Volume 2011, Article ID 601910,23pages doi:10.1155/2011/601910
Research Article
Algorithms of Common Solutions to Generalized
Mixed Equilibrium Problems and a System of
Quasivariational Inclusions for Two Difference
Nonlinear Operators in Banach Spaces
Nawitcha Onjai-uea
1, 2and Poom Kumam
1, 21Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi
(KMUTT), Bangmod, Bangkok 10140, Thailand
2Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand
Correspondence should be addressed to Poom Kumam,[email protected]
Received 11 December 2010; Accepted 3 January 2011
Academic Editor: S. Al-Homidan
Copyrightq2011 N. Onjai-uea 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 consider a new iterative algorithm for finding a common element of the set of generalized mixed equilibrium problems, the set of solutions of a system of quasivariational inclusions for two difference inverse strongly accretive operators, and common set of fixed points for strict pseudo-contraction mappings in Banach spaces. Furthermore, strong convergence theorems of this method were established under suitable assumptions imposed on the algorithm parameters. The results obtained in this paper improve and extend some results in the literature.
1. Introduction
LetF :C×C → Rbe a bifunction, letϕ :C → R∪ {∞}be a function, and letB :
C → E∗be a nonlinear mapping, whereRis the set of real numbers. The so-calledgeneralized mixed equilibrium problemis to findu∈Csuch that
Fu, yBu, y−uϕy−ϕu≥0, ∀y∈C. 1.1
The set of solutions to1.1is denoted by GMEPF, ϕ, B, that is,
GMEPF, ϕ, B u∈C:Fu, yBu, y−uϕy−ϕu≥0, ∀y∈C. 1.2
It is easy to see thatuis a solution of problem implying thatu∈domϕ {u∈C|ϕu<∞}. IfB 0, then the generalized mixed equilibrium problem1.1becomes the following
mixed equilibrium problemwhich is to findu∈Csuch that
Fu, yϕy−ϕu≥0, ∀y∈C. 1.3
The set of solutions of1.3is denoted by MEPF, ϕ.
Ifϕ 0, then the generalized mixed equilibrium problem1.1becomes the following
generalized equilibrium problemwhich is to findu∈Csuch that
Fu, yBu, y−u ≥0, ∀y∈C. 1.4
The set of solution of1.4is denoted by GEPF, B.
IfB 0, then the generalized mixed equilibrium problem1.4becomes the following
equilibrium problemis to findu∈Csuch that
Fu, y≥0, ∀y∈C. 1.5
The set of solution of1.5is denoted by EPF. The generalized mixed equilibrium problems include fixed point problems, variational inequality problems, optimization problems, Nash equilibrium problems, and the equilibrium problem as special cases. Numerous problems in physics, optimization, and economics reduce to find a solution of1.5. Some methods have been proposed to solve the equilibrium problem and variational inequality problems in Hilbert spaces and Banach spaces, see, for instance,1–22and the references therein.
Throughout this paper, let E be a real Banach space with norm · , let E∗ be the dual space of E, and letCbe a nonempty closed convex subset ofE, and ·,·denote the pairing betweenEandE∗. LetA1, A2 : E → Ebe single-valued nonlinear mappings, and
letM1, M2:E → 2Eset-valued nonlinear mappings. We considera system of quasivariational
inclusionsSQVI: findx∗, y∗∈E×Esuch that
0∈x∗−y∗ρ1
A1y∗M1x∗
,
0∈y∗−x∗ρ2
A2x∗M2y∗
. 1.6
aIf A1 A2 A andM1 M2 M, then the problem 1.6 is reduced to find
x∗, y∗∈E×Esuch that
0∈x∗−y∗ρ1
Ay∗Mx∗,
0∈y∗−x∗ρ2
Ax∗My∗.
1.7
The problem 1.7 is called system variational inclusion problem denoted by SVIE, A, M.
bFurther, ifx∗ y∗ in the problem1.7, then the problem1.7is reduced to find
x∗∈Esuch that
0∈Ax∗Mx∗. 1.8
The problem1.8is calledvariational inclusion problemdenoted by VIE, A, M. Here we have examples of the variational inclusion1.8.
IfM ∂δC, whereCis a nonempty closed convex subset ofE, andδC :E → 0,∞is
theindicator functionofC, that is,
δCx
⎧ ⎨ ⎩
0, x∈C,
∞, x /∈C,
1.9
then the variational inclusion problem1.8is equivalentsee23to findingu∈Csuch that
Au, v−u ≥0, ∀x∈C. 1.10
This problem is calledHartman-Stampacchia variational inequality problemdenoted by VIC, A. The generalized duality mappingJq:E → 2E
∗
is defined by
Jqx f∈E∗ :
x, f xq,f xq−1, ∀x∈E. 1.11
In particular, ifq 2, the mappingJ2 is called thenormalized duality mappingand, usually,
written asJ2 J.
LetU {x∈E : x 1}. A Banach spaceEis said to beuniformly convexif, for any
∈0,2, there existsδ >0 such that, for anyx, y∈U,x−y ≥impliesxy/2 ≤1−δ. It is known that a uniformly convex Banach space is reflexive and strictly convex. A Banach spaceEis said to besmoothif the limit limt→0xty − x/texists for allx, y∈U.
It is also said to beuniformly smoothif the limit is attained uniformly forx, y∈U. Themodulus of smoothnessofEis defined by
ρτ sup
1
2xyx−y−1 : x, y∈E, x 1, y τ
whereρ:0,∞ → 0,∞is a function. It is known thatEis uniformly smooth if and only if limτ→0ρτ/τ 0. Letqbe a fixed real number with 1< q≤2.
A Banach spaceEis said to beq-uniformly smoothif there exists a constantc >0 such thatρτ≤cτqfor allτ >0.
We note thatEis a uniformly smooth Banach space if and only ifJq is single valued
and uniformly continuous on any bounded subset of E. It is known that if E is smooth, thenJ is single valued, which is denoted byj. Typical examples of both uniformly convex and uniformly smooth Banach spaces areLp, wherep > 1. More precisely, Lpis min{p,2} -uniformly smooth for everyp >1.
LetT be a mapping from Einto itself. In this paper, we use FTto denote the set of fixed points of the mapping T. Recall that the mapping T is said to be nonexpansive if
Tx−Ty ≤ x−y, for allx, y∈E. Recall that a mappingf :C → Cis calledcontractiveif there exists a constantα∈0,1such thatfx−fy ≤αx−y, for allx, y∈C.
A mappingT :C → Cis said to beλ-strictly pseudocontractiveif there exists a constant
λ∈0,1such that
Tx−Ty, Jx−y≤x−y2−λI−Tx−I−Ty2, ∀x, y∈C. 1.13
Recall that an operatorAofEinto itself is said to beaccretiveif
Ax−Ay, Jx−y ≥0, ∀x, y∈E. 1.14
Forα >0, recall that an operatorAofEinto itself is said to beα-inverse strongly accretiveif
Ax−Ay, Jx−y≥αAx−Ay2, ∀x, y∈E. 1.15
The resolvent operator technique for solving variational inequalities and variational inclusions is interesting and important. The resolvent equation technique is used to develop powerful and efficient numerical techniques for solving various classes of variational inequalities, inclusions, and related optimization problems.
Definition 1.1. Let M : E → 2E be a multivalued maximal accretive mapping. The
single-valued mappingJM,ρ:E → E, defined by
JM,ρu IρM−1u, ∀u∈E, 1.16
is called theresolventoperator associated withM, whereρis any positive number andI is the identity mapping.
LetDbe a subset ofC, and letPbe a mapping ofCintoD. Then,Pis said to besunny
if
PP xtx−P x P x, 1.17
wheneverP xtx−P x ∈ Cforx ∈ Candt ≥ 0. A mappingP ofCinto itself is called a
whereRPis the range ofP. A subsetD ofCis called asunny nonexpansive retractofCif there exists a sunny nonexpansive retraction fromContoD.
In 2006, Aoyama et al.24considered the following problem: findu∈Csuch that
Au, Jv−u ≥0, ∀v∈C. 1.18
They proved that the variational inequality1.18is equivalent to a fixed point problem. The elementu∈Cis a solution of the variational inequality1.18if and only ifu∈Csatisfies the following equation:
u PCu−λAu, 1.19
whereλ >0 is a constant andPCis asunny nonexpansive retractionfromEontoC.
In order to find a solution of the variational inequality1.18, the authors proved the following theorem in the framework of Banach spaces.
Theorem AITsee24. LetEbe a uniformly convex and 2-uniformly smooth Banach space, and let Cbe a nonempty closed convex subset ofE. LetPCbe a sunny nonexpansive retraction fromEonto
C, letα >0, and letAbe anα-inverse strongly accretive operator ofCintoEwithSC, A/∅, where
SC, A x∗∈C : Ax∗, jx−x∗≥0, x∈C. 1.20
If{λn}and{αn}are chosen such thatλn ∈a, α/K2, for somea > 0andαn∈b, c, for someb, c with0< b < c <1, then the sequence{xn}defined by the following manners:x1−x∈Cand
xn1 αnxn 1−αnPCxn−λnAxn 1.21
converges weakly to some elementzofSC, A, whereKis the 2-uniformly smoothness constant ofE
andPCis a sunny nonexpansive retraction.
Motivated by Aoyama et al. [24] and also Ceng et al. [25], Qin et al. [26] and Yao et al. [27] considered the following general system of variational inequalities: letCbe nonempty closed convex subset of a real Banach spaceE. For given two operatorsA, B : C → E, we consider the problem of findingx∗, y∗∈C×Csuch that
λAy∗x∗−y∗, jx−x∗≥0, ∀x∈C,
μBx∗y∗−x∗, jx−y∗≥0, ∀x∈C, 1.22
where λ and μ are two positive real numbers. This system is called the system of general variational inequalitiesin a real Banach space. If we add up the requirement thatA B, then the problem1.22is reduced to the system1.23below. Findx∗, y∗∈C×Csuch that
λAy∗x∗−y∗, jx−x∗≥0, ∀x∈C,
For the class of nonexpansive mappings, one classical way to study nonexpansive mappings is to use contractions to approximate a nonexpansive mapping 28, 29. More precisely, taket∈0,1and define a contractionTt:C → Cby
Ttx tu 1−tTx, ∀x∈C, 1.24
where u ∈ C is a fixed point and T : C → C is a nonexpansive mapping. The Banach contraction mapping principle guarantees thatTthas a unique fixed pointxtinC, that is,
xt tu 1−tTxt. 1.25
It is unclear, in general, what the behavior ofxt is ast → 0, even if T has a fixed point.
However, in the case ofT having a fixed point, Browder28proved that if Eis a Hilbert space, thenxtconverges strongly to a fixed point ofT. Reich29extended Browder s result
to the setting of Banach spaces and proved that ifE is a uniformly smooth Banach space, then xt converges strongly to a fixed point ofT and the limit defines the unique sunny
nonexpansive retraction fromContoFT.
Reich 29 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 fromContoD, and it can be constructed as follows.
Proposition 1.2see29. LetEbe a uniformly smooth Banach space, and letT : C → Cbe a nonexpansive mapping such thatFT/∅. For each fixedu∈Cand everyt∈0,1, the unique fixed pointxt∈Cof the contractionCx→tu 1−tTxconverges strongly ast → 0to a fixed point of T. DefineP : C → DbyP u s−limt→0xt. ThenP is the unique sunny nonexpansive retract fromContoD; that is,Psatisfies the following property:
u−P u, Jy−P u≤0, ∀u∈C, y∈D. 1.26
Note that we useP u s−limt→0xt to denote strong convergence toPu of the net{xt}as
t → 0.
In 2010, Qin et al.16considered the generalized equilibrium problem and a strictly pseudocontractive mapping to prove the following result.
Theorem QCK [see [
16
]]
LetCbe a nonempty closed convex subset of a real Hilbert spaceH. LetF be a bifunction from C×Cto R which satisfiesA1–A4, and let B : C → H be a λ-inverse strongly monotone mapping. LetS:C → Cbe ak-strict pseudocontraction, letA1:C → Hbe anα
-inverse strongly monotone mapping, and letA2 :C → Hbe aβ-inverse strongly monotone
mapping. Assume thatF : EPF, B∩VIC, A1∩VIC, A2∩FSis nonempty. Let{αn}
0,2β, and let{rn}be a sequence in0,2λ. Let{xn}be a sequence generated in the following
manner:
x1∈C, chosen arbitrary,
un∈Csuch thatFun, u Bxn, u−un
1
rnu−un, un−xn ≥
0, ∀u∈C,
zn QCun−snA2un,
yn QCzn−tnA1zn,
xn1 αnxn 1−αn
βnyn
1−βn
Syn
, ∀n≥1.
1.27
Assume that the sequences{αn},{βn},{tn},{sn}, and{rn}satisfy the following restrictions: a0< a≤αn≤a<1;
b0< k≤βn≤b <1;
c0< c≤rn≤d <2λ, 0< c≤sn≤d<2β, and 0< c≤tn≤d <2α.
Then the sequence{xn} generated in1.27converges weakly to some pointx ∈ F, where
x limn→ ∞QFxnandQFis the projection ofHonto setF.
Recently, W. Kumam and P. Kumam 12 introduced a new viscosity relaxed extragradient approximation method which is based on the so-called relaxed extragradient method and viscosity approximation method for finding the common element of the set of fixed points of a nonexpansive mapping, the set of solutions of an equilibrium problem, and the solutions of the variational inequality problem for two inverse strongly monotone mappings in Hilbert spaces. Katchang et al. 13 introduced a new iterative scheme for finding solutions of a variational inequality for inverse strongly accretive mappings with a viscosity approximation method in Banach spaces. They prove a strong convergence theorem in Banach spaces under some parameters controlling conditions. Katchang and Kumam30, further extended the work of26and constructed a viscosity iterative scheme for finding solutions of a general system of variational inequalities 1.22 for two inverse-strongly accretive operators with a viscosity of modified extragradient methods and solutions of fixed point problems involving the nonexpansive mapping in Banach spaces. Then, they obtained strong convergence theorems for a solution of the system of general variational inequalities
1.22in the frame work of Banach spaces.
Very recently, Qin et al. 31 considered the problem of finding the solutions of a general system of variational inclusion1.6withα-inverse strongly accretive mappings. To be more precise, they obtained the following results.
Lemma 1.3see31. For givenx∗, y∗∈E×E, wherey∗ JM2,ρ2x∗−ρ2A2x∗,x∗, y∗is a
solution of the problem1.1if and only ifx∗is a fixed point of the mappingQ defined by
Qx JM1,ρ1
JM2,ρ2
x−ρ2A2x
−ρ1A1JM2,ρ2
x−ρ2A2x
. 1.28
pseudocontraction with fixed point. Define a mappingSbySx 1−λ/K2x λ/K2Tx,for
allx∈E. Assume thatΘ FT∩FQ/∅, whereQis defined asLemma 1.3. Letx1 u∈E, and
let{xn}be a sequence generated by
zn JM2,ρ2
xn−ρ2A2xn
,
yn JM1,ρ1
zn−ρ1A1zn
,
xn1 αnuβnxn
1−βn−αn
μSxn
1−μyn
, ∀n≥1,
1.29
whereμ∈0,1,ρ1 ∈0, γ1/K2, ρ2∈0, γ2/K2and{αn}and{βn}are sequences in (0,1). If the control consequences{αn}and{βn}satisfy the following restrictions
C10<lim infn→ ∞βn≤lim supn→ ∞βn<1and C2limn→ ∞αn 0and∞n 0αn ∞,
then{xn}converges strongly tox∗ PΘu, wherePΘis the sunny nonexpansive retraction fromE
ontoΘandx∗, y∗, wherey∗ JM2,ρ2x∗−ρ2A2x∗, is solution to the problem1.6.
In this paper, motivated by the above results and the iterative schemes considered in Qin et al.31,32and Katchang and Kumam30, we present a new general iterative scheme so call a relaxed extragradient-type method for finding a common element of the set of solutions for generalized mixed equilibrium problems, the set of solutions of common system of variational inclusions for two inverse-strongly accretive operators and common set of fixed points for a strict pseudocontraction in 2-uniformly smooth Banach spaces. Then, we prove the strong convergence of the proposed iterative method under some suitable conditions. The results presented in this paper extend and improve the results of Qin et al.31,32and many authors.
2. Preliminaries
First, we recall some definitions and conclusions.
For solving the generalized mixed equilibrium problem, let us give the following assumptions for the bifunction F : C× C → R; ϕ : C → R is convex and lower semicontinuous; the nonlinear mappingB:C → E∗is continuous and monotone satisfying the following conditions:
A1Fx, x 0 for allx∈C;
A2Fis monotone, that is,Fx, y Fy, x≤0 for allx, y∈C;
A3for eachx, y, z∈C, limt↓0Ftz 1−tx, y≤Fx, y;
A4for eachx∈C, y→Fx, yis convex and lower semicontinuous;
B1for eachx∈Eandr >0, there exist abounded subsetDx⊆Candyx∈Csuch that
for anyz∈C\Dx,
Fz, yx
ϕyx
−ϕz 1 r
yx−z, Jz−Jx
<0; 2.1
Lemma 2.1see33, Lemma 2.7. LetCbe a closed convex subset of smooth, strictly convex, and reflexive Banach spaceE, letF:C×C → Rbe a bifunction satisfying (A1)–(A4), and letr >0and
x∈E. Then, there existsz∈Csuch that
Fz, y1
ry−z, Jz−Jx ≥0, ∀y∈C. 2.2
Motivated by the work of Combettes and Hirstoaga 34 in a Hilbert space and Takahashi and Zembayashi 33 in a Banach space, Zhang 35 and also authors of 36
obtained the following lemma.
Lemma 2.2 see 35. Let C be nonempty closed convex subset of a uniformly smooth, strictly convex and reflexive Banach spaceE. LetB : C → E∗be a continuous and monotone mapping, let
ϕ:C → Rbe a lower semicontinuous and convex function, and letF : C×C → Rbe a bifunction satisfying (A1)–(A4). Forr >0andx∈E, there existsu∈Csuch that
Fu, yBu, y−uϕy−ϕu 1
r
y−u, Ju−Jx, ∀y∈C. 2.3
Define a mappingKr : C → Cas follows:
Krx
u∈C:Fu, yBu, y−uϕy−ϕu 1
r
y−u, Ju−Jx≥0, ∀y∈C
2.4
for allx∈C. Then, the following conclusions hold:
1Kris single valued;
2Kris firmly nonexpansive; that is, for anyx, y∈E,Krx−Kry, JKrx−JKry ≤ Krx−
Kry, Jx−Jy;
3FKr GMEPF, ϕ, B;
4GMEPF, ϕ, Bis closed and convex.
Lemma 2.3see37. Assume that{an}is a sequence of nonnegative real numbers such that
an1≤1−αnanδn, n≥0, 2.5
where{αn}is a sequence in0,1and{δn}is a sequence inRsuch that 1∞n 1αn ∞;
2lim supn→ ∞δn/αn≤0or
∞
n 1|δn|<∞.
Then,limn→ ∞an 0.
Lemma 2.4see38. Let{xn}and{yn}be bounded sequences in a Banach spaceX, and let{βn}be a sequence in0,1with0<lim infn→ ∞βn≤lim supn→ ∞βn<1. Suppose thatxn1 1−βnyn
βnxnfor all integersn≥0andlim supn→ ∞yn1−yn−xn1−xn≤0. Then,limn→ ∞yn−xn
Lemma 2.5 see 23. The resolvent operator JM,ρ associated with M is single valued and nonexpansive for allρ >0.
Lemma 2.6see23. Letu∈ E. Thenuis a solution of variational inclusion1.6if and only if
u JM,ρu−ρAu,for allρ >0, that is,
VIE, A, M FJM,ρI−ρA, ∀ρ >0, 2.6
where VIE, A, Mdenotes the set of solutions to the problem1.8.
The following results describe a characterization of sunny nonexpansive retractions on a smooth Banach space.
Proposition 2.7see39. LetEbe a smooth Banach space, and letCbe a nonempty subset ofE. LetP:E → Cbe a retraction, and letJbe the normalized duality mapping onE. Then the following are equivalent:
1Pis sunny and nonexpansive;
2P x−P y2 ≤ x−y, JP x−P y, for allx, y∈C;
3x−P x, Jy−P x ≤0, for allx∈E, y∈C.
Proposition 2.8see40. LetCbe a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach spaceE, and let T be a nonexpansive mapping ofCinto itself withFT/∅. Then the setFTis a sunny nonexpansive retract ofC.
Lemma 2.9see31. LetEbe a strictly convex Banach space. LetT1andT2be two nonexpansive
mappings fromEinto itself with a common fixed point. Define a mappingSby
Sx λT1x 1−λT2x, ∀x∈E, 2.7
whereλis a constant in0,1. ThenSis nonexpansive andFS FT1∩FT2.
Lemma 2.10see28. Let Ebe a uniformly convex Banach space, and letS be a nonexpansive mapping onE. ThenI−Sis demiclosed at zero.
Lemma 2.11see31. LetEbe a real 2-uniformly smooth Banach space, and letT :E → Ebe a
λ-strict pseudocontraction. ThenS: 1−λ/K2Iλ/K2Tis nonexpansive andFT FS.
Lemma 2.12see 41. Let E be a real 2-uniformly smooth Banach space with the best smooth constantK. Then the following inequality holds:
xy2
≤ x22y, Jx2Ky2, ∀x, y∈E. 2.8
Lemma 2.13. In a real Banach spaceE, the following inequality holds:
xy2
Lemma 2.14. LetCbe a nonempty closed convex subset of a real 2-uniformly smooth Banach spaceE
with the smooth constantK. Let the mappingA:E → Ebe aγ-inverse-strongly accretive mapping. Ifρ∈0, γ/K2, thenI−ρAis nonexpansive.
Proof. For anyx, y∈C, fromLemma 2.12, one has
I−ρAx−I−ρAy2 x−y−ρAx−Ay2
≤x−y2−2ρAx−Ay, Jx−y2K2ρ21Ax−Ay2
≤x−y2−2ργAx−Ay22K2ρ2Ax−Ay2
x−y2−
2ργ−K2ρAx−Ay2
≤x−y2,
2.10
which implies that the mappingI−ρAis nonexpansive.
3. Main Result
In this section, we prove a strong convergence theorem for finding a common element of the set of fixed points of strict pseudocontraction mappings, the set of solutions of a generalized mixed equilibrium problem, and the set of solutions of system of quasivariational inclusion problem for an inverse-strongly monotone mapping in a uniformly convex and 2-uniformly smooth Banach space.
Theorem 3.1. LetEbe a uniformly convex and 2-uniformly smooth Banach space with the smooth constantK. Let Mi : E → 2E be a maximal monotone mapping, and letAi : E → E be aγi -inverse strongly accretive mapping, respectively, for eachi 1,2. LetF be a bifunction of C×C
into real numbersRsatisfying (A1)–(A4). LetB:E → E∗be a continuous and monotone mapping and let ϕ : C → R∪ {∞} be a proper lower semicontinuous and convex function. Let f be a contraction ofEinto itself with coefficientα∈0,1. LetS:E → Ebe aλ-strict pseudocontraction with a fixed point. Define a mappingSk by Skx kx 1−kSx, for all x ∈ E. Assume that
Ω : FS∩FQ∩GMEPF, ϕ, B/∅, whereQ is defined as inLemma 1.3. Assume that either
(B1) or (B2) holds. Let{xn}be a sequence generated byx1 ∈Eand
Fun, y
Bun, y−unϕ
y−ϕun 1
r
y−un, Jun−Jxn
≥0, ∀y∈C,
yn JM2,ρ2
un−ρ2A2un
, vn JM1,ρ1
yn−ρ1A1yn
,
xn1 αnfxn βnxnγn
μ1Skxn
1−μ1
vn
,
3.1
for everyn ≥ 1, where{αn},{βn}and{γn}are sequences in 0,1, μ1 ∈ 0,1,ρ1 ∈ 0, γ1/K2,
iαnβnγn 1,
ii∞n 0αn ∞andlimn→ ∞αn 0,
iii0<lim infn→ ∞βn≤lim supn→ ∞βn<1,
then{xn}converges strongly tox∈Ω, wherex PΩfx,PΩis the sunny nonexpansive retraction fromEontoΩandx, yis solution to the problem1.6, wherey JM2,ρ2x−ρ2A2x.
Proof. LetHun, y Fun, y Bun, y−unϕy−ϕun, y∈C,
Kr
u∈C : Hun, y
1
r
y−un, Jun−Jxn
≥0, ∀y∈C
. 3.2
First, from condition ρ1 ∈ 0, γ1/K2,ρ2 ∈ 0, γ2/K2 and Lemma 2.14, we have that the
mappingsI−ρ1A1andI−ρ2A2are nonexpansive.
We claim that{xn}is bounded. Takingx∈Ω, one has
x JM1,ρ1
JM2,ρ2
x−ρ2A2x
−ρ1A1JM2,ρ2
x−ρ2A2x
. 3.3
Puttingy JM2,ρ2x−ρ2A2x, one sees that
x JM1,ρ1
y−ρ1A1y
. 3.4
Sincex KrxandKris nonexpansive mapping, we have
un−x ≤ Krxn−Krx ≤ xn−x. 3.5
From the fact thatJM2,ρ2andI−ρ2A2are nonexpansive mappings, we get
yn−y JM
2,ρ2
un−ρ2A2un
−JM2,ρ2
x−ρ2A2x ≤un−ρ2A2un
−x−ρ2A2x
I−ρ2A2
un−
I−ρ2A2
x
≤ un−x ≤ xn−x.
3.6
Similar to the above, from the fact thatJM1,ρ1andI−ρ1A1are nonexpansive mappings, we
also have
FromSkbeing nonexpansive and puttingen μ1Skxn 1−μ1vn, we have
en−x μ1Skxn−x
1−μ1
vn−x
≤μ1Skxn−x
1−μ1
vn−x
μ1Skxn−Skx
1−μ1
xn−x
≤μ1xn−x
1−μ1
xn−x xn−x.
3.8
From3.1,3.8, andαnβnγn 1, we note that
xn1−x αn
fxn−xβnxn−x γnen−x
≤αnfxn−xβnxn−xγnen−x
≤αnfxn−fxαnfx−xβnxn−xγnen−x
≤αnαxn−xαnfx−xβnxn−xγnxn−x
αnαxn−xαnfx−x 1−αnxn−x
1−1−ααnxn−x 1−ααn
fx−x 1−α ,
3.9
for everyn∈N. It follows by mathematical induction that
xn1−x ≤max
x1−x,
fx−x 1−α
. 3.10
This shows that the sequence{xn}is bounded, so are{un},{vn}, and{yn}.
We claim thatxn1−xn → 0 asn → ∞.
From algorithm3.1, we have
yn1−yn JM
2,ρ2
un1−ρ2A2un1
−JM2,ρ2
un−ρ2A2un
≤un1−ρ2A2un1
−un−ρ2A2un
≤ un1−un
Krxn1−Krxn ≤ xn1−xn.
3.11
Fromen μ1Skxn 1−μ1vn, we have
en1−en μ1Skxn1
1−μ1
vn1−
μ1Skxn
1−μ1
vn
μ1Skxn1−Skxn 1−μ1
vn1−vn ≤μ1Skxn1−Skxn
1−μ1
vn1−vn
≤μ1xn1−xn
1−μ1
xn1−xn xn1−xn.
3.12
Puttingln xn1−βnxn/1−βn, for alln≥1. That is,
xn1
1−βn
lnβnxn. 3.13
One sees that
ln1−ln
αn1fxn1 γn1en1
1−βn1 −
αnfxn γnen
1−βn
αn1
1−βn1fxn1
1−βn1−αn1
1−βn1 en1−
αn
1−βnfxn−
1−βn−αn
1−βn en
αn1
1−βn1
fxn1−en1
αn
1−βn
en−fxn
en1−en.
3.14
It follows that
ln1−ln ≤ 1α−n1
βn1
fxn1−en1 αn
1−βn
en−fxnen1−en. 3.15
Substituting3.12into3.15, we acheive
ln1−ln − xn1−xn ≤ 1α−n1
βn1
fxn1−en1 αn
1−βn
en−fxn. 3.16
It follows from the conditionsiiandiiithat
lim sup
n→ ∞
ln1−ln − xn1−xn ≤0. 3.17
FromLemma 2.4, we obtain
lim
From3.13, we see
xn1−xn
1−βn
ln−xn. 3.19
In view of conditioniii, we have
lim
n→ ∞xn1−xn 0. 3.20
On the other hand, one has
xn1−xn αnfxn βnxn
1−αn−βn
en−xn
αn
fxn−en
1−βn
en−xn.
3.21
It follows that
1−βn
en−xn ≤αnfxn−enxn1−xn. 3.22
From conditionsii,iiiand3.20, one sees that
lim
n→ ∞en−xn 0. 3.23
Next, we show that limn→ ∞un−xn 0.
Lettingp∈Ω, we get thatp Krp. ByLemma 2.2; that is,Kris firmly nonexpansive,
we have
un−p2 Krxn−Krp2
≤Krxn−Krp, Jxn−Jp
un−p, Jxn−Jp
≤un−pJxn−Jp
≤un−pxn−p
≤ 1
2
un−p2xn−p2− xn−un2
.
3.24
It follows that
un−p2≤
Observe that
vn−p2 JM1,ρ1yn−ρ1A1yn−JM1,ρ1p−ρ1A1p
2
≤yn−ρ1A1yn−p−ρ1A1p2
≤yn−p2
JM2,ρ2un−ρ2A2un−JM2,ρ2p−ρ2A2p2
≤un−ρ2A2un−p−ρ2A2p2
≤un−p2.
3.26
From3.25and3.26, we have
en−p2 μ1Skxn 1−μ1vn−p2
≤μ1Skxn−p2
1−μ1vn−p2
≤μ1xn−p2
1−μ1un−p2
≤μ1xn−p2
1−μ1xn−p2− xn−un2
xn−p2−
1−μ1
xn−un2.
3.27
From3.1and3.27, we obtain xn1−p2
αnfxn βnxn 1−αn−βnen−p2
≤αnfxn−p2βnxn−p2
1−αn−βnen−p2
≤αnfxn−p2βnxn−p2
1−αn−βn xn−p2−
1−μ1
xn−un2
≤αnfxn−p2 1−αnxn−p2−
1−αn−βn
1−μ1
xn−un2
≤αnfxn−p2xn−p2−
1−αn−βn
1−μ1
xn−un2.
3.28
It follows that
1−αn−βn
1−μ1
xn−un2≤αnfxn−p2xn1−xnxn−pxn1−p.
3.29
Fromi–iii,μ1∈0,1, andxn1−xn → 0 asn → ∞, we have
lim
Next, we prove that
p∈Ω: FS∩FJM1,ρ1
I−ρ1A1
JM2,ρ2
I−ρ2A2
∩GMEPF, ϕ, B. 3.31
iWe will show thatp∈GMEPF, ϕ, B.
SinceJis uniformly norm-to-norm continuous on bounded sets, we have
lim
n→ ∞Jxn−Jun 0. 3.32
We obtain
lim
n→ ∞
Jxn−Jun
r 0. 3.33
Noticing thatun Krxn, we have
Hun, y
1
r
y−un, Jun−Jxn
≥0, ∀y∈C. 3.34
FromA2, we note that
y−unJun−Jxn
r ≥
1
r
y−un, Jun−Jxn
≥ −Hun, y
≥Hy, un
, ∀y∈C. 3.35
Taking the limit asn → ∞in the above inequality, fromA4andun → p, we have
Hy, p≤0, y ∈C. For 0< t <1 andy∈C, defineyt ty 1−tp. Noticing thaty, p∈C,
we obtainyt∈C, which yieldsHyt, p≤0. It follows fromA1that
0 Hyt, yt
≤tHyt, y
1−tHyt, p
≤tHyt, y
, 3.36
that is,Hyt, y≥0.
Lett ↓ 0; fromA3, we obtainHp, y ≥ 0, y ∈ C. This implies thatp ∈ GMEPF, ϕ, B.
iiNext, we will show thatp∈FS∩FJM1,ρ1I−ρ1A1JM2,ρ2I−ρ2A2.
Define a mappingG:E → Eby
Gx μ1Skx
1−μ1
JM1,ρ1
I−ρ1A1
JM2,ρ2
I−ρ2A2
x, x∈E. 3.37
FromLemma 2.9, we see thatGis nonexpansive mapping such that
FG FS∩FJM1,ρ1
I−ρ1A1
JM2,ρ2
I−ρ2A2
. 3.38
We define a mappingG:E → EbyGx σGx 1−σKrx, x∈E, σ∈0,1.
Again fromLemma 2.9, we see thatGis nonexpansive mapping such that
FG FG∩GMEPF, ϕ, B
FS∩FJM1,ρ1
I−ρ1A1
JM2,ρ2
I−ρ2A2
∩GMEPF, ϕ, B.
3.39
Hence,p∈Ω. Next, we show that lim supn→ ∞fx−x, Jxn−x ≤0, wherex PΩfx.
Since{xn}is bounded, we can choose a sequence {xni}of {xn}whichxni psuch
that
lim sup
n→ ∞ fx−x, Jxn−x ilim→ ∞fx−x, Jxni−x. 3.40
Now, from3.40andProposition 2.7iiiand sinceJis strong to weak∗uniformly continuous on bounded subset ofE, we have
lim sup
n→ ∞
fx−x, Jxn−x
lim
i→ ∞fx−x, Jxni−x
fx−x, Jp−x≤0.
3.41
From3.20, it follows that
lim sup
n→ ∞
fx−x, Jxn1−x
≤0. 3.42
Finally, we show thatxn → xasn → ∞.
Notice that
xn1−x2 αnfxn−x βnxn−x 1−αn−βnen−x2
≤βnxn−x
1−αn−βn
en−x22αnfxn−x, Jxn1−x
≤βnxn−x
1−αn−βn
en−x
2
2αnfxn−fx, Jxn1−x
2αn
fx−x, Jxn1−x
≤βnxn−x
1−αn−βn
xn−x
2
2αnαxn−x, Jxn1−x
2αn
fx−x, Jxn1−x
≤1−αn2xn−x2αnα
xn−x2xn1−x2
2αn
fx−x, Jxn1−x
,
which implies that
xn1−x2≤
1−αn2αnα
1−αnα xn−x
2 2αn
1−αnαfx−x, Jxn1−x
≤
1−2αn1−α 1−αnα
xn−x2
2αn1−α
1−αnα
×
1
1−α
fx−x, Jxn1−x
αn
21−αM2
,
3.44
whereM2is an appropriate constant such thatM2≥supn≥1{xn−x2}.
Setbn 2αn1−α/1−αnαandcn 1/1−αfx−x, Jxn1−xαn/21−αM2.
Then, we have
xn1−x2≤1−bnxn−x2bncn, ∀n≥0. 3.45
From conditioniiand3.42, we see that
lim
n→ ∞bn 0, ∞
n 0
bn ∞, lim sup
n→ ∞ cn≤0. 3.46
Therefore, applyingLemma 2.3to3.45, we have
lim
n→ ∞xn−x 0. 3.47
This completes the proof.
UsingTheorem 3.1, we obtain the following corollaries.
Corollary 3.2. LetEbe a uniformly convex and 2-uniformly smooth Banach space with the smooth constantK. LetMi :E → 2Ebe a maximal monotone mapping, and letAi:E → Ebe aγi-inverse strongly accretive mapping, respectively, for eachi 1,2. LetF be a bifunction ofC×Cinto real numbersRsatisfying (A1)–(A4). Letf be a contraction ofEinto itself with coefficientα ∈ 0,1. LetS:E → Ebe anλ-strict pseudocontraction with a fixed point. Define a mappingSkbySkx
kx 1−kSx, for allx∈E. Assume thatΩ: FS∩FQ∩EPF/∅, whereQ is defined as Lemma 1.3. Let{xn}be a sequence generated byx1∈Eand
Fun, y
1
r
y−un, Jun−Jxn
≥0, ∀y∈C,
yn JM2,ρ2
un−ρ2A2un
, vn JM1,ρ1
yn−ρ1A1yn
,
xn1 αnfxn βnxnγn
μ1Skxn
1−μ1
vn
,
for everyn ≥ 1, where{αn},{βn}, and{γn}are sequences in0,1,μ1 ∈ 0,1,ρ1 ∈ 0, γ1/K2,
ρ2∈0, γ2/K2, andr >0. If the control sequences satisfy the following restrictions:
iαnβnγn 1,
ii∞n 0αn ∞andlimn→ ∞αn 0, iii0<lim infn→ ∞βn≤lim supn→ ∞βn<1,
then{xn}converges strongly tox PΩfx, wherePΩis the sunny nonexpansive retraction fromE
ontoΩandx, yis a solution to the problem1.6, wherey JM2,ρ2x−ρ2A2x.
Proof. PutB ϕ 0, inTheorem 3.1. The conclusion ofCorollary 3.2can be obtained with the desired result easily.
Corollary 3.3. LetEbe a uniformly convex and 2-uniformly smooth Banach space with the smooth constantK. Let Mi : E → 2E be a maximal monotone mapping, and letAi : E → E be aγi -inverse strongly accretive mapping, respectively, for each i 1,2. LetS : E → E be a λ-strict pseudocontraction with a fixed point, and letf be a contraction of Einto itself with coefficientα ∈
0,1. Define a mappingSkbySkx kx 1−kSx,∀x∈E. Assume thatΩ: FS∩FQ/∅, whereQis defined as inLemma 1.3. Let{xn}be a sequence generated byx1 ∈Eand
yn JM2,ρ2
xn−ρ2A2xn
, vn JM1,ρ1
yn−ρ1A1yn
,
xn1 αnfxn βnxnγn
μ1Skxn
1−μ1
vn
,
3.49
for everyn ≥ 1, where{αn},{βn}, and{γn}are sequences in0,1,μ1 ∈ 0,1,ρ1 ∈ 0, γ1/K2,
ρ2∈0, γ2/K2. If the control sequences satisfy the following restrictions:
iαnβnγn 1,
ii∞n 0αn ∞andlimn→ ∞αn 0, iii0<lim infn→ ∞βn≤lim supn→ ∞βn<1,
then{xn}converges strongly tox PΩfx, wherePΩis the sunny nonexpansive retraction fromE ontoΩandx, yis a solution to the problem1.6, wherey JM2,ρ2x−ρ2A2x.
Proof. PutFx, y 0, for all x, y ∈ C, andB ϕ 0, inTheorem 3.1. The conclusion of
Corollary 3.3can be obtained with the desired result easily.
Remark 3.4. Corollary 3.3extends and improves the results in31.
Corollary 3.5. LetEbe a uniformly convex and 2-uniformly smooth Banach space with the smooth constantK. LetM:E → 2Ebe a maximal monotone mapping, and letA:E → Ebe aγ -inverse-strongly accretive mapping. LetS:E → Ebe aλ-strict pseudocontraction with a fixed point. Define a mappingSkbySkx kx 1−kS, for allx∈E. Assume thatΩ: FS∩SV IE, A, M/∅. Let{xn}be a sequence generated byx1 u∈Eand
yn JM,ρ
xn−ρAxn
,
vn JM,ρ
yn−ρAyn
, xn1 αnuβnxnγn
μ1Skxn
1−μ1
vn
,
for everyn≥1, where{αn},{βn}, and{γn}are sequences in0,1,μ1 ∈0,1,ρ∈0, γ/K2. If the
control sequences satisfy the following restrictions:
iαnβnγn 1,
ii∞n 0αn ∞andlimn→ ∞αn 0, iii0<lim infn→ ∞βn≤lim supn→ ∞βn<1,
then{xn}converges strongly tox PΩu, wherePΩis the sunny nonexpansive retraction fromEonto Ωandx, yis a solution to the problem1.7, wherey JM,ρx−ρAx.
Proof. PutFx, y 0, for allx, y∈C,B ϕ 0,M1 M2 M,A1 A2 A, andfx u
for allx∈EinTheorem 3.1. The conclusion ofCorollary 3.5can be obtained with the desired result easily.
Acknowledgments
This research is supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand. Also, the authors would like to thank the referees for their careful readings and valuable suggestions to improve the writing of this paper.
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