Volume 2010, Article ID 734181,21pages doi:10.1155/2010/734181
Research Article
Robustness of Mann Type Algorithm with
Perturbed Mapping for Nonexpansive Mappings
in Banach Spaces
L. C. Ceng,
1, 2Y. C. Liou,
3and J. C. Yao
41Department of Mathematics, Shanghai Normal University, Shanghai 200234, China 2Scientific Computing Key Laboratory, Shanghai Universities, Shanghai, China
3Department of Information Management, Cheng Shiu University, no.840, Chengcing Road, Niaosong Township, Kaohsiung County 833, Taiwan
4Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung 804, Taiwan
Correspondence should be addressed to J. C. Yao,[email protected]
Received 30 October 2009; Accepted 10 January 2010
Academic Editor: Simeon Reich
Copyrightq2010 L. C. Ceng 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.
The purpose of this paper is to study the robustness of Mann type algorithm in the sense that approximately perturbed mapping does not alter the convergence of Mann type algorithm. It is proven that Mann type algorithm with perturbed mappingxn1 λnxn 1−λnTxnen− λnμnFxnremains convergent in a Banach space setting whereλn, μn ∈0,1,T a nonexpansive mapping,en,n0,1, . . ., errors andFa strongly accretive and strictly pseudocontractive mapping.
1. Introduction
Let C be a nonempty closed convex subset of a real Banach spaceX, and T : C → Ca
nonexpansive mappingi.e.,Tx−Ty ≤ x−yfor allx, y∈C. We use FixTto denote the
set of fixed points ofT; that is, FixT {x∈C:Txx}. Throughout this paper it is assumed
that FixT/∅. 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 vast investigation since these methods find applications in a variety of applied areas of variational inequality problems, equilibrium problems, inverse problems,
partial differential equations, image recovery, and signal processingsee, e.g.,1–17.
In 1953, Mann 18 introduced an iterative algorithm which is now referred to as
algorithm; that is, for an arbitrary initial guessx0∈C, the sequence{xn}is generated by the
recursive manner
xn1λnxn 1−λnTxn, ∀n≥0, 1.1
where C is a convex subset of a Banach spaceX, T : C → C is a mapping and {λn} is
a sequence in the interval0,1. It is well known that Mann’s algorithm can be employed
to approximate fixed points of nonexpansive mappings and zeros of strongly accretive
mappings in Hilbert spaces and Banach spaces. Many convergence theorems have been announced and published by a large number of authors. A typical convergence result in
connection with the Mann’s algorithm is the following theorem proved by Ishikawa19.
Theorem IS (see [
19
])
Let Cbe a nonempty subset of a Banach spaceX and letT : C → X be a nonexpansive
mapping. Let{λn}be a real sequence satisfying the following control conditions:
a∞n0λn∞;
b0≤λn ≤λ <1.
Let{xn}be defined by1.1such thatxn∈Cfor alln≥0. If{xn}is bounded thenxn−Txn → 0
asn → ∞.
The interest and importance of Theorem IS lie in the fact that strong or weak
convergence of the sequence{xn} can be achieved under certain appropriate assumptions
imposed on the mappingT, the domainDTor the spaceX. In an infinite-dimensional space
X, Mann’s algorithm has only weak convergence, in general. In fact, it is known that if the
sequence{λn}is such that∞n0λn1−λn ∞, then Mann’s algorithm converges weakly to
a fixed point ofT provided that the underlying spaceXis a Hilbert space or more general, a
uniformly convex Banach space which has a Fr´echet differentiable norm or satisfies Opial’s
property. See, for example,20,21.
The study of the robustness of Mann’s algorithm is initiated by Combettes 22
where he considered a parallel projection method algorithm in signal synthesisdesign and
recoveryproblems in a real Hilbert spaceHas follows:
xn1xnλn
m
i1
wiPixnci,n−xn
, 1.2
where for each i, Pix is the nearest point projection of a signal x ∈ H onto a closed
convex subsetSiofH23 Si is interpreted as theith constraint set of the signals,{λn}n≥0
is a sequence of relaxation parameters in0,2,{wi}mi1are strictly positive weights such that
m
i1wi1, andci,nstands for the error made in computing the projection ontoSiat iteration
n. Then he proved the following robustness result of algorithm1.2.
Theorem 1.1see22. AssumeG:mi1Si/∅. Assume also
i∞n0λn2−λn ∞,
ii∞n0λnmi1wici,n<∞.
Define a mappingT :H → Hby
Tx:2
m
i1
wiPix−x, ∀x∈H, 1.3
and put
en:2 m
i1
wici,n, αn: λ2n ∈0,1, ∀n≥0. 1.4
SincePiis a projection, the mappingVi:2Pi−Iis nonexpansive. ThusPi IVi/2and algorithm
1.2can be rewritten as
xn1 1−αnxnαnTxnen, 1.5
whereTis given by1.3. Note thatT can be written asT mi1wiViand thusT is nonexpansive. Note also thatFixT mi1FixVi G. Furthermore, conditions (i) and (ii) inTheorem 1.1can be stated as
i∞n0αn1−αn ∞
ii∞n0αnen<∞.
Very early, some authors had considered Mann iterations in the setting of uniformly convex Banach spaces and established strong and weak convergence results for Mann
iterations; see, e.g., 24, 25. Recently, Kim and Xu26 studied the robustness of Mann’s
algorithm for nonexpansive mappings in Banach spaces and extended Combettes’ robustness
result Theorem 1.1abovefor projections from Hilbert spaces to the setting of uniformly
convex Banach spaces.
Theorem 1.2see26, Theorem 3.3. Assume thatXis a uniformly convex Banach space. Assume, in addition, that eitherX∗has the KK- property orX satisfies Opial’s property. LetT :X → Xbe a nonexpansive mapping such thatFixT/∅. Given an initial guessx0∈X. Let{xn}be generated by the following perturbed Mann’s algorithm:
xn1 1−αnxnαnTxnen, ∀n≥0, 1.6
where{αn} ⊂0,1and{en} ⊂Xsatisfy the following properties:
i∞n0αn1−αn ∞,
ii∞n0αnen<∞.
Then the sequence{xn}converges weakly to a fixed point ofT.
Further, Kim and Xu 26also extended the robustness to nonexpansive mappings
Theorem 1.3see26, Theorem 4.1. LetCbe a nonempty closed convex subset of a Hilbert space HandT :C → Ca nonexpansive mapping withFixT/∅. Let{xn}be generated from an arbitrary x0∈Cvia one of the following algorithms1.7and (1.7):
xn1 1−αnxnαnPCTxnen, ∀n≥0, xn1PC1−αnxnαnTxnen, ∀n≥0,
1.7
where the sequences{αn} ⊂0,1and{en} ⊂Xare such that
i∞n0αn1−αn ∞,
ii∞n0αnen<∞.
Then{xn}converges weakly to a fixed point ofT.
Theorem 1.4 see 26, Theorem 5.1. Let X be a uniformly convex Banach space. Assume in addition that eitherX∗has the KK- property orX satisfies Opial’s property. LetAbe anm-accretive operator in X such that A−10/∅. Moreover, assume that {α
n} ⊂ 0,1, {cn} ⊂ 0,∞, and
{en} ⊂Xsatisfy the following properties:
i∞n0αn1−αn ∞;
ii∞n0αnen<∞;
iii0< c∗< cn< c∗<∞, wherec∗andc∗are two constants;
iv∞n0|cn1−cn|<∞.
Then the sequence{xn}generated from an arbitraryx0∈Xby
xn1 1−αnxnαnJcnxnen, ∀n≥0, 1.8
converges weakly to a point ofA−10.
Let X be a real reflexive Banach space. Let T : X → X be a nonexpansive
mapping with FixT/∅. Assume that F : X → X is δ-strongly accretive and λ-strictly
pseudocontractive withδλ ≥ 1 where δ, λ∈ 0,1. In this paper, inspired by Combettes’
robustness result Theorem 1.1 above and Kim and Xu’s robustness result Theorem 1.2
above we will consider the robustness of Mann type algorithm with perturbed mapping,
which generates, from an arbitrary initial guess x0 ∈ X, a sequence{xn} by the recursive
manner
ynλnxn 1−λnTxnen, xn1yn−λnμnFxn, ∀n≥0,
1.9
where{λn},{μn},and{en}are sequences in0,1and inX, respectively, such that
i∞n0λn1−λn ∞;
ii∞n01−λnen<∞;
More precisely, we will prove under conditionsi–iiithe weak convergence of the
algorithm1.9in a uniformly convex Banach spaceXwhich either has the KK-property or
satisfies Opial’s property. This theorem extends Kim and Xu’s robustness resultTheorem 1.2
abovefrom Mann’s algorithm1.6with errors to Mann type algorithm1.9with perturbed
mappingF. On the other hand, we also extend Kim and Xu’s robustness resultsTheorems
1.3and 1.4 above for nonexpansive mappings which are defined on subsets of a Hilbert
space and accretive operators in a uniformly convex Banach space from Mann’s algorithm with errors to Mann type algorithm with perturbed mapping.
Throughout this paper, we use the following notations:
istands for weak convergence and → for strong convergence,
iiωw{xn} {x:∃xnk x}denotes the weakω-limit set of{xn}.
2. Preliminaries
LetXbe a real Banach space. Recall that the norm ofXis said to be Fr´echet differentiable if,
for eachx∈SX, the unit sphere ofX, the limit
lim
t→0
xty− x
t 2.1
exists and is attained uniformly iny∈SX. In this case, we have
1
2x
2h, Jx ≤ 1
2xh
2≤ 1
2x
2h, Jxbh 2.2
for allx, h∈X, whereJis the normalized duality map fromXtoX∗defined by
jx x∗∈X∗:x, x∗x2x∗2 , 2.3
·,·is the duality pairing betweenXand X∗, andbis a function defined on0,∞such that
limt↓0bt/t0. Examples of Banach spaces which have a Fr´echet differentiable norm include
lpandLpfor 1< p <∞these spaces are actually uniformly smooth.
It is known that a Banach spaceX is Fr´echet differentiable if and only if the duality
mapJis single-valued and norm-to-norm continuous.
We need the concept of the KK-property. A Banach spaceX is said to have the KK
-propertythe Kadec-Klee propertyif, for any sequence{zn}inX, the conditionszn zand
zn → zimply thatzn → z. It is known27, Remark 3.2that the dual space of a reflexive
Banach space with a Fr´echet differentiable norm has the KK-property.
Recall now thatXsatisfies Opial’s property28provided that, for each sequence{xn}
inX, the conditionxn ximplies
lim sup
n→ ∞ xn−x<lim supn→ ∞
It is known28that eachlp1≤p <∞enjoys this property, whileLpdoes not unlessp2.
It is known29 that any separable Banach space can be equivalently renormed so that it
satisfies Opial’s property.
Recall that a Banach spaceXis said to be uniformly convex if, for each 0< ε≤ 2, the
modulus of convexityδXεofXdefined by
δXε:inf
1−xy
2 :x, y∈X, x ≤1, y≤1, andx−y≥ε
2.5
is positive.
We need an inequality characterization of uniform convexity.
Lemma 2.1see30. Given a numberr > 0. A real Banach spaceXis uniformly convex if and only if there exists a continuous strictly increasing functionϕ:0,∞ → 0,∞, ϕ0 0, such that
λx 1−λy2 ≤λx2 1−λy2−λ1−λϕx−y 2.6
for allλ∈0,1andx, y∈Xsuch thatx ≤randy ≤r.
A mappingFwith domainDFand rangeRFinXis calledδ-strongly accretive if
for eachx, y∈DF,
Fx−Fy, jx−y≥δx−y2, ∀jx−y∈Jx−y 2.7
for someδ∈0,1.Fis calledλ-strictly pseudocontractive if for eachx, y∈DF,
Fx−Fy, jx−y≥x−y2−λx−y−Fx−Fy2, ∀jx−y∈Jx−y 2.8
for someλ∈0,1. It is easy to see that2.8can be rewritten as
I−Fx−I−Fy, jx−y≥λI−Fx−I−Fy2. 2.9
The following proposition will be used frequently throughout this paper. For the sake of completeness, we include its proof.
Proposition 2.2. LetXbe a real Banach space andF:DF → Xa mapping.
iIfF is aλ-strictly pseudocontractive, thenF is Lipschitz continuous with constant1 1/λ.
iiIfFisδ-strongly accretive andλ-strictly pseudocontractive withδλ≥1, then for each fixedμ∈0,1, the mappingI−μFhas the following property:
I−μF
x−I−μFy≤
⎛
⎝1−μ
⎛
⎝1−
1−δ
λ
⎞ ⎠
⎞
Proof. iFrom2.9, we derive
λI−Fx−I−Fy2≤ I−Fx−I−Fy, jx−y
≤I−Fx−I−Fyx−y, ∀jx−y∈Jx−y 2.11
which implies that
I−Fx−I−Fy≤ 1
λx−y. 2.12
Thus
Fx−Fy≤I−Fx−I−Fyx−y≤1 1
λ
x−y, 2.13
and soFis Lipschitz continuous with constant11/λ.
iiFrom2.8and2.9, we obtain
λI−Fx−I−Fy2≤I−Fx−I−Fy, jx−y
x−y2−Fx−Fy, jx−y
≤1−δx−y2.
2.14
Sinceδλ≥1⇔1−δ/λ∈0,1, we have
I−Fx−I−Fy≤⎛⎝
1−δ
λ
⎞
⎠x−y, ∀x, y∈DF. 2.15
Consequently, for each fixedμ∈0,1, we have
x−y−μ
Fx−Fy1−μx−yμI−Fx−I−Fy
≤1−μx−yμI−Fx−I−Fy
≤1−μx−yμ ⎛ ⎝
1−δ
λ
⎞
⎠x−y
⎛
⎝1−μ
⎛
⎝1−
1−δ
λ
⎞ ⎠
⎞
⎠x−y, ∀x, y∈DF.
2.16
Proposition 2.3. LetXbe a uniformly convex Banach space andCa nonempty closed convex subset ofX.
iReference [31] (demiclosedness principle). IfT :C → Cis a nonexpansive mapping and if
{xn}is a sequence inCsuch thatxn xandI−Txn → y, thenI−Txy.
iiReference [32]. IfCis also bounded, then there exists a continuous, strictly increasing, and convex functionγ:0,∞ → 0,∞(depending only on the diameter ofC) withγ0 0
and such that
γTλx 1−λy−λTx 1−λTy≤x−y−Tx−Ty 2.17
for allx, y∈C, λ∈0,1, and nonexpansive mappingsT :C → X.
We also use the following elementary lemma.
Lemma 2.4 see 33. Let {an} and {bn} be sequences of nonnegative real numbers such that
∞
n0bn<∞andan1≤anbnfor alln≥1. Thenlimn→ ∞anexists.
3. Robustness of Mann Type Algorithm with Perturbed Mapping
LetX be a real reflexive Banach space. LetT : X → X be a nonexpansive mapping with
FixT/∅. Assume thatF :X → X isδ-strongly accretive andλ-strictly pseudocontractive
with δ λ ≥ 1. We now discuss the robustness of Mann type algorithm with perturbed
mapping, which generates, from an initial guessx0∈X, a sequence{xn}as follows:
ynλnxn 1−λnTxnen, xn1yn−λnμnFxn, ∀n≥0,
3.1
where{λn},{μn},and{en}are sequences in0,1and inX, respectively, such that
i∞n0λn1−λn ∞;
ii∞n01−λnen<∞;
iii∞n0λnμn<∞.
We remark that Mann type algorithm with perturbed mapping is based on Mann
“ynλnxn 1−λnTxnen” is taken from Mann iteration method, and another iteration
step “xn1yn−λnμnFxn” is taken from steepest-descent method.
We first discuss some properties of algorithm3.1.
Lemma 3.1. Let{xn}be generated by algorithm3.1and let p ∈ FixT.Thenlimn→ ∞xn−p exists.
Proof. We have
xn1−p
yn−λnμnFxn−p
λnxn 1−λnTxnen−λnμnFxn−p
≤λnI−μnFxn−p 1−λnTxn−p 1−λnen
≤1−λnTxn−pλnI−μnFxn−p 1−λnen
≤1−λnxn−pλnI−μnFxn−I−μnFp
I−μnFp−p 1−λnen
≤1−λnxn−pλn
⎡ ⎣1−μn
⎛
⎝1−
1−δ
λ
⎞ ⎠ ⎤
⎦xn−p
λnμnFp 1−λnen
⎛
⎝1−λnμn ⎛
⎝1−
1−δ
λ
⎞ ⎠
⎞
⎠xn−pλnμnFp 1−λnen
≤xn−pλnμnFp 1−λnen.
3.2
The conclusion of the lemma is a consequence ofLemma 2.4.
Proposition 3.2. LetXbe a uniformly convex Banach space.
iFor allp, q∈FixTand0≤t≤1,limn→ ∞txn 1−tp−qexists.
iiIf, in addition, the dual spaceX∗ofX has theKK-property, then the weakω-limit set of
Proof. iFor integersn, m≥1, define the mappingsTnandSn,mas follows:
Tnx:λnx 1−λnTx 1−λnen−λnμnFx, ∀x∈X, 3.3
andSn,m : Tnm−1Tnm−2· · ·Tn. It is easy to see thatxnm Sn,mxn. First, let us show thatTn
andSn,mare nonexpansive. Indeed, for allx, y∈X, usingProposition 2.2no.iiwe have
Tnx−Tnyλnx 1−λnTx 1−λnen−λnμnFx
−λny 1−λnTy 1−λnen−λnμnFy
λnI−μnFx 1−λnTx−λnI−μnFy 1−λnTy
≤λnI−μnFx−I−μnFy 1−λnTx−Ty
≤λn
⎡ ⎣1−μn
⎛
⎝1−
1−δ
λ
⎞ ⎠ ⎤
⎦x−y 1−λnx−y
⎡
⎣1−λnμn
⎛
⎝1−
1−δ
λ
⎞ ⎠ ⎤
⎦x−y.
3.4
ThusTn:X → Xis nonexpansivedue toλnμn∈0,1and so isSn,m.
Second, let us show that for eachv∈FixT,
Sn,mv−v ≤ nm−1
jn
1−λjejλjμjFv. 3.5
Indeed, wheneverm1, we have
Sn,1v−vTnv−v
λnv 1−λnTv 1−λnen−λnμnFv−v
≤1−λnenλnμnFv
n1−1
jn
1−λjejλjμjFv.
This implies that inequality3.5holds form1. Assume that inequality3.5holds for some
m≥1. Consider the case ofm1. Observe that
Sn,m1v−vTnmSn,mv−v
λnmSn,mv 1−λnmTSn,mv 1−λnmenm−λnmμnmFSn,mv−v
λnmI−μnmFSn,mv−v 1−λnmTSn,mv−v 1−λnmenm
≤1−λnmTSn,mv−vλnmI−μnmFSn,mv−v 1−λnmenm
≤1−λnmSn,mv−vλnmI−μnmFSn,mv−I−μnmFv
I−μnmFv−v 1−λnmenm
≤1−λnmSn,mv−vλnm ⎡
⎣1−μnm
⎛
⎝1−
1−δ
λ
⎞ ⎠ ⎤
⎦Sn,mv−v
λnmμnmFv 1−λnmenm
⎡
⎣1−λnmμnm
⎛
⎝1−
1−δ
λ
⎞ ⎠ ⎤
⎦Sn,mv−vλnmμnmFv 1−λnmenm
≤ Sn,mv−vλnmμnmFv 1−λnmenm
≤nm−1
jn
1−λjejλjμjFvλnmμnmFv 1−λnmenm
nm1−1
jn
1−λjejλjμjFv.
3.7
This shows that inequality3.5holds for the case ofm1. Thus, by induction, we know that
inequality3.5holds for allm≥1.
Now set
antxn 1−tp−q,
bn,mSn,mtxn 1−tp−txnm 1−tp.
3.8
ByProposition 2.3no.iiand noticing inequality3.5we deduce that
bn,m≤Sn,mtxn 1−tp−tSn,mxn 1−tSn,mp 1−tSn,mp−p
≤γ−1x
n−p−xnm−Sn,mp nm−1
jn
1−λjejλjμjFp
≤γ−1x
n−p−xnm−pp−Sn,mp nm−1
jn
1−λjejλjμjFp.
Therefore,
bn,m≤γ−1
⎛
⎝xn−p−xnm−pnm−1
jn
1−λjejλjμjFp
⎞ ⎠
nm−1
jn
1−λjejλjμjFp.
3.10
Since limn→ ∞xn−pexists and∞n01−λnenand∞n0λnμnare convergent, we conclude
from3.10that
lim
n,m→ ∞bn,m0. 3.11
Also, since, for alln, m≥1,
anmtxnm 1−tp−q
≤bn,mSn,mtxn 1−tp−Sn,mqSn,mq−q
≤anbn,m nm−1
jn
1−λjejλjμjFq,
3.12
it follows from3.11and3.12that limn→ ∞anexists.
iiThis is Lemma 3.2 of27.
Now we can state and prove the main result of this section.
Theorem 3.3. Assume thatXis a uniformly convex Banach space. Assume, in addition, that either X∗has theKK-property orXsatisfies Opial’s property. LetT :X → Xbe a nonexpansive mapping
such that FixT/∅ and F : X → Xδ-strongly accretive and λ-strictly pseudocontractive with δλ≥1. Given an initial guessx0∈X. Let{xn}be generated by the following Mann type algorithm with perturbed mappingF:
xn1λnxn 1−λnTxnen−λnμnFxn, ∀n≥0, 3.13
where{λn}∞n0,{μn}∞n0,and{en}∞n0satisfy the following properties:
i∞n0λn1−λn ∞;
ii∞n01−λnen<∞;
iii∞n0λnμn<∞.
Proof. Fixp∈FixTand select a numberr >0 large enough so thatxn−p ≤rfor alln≥0.
LetM >0 satisfyM >2r 1−λnenλnμnFxnfor alln≥0. ByLemma 2.1, we have
xn1−p2λ
nxn−p 1−λnTxn−p 1−λnen−λnμnFxn2
≤λnxn−p 1−λnTxn−p2
21−λnen−λnμnFxnλnxn−p 1−λnTxn−p
1−λnen−λnμnFxn2
≤λnxn−p2 1−λnTxn−p2−λn1−λnφxn−Txn
M1−λnenλnμnFxn
≤xn−p2−λn1−λnφxn−Txn M1−λnenλnμnFxn.
3.14
It follows that
λn1−λnφxn−Txn≤xn−p2−xn1−p2M
1−λnenλnμnFxn.
3.15
This implies that
∞
n0
λn1−λnφxn−Txn<∞. 3.16
In particular, limn→ ∞λn1−λnφxn−Txn 0. Due to conditioni, we must have that
lim infn→ ∞φxn−Txn 0. Hence
lim inf
n→ ∞ xn−Txn0. 3.17
However, since
Txn1−xn1 Txn1−Txn λnTxn−xn−1−λnenλnμnFxn, 3.18
we have
Txn1−xn1 ≤ Txn1−TxnλnTxn−xn 1−λnenλnμnFxn
≤ xn1−xnλnTxn−xn 1−λnenλnμnFxn
≤ Txn−xn21−λnen2λnμnFxn,
and, byLemma 2.4, limn→ ∞Txn−xnexists and hence, by3.17,
lim
n→ ∞Txn−xn0. 3.20
Notice that, by the demiclosedness principle ofI−T, we obtain
ωwxn⊂FixT. 3.21
Hence to prove that{xn} converges weakly to a fixed point of T, it suffices to show that
ωwxnis a singleton. We distinguish two cases. First assume thatX∗ has the KK-property.
Then thatωwxnis a singleton is guaranteed byProposition 3.2no.ii.
Next assume thatXsatisfies Opial’s property. Takep1, p2 ∈ ωwxnand let{xni}and
{xmj}be subsequences of{xn}such thatxni p1 andxmj p2, respectively. Ifp1/p2, we
reach the following contradiction:
lim
n→ ∞xn−p1ilim→ ∞xni−p1
< lim
i→ ∞xni−p2jlim→ ∞
xmj−p2
< lim
j→ ∞
xmj−p1
lim
n→ ∞xn−p1.
3.22
This shows thatωwxnis a singleton. The proof is therefore complete.
4. The Case Where Mappings Are Defined on Subsets
We observe that if the domainDTis a proper closed convex subsetCofX, then the vectors
TxnenandI−μnFxnmay not belong toC. In this case the next iteratexn1may not be well
defined by3.13. In order to consider this situation, we will use the nearest projectionPCand
for the projection to be nonexpansive, we have to restrict our spaces to be Hilbert spaces.
LetH be a real Hilbert space with inner product·,·and norm · . Given a closed
convex subsetCofH. Recall that thenearest pointprojectionPC fromHontoCassigns
each point x ∈ H with itsuniquenearest point in Cwhich is denoted by PCx. Namely,
PCx∈Cis the unique point inCwith the property
x−PCxinfx−y:y∈C. 4.1
Note thatPCis nonexpansive.
LetT :C → Cbe a nonexpansive mapping with FixT/∅andF:C → Hδ-strongly
monotone andλ-strictly pseudocontractive withδλ≥1. Starting withx0 ∈Cand afterxnin
Cis defined, we have two ways to define the next iteratexn1: either applying the projection
PCI −μnFxnandPCTxnen, or projecting a convex combination ofI−μnFxn and
TxnenontoCto definexn1. More precisely, we definexn1as follows:
xn1λnPCI−μnFxn 1−λnPCTxnen, ∀n≥0, 4.2
or
xn1PCλnI−μnFxn 1−λnTxnen, ∀n≥0. 4.3
Theorem 4.1. LetC be a nonempty closed convex subset of a Hilbert space H. Let T : C → C
be a nonexpansive mapping with FixT/∅and F : C → Hδ-strongly monotone and λ-strictly pseudocontractive withδλ≥1. Let{xn}be generated by either4.2or4.3where the sequences
{λn},{μn}and{en}are such that
i∞n0λn1−λn ∞;
ii∞n01−λnen<∞;
iii∞n0λnμn<∞.
Then{xn}converges weakly to a fixed point ofT.
Proof. Givenp∈FixT. Assume that{xn}is generated by4.2. Then
xn1−pλnPC
I−μnFxn 1−λnPCTxnen−p
≤λnPCI−μnFxn−p 1−λnPCTxnen−p
≤λnI−μnFxn−p 1−λnTxnen−p
≤λnI−μnFxn−I−μnFpI−μnFp−p
1−λnTxnen−p
≤λn
⎡ ⎣
⎛ ⎝1−μn
⎛
⎝1−
1−δ
λ
⎞ ⎠
⎞
⎠xn−pI−μnFp−p
⎤ ⎦
1−λnxn−p 1−λnen
≤λnxn−pλnμnFp 1−λnxn−p 1−λnen
xn−p 1−λnenλnμnFp.
4.4
Hence limn→ ∞xn−pexists; in particular,{xn}is bounded. LetM > 0 be a constant such
We compute
xn1−p2
λnPCI−μnFxn 1−λnPCTxnen−p2
λnxn−p 1−λnTxn−pλnPCI−μnFxn−xn
1−λnPCTxnen−Txn2
≤λnxn−p 1−λnTxn−p2
λnPCI−μnFxn−xn 1−λnPCTxnen−Txn2
2λnxn−p 1−λnTxn−pλnPCI−μnFxn−xn
× 1−λnPCTxnen−Txn
≤λnxn−p2 1−λnTxn−p2−λn1−λnxn−Txn2
M1−λnenλnμnFxn
≤xn−p2−λn1−λnxn−Txn2M1−λnenλnμnFxn.
4.5
That is,
λn1−λnxn−Txn2≤xn−p2−xn1−p2M
1−λnenλnμnFxn. 4.6
This implies that
∞
n0
λn1−λnxn−Txn2 <∞. 4.7
In particularnoticing assumptioni,
lim inf
n→ ∞ xn−Txn0. 4.8
We also have
xn1−xnλnPCI−μnFxn 1−λnPCTxnen−xn
≤λnPCI−μnFxn−xn 1−λnPCTxnen−xn
≤λnμnFxn 1−λnTxn−xnen.
4.9
Moreover, noticing
Txn1−xn1 Txn1−Txn λnTxn−PCI−μnFxn 1−λnTxn−PCTxnen,
we have
Txn1−xn1 ≤ Txn1−TxnλnTxn−PCI−μnFxn
1−λnTxn−PCTxnen
≤ xn1−xnλnTxn−xnμnFxn 1−λnen
≤λnμnFxn 1−λnTxn−xnen
λnTxn−xnμnFxn 1−λnen
Txn−xn21−λnen2λnμnFxn.
4.11
Similarly, if{xn}is generated by algorithm4.3, then relations4.4–4.11still hold.
It is now readily seen that4.11together withLemma 2.4implies that limn→ ∞xn−
Txnexists, which together with4.8further implies that
lim
n→ ∞xn−Txn0. 4.12
Equation4.12implies thatωwxn ⊂ FixT, due to the demiclosedness principle. Finally,
repeating the last part of the proof ofTheorem 3.3in the case of Opial’s property, we see that
{xn}converges weakly to a fixed point ofT. The proof is therefore complete.
Finally in this section, we consider the case of accretive operators. Recall that a
multivalued operatorAwith domainDAand rangeRAin a Banach spaceXis said to be
accretive if, for eachxi∈DAandyi ∈Axi i1,2, there isjx2−x1∈Jx2−x1such that
y2−y1, jx2−x1 ≥0, 4.13
whereJis the duality map fromXto the dual spaceX∗. An accretive operatorAism-accretive
ifRIλA Xfor allλ >0.
Denote byΩthe zero set ofA; that is,
Ω:A−10 {x∈DA: 0∈Ax}. 4.14
Throughout the rest of this paper it is always assumed that A is m-accretive and Ω is
nonempty.
Denote byJr the resolvent ofAforr >0:
Jr IrA−1. 4.15
It is known thatJr is a nonexpansive mapping fromXtoC:DAwhich will be assumed
convexthis is so ifXis uniformly convex. It is also known that FixJr Ωforr >0.
Now consider the problem of finding a zero of anm-accretive operatorAin a Banach
spaceX,
We will study the convergence of the following algorithm:
xn1λnxn 1−λnJcnxnen−λnμnFxn, 4.17
whereF : X → X is a perturbed mapping, the initial guessx0 ∈ X is arbitrary, {λn}and
{μn}are two sequences in0,1,{cn}is a sequence of positive numbers, and{en}is an error
sequence inX.
Theorem 4.2. LetXbe a uniformly convex Banach space. Assume in addition that eitherX∗has the KK-property orXsatisfies Opial’s property. LetAbe anm-accretive operator inX such thatΩ/∅
and letF:X → Xbeδ-strongly accretive andλ-strictly pseudocontractive withδλ≥1. Moreover, assume that{λn},{μn},{cn},and{en}satisfy the following properties:
i∞n0λn1−λn ∞;
ii∞n01−λnen<∞;
iii∞n0λnμn<∞;
iv0< c∗< cn< c∗<∞, wherec∗andc∗are two constants;
v∞n0|cn1−cn|<∞.
Then the sequence{xn}generated by algorithm4.17converges weakly to a point ofΩ.
Proof. The proof is a refinement of that of Theorem 3.3 given in Section 3 and 34,
Theorem 3.3together withProposition 3.2. So we only sketch it.
Letp∈Ω. By4.17, we have
xn1−pλnxn 1−λnJcnxnen−λnμnFxn−p
≤λnI−μnFxn−p 1−λnJcnxn−p 1−λnen
≤λnI−μnFxn−I−μnFpI−μnFp−p
1−λnxn−p 1−λnen
≤λn
⎡ ⎣1−μn
⎛
⎝1−
1−δ
λ
⎞ ⎠ ⎤
⎦xn−pλnμnFp
1−λnxn−p 1−λnen
≤λnxn−pλnμnFp 1−λnxn−p 1−λnen
xn−pλnμnFp 1−λnen.
4.18
ByLemma 2.4, we see that limn→ ∞xn−pexists.
With slight modifications of the proof ofTheorem 3.3replacingTxnbyJcnxn, we can
obtain that
lim inf
Now noticing
Jcn1xn1−xn1Jcn1xn1−Jcnxnλn
Jcnxn−
I−μnFxn−1−λnen, 4.20
and lettingsn :Jcnxn−xnfor alln≥0, we deduce that
sn1≤λnJcnxn−
I−μnFxnJcn1xn1−Jcnxn 1−λnen
≤λnsnJcn1xn1−Jcnxn 1−λnenλnμnFxn.
4.21
By mimicking the proof of Theorem 3.3 in34, we can show that, in the case ofcn≤cn1,
sn1
cn1 ≤
sn cn
2
cn
1−λnenλnμnFxn, 4.22
and in the case ofcn> cn1,
sn1
cn1 ≤
sn cn
2s∗
c2
∗ |cn−cn1|
2
c∗
1−λnenλnμnFxn, 4.23
wheres∗is such thatsn ≤s∗for alln≥ 0. In either case we conclude from4.22and4.23
that{sn}satisfies
sn1
cn1 ≤
sn
cn σn, ∀n≥0, 4.24
where σn : 2s∗|cn1−cn|/c∗221 −λnenλnμnFxn/c∗ fulfills ∞n0σn < ∞. By
Lemma 2.4,4.24implies that limn→ ∞sn/cnexists. This together with the assumptioniv
and4.19implies that lim supn→ ∞xn−Jcnxn0. So, by Lemma 3.3 in34, we have
xn−Jc∗xn ≤2xn−Jcnxn −→0. 4.25
By the demiclosedness principle,4.25ensures that ωwxn ⊂ FixJc∗ Ω. Repeating the
last part of the proof ofTheorem 3.3, we conclude that{xn}converges weakly to a point of
Ω.
Acknowledgments
This research was partially supported by Grant no. NSC 98-2923-E-110-003-MY3 and was also partially supported by the Leading Academic Discipline Project of Shanghai Normal
University DZL707, Innovation Program of Shanghai Municipal Education Commission
Grant 09ZZ133, National Science Foundation of China 10771141, Ph.D. Program
Foundation of Ministry of Education of China 20070270004, Science and Technology
Commission of Shanghai Municipality Grant075105118, and Shanghai Leading Academic
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