Parallel hybrid iterative methods for variational inequalities, equilibrium problems and common fixed point problems

# Parallel hybrid iterative methods for variational inequalities, equilibrium problems and common fixed point problems

P. K. Anh P. K. Anh (Corresponding author)D.V. Hieu College of Science, Vietnam National University, Hanoi, Nguyen Trai, Thanh Xuan, Hanoi, Vietnam
33email: anhpk@vnu.edu.vn, dv.hieu83@gmail.com
D.V. Hieu P. K. Anh (Corresponding author)D.V. Hieu College of Science, Vietnam National University, Hanoi, Nguyen Trai, Thanh Xuan, Hanoi, Vietnam
33email: anhpk@vnu.edu.vn, dv.hieu83@gmail.com
###### Abstract

In this paper we propose two strongly convergent parallel hybrid iterative methods for finding a common element of the set of fixed points of a family of quasi -asymptotically nonexpansive mappings , the set of solutions of variational inequalities and the set of solutions of equilibrium problems in uniformly smooth and 2-uniformly convex Banach spaces. A numerical experiment is given to verify the efficiency of the proposed parallel algorithms.

###### Keywords:
Quasi -asymptotically nonexpansive mapping Variational inequality Equilibrium problem Hybrid method Parallel computation
###### Msc:
47H05 47H09 47H10 47J25 65J15 65Y05
journal: Vietnam Journal of Mathematicsdedicatory: Dedicated to Professor Nguyen Khoa Son’s 65th Birthday

## 1 Introduction

Let be a nonempty closed convex subset of a Banach space . The variational inequality for a possibly nonlinear mapping , consists of finding such as

 ⟨Ap∗,p−p∗⟩≥0,∀p∈C. (1.1)

The set of solutions of (1.1) is denoted by .
Takahashi and Toyoda TT2003 () proposed a weakly convergent method for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality for an - inverse strongly monotone mapping in a Hilbert space.

###### Theorem 1.1

TT2003 () Let be a closed convex subset of a real Hilbert space . Let . Let be an - inverse strongly-monotone mapping of into , and let be a nonexpansive mapping of into itself such that . Let be a sequence generated by

for every , where for some and for some . Then, converges weakly to , where .

In 2008, Iiduka and Takahashi IT2008 () considered problem in a 2-uniformly convex, uniformly smooth Banach space under the following assumptions:

• is -inverse-strongly-monotone.

• .

• for all and .

###### Theorem 1.2

IT2008 () Let be a -uniformly convex, uniformly smooth Banach space whose duality mapping is weakly sequentially continuous, and let be a nonempty, closed convex subset of . Assume that is a mapping of into satisfing conditions . Suppose that and is given by

 xn+1=ΠCJ−1(Jxn−λnAxn)

for every where is a sequence of positive numbers. If is chosen so that for some with , then the sequence converges weakly to some element in Here is the -uniform convexity constant of , and .

In 2009, Zegeye and Shahzad ZS2009 () studied the following hybrid iterative algorithm in a 2-uniformly convex and uniformly smooth Banach space for finding a common element of the set of fixed points of a weakly relatively nonexpansive mapping and the set of solutions of a variational inequality involving an -inverse strongly monotone mapping :

 ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩yn=ΠC(J−1(Jxn−λnAxn)),zn=Tyn,H0={v∈C:ϕ(v,z0)≤ϕ(v,y0)≤ϕ(v,x0)},Hn={v∈Hn−1⋂Wn−1:ϕ(v,zn)≤ϕ(v,yn)≤ϕ(v,xn)},W0=C,Wn={v∈Hn−1⋂Wn−1:⟨xn−v,Jx0−Jxn⟩≥0},xn+1=PHn⋂Wnx0,n≥1,

where is the normalized duality mapping on . The strong convergence of to has been established.
Kang, Su, and Zhang KSZ2010 () extended this algorithm to a weakly relatively nonexpansive mapping, a variational inequality and an equilibrium problem. Recently, Saewan and Kumam SK2012 () have constructed a sequential hybrid block iterative algorithm for an infinite family of closed and uniformly quasi - asymptotically nonexpansive mappings, a variational inequality for an -inverse-strongly monotone mapping, and a system of equilibrium problems.

Qin, Kang, and Cho QKC2009 () considered the following sequential hybrid method for a pair of inverse strongly monotone and a quasi -nonexpansive mappings in a 2-uniformly convex and uniformly smooth Banach space:

 ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩x0=E,C1=C,x1=ΠC1x0,un=ΠC(J−1(Jxn−ηnBxn)),zn=ΠC(J−1(Jun−λnAun)),yn=Tzn,Cn+1={v∈Cn:ϕ(v,yn)≤ϕ(v,zn)≤ϕ(v,un)≤ϕ(v,xn)},xn+1=ΠCn+1x0,n≥0.

They proved that the sequence converges strongly to , where

Let be a bifunction from to a set of real numbers . The equilibrium problem for consists of finding an element , such that

 f(ˆx,y)≥0,∀y∈C. (1.2)

The set of solutions of the equilibrium problem is denoted by . Equilibrium problems include several problems such as: variational inequalities, optimization problems, fixed point problems, ect. In recent years, equilibrium problems have been studied widely and several solution methods have been proposed (see ABH2014 (); KSZ2010 (); SK2012 (); SLZ2011 (); TT2007 ()). On the other hand, for finding a common element in , Takahashi and Zembayashi TZ2009 () introduced the following algorithm in a uniformly smooth and uniformly convex Banach space:

 ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩x0∈C,yn=J−1(αnJxn+(1−αn)JTyn),un∈C,such that,f(un,y)+1rn⟨y−un,Jun−Jyn⟩≥0∀y∈C,Hn={v∈C:ϕ(v,un)≤ϕ(v,xn)},Wn={v∈C:⟨xn−v,Jx0−Jxn⟩≥0},xn+1=PHn⋂Wnx0,n≥1.

The strong convergence of the sequences and to has been established.
Recently, the above mentioned algorithms have been generalized and modified for finding a common point of the set of solutions of variational inequalities, the set of fixed points of quasi - (asymptotically) nonexpansive mappings, and the set of solutions of equilibrium problems by several authors, such as Takahashi and Zembayashi TZ2009 (), Wang et al. WKC2012 () and others.
Very recently, Anh and Chung AC2013 () have considered the following parallel hybrid method for a finite family of relatively nonexpansive mappings :

 ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩x0∈C,yin=J−1(αnJxn+(1−αn)JTixn),i=1,…,N,in=argmax1≤i≤N{∥∥yin−xn∥∥},¯yn:=yinn,Cn={v∈C:ϕ(v,¯yn)≤ϕ(v,xn)},Qn={v∈C:⟨Jx0−Jxn,xn−v⟩≥0},xn+1=ΠCn⋂Qnx0,n≥0.

This algorithm was extended, modified and generelized by Anh and Hieu AH2014 () for a finite family of asymptotically quasi -nonexpansive mappings in Banach spaces. Note that the proposed parallel hybrid methods in AC2013 (); AH2014 () can be used for solving simultaneuous systems of maximal monotone mappings. Other parallel methods for solving accretive operator equations can be found in ABH2014 ().
In this paper, motivated and inspired by the above mentioned results, we propose two novel parallel iterative methods for finding a common element of the set of fixed points of a family of asymptotically quasi -nonexpansive mappings , the set of solutions of variational inequalities , and the set of solutions of equilibrium problems in uniformly smooth and 2-uniformly convex Banach spaces, namely:
Method A

 ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩x0∈Cchosen arbitrarily,yin=ΠC(J−1(Jxn−λnAixn)),i=1,2,…M,in=argmax{||yin−xn||:i=1,…,M},¯yn=yinn,zjn=J−1(αnJxn+(1−αn)JSnj¯yn),j=1,…,N,jn=argmax{||zjn−xn||:j=1,…,N},¯zn=zjnn,ukn=Tkrn¯zn,k=1,…,K,kn=argmax{||ukn−xn||:k=1,2,…K},¯un=uknn,Cn+1={z∈Cn:ϕ(z,¯un)≤ϕ(z,¯zn)≤ϕ(z,xn)+ϵn},xn+1=ΠCn+1x0,n≥0, (1.3)

where, is a unique solution to a regularized equlibrium problem

Further, the control parameter sequences satisfy the conditions

 0≤αn≤1,limsupn→∞αn<1,λn∈[a,b],rn≥d, (1.4)

for some with being the 2-uniform convexity constant of Concerning the sequence , we consider two cases. If the mappings are quasi -asymptotically nonexpansive, we assume that the solution set is bounded, i.e., there exists a positive number , such that and put . If the mappings are quasi -nonexapansive, then , and we put .
Method B

 ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩x0∈Cchosen arbitrarily,yin=ΠC(J−1(Jxn−λnAixn)),i=1,…,M,in=argmax{||yin−xn||:i=1,…,M},¯yn=yinn,zn=J−1(αn,0Jxn+∑Nj=1αn,jJSnj¯yn),ukn=Tkrnzn,k=1,…,K,kn=argmax{||ukn−xn||:k=1,…,K},¯un=uinn,Cn+1={z∈Cn:ϕ(z,¯un)≤ϕ(z,xn)+ϵn},xn+1=ΠCn+1x0,n≥0, (1.5)

where, the control parameter sequences satisfy the conditions

 0≤αn,j≤1,N∑j=0αn,j=1,limn→∞infαn,0αn,j>0,λn∈[a,b],rn≥d. (1.6)

In Method A , knowing we find the intermediate approximations in parallel. Using the farthest element among from , we compute in parallel. Further, among , we choose the farthest element from and determine solutions of regularized equilibrium problems in parallel. Then the farthest from element among denoted by is chosen. Based on , a closed convex subset is constructed. Finally, the next approximation is defined as the generalized projection of onto .
A similar idea of parallelism is employed in Method B . However, the subset in Method B is simpler than that in Method A.
The results obtained in this paper extend and modify the corresponding results of Zegeye and Shahzad ZS2009 (), Takahashi and Z embayashi TZ2009 (), Anh and Chung AC2013 (), Anh and Hieu AH2014 () and others.
The paper is organized as follows: In Section 2, we collect some definitions and results needed for further investigtion. Section 3 deals with the convergence analysis of the methods and . In the last section, a novel parallel hybrid iterative method for variational inequalities and closed, quasi - nonexpansive mappings is studied.

## 2 Preliminaries

In this section we recall some definitions and results which will be used later. The reader is refered to AR2006 () for more details.

###### Definition 1

A Banach space is called

• strictly convex if the unit sphere is strictly convex, i.e., the inequality holds for all

• uniformly convex if for any given there exists such that for all with the inequality holds;

• smooth if the limit

 limt→0∥x+ty∥−∥x∥t (2.1)

exists for all ;

• uniformly smooth if the limit exists uniformly for all .

The modulus of convexity of is the function defined by

for all . Note that is uniformly convex if only if for all and . Let , is said to be -uniformly convex if there exists some constant such that . It is well-known that spaces and are -uniformly convex if and -uniformly convex if and a Hilbert space is uniformly smooth and -uniformly convex.

Let be a real Banach space with its dual . The dual product of and is denoted by or . For the sake of simpicity, the norms of and are denoted by the same symbol . The normalized duality mapping is defined by

 J(x)={f∈E∗:⟨f,x⟩=∥x∥2=∥f∥2}.

The following properties can be found in C1990 ():

• If is a smooth, strictly convex, and reflexive Banach space, then the normalized duality mapping is single-valued, one-to-one, and onto;

• If is a reflexive and strictly convex Banach space, then is norm to weak continuous;

• If is a uniformly smooth Banach space, then is uniformly continuous on each bounded subset of ;

• A Banach space is uniformly smooth if and only if is uniformly convex;

• Each uniformly convex Banach space has the Kadec-Klee property, i.e., for any sequence , if and , then .

###### Lemma 2.1

ZS2009 () If is a 2-uniformly convex Banach space, then

 ||x−y||≤2c2||Jx−Jy||,∀x,y∈E,

where is the normalized duality mapping on and .

The best constant is called the -uniform convexity constant of .
Next we assume that is a smooth, strictly convex, and reflexive Banach space. In the sequel we always use to denote the Lyapunov functional defined by

From the definition of , we have

 (2.2)

Moreover, the Lyapunov functional satisfies the identity

 ϕ(x,y)=ϕ(x,z)+ϕ(z,y)+2⟨z−x,Jy−Jz⟩ (2.3)

for all .
The generalized projection is defined by

 ΠC(x)=argminy∈Cϕ(x,y).

In what follows, we need the following properties of the functional and the generalized projection .

###### Lemma 2.2

A1996 () Let E be a smooth, strictly convex, and reflexive Banach space and a nonempty closed convex subset of . Then the following conclusions hold:

• ;

• if , then iff ;

• iff .

###### Lemma 2.3

KL2008 () Let C be a nonempty closed convex subset of a smooth Banach , and . For a given real number , the set

 D:={v∈C:ϕ(v,z)≤λϕ(v,x)+(1−λ)ϕ(v,y)+a}

is closed and convex.

###### Lemma 2.4

A1996 () Let and be two sequences in a uniformly convex and uniformly smooth real Banach space . If and either or is bounded, then as .

###### Lemma 2.5

CKW2010 () Let E be a uniformly convex Banach space, be a positive number and be a closed ball with center at origin and radius . Then, for any given subset and for any positive numbers with , there exists a continuous, strictly increasing, and convex function with such that, for any with ,

 ∥∥ ∥∥N∑k=1λkxk∥∥ ∥∥2≤N∑k=1λk∥xk∥2−λiλjg(||xi−xj||).
###### Definition 2

A mapping is called

• monotone, if

 ⟨A(x)−A(y),x−y⟩≥0∀x,y∈E;
• uniformly monotone, if there exists a strictly increasing function such that

 ⟨A(x)−A(y),x−y⟩≥ψ(||x−y||)∀x,y∈E; (2.4)
• -strongly monotone, if there exists a positive constant such that in ,

• -inverse strongly monotone, if there exists a positive constant such that

 ⟨A(x)−A(y),x−y⟩≥α||A(x)−A(y)||2∀x,y∈E.
• -Lipschitz continuous if there exists a positive constant , such that

 ||A(x)−A(y)||≤L||x−y||∀x,y∈E.

If is -inverse strongly monotone then it is -Lipschitz continuous. If is -strongly monotone and -Lipschitz continuous then it is -inverse strongly monotone.

###### Lemma 2.6

T2000 () Let be a nonempty, closed convex subset of a Banach space and be a monotone, hemicontinuous mapping of into . Then

 VI(C,A)={u∈C:⟨v−u,A(v)⟩≥0,∀v∈C}.

Let be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space , be a mapping. The set

 F(T)={x∈E:Tx=x}

is called the set of fixed points of . A point is said to be an asymptotic fixed point of if there exists a sequence such that and as . The set of all asymptotic fixed points of will be denoted by .

###### Definition 3

A mapping is called

• relatively nonexpansive mapping if , and

 ϕ(p,Tx)≤ϕ(p,x),∀p∈F(T),∀x∈C;
• closed if for any sequence and , then ;

• quasi - nonexpansive mapping (or hemi-relatively nonexpansive mapping) if and

 ϕ(p,Tx)≤ϕ(p,x),∀p∈F(T),∀x∈C;
• quasi - asymptotically nonexpansive if and there exists a sequence with as such that

 ϕ(p,Tnx)≤knϕ(p,x),∀n≥1,∀p∈F(T),∀x∈C;
• uniformly - Lipschitz continuous, if there exists a constant such that

 ∥Tnx−Tny∥≤L∥x−y∥,∀n≥1,∀x,y∈C.

The reader is refered to CKW2010 (); SWX2009 () for examples of closed and asymptotically quasi -nonexpansive mappings. It has been shown that the class of asymptotically quasi -nonexpansive mappings contains properly the class of quasi -nonexpansive mappings, and the class of quasi -nonexpansive mappings contains the class of relatively nonexpansive mappings as a proper subset.

###### Lemma 2.7

CKW2010 () Let be a real uniformly smooth and strictly convex Banach space with Kadec-Klee property, and be a nonempty closed convex subset of . Let be a closed and quasi -asymptotically nonexpansive mapping with a sequence . Then is a closed convex subset of .

Next, for solving the equilibrium problem , we assume that the bifunction satisfies the following conditions:

1. for all ;

2. is monotone, i.e., for all ;

3. For all ,

 limt→0+supf(tz+(1−t)x,y)≤f(x,y);
4. For all , is convex and lower semicontinuous.

The following results show that in a smooth (uniformly smooth), strictly convex and reflexive Banach space, the regularized equilibrium problem has a solution (unique solution), respectively.

###### Lemma 2.8

TZ2009 () Let be a closed and convex subset of a smooth, strictly convex and reflexive Banach space , be a bifunction from to satisfying conditions - and let , . Then there exists such that

 f(z,y)+1r⟨y−z,Jz−Jx⟩≥0,∀y∈C.
###### Lemma 2.9

TZ2009 () Let be a closed and convex subset of a uniformly smooth, strictly convex and reflexive Banach spaces , be a bifunction from to satisfying conditions -. For all and , define the mapping

 Trx={z∈C:f(z,y)+1r⟨y−z,Jz−Jx⟩≥0,∀y∈C}.

Then the following hold:

(B1) is single-valued;

(B2) is a firmly nonexpansive-type mapping, i.e., for all

 ⟨Trx−Try,JTrx−JTry⟩≤⟨Trx−Try,Jx−Jy⟩;

(B3)

(B4) is closed and convex and is relatively nonexpansive mapping.

###### Lemma 2.10

TZ2009 () Let be a closed convex subset of a smooth, strictly convex and reflexive Banach space . Let be a bifunction from to satisfying and let . Then, for and ,

 ϕ(q,Trx)+ϕ(Trx,x)≤ϕ(q,x).

Let be a real Banach space. Alber A1996 () studied the function defined by

 V(x,x∗)=||x||2−2⟨x,x∗⟩+||x∗||2.

Clearly, .

###### Lemma 2.11

A1996 () Let E be a refexive, strictly convex and smooth Banach space with as its dual. Then

 V(x,x∗)+2⟨J−1x−x∗,y∗⟩≤V(x,x∗+y∗),∀x∈Eand∀x∗,y∗∈E∗.

Consider the normal cone to a set at the point defined by

 NC(x)={x∗∈E∗:⟨x−y,x∗⟩≥0,∀y∈C}.

We have the following result.

###### Lemma 2.12

R1970 () Let be a nonempty closed convex subset of a Banach space and let be a monotone and hemi-continuous mapping of into with . Let be a mapping defined by:

 Q(x)={Ax+NC(x) if x∈C,∅ifx∉C.

Then is a maximal monotone and .

## 3 Convergence analysis

Throughout this section, we assume that is a nonempty closed convex subset of a real uniformly smooth and 2-uniformly convex Banach space . Denote

 F=(M⋂i=1VI(Ai,C))⋂(N⋂j=1F(Sj))⋂(K⋂k=1EP(fk))

and assume that the set is nonempty.
We prove convergence theorems for methods and with the control parameter sequences satisfying conditions and , respectively. We also propose similar parallel hybrid methods for quasi -nonexpansive mappings, variational inequalities and equilibrium problems.

###### Theorem 3.1

Let be a finite family of mappings from to satisfying conditions (V1)-(V3). Let be a finite family of bifunctions satisfying conditions (A1)-(A4). Let be a finite family of uniform -Lipschitz continuous and quasi--asymptotically nonexpansive mappings with the same sequence . Assume that there exists a positive number such that . If the control parameter sequences satisfy condition , then the sequence generated by converges strongly to .

###### Proof

We divide the proof of Theorem 3.1 into seven steps.
Step 1. Claim that are closed convex subsets of .
Indeed, since each mapping is uniformly -Lipschitz continuous, it is closed. By Lemmas and 2.9, and are closed convex sets, therefore, and are also closed and convex. Hence is a closed and convex subset of . It is obvious that is closed for all . We prove the convexity of by induction. Clearly, is closed convex. Assume that is closed convex for some . From the construction of , we find

Lemma 2.3 ensures that is convex. Thus, is closed convex for all . Hence, and are well-defined.
Step 2. Claim that for all .
By Lemma 2.10 and the relative nonexpansiveness of , we obtain for all . From the convexity of and the quasi -asymptotical nonexpansiveness of , we find

 ϕ(u,¯zn) = ϕ(u,J−1(αnJxn+(1−αn)JSnjn¯yn)) (3.1) = ||u||2−2αn⟨u,xn⟩−2(1−αn)⟨u,JSnjn¯yn⟩ +||αnJxn+(1−αn)JSnjn¯yn||2 ≤ ||u||2−2αn⟨u,xn⟩−2(1−αn)⟨u,JSnjn¯yn⟩ +αn||xn||2+(1−αn)||Snjn¯yn||2 = αnϕ(u,xn)+(1−αn)ϕ(u,Snjn¯yn) ≤ αnϕ(u,xn)+(1−αn)knϕ(u,¯yn)

for all . By the hypotheses of Theorem 3.1, Lemmas and , we have

 ϕ(u,¯yn) = ϕ(u,ΠC(J−1(Jxn−λnAinxn))) (3.2) ≤ ϕ(u,J−1(Jxn−λnAinxn)) = V(u,Jxn−λnAinxn) ≤ V(u,Jxn−λnAinxn+λnAinxn) = ϕ(u,xn)−2λn⟨J−1(Jxn−λnAinxn)−J−1(Jxn),Ainxn⟩ −2λn⟨xn−u,Ainxn−Ain(u)⟩−2λn⟨xn−u,Ainu⟩ ≤ ϕ(u,xn)+4λnc2||Jxn−λnAinxn−Jxn