A strong convergence theorem for solutions of equilibrium problems and asymptotically quasi-ϕ-nonexpansive mappings in the intermediate sense

(1)

R E S E A R C H

Open Access

A strong convergence theorem for solutions

of equilibrium problems and asymptotically

quasi-

φ

-nonexpansive mappings in the

intermediate sense

Yuan Qing

1

_{and Songtao Lv}

2*

*_{Correspondence: [email protected]} 2_{School of Mathematics and}

Information Science, Shangqiu Normal University, Shangqiu, 476000, China

Full list of author information is available at the end of the article

Abstract

In this paper, common solutions to an equilibrium problem and a nonlinear operator equation involving a ﬁnite family of asymptotically quasi-φ-nonexpansive mappings in the intermediate sense are discussed. Strong convergence theorems of common solutions are established in a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property.

MSC: 47H09; 47J25

Keywords: asymptotically quasi-φ-nonexpansive mapping; asymptotically quasi-φ-nonexpansive mapping in the intermediate sense; generalized projection; equilibrium problem; ﬁxed point

1 Introduction

Equilibrium problems have been revealed as a very powerful and important tool in the study of nonlinear phenomena. They have turned out to be very useful in studying opti-mization problems, differential equations, and minimax theorems and in certain applica-tions to mechanics and economic theory; see [–] and the references therein. Important practical situations motivate the study of a system of equilibrium problems. For instance, the flow of fluid through a fissured porous medium and certain models of plasticity led to such problems; see, for instance, []. Because of their importance, they have been exten-sively analyzed. The aim of this paper is to present an iterative method for solving solutions of an equilibrium problem, which is known as the Ky Fan inequality, and a nonlinear oper-ator equation involving a finite family of asymptotically quasi-φ-nonexpansive mappings in the intermediate sense.

The organization of this paper is as follows. In Section , we provide some necessary pre-liminaries. In Section , an iterative algorithm is presented. A strong convergence theorem is established in a reﬂexive Banach space. Some results in Hilbert spaces are also discussed.

2 Preliminaries

LetEbe a real Banach space. Recall that the normalized duality mappingJfromEto E∗

is deﬁned by

Jx=f∗∈E∗:x,f∗=x=f∗,

(2)

where·,·denotes the generalized duality pairing. Recall thatEis said to be strictly convex ifx+_y<  for allx,y∈Ewithx=y=  andx=y. It is said to be uniformly convex if

lim_n_→∞xn–yn=  for any two sequences{xn}and{yn}inEsuch thatxn=yn=  and

limn→∞xn+_yn= . LetUE={x∈E:x= }be the unit sphere ofE. Then the Banach

space Eis said to be smooth iflimt→x+ty_t–x exists for eachx,y∈UE. It is said to be

uniformly smooth if the above limit is attained uniformly forx,y∈UE. It is well known that

ifEis uniformly smooth, thenJis uniformly norm-to-norm continuous on each bounded subset ofE. It is also well known that ifEis uniformly smooth if and only ifE∗is uniformly convex.

Recall thatE enjoys the Kadec-Klee property if for any sequence{xn} ⊂E, andx∈E

withxnx, andxn → x, thenxn–x → asn→ ∞. It is well known that ifEis a

uniformly convex Banach space, thenEenjoys the Kadec-Klee property.

Next, we assume thatEis a smooth Banach space. Consider the functional deﬁned by

φ(x,y) =x– x,Jy+y, ∀x,y∈E.

Observe that in a Hilbert spaceH, the equality is reduced toφ(x,y) =x–y_,_x_,_y_∈_H_{. As} we all know, ifCis a nonempty closed convex subset of a Hilbert spaceHandPC:H→C

is the metric projection ofHontoC, thenPCis nonexpansive. This fact actually

charac-terizes Hilbert spaces and, consequently, it is not available in more general Banach spaces. In this connection, Alber [] recently introduced a generalized projection operatorC

in a Banach spaceEwhich is an analogue of the metric projectionPCin Hilbert spaces.

Recall that the generalized projectionC:E→Cis a map that assigns to an arbitrary

pointx∈Ethe minimum point of the functionalφ(x,y), that is,Cx=x¯, wherex¯ is the

solution to the minimization problem

φ(x¯,x) =min

y∈Cφ(y,x).

Existence and uniqueness of the operatorCfollows from the properties of the functional

φ(x,y) and strict monotonicity of the mappingJ. In Hilbert spaces,C=PC. It is obvious

from the deﬁnition of functionφthat

x–y≤φ(x,y)≤y+x, ∀x,y∈E (.)

and

φ(x,y) =φ(x,z) +φ(z,y) + x–z,Jz–Jy, ∀x,y,z∈E. (.)

Remark . IfEis a reﬂexive, strictly convex and smooth Banach space, thenφ(x,y) =  if and only ifx=y; for more details, see [] and the reference therein.

Letf be a bifunction fromC×CtoR, whereRdenotes the set of real numbers and let

A:C→E∗be a mapping. Consider the following equilibrium problem. Findp∈Csuch that

(3)

We useEP(f,A) to denote the solution set of inequality (.). That is,

EP(f) =p∈C:f(p,q) +Ap,q–p ≥,∀q∈C.

IfA= , then problem (.) is reduced to the following Ky Fan inequality. Findp∈C

such that

f(p,q)≥, ∀q∈C. (.)

We useEP(f) to denote the solution set of inequality (.). That is,

EP(f) =p∈C:f(p,q)≥,∀q∈C.

Iff = , then problem (.) is reduced to the classical variational inequality. Findp∈C

such that

Ap,q–p ≥, ∀q∈C. (.)

We useVI(C,A) to denote the solution set of inequality (.). That is,

VI(C,A) =p∈C:Ap,q–p ≥,∀q∈C.

Recall that a mappingA:C→E∗ is said to beα-inverse-strongly monotone if there existsα>  such that

Ax–Ay,x–y ≥αAx–Ay.

For solving problem (.), let us assume that the nonlinear mapping A:C→E∗ is α-inverse-strongly monotone and the bifunctionf :C×C→Rsatisﬁes the following conditions:

(A) F(x,x) = ,∀x∈C;

(A) Fis monotone,i.e.,F(x,y) +F(y,x)≤,∀x,y∈C; (A)

lim sup

t↓

Ftz+ ( –t)x,y≤F(x,y), ∀x,y,z∈C;

(A) for eachx∈C,y→F(x,y)is convex and weakly lower semi-continuous.

LetCbe a nonempty subset ofEandT :C→Cbe a mapping. In this paper, we use

F(T) to denote the ﬁxed point set ofT.Tis said to be asymptotically regular onCif for any bounded subsetKofC,

lim sup

n→∞

Tn+x–Tnx:x∈K= .

T is said to be closed if for any sequence {xn} ⊂ C such that limn→∞xn= x and

(4)

Recall that a pointpinCis said to be an asymptotic ﬁxed point ofT iﬀCcontains a sequence{xn}which converges weakly topsuch thatlimn→∞xn–Txn= . The set of

asymptotic ﬁxed points ofTwill be denoted byF(T). A mappingT is said to be relatively nonexpansive iﬀ

F(T) =F(T)= ∅, φ(p,Tx)≤φ(p,x), ∀x∈C,∀p∈F(T). A mappingT is said to be relatively asymptotically nonexpansive iﬀ

F(T) =F(T)=∅, φp,Tnx≤( +μn)φ(p,x), ∀x∈C,∀p∈F(T),∀n≥,

where{μn} ⊂[,∞) is a sequence such thatμn→ asn→ ∞.

Remark . The class of relatively asymptotically nonexpansive mappings was ﬁrst

con-sidered in Agarwalet al.[].

Recall that a mappingTis said to be quasi-φ-nonexpansive iﬀ

F(T)=∅, φ(p,Tx)≤φ(p,x), ∀x∈C,∀p∈F(T).

Recall that a mappingTis said to be asymptotically quasi-φ-nonexpansive iﬀ there exists a sequence{μn} ⊂[,∞) withμn→ asn→ ∞such that

F(T)=∅, φp,Tnx≤( +μn)φ(p,x), ∀x∈C,∀p∈F(T),∀n≥.

Remark . The class of quasi-φ-nonexpansive mappings and the class of asymptotically

quasi-φ-nonexpansive mappings are more general than the class of relatively nonexpan-sive mappings and the class of relatively asymptotically nonexpannonexpan-sive mappings. Quasi-φ-nonexpansive mappings and asymptotically quasi-φ-nonexpansive mappings do not re-quire the restrictionF(T) =F(T); for more details, see [–] the reference therein.

Remark . The class of quasi-φ-nonexpansive mappings and the class of asymptotically

quasi-φ-nonexpansive mappings are generalizations of the class of quasi-nonexpansive mappings and the class of asymptotically quasi-nonexpansive mappings in Banach spaces.

Recall that T is said to be asymptotically quasi-φ-nonexpansive in the intermediate sense iﬀF(T)=∅and the following inequality holds:

lim sup

n→∞

sup

p∈F(T),x∈C

φp,Tnx–φ(p,x)≤. (.)

Put

ξn=max

, sup

p∈F(T),x∈C

φp,Tnx–φ(p,x).

It follows thatξn→ asn→ ∞. Then (.) is reduced to the following:

(5)

Remark . The class of asymptotically quasi-φ-nonexpansive mappings in the interme-diate sense was ﬁrst considered by Qin and Wang []; see also [].

Remark . The class of asymptotically quasi-φ-nonexpansive mappings in the

interme-diate sense is a generalization of the class of asymptotically quasi-nonexpansive mappings in the intermediate sense, which was considered by Kirk [] in the framework of Banach spaces.

In order to state our main results, we also need the following lemmas.

Lemma .[] Let C be a nonempty closed convex subset of a smooth Banach space E

and x∈E.Then x=Cx if and only if

x–y,Jx–Jx ≥, ∀y∈C.

Lemma .[] Let E be a reﬂexive,strictly convex,and smooth Banach space,let C be a nonempty closed convex subset of E and x∈E.Then

φ(y,Cx) +φ(Cx,x)≤φ(y,x), ∀y∈C.

Lemma .[] Let E be a smooth,strictly convex,and reﬂexive Banach space,and let C be a nonempty closed convex subset of E.Let A:C→E∗be anα-inverse-strongly mono-tone mapping and f be a bifunction satisfying conditions(A)-(A).Let r> be any given number,and let x∈E deﬁne a mapping Kr:C→C as follows:for any x∈C,

Krx=

p∈C:f(p,q) +Ap,q–p+

rq–p,Jp–Jx ≥

, ∀q∈C.

Then the following conclusions hold: () Kris single-valued;

() Kris a ﬁrmly nonexpansive-type mapping,i.e.,for allx,y∈E,

Krx–Kry,JKrx–JKry ≤ Srx–Sry,Jx–Jy;

() F(Kr) =EP(f,A);

() Kris quasi-φ-nonexpansive;

()

φ(q,Krx) +φ(Krx,x)≤φ(q,x), ∀q∈F(Kr);

() EP(f,A)is closed and convex.

Lemma .[] Let E be a smooth and uniformly convex Banach space,and let r> .

Then there exists a strictly increasing,continuous,and convex function g: [, r]→R such that g() = and

_tx_{+ ( –}_t₎_y

≤tx+ ( –t)y–t( –t)gx–y

(6)

3 Main results

Theorem . Let E be a uniformly smooth and strictly convex Banach space which also

enjoys the Kadec-Klee property,and let C be a nonempty closed and convex subset of E.Let f be a bifunction from C×C toRsatisfying(A)-(A),and let N be some positive integer.

Let A:C→E∗ be aκi-inverse-strongly monotone mapping.Let Ti:C→C be an

asymp-totically quasi-φ-nonexpansive mapping in the intermediate sense for every≤i≤N. As-sume that Tiis closed asymptotically regular on C and

N

i=F(Ti)∩EP(f,A)is nonempty

and bounded.Let{xn}be a sequence generated in the following manner: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

x∈C chosen arbitrarily,

yn=J–(αn,Jxn+ N

i=αn,iJTinxn),

un∈C such that f(un,y) +Aun+y–un+_r_ny–un,Jun–Jyn ≥, ∀y∈C,

Hn={z∈C:φ(z,un)≤φ(z,xn) +Nξn},

Wn={z∈C:xn–z,Jx–Jxn ≥},

xn+=Hn∩Wnx,

whereξn=max{,sup_p_∈N

i=F(Ti),x∈C(φ(p,T n

ix) –φ(p,x))},{αn,i}is a real number sequence in

(, )for every≤i≤N,{rn}is a real number sequence in[k,∞),where k is some positive

real number.Assume thatN_i_=αn,i= andlim infn→∞αn,αn,i>  for every≤i≤N.

Then the sequence{xn}converges strongly toN

i=F(Ti)∩EP(f,A)x,whereNi=F(Ti)∩EP(f,A)is

the generalized projection from E ontoN_i_=F(Ti)∩EP(f,A).

Proof SinceFiis closed and convex for every ≤i≤N, we obtain from Lemma . that

the common element setN_i_=F(Ti)∩EP(f,A) is closed and convex. Next, we show that

bothHnandWnare closed and convex. From the deﬁnition ofHnandWn, it is obvious

thatHnis closed andWnis closed and convex. We show thatHnis convex sinceφ(z,un)≤

φ(z,xn) +Nξnis equivalent to

z,Jxn–Jun ≤ xn–un+Nξn.

It follows thatHnis convex. This in turn shows thatHn∩Wnx is well deﬁned. Putting

un=krnyn, from Lemma . we see thatKrn is quasi-φ-nonexpansive. Now, we are in a

position to prove thatN_i_=F(Ti)∩EP(f,A)⊂Hn∩Wn. Letw∈ N

i=F(Ti)∩EP(f,A),

φ(w,un) =φ(w,Krnyn)

≤φ(w,yn)

=φ

w,J–

αn,Jxn+ N

i=

αn,iJTinxn

=w– 

w,αh,Jxn+ N

i=

αn,iJTinxn

+

αn,Jxn+ N

i=

αn,iJTinxn



≤ w– αn,w,Jxn–  N

i= αn,i

w,JT_inxn

+αn,xn+ N

i=

(7)

=αn,φ(w,xn) + N

i= αn,iφ

w,T_inxn

≤αn,φ(w,xn) + N

i=

αn,iφ(w,xn) + N

i= αn,iξn

=φ(w,xn) + N

i= αn,iξn

≤φ(w,xn) +Nξn. (.)

We havew∈Hn. This implies that N

i=F(Ti)∩EP(f,A)⊂Hn. On the other hand, we see

thatN_i_=F(Ti)∩EP(f,A)⊂H∩W. Suppose that

N

i=F(Ti)∩EP(f,A)⊂Hm∩Wmfor

somem. There exists an elementxm+∈Hm∩Wmsuch thatxm+=Hm∩Wmx. In view of

Lemma ., we ﬁnd that

xm+–w,Jx–Jxm+ ≥, w∈Hm∩Wm.

SinceN_i_=F(Ti)∩EP(f,A)⊂Hm∩Wm, we arrive at

xm+–w,Jx–Jxm+ ≥ (.)

for everyw∈N_i_=F(Ti)∩EP(f,A). We therefore ﬁnd that N

i=F(Ti)∩EP(f,A)⊂Wm+. It follows thatN_i_=F(Ti)∩EP(f,A)⊂Hm+∩Wm+. This shows that the sequence{xn}is

well deﬁned.

Next, we prove that the sequence{xn}is bounded. It follows from the deﬁnition ofWn

and Lemma . thatxn=Wnx. In view of Lemma ., we ﬁnd that

φ(xn,x) =φ(Wnx,x)≤φ(w,x) –φ(w,xn)≤φ(w,x)

for eachw∈N_i_=F(Ti)∩EP(f,A)⊂Wn. This implies that{φ(xn,x)}is bounded. It follows

from (.) that{xn}is also bounded. Sincexn+=Hn∩Wnx∈Wn, we ﬁnd from Lemma .

thatφ(xn,x)≤φ(xn+,x). Therefore, we obtain that{φ(xn,x)}is nondecreasing. So there

exists the limit ofφ(xn,x). It follows from Lemma . that

φ(xn+,xn) =φ(xn+,Wnx)

≤φ(xn+,x) –φ(Wnx,x)

=φ(xn+,x) –φ(xn,x).

This shows that limn→∞φ(xn+,xn) = . Since xn+ = Hn∩Wnx ∈ Hn, we ﬁnd that

limn→∞φ(xn+,un) = . Since the space is reﬂexive, we may assume, without loss of

gener-ality, thatxnx¯. SinceWnis closed and convex, we ﬁnd thatx¯∈Wn. This implies from

xn=Wnxthatφ(xn,x)≤φ(x¯,x). On the other hand, we see from the weakly lower

semicontinuity of · that

φ(x¯,x) =¯x– ¯x,Jx+x ≤lim inf

n→∞

xn– xn,Jx+x

(8)

=lim inf

n→∞ φ(xn,x)

≤lim sup

n→∞

φ(xn,x)

≤φ(x¯,x),

which implies thatlim_n_→∞φ(xn,x) =φ(x¯,x). Hence, we havelimn→∞xn=¯x. In view

of the Kadec-Klee property ofE, we ﬁnd thatxn→ ¯xasn→ ∞. In view of (.), we see

thatlimn→∞(xn+–un) = . This implies thatlimn→∞un=¯x. That is,

lim

n→∞Jun=nlim→∞un=Jx¯. (.)

This implies that{Jun}is bounded. Note that bothEandE∗are reﬂexive. We may assume,

without loss of generality, thatJunu∗∈E∗. In view of the reﬂexivity ofE, we see that

J(E) =E∗. This shows that there exists an elementu∈Esuch thatJu=u∗. It follows that φ(xn+,un) =xn+– xn+,Jun+un

=xn+– xn+,Jun+Jun.

Takinglim infn→∞on the both sides of the equality above yields that

≥ ¯x– x¯,u∗+u∗

=¯x– ¯x,Ju+Ju

=¯x– ¯x,Ju+u

=φ(x¯,u).

That is,x¯ =u, which in turn implies thatu∗=Jx¯. It follows thatJunJx¯∈E∗. Since

E∗enjoys the Kadec-Klee property, we obtain from (.) thatlimn→∞Jun=Jx¯. SinceJ–:

E∗→Eis demi-continuous andEenjoys the Kadec-Klee property, we obtain thatun→ ¯x,

asn→ ∞. Note thatxn–un ≤ xn–x¯+¯x–un. It follows that

lim

n→∞xn–un= . (.)

SinceJis uniformly norm-to-norm continuous on any bounded sets, we have

lim

n→∞Jxn–Jun= . (.)

On the other hand, we have

φ(w,xn) –φ(w,un) =xn–un– w,Jxn–Jun

≤ xn–un

xn+un

+ wJxn–Jun.

We, therefore, ﬁnd that

lim

n→∞

φ(w,xn) –φ(w,un)

(9)

SinceEis uniformly smooth, we know thatE∗is uniformly convex. In view of Lemma ., we ﬁnd that

φ(w,un) =φ(w,Krnyn)

≤φ(w,yn)

=φ

w,J–

αn,Jxn+ N

i=

αn,iJTinxn

=w– 

w,αn,Jxn+ N

i=

αn,iJTinxn

+

αn,Jxn+ N

i=

αn,iJTinxn



≤ w– αn,w,Jxn–  N

i= αn,i

w,JT_inxn

+αn,xn

+

N

i=

αn,iTinxn



–αn,αn,gJxn–JTnxn

=αn,φ(w,xn) + N

i= αn,iφ

w,T_inxn

≤αn,φ(w,xn) + N

i=

αn,iφ(w,xn) + N

i= αn,iξh

=φ(w,xn) + N

i=

αn,iξn–αn,αn,gJxn–JTnxn

≤φ(w,xn) +Nξn–αn,αn,gJxn–JTnxn.

It follows thatαn,αn,g(Jxn–JTnxn)≤φ(w,xn) –φ(w,un) +ξn. In view of the

restric-tion on the sequences, we ﬁnd from (.) that limn→∞g(Jxn–JTnxn) = . It follows

thatlimn→∞Jxn–JTnxn= . In the same way, we obtain thatlimn→∞Jxn–JTinxn=

, ∀≤i≤N. Notice that JT_inxn–Jx¯ ≤ JTinxn–Jxn+Jxn–Jx¯. It follows that

limn→∞JTinxn–Jx¯= . The demicontinuity ofJ–:E∗→Eimplies thatTinxnx¯. Note

that

T_inxn–¯x=JTinxn–Jx¯≤JTinxn–Jx¯.

This implies thatlimn→∞Tinxn=¯x. SinceEhas the Kadec-Klee property, we obtain

thatlimn→∞Tinxn–x¯= . On the other hand, we have _Tn+

i xn–x¯≤Tin+xn–Tinxn+Tinxn–x¯.

It follows from the asymptotic regularity of Ti that limn→∞Tin+xn–x¯= . That is,

TiTinxn→ ¯x. From the closedness ofTi, we ﬁndx¯=Tix¯ for every ≤i≤N. This proves

¯

x∈N_i_=F(Ti). Now, we state thatx¯∈EP(f,A). In view of Lemma ., we ﬁnd that

(10)

It follows from (.) thatlimn→∞φ(un,yn) = . This implies thatlimn→∞(un–yn) = .

It follows from (.) thatlimn→∞yn=¯x. It follows that

lim

n→∞Jyn=nlim→∞yn=¯x=Jx¯.

This shows that{Jyn}is bounded. SinceE∗is reﬂexive, we may assume thatJyny∗∈E∗.

In view ofJ(E) =E∗, we see that there existsy∈Esuch thatJy=y∗. It follows thatφ(un,yn) =

un– un,Jyn+Jyn. Takinglim infn→∞on the both sides of the equality above yields

that

≥ ¯x– x¯,y∗+y∗

=¯x– ¯x,Jy+Jy

=¯x– ¯x,Jy+y

=φ(x¯,y).

That is,x¯=y, which in turn implies thaty∗=Jx¯. It follows thatJynJx¯∈E∗. SinceE∗

enjoys the Kadec-Klee property, we obtain thatJyn–Jx¯→ asn→ ∞. Note thatJ–:

E∗→Eis demi-continuous. It follows thatynx¯. SinceEenjoys the Kadec-Klee property,

we obtain thatyn→ ¯xasn→ ∞. Note thatun–yn ≤ un–x¯+¯x–yn. This implies

thatlimn→∞un–yn= . SinceJis uniformly norm-to-norm continuous on any bounded

sets, we havelimn→∞Jun–Jyn= . In view of the assumptionrn≥k, we see that

lim

n→∞

Jun–Jyn

rn

= . (.)

Sinceun=Krnyn, we ﬁnd that

F(un,y) +



rn

y–un,Jun–Jyn ≥, ∀y∈C,

whereF(un,y) =f(un,y) +Aun+y–unfor everyy∈C. It follows from (A) that

y–un

Jun–Jyn

rn

≥ 

rn

y–un,Jun–Jyn ≥F(y,un), ∀y∈C.

It follows from (.) that

F(y,x¯)≤, ∀y∈C.

For  <t<  andy∈C, deﬁneyt =ty+ ( –t)x¯. It follows thatyt ∈C, which yields that

F(yt,x¯)≤. It follows from (A) and (A) that

 =F(yt,yt)≤tF(yt,y) + ( –t)F(yt,x¯)≤tF(yt,y).

That is,

(11)

Lettingt↓, we obtain from (A) thatF(x¯,y)≥,∀y∈C. That is,f(un,y) +Aun+y–

un ≥ for everyy∈C. This implies thatx¯∈EP(f,A). This completes the proof thatx¯∈ N

i=F(Ti)∩EP(f,A). Sincexn+=Hn∩Wnx. In view of Lemma ., we ﬁnd that

xn+–w,Jx–Jxn+ ≥, w∈Hn∩Wn.

SinceN_i_=F(Ti)∩EP(f,A)⊂Hn∩Wn, we arrive at

xn+–w,Jx–Jxn+ ≥.

Lettingn→ ∞in the above inequality, we obtain that

¯x–w,Jx–Jx¯ ≥, ∀w∈

N

i=

F(Ti)∩EP(f,A).

It follows from Lemma . thatx¯=N

i=F(Ti)∩EP(f,A)x. This completes the proof.

Remark . Theorem . includes the corresponding results in the literature as special

cases. Since every uniformly convex Banach space enjoys the Kadec-Klee property, the framework of the space can be applicable toLp_,_p_≥_.

Next, we state a result on Ky Fan inequality (.) and a single asymptotically quasi-φ-nonexpansive mapping in the intermediate sense.

Corollary . Let E be a uniformly smooth and strictly convex Banach space which also

enjoys the Kadec-Klee property,and let C be a nonempty closed and convex subset of E.Let f be a bifunction from C×C toRsatisfying(A)-(A).Let T:C→C be an asymptotically quasi-φ-nonexpansive mapping in the intermediate sense.Assume that T is closed asymp-totically regular on C and F(T)∩EP(f)is nonempty and bounded.Let{xn}be a sequence

generated in the following manner:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

yn=J–(αnJxn+ ( –αn)JTnxn),

un∈C such that f(un,y) +_r_ny–un,Jun–Jyn ≥, ∀y∈C,

Hn={z∈C:φ(z,un)≤φ(z,xn) +Nξn},

whereξn=max{,supp∈F(T),x∈C(φ(p,Tnx)–φ(p,x))},{αn}is a real number sequence in(, ),

{rn}is a real number sequence in[k,∞),where k is some positive real number.Assume that

lim inf_n_→∞αn( –αn) > .Then the sequence{xn}converges strongly toF(T)∩EP(f)x,where F(T)∩EP(f)is the generalized projection from E onto F(T)∩EP(f).

(12)

Corollary . Let E be a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property,and let C be a nonempty closed and convex subset of E.

Let f be a bifunction from C×C toRsatisfying(A)-(A).Let T :C→C be a

quasi-φ-nonexpansive mapping.Assume that F(T)∩EP(f)is nonempty.Let{xn}be a sequence

generated in the following manner:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

yn=J–(αnJxn+ ( –αn)JTxn),

un∈C such that f(un,y) +_r_ny–un,Jun–Jyn ≥, ∀y∈C,

Hn={z∈C:φ(z,un)≤φ(z,xn)},

where{αn}is a real number sequence in(, ),{rn}is a real number sequence in[k,∞),

where k is some positive real number.Assume thatlim inf_n_→∞αn( –αn) > .Then the

sequence{xn}converges strongly toF(T)∩EP(f)x,whereF(T)∩EP(f)is the generalized

pro-jection from E onto F(T)∩EP(f).

Finally, we give a result in the framework of Hilbert spaces based on Theorem ..

Corollary . Let E be a Hilbert space,and let C be a nonempty closed and convex subset of E.Let f be a bifunction from C×C toRsatisfying(A)-(A),and let N be some positive integer.Let A:C→E be aκi-inverse-strongly monotone mapping.Let Ti:C→C be an

asymptotically quasi-nonexpansive mapping in the intermediate sense for every≤i≤N.

Assume that Tiis closed asymptotically regular on C and N

i=F(Ti)∩EP(f,A)is nonempty

and bounded.Let{xn}be a sequence generated in the following manner: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

yn=αn,xn+ N

i=αn,iTinxn,

un∈C such that f(un,y) +Aun+y–un+_r_ny–un,un–yn ≥, ∀y∈C,

Hn={z∈C:z–un≤ z–xn+Nξn},

Wn={z∈C:xn–z,x–xn ≥},

xn+=ProjHn∩Wnx,

whereξn=max{,supp∈N

i=F(Ti),x∈C(p–T n

ix–p–x)},{αn,i}is a real number sequence

in(, )for every≤i≤N,{rn}is a real number sequence in[k,∞),where k is some positive

real number.Assume thatN_i_=αn,i= andlim infn→∞αn,αn,i> for every≤i≤N.Then

the sequence{xn} converges strongly toProjN

i=F(Ti)∩EP(f,A)x,whereProj N

i=F(Ti)∩EP(f,A) is

the metric projection from E ontoN_i_=F(Ti)∩EP(f,A).

Proof Sinceφ(x,y) =x–y_and_J₌_I_{in the framework of Hilbert spaces, we draw the}

desired conclusion immediately.

Competing interests

(13)

Authors’ contributions

Both authors contributed equally to this manuscript. Both authors read and approved the ﬁnal manuscript.

Author details

1_{Department of Mathematics, Hangzhou Normal University, Hangzhou, China.}2_{School of Mathematics and Information}

Science, Shangqiu Normal University, Shangqiu, 476000, China.

Acknowledgements

The authors are grateful to the two anonymous reviewers’ suggestions which improved the contents of the article.

Received: 9 September 2013 Accepted: 25 October 2013 Published:21 Nov 2013

References

1. Park, S: A review of the KKM theory onφA-space or GFC-spaces. Adv. Fixed Point Theory3, 355-382 (2013)

2. Cho, SY, Li, W, Kang, SM: Convergence analysis of an iterative algorithm for monotone operators. J. Inequal. Appl.

2013, 199 (2013)

3. Kim, JK: Strong convergence theorems by hybrid projection methods for equilibrium problems and ﬁxed point problems of the asymptotically quasi-φ-nonexpansive mappings. Fixed Point Theory Appl.2011, 10 (2011) 4. Wang, ZM, Lou, W: A new iterative algorithm of common solutions to quasi-variational inclusion and ﬁxed point

problems. J. Math. Comput. Sci.3, 57-72 (2013)

5. Wang, G, Sun, S: Hybrid projection algorithms for ﬁxed point and equilibrium problems in a Banach space. Adv. Fixed Point Theory3, 578-594 (2013)

6. Zegeye, H, Shahzad, N: Strong convergence theorem for a common point of solution of variational inequality and ﬁxed point problem. Adv. Fixed Point Theory2, 374-397 (2012)

7. Cho, SY, Kang, SM: Approximation of ﬁxed points of pseudocontraction semigroups based on a viscosity iterative process. Appl. Math. Lett.24, 224-228 (2011)

8. Qin, X, Cho, SY, Kang, SM: Convergence of an iterative algorithm for systems of variational inequalities and nonexpansive mappings with applications. J. Comput. Appl. Math.233, 231-240 (2009)

9. Lv, S: Strong convergence of a general iterative algorithm in Hilbert spaces. J. Inequal. Appl.2013, 19 (2013) 10. Yuan, Q: Some results on asymptotically quasi-φ-nonexpansive mappings in the intermediate sense. J. Fixed Point

Theory2012, 1 (2012)

11. Cho, SY, Qin, X, Kang, SM: Iterative processes for common ﬁxed points of two diﬀerent families of mappings with applications. J. Glob. Optim. (2013). doi:10.1007/s10898-012-0017-y

12. He, RH: Coincidence theorem and existence theorems of solutions for a system of Ky Fan type minimax inequalities inFC-spaces. Adv. Fixed Point Theory2, 47-57 (2012)

13. Tanaka, Y: A constructive version of Ky Fan’s coincidence theorem. J. Math. Comput. Sci.2, 926-936 (2012) 14. Qin, X, Cho, SY, Kang, SM: An extragradient-type method for generalized equilibrium problems involving strictly

pseudocontractive mappings. J. Glob. Optim.49, 679-693 (2011)

15. Al-Bayati, AY, Al-Kawaz, RZ: A new hybrid WC-FR conjugate gradient-algorithm with modiﬁed secant condition for unconstrained optimization. J. Math. Comput. Sci.2, 937-966 (2012)

16. Qin, X, Cho, SY, Kang, SM: Iterative algorithms for variational inequality and equilibrium problems with applications. J. Glob. Optim.48, 423-445 (2010)

17. Chen, JH: Iterations for equilibrium and ﬁxed point problems. J. Nonlinear Funct. Anal.2013, 4 (2013) 18. Yang, S: A proximal point algorithm for zeros of monotone operators. Math. Finance Lett.2013, 7 (2013) 19. Song, J, Chen, M: On generalized asymptotically quasi-φ-nonexpansive mappings and a Ky Fan inequality. Fixed

Point Theory Appl.2013, 237 (2013)

20. Cho, SY, Kang, SM: Approximation of common solutions of variational inequalities via strict pseudocontractions. Acta Math. Sci.32, 1607-1618 (2012)

21. Takahashi, W, Yao, JC: Nonlinear operators of monotone type and convergence theorems with equilibrium problems in Banach spaces. Taiwan. J. Math.15, 787-818 (2011)

22. Takahashi, W, Wong, NC, Yao, JC: Iterative common solutions for monotone inclusion problems, ﬁxed point problems and equilibrium problems. Fixed Point Theory Appl.2012, 181 (2012)

23. Noor, MA, Noor, KI, Waseem, M: Decomposition method for solving system of linear equations. Eng. Math. Lett.2, 34-41 (2013)

24. Qing, Y, Kim, JK: Weak convergence of algorithms for asymptotically strict pseudocontractions in the intermediate sense and equilibrium problems. Fixed Point Theory Appl.2012, 132 (2012)

25. Kim, JK: A new iterative algorithm of pseudomonotone mappings for equilibrium problems in Hilbert spaces. J. Inequal. Appl.2013, 128 (2013)

26. Kang, SM, Cho, SY, Liu, Z: Convergence of iterative sequences for generalized equilibrium problems involving inverse-strongly monotone mappings. J. Inequal. Appl.2010, 827082 (2010)

27. Wu, C: Mann iteration for zero theorems of accretive operators. J. Fixed Point Theory2013, 3 (2013)

28. Showalter, RE: Monotone Operators in Banach Space and Nonlinear Partial Diﬀerential Equations. Math. Surveys Monogr., vol. 49. Am. Math. Soc., Providence (1997)

29. Alber, YI: Metric and generalized projection operators in Banach spaces: properties and applications. In: Kartsatos, AG (ed.) Theory and Applications of Nonlinear Operators of Accretive and Monotone Type. Dekker, New York (1996) 30. Agarwal, RP, Cho, YJ, Qin, X: Generalized projection algorithms for nonlinear operators. Numer. Funct. Anal. Optim.28,

1197-1215 (2007)

31. Qin, X, Cho, YJ, Kang, SM: Convergence theorems of common elements for equilibrium problems and ﬁxed point problems in Banach spaces. J. Comput. Appl. Math.225, 20-30 (2009)

32. Zhou, H, Gao, G, Tan, B: Convergence theorems of a modiﬁed hybrid algorithm for a family of quasi-φ-asymptotically nonexpansive mappings. J. Appl. Math. Comput.32, 453-464 (2010)

(14)

34. Qin, X, Wang, L: On asymptotically quasi-φ-nonexpansive mappings in the intermediate sense. Abstr. Appl. Anal.

2012, 636217 (2012)

35. Hecai, Y, Aichao, L: Projection algorithms for treating asymptotically quasi-φ-nonexpansive mappings in the intermediate sense. J. Inequal. Appl.2013, 265 (2013)

36. Kirk, WA: Fixed point theorems for non-Lipschitzian mappings of asymptotically nonexpansive type. Isr. J. Math.17, 339-346 (1974)

37. Chang, SS, Chan, CK, Lee, HWJ: A new hybrid method for solving a generalized equilibrium problem, solving a variational inequality problem and obtaining common ﬁxed points in Banach spaces, with applications. Nonlinear Anal.73, 2260-2270 (2010)

38. Zˇalinescu, C: On uniformly convex functions. J. Math. Anal. Appl.95, 344-374 (1983) 10.1186/1687-1812-2013-305

Cite this article as:Qing and Lv:A strong convergence theorem for solutions of equilibrium problems and asymptotically quasi-φ-nonexpansive mappings in the intermediate sense.Fixed Point Theory and Applications