R E S E A R C H
Open Access
Existence and approximation for a solution of
a generalized equilibrium problem on the
dual space of a Banach space
Pongrus Phuangphoo
1,2and Poom Kumam
1*Dedicated to Prof. W Takahashi on the occasion of his 70th birthday
*Correspondence:
1Department of Mathematics,
Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bang Mod, Thung Khru, Bangkok, 10140, Thailand Full list of author information is available at the end of the article
Abstract
In this paper, we first prove the existence of a solution for a generalized equilibrium problem with a bifunction defined on the dual space in a Banach space setting. Second, by the virtue of this result, we construct the hybrid projection method for solving a solution of a generalized equilibrium problem. Consequently, we establish the strong convergence theorem by using sunny generalized nonexpansive retraction in the dual of Banach spaces.
MSC: 47H09; 47H10; 47J25
Keywords: a generalized equilibrium problem; a bifunction defined on the dual space; the hybrid projection method
1 Introduction
LetRbe the set of real numbers. LetEbe a real Banach space with the norm · , and·,· is the dual pair betweenEandE∗, whereE∗is the dual space ofE. LetCbe a closed and convex subset of a real Banach spaceEwith the dual spaceE∗, and letC∗be a closed and convex subset ofE∗. We recall the following definitions:
() A mappingA:C→E∗is said to bemonotoneif for eachx,y∈Csuch that
x–y,Ax–Ay ≥.
() A mappingA:C→E∗is said to beδ-strongly monotoneif there exists a constant
δ> such that
x–y,Ax–Ay ≥δx–y, ∀x,y∈C.
() A mappingA:C→E∗is said to beδ-inverse strongly monotoneif there exists a constantδ> such that
x–y,Ax–Ay ≥δAx–Ay, ∀x,y∈C.
() A mappingA:C∗→Eis said to beskew monotoneif for eachx∗,y∗∈C∗such that
Ax∗–Ay∗,x∗–y∗≥.
() A mappingA:D(A)⊂E∗→Eis said to beα-inverse strongly skew monotoneif there exists a constantα> such that
Ax∗–Ay∗,x∗–y∗≥αAx∗–Ay∗, ∀x∗,y∗∈D(A).
() A mappingA:D(A)⊂E∗→Eis said to behemicontinuousif for allx∗,y∗∈D(A), the mappingf of[, ]intoEdefined byf(t) =A(tx∗+ ( –t)y∗)is continuous.
LetCbe a nonempty, closed and convex subset ofE, and letJbe the duality mapping fromEintoE∗such thatJ(C) is closed and convex ofE∗, let us assume that a bifunction F:J(C)×J(C)→Rsatisfies suitable conditions, andA:C∗ →E is a skew monotone operator fromJ(C) intoE.
Thegeneralized equilibrium problemis to findzˆ∈Csuch that
F(Jˆz,Jy) +AJˆz,Jy–Jzˆ ≥, ∀y∈C. (.)
The set of solutions of (.) is denoted byGEP(F,A), that is,
GEP(F,A) =zˆ∈C:F(Jz,ˆ Jy) +AJz,ˆ Jy–Jzˆ ≥,∀y∈C. (.)
IfA≡, then problem (.) reduces to theequilibrium problem, which is to findzˆ∈C such that
F(Jˆz,Jy)≥, ∀y∈C. (.)
The set of solutions of problem (.) is denoted byEP(F), that is,
EP(F) =zˆ∈C:F(Jz,ˆ Jy)≥,∀y∈C. (.)
The above formulation (.) was considered in Takahashi and Zembayashi [], and they proved a strong convergence theorem for finding a solution of the equilibrium problem (.) in Banach spaces.
IfF≡, then problem (.) reduces tovariational inequality, which is to findzˆ∈Csuch that
AJz,ˆ Jy–Jzˆ ≥, ∀y∈C. (.)
The set of solutions of problem (.) is denoted byVI(J(C),A), that is,
VIJ(C),A=zˆ∈C:AJz,ˆ Jy–Jzˆ ≥,∀y∈C. (.)
In the sequel, letT:C→Cbe a mapping, we denote byFix(T) the set of fixed points of T, that is,
Fix(T) ={x∈C:Tx=x}. (.)
LetCbe a nonempty, closed subset of a smooth, strictly convex and reflexive Banach spaceEsuch thatJ(C) is closed and convex. For solving the equilibrium problem, let us assume that a bifunctionF:J(C)×J(C)→Rsatisfies the following conditions:
(DA) F(x∗,x∗) = for allx∗∈J(C);
(DA) Fis monotone,i.e.,F(x∗,y∗) +F(y∗,x∗)≤for allx∗,y∗∈J(C); (DA) for allx∗,y∗,z∗∈J(C),
lim sup t↓
Ftz∗+ ( –t)x∗,y∗≤Fx∗,y∗;
(DA) for allx∗∈J(C),F(x∗,·)is convex and lower semicontinuous.
The following result is in Blum and Oettli [], and see the proof in [].
Definition . LetEbe a Banach space. Then,
() Eis said to bestrictly convexif x+y< for allx,y∈UE={z∈E:z= }withx=y.
() Eis said to beuniformly convexif for each∈(, ], there existsδ> such that
x+y
≤ –δfor allx,y∈UEwithx–y>.
() Eis said to besmoothif the limit(.)
lim t−→
x+ty–x
t (.)
exists for eachx,y∈UE.
() Eis said to beuniformly smoothif the limit(.)is attained uniformly for all
x,y∈UE.
() Eis said to haveuniformly Gâteaux differentiable normif for ally∈U(E), the limit
(.)converges uniformly forx∈UE.
Definition . LetEbe a Banach space. Then a functionρE:R+→R+is said to be the modulus of smoothnessofEif
ρE(t) =sup
x+y+x–y
– :x= ,y=t
.
() Eis said to besmoothifρE(t) > ,∀t> .
() Eis said to beuniformly smoothif and only iflimt→+ρE(t) t = .
Definition . LetEbe a Banach space. Then themodulus of convexityofEis the function δE: [, ]→[, ] defined by
δE() =inf –x+y
:x ≤,y ≤;x–y ≥
.
() Eis said to beuniformly convexif and only ifδE() > for all∈(, ].
Observe that everyp-uniformly convex is uniformly convex. One should note that no Banach space isp-uniformly convex for <p< . It is well known that a Hilbert space is -uniformly convex and uniformly smooth.
For anyp> , thegeneralized duality mapping Jp:E→E∗is defined by
Jpx=
f∗∈E∗:x,f∗=xp,f∗=xp–, ∀x∈E. (.)
In particular,J=Jis called thenormalized duality mapping. IfEis a Hilbert space, then
J=I, whereIis the identity mapping. That is,
Jx=Jx=
f∗∈E∗:x,f∗=x=f∗, ∀x∈E. (.)
Remark . The basic properties below hold, see [–].
() IfEis a uniformly smooth real Banach space, thenJis uniformly continuous on each bounded subset ofE.
() IfEis a uniformly smooth real Banach space, thenJ∗:E∗→Eis a normalized
duality mapping onE∗, thenJ–=J∗,(J∗)J=IEandJ(J∗) =IE∗, where onIEandIE∗
are the identity mappings onEandE∗, respectively.
() LetEbe a smooth, strictly convex reflexive Banach space andJbe the duality mapping fromEintoE∗. ThenJ–is also single-valued, one-to-one, onto, and it is
also the duality mapping fromE∗intoE.
() IfEis a reflexive, strictly convex Banach space, thenJ–is hemicontinuous, that is,
J–is norm-to-weak∗-continuous.
() IfEis a reflexive, smooth and strictly convex Banach space, thenJis single-valued, one-to-one, and onto.
() A Banach spaceEis uniformly smooth if and only ifE∗is uniformly convex. () Each uniformly convex Banach spaceEhas theKadec-Klee property, that is, for any
sequence{xn} ⊂E, ifxnx∈E, andxn → x, thenxn→x.
() A Banach spaceEis strictly convex if and only ifJis strictly monotone, that is,
x–y,x∗–y∗> , wheneverx,y∈E,x=y, andx∗∈Jx,y∗∈Jy.
() Both, uniformly smooth Banach space and uniformly convex Banach space, are reflexive.
() IfE∗is uniformly convex, thenJis uniformly norm-to-norm continuous on each bounded subset ofE.
() IfE∗is a strictly convex Banach space, thenJis one-to-one, that is,x=yimplies thatJx∩Jy=∅.
LetJbe thenormalized duality mapping, thenJis said to beweakly sequentially contin-uousif the strong convergence of a sequence{xn}tox∈Eimplies the weak∗convergence of a sequence{Jxn}toJxinE∗.
LetEbe a smooth and strictly convex reflexive Banach space, and letCbe a nonempty, closed and convex subset ofE. We assume that theLyapunov functionalφ:E×E→R+ is defined by [, ]
LetCbe a nonempty, closed and convex subset of a Banach spaceE. Thegeneralized projection[]C:E→Cis defined by, for eachx∈E,
C(x) =arg miny∈Cφ(x,y).
Remark . From the definition ofφ. It is easy to see that
() (x–y)≤φ(x,y)≤(x+y)for allx,y∈E.
() φ(x,y) =φ(x,z) +φ(z,y) + x–z,Jz–Jyfor allx,y,z∈E.
() φ(x,y) =x,Jx–Jy+y–x,Jy ≤ xJx–Jy+y–xyfor allx,y,z∈E. () IfEis a real Hilbert spaceH, thenφ(x,y) =x–y, and
C=PC(the metric
projection ofHontoC).
Lemma .[, ] If C is a nonempty,closed and convex subset of a smooth and strictly convex reflexive real Banach space E,then
() forx∈E,andu∈C,one has
u=C(x) ⇔ u–y,Jx–Ju ≥, ∀y∈C.
() φ(x,C(y)) +φ(C(y),y)≤φ(x,y),∀x∈C,y∈E. () φ(x,y) = if and only ifx=y,∀x,y∈E.
In , Takahashi and Zembayashi [] introduced an iterative algorithm for finding a solution of an equilibrium problem with a bifunction defined on the dual space of a Banach space by using the shrinking projection method, and they established the strong convergence as follows.
Theorem TZ Let E be a uniformly convex Banach space whose norm is uniformly Gâteaux differentiable,and let C be a nonempty,closed and convex subset of E such that J(C)is closed and convex of E∗.Assume that a mapping F:J(C)×J(C)→Rsatisfies the condi-tion(DA)-(DA)such thatEP(F)=∅.Let{xn}be a sequence generated by the following algorithm:
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈C chosen arbitrary, C=C,
un∈C such that F(Jun,Jy) +rnun–xn,Jy–Jun ≥, ∀y∈C,
yn=αnxn+ ( –αn)un,
Cn+={z∈Cn:φ(yn,z)≤φ(xn,z)},
xn+=RCn+(x), ∀n∈N∪ {},
(.)
where J is the duality mapping on E,the sequence{αn} ⊂[, ]such thatlim supn→∞αn< , rn⊂[a,∞)for some a> ,and RCn+is the sunny generalized nonexpansive retraction from E onto Cn+.
Then the sequence{xn}converges strongly to some point p=REP(F)(x),where REP(F)is the
sunny generalized nonexpansive retraction from E toEP(F).
variational inequality problem using the projection algorithm method with a new projec-tion, which was introduced by Ibaraki and Takahashi [] and Iiduka and Takahashi [] in Banach spaces.
Theorem PS Let E be a uniformly convex and-uniformly smooth Banach space whose duality mapping J is weakly sequentially continuous.Let C be a nonempty,closed and con-vex subset of E such that J(C)is closed and convex,and let A be anα -inverse-strongly-skew-monotone operator of J(C)into E such thatVI(J(C),A)=∅andAJz ≤ AJy–AJz,for all y∈C and z∈VI(J(C),A).Let{xn}be a sequence defined by x=x∈C and
xn+=RC(xn–λnAJxn) (.)
for every n= , , , . . . , where RC is the sunny generalized nonexpansive retraction of E into C,{αn} ⊂[a,b],for some a,b with <a<b< αc,where c> is a constant satisfying
Jx–Jy ≤cx–yfor all x,y∈C.Then the sequence{xn}converges weakly to some element z∈VI(J(C),A).Further z=limn→∞RVI(J(C),A)(xn).
In this paper, motivated and inspired by the previously mentioned results, we study the existence theorem for a generalized equilibrium problem with a bifunction defined on the dual space of a Banach space, and we also construct an iterative procedure generated by a hybrid method for solving the solution of a generalized equilibrium problem by using the sunny generalized nonexpansive retraction. Under some suitable assumptions, the strong convergence theorems are established in Banach spaces. The results obtained in this paper extend and improve several recent results in this area.
2 Preliminaries
Definition . LetCbe a nonempty and closed subset of a smooth Banach space.
() A mappingT:C→Cis said to beclosedif for each{xn} ⊂C,xn→xandTxn→y
imply thatTx=y.
() A mappingT:C→Cis said to benonexpansiveif
Tx–Ty ≤ x–y, ∀x,y∈C.
() A mappingT:C→Cis said to beφ-nonexpansiveifFix(T)=∅, and
φ(Tx,Ty)≤φ(x,y), ∀x,y∈C.
() A mappingT:C→Cis said to begeneralized nonexpansive[] ifFix(T)=∅, and
φ(Tx,p)≤φ(x,p), ∀x∈C,p∈Fix(T).
Definition .[] LetCbe a nonempty and closed subset of a smooth Banach spaceE. A mappingR:E→Cis called
() a retractionifR=R;
() a sunnyifR(Rx+t(x–Rx)) =Rxfor allx∈Eandt≥.
nonex-pansive retractionRCofEontoCif and only ifJ(C) is closed and convex. In this case,RC is given by
RC=J–J(C)J.
Definition .[] LetCbe a nonempty and closed subset of a smooth Banach spaceE. The setCis called asunny generalized nonexpansive retractionofEif there exists a sunny generalized nonexpansiveRfromEontoC.
Lemma .[] Let C be a nonempty and closed subset of a smooth and strictly convex Banach space E,and let R be a retraction from E onto C.Then the following are equivalent:
() Ris sunny generalized nonexpansive; () x–Rx,Jy–JRx ≤for allx∈Eandy∈C.
Lemma .[] Let C be a nonempty,closed and sunny generalized nonexpansive retrac-tion of a smooth and strictly convex Banach space E.Then the sunny generalized nonex-pansive retraction from E onto C is uniquely determined.
Lemma .[] Let C be a nonempty and closed subset of a smooth and strictly convex Banach space E such that there exists a sunny generalized nonexpansive retraction R from E onto C.Let x∈E and z∈C.Then the following hold:
() z=Rxif and only ifx–z,Jy–Jz ≤for ally∈C; () φ(x,Rx) +φ(Rx,z)≤φ(x,z).
Lemma .[] Let C be a nonempty and closed subset of a smooth,strictly convex and reflexive Banach space E.Then the following are equivalent:
() Cis a sunny generalized nonexpansive retraction ofE; () J(C)is closed and convex.
Remark . From Lemmas . and .. IfEis a Hilbert space, then a sunny generalized nonexpansive retraction fromEontoCreduces to a metric projection operatorPfromE ontoC.
Lemma .[] Let E be a smooth,strictly convex and reflexive Banach space,let C be a nonempty,closed and sunny generalized nonexpansive retraction of E,and let R be the sunny generalized nonexpansive retraction from E onto C.Let x∈E and z∈C.Then the following are equivalent:
() z=Rx;
() φ(x,z) =miny∈Cφ(x,y).
Lemma .[] Let E be a uniformly smooth and strictly convex real Banach space,and let{xn}and{yn}be two sequences of E.Ifφ(xn,yn)→and either{xn}or{yn}is bounded, thenxn–yn →.
Lemma .[] Let{an}and{bn}be two sequences of nonnegative real numbers satisfy-ing the inequality
an+≤an+bn, ∀n≥.
If∞n=bn<∞,thenlimn→∞anexists.
Now, let us recall the following well-known concept and the result.
Definition .[] Let Bbe a subset of a topological vector spaceX. A mappingG: B→X is called a KKM mapping ifconv{x,x,x, . . . ,xm} ⊂
m
i=G(xi), forxi∈Band
i= , , , . . . ,m, whereconvAdenotes the convex hull of the setA.
In [], Ky Fan gave the following famous infinite-dimensional generalization of Knaster, Kuratowski and Mazurkiewicz’s classical finite-dimensional result.
Lemma .[] Let B be a subset of a Hausdorff topological vector space X,and let G: B→Xbe a KKM mapping.If G(x)is closed for all x∈B and is compact for at least one x∈B,thenx∈BG(x)=∅.
3 Existence theorem
In this section, we prove the existence theorem of a solution for a generalized equilibrium problem with a bifunction defined on the dual space of a Banach space. Now, we use the concept of KKM mapping to prove the lemma for our main result.
Lemma . Let C be a nonempty, compact and convex subset of a uniformly smooth, strictly convex and reflexive Banach space E,and let J be the duality mapping from E into E∗such that J(C)is closed and convex,let us assume that a bifunction F:J(C)×J(C)→R satisfies the following conditions(DA)-(DA),let C∗ be a nonempty,closed and convex subset of E∗,and let A:C∗→E be anα-inverse strongly skew monotone.Let any r> be a given real number,and let x∈E be any point.Then there exists z∈C such that
F(Jz,Jy) +AJz,Jy–Jz+ r
z–x,J(y–z)≥, ∀y∈C.
Proof Letxbe any point in E. For eachy∈C, we define the mappingH:C→E as
follows:
H(y) = z∈C:F(Jz,Jy) +AJz,Jy–Jz+ r
z–x,J(y–z)
≥,∀y∈C
.
It is easy to see thaty∈H(y), and henceH(y)=∅. (a) First, we show thatHis a KKM mapping.
Suppose thatHis not a KKM mapping. Then there exists a finite subset{y,y, . . . ,ym} ofCandαi> with
m
i=αi= such thatxˆ=
m i=αiyi∈/
m
i=H(yi) for alli= , , , . . . ,m.
It follows from the definition of a mappingHthat
F(Jx,ˆ Jyi) +AJx,ˆ Jyi–Jxˆ+ r
ˆ
x–x,J(yi–x)ˆ
By the assumptions of (DA) and (DA), we get
= F(Jx,ˆ Jx) +ˆ AJx,ˆ Jxˆ–Jxˆ+ r
ˆ
x–x,J(xˆ–x)ˆ ≤ m i= αi
F(Jx,ˆ Jyi) +AJx,ˆ Jyi–Jxˆ+ r
ˆ
x–x,J(yi–x)ˆ
< ,
which is a contradiction. Thus,His a KKM mapping onC. (b) Next, we show thatH(y) is closed for ally∈C.
Let{zn}be a sequence inH(y) such thatzn→z, asn→ ∞.
It then follows fromzn∈H(y) that
F(Jzn,Jy) +AJzn,Jy–Jzn+ r
zn–x,J(y–zn)
≥. (.)
By assumption (DA), the continuity ofJand the lower semicontinuity of · , we obtain
from (.) that
≤lim inf n→∞
F(Jzn,Jy) +AJzn,Jy–Jzn+ r
zn–x,J(y–zn)
≤lim sup n→∞
F(Jzn,Jy) +AJzn,Jy–Jzn+ r
zn–x,J(y–zn)
=lim sup n→∞
F(Jzn,Jy) +AJzn,Jy–Jzn+ r
(zn–y) + (y–x),J(y–zn)
=lim sup n→∞
F(Jzn,Jy) +AJzn,Jy–Jzn+ r
zn–y,J(y–zn)+ r
y–x,J(y–zn)
=lim sup n→∞
F(Jzn,Jy) +AJzn,Jy–Jzn+ r
y–x,J(y–zn)
– r
y–zn,J(y–zn)
=lim sup n→∞
F(Jzn,Jy) +AJzn,Jy–Jzn+ r
y–x,J(y–zn)
– ry–zn
≤lim sup
n→∞ F(Jzn,Jy) +lim supn→∞ AJzn,Jy–Jzn
+ rlim supn→∞
y–x,J(y–zn)
–lim inf n→∞
ry–zn
≤F(Jz,Jy) +AJz,Jy–Jz+
r
y–x,J(y–z)
– ry–z
=F(Jz,Jy) +AJz,Jy–Jz+
r
y–x,J(y–z)
– r
y–z,J(y–z)
=F(Jz,Jy) +AJz,Jy–Jz+
r
y–x,J(y–z)
+ r
z–y,J(y–z)
=F(Jz,Jy) +AJz,Jy–Jz+
r
(y–x) + (z–y),J(y–z)
=F(Jz,Jy) +AJz,Jy–Jz+
r
z–x,J(y–z)
Now, we get
F(Jz,Jy) +AJz,Jy–Jz+
r
z–x,J(y–z)
≥.
Therefore,z∈H(y), and soH(y) is closed for ally∈C.
(c) We show thatH(y) is weakly compact.
Now, we know thatH(y) is closed and subset ofC.
SinceCis compact. Therefore,H(y) is compact, and thenH(y) is weakly compact. By using (a), (b), (c) and Lemma ., we can conclude thaty∈CH(y)=∅. Therefore, there existsz∈Csuch that
F(Jz,Jy) +AJz,Jy–Jz+ r
z–x,J(y–z)≥, ∀y∈C.
Theorem . Let C be a nonempty,closed and convex subset of a uniformly smooth,strictly convex and reflexive Banach space E,and let J be the duality mapping from E into E∗such that J(C)is closed and convex,let us assume that a bifunction F:J(C)×J(C)→Rsatisfies the following conditions(DA)-(DA),let C∗be a nonempty,closed and convex subset of E∗, and let A:C∗→E be anα-inverse strongly skew monotone and hemicontinuous.Let any r> be a given real number,and let x∈E be any point.We define a mapping Tr:E→C as follows:
Tr(x) = z∈C:F(Jz,Jy) +AJz,Jy–Jz+ r
z–x,J(y–z)≥,∀y∈C
. (.)
Then the following conclusions hold:
() Tris single-valued;
() Trx–Try,J(Trx–Try) ≤ x–y,J(Trx–Try),∀x,y∈E; () Fix(Tr) =GEP(F,A);
() J(GEP(F,A))is closed and convex.
Proof We complete this proof by four items below. () We show thatTris single-valued.
From the definition ofTr(x), it is easy to see that
F(Jy,Jy) +AJy,Jy–Jy+ r
z–y,J(y–y)= ≥.
Therefore,y∈Tr(x). Hence,Tr(x)=∅.
Indeed, for anyx∈Eandr> , letz,z∈Tr(x). Then
F(Jz,Jz) +AJz,Jz–Jz+
r
z–x,J(z–z)
≥,
and
F(Jz,Jz) +AJz,Jz–Jz+
r
z–x,J(z–z)
Adding the two inequalities, we have
≤F(Jz,Jz) +F(Jz,Jz) +AJz,Jz–Jz+AJz,Jz–Jz
+ r
z–x,J(z–z)
+ r
z–x,J(z–z)
=F(Jz,Jz) +F(Jz,Jz) +AJz,Jz–Jz–AJz,Jz–Jz
+ r
z–x,J(z–z)
– r
z–x,J(z–z)
=F(Jz,Jz) +F(Jz,Jz) +AJz–AJz,Jz–Jz+
r
(z–x) – (z–x),J(z–z)
=F(Jz,Jz) +F(Jz,Jz) +AJz–AJz,Jz–Jz+
r
z–z,J(z–z)
=F(Jz,Jz) +F(Jz,Jz) –AJz–AJz,Jz–Jz+
r
z–z,J(z–z)
.
Therefore, we obtain
F(Jz,Jz) +F(Jz,Jz) –AJz–AJz,Jz–Jz+
r
z–z,J(z–z)
≥. (.)
From condition (DA) and the fact thatAis anα-inverse strongly skew monotone, we have
≤F(Jz,Jz) +F(Jz,Jz) –AJz–AJz,Jz–Jz+
r
z–z,J(z–z)
≤–αAJz–AJz+
r
z–z,J(z–z)
≤
r
z–z,J(z–z)
.
Sincer> ,Jis monotone, andEis strictly convex, we obtain
z=z.
This implies thatTris single-valued.
() We show thatTrx–Try,J(Trx–Try) ≤ x–y,J(Trx–Try)for allx,y∈E. Indeed, for anyx,y∈Candr> , we have
F(JTrx,JTry) +AJTrx,JTry–JTrx+ r
Trx–x,J(Try–Trx)
≥,
and
F(JTry,JTrx) +AJTry,JTrx–JTry+ r
Try–y,J(Trx–Try)≥.
Adding the two inequalities, we have
≤F(JTrx,JTry) +F(JTry,JTrx) +AJTrx,JTry–JTrx+AJTry,JTrx–JTry
+ r
Trx–x,J(Try–Trx)+ r
Try–y,J(Trx–Try)
+ r
Trx–x,J(Try–Trx)– r
Try–y,J(Try–Trx)
=F(JTrx,JTry) +F(JTry,JTrx) +AJTrx–AJTry,JTry–JTrx
+ r
(Trx–x) – (Try–y),J(Try–Trx)
=F(JTrx,JTry) +F(JTry,JTrx) +AJTrx–AJTry,JTry–JTrx
+ r
(Trx–Try) – (x–y),J(Try–Trx)
=F(JTrx,JTry) +F(JTry,JTrx) –AJTrx–AJTry,JTrx–JTry
+ r
(Trx–Try) – (x–y),J(Try–Trx). (.)
It follows that
F(JTrx,JTry) +F(JTry,JTrx) –AJTrx–AJTry,JTrx–JTry
+ r
(Trx–Try) – (x–y),J(Try–Trx)
≥. (.)
From condition (DA) and the fact thatAis anα-inverse strongly skew monotone, we get
≤F(JTrx,JTry) +F(JTry,JTrx) –AJTrx–AJTry,JTrx–JTry
+ r
(Trx–Try) – (x–y),J(Try–Trx)
≤–αAJTrx–AJTry+ r
(Trx–Try) – (x–y),J(Try–Trx)
≤
r
(Trx–Try) – (x–y),J(Try–Trx)
≤– r
(Trx–Try) – (x–y),J(Trx–Try).
Sincer> , we have
(Trx–Try) – (x–y),J(Trx–Try)≤.
Therefore, we also have
Trx–Try,J(Trx–Try)–x–y,J(Trx–Try)≤.
This implies that
Trx–Try,J(Trx–Try)≤x–y,J(Trx–Try).
() We show thatFix(Tr) =GEP(F,A). It is easy to see that
z∈Fix(Tr) ⇔ z=Trz
⇔ F(Jz,Jy) +AJz,Jy–Jz+ r
⇔ F(Jz,Jy) +AJz,Jy–Jz ≥
⇔ z∈GEP(F,A).
This implies thatFix(Tr) =GEP(F,A).
() We show thatJ(GEP(F,A)) is closed and convex.
For eachy∈C, we define the mappingG:C→Eas follows:
G(y) =z∈C:F(Jz,Jy) +AJz,Jy–Jz ≥.
It is easy to see thaty∈G(y), so thatG(y)=∅. Next, we show thatGis a KKM mapping.
Suppose thatGis not a KKM mapping. Then there exists a finite subset{z,z, . . . ,zm}of
C, andβi> with
m
i=βi= such thatzˆ=
m
i=βizi∈/G(zi) for alli= , , , . . . ,m. Then
we have
F(Jˆz,Jzi) +AJz,ˆ Jzi–Jˆz< , i= , , , . . . ,m.
It follows from (DA) and (DA) that
=F(Jz,ˆ Jz) +ˆ AJˆz,Jzˆ–Jzˆ ≤ m
i=
βi
F(Jz,ˆ Jzi) +AJz,ˆ Jzi–Jˆz< ,
which is the contradiction. Hence,Gis a KKM mapping onC. (.) Next, we show thatG(y) is closed for eachy∈C.
For anyy∈C, let{zn}be any sequence inG(y) such thatzn→z, asn→ ∞.
Hence,zn–x→z–x, asn→ ∞.
Next, we show thatz∈G(y). Then for eachy∈C, we have
F(Jzn,Jy) +AJzn,Jy–Jzn ≥. (.)
It follows from assumption (DA) that
F(Jz,Jy) +AJz,Jy–Jz ≥lim sup
n→∞ F(Jzn,Jy) +n→∞lim AJzn,Jy–Jzn ≥.
This implies thatz∈G(y), and henceG(y) is closed for eachy∈C.
SinceJis continuous. Therefore,y∈CG(y) =J(GEP(F,A)) is closed. (.) Next, we show thatJ(GEP(F,A)) is convex.
Let z∗,z∗ ∈J(GEP(F,A)), then we have z∗ =Jz ∈J(C) and z∗ =Jz ∈ J(C), where
z,z∈C.
Fork,t∈(, ), letz∗=kz∗+ ( –k)z∗and for anyy∈C, we setx∗t =tJy+ ( –t)z∗. It follows from (DA) and (DA) that
=Fx∗t,x∗t
≤Fx∗t,tJy+ ( –t)z∗
≤tFx∗t,Jy+ ( –t)Fx∗t,z∗
and
= Ax∗t,x∗t –x∗t
=Ax∗t,x∗t –Jy+Jy–x∗t
=Ax∗t,x∗t –Jy+Ax∗t,Jy–x∗t
=Ax∗t,x∗t –Jy–Ax∗t,x∗t–Jy
≤Ax∗t,x∗t –Jy
=Ax∗t,tJy+ ( –t)z∗–Jy
=Ax∗t,tJy+ ( –t)z∗–t+ ( –t)Jy
=Ax∗t, ( –t)z∗–Jy
= ( –t)Ax∗t,z∗–Jy
= (t– )Ax∗t,Jy–z∗
≤tAx∗t,Jy–z∗. (.)
Adding two inequalities (.) and (.) and dividing byt> , we get
≤tFx∗t,Jy+tAx∗t,Jy–z∗
≤Fx∗t,Jy+Ax∗t,Jy–z∗.
Lettingtto by (DA) and the hemicontinuous ofA, we obtain
Fz∗,Jy+Az∗,Jy–z∗≥ for ally∈C.
Hence,z∗∈J(GEP(F,A)), and thus,J(GEP(F,A)) is convex.
This completes the proof.
4 Convergence theorem
In this section, we use the hybrid projection method for finding a solution of a generalized equilibrium problem in the dual space of Banach spaces.
Theorem . Let E be a uniformly smooth,strictly convex and reflexive real Banach space,
which has a Kadec-Klee property, let C be a nonempty, closed and convex subset of E, and let J be the duality mapping from E into E∗ such that J(C)is closed and convex of E∗,let us assume that a bifunction F:J(C)×J(C)→Rsatisfies the following conditions (DA)-(DA),let C∗ be a nonempty,closed and convex subset of E∗,and let A:C∗→E be anα-inverse strongly skew monotone.For x∈E,we define a mapping Trn:E→C as follows:
Trn(x) = z∈C:F(Jz,Jy) +AJz,Jy–Jz+ rn
z–x,J(y–z)≥,∀y∈C
Suppose thatGEP(F,A)=∅.Let{xn}be a sequence generated by
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈C chosen arbitrary, C=C,
un=Trnxn,
yn=αnxn+ ( –αn)un,
Cn+={z∈Cn:φ(yn,z)≤φ(xn,z)},
xn+=RCn+(x), ∀n≥,
(.)
where J is the duality mapping on E,{αn}is the sequence in[, ]such thatlim supn→∞αn< ,{rn} ⊂[a,∞)for some a> and RCn+is the sunny generalized nonexpansive retraction from E onto Cn+.Then the sequence{xn}converges strongly to RGEP(F,A)(x),where RGEP(F,A)
is the sunny generalized nonexpansive retraction from E ontoGEP(F,A).
Proof We complete this proof by seven steps below.
Step . We show thatJ(Cn) are closed and convex subsets ofEfor eachn∈N∪ {}. It is obvious thatJ(C) is closed and convex. Suppose thatJ(Ck) is closed and convex for
somek∈N∪ {}.
For eachz∈Cn, we see that
φ(yn,z)≤φ(xn,z) ⇔ yn– yn,Jz+z≤ xn– xn,Jz+z
⇔ yn–xn– yn,Jz+ xn,Jz ≤
⇔ yn–xn– yn–xn,Jz ≤.
Hence,J(Ck+) is closed and convex. Therefore,J(Cn) are closed and convex subsets ofE for eachn∈N∪ {}.
Step . We show thatGEP(F,A)⊂Cnfor alln∈N∪ {}. Note thatun=TrnxnandTrnis generalized nonexpansive.
From C=C, we haveGEP(F,A)⊂C. Suppose thatGEP(F,A)⊂Ck, for some k∈
N∪ {}.
For anyp∈GEP(F,A), from algorithm (.) and the fact thatTrk is generalized nonex-pansive, we compute
φ(yk,p) =φαnxk+ ( –αn)uk,p
=αkxk+ ( –αk)uk– αkxk+ ( –αk)uk,Jp+p
≤αkxk+ ( –αk)uk– αkxk,Jp– ( –αk)uk,Jp+p
=αkxk+ ( –αk)uk– αkxk,Jp
– ( –αk)uk,Jp+αk+ ( –αk)
p
=αkxk– αkxk,Jp+αkp+ ( –αk)uk
– ( –αk)uk,Jp+ ( –αk)p
=αkφ(xk,p) + ( –αk)φ(uk,p)
≤αkφ(xk,p) + ( –αk)φ(xk,p)
=φ(xk,p).
Therefore,p∈Ck. Hence, we getp∈Ck+.
This implies thatGEP(F,A)⊂Cn,∀n∈N∪ {}, and also the sequence{xn}is well de-fined.
Step . We show that{xn}is bounded.
From the definition of{xn}, we know thatxn=RCn(x) andxn+=RCn+(x)∈Cn+⊂Cn. For allp∈GEP(F,A)⊂Cn, we have
φ(xn,x) =φ
RCn(x),x
≤φ(p,x) –φ
p,RCn(x)
≤φ(p,x).
Thenφ(x,xn) is bounded. Therefore,{xn}is bounded, and also{Trnxn}is bounded.
Step . We show that there existspˆ∈Csuch thatxn→ ˆp, asn→ ∞. Sincexn=RCn(x) andxn+=RCn+(x)∈Cn+⊂Cn, we have
φ(xn,x)≤φ(xn+,x), ∀n∈N∪ {}.
Therefore, the sequence{φ(xn,x)}is nondecreasing. Hence,limn→∞φ(xn,x) exists.
By the definition ofCn, one has thatCm⊂Cnandxm=RCm(x)∈Cnfor any positive integerm≥n. It follows that
φ(xm,xn) = φxm,RCn(x)
≤ φ(xm,x) –φ
RCn(x),x
≤ φ(xm,x) –φ(xn,x) →, asm,n→ ∞.
It follows from Lemma . thatxm–xn →, asm,n→ ∞. Thus, the sequence{xn}is a Cauchy sequence.
Without loss of generality, we can assume thatxnp. Sinceˆ {xn}is bounded, andEis reflexive.
We know thatCn+⊂Cn, andCnis closed and convex, forpˆ∈Cn, we have
lim inf
n→∞ φ(xn,x) =lim infn→∞
xn– xn,Jx+x
≥ ˆp– ˆp,Jx+x
=φ(p,ˆ x).
It follows that
This implies that
lim
n→∞φ(xn,x) =φ(p,ˆ x).
Soxn → ˆp. Sincexnp. By the Kadec-Klee property ofˆ E, we obtain that
xn→ ˆp, asn→ ∞. (.)
FromJis uniformly norm-to-norm continuous on bounded subset ofE, we also have
Jxn→Jp,ˆ asn→ ∞. (.)
Step . We show thatxn–yn → andxn–un →, asn→ ∞. Sincexn=RCn(x) andlimn→∞φ(xn,x) exists. We get
φ(xn,xn+) = φ
RCn(x),xn+
≤ φ(x,xn+) –φ
x,RCn(x)
= φ(x,xn+) –φ(x,xn)
→, asn→ ∞.
Fromxn+=RCn+(x)∈Cn+⊂Cn, we have
φ(yn,xn+)≤φ(xn,xn+)→, asn→ ∞.
By Lemma ., we obtain
xn–xn+ → and yn–xn+ →, asn→ ∞.
Therefore,
xn–yn →, asn→ ∞. (.)
Since
= lim
n→∞xn–yn
= lim
n→∞xn–
αnxn+ ( –αn)un
= lim
n→∞( –αn)xn–un.
By the assumption, we havelim supn→∞αn< , we obtain
xn–un →, asn→ ∞. (.)
From (.), (.) and (.), it follows that
un→ ˆp and yn→ ˆp, asn→ ∞. (.)
FromJis uniformly norm-to-norm continuous on bounded subset ofE, we also have
Jun→Jpˆ and Jyn→Jp,ˆ asn→ ∞. (.)
It follows from (.) and (.) and the property ofJthat
Jxn–Jyn → and Jxn–Jun → asn→ ∞. (.)
Since{xn}is bounded. Therefore,{un}and{yn}are bounded. Hence,{Jxn},{Jun}and{Jyn}are also bounded.
So, there exists a subsequence {Jxnk} of{Jxn}such thatJxnk pˆ∗, and there exists a subsequence{Junk}of{Jun}such thatJunkpˆ∗.
From (.) andrn⊂[a,∞), we have
lim n→∞
rnxn–un= .
Sinceun=Trnxn, we have
F(Jun,Jy) +AJun,Jy–Jun+ rn
un–xn,J(y–un)≥, ∀y∈C.
It follows from (DA) that
AJun,Jy–Jun+ rn
un–xn,J(y–un)
≥–F(Jun,Jy)≥F(Jy,Jun), ∀y∈C.
Fromxn→ ˆpandun→ ˆp, we get
AJp,ˆ Jy–Jpˆ+ rn
ˆ
p–p,ˆ J(y–p)ˆ ≥F(Jy,Jp),ˆ ∀y∈C.
Therefore,
AJp,ˆ Jy–Jpˆ ≥F(Jy,Jp),ˆ ∀y∈C.
For any <t< ,y∈Cand settingy∗t =tJy+ ( –t)Jp. Then we getˆ y∗t ∈J(C), and so
AJp,ˆ y∗t –Jpˆ≥Fy∗t,Jpˆ, ∀y∈C.
It follows from (DA) and (DA) that
=Fy∗t,y∗t
≤tFy∗t,Jy+ ( –t)Fy∗t,Jpˆ
Lettingt↓, we obtain from assumption (DA) that
F(Jp,ˆ Jy) +AJp,ˆ y∗t –Jpˆ≥.
This implies that
Jpˆ∈JGEP(F,A).
Step . We show that the sequence{xn}converges strongly toRGEP(F,A)(x).
We know thatxn=RCn(x)∈GEP(F,A)⊂Cn. Letw=RGEP(F,A)(x). It follows from Lemma . that
φ(x,xn)≤φ(x,w).
Since the norm is weakly lower semicontinuous, and from (DA), we have
φx,J–pˆ∗
=x–
x,J
J–pˆ∗+ˆp
=x–
x,pˆ∗
+ˆp
≤lim inf k→∞
x– x,Jxnk+xnk
=lim inf
k→∞ φ(x,xnk)
≤lim sup
k→∞ φ(x,xnk)
≤φ(x,w).
From the definition ofRGEP(F,A), we haveJ–pˆ∗=w.
Finally, we show thatxn→w, wherew=RGEP(F,A)(x).
Now, we have
φ(x,xn)≤φ(x,w) =φ
xn,RGEP(F,A)(x)
,
and from Remark .(), we get
φ(w,xn) =φ(w,x) +φ(x,xn) + w–x,Jx–Jxn.
Therefore,
lim sup
n→∞ φ(w,xn) =lim supn→∞
φ(w,x) +φ(x,xn) + w–x,Jx–Jxn
≤lim sup n→∞
φ(w,x) +φ(x,w) + w–x,Jx–Jxn
=φ(w,x) +φ(x,w) + w–x,Jx–Jw
=φ(w,w) = .
This implies that
lim
By Lemma ., we get
lim
n→∞w–xn=n→∞limRGEP(F,A)(x) –xn= .
Hence,
xn→RGEP(F,A)(x), asn→ ∞.
Therefore, the sequence {xn} converges strongly to RGEP(F,A)(x). This completes the
proof.
If we substitutex=xnandz=unin equation (.), then we obtain the following result which extends the following results by Takahashi and Zembayashi [] from an equilibrium problem to a generalized equilibrium problem.
Corollary . Let E be a uniformly smooth,strictly convex and reflexive real Banach space which has a Kadec-Klee property,let C be a nonempty,closed and convex subset of E,and let J be the duality mapping from E into E∗such that J(C)is closed and convex of E∗,let us assume that a bifunction F:J(C)×J(C)→Rsatisfies the following conditions (DA)-(DA),let C∗ be a nonempty,closed and convex subset of E∗,and let A:C∗→E be an α-inverse strongly skew monotone.Suppose thatGEP(F,A)=∅.Let{xn}be a sequence gen-erated by
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈C chosen arbitrary, C=C,
un∈C such that
F(Jun,Jy) +AJun,Jy–Jun+rnun–xn,J(y–un) ≥, ∀y∈C,
yn=αnxn+ ( –αn)un,
Cn+={z∈Cn:φ(yn,z)≤φ(xn,z)},
xn+=RCn+(x), ∀n≥,
(.)
where J is the duality mapping on E,{αn}is the sequence in[, ]such thatlim supn→∞αn< ,{rn} ⊂[a,∞)for some a> and RCn+is the sunny generalized nonexpansive retraction from E onto Cn+.Then the sequence{xn}converges strongly to RGEP(F,A)(x),where RGEP(F,A)
is the sunny generalized nonexpansive retraction from E ontoGEP(F,A).
If we setA≡ in Corollary ., then we obtain the following result which extends the following results by Takahashi and Zembayashi [].
by
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈C chosen arbitrary, C=C,
un∈C such that F(Jun,Jy) +rnun–xn,J(y–un) ≥, ∀y∈C,
yn=αnxn+ ( –αn)un,
Cn+={z∈Cn:φ(yn,z)≤φ(xn,z)},
xn+=RCn+(x), ∀n≥,
(.)
where J is the duality mapping on E,{αn}is the sequence in[, ]such thatlim supn→∞αn< ,{rn} ⊂[a,∞)for some a> and RCn+is the sunny generalized nonexpansive retraction from E onto Cn+.Then the sequence{xn}converges strongly to REP(F)(x),where REP(F)is
the sunny generalized nonexpansive retraction from E ontoEP(F).
If we setF≡ in Corollary ., then Corollary . is reduced to the following corollary.
Corollary . Let E be a uniformly smooth,strictly convex and reflexive real Banach space which has a Kadec-Klee property,let C be a nonempty,closed and convex subset of E such that J(C)is closed and convex of E∗,let C∗be a nonempty,closed and convex subset of E∗ and A:C∗→E be anα-inverse strongly skew monotone.Suppose thatVI(J(C),A)=∅.Let
{xn}be a sequence generated by
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈C chosen arbitrary, C=C,
un∈C such that AJun,Jy–Jun+rnun–xn,J(y–un) ≥, ∀y∈C,
yn=αnxn+ ( –αn)un,
Cn+={z∈Cn:φ(yn,z)≤φ(xn,z)},
xn+=RCn+(x), ∀n≥,
(.)
where J is the duality mapping on E,{αn}is the sequence in[, ]such thatlim supn→∞αn< ,{rn} ⊂[a,∞)for some a> ,and RCn+ is the sunny generalized nonexpansive retrac-tion from E onto Cn+.Then the sequence{xn} converges strongly to RVI(J(C),A)(x),where
RVI(J(C),A)is the sunny generalized nonexpansive retraction from E ontoVI(J(C),A).
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Author details
1Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bang Mod,
Thung Khru, Bangkok, 10140, Thailand.2Department of Mathematics, Faculty of Education, Bansomdejchaopraya
Rajabhat University (BSRU), Thonburi, Bangkok, 10600, Thailand.
Acknowledgements
The first author would like to thank the Bansomdejchaopraya Rajabhat University for financial support. The authors would like to thank the Higher Education Research Promotion and National Research University Project of Thailand’s Office of the Higher Education Commission for financial support (Under NRU-CSEC Project No. NRU56000508).
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10.1186/1687-1812-2013-264
