R E S E A R C H
Open Access
Hybrid algorithms for equilibrium and
common fixed point problems with
applications
Qingnian Zhang
1*and Huilian Wu
2*Correspondence:
1School of Mathematics and
Information Science, North China University of Water Resources and Electric Power, Zhengzhou, 450011, China
Full list of author information is available at the end of the article
Abstract
In this paper, hybrid algorithms are investigated for equilibrium and common fixed point problems. Strong convergence of the algorithms is obtained in the framework of reflexive Banach spaces.
Keywords: asymptotically quasi-φ-nonexpansive mapping; quasi-φ-nonexpansive mapping; algorithm; equilibrium problem; fixed point
1 Introduction
Iterative algorithms have emerged as an effective and powerful tool for studying a wide class of problems which arise in economics, finance, image reconstruction, ecology, trans-portation, network, elasticity and optimization; see [–] and the references therein. The computation of solutions of nonlinear operator equations (inequalities) is important in the study of many real-world problems. Recently, the study of the convergence of various iterative algorithms for solving various nonlinear mathematical models has formed a ma-jor part of numerical mathematics. Among these iterative algorithms, the Krasnoselski-Mann iterative algorithm and the Ishikawa iterative algorithm are popular and much dis-cussed. It is well known that both the Krasnoselski-Mann iterative algorithm and the Ishikawa iterative algorithm only have weak convergence even for nonexpansive mappings in infinite-dimensional Hilbert spaces. In many disciplines, including economics, image recovery, quantum physics, and control theory, problems arise in infinite-dimensional spaces. In such problems, strong convergence is often much more desirable than weak convergence. To improve the weak convergence of the Krasnoselski-Mann iterative al-gorithm and the Ishikawa iterative alal-gorithm, so-called projection alal-gorithms have been investigated in different frameworks of spaces; see [–] and the references therein.
The aim of this article is to investigate solutions of equilibrium and common fixed point problems in Hilbert spaces. In Section , we provide some necessary preliminaries. In Section , a projection algorithm is investigated. Strong convergence of the algorithm is obtained in a uniformly smooth and strictly convex Banach space which also has the Kadec-Klee property. In Section , applications are provided to support the main results of this paper.
2 Preliminaries
LetEbe a real Banach space with the dualE∗. Recall that the normalized duality mapping JfromEto E∗is defined byJx={f∗∈E∗:x,f∗=x=f∗}, where·,·denotes the
generalized duality pairing. LetUE={x∈E:x= }be the unit sphere ofE.Eis said
to be smooth ifflimt→x+tyt–x exists for eachx,y∈UE. It is also said to be uniformly
smooth iff the above limit is attained uniformly forx,y∈UE.Eis said to be strictly convex
iffx+y< for allx,y∈Ewithx=y= andx=y. It is said to be uniformly convex iff limn→∞xn–yn= for any two sequences{xn}and{yn}inEsuch thatxn=yn=
andlimn→∞xn+yn= . 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.
In what follows, we useand→to stand for the weak and strong convergence, re-spectively. Recall that a spaceEhas the Kadec-Klee property iff for any sequence{xn} ⊂E,
x∈Ewithxnx, andxn → x, we havexn–x → asn→ ∞. It is known that ifE
is a uniformly convex Banach spaces, thenEhas the Kadec-Klee property.
LetE be a smooth Banach space. Let us consider the functional defined byφ(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 know, ifCis a nonempty, closed, and convex subset of a Hilbert spaceHandPC:H→Cis the metric projection ofHontoC, thenPCis
non-expansive. This fact actually characterizes Hilbert spaces, and consequently it is not avail-able in more general Banach spaces. In this connection, Alber [] recently introduced a generalized projection operatorCin a Banach spaceEwhich is an analog 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, where¯ x¯is the solution to the minimization problemφ(x,¯ x) =miny∈Cφ(y,x).
Existence and uniqueness of the operatorCfollow from the properties of the functional φ(x,y) and strict monotonicity of the mappingJ. IfEis a reflexive, strictly convex, and smooth Banach space, thenφ(x,y) = if and only ifx=y; for more details, see [] and the references therein. In Hilbert spaces,C=PC. From the definition of the functionφ,
we also have
x–y≤φ(x,y)≤y+x, ∀x,y∈E (.)
and
x–y≤φ(x,y)≤y+x, ∀x,y∈E. (.)
LetFbe a bifunction fromC×CtoR, whereRdenotes the set of real numbers. Recall the following equilibrium problem. Findp∈Csuch thatF(p,y)≥,∀y∈C. We useEP(F) to denote the solution set of the equilibrium problem. Numerous problems in physics, optimization and economics reduce to finding a solution of the equilibrium problem.
Next, we give the following assumptions:
(A) F(x,x) = ,∀x∈C;
(A) Fis monotone,i.e.,F(x,y) +F(y,x)≤,∀x,y∈C; (A) lim supt↓F(tz+ ( –t)x,y)≤F(x,y),∀x,y,z∈C;
LetCbe a nonempty subset ofE and letT:C→Cbe a mapping. In this paper, we useF(T) to stand for the fixed point set ofT. Recall thatT is said to be asymptotically regular onCiff for any bounded subsetK ofC,lim supn→∞{Tn+x–Tnx:x∈K}= .
Recall thatT is said to be closed iff for any sequence{xn} ⊂Csuch thatlimn→∞xn=x
andlimn→∞Txn=y, thenTx=y. Recall that a pointpinCis said to be an asymptotic
fixed point of T iffC contains a sequence{xn} which converges weakly topsuch that
limn→∞xn–Txn= . The set of asymptotic fixed points ofTwill be denoted byF(T).
T is said to be relatively nonexpansive iffF(T) =F(T)=∅and
φ(p,Tx)≤φ(p,x), ∀x∈C,∀p∈F(T).
T is said to be relatively asymptotically nonexpansive iffF(T) =F(T)=∅and
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, which is an
extension of the class of relatively nonexpansive mappings, was first introduced in [].
Recall thatTis said to be quasi-φ-nonexpansive iffF(T)=∅and
φ(p,Tx)≤φ(p,x), ∀x∈C,∀p∈F(T).
Recall that T is said to be asymptotically quasi-φ-nonexpansive iff there exists a se-quence{μn} ⊂[,∞) withμn→ asn→ ∞such that
F(T)=∅, φp,Tnx≤( +μ
n)φ(p,x), ∀x∈C,∀p∈F(T),∀n≥.
Remark . The class of asymptotically quasi-φ-nonexpansive mappings [, ] which
is an extension of the class of quasi-φ-nonexpansive mappings []. The class of
quasi-φ-nonexpansive mappings and the class of asymptotically quasi-φ-nonexpansive map-pings are more general than the class of relatively nonexpansive mapmap-pings and the class of relatively asymptotically nonexpansive mappings. Quasi-φ-nonexpansive mappings and asymptotically quasi-φ-nonexpansive ones do not require the restrictionF(T) =F(T).
In order to prove our main results, we need the following lemmas.
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
∞
i=
(αixi)
≤
∞
i=
αixi
–αiαjg
xi–xj
,
for all x,x, . . . ,xN, . . .∈Br:={x∈E:x ≤r} and α,α, . . . ,αN, . . .∈[, ] such that
∞
Lemma .[] Let E be a reflexive,strictly convex,and smooth Banach space.Let C be a nonempty,closed,and convex subset of E and let x∈E.Then
φ(y,Cx) +φ(Cx,x)≤φ(y,x), ∀y∈C.
Lemma .[] Let C be a nonempty,closed,and convex subset of a smooth Banach space
E and let x∈E.Then x=Cx if and only if
x–y,Jx–Jx ≥ ∀y∈C.
Lemma .[] 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 T:C→C be an asymptotically quasi-φ-nonexpansive mapping.Then F(T)is closed and convex.
Lemma . [, ] Let C be a closed and convex subset of a smooth, strictly
con-vex, and reflexive Banach space E. Let F be a bifunction from C×C to R satisfying the assumptions (A)-(A). Let r> and let x∈E. Then there exists z∈C such that F(z,y) +
ry–z,Jz–Jx ≥,∀y∈C.Define a mapping Tr:E→C by
Srx=
z∈C:f(z,y) +
ry–z,Jz–Jx,∀y∈C . Then the following conclusions hold:
() Sris a single-valued firmly nonexpansive-type mapping,i.e.,for allx,y∈E,
Srx–Sry,JSrx–JSry ≤ Srx–Sry,Jx–Jy;
() F(Sr) =EP(F)is closed and convex;
() Sris quasi-φ-nonexpansive;
() φ(q,Srx) +φ(Srx,x)≤φ(q,x),∀q∈F(Sr).
3 Main results
Theorem . Let E be a uniformly smooth and strictly convex Banach space which also
has the Kadec-Klee property.Let C be a nonempty,closed,and convex subset of E and let
be an index set.Let Fjbe a bifunction from C×C toRsatisfying(A)-(A)for every j∈.
Let Ti:C→C an asymptotically quasi-φ-nonexpansive mapping for every i≥.Assume
that Tiis closed asymptotically regular on C and
∞
i=F(Ti)∩
j∈EP(Fj)is nonempty and
bounded.Let{xn}be a sequence generated in the following manner: ⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈E, chosen arbitrarily,
C,j=C,
C=j∈C,j,
x=Cx,
yn=J–(αn,Jxn+
∞
i=αn,iJTinxn),
un,j∈C such that Fj(un,j,y) +rn,jy–un,j,Jun,j–Jyn ≥, ∀y∈C,
Cn+,j={z∈Cn:φ(z,un,j)≤φ(z,xn) +
∞
i=μn,iMn},
Cn+=j∈Cn+,j,
where{αn,i}is a real number sequence in(, )for every i≥,{rn,j}is a real number sequence
in[r,∞),where r is some positive real number and Mn:=sup{φ(z,xn) :z∈
∞
i=F(Ti)∩
j∈EP(Fj)}.Assume that
∞
i=αn,i= andlim infn→∞αn,αn,i> for every i≥.Then
the sequence{xn}converges strongly to∞i=F(Ti)∩j∈EP(Fj)x,where
∞
i=F(Ti)∩
j∈EP(Fj)
is the generalized projection from E onto∞i=F(Ti)∩
j∈EP(Fj).
Proof Using Lemma . and Lemma ., we find that∞i=F(Ti)∩
j∈EP(Fj) is closed
and convex so that∞
i=F(Ti)∩
j∈EP(Fj)xis well defined. By induction, we easily find that
the setsCnare convex and closed.
Next, we show that∞i=F(Ti)∩
j∈EF(Fj)⊂Cn. It suffices to claim that
∞
i=
F(Ti)∩
j∈
EF(Fj)⊂Cn,j,
for everyj∈. It is clear that∞i=F(Ti)∩
j∈EF(Fj)⊂C,j=C. Now, we assume that
∞
i=F(Ti)∩
j∈EF(Fj)⊂Cm,jfor somemand for everyj∈. Since∀z∈
∞
i=F(Ti)∩
j∈EF(Fj)⊂Cm,j, one finds that φ(z,um,j) =φ(z,Srm,jym)
≤φ(z,ym)
=φ
z,J–
αm,Jxm+
∞
i=
αm,iJTimxm
=z–
z,αm,Jxm+
∞
i=
αm,iJTimxm
+
αm,Jxm+
∞
i=
αm,iJTimxm
≤ z– αm,z,Jxm–
∞
i= αm,i
z,JTimxm
+αm,xm+
∞
i=
αm,iTimxm
=αm,φ(z,xm) +
∞
i= αm,iφ
z,Timxm
≤αm,φ(z,xm) +
∞
i=
αm,iφ(z,xm) +
∞
i=
αm,iμm,iφ(z,xm)
≤φ(z,xm) +
∞
i=
μm,iφ(z,xm)
≤φ(z,xm) +
∞
i=
μm,iMm, (.)
which yieldsz∈Cm+,j, that is,∞i=F(Ti)∩j∈EF(Fj)⊂Cn.
Next, we prove that xn → p, where p ∈
∞
i=F(Ti) ∩
j∈EF(Fj). It follows
from Lemma . that φ(xn,x)≤φ(w,x) –φ(w,xn)≤φ(w,x), for∀w∈
N
i=F(Ti)∩
bounded. Since the space is reflexive, we may, without loss of generality, assume that xnp, wherep∈Cn. Usingφ(xn,x)≤φ(p,x), we find that
φ(p,x)≤lim inf
n→∞ φ(xn,x)≤lim supn→∞ φ(xn,x)≤φ(p,x).
It follows that limn→∞φ(xn,x) =φ(p,x). Hence, we have limn→∞xn=p. Since
E has the Kadec-Klee property, one sees that xn → p as n→ ∞. Next, we prove
p∈j∈EF(Fj). In view ofCn+⊂Cn andxn+ =Cn+x∈Cn, we have φ(xn+,xn)≤ φ(xn+,x) –φ(xn,x). Lettingn→ ∞, we obtainφ(xn+,xn)→. Sincexn+∈Cn+, we see
thatφ(xn+,un,j)≤φ(xn+,xn) +
∞
i=μn,iMn. We, therefore, obtainlimn→∞φ(xn+,un,j) = .
Hencelimn→∞un,j=p. It follows thatlimn→∞Jun,j=Jp. This shows that{Jun,j}is
a bounded sequence. SinceEis reflexive (E∗is also reflexive), we may assume thatJun,j
u∗,j∈E∗. Using the reflexivity ofE, we also see thatJ(E) =E∗. This shows that there exists anuj∈Esuch thatJuj=u∗,j. It follows thatφ(x
n+,un) =xn+– xn+,Jun+Jun.
Takinglim infn→∞in both sides of the equality above yields
≥ p– p,u∗,j+u∗,j =p– p,Juj+Juj =φp,uj,
that is,p=uj. This yieldsJp=u∗,j. It follows thatJu
n,jJp∈E∗. SinceE∗has the
Kadec-Klee property, we obtainJun,j–Jp→ asn→ ∞. SinceJ–:E∗→Eis demicontinuous. It
follows thatun,jp. Since the spaceEhas the Kadec-Klee property, one finds thatun,j→p
asn→ ∞. In view ofxn–un,j ≤ xn–p+p–un,j, we havelimn→∞xn–un,j= . It
follows thatlimn→∞Jxn–Jun,j= . Notice that
φ(z,xn) –φ(z,un,j) =xn–un,j– z,Jxn–Jun,j
≤ xn–un,j
xn+un,j
+ zJxn–Jun,j.
It follows that limn→∞φ(z,xn) –φ(z,un,j) = . By virtue of (.), we find thatφ(z,yn)≤ φ(z,xn) +Ni=μn,jMn, wherez∈∞i=F(Ti)∩j∈EF(Fj). In view ofun,j=Srn,jyn, we find
from Lemma . that
φ(un,j,yn) =φ(Srn,jyn,yn)
≤φ(z,yn) –φ(z,Srn,jyn)
≤φ(z,xn) –φ(z,Srn,jyn) +
∞
i= μn,jMn
=φ(z,xn) –φ(z,un,j) +
∞
i= μn,jMn.
It follows that limn→∞φ(un,j,yn) = . This in turn yields that un,j –yn → as
n→ ∞. Sinceun,j→p asn→ ∞, one finds that limn→∞yn=p. It follows that
ofJ(E) =E∗, we see that there existsy∈Esuch thatJy=y∗. It follows that
φ(un,j,yn) =un,j– un,j,Jyn+Jyn.
Takinglim infn→∞in both sides of the equality above yields
≥ p– p,y∗+y∗ =p– p,Jy+Jy =φ(p,y).
That is, p=y, which in turn implies thaty∗ =Jp. It follows thatJynJp∈E∗. Since
the space E∗ has the Kadec-Klee property, we arrive at Jyn–Jp→ asn→ ∞. Note
thatJ–:E∗→Eis demicontinuous. It follows thaty
np. SinceEhas the Kadec-Klee
property, we obtainyn→pasn→ ∞. In view ofun,j–yn ≤ un,j–p+p–yn, we
find thatlimn→∞un,i–yn= . SinceJ is uniformly norm-to-norm continuous on any
bounded sets, we havelimn→∞Jun,j–Jyn= . From the assumptionrn,j≥rj, we see that
limn→∞Junr,jn–,jJyn= . Notice that
Fj(un,j,y) +
rn,j
y–un,j,Jun,j–Jyn ≥, ∀y∈C.
From the condition (A), we find that
y–un,j
Jun,j–Jyn
rn,j
≥
rn,j
y–un,j,Jun,j–Jyn ≥Fj(y,un,j), ∀y∈C.
Taking the limit asn→ ∞, we find thatFj(y,p)≤,∀y∈C. For <tj< andy∈C, define
ytj =tjy+ ( –tj)p. It follows thatyt,j∈C, which yieldsFj(yt,j,p)≤. It follows from the
conditions (A) and (A) that
=Fj(yt,j,yt,j)≤tjFj(yt,j,y) + ( –tj)Fj(yt,j,p)≤tjFj(yt,j,y).
This yieldsFj(yt,j,y)≥. Lettingtj→, we find from the condition (A) thatFj(p,y)≥,
∀y∈C. This implies thatp∈EP(Fj) for everyj∈.
Now, we are in a position to show thatp∈∞i=F(Ti). It follows from Lemma . that φ(z,un,j)≤φ(z,yn)
=φ
z,J–
αn,Jxn+
∞
i=
αn,iJTinxn
=z–
z,αn,Jxn+
∞
i=
αn,iJTinxn
+
αn,Jxn+
∞
i=
αn,iJTinxn
≤ z– αn,z,Jxn–
∞
i= αn,i
z,JTn ixn
+αn,xn+
∞
i=
=αn,φ(z,xm) +
∞
i= αn,iφ
z,Tinxn
–αn,( –αn,i)gJxn–JTinxn
≤αn,φ(z,xm) +
∞
i=
αn,iφ(z,xm) +
∞
i=
αn,iμn,iφ(z,xn)
–αn,( –αn,i)gJxn–JTinxn
≤φ(z,xn) +
∞
i=
μn,iφ(z,xm) –αn,( –αn,i)gJxn–JTinxn
≤φ(z,xn) +
∞
i=
μn,iMn–αn,( –αn,i)gJxn–JTinxn.
Fromlim infn→∞αn,( –αn,i) > , we find thatlimn→∞g(Jxn–JTinxn) = . Hence, we
have
lim
n→∞Jxn–JT
n
ixn= . (.)
Sincexn→pasn→ ∞andJ:E→E∗is demicontinuous, we obtainJxnJp∈E∗. Note
that|Jxn–Jp|=|xn–p| ≤ xn–p. This implies thatJxn → Jpasn→ ∞.
SinceE∗has the Kadec-Klee property, we see that
lim
n→∞Jxn–Jp= . (.)
On the other hand, we haveJTn
ixn–Jp ≤ JTinxn–Jxn+Jxn–Jp. Combining (.) with
(.), one obtains thatlimn→∞JTinxn–Jp= . SinceJ–:E∗→Eis demicontinuous, one
sees thatTn
ixnp. Notice that
Tn
ixn–p=JTinxn–Jp≤JTinxn–Jp.
This yieldslimn→∞Tinxn=p. Since the spaceEhas the Kadec-Klee property, we
ob-tainlimn→∞Tn
ixn–p= . SinceTn+xn–p ≤ Tn+xn–Tnxn+Tnxn–pandTis
asymptotically regular, we find thatlimn→∞Tin+xn–p= . That is,TiTinxn–p→ as
n→ ∞. It follows from the closedness ofTithatTip=pfor everyi≥.
Finally, we prove thatp=∞
i=F(Ti)∩
j∈EF(Fj)x. Sincexn=Cnx, we see that
xn–z,Jx–Jxn ≥, ∀z∈Cn.
Since∞i=F(Ti)∩
j∈EP(Fj)⊂Cn, we find that
xn–w,Jx–Jxn ≥, ∀w∈ N
i=
F(Ti)∩
j∈ EF(Fj).
Lettingn→ ∞, we arrive atp–w,Jx–Jp ≥,∀w∈
∞
i=F(Ti)∩
j∈EP(Fj). In view
of Lemma ., we find thatp=∞
i=F(Ti)∩
j∈EF(Fj)x. This completes the proof.
Remark . Theorem . mainly improves the corresponding results in Kim [] and Qin
et al.[].
Corollary . Let E be a uniformly smooth and strictly convex Banach space which also has the Kadec-Klee property.Let C be a nonempty,closed,and convex subset of E and let
be an index set.Let Fjbe a bifunction from C×C toRsatisfying(A)-(A)for every
j∈.Let Ti:C→C a quasi-φ-nonexpansive mapping for every i≥.Assume that Tiis
closed on C and∞i=F(Ti)∩
j∈EP(Fj)is nonempty.Let{xn}be a sequence generated in
the following manner:
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈E, chosen arbitrarily,
C,j=C,
C=
j∈C,j,
x=Cx,
yn=J–(αn,Jxn+
∞
i=αn,iJTixn),
un,j∈C such that Fj(un,j,y) +rn,jy–un,j,Jun,j–Jyn ≥, ∀y∈C,
Cn+,j={z∈Cn:φ(z,un,j)≤φ(z,xn)},
Cn+=
j∈Cn+,j,
xn+=Cn+x,
where {αn,i} is a real number sequence in (, ) for every i≥, {rn,j} is a real
num-ber sequence in[r,∞),where r is some positive real number. Assume thatNi=αn,i=
andlim infn→∞αn,αn,i> for every i≥.Then the sequence{xn} converges strongly to
∞
i=F(Ti)∩
j∈EP(Fj)x,where
∞
i=F(Ti)∩
j∈EP(Fj)is the generalized projection from E onto
∞
i=F(Ti)∩
j∈EP(Fj).
Remark . Corollary . mainly improves the corresponding results in Qinet al.[]. In-deed, the space is extended from a uniformly smooth and uniformly convex Banach space to a uniformly smooth and strictly convex Banach space; the mapping is extended from a pari of mappings to infinite many mappings. We also remark here that the framework of the space in this paper can be applicable toLp,p≥.
Finally for a single mapping and bifunction, we have the following result.
Corollary . Let E be a uniformly smooth and strictly convex Banach space which also
has the Kadec-Klee property.Let C be a nonempty,closed,and convex subset of E and let F be a bifunction from C×C toRsatisfying(A)-(A).Let T:C→C an asymptotically quasi-φ-nonexpansive mapping.Assume that T is closed asymptotically regular on C and F(T)∩EP(F)is nonempty and bounded.Let{xn}be a sequence generated in the following
manner: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈E, chosen arbitrarily,
C=C,
x=Cx,
yn=J–(αnJxn+ ( –αn)JTnxn),
un∈C such that F(un,y) +rny–un,Jun–Jyn ≥, ∀y∈C,
Cn+={z∈Cn:φ(z,un)≤φ(z,xn) +μnMn},
where{αn}is a real number sequence in(, ),{rn}is a real number sequence in[r,∞),
where r is some positive real number and Mn:=sup{φ(z,xn) :z∈F(T)∩EP(F)}.Assume
lim infn→∞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).
4 Applications
First, we give the corresponding results in the framework of Hilbert spaces.
Theorem . Let E be a Hilbert space.Let C be a nonempty,closed,and convex subset of E and letbe an index set.Let Fjbe a bifunction from C×C toRsatisfying(A)-(A)
for every j∈.Let Ti:C→C an asymptotically quasi-nonexpansive mapping for every
i≥.Assume that Tiis closed asymptotically regular on C and
∞
i=F(Ti)∩
j∈EP(Fj)is
nonempty and bounded.Let{xn}be a sequence generated in the following manner: ⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈E, chosen arbitrarily,
C,j=C,
C=
j∈C,j,
x=ProjCx,
yn=αn,xn+
∞
i=αn,iTinxn,
un,j∈C such that Fj(un,j,y) +rn,jy–un,j,un,j–yn ≥, ∀y∈C,
Cn+,j={z∈Cn:z–un,j≤ z–xn+∞i=μn,iMn},
Cn+=
j∈Cn+,j,
xn+=ProjCn+x,
where{αn,i}is a real number sequence in(, )for every i≥,{rn,j}is a real number sequence
in[r,∞),where r is some positive real number and Mn:=sup{z–xn:z∈
∞
i=F(Ti)∩
j∈EP(Fj)}.Assume that
∞
i=αn,i= andlim infn→∞αn,αn,i> for every i≥.Then the
sequence{xn}converges strongly toProj∞i=F(Ti)∩
j∈EP(Fj)x,whereProj
∞
i=F(Ti)∩
j∈EP(Fj)
is the metric projection from E onto∞i=F(Ti)∩
j∈EP(Fj).
Proof Note thatφ(x,y) =x–y,J=I, the identity mapping, and the generalized
pro-jection is reduced to the metric propro-jection. In the framework of Hilbert spaces, the class of asymptotically quasi-φ-nonexpansive mappings is reduced to the class of asymptoti-cally quasi-nonexpansive mappings. Using Theorem ., we easily find the desired
con-clusion.
Next, we give a results on common solutions of a family of equilibrium problems.
Corollary . Let E be a Hilbert space.Let C be a nonempty,closed,and convex subset of E and letbe an index set.Let Fjbe a bifunction from C×C toRsatisfying(A)-(A)for
following manner:
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈E, chosen arbitrarily,
C,j=C,
C=
j∈C,j,
x=ProjCx,
un,j∈C such that Fj(un,j,y) +rn,jy–un,j,un,j–xn ≥, ∀y∈C,
Cn+,j={z∈Cn:z–un,j ≤ z–xn},
Cn+=
j∈Cn+,j,
xn+=ProjCn+x,
where{rn,j}is a real number sequence in[r,∞),where r is some positive real number.Then
the sequence{xn}converges strongly toProjj∈EP(Fj)x,whereProjj∈EP(Fj)is the metric
projection from E ontoj∈EP(Fj).
IfFj(x,y) = , then we find from Theorem . the following result.
Corollary . Let E be a Hilbert space and let C be a nonempty,closed,and convex subset of E.Let Ti:C→C an asymptotically quasi-nonexpansive mapping for every i≥.Assume
that Tiis closed asymptotically regular on C and
∞
i=F(Ti)is nonempty and bounded.Let
{xn}be a sequence generated in the following manner: ⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈E, chosen arbitrarily,
C=C,
x=ProjCx,
yn=αn,xn+
∞
i=αn,iTinxn,
Cn+={z∈Cn:z–yn≤ z–xn+
∞
i=μn,iMn},
xn+=ProjCn+x,
where{αn,i}is a real number sequence in(, )for every i≥,and Mn:=sup{z–xn:z∈
∞
i=F(Ti)}.Assume that
∞
i=αn,i= andlim infn→∞αn,αn,i> for every i≥.Then the
sequence{xn}converges strongly toProj∞i=F(Ti)x,whereProj∞i=F(Ti)is the metric
projec-tion from E onto∞i=F(Ti).
Remark . Comparing with Theorem . in Kim and Xu [], we have the following:
() one mapping is extended to an infinite family of mappings; () the setQnis relaxed;
() the mapping is extended from asymptotically nonexpansive mappings to asymptoti-cally quasi-nonexpansive mappings, which may not be continuous.
From Corollary ., the following result is not hard to derive.
Corollary . Let E be a Hilbert space.Let C be a nonempty,closed,and convex subset
gen-erated in the following manner:
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈E, chosen arbitrarily,
C=C, x=ProjCx,
yn=αnxn+ ( –αn)Txn,
Cn+={z∈Cn:z–yn ≤ z–xn},
xn+=ProjCn+x,
where{αn}is a real number sequence in(, )such thatlim infn→∞αn( –αn) > .Then the
sequence{xn}converges strongly toProjF(T)x,whereProjF(T)is the metric projection from
E onto F(T).
Remark . Corollary . is a shrinking version of the corresponding results in Nakajo
and Takahashi []. It deserves mentioning that the mapping in our result is quasi-nonexpansive. The restriction of the demiclosed principal is relaxed.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Both authors contributed equally to this manuscript. Both authors read and approved the final manuscript.
Author details
1School of Mathematics and Information Science, North China University of Water Resources and Electric Power,
Zhengzhou, 450011, China. 2College of Science, Hebei University of Engineering, Handan, 056038, China.
Acknowledgements
This paper is dedicated to Professor Chang with respect and admiration. The authors are grateful to the editor and the three anonymous reviewers for useful suggestions which improved the contents of the article.
Received: 13 March 2014 Accepted: 25 May 2014 Published:03 Jun 2014
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10.1186/1029-242X-2014-221
