Parallel hybrid iterative methods for
variational inequalities, equilibrium problems
and common fixed point problems
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 2uniformly 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 computationMsc:
47H05 47H09 47H10 47J25 65J15 65Y051 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
(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 stronglymonotone 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 2uniformly convex, uniformly smooth Banach space under the following assumptions:

is inversestronglymonotone.

.

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
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 2uniformly 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 :
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 inversestrongly 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 2uniformly convex and uniformly smooth Banach space:
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
(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:
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 :
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 2uniformly convex Banach spaces, namely:
Method A
(1.3) 
where, is a unique solution to a regularized equlibrium problem
Further, the control parameter sequences satisfy the conditions
(1.4) 
for some with being the 2uniform 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
(1.5) 
where, the control parameter sequences satisfy the conditions
(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
(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 wellknown 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
The following properties can be found in C1990 ():

If is a smooth, strictly convex, and reflexive Banach space, then the normalized duality mapping is singlevalued, onetoone, 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 KadecKlee property, i.e., for any sequence , if and , then .
Lemma 2.1
ZS2009 () If is a 2uniformly convex Banach space, then
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
(2.3) 
for all .
The generalized projection is defined by
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
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 ,
Definition 2
A mapping is called

monotone, if

uniformly monotone, if there exists a strictly increasing function such that
(2.4) 
strongly monotone, if there exists a positive constant such that in ,

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

Lipschitz continuous if there exists a positive constant , such that
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
Let be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space , be a mapping. The set
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

closed if for any sequence and , then ;

quasi  nonexpansive mapping (or hemirelatively nonexpansive mapping) if and

quasi  asymptotically nonexpansive if and there exists a sequence with as such that

uniformly  Lipschitz continuous, if there exists a constant such that
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 KadecKlee 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:

for all ;

is monotone, i.e., for all ;

For all ,

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
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
Then the following hold:
(B1) is singlevalued;
(B2) is a firmly nonexpansivetype mapping, i.e., for all
(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 ,
Lemma 2.11
A1996 () Let E be a refexive, strictly convex and smooth Banach space with as its dual. Then
Consider the normal cone to a set at the point defined by
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 hemicontinuous mapping of into with . Let be a mapping defined by:
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 2uniformly convex Banach space . Denote
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 quasiasymptotically 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 welldefined.
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
(3.1)  
for all . By the hypotheses of Theorem 3.1, Lemmas and , we have
(3.2)  