Volume 2011, Article ID 174796,31pages doi:10.1155/2011/174796
Research Article
Hybrid Proximal-Point Methods for
Zeros of Maximal Monotone Operators, Variational
Inequalities and Mixed Equilibrium Problems
Kriengsak Wattanawitoon
1, 2and Poom Kumam
2, 31Department of Mathematics and Statistics, Faculty of Science and Agricultural Technology, Rajamangala University of Technology Lanna Tak, Tak 63000, Thailand
2Centre of Excellence in Mathematics, CHE, Si Ayuthaya Road, Bangkok 10400, Thailand 3Department of Mathematics, Faculty of Science, King Mongkut’s University of
Technology Thonburi (KMUTT), Bang Mod, Bangkok 10140, Thailand
Correspondence should be addressed to Poom Kumam,[email protected]
Received 15 November 2010; Accepted 29 December 2010
Academic Editor: Yonghong Yao
Copyrightq2011 K. Wattanawitoon and P. Kumam. 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.
We prove strong and weak convergence theorems of modified hybrid proximal-point algorithms for finding a common element of the zero point of a maximal monotone operator, the set of solutions of equilibrium problems, and the set of solution of the variational inequality operators of an inverse strongly monotone in a Banach space under different conditions. Moreover, applications to complementarity problems are given. Our results modify and improve the recently announced ones by Li and Song2008and many authors.
1. Introduction
LetEbe a Banach space with norm·,Ca nonempty closed convex subset ofE, letE∗denote the dual ofEand<·,·>is the pairing betweenEandE∗.
Consider the problem of finding
v∈E such that 0∈Tv, 1.1
whereT is an operator fromEintoE∗. Suchv ∈ Eis called a zero point of T. When T is a maximal monotone operator, a well-known method for solving1.1in a Hilbert spaceHis the proximal point algorithmx1x∈Hand
where{rn} ⊂0,∞andJrn IrnT−1, then Rockafellar1proved that the sequence{xn}
converges weakly to an element ofT−10.
In 2000, Kamimura and Takahashi 2 proved the following strong convergence theorem in Hilbert spaces, by the following algorithm:
xn1 αnx 1−αnJrnxn, n1,2,3, . . . , 1.3
whereJr IrT−1J, then the sequence{xn}converges strongly toPT−10x, wherePT−10
is the projection fromHontoT−10. These results were extended to more general Banach
spaces see3,4.
In 2004, Kohsaka and Takahashi4introduced the following iterative sequence for a maximal monotone operatorT in a smooth and uniformly convex Banach space:x1 x∈E
and
xn1J−1αnJx 1−αnJJrnxn, n1,2,3, . . . , 1.4
whereJis the duality mapping fromEintoE∗andJr IrT−1J.
Recently, Li and Song5proved a strong convergence theorem in a Banach space, by the following algorithm:x1 x∈Eand
ynJ−1βnJxn 1−βnJJrnxn,
xn1J−1αnJx1 1−αnJyn, 1.5
with the coefficient sequences{αn},{βn} ⊂0,1and{rn} ⊂0,∞satisfying limn→ ∞αn 0,
∞
n1αn∞, limn→ ∞βn0, and limn→ ∞rn∞. WhereJis the duality mapping fromEinto
E∗andJr IrT−1J. Then, they proved that the sequence{xn}converges strongly toΠ
Cx,
whereΠCis the generalized projection fromEontoC.
LetCbe a nonempty closed convex subset ofE, and letAbe a monotone operator ofC intoE∗. The variational inequality problem is to find a pointx∗∈Csuch that
v−x∗, Ax∗ ≥0, ∀v∈C. 1.6
The set of solutions of the variational inequality problem is denoted by VIC, A. Such a problem is connected with the convex minimization problem, the complementarity problem, the problem of finding a pointu∈Esatisfying 0Au, and so on. An operatorAofCintoE∗ is said to be inverse-strongly monotone if there exists a positive real numberαsuch that
x−y, Ax−Ay≥αAx−Ay2, 1.7
for allx, y∈C. In such a case,Ais said to beα-inverse-strongly monotone. If an operatorAof
CintoE∗isα-inverse-strongly monotone, thenAis Lipschitz continuous, that is,Ax−Ay ≤
In a Hilbert spaceH, Iiduka et al.6proved that the sequence{xn}defined by:x1
x∈Cand
xn1PCxn−λnAxn, 1.8
wherePCis the metric projection ofHontoCand{λn}is a sequence of positive real numbers,
converges weakly to some element of VIC, A.
In 2008, Iiduka and Takahashi7introduced the folowing iterative scheme for finding a solution of the variational inequality problem for an inverse-strongly monotone operatorA in a Banach spacex1x∈Cand
xn1 ΠCJ−1Jxn−λnAxn, 1.9
for everyn1,2,3, . . ., whereΠC is the generalized metric projection fromEontoC,Jis the
duality mapping fromEintoE∗and{λn}is a sequence of positive real numbers. They proved that the sequence{xn}generated by1.9converges weakly to some element of VIC, A.
LetΘbe a bifunction ofC×CintoRandϕ:C → Ra real-valued function. The mixed
equilibrium problem, denoted by MEPΘ, ϕ, is to findx∈Csuch that
Θx, yϕy−ϕx≥0, ∀y∈C. 1.10
Ifϕ≡0, the problem1.10reduces into the equilibrium problem forΘ, denoted by EPΘ, is to findx∈Csuch that
Θx, y≥0, ∀y∈C. 1.11
IfΘ≡ 0, the problem1.10reduces into the minimize problem, denoted by Argminϕ, is to findx∈Csuch that
ϕy−ϕx≥0, ∀y∈C. 1.12
Recall, a mappingS:C → Cis said to be nonexpansive if
Sx−Sy≤x−y, 1.13
for all x, y ∈ C. We denote byFS the set of fixed points of S. If C is bounded closed convex andSis a nonexpansive mapping ofCinto itself, thenFSis nonemptysee22. A mapping S is said to be quasi-nonexpansive if FS/∅ and Sx −y ≤ x− y for all
x∈Candy ∈FS. It is easy to see that ifSis nonexpansive withFS/∅, then it is quasi-nonexpansive. We writexn → xxn x, resp.if{xn}convergesweakly, resp.tox. LetE
be a real Banach space with norm · and letJbe the normalized duality mapping fromEinto 2E∗given by
Jx{x∗∈E∗:x, x∗ xx∗,xx∗}, 1.14
for allx∈E, whereE∗denotes the dual space ofEand·,· the generalized duality pairing between E and E∗. It is well known that if E∗ is uniformly convex, then J is uniformly continuous on bounded subsets ofE.
LetCbe a closed convex subset ofE, and letSbe a mapping fromCinto itself. A point
pinCis said to be an asymptotic fixed point ofS23ifCcontains a sequence{xn}which converges weakly topsuch that limn→ ∞xn−Sxn0. The set of asymptotic fixed points of
Swill be denoted byFS. A mappingSfromCinto itself is said to be relatively nonexpansive
24–26if FS FSandφp, Sx ≤ φp, xfor allx ∈ Cand p ∈ FS. The asymptotic behavior of a relatively nonexpansive mapping was studied in 27,28.S is said to beφ -nonexpansive, ifφSx, Sy≤φx, yforx, y∈C.Sis said to be relatively quasi-nonexpansive ifFS/∅andφp, Sx≤φp, xforx∈Candp∈FS.
In 2009, Takahashi and Zembayashi29introduced the following shrinking projection method of closed relatively nonexpansive mappings as follows:
x0x∈C, C0C,
ynJ−1αnJxn 1−αnJSxn,
un∈C such thatΘun, yr1 n
y−un, Jun−Jyn≥0, ∀y∈C,
Cn1z∈Cn:φz, un≤φz, xn ,
xn1 ΠCn1x,
1.15
for every n ∈ N ∪ {0}, where J is the duality mapping on E, {αn} ⊂ 0,1 satisfies lim infn→ ∞αn1−αn > 0 and{rn} ⊂ a,∞ for some a > 0. Then, they proved that the
In 2009, Qin et al.30modified the Halpern-type iteration algorithm for closed
quasi-φ-nonexpansive mappingsor relatively quasi-nonexpansivedefined by
x0 ∈Echosen arbitrarily,
C1C,
x1 ΠC1x0,
ynJ−1αnJx1 1−αnJTxn,
Cn1z∈Cn:φz, yn≤αnφz, x1 1−αnφz, xn ,
xn1 ΠCn1x1, ∀n≥1.
1.16
Then, they proved that under appropriate control conditions the sequence{xn}converges strongly toΠFTx1.
Recently, Ceng et al.31proved the following strong convergence theorem for finding a common element of the set of solutions for an equilibrium and the set of a zero point for a maximal monotone operatorTin a Banach spaceE
ynJ−1αnJx0 1−αn
βnJxn1−βnJJrnxn,
Hnz∈C:φz, Trnyn≤αnφz, x0 1−αnφz, xn ,
Wn {z∈C:xn−z, Jx0−Jxn ≥0},
xn1 ΠHn∩Wnx0.
1.17
Then, the sequence {xn} converges strongly to ΠT−10∩EPΘx0, where ΠT−10∩EPΘ is the
generalized projection ofEontoT−10∩EPΘ.
In this paper, motivated and inspired by Li and Song5, Iiduka and Takahashi7, Takahashi and Zembayashi 29, Ceng et al. 31 and Qin et al. 30, we introduce the following new hybrid proximal-point algorithms defined byx1x∈C:
wn ΠCJ−1Jxn−λnAxn,
znJ−1βnJxn 1−βnJJrnwn,
ynJ−1αnJx1 1−αnJzn,
un∈C such thatΘun, yϕy−ϕun r1 n
y−un, Jun−Jyn≥0, ∀y∈C,
Cn1z∈Cn :φz, un≤αnφz, x1 1−αnφz, xn ,
xn1 ΠCn1x
and
un∈C such thatΘun, yϕy−ϕun r1 n
y−un, Jun−Jyn≥0, ∀y∈C,
zn ΠCJ−1Jun−λnAun,
ynJ−1βnJxn 1−βnJJrnzn,
xn1 ΠCJ−1αnJx1 1−αnJyn.
1.19
Under appropriate conditions, we will prove that the sequence{xn}generated by algorithms
1.18 and 1.19 converges strongly to the point ΠVIC,A∩T−10∩MEPΘ,ϕx and converges
weakly to the point limn→ ∞ΠVIC,A∩T−10∩MEPΘ,ϕxn, respectively. The results presented in
this paper extend and improve the corresponding ones announced by Li and Song5and many authors in the literature.
2. Preliminaries
A Banach spaceEis said to be strictly convex ifxy/2 < 1 for allx, y ∈Ewithx
y1 andx /y. LetU{x∈E:x1}be the unit sphere ofE. Then, the Banach space
Eis said to be smooth provided
lim
t→0
xty− x
t 2.1
exists for eachx, y∈U. It is also said to be uniformly smooth if the limit is attained uniformly forx, y∈E. The modulus of convexity ofEis the functionδ:0,2 → 0,1defined by
δε inf
1−xy 2
:x, y∈E,xy1,x−y≥ε
. 2.2
A Banach spaceEis uniformly convex if and only ifδε>0 for allε∈0,2. Letpbe a fixed real number withp ≥ 2. A Banach spaceEis said to be p-uniformly convex if there exists a
constantc >0 such thatδε ≥ cεpfor allε ∈ 0,2; see32,33for more details. Observe that everyp-uniform convex is uniformly convex. One should note that no Banach space is
p-uniform convex for 1< p <2. It is well known that a Hilbert space is 2-uniformly convex and uniformly smooth. For eachp >1, the generalized duality mappingJp:E → 2E∗is defined
by
Jpx
x∗∈E∗:x, x∗ xp,x∗xp−1, 2.3
for allx∈E. In particular,JJ2is called the normalized duality mapping. IfEis a Hilbert space,
We know the followingsee34:
1ifEis smooth, thenJis single-valued,
2if Eis strictly convex, thenJ is one-to-one andx−y, x∗−y∗ > 0 holds for all
x, x∗,y, y∗∈Jwithx /y,
3ifEis reflexive, thenJis surjective,
4ifEis uniformly convex, then it is reflexive,
5if E∗ is uniformly convex, thenJ is uniformly norm-to-norm continuous on each bounded subset ofE.
The dualityJ from a smooth Banach spaceEintoE∗ is said to be weakly sequentially
continuous35ifxn ximpliesJxn∗Jx, where∗implies the weak∗convergence.
Lemma 2.1see36,37. IfEbe a 2-uniformly convex Banach space. Then, for allx, y∈Eone has
x−y≤ 2
c2Jx−Jy, 2.4
whereJis the normalized duality mapping ofEand 0< c≤1.
The best constant 1/cin Lemma is called the 2-uniformly convex constant ofE; see
32.
Lemma 2.2see36,38. IfEa p-uniformly convex Banach space and letpbe a given real number
withp≥2. Then, for allx, y∈E,Jx∈JpxandJy ∈Jpy
x−y, Jx−Jy≥ cp
2p−2px−y
p,
2.5
whereJpis the generalized duality mapping ofEand 1/cis the p-uniformly convexity constant ofE.
Lemma 2.3see Xu37. LetEbe a uniformly convex Banach space. Then, for eachr > 0, there
exists a strictly increasing, continuous, and convex functionK:0,∞ → 0,∞such thatK0 0
and
λx1−λy2≤λx2 1−λy2−λ1−λKx−y, 2.6
for allx, y∈ {z∈E:z ≤r}andλ∈0,1.
LetEbe a smooth, strictly convex, and reflexive Banach space and letCbe a nonempty closed convex subset ofE. Throughout this paper, we denote byφthe function defined by
φx, yx2−2x, Jyy2,
Following Alber 39, the generalized projection ΠC : E → C is a map that assigns to an
arbitrary pointx∈Ethe minimum point of the functionalφx, y, that is,ΠCxx, wherex
is the solution to the minimization problem
φx, x inf
y∈Cφ
y, x 2.8
existence and uniqueness of the operatorΠC follows from the properties of the functional
φx, yand strict monotonicity of the mappingJ. It is obvious from the definition of function
φthatsee39
y− x2≤φy, x≤yx2, ∀x, y∈E. 2.9
IfEis a Hilbert space, thenφx, y x−y2.
IfEis a reflexive, strictly convex and smooth Banach space, then forx, y∈E,φx, y 0 if and only ifx y. It is sufficient to show that ifφx, y 0, thenx y. From2.9, we havex y. This implies thatx, Jy x2 Jy2. From the definition ofJ, one has
JxJy. Therefore, we havexy; see34,40for more details.
Lemma 2.4see Kamimura and Takahashi 3. LetE be a uniformly convex and smooth real
Banach space and let{xn},{yn}be two sequences ofE. Ifφxn, yn → 0 and either{xn}or{yn}is
bounded, thenxn−yn → 0.
Lemma 2.5see Alber39. LetCbe a nonempty, closed, convex subset of a smooth Banach space
Eandx∈E. Then,x0 ΠCxif and only if
x0−y, Jx−Jx0
≥0, ∀y∈C. 2.10
Lemma 2.6see Alber39. LetEbe a reflexive, strictly convex, and smooth Banach space, letC
be a nonempty closed convex subset ofEand letx∈E. Then,
φy,ΠCxφΠCx, x≤φy, x, ∀y∈C. 2.11
Let Ebe a strictly convex, smooth, and reflexive Banach space, letJ be the duality mapping fromEintoE∗. Then,J−1is also single-valued, one-to-one, and surjective, and it is the duality mapping fromE∗intoE. Define a functionV :E×E∗ → Ras followssee4:
Vx, x∗ x2−2x, x∗ x∗2, 2.12
for all x ∈ E, x ∈ E and x∗ ∈ E∗. Then, it is obvious thatVx, x∗ φx, J−1x∗ and
Lemma 2.7see Kohsaka and Takahashi4, Lemma 3.2. LetEbe a strictly convex, smooth, and
reflexive Banach space, and letV be as in2.12. Then,
Vx, x∗ 2J−1x∗−x, y∗≤Vx, x∗y∗, 2.13
for allx∈Eandx∗, y∗∈E∗.
Let E be a reflexive, strictly convex, and smooth Banach space. Let C be a closed convex subset of E. Because φx, y is strictly convex and coercive in the first variable, we know that the minimization problem infy∈Cφx, yhas a unique solution. The operator
ΠCx:arg miny∈Cφx, yis said to be the generalized projection ofxonC.
A set-valued mappingT :E → E∗with domainDT {x∈E:Tx/∅}and range
RT {x∗ ∈ E∗ : x∗ ∈ Tx, x ∈ DT}is said to be monotone ifx−y, x∗−y∗ ≥ 0 for all
x∗ ∈Tx, y∗ ∈Ty. We denote the set{s∈E: 0∈Tx}byT−10.T is maximal monotone if
its graphGTis not properly contained in the graph of any other monotone operator. IfTis maximal monotone, then the solution setT−10 is closed and convex.
LetEbe a reflexive, strictly convex, and smooth Banach space, it is known thatTis a maximal monotone if and only ifRJrT E∗for allr >0.
Define the resolvent ofT byJrxxr. In other words,Jr JrT−1Jfor allr >0.Jr is
a single-valued mapping fromEtoDT. Also,T−10 FJrfor allr >0, whereFJris the
set of all fixed points ofJr. Define, forr >0, the Yosida approximation ofT byAr J−JJr/r.
We know thatArx∈TJrxfor allr >0 andx∈E.
Lemma 2.8 see Kohsaka and Takahashi4, Lemma 3.1. LetEbe a smooth, strictly convex,
and reflexive Banach space, T ⊂ E×E∗ a maximal monotone operator withT−10/∅, r > 0 and
Jr JrT−1J. Then,
φx, JryφJry, y≤φx, y, 2.14
for allx∈T−10 andy∈E.
Let A be an inverse-strongly monotone mapping of C into E∗ which is said to be
hemicontinuous if for all x, y ∈ C, the mapping F of 0,1 into E∗, defined by Ft
Atx 1−ty, is continuous with respect to the weak∗topology ofE∗. We define byNCv
the normal cone forCat a pointv∈C, that is,
NCv x∗∈E∗:v−y, x∗≥0, ∀y∈C . 2.15
Theorem 2.9see Rockafellar1. LetCbe a nonempty, closed, convex subset of a Banach space
EandAa monotone, hemicontinuous operator ofCintoE∗. LetT⊂E×E∗be an operator defined as
follows:
Tv
⎧ ⎨ ⎩
AvNCv, v∈C,
∅, otherwise.
2.16
Lemma 2.10see Tan and Xu41. Let{an}and{bn}be two sequence of nonnegative real numbers satisfying the inequality
an1 anbn, ∀n≥0. 2.17
If∞n1bn<∞, then limn→ ∞anexists.
For solving the mixed equilibrium problem, let us assume that the bifunction Θ :
C×C → Rand ϕ : C → Ris convex and lower semicontinuous satisfies the following conditions:
A1 Θx, x 0 for allx∈C,
A2 Θis monotone, that is,Θx, y Θy, x≤0 for allx, y∈C,
A3for eachx, y, z∈C,
lim sup
t↓0
Θtz 1−tx, y≤Θx, y, 2.18
A4for eachx∈C,y→Θx, yis convex and lower semicontinuous.
Motivated by Blum and Oettli 8, Takahashi and Zembayashi 29, Lemma 2.7 obtained the following lemmas.
Lemma 2.11see29, Lemma 2.7. LetCbe a nonempty closed convex subset of a smooth, strictly
convex, and reflexive Banach spaceE, letθbe a bifunction fromC×CtoRsatisfying (A1)–(A4), let
r >0, and letx∈E. Then, there existsz∈Csuch that
Θz, y 1ry−z, Jz−Jx≥0, ∀y∈C. 2.19
Lemma 2.12see Takahashi and Zembayashi29. LetCbe a closed convex subset of a uniformly
smooth, strictly convex, and reflexive Banach spaceE and letΘ be a bifunction fromC×C toR
satisfying (A1)–(A4). For allr >0 andx∈E, define a mappingTr :E → Cas follows:
Trx
z∈C:Θz, y1
r
y−z, Jz−Jx≥0, ∀y∈C
, 2.20
for allx∈E. Then, the followings hold:
1Tr is single-valued,
2Tr is a firmly nonexpansive-type mapping, that is, for allx, y∈E,
Trx−Try, JTrx−JTry≤Trx−Try, Jx−Jy, 2.21
3FTr EPΘ,
Lemma 2.13see Takahashi and Zembayashi29. LetCbe a closed, convex subset of a smooth,
strictly convex, and reflexive Banach spaceE, letΘa bifunction fromC×CtoRsatisfying (A1)–(A4)
and letr >0. Then, forx∈Eandq∈FTr,
φq, TrxφTrx, x≤φq, x. 2.22
Lemma 2.14. LetCbe a closed convex subset of a smooth, strictly convex and reflexive Banach space
E. Let ϕ : C → R is convex and lower semicontinuous andΘbe a bifunction from C×CtoR
satisfying (A1)–(A4). Forr >0 andx∈E, then there existsu∈Csuch that
Θu, yϕy−ϕu 1 r
y−u, Ju−Jx. 2.23
Define a mappingKr :E → Cas follows:
Krx
u∈C:Θu, yϕy−ϕu 1
r
y−u, Ju−Jx≥0, ∀y∈C
2.24
for allx∈E. Then, the followings hold:
1Kris single-valued,
2Kris firmly nonexpansive, that is, for allx, y∈E,Krx−Kry, JKrx−JKry ≤ Krx−
Kry, Jx−Jy ,
3FKr MEPΘ, ϕ,
4MEPΘ, ϕis closed and convex.
Proof. Define a bifunctionF:C×C → Ras follows:
Fu, y Θu, yϕy−ϕu, ∀u, y∈C. 2.25
It is easily seen thatFsatisfiesA1–A4. Therefore,KrinLemma 2.14can be obtained
fromLemma 2.12immediately.
3. Strong Convergence Theorem
In this section, we prove a strong convergence theorem for finding a common element of the zero point of a maximal monotone operator, the set of solutions of equilibrium problems, and the set of solution of the variational inequality operators of an inverse strongly monotone in a Banach space by using the shrinking hybrid projection method.
Theorem 3.1. LetE be a 2-uniformly convex and uniformly smooth Banach space and letCbe a
nonempty closed convex subset ofE. LetΘbe a bifunction fromC×CtoRsatisfying (A1)–(A4) let
ϕ : C → Rbe a lower semicontinuous and convex function, and let T : E → E∗ be a maximal
operator ofCintoE∗withF:VIC, A∩T−10∩MEPΘ, ϕ/∅andAy ≤ Ay−Aufor all
y∈Candu∈F. Let{xn}be a sequence generated byx0∈Ewithx1 ΠC1x0andC1C,
wn ΠCJ−1Jxn−λnAxn,
znJ−1βnJxn 1−βnJJr nwn,
ynJ−1αnJx1 1−αnJzn,
un∈ C such thatΘun, yϕy−ϕun r1 n
y−un, Jun−Jyn≥0, ∀y C,
Cn1z∈Cn:φz, un≤αnφz, x1 1−αnφz, xn ,
xn1 ΠCn1x0,
3.1
forn∈N, whereΠCis the generalized projection fromEontoC,Jis the duality mapping onE. The
coefficient sequence{αn},{βn} ⊂ 0,1,{rn} ⊂ 0,∞satisfying limn→ ∞αn 0,lim supn→ ∞βn <
1, lim infn→ ∞rn>0, and{λn} ⊂a, bfor somea, bwith 0< a < b < c2α/2, 1/cis the 2-uniformly
convexity constant ofE. Then, the sequence{xn}converges strongly toΠFx0.
Proof. We first show that{xn}is bounded. Putvn J−1Jxn−λnAxn, letp∈F:VIC, A∩
T−10∩MEPΘ, ϕ, and let{K
rn}be a sequence of mapping define asLemma 2.14andun
Krnyn. By 3.1andLemma 2.7, the convexity of the functionV in the second variable, we
have
φp, wnφp,ΠCvn
≤φp, vnφ
p, J−1Jx
n−λnAxn
≤Vp, Jxn−λnAxnλnAxn−2
J−1Jx
n−λnAxn−p, λnAxn
Vp, Jxn−2λnvn−p, Axn
φp, xn−2λnxn−p, Axn2vn−xn,−λnAxn .
3.2
Sincep∈VIA, CandAisα-inverse-strongly monotone, we have
−2λnxn−p, Axn−2λnxn−p, Axn−Ap−2λnxn−p, Ap
≤ −2αλnAxn−Ap2,
and byLemma 2.1, we obtain
2vn−xn,−λnAxn 2
J−1Jx
n−λnAxn−xn,−λnAxn
≤2J−1Jx
n−λnAxn−xnλnAxn
≤ 4
c2Jxn−λnAxn−JxnλnAxn
4
c2λ 2
nAxn2 ≤ c42λ2nAxn−Ap2.
3.4
Substituting3.3and3.4into3.2, we get
φp, wn≤φp, xn−2αλnAxn−Ap2c42λ2nAxn−Ap2
≤φp, xn2λn
2
c2λn−α
Axn−Ap2
≤φp, xn.
3.5
By Lemmas2.7,2.8and3.5, we have
φp, znφ
p, J−1β
nJxn 1−βnJJrnwn
Vp, βnJxn 1−βnJJrnwn
≤βnVp, Jxn1−βnVp, JJrnwn
βnφp, xn1−βnφp, Jrnwn
≤βnφp, xn1−βnφp, wn−φJrnwn, wn
≤βnφp, xn1−βnφp, wn
≤βnφp, xn1−βnφp, xn
φp, xn.
3.6
It follows that
φp, ynφ
p, J−1α
nJx1 1−αnJzn
Vp, αnJx1 1−αnJzn≤αnVp, Jx1
1−αnVp, Jzn
αnφp, x1
1−αnφp, zn≤αnφp, x1
1−αnφp, xn.
From3.1and3.7, we obtain
φp, unφp, Krnyn≤φp, yn≤φp, x1
1−αnφp, xn. 3.8
So, we havep∈Cn1. This implies thatF⊂Cn, for alln∈N.
FromLemma 2.5andxn ΠCnx0, we have
xn−z, Jx0−Jxn ≥0, ∀z∈Cn,
xn−p, Jx0−Jxn
≥0, ∀p∈F. 3.9
FromLemma 2.6, one has
φxn, x0 φΠCnx0, x0≤φ
p, x0
−φp, xn≤φp, x0
, 3.10
for allp ∈ F ⊂ Cn andn≥ 1. Then, the sequence{φxn, x0}is bounded. Sincexn ΠCnx0
andxn1 ΠCn1x0∈Cn1⊂Cn, we have
φxn, x0≤φxn1, x0, ∀n∈N. 3.11
Therefore, {φxn, x0} is nondecreasing. Hence, the limit of {φxn, x0} exists. By the
construction ofCn, one has thatCm ⊂ Cn and xm ΠCmx0 ∈ Cn for any positive integer
m≥n. It follows that
φxm, xn φxm,ΠCnx0≤φxm, x0−φΠCnx0, x0 φxm, x0−φxn, x0. 3.12
Lettingm, n → ∞in3.12, we get φxm, xn → 0. It follows fromLemma 2.4, thatxm−
xn → 0 asm, n → ∞, that is,{xn}is a Cauchy sequence. SinceEis a Banach space andCis closed and convex, we can assume thatxn → u∈C, asn → ∞. Since
φxn1, xn φxn1,ΠCnx0≤φxn1, x0−φΠCnx0, x0 φxn1, x0−φxn, x0, 3.13
for alln∈ N, we also have limn→ ∞φxn1, xn 0. FromLemma 2.4, we get limn→ ∞xn1−
xn0. Sincexn1 ΠCn1x0∈Cn1and by definition ofCn1, we have
φxn1, un≤αnφxn1, x1 1−αnφxn1, xn. 3.14
Noticing the conditions limn→ ∞αn0 and limn→ ∞φxn1, xn 0, we obtain
lim
n→ ∞φxn1, un 0. 3.15
From againLemma 2.4,
lim
So, by the triangle inequality, we get
lim
n→ ∞xn−un0. 3.17
SinceJis uniformly norm-to-norm continuous on bounded sets, we have
lim
n→ ∞Jxn−Jun0. 3.18
On the other hand, we observe that
φp, xn−φp, unxn2− un2−2p, Jxn−Jun
≤ xn−unxnun 2pJxn−Jun.
3.19
It follows that
φp, xn−φp, un−→0, asn−→ ∞. 3.20
From3.1,3.5,3.6,3.7, and3.8, we have
φp, un≤φp, yn≤αnφp, x1
1−αnφp, zn
≤αnφp, x1
1−αnβnφp, xn1−βnφp, wn−φJrnwn, wn
≤αnφp, x1
1−αnβnφp, xn1−βnφp, xn−φJrnwn, wn
≤αnφp, x1
1−αnφp, xn−1−αn1−βnφJrnwn, wn
3.21
and then
1−αn1−βnφJrnwn, wn≤αnφp, x1
1−αnφp, xn−φp, un. 3.22
From conditions limn→ ∞αn0, lim supn→ ∞βn<1 and3.20, we obtain
lim
n→ ∞φJrnwn, wn 0. 3.23
By againLemma 2.4, we have limn→ ∞Jrnwn−wn0.
SinceJis uniformly norm-to-norm continuous on bounded sets, we obtain
lim
Applying3.5and3.6, we observe that
φp, un≤φp, yn≤αnφp, x1
1−αnφp, zn
≤αnφp, x1
1−αnβnφp, xn1−βnφp, wn≤αnφp, x1
1−αn
βnφp, xn1−βnφp, xn−2λn
α− 2
c2λn
Axn−Ap2
≤αnφp, x1
1−αnφp, xn−1−αn1−βn2λn
α− 2
c2λn
Axn−Ap2
3.25
and, hence,
2λn
α− 2
c2λn
Axn−Ap2≤ 1
1−αn1−βn
αnφp, x1
1−αnφp, xn−φp, un,
3.26
for alln∈N. Since 0< a≤λn ≤b < c2α/2, limn→ ∞αn 0, lim supn→ ∞βn <1 and3.20, we
have
lim
n→ ∞Axn−Ap0. 3.27
From Lemmas2.6,2.7, and3.4, we get
φxn, wn φxn,ΠCvn≤φxn, vn φ
xn, J−1Jxn−λnAxn
Vxn, Jxn−λnAxn
≤Vxn,Jxn−λnAxn λnAxn
−2J−1Jxn−λnAxn−xn, λnAxn
φxn, xn 2vn−xn,−λnAxn
2vn−xn,−λnAxn ≤ 4λ 2 n
c2 Axn−Ap
2.
3.28
FromLemma 2.4and3.27, we have
lim
n→ ∞xn−wn0. 3.29
SinceJis uniformly norm-to-norm continuous on bounded sets, we obtain
lim
Since{xn}is bounded, there exists a subsequence{xni}of{xn}such thatxni u∈E. Since
xn−wn → 0, then we getwni uasi → ∞.
Now, we claim thatu∈F. First, we show thatu∈T−10. Indeed, since lim infn
→ ∞rn>0,
it follows from3.24that
lim
n→ ∞Arnwnnlim→ ∞
1
rnJwn−JJrnwn0. 3.31
Ifz, z∗∈T, then it holds from the monotonicity ofTthat
z−wni, z∗−Arniwni
≥0, 3.32
for alli∈N. Lettingi → ∞, we getz−u, z∗ ≥0. Then, the maximality ofTimpliesu∈T−10.
Next, we show thatu∈VIC, A. LetB⊂E×E∗be an operator as follows:
Bv
⎧ ⎨ ⎩
AvNCv, v∈C,
∅, otherwise. 3.33
ByTheorem 2.9,Bis maximal monotone andB−10 VIA, C. Letv, w ∈GB. Sincew ∈
BvAvNCv, we getw−Av∈NCv. Fromwn∈C, we have
v−wn, w−Av ≥0. 3.34
On the other hand, sincewn ΠCJ−1Jxn−λnAxn, then byLemma 2.5, we have
v−wn, Jwn−Jxn−λnAxn ≥0. 3.35
Thus,
v−wn,Jxnλ−Jwn
n −Axn
It follows from3.34and3.36that
v−wn, w ≥ v−wn, Av ≥ v−wn, Av
v−wn,Jxnλ−Jwn
n −Axn
v−wn, Av−Axn
v−wn,Jxnλ−Jwn n
v−wn, Av−Awn v−wn, Awn−Axn
v−wn,Jxnλ−Jwn n
≥ −v−wnwnα−xn− v−wnJxn−aJwn
≥ −M
w
n−xn
α
Jxn−Jwn
a
,
3.37
where M supn≥1{v−wn}. From 3.29 and 3.30, we obtain v−u, w ≥ 0. By the maximality ofB, we haveu∈B−10 and, hence,u∈VIC, A.
Next, we show thatu∈MEPΘ, ϕ. SinceunKrnyn. From Lemmas2.13and2.14, we
have
φun, ynφKrnyn, yn≤φu, yn−φu, Krnyn≤φu, xn−φu, un. 3.38
Similarly by3.20,
lim
n→ ∞φ
un, yn0, 3.39
and so
lim
n→ ∞un−yn0. 3.40
SinceJis uniformly norm-to-norm continuous on bounded sets, we obtain
lim
n→ ∞Jun−Jyn0. 3.41
From3.1andA2, we also have
ϕy−ϕun r1 n
y−un, Jun−Jyn≥Θy, un, ∀y∈C. 3.42
Hence,
ϕy−ϕuni
y−uni,Junir−Jyni ni
Fromxn−un → 0,xn−wn → 0, we getuni u. SinceJuni−Jyni/rni → 0, it follows
byA4and the weak, lower semicontinuous ofϕthat
Θy, uϕu−ϕy≤0, ∀y∈C. 3.44
Fortwith 0< t≤1 andy∈C, letytty 1−tu. Sincey ∈Candu∈C, we haveyt∈C
and henceΘyt, u ϕu−ϕyt≤0. So, fromA1,A4, and the convexity ofϕ, we have
0 Θyt, ytϕyt−ϕyt≤tΘyt, y 1−tΘyt, utϕy 1−tϕy−ϕyt
≤tΘyt, yϕy−ϕyt.
3.45
Dividing by t, we get Θyt, y ϕy − ϕyt ≥ 0. From A3 and the weakly lower
semicontinuity ofϕ, we haveΘu, y ϕy−ϕu≥0 for ally∈Cimpliesu∈MEPΘ, ϕ. Hence,u∈F :VIC, A∩T−10∩MEPΘ, ϕ.
Finally, we show thatu ΠFx. Indeed, fromxn ΠCnxandLemma 2.5, we have
Jx−Jxn, xn−z ≥0, ∀z∈Cn. 3.46
SinceF ⊂Cn, we also have
Jx−Jxn, xn−p≥0, ∀p∈F. 3.47
Taking limitn → ∞, we have
Jx−Ju, u−p≥0, ∀p∈F. 3.48
By againLemma 2.5, we can conclude thatu ΠFx0. This completes the proof.
Corollary 3.2. Let E be a 2-uniformly convex and uniformly smooth Banach space, let C be a
nonempty, closed, convex subset ofE. LetΘbe a bifunction fromC×CtoRsatisfying (A1)–(A4)
letϕ : C → Rbe a lower semicontinuous and convex function, and letT : E → E∗be a maximal
monotone operator. LetJr JrT−1J forr >0 withF :T−10∩MEPΘ, ϕ/∅. Let{xn}be a
sequence generated byx0∈Ewithx1 ΠC1x0andC1C,
znJ−1βnJxn 1−βnJJrnxn,
ynJ−1αnJx1 1−αnJzn,
un∈C such thatΘun, yϕy−ϕun r1 n
y−un, Jun−Jyn≥0, ∀y∈C,
Cn1z∈Cn :φz, un≤αnφz, x1 1−αnφz, xn ,
xn1 ΠCn1x0,
forn∈N, whereΠCis the generalized projection fromEontoC,Jis the duality mapping onE. The
coefficient sequence{αn},{βn} ⊂0,1,{rn} ⊂0,∞satisfying limn→ ∞αn 0, lim supn→ ∞βn<1
and lim infn→ ∞rn >0. Then, the sequence{xn}converges strongly toΠFx0.
Proof. InTheorem 3.1ifA≡0, then3.1reduced to3.49.
4. Weak Convergence Theorem
In this section, we first prove the following strong convergence theorem by using the idea of Plubtieng and Sriprad42.
Theorem 4.1. Let E be a 2-uniformly convex and uniformly smooth Banach space whose duality
mappingJis weak sequentially continuous. LetT :E → E∗be a maximal monotone operator and let
Jr JrT−1J forr > 0. LetCbe a nonempty, closed, convex subset ofEsuch thatDT⊂C ⊂
J−1
r>0RJrT, letΘbe a bifunction fromC×CtoRsatisfying (A1)–(A4), letϕ:C → Rbe a
lower semicontinuous and convex function, and letAbe anα-inverse-strongly monotone operator of
CintoE∗withF :VIC, A∩T−10∩MEPΘ, ϕ/∅andAy ≤ Ay−Aufor ally∈Cand
u∈F. Let{xn}be a sequence generated byx1x∈Cand
unKrnxn,
zn ΠCJ−1Jun−λnAun,
ynJ−1βnJxn 1−βnJJrnzn,
xn1 ΠCJ−1αnJx1 1−αnJyn,
4.1
forn ∈ N∪ {0}, whereΠC is the generalized projection fromE ontoC,J is the duality mapping
on E. The coefficient sequence {αn},{βn} ⊂ 0,1, {rn} ⊂ 0,∞ satisfying ∞n0αn < ∞,
lim supn→ ∞βn < 1 lim infn→ ∞rn > 0 and{λn} ⊂ a, bfor somea, b with 0< a < b < c2α/2,
1/cis the 2-uniformly convexity constant ofE. Then, the sequence{ΠFxn}converges strongly to an
element ofF, which is a unique elementv∈Fsuch that
lim
n→ ∞φv, xn miny∈Fnlim→ ∞φ
y, xn, 4.2
whereΠFis the generalized projection fromContoF.
Proof. PutvnJ−1Jun−λnAun. Letp∈F:VIC, A∩T−10∩MEPΘ, ϕ, byLemma 2.14
and nonexpansiveness ofKr, we have
By4.1andLemma 2.7, the convexity of the functionVin the second variable, we obtain
φp, znφp,ΠCvn≤φp, vnφ
p, J−1Ju
n−λnAun
≤Vp, Jun−λnAunλnAun−2
J−1Ju
n−λnAun−p, λnAun
Vp, Jxun−2λnvn−p, Aun
φp, un−2λnun−p, Aun2vn−un,−λnAun .
4.4
Sincep∈VIA, CandAisα-inverse-strongly monotone, we also have
−2λnun−p, Aun−2λnun−p, Aun−Ap−2λnun−p, Ap≤ −2αλnAun−Ap2,
4.5
2vn−un,−λnAun 2
J−1Ju
n−λnAun−xn,−λnAun
≤2J−1Jun−λnAun−xnλnAun
≤ c42Jun−λnAun−JunλnAun ≤ c42λ2nAun−Ap2.
4.6
Substituting4.5and4.6into4.4and4.3, we get
φp, zn≤φp, un−2αλnAun−Ap2c42λ2nAun−Ap2
≤φp, un−2λn
α− 2
c2λn
Aun−Ap2≤φp, un≤φp, xn.
4.7
By Lemmas2.7,2.8,4.7, and using the same argument inTheorem 3.1,3.6, we obtain
φp, yn≤φp, xn, 4.8
and hence byLemma 2.6and4.7, we note that
φp, xn1φp, J−1α
nJx1 1−αnJyn
Vp, αnJx1 1−αnJ
yn≤αnVp, Jx1
1−αnVp, Jyn
αnφp, x1
1−αnφp, yn≤αnφp, x1
1−αnφp, xn,
4.9
for alln≥0. So, from∞n0αn <∞andLemma 2.10, we deduce that limn→ ∞φp, xnexists.
This implies that {φp, xn} is bounded. It implies that {xn}, {yn}, {zn}, and {Jrnzn} are
bounded. Define a functiong:F → 0,∞as follows:
gp lim
n→ ∞φ
Then, by the same argument as in proof of 43, Theorem 3.1, we obtaing is a continuous convex function and ifzn → ∞, thengzn → ∞. Hence, by34, Theorem 1.3.11, there
exists a pointv∈Fsuch that
gv min
y∈F g
y:l. 4.11
Putwn ΠFxnfor alln≥0. We next prove thatwn → vasn → ∞. Suppose on the contrary
that there exists0>0 such that, for eachn∈N, there isn≥nsatisfyingwn−v ≥0. Since
v∈F, we have
φwn, xn φΠFxn, xn≤φv,ΠFxn φΠFxn, xn≤φv, xn, 4.12
for alln≥0. This implies that
lim sup
n→ ∞ φwn, xn≤nlim→ ∞φv, xn l. 4.13
Sincev − ΠFxn2 ≤ φv, wn ≤ φv, xn for all n ≥ 0 and{xn} is bounded, {wn} is
bounded. ByLemma 2.3, there exists a stricly increasing, continuous, and convex function
K:0,∞ → 0,∞such thatK0 0 and
wnv
2
2≤ 1
2wn
21
2v
2−1
4Kwn−v, 4.14
for alln≥ 0. Now, chooseσsatisfying 0 < σ <1/4K0. Hence, there existsn0 ∈Nsuch
that
φwn, xn≤lσ, φv, xn≤lσ, 4.15
for alln≥0. Thus, there existsk≥n0satisfying the following:
φwk, xk≤lσ, φv, xk≤lσ, wk−v ≥0. 4.16
From4.9,4.14, and4.16, we obtain
φwkv
2 , xnk
≤φwkv
2 , xk
wkv
2
2−2wkv
2 , Jxk
xk2
≤ 1
2wk
21
2v
2−1
4Kwk−v− wkv, Jxk xk
2
1
2φwk, xk 1
2φv, xk− 1
4Kwk−v≤lσ− 1 4K0,
4.17
for alln≥0. Hence,
l≤ lim
n→ ∞φ
wkv
2 , xn
lim
n→ ∞φ
wkv
2 , xnk
≤lσ−1
This is a contradiction. So,{wn}converges strongly tov∈F:VIC, A∩T−10∩MEPΘ, ϕ.
Consequently,v∈Fis the unique element ofFsuch that
lim
n→ ∞φv, xn miny∈Fnlim→ ∞φ
y, xn. 4.19
This completes the proof.
Now, we prove a weak convergence theorem for the algorithm4.20 below under different condition on data.
Theorem 4.2. Let E be a 2-uniformly convex and uniformly smooth Banach space whose duality
mappingJis weakly sequentially continuous. LetT :E → E∗be a maximal monotone operator and
letJr JrT−1Jforr >0. LetCbe a nonempty closed convex subset ofEsuch thatDT⊂C⊂
J−1
r>0RJrT, letΘbe a bifunction fromC×CtoRsatisfying (A1)–(A4), letϕ:C → Rbe a
lower semicontinuous and convex function, and letAbe anα-inverse-strongly monotone operator of
CintoE∗withF :VIC, A∩T−10∩MEPΘ, ϕ/∅andAy ≤ Ay−Aufor ally∈Cand
u∈F. Let{xn}be a sequence generated byx1x∈Cand
unKrnxn,
zn ΠCJ−1Jun−λnAun,
ynJ−1βnJxn 1−βnJJrnzn,
xn1 ΠCJ−1αnJx1 1−αnJyn,
4.20
forn ∈ N∪ {0}, whereΠC is the generalized projection fromE ontoC,J is the duality mapping
on E. The coefficient sequence {αn},{βn} ⊂ 0,1, {rn} ⊂ 0,∞ satisfying ∞n0αn < ∞,
lim supn→ ∞βn<1 lim infn→ ∞rn>0 and{λn} ⊂a, bfor somea, bwith 0< a < b < c2α/2, 1/c
is the 2-uniformly convexity constant ofE. Then, the sequence{xn}converges weakly to an elementv
ofF, wherevlimn→ ∞ΠFxn.
Proof. ByTheorem 4.1, we have{xn}is bounded and so are{zn},{Jrnzn}.
From4.9, we obtain
φp, xn1≤αnφp, x1
1−αnφp, yn
≤αnφp, x1
1−αnβnφp, xn1−βnφp, zn−φJrnzn, zn
≤αnφp, x1
1−αnβnφp, xn1−βnφp, xn−φJrnzn, zn
≤αnφp, x1
1−αnβnφp, xn−1−αn1−βnφJrnzn, zn,
4.21
and then
1−αn1−βnφJrnzn, zn≤αnφp, x1
Since limn→ ∞αn0, lim supn→ ∞βn<1 and{φp, xn}exists, then we have
lim
n→ ∞φJrnzn, zn 0. 4.23
By againLemma 2.4, we have limn→ ∞Jrnzn−zn 0. SinceJ is uniformly norm-to-norm
continuous on bounded sets, we obtain
lim
n→ ∞JJrnzn−Jzn0. 4.24
Apply4.7,4.8, and4.9, we observe that
φp, xn1
≤αnφp, x1
1−αnφp, yn≤αnφp, x1
1−αnβnφp, xn1−βnφp, zn
≤αnφp, x1
1−αn
βnφp, xn1−βnφp, un−2λn
α− 2
c2λn
Aun−Ap2
≤αnφp, x1
1−αn
βnφp, xn1−βnφp, xn−2λn
α−c22λnAun−Ap2
≤αnφp, x1
1−αnφp, xn−1−αn1−βn2λn
α− 2
c2λn
Aun−Ap2,
4.25
and hence
2λn
α− 2
c2λn
Aun−Ap2≤ 1
1−αn1−βn
αnφp, x1
1−αnφp, xn−φp, xn1,
4.26
for alln∈N. Since 0< a≤λn≤b < c2α/2, limn→ ∞αn0 and lim supn→ ∞βn <1, we have
lim
n→ ∞Aun−Ap0. 4.27
From Lemmas2.6,2.7, and4.7, we get
φun, zn φun,ΠCvn≤φun, vn φ
un, J−1Jun−λnAun
Vun, Jun−λnAun
≤Vun,Jun−λnAun λnAun−2
J−1Ju
n−λnAun−xn, λnAun
φun, un 2vn−un, λnAun 2vn−un, λnAun
≤ 4cλ22nAun−Ap2.
FromLemma 2.4and4.27, we have
lim
n→ ∞un−zn0. 4.29
SinceJis uniformly norm-to-norm continuous on bounded sets, we obtain
lim
n→ ∞Jun−Jzn0. 4.30
Since{zn}is bounded, there exists a subsequence{zni}of{zn}such thatzni u∈C.
It follows thatJrnizni uanduni u∈Casi → ∞.
Now, we claim thatu∈F. First, we show thatu∈T−10. Indeed, since lim infn
→ ∞rn>0,
it follows that
lim
n→ ∞Arnznnlim→ ∞
1
rnJzn−JJrnzn0. 4.31
Ifz, z∗∈T, then it holds from the monotonicity ofTthat
z−Jrnizni, z∗−Arnizni
≥0, 4.32
for alli∈N. Lettingi → ∞, we getz−u, z∗ ≥0. Then, the maximality ofTimpliesu∈T−10.
Next, we show thatu∈VIC, A. LetB⊂E×E∗be an operator as follows:
Bv
⎧ ⎨ ⎩
AvNCv, v∈C,
∅, otherwise.
4.33
ByTheorem 2.9,Bis maximal monotone andB−10 VIA, C. Letv, w ∈GB. Sincew ∈
BvAvNCv, we getw−Av∈NCv. Fromzn∈C, we have
v−zn, w−Av ≥0. 4.34
On the other hand, sincezn ΠCJ−1Jun−λnAun. Then, byLemma 2.5, we have
v−zn, Jwn−Jun−λnAun ≥0. 4.35
Thus,
v−zn,Junλ−Jzn
n −Aun
It follows from4.34and4.36that
v−zn, w ≥ v−zn, Av ≥ v−zn, Av
v−zn,Junλ−Jzn
n −Axn
v−zn, Av−Aun
v−zn,Junλ−Jzn n
v−zn, Av−Azn v−zn, Azn−Aun
v−zn,Junλ−Jzn n
≥ −v−znzn−un
α − v−zn
Jun−Jzn
a
≥ −M
z
n−un
α
Jun−Jzn
a
,
4.37
where M supn≥1{v−zn}. From 4.29 and 4.30, we obtain v− u, w ≥ 0. By the maximality ofB, we haveu∈B−10 and henceu∈VIC, A.
Next, we showu∈MEPf FKr. FromunKrnxn. It follows from4.7,4.8, and
4.9that
φp, xn1≤αnφp, x1
1−αnφp, yn
≤αnφp, x1
1−αnβnφp, xn1−βnφp, zn
≤αnφp, x1
1−αnβnφp, xn1−βnφp, un
≤αnφp, x1
1−αnβnφp, xn1−βnφp, xn,
4.38
or, equivalently,
φp, xn1−αnφp, x1
≤1−αnβnφp, xn1−βnφp, un≤1−αnφp, xn,
4.39
with limn→ ∞αn0 and lim supn→ ∞βn<1, yield that limn→ ∞φp, un limn→ ∞φp, xn.
From Lemmas2.13and2.14, forp∈F,
φun, xn≤φp, xn−φp, un. 4.40
This implies that limn→ ∞φun, xn 0. NoticingLemma 2.4, we get
un−xn −→0, asn−→ ∞. 4.41
SinceJis uniformly norm-to-norm continuous on bounded sets, we obtain
lim
From4.20andA2, we also have
ϕy−ϕun r1 n
y−un, Jun−Jxn≥Θy, un, ∀y∈C. 4.43
Hence,
ϕy−ϕuni
y−uni,Junir−Jxni ni
≥Θy, uni, ∀y∈C. 4.44
Fromun−zn → 0, we getuni u. SinceJuni −Jxni/rni → 0, it follows byA4and the
weakly lower semicontinuous ofϕthat
Θy, uϕu−ϕy≤0, ∀y∈C. 4.45
Fortwith 0< t≤1 andy∈C, letytty 1−tu. Sincey ∈Candu∈C, we haveyt∈C
and henceΘyt, u ϕu−ϕyt≤0. So, fromA1,A4, and the convexity ofϕ, we have
0 Θyt, ytϕyt−ϕyt
≤tΘyt, y 1−tΘyt, utϕy 1−tϕy−ϕyt
≤tΘyt, yϕy−ϕyt.
4.46
Dividing by t, we get Θyt, y ϕy − ϕyt ≥ 0. From A3 and the weakly lower
semicontinuity ofϕ, we haveΘu, y ϕy−ϕu≥0 for ally∈Cimpliesu∈MEPΘ, ϕ. Hence,u∈F :VIC, A∩T−10∩MEPΘ, ϕ.
ByTheorem 4.1, the{ΠFxn}converges strongly to a pointv ∈ F which is a unique
element ofFsuch that
lim
n→ ∞φv, xn miny∈Fnlim→ ∞φ
y, xn. 4.47
By the uniform smoothness ofE, we also have limn→ ∞JΠFxni−Jv0.
Finally, we proveuv. FromLemma 2.5andu∈F, we have
ΠFxni−u, Jxni−JΠFxni ≥0. 4.48
SinceJis weakly sequentially continuous,uni uandun−xn → 0. Then,
v−u, Ju−Jv ≥0. 4.49
On the other hand, sinceJis monotone, we have
Hence,
v−u, Ju−Jv 0. 4.51
Since E is strict convexity, it follows that u v. Therefore, the sequence{xn} converges weakly tovlimn→ ∞ΠFxn. This completes the proof.
5. Application to Complementarity Problems
LetCbe a nonempty, closed convex cone inEand A an operator ofCintoE∗. We define its
polar inE∗to be the set
K∗y∗∈E∗:x, y∗≥0, ∀x∈C . 5.1
Then, the elementu∈Cis called a solution of the complementarity problem if
Au∈K∗, u, Au 0. 5.2
The set of solutions of the complementarity problem is denoted by CPK, A; see34, for more detial.
Theorem 5.1. LetEbe a 2-uniformly convex and uniformly smooth Banach space and let K be a
nonempty closed convex subset ofE. LetΘbe a bifunction fromK×KtoRsatisfying (A1)–(A4)
letϕ :K → Rbe a lower semicontinuous and convex function, and letT :E → E∗be a maximal
monotone operator. Let Jr JrT−1J forr > 0 and letA be anα-inverse-strongly monotone
operator ofKintoE∗withF:T−10∩CPK, A∩MEPΘ, ϕ/∅andAy ≤ Ay−Aufor
ally∈Kandu∈F. For an initial pointx0∈Ewithx1 ΠC1x0andK1K,
wn ΠKJ−1Jxn−λnAxn,
znJ−1βnJxn 1−βnJJr nwn,
ynJ−1αnJx1 1−αnJzn,
un∈K such that Θun, yϕy−ϕun r1 n
y−un, Jun−Jyn≥0, ∀y∈K,
Kn1z∈Kn: φz, un≤αnφz, x1 1−αnφz, xn ,
xn1 ΠKn1x0,
5.3
forn ∈ N, where ΠK is the generalized projection from E ontoK and J is the duality mapping
on E. The coefficient sequence {αn},{βn} ⊂ 0,1, {rn} ⊂ 0,∞ satisfying limn→ ∞αn 0,
lim supn→ ∞βn<1, lim infn→ ∞rn>0 and{λn} ⊂a, bfor somea, bwith 0< a < b < c2α/2, 1/c
is the 2-uniformly convexity constant ofE. Then, the sequence{xn}converges strongly toΠFx0.
Proof. As in the proof Lemma 7.1.1 of Takahashi in44, we have VIC, A CPK, A. So, we
