Convergence theorems for mixed equilibrium problems, variational inequality problem and uniformly quasi-ϕ-asymptotically nonexpansive mappings

Applied Mathematics and Computation 218 (2011) 3522–3538

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate /amc

Convergence theorems for mixed equilibrium problems, variationalinequality problem and uniformly quasi-/-asymptoticallynonexpansive mappings q

Siwaporn Saewan, Poom Kumam ⇑Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangmod, Bangkok 10140, Thailand

a r t i c l e i n f o

Keywords:Modified block iterative algorithmInverse-strongly monotone operatorVariational inequalityMixed equilibrium problemUniformly quasi-/-asymptoticallynonexpansive mapping

0096-3003/$ - see front matter � 2011 Elsevier Incdoi:10.1016/j.amc.2011.08.099

q The second author was supported by the ThailanThonburi under Grant MRG5380044.⇑ Corresponding author.

E-mail addresses: [email protected] (S. S

a b s t r a c t

We introduce a modified block hybrid projection algorithm for finding a common elementof the set of common fixed points of an infinite family of closed and uniformly quasi-/-asymptotically nonexpansive mappings, the set of the variational inequality for ana-inverse-strongly monotone operator, the set of solutions of the mixed equilibriumproblems. Then, we obtain strong convergence theorems for the sequences generated bythis process in a 2-uniformly convex and uniformly smooth Banach space. Our resultsextend and improve ones from several earlier works.

� 2011 Elsevier Inc. All rights reserved.

1. Introduction

Let C be a nonempty closed convex subset of a real Banach space E with k � k and E⁄ the dual space of E. Let f be a bifunctionof C � C into R and u : C ! R be a real-valued function. The mixed equilibrium problem, denoted by MEP(f,u), is to find x 2 Csuch that

f ðx; yÞ þuðyÞ �uðxÞP 0; 8y 2 C: ð1:1Þ

If u � 0, the problem (1.1) reduce into the equilibrium problem for f, denoted by EP(f), is to find x 2 C such that

f ðx; yÞP 0; 8y 2 C: ð1:2Þ

If f � 0, the problem (1.1) reduce into the minimize problem, denoted by Argmin(u), is to find x 2 C such that

uðyÞ �uðxÞP 0; 8y 2 C: ð1:3Þ

The above formulation (1.2) was shown in [5] to cover monotone inclusion problems, saddle point problems, variationalinequality problems, minimization problems, variational inequality problems, vector equilibrium problems, Nash equilibriain noncooperative games. In addition, there are several other problems, for example, the complementarity problem, fixedpoint problem and optimization problem, which can also be written in the form of an EP(f). In other words, the EP(f) is anunifying model for several problems arising in physics, engineering, science, optimization, economics, etc. In the last twodecades, many papers have appeared in the literature on the existence of solutions of EP(f); see, for example [5,16,22]

. All rights reserved.

d Research Fund, the Higher Education Commission and the King Mongkut’s University of Technology

aewan), [email protected] (P. Kumam).

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S. Saewan, P. Kumam / Applied Mathematics and Computation 218 (2011) 3522–3538 3523

and references therein. Some solution methods have been proposed to solve the EP(f) in Banach spaces; see, for example,[5,12,15,21,24–26,29–33,41,45] and references therein.

For each p > 1, the generalized duality mapping Jp : E! 2E� is defined by

JpðxÞ ¼ fx� 2 E� : hx; x�i ¼ kxkp; kx�k ¼ kxkp�1g

for all x 2 E. In particular, J = J2 is called the normalized duality mapping. If E is a Hilbert space, then J = I, where I is the identitymapping. Consider the functional defined by

/ðx; yÞ ¼ kxk2 � 2hx; Jyi þ kyk2; 8x; y 2 E: ð1:4Þ

As well know that if C is a nonempty closed convex subset of a Hilbert space H and PC :H ? C is the metric projection of Honto C, then PC is nonexpansive. This fact actually characterizes Hilbert spaces and consequently, it is not available in moregeneral Banach spaces. It is obvious from the definition of function / that

ðkxk � kykÞ2 6 /ðx; yÞ 6 ðkxk þ kykÞ2; 8x; y 2 E: ð1:5Þ

If E is a Hilbert space, then /(x,y) = kx � yk2, for all x,y 2 E. On the author hand, the generalized projection (Alber [2]) PC :E ? Cis a map that assigns to an arbitrary point x 2 E the minimum point of the functional /(x,y), that is, PCx ¼ �x, where �x is thesolution to the minimization problem

/ð�x; xÞ ¼ infy2C

/ðy; xÞ; ð1:6Þ

existence and uniqueness of the operator PC follows from the properties of the functional /(x,y) and strict monotonicity ofthe mapping J (see, for example, [1,2,11,14,36]).

Remark 1.1. If E is a reflexive, strictly convex and smooth Banach space, then for x,y 2 E,/(x,y) = 0 if and only if x = y. It issufficient to show that if /(x,y) = 0 then x = y. From (1.4), we have kxk = kyk. This implies that hx, Jyi = kxk2 = kJyk2. From thedefinition of J, one has Jx = Jy. Therefore, we have x = y; see [11,36] for more details.

Let C be a closed convex subset of E, a mapping T :C ? C is said to be L-Lipschitz continuous if kTx � Tyk 6 Lkx � yk, "x,y 2 Cand a mapping T is said to be nonexpansive if kTx � Tyk 6 kx � yk, "x,y 2 C. A point x 2 C is a fixed point of T provided Tx = x.Denote by F(T) the set of fixed points of T; that is, F(T) = {x 2 C :Tx = x}.

Recall that a point p in C is said to be an asymptotic fixed point of T [27] if C contains a sequence {xn} which convergesweakly to p such that limn?1kxn � Txnk = 0. The set of asymptotic fixed points of T will be denoted by gFðTÞ. A mapping T fromC into itself is said to be relatively nonexpansive [35,44] if gFðTÞ ¼ FðTÞ and /(p,Tx) 6 /(p,x) for all x 2 C and p 2 F(T). Theasymptotic behavior of a relatively nonexpansive mapping was studied in [6–8]. T is said to be /-nonexpansive, if /(Tx,Ty) 6 /(x,y) for x,y 2 C. T is said to be relatively quasi-nonexpansive if F(T) – ; and /(p,Tx) 6 /(p,x) for all x 2 C andp 2 F(T). T is said to be quasi-/-asymptotically nonexpansive [15] if F(T) – ; and there exists a real sequence {kn} � [1,1) withkn ? 1 such that /(p,Tnx) 6 kn/(p,x) for all n P 1 x 2 C and p 2 F(T).

We note that the class of relatively quasi-nonexpansive mappings is more general than the class of relatively nonexpan-sive mappings [6–8,19,34] which requires the strong restriction FðTÞ ¼ gFðTÞ. A mapping T is said to be closed if for any se-quence {xn} � C with xn ? x and Txn ? y, then Tx = y. It is easy to know that each relatively nonexpansive mapping is closed.

Recall that let A :C ? E⁄ be a mapping. Then A is called

(i) monotone if

hAx� Ay; x� yiP 0; 8x; y 2 C;

(ii) a � inverse-strongly monotone if there exists a constant a > 0 such that

hAx� Ay; x� yiP akAx� Ayk2; 8x; y 2 C:

The classical variational inequality problem for an operator A is to find x⁄ 2 C such that

hAx�; y� x�iP 0; 8y 2 C: ð1:7Þ

The set of solution of (1.7) is denote by VI(A,C).

Remark 1.2. It is easy to see that an a � inverse-strongly monotone is monotone and 1a-Lipschitz continuous.

In 2004, Matsushita and Takahashi [20] introduced the following iteration: a sequence {xn} defined by

xnþ1 ¼ PCJ�1ðanJxn þ ð1� anÞJTxnÞ; ð1:8Þ

where the initial guess element x0 2 C is arbitrary, {an} is a real sequence in [0,1], T is a relatively nonexpansive mapping andPC denotes the generalized projection from E onto a closed convex subset C of E. They proved that the sequence {xn} con-verges weakly to a fixed point of T.

In 2005, Matsushita and Takahashi [19] proposed the following hybrid iteration method (it is also called the CQ method)with generalized projection for relatively nonexpansive mapping T in a Banach space E:

3524 S. Saewan, P. Kumam / Applied Mathematics and Computation 218 (2011) 3522–3538

x0 2 C chosen arbitrarily;yn ¼ J�1ðanJxn þ ð1� anÞJTxnÞ;Cn ¼ fz 2 C : /ðz; ynÞ 6 /ðz; xnÞg;Q n ¼ fz 2 C : hxn � z; Jx0 � JxniP 0g;xnþ1 ¼ PCn\Qn x0:

8>>>>>><>>>>>>:ð1:9Þ

They proved that {xn} converges strongly to PF(T)x0, where PF(T) is the generalized projection from C onto F(T). In 2008, Iidukaand Takahashi [13] introduced the following iterative scheme for finding a solution of the variational inequality problem foran inverse-strongly monotone operator A in a 2-uniformly convex and uniformly smooth Banach space E: for an initial pointx1 = x 2 C and

xnþ1 ¼ PCJ�1ðJxn � knAxnÞ; ð1:10Þ

for every n = 1,2,3, . . . , where PC is the generalized metric projection from E onto C, J is the duality mapping from E into E⁄

and {kn} is a sequence of positive real numbers. They proved that the sequence {xn} generated by (1.10) converges weakly tosome element of VI(A,C).

Recently, Takahashi and Zembayashi [37,38], studied the problem of finding a common element of the set of fixed pointsof a nonexpansive mapping and the set of solutions of an equilibrium problem in the framework of Banach spaces. In 2009,Wattanawitoon and Kumam [40] using the idea of Takahashi and Zembayashi [37] extend the notion from relatively non-expansive mappings or /-nonexpansive mappings to two relatively quasi-nonexpansive mappings and also proved somestrong convergence theorems to approximate a common fixed point of relatively quasi-nonexpansive mappings and theset of solutions of an equilibrium problen in the framework of Banach spaces. Later, Cholamjiak [10], proved the followingiteration:

zn ¼ PCJ�1ðJxn � knAxnÞ;yn ¼ J�1ðanJxn þ bnJTxn þ cnJSznÞ;un 2 C such that f ðun; yÞ þ 1

rnhy� un; Jun � JyniP 0; 8y 2 C;

Cnþ1 ¼ fz 2 Cn : /ðz; unÞ 6 /ðz; xnÞ;xnþ1 ¼ PCnþ1 x0;

8>>>>>><>>>>>>:ð1:11Þ

where J is the duality mapping on E. Assume that {an}, {bn} and {cn} are sequences in [0,1]. Then {xn} converges strongly toq = PFx0, where F:¼F(T) \ F(S) \ EP(f) \ VI(A,C). In 2010, Saewan et al. [34] introduced a new hybrid projection iterativescheme which is difference from the algorithm (1.11) of Cholamjiak in [10, Theorem 3.1] for two relatively quasi-nonexpan-sive mappings in a Banach space. For an initial point x0 2 E with x1 ¼ PC1 x0 and C1 = C, define a sequence {xn} as follows:

wn ¼ PCJ�1ðJxn � knAxnÞ;zn ¼ J�1ðbnJxn þ ð1� bnÞJTwnÞ;yn ¼ J�1ðanJxn þ ð1� anÞJSznÞ;un 2 C such that f ðun; yÞ þ 1


Cnþ1 ¼ fz 2 Cn : /ðz; unÞ 6 an/ðz; xnÞ þ ð1� anÞ/ðz; znÞ 6 /ðz; xnÞg;xnþ1 ¼ PCnþ1 x0; 8n P 1

8>>>>>>>>><>>>>>>>>>:ð1:12Þ

where J is the duality mapping on E. Then, they proved that under certain appropriate conditions on the paramiters the se-quences {xn} and {un} generated by (1.12) converge strongly to PF(S)\F(T)\EP(f)\VI(A, C).

We note that the block iterative method is a method which often used by many authors to solve the convex feasibilityproblem (CFP) (see, [17,18,29,30], etc.). In 2008, Plubtieng and Ungchittrakool [23] established strong convergence theoremsof block iterative methods for a finite family of relatively nonexpansive mappings in a Banach space by using the hybridmethod in mathematical programming. Chang et al. [9] proposed the modified block iterative algorithm for solving the con-vex feasibility problems for an infinite family of closed and uniformly quasi-/-asymptotically nonexpansive mapping, theyobtained the strong convergence theorems in a Banach space.

Very recently, Qin et al. [26] purposed the problem of approximating a common fixed point of two asymptotically qua-si-/-nonexpansive mappings based on hybrid projection methods. Strong convergence theorems are established in a realBanach space. Zegeye et al. [45] introduced an iterative process which converges strongly to a common element of set ofcommon fixed points of countably infinite family of closed relatively quasi-nonexpansive mappings, the solution set of gen-eralized equilibrium problem and the solution set of the variational inequality problem for an a-inverse strongly monotonemapping in Banach spaces.

In this paper, motivated and inspired by the work of Chang et al. [9], Qin et al. [24], Takahashi and Zembayashi [37],Wattanawitoon and Kumam [40] and Zegeye [43], we introduce a new modified block hybrid projection algorithm for find-ing a common element of the set of the variational inequality for an a-inverse-strongly monotone operator, the set of solu-tions of the mixed equilibrium problems and the set of common fixed points of an infinite family of closed and uniformlyquasi-/-asymptotically nonexpansive mappings in the framework Banach spaces. The results presented in this paper


improve and generalize the main results of Chang et al. [9], Zegeye [43], Wattanawitoon and Kumam [40] and some well-known results in the literature.

2. Preliminaries

A Banach space E is said to be strictly convex if k xþy2 k < 1 for all x,y 2 E with kxk = kyk = 1 and x – y. Let U = {x 2 E :kxk = 1}

be the unit sphere of E. Then a Banach space E is said to be smooth if the limit

limt!0

kxþ tyk � kxkt

exists for each x,y 2 U. It is also said to be uniformly smooth if the limit is attained uniformly for x,y 2 U. Let E be a Banachspace. The modulus of convexity of E is the function d: [0,2] ? [0,1] defined by

dðeÞ ¼ inf 1� k xþ y2k : x; y 2 E; kxk ¼ kyk ¼ 1; kx� ykP e

n o:

A Banach space E is uniformly convex if and only if d(e) > 0 for all e 2 (0,2]. Let p be a fixed real number with p P 2. A Banachspace E is said to be p-uniformly convex if there exists a constant c > 0 such that d(e) P cep for all e 2 [0,2]; see [3,39] for moredetails. Observe that every p-uniformly convex is uniformly convex. One should note that no a Banach space is p-uniformlyconvex for 1 < p < 2. It is well known that a Hilbert space is 2-uniformly convex, uniformly smooth.

Remark 2.1. The following basic properties can be found in Cioranescu [11].

(i) If E is a uniformly smooth Banach space, then J is uniformly norm-to-norm continuous on each bounded subset of E.(ii) If E is a reflexive and strictly convex Banach space, then J�1 is norm-weak⁄-continuous.

(iii) If E is a smooth, strictly convex, and reflexive Banach space, then the normalized duality mapping J : E! 2E� is single-valued, one-to-one, and onto.

(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, that is, for any sequence {xn} � E, if xn N x 2 E andkxnk? kxk, then xn ? x.

We also need the following lemmas for the proof of our main results.

Lemma 2.2 (Beauzamy [4] and Xu [42]). If E be a 2-uniformly convex Banach space. Then for all x,y 2 E we have

kx� yk 6 2c2 kJx� Jyk;

where J is the normalized duality mapping of E and 0 < c 6 1.The best constant 1

c in Lemma is called the p-uniformly convex constant of E.

Lemma 2.3 (Beauzamy [4]). If E be a p-uniformly convex Banach space and let p be a given real number with p P 2. Then for allx,y 2 E, jx 2 Jp(x) and jy 2 Jp(y)

hx� y; jx � jyiPcp

2p�2pkx� ykp

;

where Jp is the generalized duality mapping of E and 1c is the p-uniformly convexity constant of E.

Lemma 2.4 (Kamimura and Takahashi [14]). Let E be a uniformly convex and smooth Banach space and let {xn} and {yn} be twosequences of E. If /(xn,yn) ? 0 and either {xn} or {yn} is bounded, then kxn � ynk? 0.

Lemma 2.5 (Alber [2]). Let C be a nonempty closed convex subset of a smooth Banach space E and x 2 E. Then x0 = PCx if and onlyif

hx0 � y; Jx� Jx0iP 0; 8y 2 C:

Lemma 2.6 (Alber [2, Lemma 2.4]). Let E be a reflexive, strictly convex and smooth Banach space, let C be a nonempty closed con-vex subset of E and let x 2 E. Then

/ðy;PCxÞ þ /ðPCx; xÞ 6 /ðy; xÞ; 8y 2 C:
For solving the equilibrium problem for a bifunction f : C � C ! R, let us assume that f satisfies the following conditions:


(A1) f(x,x) = 0 for all x 2 C;(A2) f is monotone, i.e., f(x,y) + f(y,x) 6 0 for all x,y 2 C;(A3) for each x,y,z 2 C,

limt#0

f ðtzþ ð1� tÞx; yÞ 6 f ðx; yÞ;

(A4) for each x 2 C,y ´ f(x,y) is convex and lower semi-continuous.

Lemma 2.7 (Blum and Oettli [5]). Let C be a closed convex subset of a smooth, strictly convex and reflexive Banach space E, let f bea bifunction from C � C to R satisfying (A1)–(A4), and let r > 0 and x 2 E. Then there exists z 2 C such that

f ðz; yÞ þ 1rhy� z; Jz� JxiP 0; 8y 2 C:

Lemma 2.8 (Zhang [46]). Let C be a closed convex subset of a uniformly smooth, strictly convex and reflexive Banach space E andlet f be a bifunction from C � C to R satisfying (A1)–(A4) and u : C ! R is convex and lower semi-continuous. For r > 0 and x 2 E,define a mapping Tr :C ? C as follows:

Trx ¼ z 2 C : f ðz; yÞ þuðyÞ �uðzÞ þ 1rhy� z; Jz� JxiP 0; 8y 2 C

� �
for all x 2 C. Then the following hold:
(1) Tr is single-valued;(2) Tr is a firmly nonexpansive-type mapping, for all x,y 2 E,

hTrx� Try; JTrx� JTryi 6 hTrx� Try; Jx� Jyi;

(3) F(Tr) = MEP(f,u);(4) MEP (f,u) is closed and convex.

Lemma 2.9 (Zhang [46]). Let C be a closed convex subset of a smooth, strictly convex, and reflexive Banach space E, let f be abifunction from C � C to R satisfying (A1)–(A4). For r > 0, x 2 E and q 2 F(Tr), we have that

/ðq; TrxÞ þ /ðTrx; xÞ 6 /ðq; xÞ:

Let E be a reflexive, strictly convex, smooth Banach space and J be the duality mapping from E into E⁄. Then J�1 is alsosingle value, one-to-one, surjective, and it is the duality mapping from E⁄ into E. We make use of the following mappingV studied in Alber [2]

Vðx; x�Þ ¼ kxk2 � 2hx; x�i þ kx�k2; ð2:1Þ

for all x 2 E and x⁄ 2 E⁄, that is, V(x,x⁄) = /(x, J�1(x⁄)).

Lemma 2.10 (Alber [2]). Let E be a reflexive, strictly convex smooth Banach space and let V be as in (2.1). Then

Vðx; x�Þ þ 2hJ�1ðx�Þ � x; y�i 6 Vðx; x� þ y�Þ;

for all x 2 E and x⁄, y⁄ 2 E⁄.A set valued mapping B :E� E⁄ with graph G(B) = {(x,x⁄) :x⁄ 2 Bx}, domain D(B) = {x 2 E :Bx – ;}, and rang

R(B) = [{Bx :x 2 D(B)}. B is said to be monotone if hx � y,x⁄ � y⁄iP 0 whenever (x,x⁄) 2 G(B), (y,y⁄) 2 G(B). We denote a set val-ued operator B form E to E⁄ by B � E � E⁄. A monotone B is said to be maximal if its graph is not property contained in thegraph of any other monotone operator. If B is maximal monotone, then the solution set B�10 is closed and convex. Let Ebe a reflexive, strictly convex and smooth Banach space, it is knows that B is a maximal monotone if and only if R(J + rB) = E⁄

for all r > 0.Let A be an inverse-strongly monotone mapping of C into E⁄ which is said to be hemicontinuous if for all x,y 2 C, the map-

ping F of [0,1] into E⁄, defined by F(t) = A(tx + (1 � t)y), is continuous with respect to the weak⁄ topology of E⁄. We define byNC(v) the normal cone for C at a point v 2 C, that is,

NCðvÞ ¼ fx� 2 E� : hv � y; x�iP 0; 8y 2 Cg: ð2:2Þ

Lemma 2.11 (Rockafellar [28]). Let C be a nonempty, closed convex subset of a Banach space E and A be a monotone,hemicontinuous operator of C into E⁄. Let B � E � E⁄ be an operator defined as follows:

Bv ¼Av þ NCðvÞ; v 2 C;

;; otherwise:

�ð2:3Þ


Then B is maximal monotone and B�10 = VI(A,C).

Lemma 2.12 (Chang et al. [9]). Let E be a uniformly convex Banach space, r > 0 be a positive number and Br(0) be a closed ball ofE. Then, for any given sequence fxig1i¼1 � Brð0Þ and for any given sequence fkig1i¼1 of positive number with

P1n¼1kn ¼ 1, there exists

a continuous, strictly increasing, and convex function g: [0,2r) ? [0,1) with g(0) = 0 such that, for any positive integer i, j with i < j,

kX1n¼1

knxnk26

X1n¼1

knkxnk2 � kikjgðkxi � xjkÞ: ð2:4Þ

Lemma 2.13 (Chang et al. [9]). Let E be a real uniformly smooth and strictly convex Banach space, and C be a nonempty closedconvex subset of E. Let T :C ? C be a closed and quasi-/-asymptotically nonexpansive mapping with a sequence{kn} � [1,1),kn ? 1. Then F(T) is a closed convex subset of C.

Definition 2.14 [9]. (1) Let fSig1i¼1 : C ! C be a sequence of mapping. fSig1i¼1 is said to be a family of uniformly quasi-/-asymp-totically nonexpansive mappings, if F :¼ \1i¼1FðSiÞ–;, and there exists a sequence {kn} � [1,1) with kn ? 1 such that for eachi P 1

/ p; Sni x

� �6 kn/ðp; xÞ; 8p 2 F; x 2 C; 8n P 1: ð2:5Þ

(2) A mapping S :C ? C is said to be uniformly L-Lipschitz continuous, if there exists a constant L > 0 such that

kSnx� Snyk 6 Lkx� yk; 8x; y 2 C: ð2:6Þ

3. Main results

In this section, we prove the new convergence theorems for finding the set of solutions of a mixed equilibrium problem,the common fixed point set of a family of closed and uniformly quasi-/-asymptotically nonexpansive mappings, and thesolution set of variational inequalities for an a-inverse strongly monotone mapping in a 2-uniformly convex and uniformlysmooth Banach space.

Theorem 3.1. Let C be a nonempty closed and convex subset of a 2-uniformly convex and uniformly smooth Banach space E. Let fbe a bifunction from C � C to R satisfying (A1)–(A4) and u : C ! R be convex and lower semi-continuous. Let A be an a-inverse-strongly monotone mapping of C into E⁄ satisfying kAyk 6 kAy � Auk, "y 2 C and u 2 VI(A,C) – ;. Let fSig1i¼1 : C ! C be aninfinite family of closed uniformly Li-Lipschitz continuous and uniformly quasi-/-asymptotically nonexpansive mappings with asequence {kn} � [1,1),kn ? 1 such that F :¼ \1i¼1FðSiÞ \MEPðf ;uÞ \ VIðA;CÞ is a nonempty and bounded subset in C. For an initialpoint x0 2 E and C1 = C, we define the sequence {xn} as follows:

vn ¼ PCJ�1ðJxn � knAxnÞ;

zn ¼ J�1 an;0Jxn þP1i¼1

an;iJSni vn

� �;

yn ¼ J�1ðbnJxn þ ð1� bnÞJznÞ;f ðun; yÞ þuðyÞ �uðunÞ þ 1


Cnþ1 ¼ fz 2 Cn : /ðz;unÞ 6 /ðz; znÞ 6 /ðz; xnÞ þ hng;xnþ1 ¼ PCnþ1 x0; 8n P 0;

8>>>>>>>>>>><>>>>>>>>>>>:ð3:1Þ

where hn = supq2F(kn � 1)/(q,xn), for each i P 0,{an,i}, {bn} are sequences in [0,1], {rn} � [d,1) for some d > 0 and {kn} � [a,b] forsome a,b with 0 < a 0 and liminfn?1an,0an,i > 0 for all i P 1, then {xn} converges strongly to p 2 F, where p = PFx0.

Proof. . We first show that Cn+1 is closed and convex for each n P 0. Clearly C1 = C is closed and convex. Suppose that Cn isclosed and convex for each n 2 N. Since for any z 2 Cn, we known /(z,un) 6 /(z,xn) + hn is equivalent to2hz, Jxn � Juni 6 kxnk2 � kunk2 + hn. So, Cn+1 is closed and convex.

Next, we show that F � Cn for all n P 0. Indeed, put un ¼ Trn yn for all n P 0. On the other hand, from Lemma 2.8 one hasTrn is relatively quasi-nonexpansive mappings and F � C1 = C. Suppose F � Cn for n 2 N, by the convexity of k � k2, property of/, Lemma 2.12 and by uniformly quasi-/-asymptotically nonexpansive of Sn for each q 2 F � Cn, we have

/ðq;unÞ ¼ /ðq; Trn ynÞ 6 /ðq; ynÞ ¼ /ðq; J�1ðbnJxn þ ð1� bnÞJznÞ

¼ kqk2 � 2hq;bnJxn þ ð1� bnÞJzni þ kbnJxn þ ð1� bnÞJznk2

6 kqk2 � 2bnhq; Jxni � 2ð1� bnÞhq; Jzni þ bnkxnk2 þ ð1� bnÞkznk2 ¼ bn/ðq; xnÞ þ ð1� bnÞ/ðq; znÞ ð3:2Þ


and

/ðq; znÞ ¼ / q; J�1 an;0Jxn þX1i¼1

an;iJSni vn

! !¼ kqk2 � 2hq;an;0Jxn þ

X1i¼1

an;iJSni vni þ kan;0Jxn þ

X1i¼1

an;iJSni vnk2

¼ kqk2 � 2an;0hq; Jxni � 2X1i¼1

an;ihq; JSni vni þ kan;0Jxn þ

X1i¼1

an;iJSni vnk2

6 kqk2 � 2an;0hq; Jxni � 2X1i¼1

an;ihq; JSni vni þ an;0kJxnk

2 þX1i¼1

an;ikJSni vnk2 � an;0an;jgkJvn � JSn

j vnk

¼ kqk2 � 2an;0hq; Jxni þ an;0kJxnk2 � 2

X1i¼1

an;ihq; JSni vni þ

X1i¼1

an;ikJSni vnk2 � an;0an;jgkJvn � JSn

j vnk

¼ an;0/ðq; xnÞ þX1i¼1

an;i/ q; Sni vn

� �� an;0an;jgkJvn � JSn

j vnk

6 an;0/ðq; xnÞ þX1i¼1

an;ikn/ðq;vnÞ � an;0an;jgkJvn � JSnj vnk: ð3:3Þ

It follows from Lemma 2.10, that

/ðq;vnÞ ¼ /ðq;PCJ�1ðJxn � knAxnÞÞ 6 /ðq; J�1ðJxn � knAxnÞÞ ¼ Vðq; Jxn � knAxnÞ

6 Vðq; ðJxn � knAxnÞ þ knAxnÞ � 2hJ�1ðJxn � knAxnÞ � q; knAxni

¼ Vðq; JxnÞ � 2knhJ�1ðJxn � knAxnÞ � q;Axni

¼ /ðq; xnÞ � 2knhxn � q;Axni þ 2hJ�1ðJxn � knAxnÞ � xn;�knAxni: ð3:4Þ

Since q 2 VI(A,C) and A is an a-inverse-strongly monotone mapping, we have

�2knhxn � q;Axni ¼ �2knhxn � q;Axn � Aqi � 2knhxn � q;Aqi 6 �2knhxn � q;Axn � Aqi 6 �2aknkAxn � Aqk2: ð3:5Þ

From Lemma 2.2 and kAxnk 6 kAxn � Aqk, "q 2 VI(A,C), we also have

2hJ�1ðJxn � knAxnÞ � xn;�knAxni ¼ 2hJ�1ðJxn � knAxnÞ � J�1ðJxnÞ;�knAxni 6 2kJ�1ðJxn � knAxnÞ � J�1ðJxnÞkkknAxnk

64c2 kJJ

�1ðJxn � knAxnÞ � JJ�1ðJxnÞkkknAxnk ¼4c2 kJxn � knAxk � JxnkkknAxnk

¼ 4c2 kknAxnk2 ¼ 4

c2 k2nkAxnk2

64c2 k2

nkAxn � Aqk2: ð3:6Þ

Substituting (3.5) and (3.6) into (3.4), we obtain

/ðq;vnÞ 6 /ðq; xnÞ � 2aknkAxn � Aqk2 þ 4c2 k2

nkAxn � Aqk2 ¼ /ðq; xnÞ þ 2kn2c2 kn � a� �

kAxn � Aqk26 /ðq; xnÞ: ð3:7Þ

Substituting (3.7) into (3.3), we also have

/ðq; znÞ 6 an;0/ðq; xnÞ þX1i¼1

an;ikn/ðq; xnÞ � an;0an;jgkJvn � JSnj vnk

6 an;0kn/ðq; xnÞ þX1i¼1

an;ikn/ðq; xnÞ � an;0an;jgkJvn � JSnj vnk ¼ kn/ðq; xnÞ � an;0an;jgkJvn � JSn

j vnk

6 /ðq; xnÞ þ supq2Fðkn � 1Þ/ðq; xnÞ � an;0an;jgkJvn � JSn

j vnk ¼ /ðq; xnÞ þ hn � an;0an;jgkJvn � JSnj vnk

6 /ðq; xnÞ þ hn: ð3:8Þ

and substituting (3.8) into (3.2), we also have

/ðq;unÞ 6 /ðq; xnÞ þ hn: ð3:9Þ

This show that q 2 Cn+1 implies that F � Cn+1 and hence, F � Cn for all n P 0. This implies that the generalized projection pF iswell defined. Since xn ¼ PCn x0 and xnþ1 ¼ PCnþ1 x0 � Cnþ1 � Cn, we have

/ðxn; x0Þ 6 /ðxnþ1; x0Þ; 8n P 0: ð3:10Þ

By Lemma 2.6, we get

/ðxn; x0Þ ¼ /ðPCn x0; x0Þ 6 /ðq; x0Þ � /ðq; xnÞ 6 /ðq; x0Þ; 8q 2 F: ð3:11Þ


From (3.10) and (3.11), then {/(xn,x0)} are nondecreasing and bounded. So, we obtain that limn?1/(xn,x0) exists. In partic-ular, by (1.5), the sequence {(kxnk � kx0k)2} is bounded. This implies {xn}, {vn}, {un}, {yn} and {zn} are also bounded. Denote

M ¼ supnP0fkxnkg <1: ð3:12Þ

Moreover, by the definition of {hn} and (3.12), it follows that

hn ! 0 as n!1: ð3:13Þ

Next, we show that {xn} is a Cauchy sequence in C. Since xm ¼ PCm x0 2 Cm � Cn, for m > n, by Lemma 2.6, we have

/ðxm; xnÞ ¼ /ðxm;PCn x0Þ 6 /ðxm; x0Þ � /ðPCn x0; x0Þ ¼ /ðxm; x0Þ � /ðxn; x0Þ:

Since limn?1/(xn,x0) exists and we taking m,n ?1 then, we get /(xm,xn) ? 0. From Lemma 2.4, we havelimn?1kxm � xnk = 0. Thus {xn} is a Cauchy sequence and by the completeness of E and there exist a point p 2 C such thatxn ? p as n ?1.

Now, we claim that kJun � Jxnk? 0, as n ?1. By definition of PCn x0, we have

/ðxnþ1; xnÞ ¼ /ðxnþ1;PCn x0Þ 6 /ðxnþ1; x0Þ � /ðPCn x0; x0Þ ¼ /ðxnþ1; x0Þ � /ðxn; x0Þ:

Since limn?1/(xn,x0) exists, we also have

limn!1

/ðxnþ1; xnÞ ¼ 0: ð3:14Þ

Again form Lemma 2.4, that

limn!1kxnþ1 � xnk ¼ 0: ð3:15Þ

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

limn!1kJxnþ1 � Jxnk ¼ 0: ð3:16Þ

Since xnþ1 ¼ PCnþ1 x0 2 Cnþ1 � Cn and the definition of Cn+1, we have

/ðxnþ1;unÞ 6 /ðxnþ1; xnÞ þ hn:

By (3.13) and (3.14), that

limn!1

/ðxnþ1;unÞ ¼ 0: ð3:17Þ

Again applying Lemma 2.4, we have

limn!1kxnþ1 � unk ¼ 0: ð3:18Þ

Since

kun � xnk ¼ kun � xnþ1 þ xnþ1 � xnk 6 kun � xnþ1k þ kxnþ1 � xnk:

It follows that

limn!1kun � xnk ¼ 0: ð3:19Þ

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

limn!1kJun � Jxnk ¼ 0: ð3:20Þ

Next, we will show that p 2 F :¼ MEPðf ;uÞ \ \1i¼1FðSiÞ� �

\ VIðA;CÞ.(a) First, we show that p 2MEP(f,u). From (3.2)–(3.8) and (3.13) , we get /(q,yn) 6 /(q,xn). By Lemma 2.9 and un ¼ Trn yn,

we observe that

/ðun; ynÞ ¼ /ðTrn yn; ynÞ 6 /ðq; ynÞ � /ðq; Trn ynÞ 6 /ðq; xnÞ � /ðq; Trn ynÞ ¼ /ðq; xnÞ � /ðq;unÞ

¼ kqk2 � 2hq; Jxni þ kxnk2 � ðkqk2 � 2hq; Juni þ kunk2Þ ¼ kxnk2 � kunk2 � 2hq; Jxn � Juni6 kxn � unkðkxnk þ kunkÞ þ 2kqkkJxn � Junk: ð3:21Þ

From (3.19), (3.20) and Lemma 2.4, we have

limn!1kun � ynk ¼ 0: ð3:22Þ

Again since J is uniformly norm-to-norm continuous, we also have

limn!1kJun � Jynk ¼ 0: ð3:23Þ


From (A2), that

uðyÞ �uðunÞ þ1rnhy� un; Jun � JyniP f ðy;unÞ; 8y 2 C;

uðyÞ �uðunÞ þ hy� un;ðJun � JynÞ

rniP f ðy; unÞ; 8y 2 C:

From rn > 0 then kJun�Jynkrn

! 0 and un ? p as n ?1, we obtain

f ðy;pÞ þuðpÞ �uðyÞ 6 0:

For t with 0 < t < 1 and y 2 C, let yt = ty + (1 � t)p. Then yt 2 C and hence f(yt,p) + u(p) � u(yt) 6 0. By the conditions (A1), (A4)and convexity of u, we have

0 ¼ f ðyt; ytÞ þuðytÞ �uðytÞ 6 tf ðyt ; yÞ þ ð1� tÞf ðyt; pÞ þ tuðyÞ þ ð1� tÞuðpÞ �uðytÞ6 tðf ðyt; yÞ þuðyÞ �uðytÞÞ þ ð1� tÞðf ðyt; pÞ þuðpÞ �uðytÞÞ 6 t½f ðyt; yÞ þuðyÞ �uðytÞ�:

From (A3) and the weakly lower semicontinuity of u, we also have f(p,y) + u(y) � u(p) P 0, "y 2 C. This impliesp 2MEP(f,u).

(b) We show that p 2 \1i¼1FðSiÞ. From definition of Cn+1, we have /(z,zn) 6 /(z,xn) + hn. Since xnþ1 ¼ PCnþ1 x0 2 Cnþ1, we get/(xn+1,zn) 6 /(xn+1,xn) + hn. It follows from (3.14), that

limn!1

/ðxnþ1; znÞ ¼ 0 ð3:24Þ

again form Lemma 2.4, that

limn!1kxnþ1 � znk ¼ 0: ð3:25Þ

Since J is uniformly norm-to-norm continuous, we obtain

limn!1kJxnþ1 � Jznk ¼ 0: ð3:26Þ

From (3.1), we note that

kJxnþ1 � Jznk ¼ kJxnþ1 � an;0Jxn þX1i¼1

an;iJSni vn

!k

¼ kan;0Jxnþ1 � an;0Jxn þX1i¼1

an;iJxnþ1 �X1i¼1

an;iJSni vnk

¼ kan;0ðJxnþ1 � JxnÞ þX1i¼1

an;i Jxnþ1 � JSni vn

� �k

¼ kX1i¼1

an;i Jxnþ1 � JSni vn

� �� an;0ðJxn � Jxnþ1Þk

PX1i¼1

an;ikJxnþ1 � JSni vnk � an;0kJxn � Jxnþ1k;

and hence

kJxnþ1 � JSni vnk 6

1P1i¼1an;i

ðkJxnþ1 � Jznk þ an;0kJxn � Jxnþ1kÞ: ð3:27Þ

From (3.16), (3.26) and lim infn!1P1

i¼1an;i > 0, we obtain that

limn!1kJxnþ1 � JSn

i vnk ¼ 0: ð3:28Þ

Since J�1 is uniformly norm-to-norm continuous on bounded sets, we have

limn!1kxnþ1 � Sn

i vnk ¼ 0: ð3:29Þ

Using the triangle inequality, that

kxn � Sni vnk ¼ kxn � xnþ1 þ xnþ1 � Sn

i vnk 6 kxn � xnþ1k þ kxnþ1 � Sni vnk:

From (3.15) and (3.29), we have

limn!1kxn � Sn

i vnk ¼ 0: ð3:30Þ

On the other hand, we note that

/ðq; xnÞ � /ðq;unÞ þ hn ¼ kxnk2 � kunk2 � 2hq; Jxn � Juni þ hn:


It follows from hn ? 0, kxn � unk? 0 and kJxn � Junk? 0, that

/ðq; xnÞ � /ðq;unÞ þ hn ! 0 as n!1: ð3:31Þ

From (3.2), (3.3) and (3.7), that

/ðq;unÞ 6 /ðq; ynÞ 6 bn/ðq; xnÞ þ ð1� bnÞ/ðq; znÞ

6 bn/ðq; xnÞ þ ð1� bnÞ an;0/ðq; xnÞ þX1i¼1

an;ikn/ðq;vnÞ � an;0an;jgkJvn � JSnj vnk

" #

¼ bn/ðq; xnÞ þ ð1� bnÞan;0/ðq; xnÞ þ ð1� bnÞX1i¼1

an;ikn/ðq;vnÞ � ð1� bnÞan;0an;jgkJvn � JSnj vnk

6 bn/ðq; xnÞ þ ð1� bnÞan;0/ðq; xnÞ þ ð1� bnÞX1i¼1

an;ikn/ðq;vnÞ

6 bn/ðq; xnÞ þ ð1� bnÞan;0/ðq; xnÞ þ ð1� bnÞX1i¼1

an;ikn /ðq; xnÞ � 2kn a� 2c2 kn

� �kAxn � Aqk2

�

6 bn/ðq; xnÞ þ ð1� bnÞan;0kn/ðq; xnÞ þ ð1� bnÞX1i¼1

an;ikn/ðq; xnÞ � ð1� bnÞX1i¼1

an;ikn2kn a� 2c2 kn


¼ bnkn/ðq; xnÞ þ ð1� bnÞkn/ðq; xnÞ � ð1� bnÞX1i¼1



6 kn/ðq; xnÞ � ð1� bnÞX1i¼1


� �kAxn � Aqk2�

6 /ðq; xnÞ þ supq2Fðkn � 1Þ/ðq; xnÞ � ð1� bnÞ

X1i¼1

an;ikn2knða�2c2 knÞkAxn � Aqk2

6 /ðq; xnÞ þ hn � ð1� bnÞX1i¼1



and hence

2a a� 2bc2


6 2kn a� 2c2 kn


61

ð1� bnÞP1

i¼1an;iknð/ðq; xnÞ � /ðq;unÞ þ hnÞ: ð3:32Þ

From (3.31), {kn} � [a,b] for some a,b with 0 < a 0 and liminfn?1an,0an, i > 0, for i P 0 andkn ? 1 as n ?1, we obtain that

limn!1kAxn � Aqk ¼ 0: ð3:33Þ

From Lemma 2.6, Lemma 2.10 and (3.6), we compute

/ðxn;vnÞ ¼ /ðxn;PCJ�1ðJxn � knAxnÞÞ 6 /ðxn; J�1ðJxn � knAxnÞÞ ¼ Vðxn; Jxn � knAxnÞ

6 Vðxn; ðJxn � knAxnÞ þ knAxnÞ � 2hJ�1ðJxn � knAxnÞ � xn; knAxni ¼ /ðxn; xnÞ þ 2hJ�1ðJxn � knAxnÞ � xn;�knAxni

¼ 2hJ�1ðJxn � knAxnÞ � xn;�knAxni 64k2

n

c2 kAxn � Aqk26

4b2

c2 kAxn � Aqk2:

Applying Lemma 2.4 and (3.33) that

limn!1kxn � vnk ¼ 0 ð3:34Þ

and we also obtain

limn!1kJxn � Jvnk ¼ 0: ð3:35Þ

From Sni is continuous, for any i P 1

limn!1kSn

i xn � Sni vnk ¼ 0: ð3:36Þ

Again by the triangle inequality, we get

kxn � Sni xnk 6 kxn � Sn

i vnk þ kSni vn � Sn

i xnk:


From (3.30) and (3.36), we have

limn!1kxn � Sn

i xnk ¼ 0; 8i P 1: ð3:37Þ

Since J is uniformly continuous on any bounded subset of E, we obtain

limn!1kJxn � JSn

i xnk ¼ 0; 8i P 1: ð3:38Þ

Since xn ? p and J is uniformly continuous, it yields Jxn ? Jp. Thus from (3.38), we get

JSni xn ! Jp; 8i P 1: ð3:39Þ

Since J�1 :E⁄? E is norm-weake⁄-continuous, we have

Sni xn * p; for each i P 1: ð3:40Þ

On the other hand, for each i P 1, we have

jkSni xnk � kpkj ¼ jkJ Sn

i xn� �

k � kJpkj 6 kJ Sni xn

� �� Jpk:

In view of (3.39), we obtain kSni xnk ! kpk for each i P 1. Since E is uniformly convex Banach spaces then E has the Kadec–Klee

property, we get

Sni xn ! p for each ı P 1:

Moreover, by the assumption that "i P 1, Si is uniformly Li-Lipschitz continuous, hence we have.

kSnþ1i xn � Sn

i xnk 6 kSnþ1i xn � Snþ1

i xnþ1k þ kSnþ1i xnþ1 � xnþ1k þ kxnþ1 � xnk þ kxn � Sn

i xnk

6 ðLi þ 1Þkxnþ1 � xnk þ kSnþ1i xnþ1 � xnþ1k þ kxn � Sn

i xnk: ð3:41Þ

By (3.15) and (3.37), it yields that kSnþ1i xn � Sn

i xnk ! 0. From Sni xn ! p, we have Snþ1

i xn ! p, that is SiSni xn ! p. In view of close-

ness of Si, we have Sip = p, for all i P 1. This imply that p 2 \1i¼1FðSiÞ.(c) We show that p 2 VI(A,C). Indeed, define B � E � E⁄ by

Bv ¼Av þ NCðvÞ; v 2 C;

;; v R C:

�ð3:42Þ

By Lemma 2.11, B is maximal monotone and B�10 = VI(A,C). Let (v,w) 2 G(B). Since w 2 Bv = Av + NC(v), we get w � Av 2 NC(v).From vn 2 C, we have

hv � vn;w� AviP 0: ð3:43Þ

On the other hand, since vn = PCJ�1(Jxn � knAxn). Then by Lemma 2.5, we have

hv � vn; Jvn � ðJxn � knAxnÞiP 0;

and thus

v � vn;Jxn � Jvn

kn� Axn

�6 0: ð3:44Þ

It follows from (3.43), (3.44) and A is monotone and 1a-Lipschitz continuous, that

hv � vn;wiP hv � vn;AviP hv � vn;Avi þ hv � vn;Jxn � Jvn

kn� Axni ¼ hv � vn;Av � Axni þ hv � vn;

Jxn � Jvn

kni

¼ hv � vn;Av � Avni þ hv � vn;Avn � Axni þ hv � vn;Jxn � Jvn

kni

P �kv � vnkkvn � xnk

a� kv � vnk

kJxn � Jvnka

P �Gkvn � xnk

aþ kJxn � Jvnk

a

� �;

where G = supnP1kv � vnk. By (3.34), (3.35) and take the limit as n ?1, we obtain hv � p,wiP 0. By the maximality of B wehave p 2 B�10, that is p 2 VI(A,C).

Finally, we show that p = PFx0. From xn ¼ PCn x0, we have hJx0 � Jxn,xn � ziP 0, "z 2 Cn. Since F � Cn, we also have

hJx0 � Jxn; xn � yiP 0; 8y 2 F:

Taking limit n ?1, we obtain

hJx0 � Jp; p� yiP 0; 8y 2 F:

By Lemma 2.5, we can conclude that p = PFx0 and xn ? p as n ?1. This completes the proof. h

If Si = S for each i 2 N, then Theorem 3.1 is reduced to the following corollary.


Corollary 3.2. Let C be a nonempty closed and convex subset of a 2-uniformly convex and uniformly smooth Banach space E. Let fbe a bifunction from C � C to R satisfying (A1)–(A4) and u : C ! R be convex and lower semi-continuous. Let A be an a-inverse-strongly monotone mapping of C into E⁄ satisfying kAyk 6 kAy � Auk, "y 2 C and u 2 VI(A,C) – ;. Let S :C ? C be a closeduniformly L-Lipschitz continuous and quasi-/-asymptotically nonexpansive mappings with a sequence {kn} � [1,1),kn ? 1 suchthat F:¼ F(S) \MEP(f,u) \ VI(A,C) is a nonempty and bounded subset in C. For an initial point x0 2 E and C1 = C, we define thesequence {xn} as follows:

vn ¼ PCJ�1ðJxn � knAxnÞ;zn ¼ J�1ðanJxn þ ð1� anÞJSnvnÞ;yn ¼ J�1ðbnJxn þ ð1� bnÞJznÞ;f ðun; yÞ þuðyÞ �uðunÞ þ 1



8>>>>>>>>><>>>>>>>>>:ð3:45Þ

where hn = supq2F(kn � 1)/(q,xn),{an}, {bn} are sequences in [0,1], {rn} � [d,1) for some d > 0 and {kn} � [a,b] for some a,b with0 < a 0 and liminfn?1an(1 � an) > 0 for alln P 0, then {xn} converges strongly to p 2 F, where p = PFx0.

For a special case that i = 1,2, we can obtain the following results on a pair of quasi-/-asymptotically nonexpansive map-pings immediately from Theorem 3.1.

Corollary 3.3. Let C be a nonempty closed and convex subset of a 2-uniformly convex and uniformly smooth Banach space E. Let fbe a bifunction from C � C to R satisfying (A1)–(A4) and u : C ! R be convex and lower semi-continuous. Let A be an a-inverse-strongly monotone mapping of C into E⁄ satisfying kAyk 6 kAy � Auk, "y 2 C and u 2 VI(A,C) – ;. Let S,T:C ? C be two closedquasi-/-asymptotically nonexpansive mappings and LS,LT-Lipschitz continuous, respectively with a sequence {kn} � [1,1),kn ? 1such that F:¼F(S) \ F(T) \MEP(f,u) \ VI(A,C) is a nonempty and bounded subset in C. For an initial point x0 2 E and C1 = C, wedefine the sequence {xn} as follows:

vn ¼ PCJ�1ðJxn � knAxnÞ;zn ¼ J�1ðanJxn þ bnJSnvn þ cnJTnvnÞ;yn ¼ J�1ðdnJxn þ ð1� dnÞJznÞ;f ðun; yÞ þuðyÞ �uðunÞ þ 1



8>>>>>>>>><>>>>>>>>>:ð3:46Þ

where hn = supq2F(kn � 1)/(q,xn), {an}, {bn} and {dn} are sequences in [0,1], {rn} � [d,1) for some d > 0 and {kn} � [a,b] for somea,b with 0 < a 0and liminfn?1ancn > 0, then {xn} converges strongly to p 2 F, where p = PFx0.

Corollary 3.4. Let C be a nonempty closed and convex subset of a 2-uniformly convex and uniformly smooth Banach space E. Let fbe a bifunction from C � C to R satisfying (A1)–(A4) and u : C ! R be convex and lower semi-continuous. Let A be an a-inverse-strongly monotone mapping of C into E⁄ satisfying kAyk 6 kAy � Auk, "y 2 C and u 2 VI(A,C) – ;. Let fSig1i¼1 : C ! C be an infi-nite family of closed quasi-/-nonexpansive mappings such that F :¼ \1i¼1FðSiÞ \MEPðf ;uÞ \ VIðA;CÞ–;. For an initial point x0 2 Eand C1 = C, we define the sequence {xn} as follows:

vn ¼ PCJ�1ðJxn � knAxnÞ;

zn ¼ J�1ðan;0Jxn þP1i¼1

an;iJSivnÞ;



Cnþ1 ¼ fz 2 Cn : /ðz;unÞ 6 /ðz; znÞ 6 /ðz; xnÞg;xnþ1 ¼ PCnþ1 x0; 8n P 0;

8>>>>>>>>>>><>>>>>>>>>>>:ð3:47Þ

where {an,i},{bn} are sequences in [0,1], for all i P 1,{rn} � [d,1) for some d > 0 and {kn} � [a,b] for some a,b with 0 < a 0 and liminfn?1an,0an,i > 0for all i P 1, then {xn} converges strongly to p 2 F, where p = PFx0.

Proof. Since fSig1i¼1 : C ! C is an infinite family of closed quasi-/-nonexpansive mappings, it is an infinite family of closedand uniformly quasi-/-asymptotically nonexpansive mappings with sequence kn = 1. Hence the conditions appearing in The-orem 3.1 F is a bounded subset in C and for each i P 1, Si is uniformly Li-Lipschitz continuous are of no use here. By virtue of


the closeness of mapping Si for each i P 1, it yields that p 2 F(Si) for each i P 1, that is, p 2 \1i¼1FðSiÞ. Therefore all conditionsin Theorem 3.1 are satisfied. The conclusion of Corollary 3.4 is obtained from Theorem 3.1 immediately. h

4. Deduced theorems

Corollary 4.1 ([Theorem 3.2, 43]). Let C be a nonempty closed and convex subset of a 2-uniformly convex and uniformly smoothBanach space E. Let f be a bifunction from C � C to R satisfying (A1)–(A4) and u : C ! R be convex and lower semi-continuous. LetA be an a-inverse-strongly monotone mapping of C into E⁄ satisfying kAyk 6 kAy � Auk, "y 2 C and u 2 VI(A,C) – ;. Let

fSigNi¼1 : C ! C be a finite family of closed quasi-/-nonexpansive mappings such that F :¼ \N

i¼1FðSiÞ \MEPðf ;uÞ \ VIðA; CÞ–;. Foran initial point x0 2 E and C1 = C, we define the sequence {xn} as follows:

zn ¼ PCJ�1ðJxn � knAxnÞ;

yn ¼ J�1ða0Jxn þPNi¼1

aiJSiznÞ;

f ðun; yÞ þuðyÞ �uðunÞ þ 1rnhy� un; Jun � JyniP 0; 8y 2 C;

Cnþ1 ¼ fz 2 Cn : /ðz; unÞ 6 /ðz; znÞ 6 /ðz; xnÞg;xnþ1 ¼ PCnþ1 x0; 8n P 0;

8>>>>>>>>>><>>>>>>>>>>:ð4:1Þ

where {ai} is a sequence in [0,1] for i = 1,2,3, . . . ,N, {rn} � [d,1) for some d > 0 and {kn} � [a,b] for some a,b with 0 < a < b < c2a/2,where 1

c is the 2-uniformly convexity constant of E. If ai 2 (0,1) such thatPN

i¼0ai ¼ 1, then {xn} converges strongly to p 2 F, wherep = PFx0.

Remark 4.2. Theorems 3.1, Corollary 3.4 and Corollary 4.1 improve and extend the corresponding results of Wattanawitoonand Kumam [40] and Zegeye [43] in the following senses:

From a solution of the classical equilibrium problem to the generalized equilibrium problem with an infinite family ofquasi-/-asymptotically mappings; For the mappings, we extend the mappings from nonexpansive mappings, relatively quasi-nonexpansive mappings or

quasi-/-nonexpansive mappings and a finite family of closed relatively quasi-nonexpansive mappings to an infinite fam-ily of quasi-/-asymptotically nonexpansive mappings.

Corollary 4.3. Let C be a nonempty closed and convex subset of a uniformly convex and uniformly smooth Banach space E. Let f bea bifunction from C � C to R satisfying (A1)–(A4) and u : C ! R be convex and lower semi-continuous. Let fSig1i¼1 : C ! C be aninfinite family of closed and uniformly quasi-/-asymptotically nonexpansive mappings with a sequence {kn} � [1,1),kn ? 1 anduniformly Li-Lipschitz continuous such that F :¼ \1i¼1FðSiÞ \MEPðf ;uÞ is a nonempty and bounded subset in C. For an initial pointx0 2 E and C1 = C, we define the sequence {xn} as follows:

yn ¼ J�1 an;0Jxn þP1i¼1

an;iJSni xn

� �;

ðun; yÞ þuðyÞ �uðunÞ þ 1rnhy� un; Jun � JyniP 0; 8y 2 C;

Cnþ1 ¼ fz 2 Cn : /ðz; unÞ 6 /ðz; znÞ 6 /ðz; xnÞ þ hng;xnþ1 ¼ PCnþ1 x0; 8n P 0;

8>>>>><>>>>>:ð4:2Þ

where hn = supq2F(kn � 1)/(q,xn),{an,i} is a sequence in [0,1], {rn} � [a,1) for some a > 0. IfP1

i¼0an;i ¼ 1 for all n P 0 and li-minfn?1an,0an,i > 0 for all i P 1, then {xn} converges strongly to p 2 F, where p = PFx0.

Proof. Put A � 0 and bn � 0 in Theorem 3.1 Then, we get that vn = xn. Thus, the method of proof of Theorem 3.1 gives therequired assertion without the requirement that E be 2-uniformly convex. h

If, setting u � 0 in Corollary 4.3, then we obtain the following corollary.

Corollary 4.4. Let C be a nonempty closed and convex subset of a uniformly convex and uniformly smooth Banach space E. Let f bea bifunction from C � C to R satisfying (A1)–(A4). Let fSig1i¼1 : C ! C be an infinite family of closed and uniformly quasi-/-asymptotically nonexpansive mappings with a sequence {kn} � [1,1),kn ? 1 and uniformly Li-Lipschitz continuous such thatF :¼ \1i¼1FðSiÞ \ EPðf Þ is a nonempty and bounded subset in C. For an initial point x0 2 E and C1 = C, we define the sequence {xn} asfollows:


yn ¼ J�1 an;0Jxn þP1i¼1

an;iJSni xn

� �;

f ðun; yÞ þ 1rnhy� un; Jun � JyniP 0; 8y 2 C;


8>>>>>>><>>>>>>>:ð4:3Þ

where hn = supq2F(kn � 1)/(q,xn),{an,i} is a sequence in [0,1], {rn} � [a,1) for some a > 0. IfP1

i¼0an;i ¼ 1 for all n P 0 and li-minfn?1an,0an,i > 0 for all i P 1, then {xn} converges strongly to p 2 F, where p = PFx0.

Remark 4.5. Corollary 4.3 and Corollary 4.4 improve and extend the corresponding results of Zegeye [43] and Wattanawi-toon and Kumam [40] in the sense from a finite family of closed relatively quasi-nonexpansive mappings and closed rela-tively quasi-nonexpansive mappings to more general than an infinite family of closed and uniformly quasi-/-asymptotically nonexpansive mappings.

Remark 4.6. Moreover, Our theorems improve, generalize, unify and extend Qin et al.[26], Zegeye et al. [45], Zegeye [43] andWattanawitoon and Kumam [40] and several results recently announced.

5. Applications

5.1. Application to Hilbert spaces

If E = H, a Hilbert space, then E is a 2-uniformly convex (we can choose c = 1) and uniformly smooth real Banach space andclosed relatively quasi-nonexpansive map reduces to closed quasi-nonexpansive map. Moreover, J = I, identity operator on Hand PC = PC, projection mapping from H into C. Thus, the following theorem hold.

Theorem 5.1. Let C be a nonempty closed and convex subset of a Hilbert space H. Let f be a bifunction from C � C to R satisfying(A1)–(A4) and u : C ! R is convex and lower semi-continuous. Let A be an a-inverse-strongly monotone mapping of C into Hsatisfying kAyk 6 kAy � Auk, "y 2 C and u 2 VI(A,C) – ;. Let fSig1i¼1 : C ! C be an infinite family of closed and uniformlyquasi-/-asymptotically nonexpansive mappings with a sequence {kn} � [1,1),kn ? 1 and uniformly Li-Lipschitz continuous suchthat F :¼ \1i¼1FðSiÞ \MEPðf ;uÞ \ VIðA;CÞ is a nonempty and bounded subset in C. For an initial point x0 2 H and C1 = C, we definethe sequence {xn} as follows:

zn ¼ PCðxn � knAxnÞ;

yn ¼ an;0xn þP1i¼1

an;iSni zn;

f ðun; yÞ þuðyÞ �uðunÞ þ 1rnhy� un; un � yniP 0; 8y 2 C;

Cnþ1 ¼ fz 2 Cn : kz� unk 6 kz� znk 6 kz� xnk þ hng;xnþ1 ¼ PCnþ1 x0; 8n P 0;

8>>>>>>>>><>>>>>>>>>:ð5:1Þ

where hn = supq2F(kn � 1)kq � xnk,{an,i} is a sequence in [0,1], {rn} � [d,1) for some d > 0 and {kn} � [a,b] for some a,b with0 < a 0 for all i P 1, then {xn} converges strongly to p 2 F, where

p = PFx0.

Remark 5.2. Theorem 5.1 improve and extend the Corollary 3.7 of Zegeye [43] in the aspect for the mappings, we extend themappings from a finite family of closed relatively quasi-nonexpansive mappings to more general an infinite family of closedand uniformly quasi-/-asymptotically nonexpansive mappings.

5.2. Application to zeros of an inverse-strongly monotone operator

Next, we consider the problem of finding a zero point of an inverse-strongly monotone operator of E into E⁄. Assume thatA satisfies the conditions:

(C1) A is a-inverse-strongly monotone,(C2) A�10 = {u 2 E :Au = 0} – ;.

Theorem 5.3. Let C be a nonempty closed and convex subset of a 2-uniformly convex and uniformly smooth Banach space E. Let fbe a bifunction from C � C to R satisfying (A1)–(A4) and u : C ! R be convex and lower semi-continuous. Let A be an operator of E


into E⁄ satisfying (C1) and (C2). Let fSig1i¼1 : C ! C be an infinite family of closed uniformly Li-Lipschitz continuous and uniformlyquasi-/-asymptotically nonexpansive mappings with a sequence {kn} � [1,1),kn ? 1 such that

F :¼ \1i¼1FðSiÞ \MEPðf ;uÞ \ A�10

is a nonempty and bounded subset in C. For an initial point x0 2 E and C1 = C, we define the sequence {xn} as follows:

vn ¼ J�1ðJxn � knAxnÞ


an;iJSni vn

� �;



Cnþ1 ¼ fz 2 Cn : /ðz; unÞ 6 /ðz; znÞ 6 /ðz; xnÞ þ hng;xnþ1 ¼ PCnþ1 x0; 8n P 0;

8>>>>>>>>>>>>><>>>>>>>>>>>>>:ð5:2Þ

where hn = supq2F(kn � 1)/(q,xn), for each i P 0,{an,i}, {bn} are sequences in [0,1], {rn} � [d,1) for some d > 0 and {kn} � [a,b] forsome a,b with 0 < a 0 and liminfn?1an,0an,i > 0 for all i P 1, then {xn} converges strongly to p 2 F, where p = PFx0.

Proof. Setting C = E in Corollary 3.4, we also get PE = I. We also have VI(A,C) = VI(A,E) = {x 2 E :Ax = 0} – ; and then the con-dition kAyk 6 kAy � Auk holds for all y 2 E and u 2 A�10. So, we obtain the result. h

5.3. Application to Complementarity problems

Let K be a nonempty, closed convex cone in E. We define the polar K⁄ of K as follows:

K� ¼ fy� 2 E� : hx; y�iP 0;8x 2 Kg: ð5:3Þ

If A :K ? E⁄ is an operator, then an element u 2 K is called a solution of the complementarity problem ([36]) if

Au 2 K�; and hu;Aui ¼ 0: ð5:4Þ

The set of solutions of the complementarity problem is denoted by CP(A,K).

Theorem 5.4. Let K be a nonempty closed and convex subset of a 2-uniformly convex and uniformly smooth Banach space E. Let fbe a bifunction from K � K to R satisfying (A1)–(A4) and u : K ! R be convex and lower semi-continuous. Let A be an a-inverse-strongly monotone mapping of K into E⁄ satisfying kAyk 6 kAy � Auk, "y 2 K and u 2 CP(A,K) – ;. Let fSig1i¼1 : K ! K be aninfinite family of closed uniformly Li-Lipschitz continuous and uniformly quasi-/-asymptotically nonexpansive mappings with asequence {kn} � [1,1),kn ? 1 such that F :¼ \1i¼1FðSiÞ \MEPðf ;uÞ \ CPðA;KÞ is a nonempty and bounded subset in K. For aninitial point x0 2 E with x1 ¼ PK1 x0 and K1 = K, we define the sequence {xn} as follows:

vn ¼ PK J�1ðJxn � knAxnÞ;


an;iJSni vn

� �;


rnhy� un; Jun � JyniP 0; 8y 2 K;

Knþ1 ¼ fz 2 Kn : /ðz;unÞ 6 /ðz; znÞ 6 /ðz; xnÞ þ hng;xnþ1 ¼ PKnþ1 x0; 8n P 0;

8>>>>>>>>>>><>>>>>>>>>>>:ð5:5Þ

where hn = supq2F(kn � 1)/(q,xn), for each i P 0, {an, i}, {bn} are sequences in [0,1], {rn} � [d,1) for some d > 0 and {kn} � [a,b] forsome a,b with 0 < a 0 and liminfn?1an,0an, i > 0 for all i P 1, then {xn} converges strongly to p 2 F, where p = PFx0.

Proof. As in the proof of Takahashi in [36, Lemma 7.11], we get that VI(A,K) = CP(A,K). So, we obtain the result. h

Acknowledgements

The author thanks the Office of the Higher Education Commission, Thailand for supporting by grant under the programStrategic Scholarships for Frontier Research Network for the Join Ph.D. Program Thai Doctoral degree for this research.More-over, the authors were supported by the King Mongkuts Diamond scholarship for Ph.D. program at King Mongkuts Universityof Technology Thonburi (KMUTT), under project NRU-CSEC No.54000267 and this work was partially supported by the


Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher EducationCommission. Finally, we also thank the referees for their valuable comments and suggestions.

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Convergence theorems for mixed equilibrium problems, variational inequality problem and uniformly quasi-ϕ-asymptotically nonexpansive mappings

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