The purpose of the paper is to present a new iteration method for finding a common element for the set of solutions of equilibrium problems and of operator equations with a finite family of λi-inverse-strongly monotone mappings in Hilbert spaces.
1. Introduction
Let H be a real Hilbert space with the inner product 〈·,·〉 and the norm ∥·∥, respectively. Let C be a nonempty closed convex subset of H, and let G be a bifunction from C×C into (-∞,+∞). The equilibrium problem for G is to find u*∈C such that
(1)G(u*,v)≥0,∀v∈C.
The set of solutions of (1) is denoted by EP(G).
Equilibrium problem (1) includes the numerous problems in physics, optimization, economics, transportation, and engineering, as special cases.
Assume that the bifunction G satisfies the following standard properties.
Assumption A.
Let G:C×C→(-∞,+∞) be a bifunction satisfying the conditions (A1)–(A4):
G(u,u)=0, ∀u∈C;
G(u,v)+G(v,u)≤0, ∀(u,v)∈C×C;
for each u∈C, G(u,·):C→(-∞,+∞) is lower semicontinuous and convex;
lim¯t→+0G((1-t)u+tz,v)≤G(u,v), ∀(u,z,v)∈C×C×C.
Let {Ti}, i=1,…,N, be a finite family of ki-strictly pseudocontractive mappings from C into C with the set of fixed points F(Ti); that is,
(2)F(Ti)={x∈C:Tix=x}.
Assume that
(3)S:=⋂i=1NF(Ti)∩EP(G)≠∅.
The problem of finding an element
(4)u*∈S
is studied intensively in [1–27].
Recall that a mapping T in H is said to be a k-strictly pseudocontractive mapping in the terminology of Browder and Petryshyn [28] if there exists a constant 0≤k<1 such that
(5)∥Tx-Ty∥2≤∥x-y∥2+k∥(I-T)x-(I-T)y∥2,
for all x,y∈D(T), the domain of T, where I is the identity operator in H. Clearly, if k=0, then T is nonexpansive; that is,
(6)∥T(x)-T(y)∥≤∥x-y∥.
We know that the class of k-strictly pseudocontractive mappings strictly includes the class of nonexpansive mappings.
In the case that Ti≡I, (4) is reduced to the equilibrium problem (1) and shown in [5, 23] to cover monotone inclusion problems, saddle point problems, variational inequality problems, minimization problems, Nash equilibria in noncooperative games, vector equilibrium problems, and certain fixed point problems (see also [29]). For finding approximative solutions of (1) there exist several methods: the regularization approach in [7, 9, 15, 24, 30, 31], the gap-function approach in [8, 15, 16, 18, 19], and the iterative procedure approach in [1–4, 6, 8, 11–14, 19–22, 32, 33].
In the case that G≡0 and N=1, (4) is a problem of finding a fixed point for a k-strictly pseudocontractive mapping in C and is given by Marino and Xu [17].
Theorem 1 (see [17]).
Let C be a nonempty closed convex subset of a real Hilbert space H. Let T:C→C be a k-strictly pseudocontractive mapping for some 0≤k<1, and assume that
(7)F(T)≠∅.
Let {xn} be the sequence generated by the following algorithm:
(8)x0∈C,yn=αnxn+(1-αn)Txn,Cn={∥xn-Txn∥2z∈C:∥yn-z∥2≤∥xn-z∥2iiiii+(1-αn)(k-αn)∥xn-Txn∥2},Qn={z∈C:〈xn-z,x0-xn〉≥0},xn+1=PCn∩Qnx0.
Assume that the control sequence {αn} is chosen so that αn<1 for all n. Then {xn} converges strongly to PF(T)x0, the projection of x0 onto F(T).
For the case that G≡0 and N>1, (4) is a problem of finding a common fixed point for a finite family of ki-strictly pseudocontractive mappings Ti in C and is studied in [27].
Let x0∈C and {αn}, {βn}, and {γn} three sequences in [0,1] satisfying αn+βn+γn=1 for all n≥1, and let {un} be a sequence in C. Then the sequence {xn} generated by
(9)x1=α1x0+β1T1x1+γ1u1,x2=α2x1+β2T2x2+γ2u2,⋮xN=αNxN-1+βNTNxN+γNuN,xN+1=αN+1xN+βN+1T1xN+1+γN+1uN+1,⋮
is called the implicit iteration process with mean errors for a finite family of strictly pseudocontractive mappings {Ti}i=1N.
The scheme (9) can be expressed in the compact form as
(10)xn=αnxn-1+βnTnxn+γnun,
where Tn=TnmodN.
Theorem 2 (see [27]).
Let C be a nonempty closed convex subset of a real Hilbert space H. Let {Ti}i=1N be a finite family of strictly pseudocontractive mappings of C into itself such that
(11)⋂i=1NF(Ti)≠∅.
Let x0∈C and let {un} be a bounded sequence in C; let {αn}, {βn}, and {γn} be three sequences in [0,1] satisfying the following conditions:
αn+βn+γn=1, ∀n≥1;
there exist constants σ1,σ2 such that 0<σ1≤βn≤σ2<1, ∀n≥1;
∑n=1∞γn<∞.
Then the implicit iterative sequence {xn} defined by (9) converges weakly to a common fixed point of the mappings {Ti}i=1N. Moreover, if there exists i0∈{1,2,…,N} such that Ti0 is demicompact, then {xn} converges strongly.
If G is an arbitrary bifunction satisfying Assumption A and N=1, then (4) is a problem of finding a common element of the fixed point set for a k-strictly pseudocontractive mapping in C and of the solution set of equilibrium problem for G (see [26]).
Theorem 3 (see [26]).
Let C be a nonempty closed convex subset of a real Hilbert space H. Let G be a bifunction from C×C to (-∞,+∞) satisfying Assumption A, and let T be a nonexpansive mapping of C into H such that
(12)F(T)∩EP(G)≠∅.
Let f be a contraction of H into itself and let {xn} and {un} be sequences generated by x1∈H and
(13)G(un,y)+1rn〈y-un,un-xn〉≥0,∀y∈C,xn+1=αnf(xn)+(1-αn)Tun,
for all n∈N, where {αn}⊂[0,1] and {rn}⊂(0,∞) satisfy
(14)limn→∞αn=0,∑n=1∞αn=∞,∑n=1∞|αn+1-αn|<∞,liminfn→∞rn>0,∑n=1∞|rn+1-rn|<∞.
Then, {xn} and {un} converge strongly to z∈F(T)∩EP(G), where
(15)z=PF(T)∩EP(G)f(z).
Set Ai=I-Ti. Obviously, Ai are λi-inverse-strongly monotone; that is,
(16)〈Ai(x)-Ai(y),x-y〉≥λi∥Ai(x)-Ai(y)∥2,iiiiiiiiiiiiiiiiiiiiiiiii∀x,y∈D(Ai),λi=1-ki2.
From now on, let {Ai}i=1N be a finite family of λi-inverse-strongly monotone mappings in H with C⊂⋂i=1ND(Ai) and λi>0, i=1,…,N. On the other hand, if there exists i0∈{1,2,…,N} such that λi0>1, then Ai0 is a contraction; that is, ∥Ai0(x)-Ai0(y)∥≤(1/λi0)∥x-y∥ with 1/λi0<1. And hence, Ai0 has only one solution and, consequently, the stated problem does not have sense. So, without loss of generality, assume that 0<λi≤1, i=1,…,N.
Set
(17)S=⋂i=1NSi,
where Si={x∈C:Ai(x)=0} is the solution set of Ai in C.
Assume that EP(G)∩S≠∅.
Our problem is to find an element
(18)u*∈EP(G)∩S.
Since the mapping A=I-T is (1/2)-inverse-strongly monotone for each nonexpansive mapping T, the problem of finding an element u*∈C, which is not only a solution of a variational inequality involving an inverse-strongly monotone mapping but also a fixed point of a nonexpansive mapping, is a particular case of (18).
For instance, the case that G(u,v)≡〈A(u),v-u〉, where A is some inverse-strongly monotone mapping and N=1, is studied in [25].
Theorem 4 (see [25]).
Let C be a nonempty closed convex subset of a real Hilbert space H. Let λ>0. Let A be a λ-inverse-strongly monotone mapping of C into H, and let T be a nonexpansive mapping of C into itself such that
(19)F(T)∩VI(C,A)≠∅,
where VI(C,A) denotes the solution set of the following variational inequality: find x*∈C such that
(20)〈A(x*),x-x*〉≥0,∀x∈C.
Let {xn} be a sequence defined by
(21)x0∈C,xn+1=αnxn+(1-αn)TPC(xn-λnA(xn)),
for every n=0,1,…, where {λn}⊂[a,b] for some a,b∈(0,2λ) and {αn}⊂(c,d) for some c,d∈(0,1). Then, {xn} converges weakly to z∈F(T)∩VI(C,A), where
(22)z=limn→∞PF(T)∩VI(C,A)xn.
The following theorem is an improvement of Theorem 4 for the case of nonself-mapping.
Theorem 5 (see [34]).
Let C be a nonempty closed convex subset of a real Hilbert space H. Let A be a λ-inverse-strongly monotone mapping of C into H, and let T be a nonexpansive nonself-mapping of C into H such that
(23)F(T)∩VI(C,A)≠∅.
Suppose that x1=x∈C and {xn} is given by
(24)xn+1=PC(αnx+(1-αn)TPC(xn-λnA(xn)))
for every n=1,2,…, where {αn} is a sequence in [0,1) and {λn} is a sequence in [0,2α]. If {αn} and {λn} are chosen so that λn∈[a,b] for some a, b with 0<a<b<2α,
(25)limn→∞αn=0,∑n=1∞αn=∞,∑n=1∞|αn+1-αn|<∞,∑n=1∞|λn+1-λn|<∞,
then {xn} converges strongly to PF(T)∩VI(C,A)x.
We know that λ-inverse-strongly monotone mapping is (1/λ)-Lipschitz continuous and monotone. Therefore, for the case that G(u,v)≡〈A(u),v-u〉, where A is not inverse-strongly monotone, but Lipschitz continuous and monotone, Nadezhkina and Takahashi [35] prove the following theorem.
Theorem 6 (see [35]).
Let C be a nonempty closed convex subset of a real Hilbert space H. Let A be a monotone and k-Lipschitz continuous mapping of C into H, and let T be a nonexpansive mapping of C into itself such that
(26)F(T)∩VI(C,A)≠∅.
Let {xn}, {yn}, and {zn} be sequences generated by
(27)x0=x∈C,yn=PC(xn-λnA(xn)),zn=PC(xn-λnA(yn)),Cn={z∈C:∥zn-z∥≤∥xn-z∥},Qn={z∈C:〈xn-z,x-xn〉≥0},xn+1=PCn∩Qnx
for every n=0,1,…, where {λn}⊂[a,b] for some a,b∈(0,1/k) and αn⊂[0,c] for some c∈[0,1). Then the sequences {xn}, {yn}, and {zn} converge strongly to PF(T)∩VI(C,A)x.
Some similar results are also considered in [36, 37].
Buong [38] introduced two new implicit iteration methods for solving problem (18).
We construct a regularization solution un of the following single equilibrium problem: find un∈C such that
(28)F(un,v)≥0,∀v∈C,
where
(29)F(u,v)≔G(u,v)+∑i=1NαnμiGi(u,v)+αn〈u,v-u〉,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiαn>0,Gi(u,v)=〈Ai(u),v-u〉,i=1,…,N,0<μi<μi+1<1,i=2,…,N-1,
and {αn} is the positive sequence of regularization parameters that converges to 0, as n→+∞.
The first one is the following theorem.
Theorem 7 (see [38]).
For each αn>0, problem (28) has a unique solution un such that
limn→+∞un=u*, u*∈EP(G)∩S, ∥u*∥≤∥y∥, ∀y∈EP(G)∩S;
(30)∥un-um∥≤(∥u*∥+dN)|αn-αm|αn,
where d is a positive constant.
Next, we introduce the second result. Let {c~n} and {γn} be some sequences of positive numbers, and let z0 and z1 be two arbitrary elements in C. Then, the sequence {zn} of iterations is defined by the following equilibrium problem: find zn+1∈C such that
(31)c~n(G(zn+1,v)+∑i=1NαnμiGi(zn+1,v)+αn〈zn+1,v-zn+1〉)+〈zn+1-zn,v-zn+1〉-γn〈zn-zn-1,v-zn+1〉≥0,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii∀v∈C.
Theorem 8 (see [38]).
Assume that the parameters c~n,γn, and αn are chosen such that
0<c0<c~n,0≤γn<γ0,
∑n=1∞bn=+∞, bn=c~nαn/(1+c~nαn),
∑n=1∞γnbn-1∥zn-zn-1∥<+∞,
limn→∞αn=0,limn→∞(|αn-αn+1|/αnbn)=0.
Then, the sequence {zn} defined by (31) converges strongly to the element u*, as n→+∞.
In this paper, we consider the new another iteration method: for an arbitrary element x0 in H, the sequence {xn} of iterations is defined by finding un∈C such that
(32)G(un,y)+〈un-xn,y-un〉≥0,∀y∈C,xn+1=PC(xn-βn[xn-un+∑i=1NαnμiAi(xn)+αnxn])=PC(xn-βn[∑i=1NαnμiAi(xn)+(1+αn)xn-un]),
where PC is the metric projection of H onto C and {αn} and {βn} are sequences of positive numbers.
The strong convergence of the sequence {xn} defined by (32) is proved under some suitable conditions on {αn} and {βn} in the next section.
2. Main Results
We formulate the following lemmas for the proof of our main theorems.
Lemma 9 (see [9]).
Let C be a nonempty closed convex subset of a real Hilbert space H and let G be a bifunction of C×C into (-∞,+∞) satisfying Assumption A. Let r>0 and x∈H. Then, there exists z∈C such that
(33)G(z,y)+1r〈z-x,y-z〉≥0,∀y∈C.
Lemma 10 (see [9]).
Let C be a nonempty closed convex subset of a real Hilbert space H. Assume that G:C×C→(-∞,+∞) satisfies Assumption A. For r>0 and x∈H, define a mapping Tr:H→C as follows:
(34)Tr(x)={z∈C:G(z,y)+1r〈z-x,y-z〉≥0},∀y∈C.
Then, the following statements hold:
Tr is single valued;
Tr is firmly nonexpansive; that is, for any x,y∈H,
(35)∥Tr(x)-Tr(y)∥2≤〈Tr(x)-Tr(y),x-y〉;
F(Tr)=EP(G);
EP(G) is closed and convex.
Lemma 11 (see [36]).
Let {an},{bn}, and {cn} be the sequences of positive numbers satisfying the following conditions:
an+1≤(1-bn)an+cn,
∑n=0∞bn=+∞, bn<1, limn→+∞(cn/bn)=0.
Then, limn→+∞an=0.
Lemma 12 (see [38]).
Let A be any inverse-strongly monotone mapping from C into H with the solution set SA:={x∈C:A(x)=0}, and let C0 be a closed convex subset of C such that
(36)SA∩C0≠∅.
Then, the solution set of the following variational inequality
(37)〈A(y~),x-y~〉≥0,∀x∈C0,y~∈C0,
is coincided with SA∩C0.
From Lemma 9, we can consider the firmly nonexpansive mapping T0 defined by
(38)T0(x)={z∈C:G(z,y)+〈z-x,y-z〉≥0,∀y∈C},iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii∀x∈H.
From Lemma 10, we know that T0 is nonexpansive. Consequently, A0:=I-T0 is (1/2)-inverse-strongly monotone. Let
(39)S0:={x∈C:A0(x)=0}.
Then, S0=EP(G) and problem (18) are equivalent to finding
(40)u*∈S0∩S.
Now, we construct a regularization solution yn for (40) by solving the following variational inequality problem: find yn∈C such that
(41)〈∑i=0NαnμiAi(yn)+αnyn,v-yn〉≥0,∀v∈C,μ0=0<μ1<⋯<μN<1,
where the positive regularization parameter αn→0, as n→+∞.
Now we are in a position to introduce and prove the main results.
Theorem 13.
Let C be a nonempty closed convex subset of a real Hilbert space H. Let G be a bifunction from C×C to (-∞,+∞) satisfying Assumption A and let {Ai}i=1N be a finite family of λi-inverse-strongly monotone mappings in H with C⊂⋂i=1ND(Ai) and λi>0, i=1,…,N, such that
(42)EP(G)∩S≠∅,
where EP(G) denotes the set of solutions for (1) and
(43)S=⋂i=1NSi,Si={x∈C:Ai(x)=0}.
Then, for each αn>0, problem (41) has a unique solution yn such that
limn→+∞yn=u*, u*∈EP(G)∩S,
∥u*∥≤∥y∥,
∀y∈EP(G)∩S,
(44)∥yn-ym∥≤|αn-αm|αn(∥u*∥+dN),
where d is some positive constant.
Proof.
From Lemma 12, we know that S0 is the set of solutions for the following variational inequality problem: find u*∈C such that
(45)〈A0(u*),v-u*〉≥0,∀v∈C.
If we define the new bifunction G0(u,v) by
(46)G0(u,v)=〈A0(u*),v-u*〉,
then problem (41) is the same as (28) with a new G(u,v), and the proof for the theorem is a complete repetition of the proof for Theorem 2.1 in [38].
Set
(47)L=max{2,1λi,i=1,…,N}.
Theorem 14.
Let C be a nonempty closed convex subset of a real Hilbert space H. Let G be a bifunction from C×C to (-∞,+∞) satisfying Assumption A and let {Ai}i=1N be a finite family of λi-inverse-strongly monotone mappings in H with C⊂⋂i=1ND(Ai) and λi>0, i=1,…,N, such that
(48)EP(G)∩S≠∅,
where EP(G) denotes the set of solutions for (1) and
(49)S=⋂i=1NSi,Si={x∈C:Ai(x)=0}.
Suppose that αn,βn satisfy the following conditions:
(50)αn,βn>0(αn≤1),limn→∞αn=0,limn→∞|αn-αn+1|αn2βn=0,∑n=0∞αnβn=∞,lim¯n→∞βn(L(N+1)+αn)2αn<1.
Then, the sequence {xn} defined by (32) converges strongly to u*∈EP(G)∩S; that is,
(51)limn→∞xn=u*∈EP(G)∩S.
Proof.
Let yn be the solution of (41). Then,
(52)yn=PC(yn-βn[∑i=0NαnμiAi(yn)+αnyn]).
Set Δn=∥xn-yn∥. Obviously,
(53)Δn+1=∥xn+1-yn+1∥≤∥xn+1-yn∥+∥yn+1-yn∥.
From the nonexpansivity of PC, the monotone and Lipschitz continuous properties of Ai, i=0,…,N, (41), (52), and yn=T0(xn), we have
(54)∥xn+1-yn∥≤∥xn-yn-βn∑i=0N[∑i=0Nαnμi(Ai(xn)-Ai(yn))iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii∑i=0N+αn(xn-yn)]∥,∥xn-yn-βn[∑i=0Nαnμi(Ai(xn)-Ai(yn))+αn(xn-yn)]∥2=∥xn-yn∥2+βn2∥[∑i=0Nαnμi(Ai(xn)-Ai(yn))+αn(xn-yn)]∥2-2βn〈∑i=0Nαnμi(Ai(xn)-Ai(yn))∑iiiiiiiii=0Niiiiiiiiiiiiiiiiiiiiiiiii+αn(xn-yn),xn-yn〉≤∥xn-yn∥2[1-2βnαn+βn2(2+∑i=1Nαnμi1λi+αn)2].
Thus,
(55)∥xn+1-yn∥≤Δn(1-2βnαn+βn2(L(N+1)+αn)2)1/2.
Therefore,
(56)Δn+1≤Δn(1-2βnαn+βn2(L(N+1)+αn)2)1/2+|αn-αn+1|αn(∥u*∥+dN)≤Δn(1-αnβn)1/2+|αn-αn+1|αn(∥u*∥+dN).
We note that, for ε>0, a>0, b>0, the inequality
(57)(a+b)2≤(1+ε)(a2+b2ε)
holds. Thus, applying inequality (57) for ε=αnβn/2, we obtain
(58)0≤Δn+12≤Δn2(1-αnβn)(1+12αnβn)+(αn-αn+1αn(∥u*∥+dN))22αnβn(1+12αnβn)=Δn2(1-12αnβn-12(αnβn)2)+(αn-αn+1αn2βn(∥u*∥+dN))22αnβn(1+12αnβn).
Set
(59)bn=αnβn(12+12αnβn)cn=(αn-αn+1αn2βn(∥u*∥+dN))22αnβn(1+12αnβn).
Then, it is not difficult to check that bn and cn satisfy the conditions in Lemma 11 for sufficiently large n. Hence, limn→+∞Δn2=0. Since limn→∞yn=u*, we have
(60)limn→∞xn=u*∈EP(G)∩S.
This completes the proof.
Remark 15.
The sequences αn=(1+n)-p,0<p<1/2, and βn=γ0αn with
(61)0<γ0<1(L(N+1)+α0)2
satisfy all the necessary conditions in Theorem 14.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Authors’ Contribution
The main idea of this paper was proposed by Jong Kyu Kim. Jong Kyu Kim and Nguyen Buong prepared the paper initially and performed all the steps of proof in this research. All authors read and approved the final paper.
Acknowledgment
This paper was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012R1A1A2042138).
AntipinA. S.Equilibrium programming: gradient methods199758813371347AntipinA. S.Equilibrium programming: proximal methods1997371113271339Computational Mathematics and Mathematical Physics, vol. 37, no. 11, pp. 1285–1296, 1997MR1489507ZBL0944.90083AntipinA. S.Solution methods for variational inequalities with coupled constraints200040912391254Translated from Zhurnal Vychislite'noi Matematiki i Matematicheskoi Fiziki, vol. 40, no. 9, 1291–1307, 2000MR1832269ZBL0944.90083AntipinA. S.Solving variational inequalities with coupling constraints with the use of differential equations2000361115871596Translated from Differentsial'nye Uravnenye, vol. 36, no. 11, 1443–1451, 200010.1007/BF02757358MR1841464ZBL0944.90083BlumE.OettliW.From optimization and variational inequalities to equilibrium problems1994631–4123145MR1292380ZBL0944.90083BounkhelM.Al-SenanB. R.An iterative method for nonconvex equilibrium problems200672, article 75MR2221356ZBL0944.90083ChadliO.SchaibleS.YaoJ. C.Regularized equilibrium problems with application to noncoercive hemivariational inequalities2004121357159610.1023/B:JOTA.0000037604.96151.26MR2084344ZBL1107.91067ChadliO.KonnovI. V.YaoJ. C.Descent methods for equilibrium problems in a Banach space2004483-460961610.1016/j.camwa.2003.05.011MR2091222ZBL1057.49009CombettesP. L.HirstoagaS. A.Equilibrium programming in Hilbert spaces200561117136MR2138105ZBL1109.90079AnhP. N.KimJ. K.An interior proximal cutting hyperplane method for equilibrium problems20122012, article 9910.1186/1029-242X-2012-99MR2927687ZBL1276.65032KimJ. K.NamY. M.SimJ. Y.Convergence theorems of implicit iterative sequences for a finite family of asymptotically quasi-nonxpansive type mappings200971122839284810.1016/j.na.2009.06.090MR2672053KimJ. K.ChoS. Y.QinX.Some results on generalized equilibrium problems involving strictly pseudocontractive mappings20113152041205710.1016/S0252-9602(11)60380-9MR2884969ZBL1247.47061KimJ. K.Strong convergence theorems by hybrid projection methods for equilibrium problems and fixed point problems of the asymptotically quasi-φ-nonexpansive mappings20112011, article 1010.1186/1687-1812-2011-10MR2820314KimJ. K.LimW. H.A new iterative algorithm of pseudomonotone mappings for equilibrium problems in Hilbert spaces20132013, article 12810.1186/1029-242X-2013-128MR3044683ZBL06252733KonnovI. V.PinyaginaO. V.D-gap functions and descent methods for a class of monotone equilibrium problems2003135765MR2025560ZBL06252733KonnovI. V.PinyaginaO. V.D-gap functions for a class of equilibrium problems in Banach spaces20033227428610.2478/cmam-2003-0018MR1999802ZBL1051.47046MarinoG.XuH.-K.Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces2007329133634610.1016/j.jmaa.2006.06.055MR2306805ZBL1116.47053MastroeniG.Gap functions for equilibrium problems200327441142610.1023/A:1026050425030MR2012814ZBL1061.90112MastroeniG.On auxiliary principle for equilibrium problems20003.244.1258Pisa, ItalyDepartment of Mathematics of Pisa UniversityMoudafiA.Second-order differential proximal methods for equilibrium problems200341, article 18MR1965998ZBL1061.90112MoudafiA.ThéraM.Proximal and dynamical approaches to equilibrium problems1999477Berlin, GermaySpringer187201Lecture Notes in Economics and Mathematical Systems10.1007/978-3-642-45780-7_12MR1737320ZBL1061.90112NoorM. A.NoorK. I.On equilibrium problems20044125132MR2077792ZBL1061.90112OettliW.A remark on vector-valued equilibria and generalized monotonicity1997221213221MR1479747ZBL0914.90235StukalovA. S.A regularized extragradient method for solving equilibrium programming problems in a Hilbert space200545915381554MR2216065ZBL1117.90323TakahashiW.ToyodaM.Weak convergence theorems for nonexpansive mappings and monotone mappings2003118241742810.1023/A:1025407607560MR2006529ZBL1055.47052TakahashiS.TakahashiW.Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces2007331150651510.1016/j.jmaa.2006.08.036MR2306020ZBL1055.47052WangG.PengJ.LeeH.-W. J.Implicit iteration process with mean errors for common fixed points of a finite family of strictly pseudocontrative maps200711–48999MR2340931ZBL1145.47053BrowderF. E.PetryshynW. V.Construction of fixed points of nonlinear mappings in Hilbert space196720197228MR021765810.1016/0022-247X(67)90085-6ZBL0153.45701GöpfertA.RiahiH.TammerC.ZălinescuC.2003New York, NY, USASpringerxiv+350MR1994718KimJ. K.TuyenT. M.Regularization proximal point algorithm for finding a common fixed point of a finite family of nonexpansive mappings in Banach spaces20112011, article 5210.1186/1187-1812-2011-52MR2836967ZBL06200685KimJ. K.BuongN.Regularization inertial proximal point algorithm for monotone hemicontinuous mapping and inverse strongly monotone mappings in Hilbert spaces201020101045191610.1155/2010/451916MR2600204ZBL1184.49015KimJ. K.BuongN.An iteration method for common solution of a system of equilibrium problems in Hilbert spaces201120111510.1155/2011/780764780764MR2780832ZBL1221.65153KimJ. K.AnhP. N.NamY. M.Strong convergence of an extended extragradient method for equilibrium problems and fixed point problems201249118720010.4134/JKMS.2012.49.1.187MR2907549ZBL06012359IidukaH.TakahashiW.Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings200561334135010.1016/j.na.2003.07.023MR2123081ZBL06012359NadezhkinaN.TakahashiW.Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz-continuous monotone mappings20061641230124110.1137/050624315MR2219141ZBL1143.47047NoorM. A.YaoY.ChenR.LiouY.-C.An iterative method for fixed point problems and variational inequality problems2007121121132MR2420039ZBL1149.49013ZengL.-C.YaoJ.-C.Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems200610512931303MR2253379ZBL1110.49013BuongN.Approximation methods for equilibrium problems and common solution for a finite family of inverse strongly-monotone problems in Hilbert spaces2008213–16735746MR2419252ZBL1186.47071