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Strong convergence of projectionalgorithms for a family of relativelyquasi-nonexpansive mappings and anequilibrium problem in Banach spacesPrasit Cholamjiak aa School of Science and Technology , Naresuan University atPhayao , Phayao 56000, ThailandPublished online: 07 Oct 2010.

To cite this article: Prasit Cholamjiak (2011) Strong convergence of projection algorithms for afamily of relatively quasi-nonexpansive mappings and an equilibrium problem in Banach spaces,Optimization: A Journal of Mathematical Programming and Operations Research, 60:4, 495-507,DOI: 10.1080/02331930903477192

To link to this article: http://dx.doi.org/10.1080/02331930903477192

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OptimizationVol. 60, No. 4, April 2011, 495–507

Strong convergence of projection algorithms for a family of

relatively quasi-nonexpansive mappings and an equilibrium

problem in Banach spaces

Prasit Cholamjiak*

School of Science and Technology, Naresuan University at Phayao,Phayao 56000, Thailand

(Received 20 February 2009; final version received 23 October 2009)

In this article, we introduce two kinds of new hybrid projection algorithmsfor finding a common element of the set of solutions of an equilibriumproblem and the set of common fixed points of an infinitely countablefamily of relatively quasi-nonexpansive mappings in a Banach space. Ourmain results improve and extend the result obtained byMartinez-Yanes andXu [Strong convergence of the CQ method for fixed point iteration processes,Nonlinear Anal. 64 (2006), pp. 2400–2411] and the corresponding results.

Keywords: relatively quasi-nonexpansive mapping; fixed point; projectionalgorithm; strong convergence; equilibrium problem

AMS Subject Classifications: 47H05; 47H09; 47H10

1. Introduction

Let H be a real Hilbert space and C be a nonempty closed convex subset of H.Let T :C!C be a mapping. Recall that T is said to be nonexpansive ifkTx�Txk� kx� yk for all x, y2H. Denote by F(T ) the set of fixed points of T.

Two classical iteration processes are often used to approximate a fixed point ofa nonexpansive mapping T. The first one is introduced by Mann [12] and is definedas follows: For an initial point x02C, let {xn} be defined recursively by

xnþ1 ¼ �nxn þ ð1� �nÞTxn, 8n � 0, ð1:1Þ

where {�n} is a sequence in [0,1]. The second iteration process is known as Halpern’siteration process [9] which is defined as

xnþ1 ¼ �nx0 þ ð1� �nÞTxn, 8n � 0, ð1:2Þ

where the initial point x0 is taken in C arbitrarily and {�n} is a sequence in [0,1].In 2003, Nakajo and Takahashi [16] modified the process (1.1) in order to get

strong convergence by the following modification which is the so-called CQ method

*Email: prasitch2008@yahoo.com

ISSN 0233–1934 print/ISSN 1029–4945 online

� 2011 Taylor & Francis

DOI: 10.1080/02331930903477192

http://www.informaworld.com

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for a nonexpansive mapping in a Hilbert space H:

x0 2C chosen arbitrarily,

yn ¼ �nxn þ ð1� �nÞTxn,

Cn ¼ fz2C: k yn � zk � kxn � zkg,

Qn ¼ fz2C: hxn � z, x0 � xni � 0g,

xnþ1 ¼ PCn\Qnx0,

8>>>><>>>>:

where PK is the metric projection from H onto a closed convex subset K of H.Recently, Martinez-Yanes and Xu [13] has adapted Nakajo and Takahashi’s [16]

idea to modify the process (1.2) for a nonexpansive mapping in a Hilbert space H:

x0 2C chosen arbitrarily,

yn ¼ �nx0 þ ð1� �nÞTxn,

Cn ¼

nz2C: k yn � zk2 � kxn � zk2 þ �n

�kx0k

2 þ 2hxn � x0, zi�o,

Qn ¼ fz2C: hxn � z, xn � x0i � 0g,

xnþ1 ¼ PCn\Qnx0,

8>>>>><>>>>>:

ð1:3Þ

where PK is the metric projection from H onto a closed convex subset K of H. They

proved that the sequence {xn} generated by above iterative scheme converges

strongly to PF(T )x0provided the sequence {�n}� (0, 1) satisfies limn!1�n¼ 0.

Let E be a real Banach space and let E * be the dual of E. Let C be a closed

convex subset of E. Let f be a bifunction from C�C to the set of real number R.

The equilibrium problem is to find x̂2C such that

f ðx̂, yÞ � 0, 8y2C: ð1:4Þ

The set of solutions of (1.4) is denoted by EP( f ).For solving the equilibrium problem, let us assume that a bifunction f satisfies

the following conditions:

(A1) f (x, x)¼ 0 for all x2C;(A2) f is monotone, i.e. f (x, y)þ f (y, x)� 0 for all x, y2C;(A3) for all x, y, z2C, lim supt#0 f (tzþ (1� t)x, y)� f (x, y);(A4) for all x2C, f (x, �) is convex and lower semicontinuous.

Very recently, Takahashi and Zembayashi [23] introduced the following iterative

scheme which is called the shrinking projection method for a relatively nonexpansive

mapping in a Banach space E:

x0 2C chosen arbitrarily,

C0 ¼ C,

yn ¼ J�1��nJxn þ ð1� �nÞJTxn

�,

un 2C such that f ðun, yÞ þ1rnhy� un, Jun � Jyni � 0 8y2C,

Cnþ1 ¼ fz2Cn: �ðz, unÞ � �ðz, xnÞg,

xnþ1 ¼ �Cnþ1x0, 8n � 0,

8>>>>>>><>>>>>>>:

where J is the duality mapping on E and �K is the generalized projection from E

onto K. They proved that the sequence {xn} converges strongly to �F(T )\EP( f )x0provided lim infn!1�n(1� �n)40 and {rn}� [a,1) for some a40.

The problem of finding a common element of the set of fixed points and the set of

solutions of an equilibrium problem in the framework of Hilbert spaces and Banach

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spaces has been intensively studied by many authors; for instance, see

[3,8,14,15,17,20,22,24,25,26] and the references cited therein.Motivated by Martinez-Yanes and Xu [13] and Takahashi and Zembayashi [23],

we introduce two kinds of monotone projection algorithms for an infinitely

countable family of closed relatively quasi-nonexpansive mappings and an equilib-

rium problem. Our main results improve and extend the CQ algorithm (1.3) from

Hilbert spaces to Banach spaces. By using the monotone projection algorithms, we

can easily show that the iteration sequence {xn} is a Cauchy sequence, without the

use of the Kadec–Klee property, demiclosedness principle and Opial’s condition.

2. Preliminaries

Let E be a real Banach space with k�k and let E* be the dual of E. The normalized

duality mapping J from E to E* is defined by

JðxÞ ¼ fx� 2E�: hx, x�i ¼ kxk2 ¼ kx�k2g

for all x2E. A Banach space E is said to be strictly convex if k xþy2 k5 1 for all x,

y2E with kxk¼kyk¼ 1 and x 6¼ y. It is also said to be a uniformly convex if

limn!1kxn� ynk¼ 0 for any two sequences {xn}, {yn} in E such that kxnk¼kynk¼ 1

and limn!1kxnþyn

2 k ¼ 1. Let S¼ {x2E: kxk¼ 1} be the unit sphere of E. Then the

Banach space E is said to be smooth provided that limt!0kxþtyk�kxk

t exists for each

x, y2S. It is said to be uniformly smooth if the limit is attained uniformly for

x, y2S. It is known that if E is uniformly smooth, then J is uniformly norm-to-norm

continuous on each bounded subset of E.Let E be a smooth Banach space. The function � :E�E!R is defined by

�ðx, yÞ ¼ kxk2 � 2hx, Jyi þ k yk2

for all x, y2E. It is obvious from the definition of the function � that

�k yk � kxk

�2� �ð y, xÞ �

�kxk þ k yk

�2, 8x, y2E:

Let C be a closed convex subset of E, and let T be mapping from C into itself.

A point p in C is said to be an asymptotic fixed point of T [19] if C contains

a sequence {xn} which converges weakly to p such that limn!1kxn�Txnk¼ 0. The

set of asymptotic fixed point of T will be denoted by FðT̂ Þ. A mapping T is said to be

relatively nonexpansive [4,5,6,15] if FðT̂ Þ ¼ FðTÞ and �(p,Tx)��(p, x) for all

p2F(T ) and x2C. The asymptotic behaviour of a relatively nonexpansive mapping

was studied in [4,5,6]. T is said to be �-nonexpansive, if �(Tx,Ty)��(x, y) for

x, y2C. T is said to be relatively quasi-nonexpansive or quasi-�-nonexpansive if

F(T ) 6¼ ; and �(p,Tx)��(p, x) for all p2F(T ) and x2C. It is easy to see that the

class of relatively quasi-nonexpansive mappings is more general than the class of

relatively nonexpansive mappings [4,5,6,15]. Recall that T is closed if

xn! x, Txn! y implies Tx ¼ y:

for all x, y2C.We give some examples which are closed relatively quasi-nonexpansive; see [18].

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Example 2.1 Let E be a uniformly smooth and strictly convex Banach space and

A�E�E* be a maximal monotone mapping such that its zero set A�10 6¼ ;.

Then, Jr¼ (Jþ rA)�1J is a closed relatively quasi-nonexpansive mapping from E

onto D(A) and F(Jr)¼A�10.

Example 2.2 Let �C be the generalized projection from a smooth, strictly convex,

and reflexive Banach space E onto a nonempty closed convex subset C of E. Then,

�C is a closed relatively quasi-nonexpansive mapping with F(�C)¼C.

LEMMA 2.3 [17] Let E be a uniformly convex and uniformly smooth Banach space, C a

nonempty closed convex subset of E. Then for points w, x, y, z2E and a real number

a2R, the set

K9 {v2C: �(v, y)��(v, x)þhv, Jz� Jwiþ a} is closed and convex.

LEMMA 2.4 [10] Let E be a uniformly convex and smooth Banach space and let {xn},

{yn} be two sequences of E. If �(xn, yn)! 0 and either {xn} or {yn} is bounded, then

kxn� ynk! 0 as n!1.

Let C be a nonempty closed convex subset of E. If E is reflexive, strictly convex

and smooth, then there exists x02C such that �(x0,x)¼min�(y, x) for x2E and

y2C. The generalized projection �C :E!C defined by �Cx¼ x0. The existence

and uniqueness of the operator �C follows from the properties of the functional �and strict monotonicity of the duality mapping J; see [1,2,7,10,21]. In a Hilbert space,

the mapping �C is coincident with the metric projection.

LEMMA 2.5 [1] Let C be a nonempty closed convex subset of a smooth Banach space E

and x2E. Then x0¼�Cx if and only if

hx0 � y, Jx� Jx0i � 0, 8y2C: ð2:1Þ

LEMMA 2.6 [1] Let C be a nonempty closed convex subset of a reflexive, strictly

convex and smooth Banach space E and let x2E. Then

�ð y,�CxÞ þ �ð�Cx, xÞ � �ð y, xÞ, 8y2C:

LEMMA 2.7 [18] Let E be a uniformly convex, smooth Banach space, let C be a closed

convex subset of E, let T be a closed and relatively quasi-nonexpansive mapping from C

into itself. Then F(T ) is a closed convex subset of C.

LEMMA 2.8 [3] 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 r40 and x2E. Then, there exists z2C such that

f ðz, yÞ þ1

rhy� z, Jz� Jxi � 0, 8y2C:

LEMMA 2.9 [18] Let C be a closed convex subset of a uniformly smooth, strictly

convex, and reflexive Banach space E, and let f be a bifunction from C�C to R

satisfying (A1)–(A4), and let r40 and x2E, define a mapping Tr: E!C

as follows:

Trx ¼ fz2C : f ðz, yÞ þ1

rhy� z, Jz� Jxi � 0, 8y2Cg:

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Then, the followings hold:

(1) Tr is single-valued;(2) Tr is a firmly nonexpansive-type mapping [11], i.e. for all x, y2E,

hTrx� Try, JTrx� JTryi � hTrx� Try, Jx� Jyi;

(3) F(Tr)¼EP( f );(4) EP( f ) is closed and convex.

LEMMA 2.10 [24] Let C be a closed convex subset of a smooth, strictly, and reflexiveBanach space E, let f be a bifunction from C�C to R satisfying (A1)–(A4), let r40.

Then, for all x2E and q2F(Tr),

�ðq,TrxÞ þ �ðTrx,xÞ � �ðq,xÞ:

3. Main results

In this section, we prove the strong convergence theorems for finding a commonelement of the set of common fixed points of a family of closed relativelyquasi-nonexpansive mappings and the set of solutions of an equilibrium problem ina Banach space E.

THEOREM 3.1 Let E be a uniformly convex and uniformly smooth Banach space, let Cbe a closed convex subset of E, and let f be a bifunction from C�C to R satisfying(A1)–(A4). Let fTig

1i¼1 be an infinitely countable family of closed relatively quasi-

nonexpansive mappings from C into itself such that F :¼T1

i¼1 FðTiÞ \ EPð f Þ 6¼ ;.Assume that {�n,i}� [0, 1] is such that limn!1�n,i¼ 0, 8i2N and {rn}� [a, 1) forsome a40. Let {xn} be a sequence generated by

x0 2E chosen arbitrarily,C1,i ¼ C, C1 ¼

T1i¼1 C1,i, x1 ¼ �C1

x0,yn,i ¼ J�1

��n,iJx0 þ ð1� �n,iÞJTixn

�, 8n � 1,

un,i 2C such that f ðun,i, yÞ þ1rnhy� un,i, Jun,i � Jyn,ii � 0, 8y2C,

Cnþ1,i ¼�z2Cn,i: �ðz, un,iÞ � �ðz, xnÞ þ �n,i

�kx0k

2 þ 2hJxn � Jx0, zi��,

Cnþ1 ¼T1

i¼1 Cnþ1,i,xnþ1 ¼ �Cnþ1

x0, 8n � 1,

8>>>>>>>><>>>>>>>>:

where J is the duality mapping on E.Then, the sequence {xn} converges strongly to q¼�Fx0.

Proof We divide the proof into seven steps.

Step 1 Show that �Fx0 and �Cnþ1x0 are well-defined for every x02E.

By Lemma 2.7, we know that F(Ti) is closed and convex for all i2N. Hence,T1i¼1 FðTiÞ is closed and convex. By Lemma 2.9 (4), we also know that EP( f ) is

closed and convex. Hence, F :¼T1

i¼1 FðTiÞ \ EPð f Þ is a nonempty closed and convexsubset of C; consequently, �Fx0 is well-defined. From Lemma 2.3, it is easy to seethat Cnþ1 ¼

T1i¼1 Cnþ1,i is closed and convex; consequently, �Cnþ1

x0 is well-defined.

Step 2 Show that F�Cn for all n� 1.

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Notice that un,i¼Trnyn,i for all n� 1 and i2N. F�C1,i¼C is obvious. Suppose

F�Ck,i for k2N. For each p2F�Ck,i, we have

�ð p, uk,iÞ ¼ �ð p,Trkyk,iÞ � �ð p, yk,iÞ

¼ ��p, J�1

��k,iJx0 þ ð1� �k,iÞJTixk

��

¼ k pk2 � 2 p,�k,iJx0 þ ð1� �k,iÞJTixk�

þ k�k,iJx0 þ ð1� �k,iÞJTixkÞk2

� k pk2 � 2�k,i p, Jx0�

� 2ð1� �k,iÞ p, JTixk�

þ �k,ikx0k2 þ ð1� �k,iÞkTixkk

2

¼ �k,i�ð p, x0Þ þ ð1� �k,iÞ�ð p,TixkÞ

� �k,i�ð p, x0Þ þ ð1� �k,iÞ�ð p, xkÞ

¼ �ð p, xkÞ þ �k,i��ð p, x0Þ � �ð p,xkÞ

�

¼ �ð p, xkÞ þ �k,i�kx0k

2 � kxkk2 þ 2hp, Jxk � Jx0i

�

� �ð p, xkÞ þ �k,i�kx0k

2 þ 2hp, Jxk � Jx0i�: ð3:1Þ

This implies that F�Ckþ1,i. By induction, we also get that F�Cn,i for all n� 1 and

i2N. Hence F�Cn for all n� 1.

Step 3 Show that limn!1�(xn, x0) exists.

From xn¼�Cnx0 and xnþ1¼�Cnþ1

x02Cnþ1�Cn, we have

�ðxn, x0Þ � �ðxnþ1, x0Þ 8n � 1: ð3:2Þ

From Lemma 2.6, we have

�ðxn, x0Þ ¼ �ð�Cnx0, x0Þ � �ð p, x0Þ � �ð p, xnÞ � �ð p,x0Þ: ð3:3Þ

Combining (3.2) and (3.3), we obtain that limn!1�(xn, x0) exists.

Step 4 Show that {xn} is a Cauchy sequence in C.

Since xm¼�Cmx02Cm�Cn for m4n, by Lemma 2.6, we have

�ðxm, xnÞ ¼ �ðxm,�Cnx0Þ � �ðxm, x0Þ � �ð�Cn

x0, x0Þ

¼ �ðxm, x0Þ � �ðxn,x0Þ:

Taking m, n!1, we obtain that �(xm, xn)! 0. From Lemma 2.4, we have

kxm� xnk! 0. Hence {xn} is a Cauchy sequence. By the completeness of E and the

closedness of C, we can assume that xn! q2C as n!1. Further, we obtain

limn!1

�ðxnþ1, xnÞ ¼ 0: ð3:4Þ

Note that limn!1�n,i¼ 0. Since xnþ1¼�Cnþ1x02Cnþ1, we have

�ðxnþ1, un,iÞ � �ðxnþ1, xnÞ þ �n,i�kx0k

2 þ 2hJxn � Jx0, xnþ1i�! 0, ð3:5Þ

as n!1. Applying Lemma 2.4 to (3.4) and (3.5), we get

limn!1kun,i � xnk ¼ 0: ð3:6Þ

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This implies that un,i! q as n!1 and i2N. Since J is uniformly norm-to-norm

continuous on bounded subsets of E, we also obtain

limn!1kJun,i � Jxnk ¼ 0: ð3:7Þ

Step 5 Show that q2T1

i¼1 FðTiÞ.

For each p2F, it follows from (3.1) that

�ð p, yn,iÞ � �ð p, xnÞ þ �n,i�kx0k

2 þ 2h p, Jxn � Jx0i�: ð3:8Þ

Note that un,i¼Trnyn,i, 8i2N. From (3.8) and Lemma 2.10, we have

�ðun,i, yn,iÞ ¼ �ðTrnyn,i, yn,iÞ

� �ð p, yn,iÞ � �ð p,Trnyn,iÞ

� �ð p, xnÞ þ �n,i�kx0k

2 þ 2h p, Jxn � Jx0i�� �ð p,Trnyn,iÞ

¼ �ð p, xnÞ þ �n,i�kx0k

2 þ 2h p, Jxn � Jx0i�� �ð p, un,iÞ

� kxn � un,ik�kxnk þ kun,ik

�þ 2k pkkJxn � Jun,ik

þ �n,i�kx0k

2 þ 2h p, Jxn � Jx0i�:

From (3.6) and (3.7), we get limn!1�(un,i, yn,i)¼ 0. By Lemma 2.4, we obtain

limn!1kun,i � yn,ik ¼ 0: ð3:9Þ

It follows from (3.6) and (3.9) that

limn!1kxn � yn,ik ¼ 0:

Since J is uniformly norm-to-norm continuous on bounded subsets of E, we obtain

limn!1kJxn � Jyn,ik ¼ 0: ð3:10Þ

Observing

kJyn,i � Jxnk ¼ k�n,iJx0 þ ð1� �n,iÞJTixn � Jxnk

¼ k��n,iðJTixn � Jx0Þ þ ðJTixn � JxnÞk

� ��n,ikJTixn � Jx0k þ kJTixn � Jxnk,

we obtain, by (3.10), that

kJTixn � Jxnk � kJyn,i � Jxnk þ �n,ikJTixn � Jx0k ! 0,

as n!1. Since J�1 is uniformly norm-to-norm continuous on bounded subsets

of E, we also have

limn!1kTixn � xnk ¼ 0 8i2N: ð3:11Þ

By the closedness of Ti, we get that q2F(Ti) 8i2N. Therefore q2T1

i¼1 FðTiÞ.

Step 6 Show that q2EP( f ).

From (3.9) and rn� a, we havekJun,i�Jyn,ik

rn! 0. From un,i¼Trn

yn,i we get

f ðun,i, yÞ þ1rnhy� un,i, Jun,i � Jyn,ii � 0, 8y2C:

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By (A2), we have

k y� un,ikkJun,i � Jyn,ik

rn�

1

rnhy� un,i, Jun,i � Jyn,ii,

� �f ðun,i, yÞ � f ð y, un,iÞ, 8y2C:

From (A4) and un,i! q, we get f (y, q)� 0 for all y2C. For 05t51 and y2C, define

yt¼ tyþ (1� t)q. Then yt2C, which implies that f (yt, q)� 0. From (A1), we obtain

that 0¼ f (yt, yt)� tf (yt, y)þ (1� t)f (yt, q)� tf (yt, y). Thus, f (yt, y)� 0. From (A3),

we have f (q, y)� 0 for all y2C. Hence q2EP( f ).

Step 7 Show that q¼�Fx0.

From xn¼�Cnx0, we have

�Jx0 � Jxn, xn � z

� 0 8z2Cn:

Since F�Cn, we also have�Jx0 � Jxn, xn � p

� 0 8p2F: ð3:12Þ

By taking limit in (3.12), we obtain that�Jx0 � Jq, q� p

� 0 8p2F:

By Lemma 2.5, we can conclude that q¼�Fx0. This completes the proof. g

In a Hilbert space, we obtain the following corollaries immediately.

COROLLARY 3.2 Let C be a nonempty and closed convex subset of a real Hilbert space

H. Let f be a bifunction from C�C to R satisfying (A1)–(A4). Let fTig1i¼1 be an

infinitely countable family of closed quasi-nonexpansive mappings from C into itself

such that F :¼T1

i¼1 FðTiÞ \ EPð f Þ 6¼ ;. Assume that {�n,i}� [0, 1] is such that

limn!1�n,i¼ 0, 8i2N and {rn}� [a, 1) for some a40. Let {xn} be a sequence

generated by

x0 2H chosen arbitrarily,C1,i ¼ C, C1 ¼

T1i¼1 C1,i, x1 ¼ PC1

x0,yn,i ¼ �n,ix0 þ ð1� �n,iÞTixn,un,i 2C such that f ðun,i, yÞ þ

1rnhy� un,i, un,i � yn,ii � 0, 8y2C,

Cnþ1,i ¼�z2Cn,i : kz� un,ik

2 � kz� xnk2 þ �n,i

�kx0k

2 þ 2hxn � x0, zi��,

Cnþ1 ¼T1

i¼1 Cnþ1,i,xnþ1 ¼ PCnþ1

x0, 8n � 1,

8>>>>>>>><>>>>>>>>:

where PK is the metric projection from H onto a subset K of H.Then, the sequence {xn} converges strongly to q¼PFx0.

Proof In a Hilbert space, we know that �K¼PK where K is a nonempty closed

convex subset of C. Since J is an identity operator, it follows that

�ðx, yÞ ¼ kx� yk2

for all x, y2H. Hence

kTix� pk � kx� pk , �ð p,TixÞ � �ð p, xÞ

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for all x2C and p2F(Ti) for all i2N. This implies that Ti is quasi-nonexpansive

if and only if Ti is relatively quasi-nonexpansive. By Theorem 3.1, we obtain

the result. g

If f (x, y)¼ 0 for all x, y2C and rn¼ 1 for all n� 1, then Corollary 3.2 collapses

to the following result.

COROLLARY 3.3 Let C be a nonempty and closed convex subset of a real Hilbert space

H. Let fTig1i¼1 be an infinitely countable family of closed quasi-nonexpansive mappings

from C into itself such that F :¼T1

i¼1 FðTiÞ 6¼ ;. Assume that {�n,i}� [0, 1] is such that

limn!1�n,i¼ 0, 8i2N. Let {xn} be a sequence generated by

x0 2H chosen arbitrarily,C1,i ¼ C, C1 ¼

T1i¼1 C1,i, x1 ¼ PC1

x0,yn,i ¼ �n,ix0 þ ð1� �n,iÞTixn,Cnþ1,i ¼

�z2Cn,i : kz� yn,ik

2 � kz� xnk2 þ �n,i

�kx0k

2 þ 2hxn � x0, zi��,

Cnþ1 ¼T1

i¼1 Cnþ1,i,xnþ1 ¼ PCnþ1

x0, 8n � 1,

8>>>>>><>>>>>>:

where PK is the metric projection from H onto a subset K of H.Then, the sequence {xn} converges strongly to q¼PFx0.

Remark 3.4 Corollary 3.3 improves and extends Theorem 3.1 of Martinez-Yanes

and Xu [13] in the following senses:

(1) from a single nonexpansive mapping to an infinitely countable family of

closed quasi-nonexpansive mappings;(2) from the computation point of view, in the case of Ti¼T and �n,i¼�n for all

n� 1 and i2N, the algorithm in Corollary 3.3 is also more simple and

convenient to compute than the algorithm defined by (1.3).

THEOREM 3.5 Let E be a uniformly convex and uniformly smooth Banach space, let C

be a closed convex subset of E, and let f be a bifunction from C�C to R satisfying

(A1)–(A4). Let fTig1i¼1 be an infinitely countable family of closed relatively quasi-

nonexpansive mappings from C into itself such that F :¼T1

i¼1 FðTiÞ \ EPð f Þ 6¼ ;.

Assume that {�n,i}� [0, 1] is such that limn!1�n,i¼ 0, 8i2N and {rn}� [a,1) for

some a40. Let {xn} be a sequence generated by

x0 2C chosen arbitrarily,

yn,i ¼ J�1��n,iJx0 þ ð1� �n,iÞJTixn

�, 8n � 0, i2N,

un,i 2C such that f ðun,i, yÞ þ1rnhy� un,i, Jun,i � Jyn,ii � 0, 8y2C,

Cn,i ¼�z2C :�ðz, un,iÞ � �ðz, xnÞ þ �n,i

�kx0k

2 þ 2hJxn � Jx0, zi��,

Cn ¼T1

i¼1 Cn,i,

Q0 ¼ C,

Qn ¼�z2Qn�1 : hxn � z, Jxn � Jx0i � 0

�, 8n � 1,

xnþ1 ¼ �Cn\Qnx0, 8n � 0,

8>>>>>>>>>>><>>>>>>>>>>>:

where J is the duality mapping on E.Then, the sequence {xn} converges strongly to q¼�Fx0.

Proof We divide the proof into four steps.

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Step 1 Show that F�Cn\Qn for all n� 0.

It is obvious that Qn is closed and convex for all n� 0. By Lemma 2.3, we have Cn

is closed and convex for all n� 0. Hence Cn\Qn is closed and convex for all n� 0.

For any p2F, we have

�ð p, un,iÞ ¼ �ð p,Trnyn,iÞ

� �ð p, yn,iÞ

¼ ��p, J�1

��n,iJx0 þ ð1� �n,iÞJTixn

��

¼ k pk2 � 2�p,�n,iJx0 þ ð1� �n,iÞJTixn

þ k�n,iJx0 þ ð1� �n,iÞJTixnÞk2

� k pk2 � 2�n,i p, Jx0�

� 2ð1� �n,iÞ p, JTixn�

þ �n,ikx0k2 þ ð1� �n,iÞkTixnk

2

¼ �n,i�ð p, x0Þ þ ð1� �n,iÞ�ð p,TixnÞ

� �n,i�ð p, x0Þ þ ð1� �n,iÞ�ð p, xnÞ

¼ �ð p, xnÞ þ �n,i��ð p, x0Þ � �ð p, xnÞ

�

¼ �ð p, xnÞ þ �n,i�kx0k

2 � kxnk2 þ 2h p, Jxn � Jx0i

�

� �ð p,xnÞ þ �n,i�kx0k

2 þ 2h p, Jxn � Jx0i�:

This implies that p2Cn,i for all n� 0 and i2N. Hence F�Cn for all n� 0.We observe that F�Q0¼C. Suppose that F�Qk for k2N. Since

xkþ1¼�Ck\Qkx0, we have hxkþ1� z, Jxkþ1� Jx0i� 0 for all z2Ck\Qk. As

F�Ck\Qk by the assumption, hence hxkþ1� z, Jxkþ1� Jx0i� 0 for all z2F. This

together with the definition of Qkþ1 implies that F�Qkþ1. Thus F�Ckþ1\Qkþ1.

By simple induction, we can show that F�Cn\Qn for all n� 0.

Step 2 Show that limn!1�(xn, x0) exists.

From xn¼�Qnx0 and xnþ1¼�Cn\Qn

x02Cn\Qn�Qn, we have

�ðxn, x0Þ � �ðxnþ1, x0Þ 8n � 0: ð3:13Þ

From Lemma 2.6, we have

�ðxn, x0Þ ¼ �ð�Qnx0, x0Þ � �ð p, x0Þ � �ð p, xnÞ � �ð p, x0Þ: ð3:14Þ

From (3.13) and (3.14), we get that limn!1�(xn, x0) exists.

Step 3 Show that {xn} is a Cauchy sequence in C.

Since xm¼�Qmx02Qm�Qn for m4n, by Lemma 2.6, we have

�ðxm, xnÞ ¼ �ðxm,�Qnx0Þ � �ðxm, x0Þ � �ð�Qn

x0, x0Þ

¼ �ðxm, x0Þ � �ðxn,x0Þ:

Taking m, n!1, we get that �(xm, xn)! 0, which proves that {xn} is Cauchy.

We assume that xn! q2C. It is easy to see that

limn!1

�ðxnþ1, xnÞ ¼ 0: ð3:15Þ

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Since xnþ1 2Cn \Qn � Cn ¼T1

i¼1 Cn,i, we have

�ðxnþ1, un,iÞ � �ðxnþ1, xnÞ þ �n,i�kx0k

2 þ 2hJxn � Jx0,xnþ1i�! 0, ð3:16Þ

as n!1. From (3.15) and (3.16), it is easy to see that

limn!1kun,i � xnk ¼ 0:

Hence un,i! q as n!1. Since J is uniformly norm-to-norm continuous, we get

limn!1kJun,i � Jxnk ¼ 0:

Similar to the proof in steps 5 and 6 of Theorem 3.1, we can show that q2F.

Step 4 Show that q¼�Fx0.

From xn¼�Qnx0, we have

�Jx0 � Jxn, xn � z

� 0 8z2Qn:

Since F�Qn, we also have

�Jx0 � Jq, q� p

� 0 8p2F:

This shows that q¼�Fx0 and completes the proof. g

As a direct consequence of Theorem 3.5, we obtain the following results.

COROLLARY 3.6 Let C be a nonempty and closed convex subset of a real Hilbert space

H. Let f be a bifunction from C�C to R satisfying (A1)–(A4). Let fTig1i¼1 be an

infinitely countable family of closed quasi-nonexpansive mappings from C into itself

such that F :¼T1

i¼1 FðTiÞ \ EPð f Þ 6¼ ;. Assume that {�n,i}� [0, 1] is such that

limn!1�n,i¼ 0, 8i2N and {rn}� [a,1) for some a40. Let {xn} be a sequence

generated by

x0 2C chosen arbitrarily,

yn,i ¼ �n,ix0 þ ð1� �n,iÞTixn, 8n � 0, i2N,

un,i 2C such that f ðun,i, yÞ þ1rnhy� un,i, un,i � yn,ii � 0, 8y2C,

Cn,i ¼�z2C : kz� un,ik

2 � kz� xnk2 þ �n,i

�kx0k

2 þ 2hxn � x0, zi��,

Cn ¼T1

i¼1 Cn,i,

Q0 ¼ C,

Qn ¼�z2Qn�1 : hxn � z, xn � x0i � 0

�, 8n � 1,

xnþ1 ¼ PCn\Qnx0, 8n � 0,

8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:

where PK is the metric projection from H onto a subset K of H.Then, the sequence {xn} converges strongly to q¼PFx0.

COROLLARY 3.7 Let C be a nonempty and closed convex subset of a real Hilbert space

H. Let fTig1i¼1 be an infinitely countable family of closed quasi-nonexpansive mappings

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from C into itself such that F :¼T1

i¼1 FðTiÞ 6¼ ;. Assume that {�n,i}� [0, 1] is such thatlimn!1�n,i¼ 0, 8i2N. Let {xn} be a sequence generated by

x0 2C chosen arbitrarily,yn,i ¼ �n,ix0 þ ð1� �n,iÞTixn, 8n � 0, i2N,Cn,i ¼

�z2C: kz� yn,ik

2 � kz� xnk2 þ �n,i

�kx0k

2 þ 2hxn � x0, zi��,

Cn ¼T1

i¼1 Cn,i,Q0 ¼ C,Qn ¼

�z2Qn�1: hxn � z, xn � x0i � 0

�, 8n � 1,

xnþ1 ¼ PCn\Qnx0, 8n � 0,

8>>>>>>>><>>>>>>>>:

where PK is the metric projection from H onto a subset K of H.Then, the sequence {xn} converges strongly to q¼PFx0.

Acknowledgements

The author would like to thank Prof Suthep Suantai and the referees for the valuablesuggestions on the manuscript. The author was supported by the Commission on HigherEducation and the Thailand Research Fund.

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