R E S E A R C H
Open Access
Iterative algorithms based on hybrid method
and Cesàro mean of asymptotically
nonexpansive mappings for equilibrium
problems
Jingxin Zhang
1*and Yunan Cui
2*Correspondence:
[email protected] 1Mathematics and Applied
Mathematics, Harbin University of Commerce, Harbin, 150028, P.R. China
Full list of author information is available at the end of the article
Abstract
Using Cesàro means of a mapping, we modify the progress of Mann’s iteration in hybrid method for asymptotically nonexpansive mappings in Hilbert spaces. Under suitable conditions, we prove that the iterative sequence converges strongly to a fixed point of an asymptotically nonexpansive mapping. We also introduce a new hybrid iterative scheme for finding a common element of the set of common fixed points of asymptotically nonexpansive mappings and the set of solutions of an equilibrium problem in Hilbert spaces.
MSC: 47H10; 47J25; 90C33
Keywords: strong convergence theorem; asymptotically nonexpansive mapping; hybrid method; equilibrium problem; Cesàro means
1 Introduction
LetH be a real Hilbert space with the inner product·,·and the norm · . LetCbe a nonempty closed convex subset of H. A mappingT :C→Cis said to be asymptoti-cally nonexpansive if for eachn≥, there exists a nonnegative real numberknsatisfying
limn→∞kn= such that
Tnx–Tny≤knx–y, ∀x,y∈C;
whenkn≡,T is called nonexpansive.
The concept of asymptotically nonexpansive mapping was introduced by Goebel and Kirk [] in . We denote byF(T) :={x∈C:Tx=x}the set of fixed points ofT. It is well known that ifT:H→His asymptotically nonexpansive, thenF(T) is nonempty convex.
In , Mann [] introduced the iteration as follows: a sequence{xn}defined by
xn+=αnxn+ ( –αn)Txn. (.)
In an infinite-dimensional Hilbert space, Mann iteration could conclude only weak con-vergence []. Attempts to modify the Mann iteration method (.) so that strong conver-gence is guaranteed have recently been made. Nakajo and Takahashi [] proposed the fol-lowing modification of Mann iteration method for a nonexpansive mappingTin a Hilbert
space: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈C is arbitrary,
yn=αnxn+ ( –αn)Txn,
Cn={z∈C:yn–z ≤ xn–z},
Qn={z∈C:xn–z,x–xn ≥},
xn+=PCn∩Qn(x), n= , , . . . ,
(.)
wherePKdenotes the metric projection fromHonto a closed convex subsetKofH. The
above method is also called CQ method or hybrid method.
In , Kim and Xu [] adapted the iteration (.) in a Hilbert space. More precisely, they introduced the following iteration process for asymptotically nonexpansive map-pings: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈C is arbitrary,
yn=αnxn+ ( –αn)Tnxn,
Cn={z∈C:yn–z≤ xn–z+θn},
Qn={z∈C:xn–z,x–xn ≥},
xn+=PCn∩Qn(x), n= , , . . . ,
(.)
where
θn= ( –αn)
kn– (diamC)→ asn→ ∞.
They proved that{xn}converges in norm toPF(T)xunder some conditions. Several au-thors (see [, ]) have studied the convergence of hybrid method.
Baillon [] first proved that the following Cesàro mean iterative sequence weakly con-verges to a fixed point of a nonexpansive mapping in Hilbert spaces:
Tnx=
n+
n i=
Tix.
Shimizu and Takahashi [] proved a strong convergence theorem of the above iteration for an asymptotically nonexpansive mapping in Hilbert spaces.
LetCbe a nonempty closed convex subset of a real Hilbert spaceH, letf :C×C→R be a functional, whereRis the set of real numbers. The equilibrium problem is to find
ˆ
x∈Csuch that
f(x,y)≥, ∀y∈C. (.)
The set of solutions of (.) is denoted byEP(f). Given a mappingT:C→X*, letf(x,y) =
Tx,y–x,∀x,y∈C, thenz∈EP(f) if and only ifTz,y–z ≥,∀y∈C,i.e.,zis the solution of the variational inequality.
equilibrium problem. So, equilibrium problems provide us with a systematic framework to study a wide class of problems arising in financial economics, optimization and oper-ation researchetc., which motivates the extensive concern. See, for example, [–]. In recent years, equilibrium problems have been deeply and thoroughly researched. See, for example, [–]. Some methods have been proposed to solve the equilibrium problem in a Hilbert space; see, for instance, [–]. In , Jitpeera, Katchang, and Kumam [] found a common element of the set of solutions for mixed equilibrium problem, the set of solutions of the variational inequality for aβ-inverse strongly monotone mapping, and the set of fixed points of a family of finitely nonexpansive mappings in a real Hilbert space by using the viscosity and Cesàro mean approximation method.
Motivated by the above-mentioned results, in this paper we introduce the following iteration process for asymptotically nonexpansive mappings T withC a closed convex bounded subset of a real Hilbert space:
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈C is arbitrary,
yn=αnxn+ ( –αn)n+
n
j=Tjxn,
Cn={z∈C:yn–z≤ xn–z+θn},
Qn={z∈C:xn–z,x–xn ≥},
xn+=PCn∩Qn(x), n= , , . . . ,
(.)
where
θn= ( –αn)
Ln– (diamC)→ asn→ ∞.
We shall prove that the above iterative sequence{xn}converges strongly to a fixed point
ofTunder some proper conditions. In addition, we also introduce a new hybrid iterative scheme for finding a common element of the set of common fixed points of asymptoti-cally nonexpansive mappings and the set of solutions of an equilibrium problem in Hilbert spaces.
We will use the notationfor weak convergence and→for strong convergence.
2 Preliminaries
LetHbe a real Hilbert space. Then
x–y=x–y– x–y,y (.)
and
λx+ ( –λ)y=λx+ ( –λ)y–λ( –λ)x–y (.)
for allx,y∈Handλ∈[, ]. It is also known thatHsatisfies
() Opial’s condition, that is, for any sequence{xn}withxnx, the inequality
lim inf
n→∞ xn–x<lim infn→∞ xn–y
() The Kadec-Klee property, that is, for any sequence{xn}withxnxand
xn → xtogether impliesxn–x →.
LetC be a nonempty closed convex subset ofH. Then, for anyx∈H, there exists a unique nearest point inC, denoted byPC(x), such that
x–PC(x)≤ x–y, ∀y∈C.
Such a mappingPCis called the metric projection ofHontoC. We know thatPCis
non-expansive. Furthermore, forx∈Handz∈C,
z=PC(x) if and only if x–z,z–y ≥, ∀y∈C.
We also need the following lemmas.
Lemma .(See []) Let T be an asymptotically nonexpansive mapping defined on a bounded closed convex subset C of a Hilbert space H.Assume that{xn}is a sequence in C
with the properties: (i)xnz; (ii)Txn–xn→.Then z∈F(T).
Lemma .(See []) Let C be a nonempty bounded subset of a Hilbert space.Let T be an asymptotically nonexpansive mapping from C into itself such that F(T)is nonempty. Then,for anyε> ,there exists a positive integer lεsuch that for any integer l≥lε,there is a positive integer nlsatisfying
n+
n j=
Tjx–Tl
n+
n j=
Tjx
<ε, ∀x∈C,∀n≥nl.
The equilibrium problem is to findxˆ∈Csuch that
f(xˆ,y)≥, ∀y∈C.
The set of solutions of the above inequality is denoted byEP(f). For solving the equilibrium problem, let us assume that a bifunctionf satisfies the following conditions:
(A) f(x,x) = for allx∈C;
(A) f is monotone,i.e.,f(x,y) +f(y,x)≤, for allx,y∈C; (A) for eachx,y,z∈C,
lim sup
t→+ f
tz+ ( –t)x,y≤f(x,y);
(A) for eachx∈C,f(x,·)is convex and lower semi-continuous.
The following lemma appears implicitly in [].
Lemma .([]) Let C be a nonempty closed convex subset of H,let f be a bifunction of C×C intoRsatisfying(A)-(A),and let r> and x∈H.Then there exists z∈C such that
f(z,y) +
ry–z,Jz–Jx ≥, ∀y∈C.
Lemma .([]) Assume that f :C×C→Rsatisfies(A)-(A).For r> and x∈H, define a mapping Tr:X→C as follows:
Tr(x) =
z∈C:f(z,y) +
ry–z,Jz–Jx ≥,∀y∈C
for all z∈H.Then the following hold:
() Tris single-valued;
() Tris firmly nonexpansive,i.e.,
Trx–Try≤ Trx–Try,x–y, ∀x,y∈H;
() F(Tr) =EP(f);
() EP(f)is closed and convex.
Lemma .([]) Let C be a nonempty closed convex subset of H,let f :C×C→Rbe a functional,satisfying(A)-(A),and let r> .Then,for x∈H and q∈F(Tr),
q–Trx+Trx–x≤ q–x.
3 Strong convergence theorem of modified Mann iteration based on hybrid method
Inspired by Kim and Xu’s results (see []), Mann-type iteration (.) is modified to obtain the strong convergence theorem as follows.
Theorem . Let C be a nonempty bounded closed convex subset of a real Hilbert space H and let T :C→C be an asymptotically nonexpansive mapping with kn,denote Ln=
n+
n
j=kj.Assume that{αn}∞n=is a sequence in(, )such thatαn≤a for all n and for
some <a< .Define a sequence{xn}∞n=in C by the following algorithm:
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈C is arbitrary,
yn=αnxn+ ( –αn)n+
n
j=Tjxn,
Cn={z∈C:yn–z≤ xn–z+θn},
Qn={z∈C:xn–z,x–xn ≥},
xn+=PCn∩Qn(x), n= , , . . . ,
(.)
where
θn= ( –αn)
Ln– (diamC)→ as n→ ∞.
Then{xn}converges strongly to PF(T)(x).
Proof First note thatThas a fixed point inC(see []); that is,F(T) is nonempty. We divide the proof of this theorem into four steps as below.
From the definition ofCnandQn, it is obvious thatCnis closed andQnis closed and
convex for eachn∈N∪ {}. We prove thatCnis convex. Sinceyn–z≤ xn–z+θn
is equivalent to
xn–yn,z ≤ xn–yn+θn,
it follows thatCnis convex.
Step. We show thatF(T)⊂Cn,∀n∈N∪ {}.
Letp∈F(T) andn∈N∪ {}. Then from
yn–p =
αnxn+ ( –αn)
n+
n j=
Tjxn–p
≤αnxn–p+ ( –αn)
n+
n j=
Tjxn–p
≤αnxn–p+ ( –αn)
n+
n j=
kjxn–p
≤αnxn–p+ ( –αn)Lnxn–p
=xn–p+ ( –αn)
Ln– xn–p
≤ xn–p+θn,
we havep∈Cn. Next, we show thatF(T)⊂Qn,∀n∈N∪ {}. We prove this by induction.
Forn= , we haveF(T)⊂C=Q. Suppose thatF(T)⊂Qn, then∅ =F(T)⊂Cn∩Qnand
there exists a unique elementxn+∈Cn∩Qnsuch thatxn+=PCn∩Qn(x). Then xn+–z,x–xn+ ≥, ∀z∈Cn∩Qn.
In particular,
xn+–p,x–xn+ ≥, ∀p∈F(T).
It follows thatF(T)⊂Qn+and henceF(T)⊂Qnfor eachn. Thus we obtainF(T)⊂Cn∩
Qn,∀n∈N∪ {}. This means that{xn}is well defined.
Step. We show that{xn}is bounded andlimn→∞xn–Txn= .
It follows from the definition ofQnthatxn=PQn(x). Therefore
xn–x ≤ z–x for allz∈Qnand alln∈N∪ {}.
Letz∈F(T)⊂Qnfor alln∈N∪ {}. Then
xn–x ≤ z–x for alln∈N∪ {}.
Therefore, the sequence {xn–x}is nondecreasing. Since C is bounded, we obtain that limn→∞xn–x exists. This implies that {xn} is bounded. Noticing again that
xn+=PCn∩Qn(x)∈Qnandxn=PQn(x), we havexn+–xn,xn–x ≥. It follows from
(.) that
xn+–xn=(xn+–x) – (xn–x)
=xn+–x–xn–x– xn+–xn,xn–x
≤ xn+–x–xn–x
for alln∈N∪ {}. This implies that
lim
n→∞xn+–xn= .
Since xn+ = PCn∩Qn ∈ Cn, we have yn – xn+ ≤ xn –xn+ +θn. Noticing that
limn→∞kn= , thenLn=n+
n
j=kj→ asn→ ∞. Hence
yn–xn+ ≤ xn–xn++
θn→.
LetAn=n+
n
j=Tj. Also sinceαn≤a< for alln, then
xn–Anxn=
–αn
yn–xn
≤
–a
yn–xn++xn–xn+
→.
From Lemma ., we have
lim sup
l→∞
lim sup
n→∞
Anxn–Tl(Anxn)≤lim sup l→∞
lim sup
n→∞
sup
x∈C
Anx–Tl(Anx)= .
Therefore,
lim sup
l→∞
lim sup
n→∞
Anxn–Tl(Anxn)= .
Putk∞=sup{kn:n≥}<∞, then
xn–Tlxn≤ xn–Anxn+Anxn–Tl(Anxn)+Tl(Anxn) –Tlxn
≤ xn–Anxn+Anxn–Tl(Anxn)+klxn–Anxn
≤( +k∞)xn–Anxn+Anxn–Tl(Anxn)→
asn,l→ ∞. Thus, we have
Txn–xn ≤Txn–Tl+xn+Tl+xn–Tl+xn+ +Tl+xn+–xn++xn+–xn
Step. We show that{xn}converges strongly toPF(T)(x).
Putw=PF(T)(x). Since{xn}is bounded, let{xnk} be a subsequence of{xn} such that
xnkw. By Lemma ., we getw∈F(T). Sincexn=PQn(x) andw∈F(T)⊂Qn, we have
xn–x ≤ w–x. It follows fromw=PF(T)(x) and the weak lower semicontinuity of the norm that
w–x ≤w–x≤lim inf
k→∞ xnk–x ≤lim supk→∞ xnk–x ≤ w–x.
Thus, we obtain that limk→∞xnk –x=w –x
=w–x. Using the Kadec-Klee property ofH, we getlimk→∞xnk=w=w. Since{xnk}is an arbitrary subsequence of{xn},
we can conclude that{xn}converges strongly toPF(T)(x).
Remark . It is not difficult to see from the proof above that the boundedness ofCcan be discarded ifTis a nonexpansive mapping.
4 Strong convergence theorem for equilibrium problems
In this section, we prove a strong convergence theorem for finding a common element of the set of zero points of an asymptotically nonexpansive mappingTand the set of solutions of an equilibrium problem in a Hilbert space.
Theorem . Let H be a real Hilbert space,and let C be a nonempty bounded closed convex subset of H,let f :C×C→Rbe a functional,satisfying(A)-(A).Let T:C→ C be an asymptotically nonexpansive mapping with kn,denote Ln=n+ nj=kjsuch that
F(T)∩EP(f)= ∅.Let{xn}be a sequence generated by
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈C is arbitrary,
yn=αnxn+ ( –αn)n+
n
j=Tjxn,
un∈C, such that f(un,y) +rny–un,un–yn ≥, ∀y∈C,
Cn={z∈C:un–z≤ xn–z+θn},
Qn={z∈C:xn–z,x–xn ≥},
xn+=PCn∩Qn(x), n= , , , . . . ,
(.)
where
θn= ( –αn)
Ln– (diamC)→, n→ ∞.
Suppose that{αn} ⊂[, ]and there exists a∈(, )such thatαn≤a,∀n∈N,and{rn} ⊂
(,∞)such thatlim infn→∞rn> .Then{xn}converges strongly to PF(T)∩EP(f)(x).
Proof We divide the proof of this theorem into four steps as below.
Step. Similar to the proof of Step in Theorem ., it is easy to see thatCnandQnare
closed convex sets for eachn∈N∪ {}.
Step. We show thatF(T)∩EP(f)⊂Cn∩Qn,∀n∈N∪ {}.
Letp∈F(T)∩EP(f). Puttingun=Trnyn,∀n∈N∪{}, by () of Lemma ., we haveTrnis
in Hilbert spaces, then for anyn∈N∪ {},
un–p=Trnyn–p=Trnyn–Trnp
≤ y
n–p.
From the proof of Step in Theorem ., we haveun–p≤ xn–p+θn. Thusp∈Cn,
henceF(T)∩EP(f)⊂Cn. Similar to the proof of Step in Theorem ., it is easy to see
thatF(T)∩EP(f)⊂Qn. Therefore we haveF(T)∩EP(f)⊂Cn∩Qn,∀n∈N∪ {}. This
means that{xn}is well defined.
Step. We show that{xn}is bounded andlimn→∞xn–Txn= .
Similar to the proof of Step in Theorem ., we may obtain that{xn}is bounded and
lim
n→∞xn+–xn= . (.)
Sincexn+=PCn∩Qn∈Cn, thenun–xn+≤ xn–xn++θn. Noticing thatθn→, we
have
un–xn+ ≤ xn–xn++
θn→.
Hence
lim
n→∞xn–un ≤nlim→∞xn–xn++nlim→∞xn+–un →. (.)
Forp∈F(T)∩EP(f)⊂Cn, we have
un–p≤ xn–p+θn. (.)
Sinceun=Trnyn, by (.) and Lemma ., we get that un–yn =Trnyn–yn
≤ yn–p–Trnyn–p
≤ xn–p+θn–un–p→. (.)
LetAn=n+
n
j=Tj. Sinceαn≤a< ,∀n∈N, then by (.) and (.), we have
xn–Anxn=
–αn
yn–xn
≤
–a
yn–un+un–xn
→. (.)
From Lemma ., it follows that
lim sup
l→∞
lim sup
n→∞
Anxn–Tl(Anxn)≤lim sup l→∞
lim sup
n→∞ supx∈C
Anx–Tl(Anx)= .
Therefore
lim sup
l→∞
lim sup
n→∞
Putk∞=sup{kn:n≥}<∞, then
yn–Tlxn=αnxn+ ( –αn)Anxn–Tlxn
≤αnxn–Anxn+Anxn–Tl(Anxn)+Tl(Anxn) –Tlxn
≤αnxn–Anxn+Anxn–Tl(Anxn)+klxn–Anxn
≤( +k∞)xn–Anxn+Anxn–Tl(Anxn).
Letn,l→ ∞, we get thatyn–Tlxn →. Thus, by (.) and (.), we have
xn–Tlxn≤ xn–un+un–yn+yn–Tlxn→. (.)
Therefore, by (.) and (.), we obtain that
Txn–xn ≤ Txn–Tl+xn+Tl+xn–Tl+xn++Tl+xn+–xn+ +xn+–xn
≤ k∞xn–Tlxn+ (k∞+ )xn+–xn+Tl+xn+–xn+
→.
Step. We show that{xn}converges strongly toPF(T)∩EP(f)(x).
Since{xn}is bounded, there exists a subsequence{xnk}of{xn}such thatxnkw. By
Lemma ., we havew∈F(T). Next we showw∈EP(f). From (.) and (.), we get thatunkw
,y
nkw
. Sinceu
n=Trnyn, then
f(un,y) +
rn
y–un,un–yn ≥, ∀y∈C.
Replacingnbynk, we have from Condition (A) that
rnk
y–unk,unk–ynk ≥–f(unk,y)≥f(y,unk), ∀y∈C.
Letk→ ∞, sincelim infn→∞rn> , by (.) and Condition (A), we get that
fy,w≤, ∀y∈C.
Fort∈(, ),y∈C, letyt=ty+ ( –t)w, thenyt∈C, thusf(yt,w)≤. By Condition (A),
we get that
=f(yt,yt)≤tf(yt,y) + ( –t)f
yt,w
≤tf(yt,y).
Dividing byt, we have
f(yt,y)≥, ∀y∈C.
Denotew=PF(T)∩EP(f)(x). Sincexn+=PCn∩Qn(x),w∈F(T)∩EP(f)⊂Cn∩Qn, then xn+–x ≤ w–x. Since the norm is weakly lower semicontinuous, we have
w–x ≤w–x≤lim inf
k→∞ xnk–x ≤lim supk→∞ xnk–x ≤ w–x.
Hencelimk→∞xnk–x=w –x
=w–x. Using the Kadec-Klee property ofH, we getlimk→∞xnk=w
=w. Since{x
nk}is an arbitrary subsequence of{xn}, we can conclude
that{xn}converges strongly toPF(T)∩EP(f)(x).
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The main idea of this paper is proposed by JZ and YC. JZ prepared the manuscript initially and performed all the steps of the proofs in this research. All authors read and approved the final manuscript.
Author details
1Mathematics and Applied Mathematics, Harbin University of Commerce, Harbin, 150028, P.R. China.2Department of
Mathematics, Harbin University of Science and Technology, Harbin, 150080, P.R. China.
Acknowledgements
The authors would like to thank the anonymous referee for some suggestions to improve the manuscript. This work was supported by Research Fund for the Doctoral Program of Harbin University of Commerce (13DL002), Scientific Research Project Fund of the Education Department of Heilongjiang Province and Foundation of Heilongjiang Educational Committee (12521070).
Received: 29 September 2013 Accepted: 5 January 2014 Published:21 Jan 2014 References
1. Goebel, K, Kirk, WA: A fixed point theorem for asymptotically nonexpansive mappings. Proc. Am. Math. Soc.35, 171-174 (1972)
2. Mann, WR: Mean value methods in iteration. Proc. Am. Math. Soc.4, 506-510 (1953) 3. Genel, A, Lindenstrass, L: An example concerning fixed points. Isr. J. Math.22, 81-86 (1975)
4. Nakajo, K, Takahashi, W: Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. J. Math. Anal. Appl.279, 372-379 (2003)
5. Kim, TH, Xu, HK: Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups. Nonlinear Anal.64, 1140-1152 (2006)
6. Yanes, CM, Xu, HK: Strong convergence of the CQ method for fixed point iteration processes. Nonlinear Anal.64, 2400-2411 (2006)
7. Nilsrakoo, W, Seajung, S: Weak and strong convergence theorems for countable Lipschitzian mappings and its applications. Nonlinear Anal.69, 2695-2708 (2008)
8. Baillon, JB: Un théorème de type ergodique pour les contractions nonlinéairs dans un espaces de Hilbert. C. R. Acad. Sci. Paris Sér. A-B280, 1511-1541 (1975)
9. Shimizu, T, Takahashi, W: Strong convergence theorem for asymptotically nonexpansive mappings. Nonlinear Anal.
26, 265-272 (1996)
10. Li, J: The generalized projection operator on reflexive Banach spaces and its applications. J. Math. Anal. Appl.306, 55-71 (2005)
11. Li, J, Lin, SL, Zhang, CJ: On the existence of Nash equilibriums for infinite matrix games. Nonlinear Anal.10, 42-53 (2009)
12. Xu, HK: Inequalities in Banach spaces with applications. Nonlinear Anal.16, 1127-1138 (1991)
13. Qin, X, Su, Y: Strong convergence theorems for relatively nonexpansive mappings in a Banach space. Nonlinear Anal.
67, 1958-1965 (2007)
14. Wlodarczyk, K, Klim, D: Equilibria and fixed points of set-valued maps with nonconvex and noncompact domains and ranges. Nonlinear Anal.65, 918-932 (2006)
15. Chadli, O, Chbani, Z, Riahi, H: Equilibrium problems and noncoercive variational inequalities. Optimization49, 1-12 (1999)
16. Ansari, QH, Yao, JC: An existence results for the generalized vector equilibrium problem. Appl. Math. Lett.12, 53-56 (1999)
17. Lan, H: Common fixed-point iterative processes with errors for generalized asymptotically quasi-nonexpansive mappings. Comput. Math. Appl.52, 1403-1412 (2006)
18. Zhao, J, He, S: A hybrid iteration scheme for equilibrium problems and common fixed point problems of generalized quasi-φ-asymptotically nonexpansive mappings in Banach spaces. Fixed Point Theory Appl.2012, 33 (2012) 19. Chang, S, Wang, L, Tang, YK: Strong convergence theorems of nonlinear operator equations for countable family of
20. Zegeye, H, Shahzad, N: Strong convergence theorems for variational inequality problems and quasi-φ-asymptotically nonexpansive mappings. J. Glob. Optim.54, 101-116 (2012)
21. Blum, E, Oettli, W: From optimization and variational inequalities to equilibrium problems. Math. Program.63, 123-145 (1994)
22. Combettes, PL, Hirstoaga, SA: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal.6, 117-136 (2005) 23. Takahashi, S, Takahashi, W: Viscosity approximation methods for equilibrium problems and fixed point in Hilbert
space. J. Math. Anal. Appl.331, 506-515 (2007)
24. Jitpeera, T, Katchang, P, Kumam, P: A viscosity of Cesàro mean approximation methods for a mixed equilibrium, variational inequalities, and fixed point problems. Fixed Point Theory Appl.2011, 945051 (2011)
25. Browder, FE: Nonlinear operators and nonlinear equations of evolution in Banach spaces. Proc. Symp. Pure Math.18, 78-81 (1976)
26. Takahashi, W, Zembayashi, K: Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces. Nonlinear Anal.70, 45-57 (2009)
10.1186/1687-1812-2014-16
