Parallel algorithms for variational inclusions and fixed points with applications

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R E S E A R C H

Open Access

Parallel algorithms for variational inclusions

and ﬁxed points with applications

Afrah AN Abdou

1*

_{, Badriah AS Alamri}

1

_{, Yeol Je Cho}

1,2

_{, Yonghong Yao}

3

_{and Li-Jun Zhu}

4

*_{Correspondence:}

[email protected]; [email protected]

1_{Department of Mathematics, King}

Abdulaziz University, Jeddah, 21589, Saudi Arabia

Full list of author information is available at the end of the article

Abstract

In this paper, we introduce two parallel algorithms for finding a zero of the sum of two monotone operators and a fixed point of a nonexpansive mapping in Hilbert spaces and prove some strong convergence theorems of the proposed algorithms. As special cases, we can approach the minimum-norm common element of the zero of the sum of two monotone operators and the fixed point of a nonexpansive mapping without using the metric projection. Further, we give some applications of our main results.

MSC: 49J40; 47J20; 47H09; 65J15

Keywords: monotone operator; nonexpansive mapping; zero point; ﬁxed point; resolvent

1 Introduction

LetHbe a real Hilbert space. LetA:H→Hbe a single-valued nonlinear mapping and

B:H→Hbe a set-valued mapping.

Now, we are concerned with the following variational inclusion:

Find a zerox∈Hof the sum of two monotone operatorsAandBsuch that

∈Ax+Bx, (.)

where  is the zero vector inH.

The set of solutions of the problem (.) is denoted by (A+B)–_{. If}_H₌_Rm_{, then the}

problem (.) becomes the generalized equation introduced by Robinson []. IfA= , then the problem (.) becomes the inclusion problem introduced by Rockafellar []. It is well known that the problem (.) is among the most interesting and intensively studied classes of mathematical problems and has wide applications in the ﬁelds of optimization and con-trol, economics and transportation equilibrium, engineering science, and many others. For the past years, many existence results and iterative algorithms for various variational inequality and variational inclusion problems have been extended and generalized in var-ious directions using novel and innovative techniques. A useful and important general-ization is called the general variational inclusion involving the sum of two nonlinear op-erators. Moudaﬁ and Noor [] studied the sensitivity analysis of variational inclusions by using the technique of resolvent equations. Recently much attention has been given to de-veloping iterative algorithms for solving the variational inclusions. Donget al.[] analyzed the solution’s sensitivity for variational inequalities and variational inclusions by using a

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resolvent operator technique. By using the concept and technique of resolvent operators, Agarwalet al.[] and Jeong [] introduced and studied a new system of parametric gener-alized nonlinear mixed quasi-variational inclusions in a Hilbert space. Lan [] introduced and studied a stable iteration procedure for a class of generalized mixed quasi-variational inclusion systems in Hilbert spaces. Recently, Zhanget al.[] introduced a new iterative scheme for finding a common element of the set of solutions to the problem (.) and the set of fixed points of nonexpansive mappings in Hilbert spaces. Penget al.[] introduced another iterative scheme by the viscosity approximate method for finding a common ele-ment of the set of solutions of a variational inclusion with set-valued maximal monotone mapping and inverse strongly monotone mappings, the set of solutions of an equilibrium problem, and the set of fixed points of a nonexpansive mapping. For some related work, see [–] and the references therein.

Recently, Takahashiet al.[] introduced the following iterative algorithm for ﬁnding a zero of the sum of two monotone operators and a ﬁxed point of a nonexpansive map-ping:

xn+=βnxn+ ( –βn)S

αnx+ ( –αn)JλBn(xn–λnAxn)

(.)

for all n≥. Under some assumptions, they proved that the sequence {xn} converges

strongly to a point ofF(S)∩(A+B)–_.

Remark . We note that the algorithm (.) cannot be used to find the minimum-norm element due to the facts thatx∈CandSis a self-mapping ofC. However, there exist a large number of problems for which one needs to find the minimum-norm solution (see, for ex-ample, [–]). A useful path to circumvent this problem is to use projection. Bauschke and Browein [] and Censor and Zenios [] provide reviews of the field. The main diffi-culty is in the computation. Hence it is an interesting problem to find the minimum-norm element without using the projection.

Motivated and inspired by the works in this ﬁeld, we ﬁrst suggest the following two algorithms without using projection:

xt= ( –κ)Sxt+κJλB

tγf(xt) + ( –t)xt–λAxt

for allt∈(, ) and

xn+= ( –κ)Sxn+κJλBn

αnγf(xn) + ( –αn)xn–λnAxn

for all n≥. Notice that these two algorithms are indeed well deﬁned (see the next

section). We show that the suggested algorithms converge strongly to a point x˜ =

PF(S)∩(A+B)–_(γf(x˜)) which solves the following variational inequality:

γf(x˜) –x˜,x˜–z≥

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As special cases, we can approach the minimum-norm element in F(S)∩(A+B)–_ without using the metric projection and give some applications.

2 Preliminaries

LetHbe a real Hilbert space with the inner product·,·and the norm · , respectively. LetCbe a nonempty closed convex subset ofH.

() A mappingS:C→Cis said to benonexpansiveif

Sx–Sy ≤ x–y

for allx,y∈C. We denote byF(S)the set of ﬁxed points ofS.

() A mappingA:C→His said to beα-inverse strongly monotoneif there existsα>  such that

Ax–Ay,x–y ≥αAx–Ay

for allx,y∈C.

It is well known that, if A isα-inverse strongly monotone, thenAx–Ay ≤_αx–y

for allx,y∈C.

LetBbe a mapping fromHinto H_{. The eﬀective domain of}_B_{is denoted by}_dom(_B_),

that is,dom(B) ={x∈H:Bx=∅}.

() A multi-valued mappingBis said to be amonotone operatoronHif

x–y,u–v ≥

for allx,y∈dom(B),u∈Bx, andv∈By.

() A monotone operatorBonHis said to bemaximalif its graph is not strictly contained in the graph of any other monotone operator onH.

LetBbe a maximal monotone operator onHandB– ={x∈H: ∈Bx}. For a maximal monotone operatorBonHandλ> , we may deﬁne a single-valued operatorJB

λ = (I+ λB)–_:_H_→_dom(_B_{), which is called the}_resolvent_of_B_for_λ_{. It is well known that the} resolventJB

λ is ﬁrmly nonexpansive,i.e.,

JλBx–JλBy 

≤JλBx–JλBy,x–y

for allx,y∈CandB–_{ =}_F₍_JB

λ) for allλ> . The following resolvent identity is well known: for anyλ>  andμ> , the following identity holds:

JλBx=JμB

μ λx+

 –μ

λ

JλBx

(.)

for allx∈H.

We use the notation thatxnxstands for the weak convergence of (xn) toxandxn→x

stands for the strong convergence of (xn) tox, respectively.

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Lemma .([]) Let C be a nonempty closed convex subset of a real Hilbert space H.Let A:C→H be anα-inverse strongly monotone mapping andλ> be a constant.Then we have

(I–λA)x– (I–λA)y≤ x–y+λ(λ– α)Ax–Ay

for all x,y∈C.In particular,if≤λ≤α,then I–λA is nonexpansive.

Lemma .([]) Let C be a closed convex subset of a Hilbert space H.Let S:C→C be a nonexpansive mapping.Then F(S)is a closed convex subset of C and the mapping I–S is demiclosed at,i.e. whenever{xn} ⊂C is such that xnx and(I–S)xn→,then

(I–S)x= .

Lemma .([]) Let C be a nonempty closed convex subset of a real Hilbert space H. As-sume that the mapping F :C→H is monotone and weakly continuous along segments,

that is,F(x+ty)→F(x)weakly as t→.Then the variational inequality

x∗∈C, Fx∗,x–x∗≥

for all x∈C is equivalent to the dual variational inequality

x∗∈C, Fx,x–x∗≥

for all x∈C.

Lemma .([]) Let{xn},{yn}be bounded sequences in a Banach space X and{βn}be a

sequence in[, ]with

 <lim inf

n→∞ βn≤lim sup_n_→∞ βn< .

Suppose that xn+= ( –βn)yn+βnxnfor all n≥and

lim sup

n→∞

yn+–yn–xn+–xn

≤.

Thenlimn→∞yn–xn= .

Lemma .([]) Assume that{an}is a sequence of nonnegative real numbers such that

an+≤( –γn)an+δnγn

for all n≥,where{γn}is a sequence in(, )and{δn}is a sequence such that

(a) ∞_n_=γn=∞;

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3 Main results

In this section, we prove our main results.

Theorem . Let C be a nonempty closed convex subset of a real Hilbert space H.Let A be anα-inverse strongly monotone mapping from C into H.Let f :C→H be aρ-contraction andγ be a constant such that <γ <_ρ.Let B be a maximal monotone operator on H such that the domain of B is included in C.Let JB

λ= (I+λB)–be the resolvent of B for anyλ> 

and S be a nonexpansive mapping from C into itself such that F(S)∩(A+B)–_₌_∅_._Let_λ

andκ be two constants satisfying a≤λ≤b,where[a,b]⊂(, α)andκ∈(, ).For any t∈(,  – λ

α),let{xt} ⊂C be a net generated by

tγf(xt) + ( –t)xt–λAxt. (.)

Then the net{xt}converges strongly as t→+to a pointx˜=P_F(S)∩(A+B)–_(γf(x˜)),which

solves the following variational inequality:

γf(x˜) –x˜,x˜–z≥

for all z∈F(S)∩(A+B)–_.

Proof First, we show that the net{xt} is well deﬁned. For anyt∈(,  – _αλ), we deﬁne a mappingW := ( –κ)S+κJB

λ(tγf + ( –t)I–λA). Note thatJλB, S, and I– λ –tA(see

Lemma .) are nonexpansive. For anyx,y∈C, we have

Wx–Wy

=( –κ)(Sx–Sy) +κ

JλB

tγf(x) + ( –t)

I– λ  –tA

x

–J_λB

tγf(y) + ( –t)

I– λ  –tA

y

≤( –κ)Sx–Sy

+κtγf(x) –f(y)+ ( –t)

I– λ  –tA

x–

I– λ  –tA

y

≤( –κ)x–y+κtγf(x) –f(y)

+ ( –t)κ

I– λ  –tA

x–

I– λ  –tA

y

≤( –κ)x–y+tκγρx–y+ ( –t)κx–y

= – ( –γρ)κtx–y,

which implies the mappingWis a contraction onC. We usextto denote the unique ﬁxed point ofWinC. Therefore,{xt}is well deﬁned. Setyt=JλButandut=γf(xt) + ( –t)xt– λAxtfor allt> . Takingz∈F(S)∩(A+B)–_{, it is obvious that}_z₌_Sz₌_JB

λ(z–λAz) for all λ>  and so

z=Sz=J_λB(z–λAz) =J_λB

tz+ ( –t)

I– λ  –tA

z

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for allt∈(,  –_αλ). From (.), it follows that

xt–z=( –κ)(Sxt–z) +κ(yt–z)

≤( –κ)Sxt–z+κyt–z ≤( –κ)xt–z+κyt–z.

Hence we getxt–z ≤ yt–z. SinceJλBis nonexpansive, we have

yt–z

=JλB

tγf(xt) + ( –t)

xt– λ  –tAxt

–JλB

tz+ ( –t)

z– λ  –tAz

≤

tγf(xt) + ( –t)

–

tz+ ( –t)

z– λ  –tAz

=( –t)

–

z– λ  –tAz

+tγf(xt) –z

≤( –t)

I– λ  –tA

xt–

I– λ  –tA

z+tγf(xt) –f(z)+tγf(z) –z

≤( –t)xt–z+tγρxt–z+tγf(z) –z. (.)

Thus it follows that

xt–z ≤ 

 –γργf(z) –z.

Therefore,{xt}is bounded. We deduce immediately that{f(xt)},{Axt},{Sxt},{ut}, and{yt}

are also bounded. By using the convexity of · and theα-inverse strong monotonicity of

A, from (.), we derive

xt–z

≤ yt–z

≤( –t)

–

z– λ  –tAz

+tγf(xt) –z



≤( –t)

–

z– λ  –tAz

+tγf(xt) –z

= ( –t)(xt–z) –

λ

 –t(Axt–Az)

+tγf(xt) –z



= ( –t)

xt–z– λ

 –tAxt–Az,xt–z+

λ

( –t)Axt–Az 

+tγf(xt) –z



≤( –t)

xt–z–

αλ

 –tAxt–Az

₊ λ

( –t)Axt–Az 

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= ( –t)

xt–z+ λ ( –t)

λ– ( –t)αAxt–Az

+tγf(xt) –z

≤( –t)xt–z+

λ ( –t)

λ– ( –t)αAxt–Az+tγf(xt) –z



(.)

and so

λ ( –t)

( –t)α–λAxt–Az≤tγf(xt) –z



–txt–z→.

By the assumption, we have ( –t)α–λ>  for allt∈(,  – λ

α) and so we obtain

lim

t→+Axt–Az= . (.)

Next, we showxt–Sxt →. By using the ﬁrm nonexpansivity ofJλB, we have

yt–z=JλB

tγf(xt) + ( –t)xt–λAxt–z

=JλB

tγf(xt) + ( –t)xt–λAxt–JλB(z–λAz) 

≤tγf(xt) + ( –t)xt–λAxt– (z–λAz),yt–z

=

tγf(xt) + ( –t)xt–λAxt– (z–λAz) 

+yt–z

–tγf(xt) + ( –t)xt–λ(Axt–Az) –yt.

yt–z≤tγf(xt) + ( –t)xt–λAxt– (z–λAz)

By the nonexpansivity ofI– λ

–tA, we have

tγf(xt) + ( –t)xt–λAxt– (z–λAz)

=( –t)

–

z– λ  –tAz

+tγf(xt) –z



≤( –t)

–

z– λ  –tAz

+tγf(xt) –z ≤( –t)xt–z+tγf(xt) –z



and thus

xt–z≤ yt–z

≤( –t)xt–z+tγf(xt) –z

Hence it follows that

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SinceAxt–Az →, we deducelimt→+xt–yt= , which implies that

lim

t→+xt–Sxt= . (.)

From (.), we have

yt–z

≤( –t)

–

z– λ  –tAz

+tγf(xt) –z



= ( –t)

–

z– λ  –tAz



+ t( –t)

γf(xt) –z,

xt–

λ  –tAxt

–

z– λ  –tAz

+tγf(xt) –z



≤( –t)xt–z+ t( –t)

γf(xt) –z,xt– λ

 –t(Axt–Az) –z

+t_γ_f₍_x

t) –z



= ( –t)xt–z+ t( –t)γ

f(xt) –f(z),xt– λ

+ t( –t)

γf(z) –z,xt– λ

+tγf(xt) –z.

Note thatxt–z ≤ yt–z. Then we obtain

xt–z≤( –t)xt–z+ t( –t)γf(xt) –f(z)

xt–z+

_{ –}λ_t(Axt–Az)

+ t( –t)

+tγf(xt) –z

≤( –t)xt–z+ t( –t)γρxt–z+ tλγρxt–zAxt–Az

+ t( –t)

γf(z) –z,xt–

λ

+tγf(xt) –z



≤ – ( –γρ)txt–z+ t

( –t)

+ t

γf(xt) –z 

+xt–z+λγρxt–zAxt–Az

.

xt–z≤   –γρ

+ t

γf(xt) –z 

+xt–z+tγf(z) –zxt– λ

+λγρxt–zAxt–Az

≤ 

 –γρ

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whereMis some constant such that

sup 

 –γρ



γf(xt) –z 

+xt–z

+γf(z) –zxt–

λ

,

λγρxt–z:t∈

,  – λ

α

≤M.

Next, we show that{xt}is relatively norm-compact ast→+. Assume that{tn} ⊂(,  – λ

α) is such thattn→+ asn→ ∞. Putxn:=xtn. From (.), we have

xn–z≤   –γρ

γf(z) –z,xn–z+tn+Axn–AzM. (.)

Since{xn}is bounded, without loss of generality, we may assume thatxnjx˜∈C. Hence

ynjx˜because ofxn–yn →. From (.), we have

lim

n→∞xn–Sxn= . (.)

We can use Lemma . to (.) to deduce x˜∈F(S). Further, we show thatx˜ is also in (A+B)–_{. Let}_v_∈_Bu_{. Note that}_yn₌_JB

λ(tnγf(xn) + ( –tn)xn–λAxn) for alln≥. Then we have

tnγf(xn) + ( –tn)xn–λAxn∈(I+λB)yn

⇒ tnγf(xn)

λ +

 –tn

λ xn–Axn–

yn

λ ∈Byn.

SinceBis monotone, we have, for all (u,v)∈B,

tnγf(xn)

λ +

 –tn

λ xn–Axn–

yn

λ –v,yn–u

≥

⇒ tnγf(xn) + ( –tn)xn–λAxn–yn–λv,yn–u

≥

⇒ Axn+v,yn–u ≤ 

λxn–yn,yn–u–

tn

λ

xn–γf(xn),yn–u

⇒ A˜x+v,yn–u ≤ 

λxn–yn,yn–u–

tn

λ

xn–γf(xn),yn–u

+A˜x–Axn,yn–u

⇒ Ax˜+v,yn–u ≤



λxn–ynyn–u+

tn

λxn–γf(xn)yn–u

+A˜x–Axnyn–u.

A˜x+v,x˜–u ≤ 

λxnj–ynjynj–u+

tnj

λxnj–γf(xnj)ynj–u

+A˜x–Axnjynj–u+A˜x+v,x˜–ynj. (.)

Sincexnj–x˜,Axnj–A˜x ≥αAxnj–A˜x,Axnj→Az, andxnjx˜, it follows thatAxnj→

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v,x˜–u ≤, that is,–A˜x–v,x˜–u ≥. SinceBis maximal monotone, we have –A˜x∈B˜x. This shows that ∈(A+B)x˜. Hence we havex˜∈F(S)∩(A+B)–_{. Therefore, substituting}

˜

xforzin (.), we get

xn–x˜≤

  –γρ

γf(x˜) –x˜,xn–x˜

+tn+Axn–Ax˜

M.

Consequently, the weak convergence of{xn}tox˜actually implies thatxn→ ˜x. This proved the relative norm-compactness of the net{xt}ast→+.

Now, we return to (.) and, taking the limit asn→ ∞, we have

˜x–z≤ 

 –γρ

γf(z) –z,x˜–z

for allz∈F(S)∩(A+B)–_{. In particular,}_x_˜_{solves the following variational inequality:}

˜

x∈F(S)∩(A+B)–, γf(z) –z,x˜–z≥

for allz∈F(S)∩(A+B)–_{ or the equivalent dual variational inequality (see Lemma .):}

˜

x∈F(S)∩(A+B)–, γf(x˜) –x˜,x˜–z≥

for allz∈F(S)∩(A+B)–_{. Hence} _x_˜₌_P

F(S)∩(A+B)–_(γf(x˜)). Clearly, this is suﬃcient to

conclude that the entire net{xt}converges tox˜. This completes the proof.

Theorem . Let C be a nonempty closed convex subset of a real Hilbert space H.Let A be anα-inverse strongly monotone mapping from C into H.Let f :C→H be aρ-contraction andγ be a constant such that <γ < 

ρ.Let B be a maximal monotone operator on H such

that the domain of B is included in C.Let JB

λ= (I+λB)–be the resolvent of B for anyλ> 

and S be a nonexpansive mapping from C into itself such that F(S)∩(A+B)–_₌_∅_._For

any x∈C,let{xn} ⊂C be a sequence generated by

(.)

for all n≥,whereκ∈(, ),{λn} ⊂(, α)and{αn} ⊂(, )satisfy the following condi-tions:

(a) limn→∞αn= ,limn→∞ααn+n = and nαn=∞;

(b) a( –αn)≤λn≤b( –αn),where[a,b]⊂(, α)andlimn→∞λnα+–λn+n = .

Then the sequence{xn}converges strongly to a pointx˜=PF(S)∩(A+B)–_(γf(x˜)),which solves

the following variational inequality:

γf(x˜) –x˜,x˜–z≥

for all z∈F(S)∩(A+B)–_.

Proof Setyn=JB

λnun,un=αnγf(xn) + ( –αn)xn–λnAxnfor alln≥. Pick upz∈F(S)∩

(A+B)–_{. It is obvious that}

z=Sz=J_λB_n(z–λnAz) =JλBn

αnz+ ( –αn)

z– λn  –αn

Az

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for alln≥. SinceJB

λn,S, andI–

λn

–αnAare nonexpansive for allλ>  andn≥, we have

yn–z

=JλBn

–z

=JλBn

αnγf(xn) + ( –αn)

xn– λn  –αn

Axn

–J_λB_n

αnz+ ( –αn)

z– λn  –αn

Az

≤

αnγf(xn) + ( –αn)

Axn

–

αnz+ ( –αn)

z– λn  –αn

Az



=( –αn)

Axn

–

z– λn  –αn

Az

+αn

γf(xn) –z

≤( –αn)xn–z+αnγf(xn) –γf(z)+αnγf(z) –z ≤ – ( –γρ)αn

xn–z+αnγf(z) –z. (.)

Hence we have

xn+–z ≤( –κ)Sxn–z+κyn–z ≤( –κ)xn–z+κ – ( –γρ)αn

xn–z+καnγf(z) –z

= – ( –γρ)καn

xn–z+καnγf(z) –z.

By induction, we have

xn+–z ≤max

x–z, 

 –γργf(z) –z

.

Therefore,{xn}is bounded. SinceAisα-inverse strongly monotone, it is _α-Lipschitz con-tinuous. We deduce immediately that{f(xn)},{Sxn},{Axn},{un}, and{yn}are also bounded.

By using the convexity of · and theα-inverse strong monotonicity ofA, it follows from (.) that

( –αn)

xn–

λn

 –αn Axn

–

z– λn  –αn

Az

+αn

γf(xn) –z



≤( –αn)

Axn

–

z– λn  –αn

Az



+αnγf(xn) –z



= ( –αn)

(xn–z) – λn  –αn

(Axn–Az) 

+αnγf(xn) –z



= ( –αn)

xn–z–

λn

 –αn

Axn–Az,xn–z+

λ

n

( –αn)

Axn–Az

+αnγf(xn) –z

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≤( –αn)

xn–z– αλn  –αn

Axn–Az+ λ 

n

( –αn)

Axn–Az

+αnγf(xn) –z



= ( –αn)

xn–z+ λn ( –αn)

λn– ( –αn)α

Axn–Az

+αnγf(xn) –z. (.)

By the condition (c), we getλn– ( –αn)α≤ for alln≥. Then, from (.) and (.),

we obtain

JλBnun–z



≤( –αn)

xn–z+ λn ( –αn)

λn– ( –αn)α

Axn–Az

+αnγf(xn) –z. (.)

From (.), it follows that

xn+–z=( –κ)(Sxn–z) +κ

JλBnun–z



≤( –κ)xn–z+κJλBnun–z



. (.)

Next, we estimatexn+–xn. In fact, we have

xn+–xn+=( –κ)(Sxn+–Sxn) +κ(yn+–yn)

≤( –κ)xn+–xn+κyn+–yn

and

yn+–yn=JλBn+un+–J

B

λnun

≤JλBn+un+–J

B

λn+un+J

B

λn+un–J

B

λnun

≤αn+γf(xn+) + ( –αn+)xn+–λn+Axn+

–αnγf(xn) + ( –αn)xn–λnAxn+JλBn+un–J

B

λnun

=αn+γ

f(xn+) –f(xn)+ (αn+–αn)γf(xn)

+ ( –αn+)

I– λn+  –αn+

A

xn+–

I– λn+  –αn+

A

xn

+ (αn–αn+)xn+ (λn–λn+)Axn

+JλBn+un–J

B

λnun

≤αn+γρxn+–xn+|αn+–αn|γf(xn)+xn

+ ( –αn+)xn+–xn+|λn–λn+|Axn+JλBn+un–J

B

λnun

= – ( –γρ)αn+

xn+–xn+|αn+–αn|γf(xn)+xn

+|λn–λn+|Axn+JλBn+un–J

B

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By the resolvent identity (.), we have

JλBn+un=J

B

λn

λn+

un+

 – λn λn+

JλBn+un

.

J_λB_n_+un–JλBnun=

J_λB_n

λn

λn+

un+

 – λn

λn+

J_λB_n_+un

–J_λB_nun

≤

λn

λn+

un+

 – λn λn+

JλBn+un

–un

≤|λn+–λn| λn+

un–JλBn+un

and so

xn+–xn+

≤( –κ)xn+–xn+κyn+–yn

≤( –κ)xn+–xn+κ

 – ( –γρ)αn+

xn+–xn

+κ|αn+–αn|γf(xn)+xn

+κ|λn–λn+|Axn

+κ|λn+–λn| λn+

un–JλBn+un

≤ – ( –γρ)καn+

xn+–xn+ ( –γρ)καn+ _|

αn+–αn| αn+

γf(xn)+xn  –γρ

+|λn–λn+| αn+

Axn

 –γρ +

|λn+–λn| αn+λn+

un–JB

λn+un  –γρ

.

By the assumptions, we know that |αn+–αn|

αn+ → and

|λn+–λn|

αn+ →. Then, from Lemma .,

we get

lim

n→∞xn+–xn= . (.)

Thus, from (.) and (.), it follows that

xn+–z

≤( –κ)xn–z+κJλBnun–z



≤κ

( –αn)

xn–z+ λn ( –αn)

λn– ( –αn)α

Axn–Az

+αnγf(xn) –z



+ ( –κ)xn–z

= [ –καn]xn–z+

κλn

 –αn

λn– ( –αn)α

Axn–Az+καnγf(xn) –z



≤ xn–z+ κλn  –αn

λn– ( –αn)α

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and so

κλn

( –αn)

( –αn)α–λn

Axn–Az

≤ xn–z–xn+–z+καnγf(xn) –z



≤xn–z–xn+–z

xn+–xn+καnγf(xn) –z

 .

Sincelimn→∞αn= ,limn→∞xn+–xn= , andlim infn→∞(–ακλnn)(( –αn)α–λn) > ,

we have

lim

n→∞Axn–Az= . (.)

Next, we showxn–Sxn →. By using the ﬁrm nonexpansivity ofJλBn, we have

JλBnun–z

 =JλBn

–JλBn(z–λnAz)



≤αnγf(xn) + ( –αn)xn–λnAxn– (z–λnAz),JλBnun–z

=

αnγf(xn) + ( –αn)xn–λnAxn– (z–λnAz) 

+JλBnun–z



–αnγf(xn) + ( –αn)xn–λn(Axn–Az) –JλBnun

 .

From the condition (c) and theα-inverse strongly monotonicity ofA, we know thatI– λnA/( –αn) is nonexpansive. Hence it follows that

αnγf(xn) + ( –αn)xn–λnAxn– (z–λnAz)



=( –αn)

xn–

λn

 –αn Axn

–

z– λn  –αn

Az

+αn

γf(xn) –z



≤( –αn)

Axn

–

z– λn  –αn

Az



+αnγf(xn) –z



≤( –αn)xn–z+αnγf(xn) –z

and thus

JλBnun–z



≤



( –αn)xn–z+αnγf(xn) –z



+JλBnun–z



–αnγf(xn) + ( –αn)xn–JλBnun–λn(Axn–Az)

 ,

that is,

JλBnun–z



≤( –αn)xn–z+αnγf(xn) –z

–αnγf(xn) + ( –αn)xn–JλBnun–λn(Axn–Az)



= ( –αn)xn–z+αnγf(xn) –z



–αnγf(xn) + ( –αn)xn–JλBnun

(15)

+ λn

αnγf(xn) + ( –αn)xn–JλBnun,Axn–Az

–λ_nAxn–Az

≤( –αn)xn–z+αnγf(xn) –z



–αnγf(xn) + ( –αn)xn–JλBnun



+ λnαnγf(xn) + ( –αn)xn–JλBnunAxn–Az.

This together with (.) implies that

xn+–z≤( –κ)xn–z+κ( –αn)xn–z+καnγf(xn) –z



–καnγf(xn) + ( –αn)xn–JλBnun



+ λnκαnγf(xn) + ( –αn)xn–JλBnunAxn–Az

= [ –καn]xn–z+καnγf(xn) –z



–καnγf(xn) + ( –αn)xn–JλBnun



+ λnκαnγf(xn) + ( –αn)xn–JλBnunAxn–Az

and hence

καnγf(xn) + ( –αn)xn–JλBnun



≤ xn–z–xn+–z–καnxn–z+καnγf(xn) –z



+ λnκαnγf(xn) + ( –αn)xn–JλBnunAxn–Az

≤xn–z+xn+–z

xn+–xn+καnγf(xn) –z



+ λnκαnγf(xn) + ( –αn)xn–JλBnunAxn–Az.

Sincexn+–xn →,αn→, andAxn–Az → (by (.)), we deduce

lim

n→∞αnγf(xn) + ( –αn)xn–J B

λnun= .

This implies that

lim

n→∞xn–J B

λnun= . (.)

Combining (.), (.), and (.), we get

lim

n→∞xn–Sxn= . (.)

Putx˜=lim_t_→+xt=PF(S)∩(A+B)–_(γf(x˜)), where{x_t}is the net deﬁned by (.).

Finally, we show thatxn→ ˜x. Takingz=x˜in (.), we getAxn–Ax˜ →. First, we

provelim sup_n_→∞γf(x˜) –x˜,xn–x ≤˜ . We take a subsequence{xni}of{xn}such that

lim sup

n→∞

=lim

i→∞

γf(x˜) –x˜,xni–x˜

.

There exists a subsequence{xn_ij}of{xni}which converges weakly to a pointw∈C. Hence

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principle of the nonexpansive mapping (see Lemma .) and (.), we deducew∈F(S). Furthermore, by a similar argument to that of Theorem ., we can show thatwis also in (A+B)–. Hence we havew∈F(S)∩(A+B)–. This implies that

lim sup

n→∞

γf(x˜) –x˜,xn–x˜=lim

j→∞

γf(x˜) –x˜,xn_ij–x˜=γf(x˜) –x˜,w–x˜.

Note thatx˜=PF(S)∩(A+B)–_(γf(x˜)). Then we have

γf(x˜) –x˜,w–x˜≤

for allw∈F(S)∩(A+B)–_{. Therefore, it follows that}

lim sup

n→∞

γf(x˜) –x˜,xn–x˜≤.

From (.), we have

xn+–x˜ 

≤( –κ)xn–x˜ +κJλBnun–x˜



= ( –κ)xn–x˜ +κJλBnun–J

B

λn(x˜–λnA˜x)



≤( –κ)xn–x˜ +κun– (x˜–λnA˜x)

= ( –κ)xn–x˜ +καnγf(xn) + ( –αn)xn–λnAxn– (x˜–λnA˜x)



=κ( –αn)

Axn

–

˜ x– λn

 –αn Ax˜

+αn

γf(xn) –x˜



+ ( –κ)xn–x˜

= ( –κ)xn–x˜ +κ

( –αn)

Axn

–

˜ x– λn

 –αn A˜x



+ αn( –αn)

γf(xn) –x˜,

Axn

–

˜ x– λn

 –αnA˜ x

+α_nγf(xn) –x˜



≤( –κ)xn–x˜ +κ( –αn)xn–x˜ + αnλn

γf(xn) –x˜,Axn–A˜x

+ αn( –αn)γ

f(xn) –f(x˜),xn–x˜+ αn( –αn)

+α_nγf(xn) –x˜

≤( –κ)xn–x˜+κ

( –αn)xn–x˜+ αnλnγf(xn) –x˜Axn–Ax˜

+ αn( –αn)γρxn–x˜ + αn( –αn)

γf(x˜) –x˜,xn–x˜+α_nγf(xn) –x˜

≤ – κ( –γρ)αn

xn–x˜ + αnκλnγf(xn) –x˜Axn–Ax˜

+ αnκ( –αn)

+κα_nγf(xn) –x˜



+xn–x˜

= – κ( –γρ)αn

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+ κ( –γρ)αn

λn

 –γργf(xn) –x˜Axn–A˜x

+  –αn  –γρ

γf(x˜) –x˜,xn–x˜+ αn

( –γρ)γf(xn) –x˜ 

+xn–x˜ .

It is clear that _nκ( –γρ)αn=∞and

lim sup

n→∞

λn

 –γργf(xn) –x˜Axn–Ax˜+  –αn

 –γρ

+ αn

( –γρ)γf(xn) –x˜ 

+xn–x˜ ≤.

Therefore, we can apply Lemma . to conclude thatxn→ ˜x. This completes the proof.

Remark . One quite often seeks a particular solution of a given nonlinear problem, in particular, the minimum-norm element. For instance, given a closed convex subsetCof a Hilbert spaceHand a bounded linear operatorW:H→H, whereHis another Hilbert space. TheC-constrained pseudoinverse ofW,W_C†, is then deﬁned as the minimum-norm solution of the constrained minimization problem

W_C†(b) :=arg min

x∈CWx–b,

which is equivalent to the ﬁxed point problem

u=proj_Cu–μW∗(Wu–b),

whereW∗is the adjoint ofW andμ>  is a constant, andb∈His such thatPW(C)(b)∈

W(C). From Theorems . and ., we get the following corollaries which can ﬁnd the minimum-norm element inF(S)∩(A+B)–; that is, ﬁndx˜∈F(S)∩(A+B)– such that

˜

x=arg min

x∈F(S)∩(A+B)–_x.

Corollary . Let C be a nonempty closed convex subset of a real Hilbert space H.Let A be anα-inverse strongly monotone mapping from C into H.Let B be a maximal monotone operator on H such that the domain of B is included in C.Let JλB= (I+λB)–be the resolvent

of B for anyλ> and S be a nonexpansive mapping from C into itself such that F(S)∩(A+

B)–_₌_∅_._Let_λ_and_κ _{be two constants satisfying a}_≤_λ_≤_b_,_where_[_a_,_b_]_⊂_{(, }_α₎_and κ∈(, ).For any t∈(,  – λ

α),let{xt} ⊂C be a net generated by

( –t)xt–λAxt.

Then the net{xt}converges strongly as t→+to a pointx˜=PF(S)∩(A+B)–_()which is the

minimum-norm element in F(S)∩(A+B)–_.

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of B for anyλ> such that(A+B)–_₌ _∅_._Let_λ_{be a constant satisfying a}_≤_λ_≤_b_,_where [a,b]⊂(, α).For any t∈(,  –_αλ),let{xt} ⊂C be a net generated by

xt=JB

λ

( –t)xt–λAxt

.

Then the net {xt} converges strongly as t→+to a pointx˜=P(A+B)–_(), which is the

minimum-norm element in(A+B)–_.

Corollary . Let C be a nonempty closed convex subset of a real Hilbert space H.Let A be anα-inverse strongly monotone mapping from C into H.Let B be a maximal monotone operator on H such that the domain of B is included in C.Let JB

λ= (I+λB)–be the resolvent

of B for anyλ> and let S be a nonexpansive mapping from C into itself such that F(S)∩ (A+B)–_₌_∅_._{For any x}

∈C,let{xn} ⊂C be a sequence generated by

( –αn)xn–λnAxn

for all n≥,whereκ∈(, ),{λn} ⊂(, α),and{αn} ⊂(, )satisfy the following condi-tions:

(a) limn→∞αn= ,limn→∞ααn+n = ,and nαn=∞;

(b) a( –αn)≤λn≤b( –αn),where[a,b]⊂(, α)andlimn→∞λn+–λαn n = .

Then the sequence {xn}converges strongly to a point x˜=PF(S)∩(A+B)–_(),which is the

minimum-norm element in F(S)∩(A+B)–_.

Corollary . Let C be a closed convex subset of a real Hilbert space H.Let A be anα -inverse strongly monotone mapping from C into H and let B be a maximal monotone oper-ator on H such that the domain of B is included in C.Let JλB= (I+λB)–be the resolvent of B

for anyλ> such that(A+B)–_₌_∅_._{For any x}

∈C,let{xn} ⊂C be a sequence generated

by

xn+= ( –κ)xn+κJλBn

( –αn)xn–λnAxn

for all n≥,whereκ∈(, ),{λn} ⊂(, α),and{αn} ⊂(, )satisfy the following condi-tions:

(b) a( –αn)≤λn≤b( –αn),where[a,b]⊂(, α)andlimn→∞λn+–λαn n = .

Then the sequence {xn} converges strongly to a point x˜ =P₍_A₊_B₎–_(), which is the

minimum-norm element in(A+B)–_.

Remark . The present paper provides some methods which do not use projection for ﬁnding the minimum-norm solution problem.

4 Applications

Next, we consider the problem for ﬁnding the minimum-norm solution of a mathematical model related to equilibrium problems. LetCbe a nonempty closed convex subset of a Hilbert space andG:C×C→Rbe a bifunction satisfying the following conditions:

(E) G(x,x) = for allx∈C;

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(E) for allx,y,z∈C,lim sup_t_↓_G(tz+ ( –t)x,y)≤G(x,y); (E) for allx∈C,G(x,·)is convex and lower semicontinuous.

Then the mathematical model related to the equilibrium problem (with respect toC) is as follows:

Findx˜∈Csuch that

G(x˜,y)≥ (.)

for ally∈C. The set of such solutionsx˜is denoted byEP(G). The following lemma appears implicitly in Blum and Oettli [].

Lemma . Let C be a nonempty closed convex subset of a Hilbert space H.Let G be a

bifunction from C×C into R satisfying the conditions(E)-(E).Then,for any r> and x∈H,there exists z∈C such that

G(z,y) +

ry–z,z–x ≥

for all y∈C.

The following lemma was given in Combettes and Hirstoaga [].

Lemma . Assume that G is a bifunction from C×C into R satisfying the conditions

(E)-(E).For any r> and x∈H,deﬁne a mapping Tr:H→C as follows:

Tr(x) =

z∈C:G(z,y) +

ry–z,z–x ≥,∀y∈C

for all x∈H.Then the following hold: (a) Tris single-valued;

(b) Tris a ﬁrmly nonexpansive mapping,i.e.,for allx,y∈H,

Trx–Try≤ Trx–Try,x–y;

(c) F(Tr) =EP(G);

(d) EP(G)is closed and convex.

We call such aTrthe resolvent ofGfor anyr> . Using Lemmas . and ., we have the following lemma (see [] for a more general result).

Lemma . Let C be a nonempty closed convex subset of a Hilbert space H.Let G be a bifunction from C×C into R satisfying the conditions(E)-(E).Let AGbe a multi-valued mapping from H into itself deﬁned by

AGx= ⎧ ⎨ ⎩

{z∈H:G(x,y)≥ y–x,z,∀y∈C}, x∈C,

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Then EP(G) =A–

G()and AGis a maximal monotone operator withdom(AG)⊂C.Further, for any x∈H and r> ,the resolvent Trof G coincides with the resolvent of AG,i.e.,

Trx= (I+rAG)–x.

Form Lemma . and Theorems . and ., we have the following results.

Theorem . Let C be a nonempty closed convex subset of a real Hilbert space H.Let G be a bifunction from C×C into R satisfying the conditions(E)-(E)and Tr be the re-solvent of G for any r> .Let S be a nonexpansive mapping from C into itself such that F(S)∩EP(G)= ∅.For any t∈(, ),let{xt} ⊂C be a net generated by

xt= ( –κ)Sxt+κTr( –t)xt.

Then the net{xt}converges strongly as t→+to a pointx˜=PF(S)∩EP(G)(),which is the minimum-norm element in F(S)∩EP(G).

Proof From Lemma ., we knowAG is maximal monotone. Thus, in Theorem ., we can setJB

λ=Tr. At the same time, in Theorem ., we can choosef =  andA= , and (.) reduces to

xt= ( –κ)Sxt+κTr( –t)xt.

Consequently, from Theorem ., we get the desired result. This completes the proof.

Corollary . Let C be a nonempty closed convex subset of a real Hilbert space H.Let G be a bifunction from C×C into R satisfying the conditions(E)-(E)and Trbe the resolvent of G for any r> .Suppose that EP(G)=∅.For any t∈(, ),let{xt} ⊂C be a net generated by

xt=Tr( –t)xt.

Then the net {xt} converges strongly as t→+ to a point x˜ =PEP(G)(), which is the minimum-norm element in EP(G).

Theorem . Let C be a nonempty closed and convex subset of a real Hilbert space H.

Let G be a bifunction from C×C into R satisfying the conditions(E)-(E)and Tλbe the

resolvent of G for anyλ> .Let S be a nonexpansive mapping from C into itself such that F(S)∩EP(G)=∅.For any x∈C,let{xn} ⊂C be a sequence generated by

xn+= ( –κ)Sxn+κTλn

( –αn)xn

for all n≥,whereκ∈(, ),{λn} ⊂(,∞),and{αn} ⊂(, )satisfy the conditions: (a) lim_n_→∞αn= ,limn→∞ααn+n = ,and nαn=∞;

(b) a≤λn≤b,where[a,b]⊂(,∞)andlimn→∞λn+–λαn n= .

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Proof From Lemma ., we knowAG is maximal monotone. Thus, in Theorem ., we can setJB

λn=Tλn. At the same time, in Theorem ., we can choosef =  andA= , and

(.) reduces to

xn+= ( –κ)Sxn+κTλn

( –αn)xn

for alln≥. Consequently, from Theorem ., we get the desired result. This completes

the proof.

Corollary . Let C be a nonempty closed convex subset of a real Hilbert space H.Let G be a bifunction from C×C into R satisfying the conditions(E)-(E)and Tλbe the resolvent

of G for anyλ> .Suppose EP(G)=∅.For any x∈C,let{xn} ⊂C be a sequence generated

by

xn+= ( –κ)xn+ ( –βn)Tλn

( –αn)xn

for all n≥,whereκ∈(, ),{λn} ⊂(,∞),and{αn} ⊂(, )satisfy the following condi-tions:

(b) a≤λn≤b,where[a,b]⊂(,∞)andlimn→∞λn+–λαn n= .

Then the sequence{xn}converges strongly to a pointx˜=PEP(G)(),which is the minimum-norm element in EP(G).

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

1_{Department of Mathematics, King Abdulaziz University, Jeddah, 21589, Saudi Arabia.}2_{Department of Mathematics}

Education and the RINS, Gyeongsang National University, Chinju, 660-701, Korea.3_{Department of Mathematics, Tianjin} Polytechnic University, Tianjin, 300387, China.4_{School of Mathematics and Information Science, Beifang University of} Nationalities, Yinchuan, 750021, China.

Acknowledgements

This project was funded by the Deanship of Scientiﬁc Research (DSR), King Abdulaziz University, under Grant No. (31-130-35-HiCi) and the ﬁfth author was supported in part by NNSF of China (61362033) and NGY2012097.

Received: 19 March 2014 Accepted: 27 June 2014 Published:18 Aug 2014

References

1. Lu, X, Xu, HK, Yin, X: Hybrid methods for a class of monotone variational inequalities. Nonlinear Anal.71, 1032-1041 (2009)

2. Rockafella, RT: Monotone operators and the proximal point algorithm. SIAM J. Control Optim.14, 877-898 (1976) 3. Moudaﬁ, A, Noor, MA: Sensitivity analysis of variational inclusions by the Wiener-Hopf equations technique. J. Appl.

Math. Stoch. Anal.12, 223-232 (1999)

4. Dong, H, Lee, BS, Huang, NJ: Sensitivity analysis for generalized parametric implicit quasi-variational inequalities. Nonlinear Anal. Forum6, 313-320 (2001)

5. Agarwal, RP, Huang, NJ, Tan, MY: Sensitivity analysis for a new system of generalized nonlinear mixed quasi-variational inclusions. Appl. Math. Lett.17, 345-352 (2004)

6. Jeong, JU: A system of parametric generalized nonlinear mixed quasi-variational inclusions in Lp spaces. J. Appl. Math. Comput.19, 493-506 (2005)

7. Lan, HY: A stable iteration procedure for relaxed cocoercive variational inclusion systems based on (A,η)-monotone operators. J. Comput. Anal. Appl.9, 147-157 (2007)

(22)

9. Peng, JW, Wang, Y, Shyu, DS, Yao, JC: Common solutions of an iterative scheme for variational inclusions, equilibrium problems, and ﬁxed point problems. J. Inequal. Appl.2008, Article ID 720371 (2008)

10. Ansari, QH, Balooee, J, Yao, J-C: Iterative algorithms for systems of extended regularized nonconvex variational inequalities and ﬁxed point problems. Appl. Anal.93, 972-993 (2014)

11. Bauschke, HH, Combettes, PL: A Dykstra-like algorithm for two monotone operators. Pac. J. Optim.4, 383-391 (2008) 12. Bauschke, HH, Combettes, PL, Reich, S: The asymptotic behavior of the composition of two resolvents. Nonlinear

Anal.60, 283-301 (2005)

13. Ceng, L-C, Al-Otaibi, A, Ansari, QH, Latif, A: Relaxed and composite viscosity methods for variational inequalities, ﬁxed points of nonexpansive mappings and zeros of accretive operators. Fixed Point Theory Appl.2014, 29 (2014) 14. Ceng, L-C, Ansari, QH, Schaible, S, Yao, J-C: Iterative methods for generalized equilibrium problems, systems of

general generalized equilibrium problems and ﬁxed point problems for nonexpansive mappings in Hilbert spaces. Fixed Point Theory12, 293-308 (2011)

15. Combettes, PL: Solving monotone inclusions via compositions of nonexpansive averaged operators. Optimization

53, 475-504 (2004)

16. Combettes, PL, Hirstoaga, SA: Approximating curves for nonexpansive and monotone operators. J. Convex Anal.13, 633-646 (2006)

17. Combettes, PL, Hirstoaga, SA: Visco-penalization of the sum of two monotone operators. Nonlinear Anal.69, 579-591 (2008)

18. Eckstein, J, Bertsekas, DP: On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators. Math. Program.55, 293-318 (1992)

19. Fang, YP, Huang, NJ:H-Monotone operator resolvent operator technique for quasi-variational inclusions. Appl. Math. Comput.145, 795-803 (2003)

20. Lin, LJ: Variational inclusions problems with applications to Ekeland’s variational principle, ﬁxed point and optimization problems. J. Glob. Optim.39, 509-527 (2007)

21. Lions, P-L, Mercier, B: Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal.16, 964-979 (1979)

22. Moudaﬁ, A: On the regularization of the sum of two maximal monotone operators. Nonlinear Anal.42, 1203-1208 (2009)

23. Passty, GB: Ergodic convergence to a zero of the sum of monotone operators in Hilbert spaces. J. Math. Anal. Appl.

72, 383-390 (1979)

24. Takahashi, S, Takahashi, W, Toyoda, M: Strong convergence theorems for maximal monotone operators with nonlinear mappings in Hilbert spaces. J. Optim. Theory Appl.147, 27-41 (2010)

25. Ding, F, Chen, T: Iterative least squares solutions of coupled Sylvester matrix equations. Syst. Control Lett.54, 95-107 (2005)

26. Liu, X, Cui, Y: Common minimal-norm ﬁxed point of a ﬁnite family of nonexpansive mappings. Nonlinear Anal.73, 76-83 (2010)

27. Sabharwal, A, Potter, LC: Convexly constrained linear inverse problems: iterative least-squares and regularization. IEEE Trans. Signal Process.46, 2345-2352 (1998)

28. Yao, Y, Chen, R, Xu, HK: Schemes for ﬁnding minimum-norm solutions of variational inequalities. Nonlinear Anal.72, 3447-3456 (2010)

29. Yao, Y, Liou, YC: Composite algorithms for minimization over the solutions of equilibrium problems and ﬁxed point problems. Abstr. Appl. Anal.2010, Article ID 763506 (2010)

30. Bauschke, HH, Borwein, JM: On projection algorithms for solving convex feasibility problems. SIAM Rev.38, 367-426 (1996)

31. Censor, Y, Zenios, SA: Parallel Optimization: Theory, Algorithms and Applications. Oxford University Press, New York (1997)

32. Takahashi, W, Toyoda, M: Weak convergence theorems for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl.118, 417-428 (2003)

33. Goebel, K, Kirk, WA: Topics in Metric Fixed Point Theory. Cambridge Studies in Advanced Mathematics, vol. 28. Cambridge University Press, Cambridge (1990)

34. Suzuki, T: Strong convergence theorems for inﬁnite families of nonexpansive mappings in general Banach spaces. Fixed Point Theory Appl.2005, 103-123 (2005)

35. Xu, HK: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc.2, 1-17 (2002)

36. Blum, E, Oettli, W: From optimization and variational inequalities to equilibrium problems. Math. Stud.63, 123-145 (1994)

37. Combettes, PL, Hirstoaga, SA: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal.6, 117-136 (2005) 38. Aoyama, K, Kimura, Y, Takahashi, W, Toyoda, M: On a strongly nonexpansive sequence in Hilbert spaces. J. Nonlinear

Convex Anal.8, 471-489 (2007) 10.1186/1687-1812-2014-174