# A Hybrid Extragradient-Like Method for Variational Inequalities, Equilibrium Problems, and an Infinitely Family of Strictly Pseudocontractive Mappings

Authors: Yaqin Wang; Hongkun Xu; Xiaoli Fang
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

The purpose of this paper is to consider a new scheme by the hybrid extragradient-like method for finding a common element of the set of solutions of a generalized mixed equilibrium problem, the set of solutions of a variational inequality, and the set of fixed points of an infinitely family of strictly pseudocontractive mappings in Hilbert spaces. Then, we obtain a strong convergence theorem of the iterative sequence generated by the proposed iterative algorithm. Our results extend and improve the results of Issara Inchan (2010) and many others.


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## Body

## 1. Introduction

LetH be a real Hilbert space with inner product 〈·,·〉 and norm ∥·∥. Let C be a nonempty closed convex subset of H and A be a mapping of C into H. We denote by F(A) the set of fixed points of A and by PC the metric projection of H onto C. We also denote by R the set of all real numbers.Recall the following definitions.(i)
A is called monotone if
(1.1)〈Ax-Ay,x-y〉≥0,∀x,y∈C.(ii)
A is called α-inverse-strongly monotone if there exists a positive constant α such that
(1.2)〈Ax-Ay,x-y〉≥α‖Ax-Ay‖2,∀x,y∈C.(iii)
A is called k-Lipschitz continuous if there exists a positive constant k such that(1.3)‖Ax-Ay‖≤k‖x-y‖,∀x,y∈C.
Clearly, every inverse strongly monotone mapping is Lipschitz continuous and monotone.A mappingT:C→C is said to be ξ-strictly pseudocontractive if there exists a constant ξ∈[0,1) such that(1.4)‖Tx-Ty‖2≤‖x-y‖2+ξ‖(I-T)x-(I-T)y‖2,∀x,y∈C.It is known that ifT is a 0-strictly pseudocontractive mapping, then T is a nonexpansive mapping. So the class of ξ-strictly pseudocontractive mappings includes the class of nonexpansive mappings.LetΘ:C×C→R be a bifunction. The equilibrium problem for Θ is to find that x∈C such that(1.5)Θ(x,y)≥0,∀y∈C.
The set of solutions of problem (1.5) is denoted by EP.Given a mappingA:C→H, let Θ(x,y)=〈Ax,y-x〉 for all x,y∈C. Then problem (1.5) reduces to the following classical variational inequality problem of finding x∈C such that(1.6)〈Ax,y-x〉≥0,∀y∈C.
The set of solutions of problem (1.6) is denoted by VI(C,A).Numerous problems in physics, optimization, saddle point problems, complementarity problems, mechanics, and economics reduce to find a solution of problem (1.5). Many methods have been proposed to solve problem (1.5); see, for instant, [1–3]. In 1997, Combettes and Hirstoaga [4] introduced an iterative scheme of finding the best approximation to initial data when EP is nonempty and proved a strong convergence theorem.Recently, Peng and Yao [5] introduced the following generalized mixed equilibrium problem of finding x∈C such that
(1.7)Θ(x,y)+φ(y)-φ(x)+〈Bx,y-x〉≥0,∀y∈C,
where B:C→H is a nonlinear mapping, and φ:C→R is a function. The set of solutions of problem (1.7) is denoted by GMEP.In the case ofB=0 and φ=0, then problem (1.7) reduces to problem (1.5). In the case of Θ=0,φ=0, and B=A, then problem (1.7) reduces to problem (1.6). In the case of φ=0, problem (1.7) reduces to the generalized equilibrium problem. In the case of B=0, problem (1.7) reduces to the following mixed equilibrium problem of finding x∈C such that
(1.8)Θ(x,y)+φ(y)-φ(x)≥0,∀y∈C,
which was considered by Ceng and Yao [6]. The set of sulutions of this problem is denoted by MEP.The problem (1.7) is very general in the sense that it includes, as special cases, some optimization, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games, economics, and others (see, for instant, [6])Recently, S. Takahashi and W. Takahashi [7] introduced the following iteration process:(1.9)x1=u∈C,Θ(un,y)+〈Axn,y-un〉+1r〈y-un,un-xn〉≥0,∀y∈C,xn+1=βnxn+(1-βn)S[αnu+(1-αn)un],n≥1,
and used this iteration process to find a common element of the set of fixed points of a nonexpansive mapping S and the set of solutions of a generalized equilibrium problem in a Hilbert space.In 2008, Bnouhachem et al. [8] introduced the following new extragradient iterative method. Let C be a closed convex subset of a real Hilbert H, A be an α-inverse strongly monotone mapping of C into H, and let S be a nonexpansive mapping of C into itself such that F(S)∩VI(C,A)≠∅. Let the sequences {xn}, {yn} be given by(1.10)x1,u∈Cchosen arbitrary,yn=PC(xn-λnAxn),xn+1=βnxn+(1-βn)S(αnu+(1-αn)PC(xn-λnAyn)),∀n≥1,
where {αn}, {βn}, and {λn}⊂(0,1) satisfy some parameters controlling conditions. They proved that the sequence {xn} converges strongly to a common element of F(S)∩VI(C,A).In 2010, Ceng et al. [9] introduced the following hybrid extragradient-like method. Let C be a nonempty closed convex subset of a real Hilbert space H, A:C→H be a monotone, k-Lipschitz continuous mapping, and let S:C→C be a nonexpansive mapping such that F(S)∩VI(C,A)≠∅. Let the sequences {xn}, {yn}, and {zn} be defined by(1.11)x0∈C,yn=(1-γn)xn+γnPC(xn-λnAxn),zn=(1-αn-βn)xn+αnyn+βnSPC(xn-λnAyn),Cn={z∈C:‖zn-z‖2≤‖xn-z‖2+(3-3γn+αn)b2‖Axn‖2},Qn={z∈C,〈xn-z,x0-xn〉≥0},xn+1=PCn∩Qnx0,∀n≥0.
Under the suitable conditions, they proved the sequences {xn}, {yn}, {zn} converge strongly to the same point PF(S)∩VI(C,A)x0.In 2010, Inchan [10] introduced a new iterative scheme by the hybrid extragradient method in a Hilbert space H as follows: x0∈H,C1=C⊂H, x1=PCx0, and let(1.12)un∈C,Θ(un,y)+〈Bxn,y-un〉+1rn〈y-un,un-xn〉≥0,∀y∈C,yn=PC(un-λnAun),zn=αnxn+(1-αn)S(βnxn+(1-βn)PC(un-λnAyn)),Cn+1={z∈Cn:‖zn-z‖≤‖xn-z‖},xn+1=PCn+1x0,∀n≥0,
where {αn}, {βn}, and {λn}⊂(0,1) satisfy some parameters controlling conditions. They proved that {xn} and {un} strongly converge to the same common element of the set of fixed points of a nonexpansive mapping, the set of solutions of an equilibrium problem, and the set of solutions of the variational inequality for nonexpansive mappings.Very recently, Wang [11] defined the mapping Wn as follows: (1.13)Un,n+1=I,Un,n=γnTn′Un,n+1+(1-γn)I,Un,n-1=γn-1Tn-1′Un,n+(1-γn-1)I,⋮Un,k=γkTk′Un,k+1+(1-γk)I,Un,k-1=γk-1Tk-1′Un,k+(1-γk-1)I,⋮Un,2=γ2T2′Un,3+(1-γ2)I,Wn=Un,1=γ1T1′Un,2+(1-γ1)I,
where γ1,γ2,… are real numbers such that 0≤γn≤1, Ti′=θiI+(1-θi)Ti, where Ti is a μi-strictly pseudocontractive mapping of C into itself and θi∈[μi,1). It follows from [12] that Ti′ is nonexpansive and F(Ti)=F(Ti′). Nonexpansivity of each Ti′ ensures the nonexpansivity of Wn.Motivated and inspired by the above work, in this paper, we introduced the following new iterative scheme by the extragradient-like method:C1=C⊂H,x1=PCx0,(1.14)un∈Csuch thatΘ(un,y)+〈Bxn,y-un〉+φ(y)-φ(un)+1rn〈y-un,un-xn〉≥0,y∈C,yn=(1-sn)un+snPC(un-λnAun),zn=αnxn+(1-αn)Wn(βnxn+(1-βn)PC(un-λnAyn)),Cn+1={z∈Cn:‖zn-z‖2≤‖xn-z‖2+3(1-sn)b2‖Aun‖2},xn+1=PCn+1x0,∀n≥0,
where A,B:C→H and A is a monotone, k-Lipschitz continuous mapping, and B is a β-inverse strongly monotone mapping. Then under the suitable conditions, we derive some strong convergence results.


## 2. Preliminaries

LetC be a nonempty closed and convex subset of a Hilbert space H, for any x∈H, and there exists a unique nearest point in C, denoted by PCx such that
(2.1)‖x-PCx‖≤‖x-y‖,∀y∈C.
The projection operator PC:H→C is nonexpansive. Moreover, PCx is characterized by the following properties: for every x∈H and y∈C,
(2.2)‖x-y‖2≥‖x-PCx‖2+‖y-PCx‖2,(2.3)〈x-PCx,y-PCx〉≤0.Suppose thatA is monotone and continuous. Then the solutions of the variational inequality VI(C,A) can be characterized as solutions of the so-called Minty variational inequality:
(2.4)x*∈VI⁡(C,A)⟺〈Ax,x-x*〉≥0,∀x∈C.In what follows, we shall make use of the following lemmas.Lemma 2.1.
LetH be a real Hilbert space. Then for any x,y∈H, we have(i)
∥x±y∥2=∥x∥2±2〈x,y〉+∥y∥2,(ii)
∥tx+(1-t)y∥2=t∥x∥2+(1-t)∥y∥2-t(1-t)∥x-y∥2,forallt∈[0,1].
We denote byNC(v) the normal cone for C at a point v∈C, that is NC(v):={x*∈E*:〈v-y,x*〉≥0forally∈C}. In the following, we shall use the following Lemma.Lemma 2.2 (see [13]).
LetC be a nonempty closed convex subset of a Banach space E, and let A be a monotone and hemicontinuous operator of C into E*. Let T⊂E×E* be an operator defined as follows:
(2.5)Tv={Av+NC(v),v∈C,∅,v∉C.
Then T is maximal monotone, and T-10=VI(C,A).Lemma 2.3 (see [14]).
LetC be a nonempty closed convex subset of a strictly convex Banach space E. Let T1′,T2′,… be nonexpansive mappings of C into itself such that ⋂i=1∞F(Ti′)≠∅ and γ1,γ2,… be real numbers such that 0<γi≤b<1 for every i=1,2,…. Then for any x∈C and k∈N, the limit limn→∞Un,k exists.
Using Lemma2.3, define the mapping W of C into itself as follows:
(2.6)Wx:=lim⁡n→∞Wnx=lim⁡n→∞Un,1x,x∈C.
Such a mapping W is called the modified W mapping generalized by T1,T2,…,γ1,γ2,…, and θ1,θ2,….Lemma 2.4 (see [14]).
LetC be a nonempty closed convex subset of a strictly convex Banach space E. Let T1′,T2′,… be nonexpansive mappings of C into itself such that ⋂i=1∞F(Ti′)≠∅ and γ1,γ2,… be real numbers such that 0<γi≤b<1 for every i=1,2,…. Then W is a nonexpansive mapping satisfying that F(W)=⋂i=1∞F(Ti′).Lemma 2.5 (see [15]).
LetC be a nonempty closed convex subset of a strictly convex Banach space E. Let T1′,T2′,… be nonexpansive mappings of C into itself such that ⋂i=1∞F(Ti′)≠∅ and γ1,γ2,… be real numbers such that 0<γi≤b<1 for every i=1,2,…. If K is any bounded subset of C, then
(2.7)lim⁡n→∞sup⁡x∈K‖Wx-Wnx‖=0.
For solving the equilibrium problem, let us assume thatΘ satisfies the following conditions:(H1)
Θ(x,x)=0 for all x∈C,(H2)
Θ is monotone, that is, Θ(x,y)+Θ(y,x)≤0 for all x,y∈C,(H3)
for eachy∈C,x↦Θ(x,y) is weakly upper semicontinuous,(H4)
for eachx∈C,y↦Θ(x,y) is convex and lower semicontinuous,(A1)
for eachx∈H and r>0, there exists a bounded subset Dx⊂C and yx∈C such that for any z∈C∖Dx,
(2.8)Θ(z,yx)+φ(yx)-φ(z)+1r〈yx-z,z-x〉<0,(A2)
C is a bounded set.Lemma 2.6 (see [6]).
LetC be a closed subset of H. Let φ:C→R be a lower semicontinuous and convex function, and Θ be a bifunction from C×C to R satisfying (H1)–(H4). For r>0 and x∈H, define a mapping Tr:H→C as follows:
(2.9)Tr(x)={z∈C:Θ(z,y)+φ(y)-φ(z)+1r〈y-z,z-x〉≥0,∀y∈C}
for all x∈H. Assume that either (A1) or (A2) holds. Then the following results hold: (1)
Tr(x)≠∅ for each x∈H, and Tr is single valued,(2)
Tr is firmly nonexpansive, that is, for all x,y∈H,∥Trx-Try∥2≤〈Trx-Try,x-y〉,(3)
F(Tr)=MEP,(4)
MEP is closed and convex.


## 3. Strong Convergence Theorems

Theorem 3.1.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let Θ be a bifunction from C×C into R satisfying (H1)–(H4) and φ:C→R be a lower semicontinuous and convex function with (A1) or (A2). Let A:C→H be a monotone, k-Lipschitz continuous mapping and B:C→H be a β-inverse-strongly monotone mapping. Let Ti:C→C be a μi-strictly pseudocontractive mapping with F=:⋂i=1∞F(Ti)∩VI(C,A)∩GMEP≠∅ and {γi} be a real sequence such that 0<γi≤b<1, for all i≥1. Assume that the control sequences {αn}, {βn}, {sn}⊂[0,1], {rn}⊂(0,2β),and {λn}⊂(0,1/2k) satisfy the following conditions:(i)
limsupn→∞αn<1,limsupn→∞βn<1,(ii)
0<a≤λn≤b<1/2k,0<d≤rn≤e<2β,(iii)
sn→1(n→∞) and sn>3/4 for all n≥0.
Then the sequence{xn} defined by (1.14) converges strongly to PFx0.Proof.
We divide the proof into several steps.
Step 1 ({xn} is well defined).
Indeed, for anyq∈F. Put tn=PC(un-λnAyn). Since un=Trn(xn-rnBxn), q=Trn(q-rnBq), and B is β-inverse-strongly monotone and rn∈[0,2β], for any n≥0, we have
(3.1)‖un-q‖2=‖Trn(xn-rnBxn)-Trn(q-rnBq)‖2≤‖xn-rnBxn-(q-rnBq)‖2≤‖xn-q‖2-2rn〈Bxn-Bq,xn-q〉+rn2‖Bxn-Bq‖2≤‖xn-q‖2+rn(rn-2β)‖Bxn-Bq‖2≤‖xn-q‖2.
It follows from (2.2) and (2.4) that
(3.2)∥tn-q∥2≤‖un-λnAyn-q‖2-‖un-λnAyn-tn‖2=‖un-q‖2-‖un-tn‖2-2λn〈Ayn,tn-q〉=‖un-q‖2-‖un-yn‖2-‖yn-tn‖2-2λn〈Ayn,yn-q〉-2〈un-yn,yn-tn〉+2λn〈Ayn,yn-tn〉≤‖un-q‖2-‖un-yn‖2-‖yn-tn‖2+2〈un-yn-λnAyn,tn-yn〉.
In addition, we have
(3.3)〈un-yn-λnAyn,tn-yn〉=〈un-yn-λnAun,tn-yn〉+λn〈Aun-Ayn,tn-yn〉≤sn〈un-λnAun-PC(un-λnAun),tn-yn〉+λn(sn-1)〈Aun,tn-yn〉+λnk‖un-yn‖‖tn-yn‖,
and by (2.3), we obtain
(3.4)〈un-λnAun-PC(un-λnAun),tn-yn〉=〈un-λnAun-PC(un-λnAun),(1-sn)(tn-un)+sn(tn-PC(un-λnAun))〉≤(1-sn)λn‖Aun‖(‖tn-yn‖+‖yn-un‖)+sn〈un-λnAun-PC(un-λnAun),tn-PC(un-λnAun)〉≤(1-sn)λn‖Aun‖(‖tn-yn‖+‖yn-un‖).
It follows from (3.2)–(3.4) that
(3.5)‖tn-q‖2≤‖un-q‖2-‖un-yn‖2-‖yn-tn‖2+2sn(1-sn)λn‖Aun‖(‖tn-yn‖+‖yn-un‖)+2λn(1-sn)‖Aun‖‖tn-yn‖+2λnk‖un-yn‖‖tn-yn‖≤‖un-q‖2-‖un-yn‖2-‖yn-tn‖2+sn(1-sn)[2b2‖Aun‖2+‖tn-yn‖2+‖yn-un‖2]+(1-sn)[b2∥Aun∥2+∥tn-yn∥2]+kb[∥tn-yn∥2+∥yn-un∥2]=∥un-q∥2-(sn-bk)∥un-yn∥2-(2sn-1-kb)∥tn-yn∥2+3(1-sn)b2∥Aun∥2≤∥un-q∥2+3(1-sn)b2∥Aun∥2.
Setting wn=βnxn+(1-βn)tn. Therefore, from (1.14), (3.1), and (3.5), we get the following:
(3.6)‖zn-q‖2≤αn‖xn-q‖2+(1-αn)‖Wnwn-q‖2≤αn‖xn-q‖2+(1-αn)‖wn-q‖2≤αn‖xn-q‖2+(1-αn)βn‖xn-q‖2+(1-αn)(1-βn)‖tn-q‖2≤(αn+(1-αn)βn+(1-αn)(1-βn))‖xn-q‖2+3(1-αn)(1-βn)(1-sn)b2‖Aun‖2≤‖xn-q‖2+3(1-sn)b2‖Aun‖2.
So, q∈Cn and hence F⊂Cn for all n≥1. It is easy to see that Cn is closed and convex for all n≥1.  This implies that {xn} and {un} are well defined.Step 2 ({xn} is a Cauchy sequence).
It is easy to see thatF is closed and convex. From xn+1=PCn+1x0∈Cn+1⊂Cn and xn=PCnx0, for any q∈F, we have
(3.7)‖xn-x0‖≤‖xn+1-x0‖≤‖q-x0‖.
So {xn} is bounded, and limn→∞∥xn-x0∥ exists. So it follows from (3.1), (3.6), and the continuity of A that {un}, {zn}, and {Aun} are bounded. By the construction of Cn, we have Cm⊂Cn and xm=PCmx0∈Cn for any positive integer m≥n. So from (2.2), we have
(3.8)‖xm-xn‖2≤‖xm-x0‖2-‖xn-x0‖2.
Letting m,n→∞ in (3.8), we have ∥xm-xn∥→0, which implies that {xn} is a Cauchy sequence. So there exists z∈C such that xn→z(n→∞).Step 3 (limn→∞∥wn-Wwn∥=0).
From (3.8), we have
(3.9)lim⁡n→∞‖xn-xn+1‖=0.
Since xn+1∈Cn+1, by (3.9) and condition (iii), we obtain that
(3.10)‖zn-xn+1‖≤‖xn-xn+1‖+3(1-sn)b2‖Aun‖2⟶0(n⟶∞).
So
(3.11)lim⁡n→∞‖zn-xn‖=0.
Since
(3.12)‖zn-xn‖=(1-αn)‖xn-Wnwn‖,
from (3.11) and condition (i), we have
(3.13)lim⁡n→∞‖xn-Wnwn‖=0.
For any q∈F, from (3.1) and (3.5), we obtain that
(3.14)‖wn-q‖2≤βn‖xn-q‖2+(1-βn)‖tn-q‖2≤βn‖xn-q‖2+(1-βn)‖un-q‖2+3(1-sn)b2‖Aun‖2≤‖xn-q‖2-(1-βn)rn(2β-rn)‖Bxn-Bq‖2+3(1-sn)b2‖Aun‖2.
Therefore, we have
(3.15)‖zn-q‖2≤αn‖xn-q‖2+(1-αn)‖wn-q‖2≤‖xn-q‖2-(1-αn)(1-βn)rn(2β-rn)‖Bxn-Bq‖2+3(1-sn)b2‖Aun‖2,
which implies that
(3.16)(1-αn)(1-βn)rn(2β-rn)‖Bxn-Bq‖2≤‖xn-q‖2-‖zn-q‖2+3(1-sn)b2‖Aun‖2≤‖zn-xn‖(‖zn-q‖+‖xn-q‖)+3(1-sn)b2‖Aun‖2.
Combining the above inequality, (3.11) and conditions (i)–(iii), we have
(3.17)lim⁡n→∞‖Bxn-Bq‖=0.
It follows from Lemma 2.6 that
(3.18)‖un-q‖2=‖Trn(xn-rnBxn)-Trn(q-rnBq)‖2≤〈xn-rnBxn-(q-rnBq),un-q〉=12(‖xn-rnBxn-(q-rnBq)‖2+‖un-q‖2-‖xn-rnBxn-(q-rnBq)-(un-q)‖2)≤12(‖xn-q‖2+‖un-q‖2-‖xn-un‖2+2rn〈xn-un,Bxn-Bq〉-rn2‖Bxn-Bq‖2).
Therefore,
(3.19)‖un-q‖2≤‖xn-q‖2-‖xn-un‖2+2rn〈xn-un,Bxn-Bq〉-rn2‖Bxn-Bq‖2.
By (3.5) and (3.19), we have
(3.20)‖zn-q‖2≤αn‖xn-q‖2+(1-αn)‖wn-q‖2≤αn‖xn-q‖2+(1-αn)(βn‖xn-q‖2+(1-βn)‖tn-q‖2)≤αn‖xn-q‖2+(1-αn)(βn‖xn-q‖2+(1-βn)‖un-q‖2+3(1-sn)b2‖Aun‖2)≤‖xn-q‖2-(1-αn)(1-βn)‖xn-un‖2+2rn(1-αn)(1-βn)〈xn-un,Bxn-Bq〉-(1-αn)(1-βn)rn2‖Bxn-Bq‖2+3(1-sn)b2‖Aun‖2.
It follows from (3.20) that
(3.21)(1-αn)(1-βn)‖xn-un‖2≤‖xn-q‖2-‖zn-q‖2+2rn(1-αn)(1-βn)〈xn-un,Bxn-Bq〉-(1-αn)(1-βn)rn2‖Bxn-Bq‖2+3(1-sn)b2‖Aun‖2≤‖zn-xn‖(‖zn-q‖+‖xn-q‖)+2e‖xn-un‖‖Bxn-Bq‖+e2‖Bxn-Bq‖2+3(1-sn)b2‖Aun‖2.
Therefore, from (3.11), (3.17), (3.21), and conditions (i), (iii),
(3.22)lim⁡n→∞‖xn-un‖=0,
which implies that un→z(n→∞). And from (3.1), (3.5), and (3.6), we have
(3.23)‖zn-q‖2≤αn‖xn-q‖2+(1-αn)βn‖xn-q‖2+(1-αn)(1-βn)‖tn-q‖2≤‖xn-q‖2-(1-αn)(1-βn)(sn-bk)‖un-yn‖2-(1-αn)(1-βn)(2sn-1-kb)‖tn-yn‖2+3(1-sn)b2‖Aun‖2.
Thus it follows that
(3.24)(1-αn)(1-βn)(sn-bk)‖un-yn‖2+(1-αn)(1-βn)(2sn-1-kb)‖tn-yn‖2≤‖xn-q‖2-‖zn-q‖2+3(1-sn)b2‖Aun‖2≤‖zn-xn‖(‖zn-q‖+‖xn-q‖)+3(1-sn)b2‖Aun‖2.
Therefore, from (3.11), (3.24) and conditions (i)–(iii), we obtain limn→∞∥un-yn∥=0 and limn→∞∥tn-yn∥=0. Furthermore, we have
(3.25)lim⁡n→∞‖tn-un‖=0,
which implies that tn→z(n→∞). It follows from (3.22) and (3.25) that
(3.26)lim⁡n→∞‖tn-xn‖=0.
Note that wn-xn=(1-βn)(tn-xn), so by (3.26) and condition (i), we obtain that
(3.27)lim⁡n→∞‖wn-xn‖=0,
which implies that wn→z(n→∞). Note that
(3.28)‖wn-Wwn‖≤‖wn-xn‖+‖xn-Wnwn‖+‖Wnwn-Wwn‖.
Therefore, by (3.13), (3.27), (3.28) and Lemma 2.5, we have
(3.29)lim⁡n→∞‖wn-Wwn‖=0.Step 4 (z∈F).
Sincewn→z(n→∞) and W is nonexpansive, by (3.29), we have
(3.30)‖z-Wz‖≤‖z-wn‖+‖wn-Wwn‖+‖Wwn-Wz‖⟶0(n⟶∞).
So z=Wz, that is,z∈F(W)=⋂i=1∞F(Ti).
Next we show thatz∈GMEP. Indeed, from (H2) and (1.14), we get
(3.31)〈Bxn,y-un〉+φ(y)-φ(un)+1rn〈y-un,un-xn〉≥Θ(y,un).
Put yt=ty+(1-t)z,t∈(0,1] and y∈C. So yt∈C. By (3.31), we have
(3.32)〈Byt,yt-un〉≥〈Byt,yt-un〉-〈Bxn,yt-un〉-φ(yt)+φ(un)-〈yt-un,un-xnrn〉+Θ(yt,un)=〈Byt-Bun,yt-un〉+〈Bun-Bxn,yt-un〉-φ(yt)+φ(un)-〈yt-un,un-xnrn〉+Θ(yt,un)≥-‖Bun-Bxn‖‖yt-un‖-φ(yt)+φ(un)-‖yt-un‖‖un-xnrn‖+Θ(yt,un).
Let n→∞ in (3.32), since B is nonexpansive and φ is lower semicontinuous, by (3.22), condition (ii) and (H4), we have
(3.33)〈Byt,yt-z〉≥-φ(yt)+φ(z)+Θ(yt,z).
So, from (H1), (H4), and the above inequality, we obtain
(3.34)0=Θ(yt,yt)+φ(yt)-φ(yt)≤tΘ(yt,y)+(1-t)Θ(yt,z)+tφ(y)+(1-t)φ(z)-φ(yt)=t(Θ(yt,y)+φ(y)-φ(yt))+(1-t)(Θ(yt,z)+φ(z)-φ(yt))≤t(Θ(yt,y)+φ(y)-φ(yt))+(1-t)t〈Byt,y-z〉,
that is,
(3.35)Θ(yt,y)+φ(y)-φ(yt)+(1-t)〈Byt,y-z〉≥0.
Letting t→0 in the above inequality, we obtain for each y∈C,
(3.36)Θ(z,y)+φ(y)-φ(z)+〈Bz,y-z〉≥0.
This implies that z∈GMEP.
Finally, we show thatz∈VI(C,A). Define a mapping T as Lemma 2.2. Let (v,u)∈G(T). Since u-Av∈NCv and tn∈C, we have 〈v-tn,u-Av〉≥0. Since tn=PC(un-λnAyn), we have
(3.37)〈v-tn,tn-(un-λnAyn)〉≥0,
and hence
(3.38)〈v-tn,u〉≥〈v-tn,Av〉≥〈v-tn,Av〉-〈v-tn,tn-unλn+Ayn〉=〈v-tn,Av-Ayn-tn-unλn〉=〈v-tn,Av-Atn〉+〈v-tn,Atn-Ayn〉-〈v-tn,tn-unλn〉≥-‖v-tn‖‖Atn-Ayn‖-‖v-tn‖‖tn-unλn‖.
Since limn→∞∥tn-yn∥=0, and A is Lipschitz continuous, by (3.25) and condition (ii), we deduce that 〈v-z,u〉≥0. Since T is maximal monotone, we have z∈T-10 and so z∈VI(C,A). Hence z∈F.Step 5 (z=PFx0).
Putz*=PFx0. Since xn=PCnx0, z∈F and the norm is lower semicontinuous, we have
(3.39)‖z*-x0‖≤‖z-x0‖≤lim⁡inf⁡n→∞‖xn-x0‖=lim⁡n→∞‖xn-x0‖≤‖z*-x0‖,
that is, ∥z*-x0∥=∥z-x0∥. Hence z=z*=PFx0, since z* is the unique element in F that minimizes the distance from x0.
Thus,{xn} converges strongly to PFx0.Remark 3.2.
Theorem3.1 mainly improves the results of Inchan [10]. To be more precise, Theorem 3.1 improves and extends Theorem 3.1 of [10] from the following several aspects:(i)
from a single nonexpansive mapping to an infinite family of strictly pseudocontractive mappings,(ii)
from generalized equilibrium problems to generalized mixed equilibrium problems,(iii)
from hybrid extragradient methods to hybrid extragradient-like methods,(iv)
the condition ofA relaxes to monotone, Lipschitz continuous.


## 4. Application

Theorem 4.1.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let Θ be a bifunction from C×C into R satisfying (H1)–(H4) and φ:C→R be a lower semicontinuous and convex function with (A1) or (A2). Let A:C→H be a monotone, k-Lipschitz continuous mapping and T:C→C be a ξ-strictly pseudocontractive mapping. Let Ti:C→C be a μi-strictly pseudocontractive mapping with F=:⋂i=1∞F(Ti)∩VI(C,A)∩GMEP≠∅ and {γi} be a real sequence such that 0<γi≤b<1, for all i≥1. Let the sequence {xn} be generated C1=C⊂H,x1=PCx0,
(4.1)un∈CsuchthatΘ(un,y)+〈(I-T)xn,y-un〉+φ(y)-φ(un)+1rn〈y-un,un-xn〉≥0,y∈C,yn=PC(un-λnAun),zn=αnxn+(1-αn)Wn(βnxn+(1-βn)PC(un-λnAyn)),Cn+1={z∈Cn:‖zn-z‖≤‖xn-z‖},xn+1=PCn+1x0,∀n≥0.
Assume that the control sequence {αn}, {βn}⊂[0,1], {rn}⊂(0,1-ξ) and {λn}⊂(0,1/2k) satisfy the following conditions:(i)
limsupn→∞αn<1,limsupn→∞βn<1,(ii)
0<a≤λn≤b<1/2k, 0<d≤rn≤e<1-ξ.Then {xn} converges strongly to PFx0.Proof.
Aξ-strictly pseudocontractive mapping is (1-ξ)/2-inverse-strongly monotone. Then taking sn=1, for all n≥1 in Theorem 3.1, we obtain the conclusion.



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*Source: 804642-2012-04-05.xml*