R E S E A R C H
Open Access
A relaxed hybrid shrinking iteration approach
to solving generalized mixed equilibrium
problems for totally quasi-
φ
-asymptotically
nonexpansive mappings
Wei-Qi Deng
**Correspondence:
[email protected] College of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan 650221, P.R. China
Abstract
An optimal existing method for the approximation of common fixed points of countable families of nonlinear operators is introduced, by which a relaxed hybrid shrinking iterative algorithm is developed for the class of totally
quasi-
φ
-asymptotically nonexpansive mappings, and a strong convergence theorem for solving generalized mixed equilibrium problems is established in the framework of Banach spaces. Since there is no need to impose the uniformity assumption on the involved countable family of mappings and no need to compute a complex series at each step in the iteration process, the result is more widely applicable than those of other authors with related interests.MSC: 47J06; 47J25
Keywords: totally quasi-
φ
-asymptotically nonexpansive mappings; generalized mixed equilibrium problems; generalized projections1 Introduction
Throughout this paper we assume thatEis a real Banach space with its dualE∗,Cis a nonempty closed convex subset ofEandJ:E→E∗is thenormalized duality mapping
defined by
Jx=f∈E∗:x,f=x=f, ∀x∈E.
In the sequel, we useF(T) to denote the set of fixed points of a mappingT.
Definition .[] () A mappingT:C→Cis said to betotally quasi-φ-asymptotically nonexpansive, ifF(T)=∅ and there exist nonnegative real sequences {νn}, {μn} with
νn,μn→ (asn→ ∞) and a strictly increasing continuous functionζ :R+∪ {} →
R+∪ {}withζ() = such that
φp,Tnx≤φ(p,x) +νnζ
φ(p,x)+μn, ∀n≥,x∈C,p∈F(T), (.)
whereφ:E×E→R+∪ {}denotes theLyapunov functionaldefined by
φ(x,y) =x– x,Jy+y, ∀x,y∈E. (.)
It is obvious from the definition ofφthat
x–y≤φ(x,y)≤x+y, ∀x,y∈E (.)
and
φx,J–λJy+ ( –λ)Jz≤λφ(x,y) + ( –λ)φ(x,z), ∀x,y∈E,λ∈[, ]. (.)
() A countable family of mappings {Ti}∞i= : C→C said to be uniformly quasi-φ -asymptotically nonexpansive, ifF:=∞i=F(Ti)=∅ and there exists a nonnegative real
sequence{kn} ⊂[,∞) withkn→ (asn→ ∞) such that
φp,Tn ix
≤knφ(p,x), ∀n≥,i≥,x∈C,p∈F(T). (.)
() A countable family of mappings{Ti}∞i=:C→Csaid to beuniformly totally quasi -φ-asymptotically nonexpansive, ifF:=∞i=F(Ti)=∅and there exist nonnegative real
se-quences{νn},{μn}withνn,μn→ (asn→ ∞) and a strictly increasing continuous
func-tionζ:R+∪ {} →R+∪ {}withζ() = such that
φp,Tinx≤φ(p,x) +νnζ
φ(p,x)+μn, ∀n≥,i≥,x∈C,p∈F(T). (.)
() A mappingT:C→Cis said to beuniformly L-Lipschitz continuous, if there exists a constantL> such that
Tnx–Tny≤Lx–y, ∀n≥,x,y∈C. (.)
Letθ:C×C→Rbe a bifunction,ψ:C→Ra real valued function andA:C→E∗ a nonlinear mapping. The so-calledgeneralized mixed equilibrium problem(GMEP) is to find anu∈Csuch that
θ(u,y) +Au,y–u+ψ(y) –ψ(u)≥, ∀y∈C, (.)
In , Qinet al.[] proposed the following shrinking projection method to find a common element of the set of solutions of an equilibrium problem and the set of com-mon fixed points of a finite family of quasi-φ-nonexpansive mappings in the framework of Banach spaces:
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈C; C=C, yn=J–[αn,Jxn+
N
i=αn,iJTixn],
un∈C such that∀y∈C,
θ(un,y) +rny–un,Jun–Jyn ≥,
Cn+={v∈Cn:φ(v,un)≤φ(v,xn)},
xn+=Cn+x, ∀n≥,
(.)
whereCn+is thegeneralized projection(see (.)) ofEontoCn+.
In , Saewan and Kumam [] introduced a modified new hybrid projection method to find a common element of the set of solutions of the generalized mixed equilibrium problems and the set of common fixed points of an infinite family of closed anduniformly quasi-φ-asymptotically nonexpansive mappingsin an uniformly smooth and strictly con-vex Banach spacesEwith the Kadec-Klee property:
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈C; C=C,
yn=J–[αnJxn+ ( –αn)Jzn],
zn=J–[βn,Jxn+ ∞
i=βn,iJTinxn],
un∈C such thatun=Krnyn,
Cn+={z∈Cn:φ(z,un)≤φ(z,xn) +ξn},
xn+=Cn+x, ∀n≥,
(.)
whereξn:= (kn– )supp∈Fζ(φ(p,xn)).
However, it is obviously a quite strong condition that the involved mappings are assumed to be a countable family of uniformly({νn},{μn},ζ)-quasi-φ-asymptotically nonexpansive
ones,which is a special case of totally quasi-φ-asymptotically nonexpansive mappings(see []).In addition,the accurate computation of the series∞i=βn,iJTinynat each step of the
iteration process is not easily attainable,which will lead to gradually increasing errors. Inspired and motivated by the studies mentioned above, by using a special way of choos-ing the indices, we propose a relaxed hybrid shrinkchoos-ing iteration scheme for approximatchoos-ing common fixed points of a countable family of totally quasi-φ-asymptotically nonexpan-sive mappings and obtain a strong convergence theorem for solving the generalized mixed equilibrium problems under suitable conditions, namely,there is no need to assume uni-formity for the totally quasi-φ-asymptotic property of the involved mappings,and no need to compute complex series in the iteration process. The results extend and improve those of other authors with related interests.
2 Preliminaries
We say that a Banach spaceEisstrictly convexif the following implication holds forx,y∈E:
x=y= , x=y ⇒ x+y
It is also said to beuniformly convexif for any> , there exists aδ> such that
x=y= , x–y ≥ ⇒ x+y
≤ –δ. (.)
It is well known that ifEis a uniformly convex Banach space, thenEis reflexive and strictly convex. A Banach spaceEis said to besmoothif
lim t→
x+ty–x
t (.)
exists for eachx,y∈S(E) :={x∈E:x= }.Eis said to beuniformly smoothif the limit (.) is attained uniformly forx,y∈S(E).
Following Alber [], thegeneralized projectionC:E→Cis defined by
C=arg inf
y∈Cφ(y,x), ∀x∈E. (.)
Lemma .[] Let E be a smooth,strictly convex and reflexive Banach space and C be a nonempty closed convex subset of E.Then the following conclusions hold:
() φ(x,Cy) +φ(Cy,y)≤φ(x,y)for allx∈Candy∈E; () Ifx∈Eandz∈C,thenz=Cx⇔ z–y,Jx–Jz ≥,∀y∈C; () Forx,y∈E,φ(x,y) = if and only ifx=y.
Remark . The following basic properties for a Banach space E can be found in Cioranescu [].
(i) IfEis uniformly smooth, thenJis uniformly continuous on each bounded subset ofE;
(ii) IfEis reflexive and strictly convex, thenJ–is norm-weak-continuous;
(iii) IfEis reflexive smooth and strictly convex, then the normalized duality mappingJ
is single-valued, one-to-one and onto;
(iv) A Banach spaceEis uniformly smooth if and only ifE∗is uniformly convex; (v) Each uniformly convex Banach spaceEhas theKadec-Klee property,i.e., for any
sequence{xn} ⊂E, ifxnx∈Eandxn → x, thenxn→xasn→ ∞.
Lemma .[] Let E be a real uniformly smooth and strictly convex Banach space with Kadec-Klee property,and C be a nonempty closed convex subset of E.Let{xn}and{yn}be
two sequences in C such that xn→p andφ(xn,yn)→,whereφis the function defined by
(.),then yn→p.
Lemma .[] Let E and C be the same as in Lemma..Let T:C→C be a closed and totally quasi-φ-asymptotically nonexpansive mappings with nonnegative real sequences {νn},{μn}and a strictly increasing continuous functionζ:R+∪ {} →R+∪ {}such that
νn,μn→andζ() = .Ifμ= ,then the fixed point set F(T)of T is a closed and convex subset of C.
Lemma . [] Let E be a real uniformly convex Banach space and let Br() be the
closed ball of E with center at the origin and radius r> .Then for any for any sequence {xi} ⊂Br()and for any sequence{λi}of positive numbers with
∞
continuous strictly increasing convex function g: [,∞)→[,∞)with g() = such that such that for any positive integer i= ,the following hold:
∞
i= λixi
≤
∞
i=
λixi–λλig
x–xi
, (.)
and for all x∈E,
φ
x,J–
∞
i= λiJxi
≤
∞
i=
λiφ(x,xi) –λλig
Jx–Jxi
. (.)
Assume that, to obtain the solution of GMEP, the functionψ:C→Ris convex and
lower semicontinuous, the nonlinear mappingA:C→E∗is continuous and monotone,
and the bifunctionθ:C×C→Rsatisfies the following conditions:
(A) θ(x,x) = ;
(A) θ is monotone,i.e.,θ(x,y) +θ(y,x)≤;
(A) lim supt↓θ(x+t(z–x),y)≤θ(x,y);
(A) the mappingy→θ(x,y)is convex and lower semicontinuous.
Lemma . [] Let E be a smooth,strictly convex,and reflexive Banach space,and C be a nonempty closed convex subset of E.Let A:C→E∗be a continuous and monotone mapping,ψ :C→Ra lower semicontinuous and convex function,andθ :C×C→R a bifunction satisfying the conditions(A)-(A).Let r> and x∈E.Then,the following hold:
() There exists anu∈Csuch that
θ(u,y) +Au,y–u+ψ(y) –ψ(u) +
ry–u,Ju–Jx ≥, ∀y∈C.
() A mappingκr:C→Cis defined by
κr(x) =
u∈C:θ(u,y) +Au,y–u+ψ(y) –ψ(u) +
ry–u,Ju–Jx ≥,∀y∈C
.
Then,the mappingκrhas the following properties: (i) κris single-valued;
(ii) κra firmly nonexpansive-type mapping,i.e.,
κrz–κry,Jκrz–Jκry ≤ κrz–κry,Jz–Jy;
(iii) F(κr) ==F˜(κr);
(iv) is a closed convex set ofC;
(v) φ(p,κrz) +φ(κrz,z)≤φ(p,z),∀p∈F(κr),z∈E,
whereF˜(κr)denotes the set of asymptotic fixed points ofκr,i.e.,
˜ F(κr) :=
x∈C:∃{xn} ⊂C,s.t.,xnx,xn–κrxn → (n→ ∞)
Lemma .[] The unique solutions to the positive integer equation
n=i+(m– )m
, m≥i,n= , , . . . , (.)
are
i=n–(m– )m
, m= –
–
n+
, n= , , . . . , (.)
where[x]denotes the maximal integer that is not larger than x.
3 Main results
Recall that a mappingT on a Banach space is closed ifxn→xandTxn→yasn→ ∞,
thenTx=y.
Theorem . Let E be a real uniformly smooth and strictly convex Banach space with Kadec-Klee property and C a nonempty closed convex subset of E.Letθ :C×C→Rbe a bifunction satisfying the conditions(A)-(A),A:C→E∗ a continuous and monotone mapping,andψ:C→Ra lower semicontinuous and convex function.Let{Ti}∞i=:C→C be a countable family of closed and totally quasi-φ-asymptotically nonexpansive mappings with nonnegative real sequences{νn(i)},{μ(ni)}satisfyingνn(i)→andμ(ni)→ (as n→ ∞
and for each i≥)and a sequence of strictly increasing and continuous functions{ζi}:
R+∪{} →R+∪{}satisfying condition(.).Assume that each T
iis uniformly Li-Lipschitz
continuous withμ(i)= for each i≥.Let{αn}be a sequence in[,]for some∈(, )
and{βi}be a sequence in(, ).Let{xn}be the sequence generated by
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈C; C=C,
yn=J–[αnJxn+ ( –αn)Jzn],
zn=J–[βinJxn+ ( –βin)JTinmnxn],
un∈C such that∀y∈C,
θ(un,y) +Aun,y–un+ψ(y) –ψ(un) +rny–un,Jun–Jyn ≥,
Cn+={v∈Cn:φ(v,un)≤φ(v,xn) +ξn},
xn+=Cn+x, n∈N,
(.)
whereξn:=νmn(in)supp∈Fζin(φ(p,xn)) +μ(mnin)andCn+is the generalized projection of E onto Cn+;inand mn are the solutions to the positive integer equation:n=i+(m–) m (m≥i,
n= , , . . .),that is,for each n≥,there exist unique inand mnsuch that
i= , i= , i= , i= ,
i= , i= , i= , i= , . . . ;
m= , m= , m= , m= ,
m= , m= , m= , m= , . . . .
Proof Two functionsτ:C×C→Randκr:C→Care defined by
τ(x,y) =θ(x,y) +Ax,y–x+ψ(y) –ψ(x);
κr(x) =
u∈C:τ(u,y) +
ry–u,Ju–Jx ≥,∀y∈C
.
By Lemma ., we know that the functionτ satisfies the conditions (A)-(A) andκrhas
the property (i)-(v). Therefore, (.) can be rewritten as
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈C; C=C,
yn=J–[αnJxn+ ( –αn)Jzn],
zn=J–[βinJxn+ ( –βin)JTinmnxn],
un∈C such thatτ(un,y) +rny–un,Jun–Jyn ≥,∀y∈C,
Cn+={v∈Cn:φ(v,un)≤φ(v,xn) +ξn},
xn+=Cn+x, ∀n≥.
(.)
We divide the proof into several steps.
(I)FandCn(∀n≥) both are closed and convex subsets inC.
In fact, it follows from Lemma . that eachF(Ti) is a closed and convex subset ofC, so
isF. In addition, withC(=C) being closed and convex, we may assume thatCnis closed
and convex for somen≥. In view of the definition ofφwe have
Cn+=
v∈C:ϕ(v)≤a∩Cn,
whereϕ(v) = v,Jxn–Jynanda=xn–yn+ξn. This shows thatCn+is closed and
convex.
(II)Gis a subset of∞n=Cn.
It is obvious thatG⊂C. Suppose thatG⊂Cnfor somen≥. Sinceun=κrnyn, by
Lemma ., it is easily shown thatκrnis quasi-φ-nonexpansive. Hence, for anyp∈G⊂Cn,
it follows from (.) that
φ(p,un) =φ(p,κrnyn)≤φ(p,yn)
=φp,J–αnJxn+ ( –αn)Jzn
≤αnφ(p,xn) + ( –αn)φ(p,zn). (.)
Furthermore, it follows from Lemma . that for anyp∈G⊂Cn,
φ(p,zn) =φ
p,J–βinJxn+ ( –βin)JTinmnxn
≤βinφ(p,xn) + ( –βin)φ
p,Tinmnxn
–βin( –βin)gJxn–JTinmnxn
≤βinφ(p,xn) + ( –βin)
φ(p,xn) +νmn(in)ζin
φ(p,xn)
+μ(mnin)
–βin( –βin)gJxn–JTinmnxn
≤φ(p,xn) +νmn(in)sup p∈F
ζin
φ(p,xn)
+μ(mnin)–βin( –βin)gJxn–JTinmnxn
Substituting (.) into (.) and simplifying it, we have
φ(p,un)≤φ(p,yn)≤φ(p,xn) + ( –αn)ξn– ( –αn)βin( –βin)gJxn–JTinmnxn
≤φ(p,xn) +ξn– ( –αn)βin( –βin)gJxn–JTinmnxn
≤φ(p,xn) +ξn. (.)
This implies thatp∈Cn+, and soG⊂Cn+.
(III)xn→x∗∈Casn→ ∞.
In fact, sincexn=Cnx, from Lemma .() we havexn–y,Jx–Jxn ≥,∀y∈Cn.
Again since F⊂∞n=Cn, we havexn–p,Jx–Jxn ≥,∀p∈F. It follows from
Lem-ma .() that for eachp∈Fand for eachn≥,
φ(xn,x) =φ(Cnx,x)≤φ(p,x) –φ(p,xn)≤φ(p,x),
which implies that{φ(xn,x)}is bounded, so is{xn}. Since for alln≥,xn=Cnx and xn+=Cn+x∈Cn+⊂Cn, we haveφ(xn,x)≤φ(xn+,x). This implies that{φ(xn,x)}is nondecreasing, hence the limit
lim
n→∞φ(xn,x) exists.
SinceE is reflexive, there exists a subsequence {xni} of{xn}such thatxnix∗∈Cas
i→ ∞. SinceCnis closed and convex andCn+⊂Cn, this implies thatCnis weakly closed
andx∗∈Cnfor eachn≥. In view ofxni=Cnix, we have
φ(xni,x)≤φ
x∗,x
, ∀i≥.
Since the norm · is weakly lower semicontinuous, we have
lim inf
i→∞ φ(xni,x) =lim infi→∞
xni– xni,Jx+x
≥x∗– x∗,Jx
+x
=φx∗,x
,
and so
φx∗,x
≤lim inf
i→∞ φ(xni,x)≤lim supi→∞ φ(xni,x)≤φ
x∗,x
.
This implies thatlimi→∞φ(xni,x) =φ(x∗,x), and soxni → x∗asi→ ∞. Sincexni
x∗, by virtue of theKadec-Klee propertyofE, we obtain
lim i→∞xni=x
∗.
Since{φ(xn,x)}is convergent, this, together withlimi→∞φ(xni,x) =φ(x∗,x), shows that
yasj→ ∞, then from Lemma .() we have
φx∗,y= lim
i,j→∞φ(xni,xnj) =i,limj→∞φ(xni,Cnjx)
≤ lim i,j→∞
φ(xni,x) –φ(Cnjx,x)
= lim
i,j→∞
φ(xni,x) –φ(xnj,x)
=φx∗,x
–φx∗,x
= ,
that is,x∗=yand so
lim n→∞xn=x
∗. (.)
(IV)x∗is a member ofF.
Set Ki={k∈N:k=i+ (m–) m,m≥i,m∈N} for each i∈N. Note that ν
(ik) mk =ν
(i)
mk,
μ(mkik)=μ(mki) andζik=ζiwheneverk∈Kifor eachi≥. For example, by Lemma . and the
definition ofK, we haveK={, , , , , , . . .}andi=i=i=i=i=i=· · ·= .
Then we have
ξk=νmk(i) sup p∈F
ζi
φ(p,xk)
+μ(mki) , ∀k∈Ki. (.)
Note that{mk}k∈Ki={i,i+ ,i+ , . . .},i.e.,mk↑ ∞asKik→ ∞. It follows from (.)
and (.) that
lim Kik→∞
ξk= , ∀i≥. (.)
Sincexn+∈Cn+, it follows from (.), (.), and (.) that
φ(xk+,yk)≤φ(xk+,xk) +ξk→ (Kik→ ∞). (.)
Sincexk→x∗asKik→ ∞, it follows from (.) and Lemma . that lim
Kik→∞
yk=x∗, ∀i≥. (.)
Note thatTikmk=Timkandβik=βiwheneverk∈Kifor eachi≥. From (.), for anyp∈F,
we have
φ(p,yk)≤φ(p,xk) +ξk– ( –αk)βi( –βi)gJxk–JTimkxk, ∀k∈Ki,
that is,
( –αk)βi( –βi)gJxk–JTimkxk≤φ(p,xk) +ξk–φ(p,yk)→ (Kik→ ∞).
This shows thatlimKik→∞g(Jxk–JTimkxk) = . In view of the property ofg, we have lim
Kik→∞
In addition, Jxk →Jx∗ (Ki k→ ∞) implies that limKik→∞JT mk
i xk =Jx∗. From
Re-mark .(ii) it yields that, asKik→ ∞,
Tmk
i xkx∗, ∀i≥. (.)
Again since for eachi≥, asKik→ ∞,
Timkxk–x∗=JTimkxk–Jx∗≤JTimkxk–Jx∗→,
this, together with (.) and theKadec-Klee propertyofE, shows that
lim Kik→∞
Timkxk=x∗, ∀i≥. (.)
We use the assumptions that for eachi≥,Tiis uniformlyLi-Lipschitz continuous.
Not-ing again that{mk}k∈Ki={i,i+ ,i+ , . . .},i.e.,mk+– =mkfor allk∈Ki, we then have Tmk+
i xk–Timkxk≤Timk+xk–Timk+xk++Timk+xk+–xk+
+xk+–xk+xk–Timkxk
≤(Li+ )xk+–xk+Timk+xk+–xk+
+xk–Timkxk. (.)
From (.) andxk→x∗ asKik→ ∞we havelimKik→∞T mk+
i xk–Timkxk= and limKik→∞Tmk+
i xk=x∗,i.e.,limKik→∞Ti(T mk+–
i xk) =x∗. It then follows that, for each
i≥,
lim Kik→∞
Ti
Timkxk
=x∗, ∀i≥. (.)
In view of the closedness ofTi, it follows from (.) thatTix∗=x∗,i.e., for eachi≥,
x∗∈F(Ti) and hencex∗∈F.
(V)x∗is also a member ofG.
Sincexn+=Cn+x, it follows from (.) and (.) that
φ(xk+,uk)≤φ(xk+,xk) +ξk→ (Kik→ ∞).
Sincexk→x∗asKik→ ∞, by virtue of Lemma . we have
lim Kik→∞
uk=x∗, ∀i≥. (.)
This, together with (.), shows that limKik→∞uk–yk= and limKik→∞Juk –
Jyk= . By the assumption that{rk}k∈Ki⊂[a,∞) for somea> , we have
lim Kik→∞
Juk–Jyk
rk
Sinceτ(uk,y) +r
ky–uk,Juk–Jyk ≥,∀y∈C, by condition (A), we have
rk
y–uk,Juk–Jyk ≥–τ(uk,y)≥τ(y,uk), ∀y∈C. (.)
By the assumption that the mappingy→τ(x,y) is convex and lower semicontinuous, let-tingKik→ ∞in (.), from (.) and (.), we haveτ(y,x∗)≤,∀y∈C. For any
t∈(, ] and anyy∈C, setyt=ty+ ( –t)x∗. Thenτ(yt,x∗)≤ sinceyt∈C. By conditions
(A) and (A), we have
=τ(yt,yt)≤tτ(yt,y) + ( –t)τ
yt,x∗
≤tτ(yt,y).
Dividing both sides of the above equation byt, we haveτ(yt,y)≥,∀y∈C. Lettingt↓,
from condition (A), we haveτ(x∗,y)≥,∀y∈C,i.e.,x∗∈and sox∗∈G.
(VI)x∗=Gx, and soxn→Gxasn→ ∞.
Putu=Gx. Sinceu∈G⊂Cnandxn=Cnx, we haveφ(xn,x)≤φ(u,x),∀n≥.
Then
φx∗,x
= lim
n→∞φ(xn,x)≤φ(u,x), (.)
which implies thatx∗ =u sinceu=Gx, and hence xn→x∗=Fx asn→ ∞. This
completes the proof.
We now provide a nontrivial family of mappings satisfying the conditions of Theo-rem ..
Example . LetE=Rwith the standard norm·=|·|andC= [–, ]. Let{T
i}:C→C
be a countable family of mappings defined by
Tix=
–x
i+, x∈(, ], x, x∈[–, ].
We first show that{Ti}is uniformlyL-Lipschitzian. Ifx∈(, ] andy∈[–, ], then
Tinx–Tiny= (–)
n
(i+ )nx–y
≤ |x|+|y| ≤|x–y|.
The rest is trivial. Second, we claim that{Ti}is a family of closed and totally quasi-φ
-asymptotically nonexpansive mappings. In fact, for any x∈(, ] and p∈∞i=F(Ti) =
[–, ], we have, for alln≥,
φp,Tinx=Tinx–p= (–)
n
(i+ )nx–p
≤
|p|+ (i+ )n|x|
≤
|x–p|+ (i+ )n
whereφ(x,p) =|x–p|,ν(i)
n = (i+) n,μ
(i)
n =(i+) n andζ(x) =
√
x. Note that|Tix–p|≤
|x–p|, that is,μ(i)= for eachi≥.
Next, we define a bifunctionθ:C×C→Rsatisfying the conditions (A)-(A) by
θ(x,y) =y–x.
LetA= andψ= . Then the set of solutionsto the generalized mixed equilibrium problem forθ,Aandψis obviously{}. SinceG:=∩F=∅andFis bounded, it follows from Theorem . that the sequence{xn}defined by (.) converges strongly toGx.
Competing interests
The author declares that they have no competing interests.
Acknowledgements
The author is very grateful to the referees for their useful suggestions, by which the contents of this article has been improved. This study is supported by the National Natural Science Foundation of China (Grant No. 11061037) and the General Project of Scientific Research Foundation of Yunnan University of Finance and Economics (YC2013A02).
Received: 20 August 2013 Accepted: 28 February 2014 Published:14 Mar 2014
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