Volume 2010, Article ID 190126,24pages doi:10.1155/2010/190126
Research Article
A General Iterative Process for Solving a System of
Variational Inclusions in Banach Spaces
Uthai Kamraksa
1and Rabian Wangkeeree
1, 21Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand
2Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand
Correspondence should be addressed to Rabian Wangkeeree,[email protected]
Received 6 April 2010; Revised 12 June 2010; Accepted 14 June 2010
Academic Editor: S. Reich
Copyrightq2010 U. Kamraksa and R. Wangkeeree. 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 introduce a general iterative method for finding solutions of a general system of variational inclusions with Lipschitzian relaxed cocoercive mappings. Strong convergence theorems are established in strictly convex and 2-uniformly smooth Banach spaces. Moreover, we apply our result to the problem of finding a common fixed point of a countable family of strict pseudo-contraction mappings.
1. Introduction
LetUE {x ∈ E : x 1}. A Banach spaceE is said to be uniformly convex if, for any
∈0,2, there existsδ >0 such that, for anyx, y∈UE,
x−y≥ impliesxy
2
≤1−δ. 1.1
It is known that a uniformly convex Banach space is reflexive and strictly convex. A Banach spaceEis said to be smooth if the limit
lim
t→0
xty− x
t 1.2
exists for allx, y∈UE. It is also said to be uniformly smooth if the limit is attained uniformly
above limit is attained uniformly for ally∈UE. The modulus of smoothness ofEis defined
by
ρτ sup
1
2xyx−y−1 :x, y∈E,x1,yτ
, 1.3
where ρ : 0,∞ → 0,∞. It is known that E is uniformly smooth if and only if limτ→0ρτ/τ 0. Let q be a fixed real number with 1 < q ≤ 2. A Banach space E is said to beq-uniformly smooth if there exists a constantc >0 such thatρτ≤cτqfor allτ >0.
From1, we know the following property.
Letqbe a real number with 1< q≤2 and letEbe a Banach space. ThenEisq-uniformly smooth if and only if there exists a constantK≥1 such that
xyq
x−yq≤2xqKyq, ∀x, y∈E. 1.4
The best constantKin the above inequality is called theq-uniformly smoothness constant of
Esee1for more details.
LetEbe a real Banach space andE∗ the dual space ofE. Let·,·denote the pairing betweenEandE∗. Forq >1, the generalized duality mappingJq:E → 2E∗is defined by
Jqx
f ∈E∗:x, f xq,fxq−1, ∀x∈E. 1.5
In particular,J J2is called the normalized duality mapping. It is known thatJqx
xq−2Jxfor allx∈E. IfEis a Hilbert space, thenJ Iis the identity. Note the following.
1E is a uniformly smooth Banach space if and only if J is single-valued and uniformly continuous on any bounded subset ofE.
2All Hilbert spaces,Lporlpspacesp≥2, and the Sobolev spacesWp
mp≥2are
2-uniformly smooth, whileLp orlpandWp
m spaces1 < p ≤ 2arep-uniformly
smooth.
3Typical examples of both uniformly convex and uniformly smooth Banach spaces areLp, wherep >1. More precisely,Lpis min{p,2}-uniformly smooth for anyp >1.
Further, we have the following properties of the generalized duality mappingJq:
iJqx xq−2J2xfor allx∈Ewithx /0,
iiJqtx tq−1Jqxfor allx∈Eandt∈0,∞,
iiiJq−x −Jqxfor allx∈E.
It is known that, ifX is smooth, thenJ is single valued. Recall that the duality mappingJ
is said to be weakly sequentially continuous if, for each sequence{xn} ⊂ Ewithxn → x
weakly, we haveJxn → Jxweakly-∗. We know that, ifX admits a weakly sequentially
continuous duality mapping, thenXis smooth. For the details, see2.
Definition 1.1. Given a mappingΨ:C → E.
i Ψis said to be accretive if
Ψx−Ψy, Jx−y ≥0 1.6
for allx, y∈C.
ii Ψis said to beα-strongly accretive if there exists a constantα >0 such that
Ψx−Ψy, Jx−y ≥αx−y2 1.7
for allx, y∈C.
iii Ψis said to beα-inverse-strongly accretiveorα-cocoercive if there exists a constant
α >0 such that
Ψx−Ψy, Jx−y ≥αΨx−Ψy2 1.8
for allx, y∈C.
iv Ψis said to beα-relaxed cocoercive if there exists a constantα >0 such that
Ψx−Ψy, Jx−y ≥ −αΨx−Ψy2 1.9
for allx, y∈C.
v Ψis said to beα, β-relaxed cocoercive if there exist positive constantsαandβsuch that
Ψx−Ψy, Jx−y ≥−αΨx−Ψy2βx−y2 1.10
for allx, y∈C.
Remark 1.2. 1Everyα-strongly accretive mapping is an accretive mapping.
2Everyα-strongly accretive mapping is aβ, α-relaxed cocoercive mapping for any positive constantβbut the converse is not true in general. Then the class of relaxed cocoercive operators is more general than the class of strongly accretive operators.
3Evidently, the definition of the inverse-strongly accretive operator is based on that of the inverse-strongly monotone operator in real Hilbert spacessee, e.g.,3.
4The notion of the cocoercivity is applied in several directions, especially for solving variational inequality problems using the auxiliary problem principle and projection methods
4. Several classes of relaxed cocoercive variational inequalities have been studied in5,6. Next, we consider a system of quasivariational inclusions as follows.
Findx∗, y∗∈E×Esuch that
0∈x∗−y∗ρ1
Ψ1y∗M1x∗
,
0∈y∗−x∗ρ2
Ψ2x∗M2y∗
, 1.11
As special cases of problem1.11, we have the following.
1IfΨ1 Ψ2 ΨandM1M2M, then problem1.11is reduced to the following.
Findx∗, y∗∈E×Esuch that
0∈x∗−y∗ρ1
Ψy∗Mx∗,
0∈y∗−x∗ρ2
Ψx∗My∗. 1.12
2Further, ifx∗y∗in problem1.12, then problem1.12is reduced to the following
Findx∗∈Esuch that
0∈Ψx∗Mx∗. 1.13
In 2006, Aoyama et al.7considered the following problem. Findu∈Csuch that
Ψu, Jv−u ≥0, ∀v∈C. 1.14
They proved that the variational inequality1.14is equivalent to a fixed point problem. The elementu∈Cis a solution of the variational inequality1.14if and only ifu∈Csatisfies the following equation:
uPCu−λΨu, 1.15
whereλ >0 is a constant andPCis a sunny nonexpansive retraction fromEontoC, see the
definition below.
LetDbe a subset ofC, andP be a mapping ofCintoD. ThenPis said to be sunny if
PP xtx−P x P x, 1.16
wheneverP xtx−P x ∈ Cforx ∈ Candt ≥ 0. A mappingP ofCinto itself is called a retraction ifP2 P. If a mappingPofCinto itself is a retraction, thenP zzfor allz∈RP, whereRPis the range ofP. A subsetDofCis called a sunny nonexpansive retract ofCif there exists a sunny nonexpansive retraction fromContoD.
The following results describe a characterization of sunny nonexpansive retractions on a smooth Banach space.
Proposition 1.3 see 8. Let E be a smooth Banach space and C a nonempty subset of E. Let
P : E → C be a retraction andJ the normalized duality mapping onE. Then the following are
equivalent:
1Pis sunny and nonexpansive,
Recall that a mappingf : C → Cis called contractive if there exists a constantα ∈
0,1such that
fx−f
y≤αx−y, ∀x, y∈C. 1.17
A mappingT :C → Cis said to beε-strictly pseudocontractiveif there exists a constant
ε∈0,1such that
Tx−Ty2≤
x−y2εI−Tx−I−Ty2, ∀x, y∈C. 1.18
Note that the class of ε-strictly pseudocontractive mappings strictly includes the class of nonexpansive mappings which are mappingsT onCsuch that
Tx−Ty≤x−y, 1.19
for allx, y ∈ C. That is,T is nonexpansive if and only ifT is 0-strict pseudocontractive. We denote byFT:{x∈C:Txx}the set of fixed points ofT.
Proposition 1.4see9. Let Cbe a nonempty closed convex subset of a uniformly convex and
uniformly smooth Banach spaceEandTa nonexpansive mapping ofCinto itself withFT/∅. Then
the setFTis a sunny nonexpansive retract ofC.
Definition 1.5. A countable family of mapping{Tn :C → C}∞i1is called a family of uniformly
ε-strict pseudocontractionsif there exists a constantε∈0,1such that
Tnx−Tny2≤
x−y2εI−Tnx−I−Tny2, ∀x, y∈C, ∀n≥1. 1.20
For the class of nonexpansive mappings, one classical way to study nonexpansive mappings is to use contractions to approximate a nonexpansive mapping 10, 11. More precisely, taket∈0,1and define a contractionTt:C → Cby
Ttxtu 1−tTx, ∀x∈C, 1.21
whereu∈Cis a fixed point andT :C → Cis a nonexpansive mapping. Banach’s contraction mapping principle guarantees thatTthas a unique fixed pointxtinC; that is,
xttu 1−tTxt. 1.22
It is unclear, in general, what the behavior ofxtis ast → 0,even ifThas a fixed point.
However, in the case ofT having a fixed point, Ceng et al.12proved that, ifEis a Hilbert space, thenxtconverges strongly to a fixed point ofT. Reich11extended Browder’s result
to the setting of Banach spaces and proved that, ifEis a uniformly smooth Banach space, then xt converges strongly to a fixed point of T, and the limit defines the unique sunny
Reich 11 showed that, if E is uniformly smooth and D is the fixed point set of a nonexpansive mapping from C into itself, then there is a unique sunny nonexpansive retraction fromContoDand it can be constructed as follows.
Proposition 1.6 see 11. Let E be a uniformly smooth Banach space and T : C → C a
nonexpansive mapping such thatFT/∅. For each fixedu ∈ Cand every t ∈ 0,1, the unique
fixed pointxt∈Cof the contractionCx→tu 1−tTxconverges strongly ast → 0to a fixed
point ofT. DefineP :C → D byP u s−limt→0xt. ThenP is the unique sunny nonexpansive
retract fromContoD; that is,Psatisfies the property.
u−P u, Jy−P u ≤0, ∀u∈C, y∈D. 1.23
Notation. We use P u s−limt→0xtto denote strong convergence to P uof the net{xt}as
t → 0.
Definition 1.7see13. LetM:E → 2Ebe a multivalued maximal accretive mapping. The
single-valued mappingJM,ρ:E → Edefined by
JM,ρu IρM−1u, ∀u∈E, 1.24
is called the resolvent operator associated withM, whereρis any positive number andIis the identity mapping.
Recently, many authors have studied the problems of finding a common element of the set of fixed points of a nonexpansive mapping and one of the sets of solutions to the variational inequalities 1.11–1.14 by using different iterative methods see, e.g., 7,14–
16.
Very recently, Qin et al. 16 considered the problem of finding the solutions of a general system of variational inclusion1.11with α-inverse strongly accretive mappings. To be more precise, they obtained the following results.
Lemma 1.8see16. For anyx∗, y∗∈ E×E,wherey∗ JM2,ρ2x∗−ρ2Ψ2x∗,x∗, y∗is a
solution of the problem1.11if and only ifx∗is a fixed point of the mappingQdefined by
Qx JM1,ρ1
JM2,ρ2
x−ρ2Ψ2x
−ρ1Ψ1JM2,ρ2
x−ρ2Ψ2x
. 1.25
Theorem QCCKsee16, Theorem 2.1. LetEbe a uniformly convex and2-uniformly smooth
Banach space with the smoothness constantK. LetMi:E → 2Ebe a maximal monotone mapping and
Ψi :E → Eaγi-inverse-strongly accretive mapping, respectively, for eachi1,2. LetT :E → E
be a ε-strict pseudocontraction such that FT/∅. Define a mappingS by Sx 1−ε/K2x
ε/K2Tx, for allx ∈E. Assume thatΩ FT∩FQ
/
∅, whereQis defined as inLemma 1.8.
Letx1u∈Eand let{xn}be a sequence generated by
znJM2,ρ2
xn−ρ2Ψ2xn
,
ynJM1,ρ1
zn−ρ1Ψ1zn
,
xn1αnuβnxn
1−βn−αn
μSxn
1−μyn
, ∀n≥1,
whereμ∈0,1,ρ1∈0, γ1/K2,ρ2∈0, γ2/K2, and{αn}and{βn}are sequences in0,1. If the
control consequences{αn}and{βn}satisfy the following restrictions:
C10<lim infn→ ∞βn≤lim supn→ ∞βn<1,
C2limn→ ∞αn0and∞n1αn∞,
then{xn}converges strongly tox∗ PΩu, wherePΩ is the sunny nonexpansive retraction fromE
ontoΩandx∗, y∗, wherey∗JM2,ρ2x∗−ρ2Ψ2x∗, is a solution to problem1.11.
On the other hand, we recall the following well-known definitions and results. In a smooth Banach space, a mappingA : C → Eis called strongly positive 17if there exists a constantγ >0 with property
Ax, Jx ≥γx2,aI−bA sup x≤1
|aI−bAx, Jx|, a∈0,1, b∈−1,1, 1.27
whereIis the identity mapping andJis the normalized duality mapping.
In18, Moudafi introduced the viscosity approximation method for nonexpansive mappingssee19for further developments in both Hilbert and Banach spaces. Letf be a contraction on C. Starting with an arbitrary initial pointx1 ∈ C, define a sequence{xn}
recursively by
xn1 1−σnTxnσnfxn, n≥0 1.28
where {σn} is a sequence in 0,1. It is proved 18, 19 that, under certain appropriate
conditions imposed on{σn}, the sequence {xn} generated by 1.28strongly converges to
the unique solutionqinCof the variational inequality
I−fq, p−q ≥0, ∀p∈C, 1.29
Recently, Marino and Xu20introduced the following general iterative method:
xn1 I−αnATxnαnγfxn, n≥0. 1.30
whereAis a strongly positive bounded linear operator on a Hilbert spaceH. They proved that, if the sequence{αn}of parameters satisfies appropriate conditions, then the sequence
{xn} generated by 1.30 converges strongly to the unique solution of the variational
inequality
which is the optimality condition for the minimization problem
min
x∈C
1
2Ax, x −hx, 1.32
wherehis a potential function forγfi.e., hx γfxforx∈H.
Recently, Qin et al.21introduce the following iterative algorithm scheme:
x1x∈C,
ynPC
βn
1−βn
Txn
,
xn1αnγfxn 1−αnAyn,
1.33
whereTis nonself-k-strict pseudo-contraction,fis a contraction, andAis a strongly positive bounded linear operator on a Hilbert space H. They proved, under certain appropriate conditions imposed on the sequences{αn}and{βn}, that {xn}defined by1.33converges
strongly to a fixed point ofT, which solves some variational inequality.
In this paper, motivated by Qin et al.16, Moudafi 18, Marino and Xu20, and Qin et al.21, we introduce a general iterative approximation method for finding common elements of the set of solutions to a general system of variational inclusions 1.11 with Lipschitzian and relaxed cocoercive mappings and the set common fixed points of a countable family of strict pseudocontractions. We prove the strong convergence theorems of such iterative scheme for finding a common element of such two sets which is a unique solution of some variational inequality and is also the optimality condition for some minimization problems in strictly convex and 2-uniformly smooth Banach spaces. The results presented in this paper improve and extend the corresponding results announced by Qin et al.16, Moudafi18, Marino and Xu20, Qin et al.21, and many others.
2. Preliminaries
Now we collect some useful lemmas for proving the convergence result of this paper.
Lemma 2.1 see 22. The resolvent operator JM,ρ associated with M is single valued and
nonexpansive for allρ >0.
Lemma 2.2see13. u∈Eis a solution of variational inclusion1.13if and only ifuJM,ρu−
ρΨu,for allρ >0; that is,
V IE,Ψ, M FJM,ρI−ρΨ, ∀ρ >0, 2.1
Lemma 2.3see23. LetEbe a strictly convex Banach space. LetT1andT2be two nonexpansive
mappings fromEinto itself with a common fixed point. Define a mappingSby
SxλT1x 1−λT2x, ∀x∈E, 2.2
whereλis a constant in0,1. ThenSis nonexpansive andFS FT1∩FT2.
Lemma 2.4see24. LetCbe a nonempty closed convex subset of reflexive Banach spaceEwhich
satisfies Opial’s condition, and suppose thatT :C → Eis nonexpansive. Then the mappingI−T is
demiclosed at zero, that is,xn x, xn−Txn → 0imply thatxTx.
Lemma 2.5see25. Assume that{αn}is a sequence of nonnegative real numbers such that
αn1≤
1−γn
αnδn, 2.3
where{γn}is a sequence in0,1and{δn}is a sequence such that
a∞n1γn∞,
blim supn→ ∞δn/γn≤0or∞n1|δn|<∞.
Thenlimn→ ∞αn0.
Lemma 2.6see26. Let{xn}and{yn}be bounded sequences in a Banach spaceEand{βn}a
sequence in0,1with0 <lim infn→ ∞βn ≤lim supn→ ∞βn<1. Suppose thatxn1 1−βnyn
βnxnfor alln≥0and
lim sup
n→ ∞
yn1−yn− xn1−xn
≤0. 2.4
Thenlimn→ ∞yn−xn0.
Definition 2.7see27. Let{Sn}be a family of mappings from a subsetCof a Banach spaceE
intoEwith∞n1FSn/∅. We say that{Sn}satisfies the AKTT-condition if, for each bounded
subsetBofC,
∞
n1 sup
z∈B
Sn1z−Snz<∞. 2.5
Remark 2.8. The example of the sequence of mappings{Sn} satisfying AKTT-condition is
supported byExample 3.11.
Lemma 2.9see27, Lemma 3.2. Suppose that {Sn}satisfies AKTT-condition. Then, for each
y∈C,{Sny}converses strongly to a point inC. Moreover, let the mappingSbe defined by
Sy lim
n→ ∞Sny, ∀y∈C. 2.6
Lemma 2.10see28. LetEbe a real2-uniformly smooth Banach space andT :E → Eaλ-strict
pseudocontraction. ThenS: 1−λ/K2Iλ/K2Tis nonexpansive andFT FS.
Lemma 2.11see29. LetEbe a real2-uniformly smooth Banach space with the best smoothness
constantK. Then the following inequality holds:
xy2 ≤
x22y, Jx2Ky2, ∀x, y∈E. 2.7
Lemma 2.12see17, Lemma 1.8. Assume thatAis a strongly positive linear bounded operator
on a smooth Banach spaceEwith coefficientγ >0and0< ρ≤ A−1.ThenI−ρA ≤1−ργ.
3. Main Results
In this section, we prove that the strong convergence theorem for a countable family of uniformlyε-strict pseudocontractions in a strictly convex and 2-uniformly smooth Banach space admits a weakly sequentially continuous duality mapping. Before proving it, we need the following theorem.
Theorem 3.1see17, Lemma 1.9. LetCbe a nonempty closed convex subset of a reflexive, smooth
Banach spaceEwhich admits a weakly sequentially continuous duality mappingJfromEtoE∗. Let
T :C → Cbe a nonexpansive mapping such thatFTis nonempty, letf:C → Cbe a contraction
with coefficientα ∈ 0,1, and letAbe a strongly positive bounded linear operator with coefficient
γ >0and0< γ < γ/α. Then the net{xt}defined by
xttγfxt 1−tATxt 3.1
converges strongly ast → 0to a fixed pointxofT which solves the variational inequality:
A−γfx, J x−z ≤0, ∀z∈FT. 3.2
Lemma 3.2. LetCbe a nonempty closed convex subset of a real2-uniformly smooth Banach spaceE
with the smoothness constantK. LetΨ:C → Ebe anLΨ-Lipschitzian and relaxedc, d-cocoercive
mapping. Then
I−λΨx−I−λΨy2≤12 λcL2
Ψ−2λd2λ2K2L2Ψ
x−y2. 3.3
Ifλ≤d−cL2
Proof. UsingLemma 2.11and the cocoercivity of the mappingΨ, we have, for allx, y∈C,
I−λΨx−I−λΨy2
x−y−λΨx−λΨy2
x−y2−2λΨx−Ψy, Jx−y2λ2K2Ψx−Ψy2
≤x−y2−2λ−cΨx−Ψy2dx−y22λ2K2Ψx−Ψy2
x−y2−2λdx−y22λcΨx−Ψy22λ2K2Ψx−Ψy2
≤12λcL2Ψ−2λd2λ2K2L2Ψx−y2.
3.4
Hence3.3is proved. Assume thatλ≤d−cL2Ψ/K2L2
Ψ. Then, we have12λcL2Ψ−2λd 2λ2K2L2
Ψ≤1. This together with3.3implies thatI−λΨis nonexpansive.
Lemma 3.3. Let E be a strictly convex and 2-uniformly smooth Banach space admiting a weakly
sequentially continuous duality mapping with the smoothness constant K. Let Mi : E → 2E be
a maximal monotone mapping andΨi : E → E aLi-Lipschitzian and relaxed ci, di-cocoercive
mapping withρi ∈ 0,di −ciL2i/K2Li2, respectively, for each i 1,2. Let {Tn : E → E}∞n1
be a countable family of uniformlyε-strict pseudocontractions. Define a mappingSn : E → Eand
Gn:E → Eby
Snx
1− ε
K2
x ε
K2Tnx, ∀x∈C, n≥1,
GnμSn
1−μQ,
3.5
where Qis defined as in Lemma 1.8. Assume that Ω : ∞n1FTn∩FQ/∅. Letf : E → E
be anα-contraction; letA :E → Ebe a strongly positive linear bounded self-adjoint operator with
coefficientγwith0< γ < γ/α. Then the following hold.
iFor eachn∈N,Gnis nonexpansive such that
FGn FSn∩FQ FTn∩FQ. 3.6
iiSuppose that{Gn}satisfies AKTT-condition. LetG : E → Ebe the mapping defined by
Gy limn→ ∞Gny for ally ∈ Eand suppose that FG
∞
n1FGn. The net{xt}
defined byxttγfxt I−tAGxtconverges strongly ast → 0to a fixed pointxofG,
which solves the variational inequality
A−γfx, J x−z ≤0, ∀z∈FG 3.7
andx, yis a solution of general system of variational inequality problem1.11such that
Proof .It follows fromLemma 2.10thatSnis nonexpansive such thatFSn FTnfor each
n∈N. Next, we prove thatQis nonexpansive. Indeed, we observe that
Qx JM1,ρ1
JM2,ρ2
x−ρ2Ψ2x
−ρ1Ψ1JM2,ρ2
x−ρ2Ψ2x
JM1,ρ1
I−ρ1Ψ1
JM2,ρ2
I−ρ2Ψ2
x. 3.8
The nonexpansivity ofJM1,ρ1,JM2,ρ2,I−ρ1Ψ1, andI−ρ2Ψ2implies thatQis nonexpansive.
ByLemma 2.3, we have thatGnis nonexpansive such that
FGn FSn∩FQ FTn∩FQ/∅, ∀n∈N. 3.9
Henceiis proved. It is well known that, ifEis uniformly smooth, thenEis reflexive. Hence
Theorem 3.1implies that{xt} converges strongly ast → 0 to a fixed point xof G, which
solves the variational inequality
A−γfx, J x−z ≤0, ∀z∈FG, 3.10
andx, yis a solution of problem1.11, wherey JM2,ρ2x−ρ2Ψ2x. This completes the
proof ofii.
Theorem 3.4. LetEbe a strictly convex and2-uniformly smooth Banach space which admits a weakly
sequentially continuous duality mapping and has the smoothness constantK. LetMi :E → 2Ebe
a maximal monotone mapping andΨi : E → E aLi-Lipschitzian and relaxed ci, di-cocoercive
mapping withρi ∈0,di−ciL2i/K2Li2, respectively, for eachi 1,2. Let{Tn :E → E}∞n1be a
countable family of uniformlyε-strict pseudocontractions. Define a mappingSn:E → Eby
Snx
1− ε
K2
x ε
K2Tnx, ∀x∈C, n≥1. 3.11
Assume that Ω : ∞n1FTn∩FQ/∅, where Qis defined as in Lemma 1.8. Letf : E → E
be anα-contraction; letA:E → Ebe a strongly positive linear bounded self adjoint operator with
coefficientγwith0< γ < γ/α. Letx1u∈Eand let{xn}be a sequence generated by
znJM2,ρ2
xn−ρ2Ψ2xn
,
ynJM1,ρ1
zn−ρ1Ψ1zn
,
xn1αnγfxn βnxn
1−βn
I−αnA
μSnxn
1−μyn
, ∀n≥1,
3.12
where μ ∈ 0,1, and {αn} and {βn} are sequences in0,1. Suppose that {Sn} satisfies
AKTT-condition. LetS : E → Ebe the mapping defined bySy limn→ ∞Snyfor ally ∈Eand suppose
thatFS ∞n1FSn. If the control consequences{αn}and{βn}satisfy the following restrictions
C10<lim infn→ ∞βn≤lim supn→ ∞βn<1,
then{xn}converges strongly tox, which solves the variational inequality
A−γfx, J x−z ≤0, z∈Ω, 3.13
and x, yis a solution of general system of variational inequality problem 1.11 such thaty
JM2,ρ2x−ρ2Ψ2x.
Proof. First, we show that sequences{xn},{yn}, and{zn}are bounded.
By the control conditionC2, we may assume, with no loss of generality, thatαn ≤
1−βnA−1.
SinceAis a linear bounded operator onE, by1.27, we have
Asup{|Au, Ju|:u∈E,u1}. 3.14
Observe that
1−βn
I−αnA
u, Ju 1−βn−αnAu, Ju ≥1−βn−αnA ≥0. 3.15
It follows that
1−βn
I−αnAsup
1−βn
I−αnA
u, Ju :u∈E,u1
sup1−βn−αnAu, Ju:u∈E,u1
≤1−βn−αnγ.
3.16
Therefore, takingx∈Ω,one has
xJM1,ρ1
JM2,ρ2
x−ρ2Ψ2x
−ρ1Ψ1JM2,ρ2
x−ρ2Ψ2x
. 3.17
PuttingyJM2,ρ2x−ρ2Ψ2x,one sees that
xJM1,ρ1
y−ρ1Ψ1y
. 3.18
It follows from Lemmas2.1and3.2that zn−yJM
2,ρ2
xn−ρ2Ψ2xn
−JM2,ρ2
x−ρ2Ψ2x
≤xn−ρ2Ψ2xn
−x−ρ2Ψ2x
≤ xn−x.
This implies that
yn−xJM
1,ρ1
zn−ρ1Ψ1zn
−JM1,ρ1
y−ρ1Ψ1y
≤zn−ρ1Ψ1zn
−y−ρ1Ψ1y
≤zn−y
≤ xn−x.
3.20
Settingtn μSnxn 1−μynand applyingLemma 2.10, we have thatSnis a nonexpansive
mapping such thatFSn FTnfor alln≥1 and hence
∞
n1FSn
∞
n1FTn. Then
tn−xμSnxn
1−μyn−x
≤μSnxn−x
1−μyn−x
≤ xn−x.
3.21
It follows from the last inequality that
xn1−xαnγfxn βnxn
1−βn
I−αnA
tn−x
αn
γfxn−Ax
βnxn−x
1−βn
I−αnA
tn−x
≤1−βn−αnγ
xn−xβnxn−xαnγfxn−Ax
≤1−αnγ
xn−xαnγαxn−xαnγfx−Ax
1−αn
γ−γαxn−xαnγfx−Ax.
3.22
By induction, we have
xn−x ≤max
x1−x,
γfx−Ax
γ−γα
, n≥1. 3.23
This shows that the sequence{xn}is bounded, and so are{yn},{zn}, and{tn}.
On the other hand, from the nonexpansivity of the mappingsJM2,ρ2, one sees that
yn1−ynJM
1,ρ1
zn1−ρ1Ψ1zn1
−JM1,ρ1
zn−ρ1Ψ1zn
≤zn1−ρ1Ψ1zn1
−zn−ρ1Ψ1zn
≤ zn1−zn.
3.24
In a similar way, one can obtain that
It follows that
yn1−yn≤ xn1−xn. 3.26
This implies that
tn1−tnμSn1xn1
1−μyn1−
μSnxn
1−μyn
μSn1xn1−μSn1xn
1−μyn1μSn1xn−μSnxn−
1−μyn
≤μSn1xn1−Sn1xn
1−μyn1−ynμSn1xn−Snxn
≤μxn1−xn
1−μxn1−xnμsup z∈{xn}
Sn1z−Snz
xn1−xnμsup z∈{xn}
Sn1z−Snz.
3.27
Setting
xn1
1−βn
enβnxn, ∀n≥1, 3.28
one sees that
en1−en
αn1γfxn1
1−βn1
I−αn1A
tn1 1−βn1 −
αnγfxn
1−βn
I−αnA
tn
1−βn
αn1 1−βn1
γfxn1−Atn1
tn1− 1−αn βn
γfxn−Atn
−tn,
3.29
and so it follows that
en1−en ≤ αn1
1−βn1
γfxn1−Atn1 αn
1−βn
γfxn−Atntn1−tn, 3.30
which, combined, with3.27yields that
en1−en − xn1−xn ≤ 1−αn1
βn1
γfxn1−Atn1 αn
1−βn
γfxn−Atn
μsup
z∈{xn}
Sn1z−Snz.
3.31
Using the conditionsC1andC2and AKTT-condition of{Sn}, we have
lim sup
Hence, fromLemma 2.6, it follows that
lim
n→ ∞en−xn0. 3.33
From3.28, it follows that
xn1−xn
1−βn
en−xn. 3.34
By3.33, one sees that
lim
n→ ∞xn1−xn0. 3.35
On the other hand, one has
xn1−xnαn
γfxn−Axn
1−βn
I−αnA
tn−xn. 3.36
It follows that
1−βn−αnγ
tn−xn ≤ xn−xn1αnγfxn−Axn. 3.37
From the conditionsC1,C2and from3.35, one sees that
lim
n→ ∞tn−xn0. 3.38
Define the mappingGnby
GnμSn
1−μQ, 3.39
whereQis defined as inLemma 1.8. FromLemma 3.3i, we see thatGnis nonexpansive such
that
FGn FTn∩FQ FSn∩FQ. 3.40
From3.38, it follows that
lim
n→ ∞Gnxn−xn0. 3.41
Since {Sn} satisfies AKTT-condition and S : E → E is the mapping defined by Sy
limn→ ∞Sny for all y ∈ E, we have that {Gn} satisfies AKTT-condition. Let the mapping
G : E → Ebe the mapping defined by Gy limn→ ∞Gny for ally ∈ E. It follows from
the nonexpansivity ofSand
thatGis nonexpansive such that
FG FS∩FQ
∞
n1
FSn∩FQ
∞
n1
FTn∩FQ
∞
n1
FGn. 3.43
Next, we prove that
lim sup
n→ ∞ γfx−Ax, J xn−x ≤0, 3.44
wherexlimt→0xtwithxtbe the fixed point of the contraction
x−→tγfx I−tAGx. 3.45
Thenxtsolves the fixed point equationxttγfxtI−tAGxt. It follows fromLemma 3.3ii
thatx∈FG Ω, which solves the variational inequality:
A−γfx, J x−z ≤0, ∀z∈FG, 3.46
andx, yis a solution of general system of variational inequality problem1.11such that
yJM2,ρ2x−ρ2Ψ2x. Let{xnk}be a subsequence of{xn}such that
lim
k→ ∞γfx−Ax, J xnk −xlim sup
n→ ∞ γfx−Ax, J xn−x. 3.47
If follows from reflexivity ofEand the boundedness of sequence{xnk}that there exists{xnki} which is a subsequence of{xnk}converging weakly tow∈Casi → ∞. It follows from3.41 and the nonexpansivity ofG, we havew ∈FGbyLemma 2.4. Since the duality mapJ is single valued and weakly sequentially continuous fromEtoE∗, we get that
lim sup
n→ ∞
γfx−Ax, J xn−x lim
k→ ∞γfx−Ax, J xnk−x
lim
i→ ∞
γfx−Ax, J xnki −x
A−γfx, J x−w ≤0
as required. Now fromLemma 2.11, we have
xn1−x2αnγfxn βnxn
1−βn
I−αnA
tn−x2
1−βn
I−αnA
tn−x αnγfxn−Ax βnxn−x2
≤1−βn−αnγ
2
tn−x22
αn
γfxn−Ax
βnxn−x, Jxn1−x
1−βn−αnγ
2
tn−x22βnxn−x, J xn1−x
2αn
γfxn−Ax, J xn1−x
1−βn−αnγ
2
tn−x22βnxn−x, J xn1−x
2αnγfxn−γfx, Jxn1−x2αn
γfx−Ax, J xn1−x
≤1−βn−αnγ
2
tn−x22βnxn−xxn1−x
2αnγfxn−γfxxn1−x2αnγfx−Ax, J xn1−x
≤1−βn−αnγ
2
xn−x2βn
xn1−x2xn−x2
αnγα
xn1−x2xn−x2
2αnγfx−Ax, J xn1−x
1−βn−αnγ
2
βnαnγα
xn−x2
βnαnγα
xn1−x2
2αn
γfx−Ax, J xn1−x ,
3.49
which implies that
xn1−x2≤
1−βn−αnγ
2
βnαnγα
1−βn−αnγα xn−x
2
2αn
1−βn−αnγαγfx−Ax, J xn1−x
1− 2αn
γ−γα
1−βn−αnγα
xn−x2
β2
n2βnαnγα2nγ2
1−βn−αnγα xn−x
2
2αn
1−βn−αnγαγfx−Ax, J xn1−x
1− 2αn
γ−γα
1−βn−αnγα
xn−x2
2αn
γ−γα
1−βn−αnγα
β2
n2βnαnγα2nγ2
2αn
γ−γα M3
1
γ−γα
γfx−Ax, J xn1−x
,
whereM3is an appropriate constant such thatM3≥supn≥0xn−x2. Put
jn
2αn
γ−γα
1−βn−αnγα, kn
β2n2βnαnγα2nγ2
2αn
γ−γα M3
1
γ−γαγfx−Ax, J xn1−x,
3.51
that is,
xn1−x2≤
1−jn
xn−x2jnkn. 3.52
It follows from conditionsC1,C2and from3.44that
lim
n→ ∞jn0, ∞
n1
jn∞, lim sup n→ ∞
kn≤0. 3.53
ApplyLemma 2.5to3.52to conclude thatxn → xasn → ∞. This completes the proof.
SettingA≡I, γ1, andf :u, we have the following result.
Theorem 3.5. LetEbe a strictly convex and2-uniformly smooth Banach space which admits a weakly
sequentially continuous duality mapping and has the smoothness constantK. LetMi :E → 2Ebe
a maximal monotone mapping andΨi : E → E aLi-Lipschitzian and relaxed ci, di-cocoercive
mapping withρi ∈0,di−ciL2i/K2Li2, respectively, for eachi 1,2. Let{Tn :E → E}∞n1be a
countable family of uniformlyε-strict pseudocontractions. Define a mappingSn:E → Eby
Snx
1− ε
K2
x ε
K2Tnx, ∀x∈C, n≥1. 3.54
Assume thatΩ:∞n1FTn∩FQ/∅, whereQis defined as inLemma 1.8. Letx1 u∈Eand
let{xn}be a sequence generated by
znJM2,ρ2
xn−ρ2Ψ2xn
,
ynJM1,ρ1
zn−ρ1Ψ1zn
,
xn1αnuβnxn
1−βn−αn
μSnxn
1−μyn
, ∀n≥1,
3.55
where μ ∈ 0,1, and {αn} and {βn} are sequences in0,1. Suppose that {Sn} satisfies
AKTT-condition. LetS : E → Ebe the mapping defined bySy limn→ ∞Snyfor ally ∈Eand suppose
thatFS ∞n1FSn. If the control consequences{αn}and{βn}satisfy the following restrictions
C10<lim infn→ ∞βn≤lim supn→ ∞βn<1,
C2limn→ ∞αn0and
∞
n1αn∞,
then{xn}converges strongly tox, which solves the variational inequality
and x, yis a solution of general system of variational inequality problem 1.11 such thaty JM2,ρ2x−ρ2Ψ2x.
Remark 3.6. Theorem 3.4mainly improves Theorem 2.1 of Qin et al. 16, in the following
respects:
afrom the class of inverse-strongly accretive mappings to the class of Lipchitzian and relaxed cocoercive mappings,
bfrom a ε-strict pseudocontraction to the countable family of uniformly ε-strict pseudocontractions,
cfrom a uniformly convex and 2-uniformly smooth Banach space to a strictly convex and 2-uniformly smooth Banach space which admits a weakly sequentially continuous duality mapping.
Further, if {Tn : E → E} is a countable family of nonexpansive mappings, then
Theorem 3.4is reduced to the following result.
Theorem 3.7. LetEbe a strictly convex and2-uniformly smooth Banach space which admits a weakly
sequentially continuous duality mapping and has the smoothness constantK. LetMi :E → 2Ebe
a maximal monotone mapping andΨi : E → E aLi-Lipschitzian and relaxed ci, di-cocoercive
mapping withρi ∈ 0, di−ciL2i/K2L2i, respectively, for eachi 1,2. Let{Tn : E → E}∞n1 be a
countable family of nonexpansive mappings. Assume thatΩ: ∞n1FTn∩FQ/∅, whereQis
defined as inLemma 1.8. Letf :E → Ebe anα-contraction; letA:E → Ebe a strongly positive
linear bounded self adjoint operator with coefficientγwith0< γ < γ/α. Letx1u∈Eand let{xn}
be a sequence generated by
znJM2,ρ2
xn−ρ2Ψ2xn
,
ynJM1,ρ1
zn−ρ1Ψ1zn
,
xn1 αnγfxn βnxn
1−βn
I−αnA
μTnxn
1−μyn
, ∀n≥1,
3.57
where μ ∈ 0,1, and{αn} and {βn} are sequences in 0,1. Suppose that {Tn} satisfies
AKTT-condition. LetT :E → Ebe the mapping defined byTy limn→ ∞Tnyfor ally ∈Eand suppose
thatFT ∞n1FTn. If the control consequences{αn}and{βn}satisfy the following restrictions
C10<lim infn→ ∞βn≤lim supn→ ∞βn<1,
C2limn→ ∞αn0and
∞
n1αn∞,
then{xn}converges strongly toxwhich solves the variational inequality:
A−γfx, J x−z ≤0, z∈Ω, 3.58
and x, yis a solution of general system of variational inequality problem 1.11 such thaty
JM2,ρ2x−ρ2Ψ2x.
Remark 3.8. As in 27, Theorem 4.1, we can generate a sequence {Tn} of nonexpansive
mappings satisfying AKTT-condition; that is, ∞n1sup{Tn1z−Tnz : z ∈ B} < ∞ for
nonexpansive mappings with a common fixed point. To be more precise, they obtained the following lemma.
Lemma 3.9see27. LetCbe a closed convex subset of a smooth Banach spaceE. Suppose that
{Sk}is a sequence of nonexpansive mappings of Einto inself with a common fixed point. For each
n∈N, defineTn:C → Cby
Tnx n
k1
βknSkx, ∀x∈E, 3.59
where{βk
n}is a family of nonnegative numbers with indicesn, k∈Nwithk≤nsuch that
ink1βk
n 1for alln∈N,
iilimn→ ∞βkn>0for everyk∈N,
iii∞n1nk1|βkn1−βk n|<∞.
Then the following are given.
1EachTnis a nonexpansive mapping.
2{Tn}satisfies AKTT-condition.
3IfT :C → Cis defined by
Tx
∞
k1
βnkSkx, ∀x∈C, 3.60
thenTxlimn→ ∞TnxandFT
∞
n1FTn
∞
k1FSk.
Theorem 3.10. Let Ebe a strictly convex and 2-uniformly smooth Banach space which admits a
weakly sequentially continuous duality mapping and has the smoothness constantK. LetMi :E →
2Ebe a maximal monotone mapping andΨ
i:E → EaLi-Lipschitzian and relaxedci, di-cocoercive
mapping withρi ∈0,di−ciL2i/K2Li2, respectively, for eachi1,2. Let{Sk :E → E}∞k1be a
countable family of nonexpansive mappings. Assume thatΩ :∞k1FSk∩FQ/∅, whereQis
defined as inLemma 1.8. Letf :E → Ebe anα-contraction; letA:E → Ebe a strongly positive
linear bounded self adjoint operator with coefficientγwith0< γ < γ/α. Letx1u∈Eand let{xn}
be a sequence generated by
znJM2,ρ2
xn−ρ2Ψ2xn
,
ynJM1,ρ1
zn−ρ1Ψ1zn
,
xn1αnγfxn βnxn
1−βn
I−αnA
μ
n
k1
βknSkxn
1−μyn
, ∀n≥1,
3.61
where{βk
n}satisfies conditions (i)–(iii) ofLemma 3.9,μ ∈ 0,1, and{αn}and{βn}are sequences
in 0,1. Suppose that{Tn}satisfies AKTT-condition. LetT : E → Ebe the mapping defined by
{αn}and{βn}satisfy the following restrictions:
C10<lim infn→ ∞βn≤lim supn→ ∞βn<1,
C2limn→ ∞αn0and
∞
n1αn∞,
then{xn}converges strongly tox, which solves the variational inequality
A−γfx, J x−z ≤0, z∈Ω, 3.62
and x, yis a solution of general system of variational inequality problem 1.11 such thaty
JM2,ρ2x−ρ2Ψ2x.
Proof. We write the iteration3.61as
znJM2,ρ2
xn−ρ2Ψ2xn
,
ynJM1,ρ1
zn−ρ1Ψ1zn
,
xn1 αnγfxn βnxn
1−βn
I−αnA
μTnxn
1−μyn
, ∀n≥1,
3.63
whereTnis defined by3.59. It is clear that each mappingTnis nonexpansive. ByTheorem 3.7
andLemma 3.9, the conclusion follows.
The following example appears in 27 shows that there exists {βk
n} satisfying the
conditions ofLemma 3.9.
Example 3.11. Let{βk
n}be defined by
βk n
⎧ ⎨ ⎩
2−k k < n,
21−k kn, 3.64
for alln, k∈Nwithk≤ n.In this case, the sequence{Tn}of mappings generated by{Sk}is
defined as follows: Forx∈C.
T1xS1x,
T2x 1 2S1x
1 2S2x,
T3x 1 2S1x
1 4S2x
1 4S3x,
T4x 1 2S1x
1 4S2x
1 8S3x
1 8S4x, ..
.
Tnx 1
2S1x 1 4S2x
1 8S3x
1
16S4x· · · 1
2n−1Sn−1x 1 2n−1Snx.
Acknowledgments
The first author is supported under grant from the program Strategic Scholarships for Frontier Research Network for the Ph.D. Program Thai Doctoral degree from the Office of the Higher Education Commission, Thailand and the second author is supported by the “Centre of Excellence in Mathematics” under the Commission on Higher Education, Ministry of Education, Thailand. Finally, The authors would like to thank the referees for reading this paper carefully, providing valuable suggestions and comments, and pointing out a major error in the original version of this paper.
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