteration scheme are obtained, too.
2. Preliminaries
LetT be a nonexpansive mapping fromH into itself,F:H→H anη-strongly mono-tone andk-Lipschitzian mapping,{λn} ⊂(0, 1),{λn} ⊂[0, 1), andμa fixed constant in (0, 2η/k2). Starting with an initial pointx0∈H, the explicit iteration scheme with
map-pingFis defined as follows:
xn+1=αnxn+1−αnTxn−λn+1μFTxn, n≥0. (2.1)
For simplicity, we define a mappingTλ:H→Hby
Tλx=Tx−λμF(Tx), ∀x∈H. (2.2)
Then (2.1) may be written as follows:
xn+1=αnxn+
1−αn
Tλn+1x
In fact, asλn=0,n≥1, then the iteration scheme (2.3) reduces to the famous Mann iteration scheme.
A Banach spaceEis said to satisfy Opial’s condition if for any sequence {xn}inE, xnximplies that lim supn→∞xn−x<lim supn→∞xn−yfor ally∈Ewithy=x, wherexnxdenotes that{xn}converges weakly tox. It is well known that every Hilbert space satisfies Opial’s condition.
A mappingT:K→Eis said to be semicompact if, for any sequence{xn}inKsuch that
xn−Txn →0 (n→ ∞), there exists subsequence{xnj}of{xn}such that{xnj}converges
strongly tox∗∈K.
A mappingTwith domainD(T) and rangeR(T) inEis said to be demiclosed atp; if whenever{xn}is a sequence inD(T) such that{xn}converges weakly tox∗∈D(T) and
{Txn}converges strongly top, thenTx∗=p.
Lemma2.1 [13]. Let{αn}and{tn}be two nonnegative sequences satisfying
αn+1≤
1 +an
αn+bn, ∀n≥1. (2.4)
If∞n=1an<∞andn∞=1bn<∞, thenlimn→∞αnexists.
Lemma2.2 [12]. LetTλx=Tx−λμF(Tx), whereT:H→His a nonexpansive mapping
fromHinto itself andFis anη-strongly monotone andk-Lipschitzian mapping fromHinto itself. If0≤λ <1and0< μ <2η/k2, thenTλis a contraction and satisfies
Tλx−Tλy≤(1−λτ)x−y, ∀x,y∈H, (2.5)
whereτ=1−1−μ(2η−μk2).
Lemma2.3 [14]. LetKbe a nonempty closed convex subset of a real Hilbert spaceHandT
a nonexpansive mapping fromKinto itself. IfT has a fixed point, thenI−T is demiclosed at zero, whereI is the identity mapping ofH, that is, whenever{xn} is a sequence inK
weakly converging to somex∈K and the sequence{(I−T)xn}strongly converges to some y, it follows that(I−T)x=y.
3. Main results
Lemma3.1. LetHbe a Hilbert space,T:H→Ha nonexpansive mapping withF(T)=φ, andF:H→Hanη-strongly monotone andk-Lipschitzian mapping. For any givenx0∈H,
{xn}is defined by
xn+1=αnxn+
1−αn
Tλn+1x
n, n≥0, (3.1)
where{αn}and{λn} ⊂[0, 1)satisfy the following conditions: (1)α≤αn≤βfor someα,β∈(0, 1);
(2)∞n=1λn<∞; (3) 0< μ <2η/k2.
Then,
Proof. (1) For anyq∈F(T), we have x
n+1−q 2
=αn
xn−q
+1−αn
Tλn+1x
n−q 2
=αnxn−q2+1−αnTλn+1xn−q2−αn1−αnxn−Tλn+1xn2, (3.2)
where (byLemma 2.2) Tλn+1x
n−q=Tλn+1xn−Tλn+1q+Tλn+1q−q
≤Tλn+1x
n−Tλn+1q+Tλn+1q−q
≤1−λn+1τxn−q+λn+1μF(q).
(3.3)
Furthermore,
Tλn+1x
n−q 2
≤1−λn+1τxn−q 2
+λn+1μ
2
τ F(q) 2
. (3.4)
Thus,
xn+1−q2
≤αnxn−q 2
+1−αn
1−λn+1τxn−q 2
+1−αnλn+1μ 2
τ F(q) 2
−αn1−αnxn−Tλn+1xn2
≤αnxn−q2+
1−αn
1−λn+1τxn−q2
+1−αn
λn+1μ2 τ F(q)
2
−αnxn+1−xn 2
≤xn−q 2
+λn+1μ
2
τ F(q) 2
−αnxn+1−xn 2
.
(3.5)
Since∞n=1λn<∞, it follows fromLemma 2.1that limn→∞xn−qexists for eachq∈ F(T). It also implies that{xn}is bounded.
(2) From (3.5), we have
αxn+1−xn2≤αnxn+1−xn2≤xn−q2−xn+1−q2+λn+1μ 2
τ F(q) 2
. (3.6)
Therefore, limn→∞xn+1−xn =0. In addition,
(1−β)xn−Tλn+1xn≤
1−αnxn−Tλn+1xn=xn+1−xn. (3.7)
Hence, limn→∞xn−Tλn+1 =0. Thus,
xn−Txn=xn−Tλn+1x
n+Tλn+1xn−Txn
≤xn−Tλn+1xn+λn+1μFTxn.
(3.8)
Theorem3.2. LetHbe a Hilbert space,T:H→Ha nonexpansive mapping withF(T)=
φ, andF:H→Hanη-strongly monotone andk-Lipschitzian mapping. For any givenx0∈
H,{xn}is defined by
xn+1=αnxn+1−αnTλn+1xn, n≥0, (3.9)
where{αn}and{λn} ⊂[0, 1)satisfy the following conditions: (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 iflim infn→∞d(xn,F(T))=0.
Proof. (1) It follows fromLemma 3.1that{xn}is bounded. Thus, letq1andq2be weak limits of subsequences{xnk}and{xnj}of{xn}, respectively. It follows from Lemmas2.3
and3.1thatq1,q2∈F(T). Assumeq1=q2, then by Opial’s condition, we obtain
lim
n→∞xn−q1=klim→∞
x
nk−q1<lim k→∞
x
nk−q2
=lim
j→∞xnj−q2<lim k→∞
xn
k−q1=nlim→∞xn−q1,
(3.10)
which is a contradiction; hence,q1=q2. Then,{xn}converges weakly to a common fixed point ofT.
(2) Suppose that{xn}converges strongly to a fixed pointqofT, then limn→∞xn− q =0. Since 0≤d(xn,F(T))≤ xn−q, we have lim infn→∞d(xn,F(T))=0.
Conversely, suppose that lim infn→∞d(xn,F(T))=0. For any p∈F(T), F(p) ≤
F(p)−F(xn)+F(xn) ≤kxn−p+F(xn). Since{xn}and{F(xn)}are bounded,
F(p) is bounded for any p ∈ F(T), that is, there exists constant M > 0 such thatF(p) ≤Mfor allp∈F(T). In addition, it follows from (3.5) that
xn+1−p2
≤xn−p2+λn+1μ 2
τ F(p) 2
. (3.11)
So,
x
n+1−p 2
≤xn−p 2
+λn+1μ
2
τ 2k
2x n−p
2
+ 2Fxn 2
=
1 + 2k2λn+1μ 2
τ
xn−p 2
+ 2λn+1μ
2
τ F
xn 2
.
(3.12)
Thus,
dxn+1,F(T)
2
≤1 + 2k2λn+1μ2 τ
dxn,F(T)
2
+ 2λn+1μ
2
τ F
xn 2
. (3.13)
In addition, we obtain that∞n=12k2(λn+1μ2/τ)<∞and∞n=12(λn+1μ2/τ)F(xn)2<
limn→∞d(xn,F(T)) exists. Furthermore, since lim infn→∞d(xn,F(T)) =0, we have limn→∞d(xn,F(T))=0. We now prove that{xn}is a Cauchy sequence.
TakingM1=max{2e(2μ2
k2
/τ)∞i=1λi, 4(μ2M2/τ)e(2μ2k2/τ)∞i=1λi}, for any>0, there exists
positive integerNsuch thatd(xn,F(T))<
/4M1and∞i=nλi</4M1asn≥N. Taking q∈F(T), for anyn,m≥N, it follows from (3.12) that
xn−xm2
2 ≤xn−q
2
+xm−q2
≤
1 + 2k2λnμ2 τ
xn−1−q 2
+ 2λnμ
2
τ F
xn−1 2
+
1 + 2k2λmμ2 τ
xm−1−q 2
+ 2λmμ
2
τ F
xm−1 2
≤
1 + 2k2λnμ2 τ
xn−1−q 2
+ 2λnμ
2
τ M
2
+
1 + 2k2λmμ2 τ
xm−1−q 2
+ 2λmμ
2
τ M
2
≤ n
i=N+1
1 + 2k2λiμ2 τ
xN−q2+ n−1
i=N+1
2λiμ
2
τ M
2 n
j=i+1
1 + 2k2λjμ 2
τ
+ 2λnμ
2
τ M
2+ m
i=N+1
1 + 2k2λiμ2 τ
xN−q2
+ m−1
i=N+1
2λiμ
2
τ M
2 m
j=i+1
1 + 2k2λjμ2 τ
+ 2λmμ
2
τ M
2
≤2e(2μ2k2/τ)∞
i=N+1λix N−q
2
+ 4μ
2M2
τ e
(2μ2k2/τ)∞
i=N+1λi ∞
i=N+1 λi.
(3.14)
Thus,
xn−xm2
≤2M1xN−q2+ 2M1
∞
i=N+1
λi. (3.15)
Taking the infimum for allq∈F(T), we have
xn−xm2
≤2M1dxN,F(T)
2
+ 2M1
∞
i=N+1
λi<. (3.16)
This implies that{xn}is a Cauchy sequence. Therefore, there existsp∈Hsuch that{xn} converges strongly top. It follows fromLemma 3.1that
p−T p ≤p−xn+xn−Txn−→0, asn−→ ∞. (3.17)
Corollary3.3. Under the conditions ofLemma 3.1, ifT is completely continuous, then
{xn}converges strongly to a fixed point ofT.
Proof. ByLemma 3.1,{xn}is bounded and limn→∞xn−Txn =0, then{Txn}is also bounded. SinceT is completely continuous, there exists subsequence{Txnj}of {Txn}
such thatTxnj→pas j→ ∞. It follows fromLemma 3.1that limj→∞xnj−Txnj =0.
So by the continuity ofTandLemma 2.3, we have limj→∞xnj−p =0 andp∈F(T).
Furthermore, byLemma 3.1, we get that limn→∞xn−pexists. Thus, limn→∞xn−p =
0. The proof is completed.
Corollary3.4. Under the conditions ofLemma 3.1, ifT is demicompact, then{xn}
con-verges strongly to a fixed point ofT.
Proof. SinceTis demicompact,{xn}is bounded and limn→∞xn−Txn =0, then there exists subsequence{xnj}of{xn}such that{xnj}converges strongly toq∈H. It follows
fromLemma 2.3 thatq∈F(T). Thus, limn→∞xn−qexists byLemma 3.1. Since the subsequence{xnj}of{xn}such that{xnj}converges strongly toq, then{xn}converges
strongly to the common fixed pointq∈F(T). The proof is completed.
For studying the strong convergence of fixed points of a nonexpansive mapping, Sen-ter and Dotson [9] introduced Condition (A). Later on, Maiti and Ghosh [5] well as Tan and Xu [10] studied Condition (A) and pointed out that Condition (A) is weaker than the requirement of demicompactness for nonexpansive mappings. A mappingT:K→K
withF(T)= {x∈K:Tx=x} =φis said to satisfy condition (A) if there exists a non-decreasing function f : [0,∞)→[0,∞) with f(0)=0 and f(t)>0 for allt∈(0,∞) such thatx−Tx ≥f(d(x,F(T))) for allx∈K, whered(x,F(T))=inf{x−q:q∈F(T)}.
Theorem3.5. Under the conditions ofLemma 3.1, ifT satisfies condition(A), then{xn}
converges strongly to a fixed point ofT.
Proof. SinceTsatisfies condition (A), thenf(d(xn,F(T)))≤ xn−Txn. It follows from
Lemma 3.1that lim infn→∞d(xn,F(T))=0. Thus, it follows fromTheorem 3.2that{xn} converges strongly to a fixed point ofT. The proof is completed.
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Lin Wang: Department of Mathematics, Kunming Teachers College, Kunming, Yunnan 650031, China
