DSpace at VNU: Parallel Hybrid Iterative Methods for Variational Inequalities, Equilibrium Problems, and Common Fixed Point Problems

DSpace at VNU: Parallel Hybrid Iterative Methods for Variational Inequalities, Equilibrium Problems, and Common Fixed Po...

DOI 10.1007/s10013-015-0129-z

Parallel Hybrid Iterative Methods for Variational

Inequalities, Equilibrium Problems, and Common

Fixed Point Problems

Pham Ky Anh · Dang Van Hieu

Received: 15 October 2014 / Accepted: 16 November 2014

© Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2015

Abstract In this paper, we propose two strongly convergent parallel hybrid iterative

meth-ods for finding a common element of the set of fixed points of a family of asymptotically

quasi φ-nonexpansive mappings, the set of solutions of variational inequalities and the set

of solutions of equilibrium problems in uniformly smooth and 2-uniformly convex Banachspaces A numerical experiment is given to verify the efficiency of the proposed parallelalgorithms

Keywords Asymptotically quasi φ-nonexpansive mapping· Variational inequality ·Equilibrium problem· Hybrid method · Parallel computation

Mathematics Subject Classification (2010) 47H05· 47H09 · 47H10 · 47J25 · 65J15 ·65Y05

The set of solutions of (1) is denoted by V I (A, C) Takahashi and Toyoda [19] 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 α-inverse

strongly monotone mapping in a Hilbert space

Dedicated to Professor Nguyen Khoa Son’s 65th birthday.

Theorem 1 [19] Let K be a closed convex subset of a real Hilbert space H Let α > 0 Let A be an α-inverse strongly monotone mapping of K into H, and let S be a nonexpansive mapping of K into itself such that F (S) 

V I (K, A) = ∅ Let {x n } be a sequence generated by



x0∈ K,

x n+1= α n x n + (1 − α n )SP K (x n − λ n Ax n ) for every n = 0, 1, 2, , where λ n ∈ [a, b] for some a, b ∈ (0, 2α) and α n ∈ [c, d] for some c, d ∈ (0, 1) Then, {x n } converges weakly to z ∈ F (S)  V I (K, A), where

Suppose that x1 = x ∈ C and {x n } is given by

x n+1=  C J−1(J x

n − λ n Ax n ) for every n = 1, 2, , where {λ n } is a sequence of positive numbers If λ n is chosen so that λ n ∈ [a, b] for some a, b with 0 < a < b < c2α

2 , then the sequence {x n } converges weakly to some element z in V I (C, A) Here, 1/c is the 2-uniform convexity constant of E, and z = limn→∞ V I (A,C) x n

In 2009, Zegeye and Shahzad [22] studied the following hybrid iterative algorithm in a uniformly convex and uniformly smooth Banach space for finding a common element of the

2-set of fixed points of a weakly relatively nonexpansive mapping T and the 2-set of solutions

of a variational inequality involving an α-inverse strongly monotone mapping A:

 F (T )

V I (A,C) x0has been established

Kang et al [9] extended this algorithm to a weakly relatively nonexpansive mapping, avariational inequality and an equilibrium problem Recently, Saewan and Kumam [14] haveconstructed a sequential hybrid block iterative algorithm for an infinite family of closed and

uniformly asymptotically quasi φ-nonexpansive mappings, a variational inequality for an α-inverse-strongly monotone mapping, and a system of equilibrium problems.

Qin et al [12] considered the following sequential hybrid method for a pair of inverse

strongly monotone and a quasi φ-nonexpansive mappings in a 2-uniformly convex and

uniformly smooth Banach space:

The set of solutions of the equilibrium problem (2) is denoted by EP (f ) Equilibrium

problems include several problems such as: variational inequalities, optimization problems,fixed point problems, etc In recent years, equilibrium problems have been studied widelyand several solution methods have been proposed (see [3,9,14,15,18]) On the other

hand, for finding a common element in F (T ) 

EP (f ), Takahashi and Zembayashi [20]introduced the following algorithm in a uniformly smooth and uniformly convex Banachspace:

Recently, the above-mentioned algorithms have been generalized and modified for ing a common point of the set of solutions of variational inequalities, the set of fixed points

find-of (asymptotically) quasi φ-nonexpansive mappings, and the set find-of solutions find-of equilibrium

problems by several authors, such as Takahashi and Zembayashi [20], Wang et al [21], andothers

Very recently, Anh and Chung [4] have considered the following parallel hybrid methodfor a finite family of relatively nonexpansive mappings{T i}N

This algorithm was extended, modified and generalized by Anh and Hieu [5] for a finite

family of asymptotically quasi φ-nonexpansive mappings in Banach spaces Note that the

proposed parallel hybrid methods in [4,5] can be used for solving simultaneous systems

of maximal monotone mappings Other parallel methods for solving accretive operatorequations can be found in [3].

In this paper, motivated and inspired by the above-mentioned results, we propose twonovel parallel iterative methods for finding a common element of the set of fixed points of a

family of asymptotically quasi φ-nonexpansive mappings {F (S j )}N

j=1, the set of solutions

for some a, b ∈ (0, αc2/2), d > 0, where 1/c is the 2-uniform convexity constant

of E Concerning the sequence { n }, we consider two cases If the mappings {S i} are

asymptotically quasi φ-nonexpansive, we assume that the solution set F is bounded, i.e., there exists a positive number ω, such that F

 n := (k n n 2 If the mappings{S i } are quasi φ-nonexapansive, then k n= 1,

In Method A (3), knowing x n we find the intermediate approximations y i

n , i = 1, , M

in parallel Using the farthest element among y n i from x n , we compute z j n , j = 1, , N in parallel Further, among z j n , we choose the farthest element from x nand determine solutions

of regularized equilibrium problems u k n , k = 1, , K in parallel Then the farthest from

x n element among u k n ,denoted by ¯u nis chosen Based on ¯u n , a closed convex subset C n+1

is constructed Finally, the next approximation x n+1is defined as the generalized projection

The paper is organized as follows: In Section2, we collect some definitions and resultsneeded for further investigation Section 3 deals with the convergence analysis of themethods (3) and (5) In Section4, a novel parallel hybrid iterative method for variational

inequalities and closed, quasi φ-nonexpansive mappings is studied Finally, a numerical

experiment is considered in Section5to verify the efficiency of the proposed parallel hybridmethods

2 Preliminaries

In this section, we recall some definitions and results which will be used later The reader isreferred to [2] for more details

Definition 1 A Banach space E is called

1) strictly convex if the unit sphere S1(0)

2) uniformly convex if for any given  > 0 there exists δ = δ() > 0 such that for all

exists for all x, y ∈ S1(0);

4) uniformly smooth if the limit (7) exists uniformly for all x, y ∈ S1(0).

The modulus of convexity of E is the function δ E : [0, 2] → [0, 1] defined by

δ E ()= inf





for all  ∈ [0, 2] Note that E is uniformly convex if and only if δ E () > 0 for all

0 <  ≤ 2 and δ E (0) = 0 Let p > 1, E is said to be p-uniformly convex if there exists some constant c > 0 such that δ E () ≥ c p It is well-known that the spaces L p , l p, and

W m p are p-uniformly convex if p > 2 and 2-uniformly convex if 1 < p ≤ 2 and a Hilbert

space H is uniformly smooth and 2-uniformly convex.

Let E be a real Banach space with its dual E∗ The dual product of f ∈ E∗and x ∈ E

by the same symbol E∗

is definedby

J (x)=f ∈ E∗ 2 2

.

The following properties can be found in [7]:

i) If E is a smooth, strictly convex, and reflexive Banach space, then the normalized duality mapping J : E → 2 E∗

is single-valued, one-to-one, and onto;

ii) If E is a reflexive and strictly convex Banach space, then J−1 is norm to weak∗

continuous;

iii) If E is a uniformly smooth Banach space, then J is uniformly continuous on each bounded subset of E;

iv) A Banach space E is uniformly smooth if and only if E∗is uniformly convex;

v) Each uniformly convex Banach space E has the Kadec–Klee property, i.e., for any

Lemma 1 [22] If E is a 2-uniformly convex Banach space, then

2

c2

where J is the normalized duality mapping on E and 0 < c ≤ 1.

The best constant1c is called the 2-uniform convexity constant of E Next, we assume that

E is a smooth, strictly convex, and reflexive Banach space In the sequel, we always use

φ : E × E → [0, ∞) to denote the Lyapunov functional defined by

ii) if x ∈ E, z ∈ C, then z =  C (x) if and only if

iii) φ(x, y) = 0 if and only if x = y.

Lemma 3 [10] Let C be a nonempty closed convex subset of a smooth Banach E,

x, y, z ∈ E and λ ∈ [0, 1] For a given real number a, the set

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

is closed and convex.

Lemma 4 [1] Let {x n } and {y n } be two sequences in a uniformly convex and uniformly smooth real Banach space E If φ(x n , y n ) → 0 and either {x n } or {y n } is bounded, then

n − y n

Lemma 5 [6] Let E be a uniformly convex Banach space, r be a positive number, and

B r (0) ⊂ E be the closed ball with center at origin and radius r Then, for any given subset {x1, x2, , x N } ⊂ B r (0) and for any positive numbers λ1, λ2, , λ N withN

i=1λ i = 1, there exists a continuous, strictly increasing, and convex function g : [0, 2r) → [0, ∞) with g(0) = 0 such that, for any i, j ∈ {1, 2, , N} with i < j,

ψ(t) = ηt2;

4) α-inverse strongly monotone, if there exists a positive constant α, such that

2 ∀x, y ∈ E;

5) L-Lipschitz continuous if there exists a positive constant L, such that

If A is α-inverse strongly monotone then it is 1α -Lipschitz continuous If A is η-strongly monotone and L-Lipschitz continuous then it is L η2-inverse strongly monotone

Lemma 6 [17] Let C be a nonempty, closed convex subset of a Banach space E and A be a monotone, hemicontinuous mapping of C into E∗ Then

V I (C, A)

Let C be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space E, T : C → C be a mapping The set

F (T ) = {x ∈ C : T x = x}

is called the set of fixed points of T A point p ∈ C is said to be an asymptotic fixed point

of T if there exists a sequence {x n } ⊂ C such that x n pand n − T x n

n → + ∞ The set of all asymptotic fixed points of T will be denoted by ˜F (T ).

Definition 3 A mapping T : C → C is called

i) relatively nonexpansive mapping if F (T ) = ∅, ˜F(T ) = F (T ), and

φ(p, T x) ≤ φ(p, x) ∀p ∈ F (T ), ∀x ∈ C;

ii) closed if for any sequence{x n } ⊂ C, x n → x and T x n → y, then T x = y;

iii) quasi φ-nonexpansive mapping (or hemi-relatively nonexpansive mapping) if

F (T ) = ∅ and

φ(p, T x) ≤ φ(p, x) ∀p ∈ F (T ), ∀x ∈ C;

iv) asymptotically quasi φ-nonexpansive if F (T ) = ∅ and there exists a sequence

{k n } ⊂ [1, +∞) with k n → 1 as n → + ∞ such that

φ(p, T n x) ≤ k n φ(p, x) ∀n ≥ 1, ∀p ∈ F (T ), ∀x ∈ C;

v) uniformly L-Lipschitz continuous, if there exists a constant L > 0 such that

n x − T n y

The reader is referred to [6, 16] 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 φ-nonexpansive mappings, and the class of quasi φ-nonexpansive mappings contains the class of relatively nonexpansive

φ-mappings as a proper subset

Lemma 7 [6] Let E be a real uniformly smooth and strictly convex Banach space with Kadec–Klee property, and C be a nonempty closed convex subset of E Let T :

C → C be a closed and asymptotically quasi φ-nonexpansive mapping with a sequence {k n } ⊂ [1, +∞), k n → 1 Then F (T ) is a closed convex subset of C.

Next, for solving the equilibrium problem (2), we assume that the bifunction f satisfies

the following conditions:

(A1) f (x, x) = 0 for all x ∈ C;

(A2) f is monotone, i.e., f (x, y) + f (y, x) ≤ 0 for all x, y ∈ C;

(A3) For all x, y, z ∈ C,

lim

t→0 +sup f (tz + (1 − t)x, y) ≤ f (x, y);

(A4) For all x ∈ C, f (x, ·) is convex and lower semicontinuous.

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

Lemma 8 [20] Let C be a closed and convex subset of a smooth, strictly convex and ive Banach space E, f be a bifunction from C × C to R satisfying conditions (A1)–(A4) and let r > 0, x ∈ E Then, there exists z ∈ C such that

reflex-f (z, y)+1

r

Lemma 9 [20] Let C be a closed and convex subset of a uniformly smooth, strictly convex, and reflexive Banach spaces E, f be a bifunction from C × C to R satisfying conditions (A1)–(A4) For all r > 0 and x ∈ E, define the mapping

T r x = {z ∈ C : f (z, y) +1r

Then the following hold:

(B1) T r is single-valued;

(B2) T r is a firmly nonexpansive-type mapping, i.e., for all x, y ∈ E,

T r x − T r y, J T r x − J T r y r x − T r y, J x

(B3) F (T r ) = ˜F(T ) = EP (f );

(B4) EP (f ) is closed and convex and T r is a relatively nonexpansive mapping.

Lemma 10 [20] Let C be a closed convex subset of a smooth, strictly convex and reflexive Banach space E Let f be a bifunction from C × C to R satisfying (A1)–(A4) and let r > 0 Then, for x ∈ E and q ∈ F (T r ),

φ(q, T r x) + φ(T r x, x) ≤ φ(q, x).

Let E be a real Banach space Alber [1] studied the function V : E×E∗→ R defined by

V (x, x∗) 2 − 2x, x∗ ∗ 2 Clearly, V (x, x∗) = φ(x, J−1x∗).

Lemma 11 [1] Let E be a reflexive, strictly convex and smooth Banach space with E∗as

its dual Then

V (x, x∗) + 2J−1x − x∗, y∗ ∗ + y∗) ∀x ∈ E, ∀x∗, y∗ ∈ E∗ Consider the normal cone N C to a set C at a point x ∈ C defined by

We have the following result

Lemma 12 [13] Let C be a nonempty closed convex subset of a Banach space E and let A

be a monotone and hemi-continuous mapping of C into E∗with D(A) = C Let Q be a mapping defined by:

and assume that the set F is nonempty.

We prove convergence theorems for methods (3) and (5) with the control ter sequences satisfying conditions (4) and (6), respectively We also propose similar

parame-parallel hybrid methods for quasi φ-nonexpansive mappings, variational inequalities and

equilibrium problems

satisfy condition (4), then the sequence {x n } generated by (3) converges strongly to  F x0 Proof We divide the proof of Theorem 3 into seven steps.

Step 1 Claim that F, C n are closed convex subsets of C Indeed, since each mapping S iis

uniformly L-Lipschitz continuous, it is closed By Lemmas 6, 7, and 9, F (S i ), V I (A j , C) and EP (f k ) are closed convex sets, therefore, N

j=1(F (S j )), M

i=1V I (A i , C) and

K

k=1EP (f k ) are also closed and convex Hence, F is a closed and convex subset of C It

is obvious that C n is closed for all n ≥ 0 We prove the convexity of C nby induction

Clearly, C0 := C is closed convex Assume that C n is closed convex for some n ≥ 0

From the construction of C n+1, we find

C n+1 = C n



{z ∈ E : φ(z, ¯u n ) ≤ φ(z, ¯z n ) ≤ φ(z, x n ) +  n }.

Lemma 3 ensures that C n+1is convex Thus, C n is closed convex for all n ≥ 0 Hence,

 F x0and x n+1 :=  C n+1x0are well-defined

Step 2 Claim that F ⊂ C n for all n ≥ 0 By Lemma 10 and the relative

nonexpan-siveness of T r n, we obtain

φ(u, ¯u n ) = φ(u, T r n ¯z n ) ≤ φ(u, ¯z n ) for all u ∈ F.

From the convexity of 2and the asymptotical quasi φ-nonexpansiveness of S j, we find

≤ α n φ(u, x n ) + (1 − α n )k n φ(u, ¯y n ) (11)

for all u ∈ F By the hypotheses of Theorem 3, Lemmas 1, 2, and 11, and u ∈ F , we have φ(u, ¯y n ) = φu,  C

From (11), (12) and the estimate (8), we obtain

φ(u, ¯z n ) ≤ α n φ(u, x n ) + (1 − α n )k n φ(u, x n ) − 2a(1 − α n ) α−2b

Therefore F ⊂ C n for all n ≥ 0

Step 3 Claim that the sequence {x n }, {y i

n }, {z j

n } and {u k

n } converge strongly to p ∈ C.

By Lemma 2 and x n =  C n x0, we have

φ(x n , x0) ≤ φ(u, x0) − φ(u, x n ) ≤ φ(u, x0) for all u ∈ F Hence, {φ(x n , x0)} is bounded By (8), {x n} is bounded, and so are thesequences{ ¯y n }, { ¯u n }, and {¯z n } By the construction of C n , x n+1=  C n+1x0∈ C n+1 ⊂ C n

From Lemma 2 and x n =  C n x0, we get

φ(x n , x0) ≤ φ(x n+1, x0) − φ(x n+1, x n ) ≤ φ(x n+1, x0).

Therefore, the sequence{φ(x n , x0)} is nondecreasing, hence, it has a finite limit Note that

for all m ≥ n, x m ∈ C m ⊂ C n, and by Lemma 2 we obtain

φ(x m , x n ) ≤ φ(x m , x0) − φ(x n , x0)→ 0 (14)

as m, n → ∞ From (14) and Lemma 4, we have n − x m

{x n } ⊂ C is a Cauchy sequence Since E is complete and C is a closed convex subset of E, {x n } converges strongly to p ∈ C From (14), φ(x n+1, x n ) → 0 as n → ∞ Taking into account that x n+1 ∈ C n+1, we find

This together with n+1 − x n

for all 1 ≤ k ≤ K and 1 ≤ j ≤ N By the hypotheses of Theorem 3, Lemmas 1, 2,

and 11, we also have

Using the fact that n − ¯z n

relations lim supn→∞α n < 1 and  n → 0 imply that

lim

n→∞ i n x n − A i n u (22)From (20) and (22), we obtain

parame -parallel hybrid methods for quasi φ-nonexpansive... φ-nonexpansive mappings, variational inequalities and< /i>

equilibrium problems