Linesearch algorithms for split equilibrium problems and nonexpansive mappings

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R E S E A R C H

Open Access

Linesearch algorithms for split equilibrium

problems and nonexpansive mappings

Bui Van Dinh

1,2

_{, Dang Xuan Son}

3

_{, Liguo Jiao}

2

_{and Do Sang Kim}

2*

*_{Correspondence:}

[email protected]

2_{Department of Applied}

Mathematics, Pukyong National University, Busan, Korea Full list of author information is available at the end of the article

Abstract

In this paper, we first propose a weak convergence algorithm, called the linesearch algorithm, for solving a split equilibrium problem and nonexpansive mapping (SEPNM) in real Hilbert spaces, in which the first bifunction is pseudomonotone with respect to its solution set, the second bifunction is monotone, and fixed point mappings are nonexpansive. In this algorithm, we combine the extragradient method incorporated with the Armijo linesearch rule for solving equilibrium problems and the Mann method for finding a fixed point of an nonexpansive mapping. We then combine the proposed algorithm with hybrid cutting technique to get a strong convergence algorithm for SEPNM. Special cases of these algorithms are also given.

MSC: 47H09; 47J25; 65K10; 65K15; 90C99

Keywords: split equilibrium problem; common ﬁxed point problem; nonexpansive mapping; pseudomonotonicity; projection method; linesearch rule; weak and strong convergence

1 Introduction

Throughout the paper, unless otherwise is stated, we assume thatH andH are real

Hilbert spaces endowed with inner products and induced norms denoted by·,· and

· , respectively, whereasHrefers to any of these spaces. We writexk_→_x_or_xk_x

iﬀxk_{converges strongly or weakly to}_{x, respectively, as}_k_{→ ∞}_{. Let}_C,_Q_{be nonempty}

closed convex subsets inH,H, respectively, andA:H→Hbe a bounded linear op-erator. The split feasible problem (SFP) in the sense of Censor and Elfving [] is to find x∗∈Csuch thatAx∗∈Q. It turns out that SFP provides a unified framework for the study of many significant real-world problems such as in signal processing, medical image re-construction, intensity-modulated radiation therapy,et cetera; see, for example, [–]. To find a solution of SFP in finite-dimensional Hilbert spaces, a basic scheme proposed by Byrne [], called the CQ-algorithm, is defined as follows:

xk+=PC

xk+γAT(PQ–I)Axk

,

whereIis the identity mapping, andPCis projection mapping ontoC. Xu [] investigated

the SFP setting in inﬁnite-dimensional Hilbert spaces. In this case, the CQ-algorithm

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comes

xk+=PC

xk+γA∗(PQ–I)Axk

,

whereA∗is the adjoint operator ofA.

The split feasibility problem when C orQare ﬁxed points of mappings or common

ﬁxed points of mappings and solutions of variational inequality problems was considered in some recent research papers; see, for instance, [–]. Recently, Moudaﬁ [] (see also [–]) considered the split equilibrium problems (SEP), more precisely:

Letf :C×C→R,g:Q×Q→Rbe equilibrium bifunctions, that is,f(x,x) =g(u,u) =  for allx∈Candu∈Q. The split equilibrium problem takes the form

Findx∗∈Csuch thatx∗∈Sol(C,f) andAx∗∈Sol(Q,g),

whereSol(C,f) is the solution set of the following equilibrium problem (EP(C,f)):

Findx¯∈Csuch thatf(x,¯ y)≥,∀y∈C,

andSol(Q,g) is the solution set of the equilibrium problemEP(Q,g). See [, ] for more detail on equilibrium problems.

For obtaining a solution of SEP, He [] introduced an iterative method, which generates a sequence{xk_}_by

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

x∈C,{rk} ⊂(, +∞), μ> ,

f(yk_,_{y) +}  rky–y

k_,_yk_–_xk_≥_, _∀_y_∈_C,

g(uk,v) +_rkv–uk,uk–Ayk ≥, ∀v∈Q,

xk+₌_P

C(yk+μA∗(uk–Ayk)), ∀k≥.

Under certain conditions on bifunctions and parameters, the author shows that{xk_}_,

{yk}weakly converges to a solution of SEP, provided thatf andgare monotone onCand Q, respectively.

On the other hand, many researchers have been proposed numerical algorithms for ﬁnd-ing a common element of the set of solutions of monotone equilibrium problems and the set of ﬁxed points of nonexpansive mappings; see, for example, [–] and the references therein.

This paper focuses mainly on a split equilibrium problem and nonexpansive mapping involving pseudomonotone and monotone equilibrium bifunctions in real Hilbert spaces. In detail, letf :C×C→Rbe a pseudomonotone bifunction with respect to its

solu-tion set,g:Q×Q→Rbe a monotone bifunction, andS:C→CandT :Q→Qbe

nonexpansive mappings. The problem considered in this paper can be stated as follows (SEPNM(C,Q,A,f,g,S,T) or SEPNM for short):

Findx∗∈Csuch thatx∗∈Sol(C,f)∩Fix(S) andAx∗∈Sol(Q,g)∩Fix(T),

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It should be noticed that, under the monotonicity assumption on f andg, the solu-tion sets Sol(C,f) andSol(C,g) of the equilibrium problemsEP(C,f) and EP(Q,g) are

closed convex sets whenever f and g are lower semicontinuous and convex with

re-spect to the second variables. In addition, the nonexpansiveness assumption ofSandT also implies thatFix(S) andFix(T) are closed convex sets. Hence,Sol(C,f)∩Fix(S) and

Sol(Q,g)∩Fix(T) are closed convex sets. However, the main diﬃculty is that, even if these sets are convex, they are not given explicitly as in a standard mathematical programming problem, and therefore the projection onto those sets cannot be computed, and conse-quently, available methods (see,e.g., [, , ] and the references therein) cannot be ap-plied for solving SEPNM directly.

In this paper, we ﬁrst propose a weak convergence algorithm for solving SEPNM by using a combination of the extragradient method with Armijo linesearch type rule for an equi-librium problem [] (see also [–] for more detail on extragradient algorithms) and the Mann method [] (see also [, ]) for a ﬁxed point problem. We then combine this algorithm with hybrid cutting technique [] (see also []) to get a strong convergence algorithm for SEPNM.

The paper is organized as follows. The next section presents some preliminary results. A weak convergence algorithm and its special case are presented in Section . In the last section, we combine the method presented in Section  with the hybrid projection method for obtaining a strong convergence algorithm for SEPNM.

2 Preliminaries

LetHbe a real Hilbert space, andCa nonempty closed convex subset of H. ByPC we

denote the metric projection operator ontoC, that is,

PC(x)∈C: x–PC(x)≤ x–y, ∀y∈C.

The following well-known results will be used in the sequel.

Lemma  Suppose that C is a nonempty closed convex subset inH.Then PC has the

fol-lowing properties:

(a) PC(x)is singleton and well deﬁned for everyx;

(b) z=PC(x)if and only ifx–z,y–z ≤,∀y∈C;

(c) PC(x) –PC(y)≤ PC(x) –PC(y),x–y,∀x,y∈H;

(d) PC(x) –PC(y)≤ x–y–x–PC(x) –y+PC(y),∀x,y∈H.

Lemma  LetHbe a real Hilbert space.Then,for all x,y∈Handα∈[, ],we have

αx+ ( –α)y=αx_{+ ( –}_α₎_y_–_α_{( –}_α₎_x_–_y_.

Lemma (Opial’s condition) For any sequence{xk_{} ⊂}_H_{with x}k_x,_{we have the}

inequal-ity

lim inf k→+∞

xk–x<lim inf k→+∞

xk–y

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Deﬁnition  We say that an operatorT:H→His demiclosed at  if, for any sequence {xk}such thatxkxandTxk→ ask→ ∞, we haveTx= .

It is well known that, for a nonexpansive operatorT :H→H, the operatorI–T is demiclosed at ; see [], Lemma .

Now, we assume that the equilibrium bifunctionsg:Q×Q→Randf :C×C→R

satisfy the following assumptions, respectively.

Assumption A

(A) gis monotone onQ, that is,g(u,v) +g(v,u)≤for allu,v∈Q;

(A) g(u,·)is convex and lower semicontinuous onQfor eachu∈Q;

(A) for allu,v,w∈Q,

lim sup

λ↓

gλw+ ( –λ)u,v≤g(u,v).

Assumption B

(B) f is pseudomonotone onC, that is, iff(x,y)≥impliesf(y,x)≤for allx,y∈C;

(B) f(x,·)is convex and subdiﬀerentiable onCfor allx∈C;

(B) f is jointly weakly continuous onC×Cin the sense that, ifx,y∈Cand{xk},{yk} ⊂C

converge weakly toxandy, respectively, thenf(xk_,_yk₎_→_f_(x,_y)_as_k_→₊_∞_.

Letϕbe an equilibrium bifunction deﬁned onC×C. Forx,y∈C, we denote by∂ϕ(x,y)

the subgradient of the convex functionϕ(x,·) aty, that is,

∂ϕ(x,y) := ξˆ∈H:ϕ(x,z)≥ϕ(x,y) +ˆξ,z–y,∀z∈C

.

In particular,

∂ϕ(x,x) = ξˆ∈H:ϕ(x,z)≥ ˆξ,z–x,∀z∈C

.

Let Δbe an open convex set containingC. The next lemma can be considered as an

inﬁnite-dimensional version of Theorem . in [].

Lemma ([], Proposition .) Letϕ:Δ×Δ→Rbe an equilibrium bifunction satisfy-ing conditions(A)onΔand(A)on C.Letx,¯ ¯y∈Δ,and let{xk},{yk}be two sequences in Δconverging weakly tox,¯ ¯y,respectively.Then,for anyε> ,there existη> and kε∈N such that

∂ϕ

xk,yk⊂∂ϕ(x,¯ y) +¯ ε ηB

for every k≥kε,where B denotes the closed unit ball inH.

Lemma  Let the equilibrium bifunctionϕsatisfy assumptions(A)onΔand(A)on C,

and{xk} ⊂C,  <ρ≤ ¯ρ,{ρk} ⊂[ρ,ρ¯].Consider the sequence{yk}deﬁned as

yk=arg min

ϕxk,y+  ρk

y–xk:y∈C

.

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Proof First, we show that if{xk_}_{converges weakly to}_x∗_{, then}_{_yk_}_{is bounded. Indeed,}

yk=arg min

ϕxk,y+  ρk

y–xk:y∈C

and

ϕxk,xk+  ρk

xk–xk= .

Therefore,

ϕxk,yk+  ρk

yk–xk≤, ∀k.

In addition, for allξˆk∈∂ϕ(xk,xk), we have

ϕxk,yk+  ρk

yk–xk≥ξˆk,yk–xk+  ρk

yk–xk.

This implies

–ξˆkyk–xk+  ρk

yk–xk≤.

Hence,

yk–xk≤ρkξˆk, ∀k.

Because{ρk}is bounded,{xk}converges weakly tox∗andξˆk∈∂ϕ(xk,xk). By Lemma 

the sequence{ˆξk}is bounded; combining this with the boundedness of{xk_}_{, we get that}

{yk_}_{is also bounded.}

Now let us prove Lemma . Suppose that{yk_}_{is unbounded, that is, there exists a}

subse-quence{yki_{} ⊆ {}_yk_}_{such that}_lim

i→∞yki= +∞. By the boundedness of{xk}this implies

that{xki}is also bounded, and without loss of generality, we may assume that{xki} con-verges weakly to somex∗. By the same argument as before, we get that{yki}is bounded,

a contradiction. Therefore,{yk}is bounded.

The following lemmas are well known in the theory of monotone equilibrium problems.

Lemma ([]) Let g satisfy AssumptionA.Then,for allα> and u∈H,there exists w∈Q such that

g(w,v) + 

αv–w,w–u ≥, ∀v∈Q.

Lemma ([]) Under the assumptions of Lemma,the mapping Tαgdeﬁned onHas

Tαg(u) =

w∈Q:g(w,v) + 

αv–w,w–u ≥,∀v∈Q

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(i) Tαg is single-valued;

(ii) Tαg is ﬁrmly nonexpansive,that is,for anyu,v∈H,

Tαg(u) –Tαg(v)

_≤

Tαg(u) –Tαg(v),u–v

;

(iii) Fix(Tαg) =Sol(Q,g);

(iv) Sol(Q,g)is closed and convex.

Lemma ([]) Under the assumptions of Lemma,forα,β> and u,v∈H,we have

Tαg(u) –T

g

β(v)≤ v–u+ |β–α|

β T g

β(v) –v. 3 A weak convergence algorithm

Algorithm 

Initialization. Pickx_∈_C_{and choose the parameters}_β_,_η_,_θ_∈_{(, )}_,_{ <}_ρ_{≤ ¯}_ρ_,

{ρk} ⊂[ρ,ρ¯], <γ≤ ¯γ < ,{γk} ⊂[γ,γ¯], <α,{αk} ⊂[α, +∞),μ∈(,_A). Iterationk(k= , , , . . .). Havingxk_{, do the following steps:}

Step . Solve the strongly convex program

CPxk min

fxk,y+  ρk

y–xk:y∈C

to obtain its unique solutionyk_.

Ifyk₌_xk_{, then set}_uk₌_xk_{and go to Step . Otherwise, go to Step .}

Step . (Armijo linesearch rule) Findmkas the smallest positive integer numberm

such that

⎧ ⎨ ⎩

zk,m_{= ( –}_ηm_)xk₊_ηm_ym_,

f(zk,m,xk) –f(zk,m,yk)≥_θ_ρ

kx

k_–_yk_. (.)

Setηk=ηmk,zk=zk,mk.

Step . Selectξk_∈_∂

f(zk,xk)and computeσk=f(z k_,xk₎

ξk ,u

k₌_P

C(xk–γkσkξk).

Step .

⎧ ⎨ ⎩

vk_{= ( –}_β_)uk₊_β_Suk_,

wk=TαgkAv k_.

Step . Takexk+₌_P

C(vk+μA∗(Twk–Avk))and go toiterationkwithkreplaced

byk+ .

Lemma  Suppose that p∈Sol(C,f),f(x,·)is convex and subdiﬀerentiable on C for all x∈C and that f is pseudomonotone on C.Then,we have:

(a) The Armijo linesearch rule(.)is well deﬁned; (b) f(zk_,_xk_{) > }_;

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(d)

uk–p≤xk–p–γk( –γk)

σkξk 

.

Proof The proof of Lemma  whenH is a ﬁnite-dimensional space can be found, for example, in []. When its dimension is inﬁnite, it can be done in the same way. So we

omit it.

Theorem  Let C and Q be two nonempty closed convex subsets inHandH,respectively. Let S:C→C;T:Q→Q be nonexpansive mappings,and let bifunctions g and f satisfy AssumptionsAandB,respectively.Let A:H→Hbe a bounded linear operator with its adjoint A∗.IfΩ={x∗∈Sol(C,f)∩Fix(S) :Ax∗∈Fix(Q,g)∩Fix(T)} =∅,then the sequences {xk_}_,_{_uk_}_,_{_vk_}_{converge weakly to an element p}_∈_Ω_,_and_{_wk_}_{converges weakly to Ap}_∈ Sol(Q,g)∩Fix(T).

Proof Letx∗∈Ω. Thenx∗∈Sol(C,f)∩Fix(S) andAx∗∈Sol(Q,g)∩Fix(T). From Lemma (d) we have

_uk_–_x∗_≤_xk_–_x∗_–_γ

k( –γk)

σkξk 

≤xk–x∗.

By Step  we get

vk–x∗=( –β)uk+βSuk–x∗

=( –β)uk–x∗+βSuk–Sx∗

≤( –β)uk–x∗+βSuk–Sx∗

≤( –β)uk–x∗+βuk–x∗

=uk–x∗. Thus,

vk–x∗≤uk–x∗≤xk–x∗. (.)

Assertions (iii) and (ii) in Lemma  imply that

TαgkAv

k_–_Ax∗₌_Tg

αkAv k_–_Tg

αkAx

∗

≤Tg

αkAv k_–_Tg

αkAx

∗_,_Avk_–_Ax∗

=T_αg

kAv

k_–_Ax∗_,_Avk_–_Ax∗ = 

T

g

αkAv

k_–_Ax∗₊_Avk_–_Ax∗_–_Tg

αkAv

k_–_Avk_.

Hence,

TαgkAv

k_–_Ax∗_≤_Avk_–_Ax∗_–_Tg

αkAv

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Because of the nonexpansiveness of the mappingT, we receive from the last inequality that

Twk–Ax∗=TTαgkAv

k_–_TAx∗

≤TαgkAv

k_–_Ax∗

≤Avk–Ax∗–TαgkAv

k_–_Avk_. _(.)

Using (.), we have

Avk–x∗,Twk–Avk=Avk–x∗+Twk–Avk–Twk–Avk,Twk–Avk =Twk–Ax∗,Twk–Avk–Twk–Avk

= 

Tw

k_–_Ax∗₊_Twk_–_Avk_–_Avk_–_Ax∗

–Twk–Avk = 

Tw

k_–_Ax∗_–_Avk_–_Ax∗_–_Twk_–_Avk

≤–

T

g

αkAv

k_–_Avk

–

Tw

k_–_Avk

. (.)

By the deﬁnition ofxk+we have

xk+–x∗=PC

vk+μA∗Twk–Avk–PC

x∗

≤vk–x∗+μA∗Twk–Avk

=vk–x∗+μA∗Twk–Avk+ μvk–x∗,A∗Twk–Avk ≤vk–x∗+μA∗Twk–Avk+ μAvk–x∗,Twk–Avk.

In combination with (.) and (.), the last inequality becomes

xk+–x∗≤vk–x∗+μA∗Twk–Avk

–μTwk–Avk–μT_αg

kAv

k_–_Avk

=vk–x∗–μ –μATwk–Avk–μwk–Avk

≤xk–x∗–μ –μATwk–Avk–μwk–Avk. (.)

In view of (.), (.), andμ∈(, 

A), we get

xk+–x∗≤vk–x∗≤uk–x∗≤xk–x∗ (.)

and

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Therefore,limk→+∞xk–x∗does exist, and we get from (.) and (.) that

lim k→+∞

xk–x∗= lim k→+∞

vk–x∗= lim k→+∞

uk–x∗ and

lim k→+∞Tw

k_–_Avk₌ _lim k→+∞w

k_–_Avk_{= .} (.)

From (.) and the inequality

Twk–wk≤Twk–Avk+wk–Avk

we get

lim k→+∞Tw

k_–_wk_{= .} _(.)

Besides, Lemma (d) implies

uk–x∗≤xk–x∗–γk( –γk)

σkξk.

Hence,

γk( –γk)

σkξk 

≤xk–x∗–uk–x∗

=xk–x∗–uk–x∗xk–x∗+uk–x∗. In view of (.), we get

lim k→+∞σk

ξk= . (.)

In addition, by the deﬁnition ofuk_,_uk₌_P

C(xk–γkσkξk). We have

uk–xk≤γkσkξk.

So we get from (.) that

lim k→+∞

uk–xk= . (.)

Usingvk_{= ( –}_β_)uk₊_β_Suk_{, Lemma , and the nonexpansiveness of}_{S, we have}

vk–x∗=( –β)uk+βSuk–x∗

=( –β)uk–x∗+βSuk–x∗

= ( –β)uk–x∗+βSuk–x∗–β( –β)Suk–uk = ( –β)uk–x∗+βSuk–Sx∗–β( –β)Suk–uk ≤( –β)uk–x∗+βuk–x∗–β( –β)Suk–uk

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Therefore,

β( –β)Suk_–_uk_≤_uk_–_x∗_–_vk_–_x∗_.

Combining the last inequality with (.), we obtain that

lim k→+∞Su

k_–_uk_{= .} _(.)

In addition,

vk–xk≤vk–uk+uk–xk

=αSuk–uk+uk–xk. Therefore, we get from (.) and (.) that

lim k→+∞v

k_–_xk_{= .} _(.)

Becauselimk→+∞xk–x∗exists,{xk}is bounded. By Lemma ,{yk}is bounded, and

consequently{zk_}_{is bounded. By Lemma ,}_{_ξk_}_{is bounded. Step  and (.) yield}

lim k→∞f

zk,xk= lim k→∞

σkξkξk= . (.)

We have

 =fzk,zk=fzk, ( –ηk)xk+ηkyk

≤( –ηk)f

zk,xk+ηkf

zk,yk, so, we get from (.) that

fzk,xk≥ηk

fzk,xk–fzk,yk

≥ θ

ρk

ηkxk–yk 

.

Combining this with (.), we have

lim k→∞ηk

xk–yk= . (.)

Suppose thatpis a weak accumulation point of{xk_}_{, that is, there exists a subsequence}

{xkj}of{xk_}_{such that}_xkj_{converges weakly to}_p_∈_C_as_j_→₊_∞_{. Then, it follows from (.)}

and (.) thatukjp,vkjp, andAvkjAp.

Sincelimk→+∞wk–Avk= , we deduce thatwkj Ap. Because{wk} ⊂QandQis

closed and convex, we have thatAp∈Q. From (.) we get

lim i→∞ηkix

ki_–_yki

= . (.)

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Case.lim sup_i_→∞ηki> .

In this case, there existη¯>  and a subsequence of{ηki}, denoted again by{ηki}, such

that, for somei> ,ηki>η¯for alli≥i. Using this fact and (.), we have

lim i→∞x

ki_–_yki_{= .} _(.)

Recall thatxk_{p, together with (.), implies that}_yki_p_as_i_{→ ∞}_.

By the deﬁnition ofyki_,

yki=arg min

fxki,y+  ρki

y–xki:y∈C

,

we have

∈∂f

xki,yki+ 

ρki

yki–xki+NC

yki,

so there existsξˆki_∈_∂

f(xki,yki) such that

_ˆ

ξki_,_y_–_yki₊  ρki

yki_–_xki_,_y_–_yki≥_, ∀_y∈_C.

Combining this with

fxki,y–fxki,yki≥ξˆki,y–yki, ∀y∈C, yields

fxki,y–fxki,yki+ 

ρki

yki–xki,y–yki≥, ∀y∈C. (.)

Since

yki–xki,y–yki≤yki–xkiy–yki, from (.) we get that

ρki

yki–xkiy–yki≥. (.)

Lettingi→ ∞, by the weak continuity off and (.), from (.) we obtain in the limit that

f(p,y) –f(p,p)≥.

Hence,

f(p,y)≥, ∀y∈C,

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Case.limi→∞ηki= .

From the boundedness of{yki_}_{, without loss of generality, we may assume that}_yki_y_¯

asi→ ∞.

Replacingybyxki_{in (.), we get}

fxki,yki≤– 

ρki

yki–xki. (.)

On the other hand, by the Armijo linesearch rule (.), formki– , we have

fzki,m_ki–_,_xki_–_f_zki,m_ki–_,_yki_< θ

ρki

yki–xki.

Combining this with (.), we get

fxki,yki≤– 

ρki

yki–xki≤

θ

fzki,m_ki–_,_yki_–_f_zki,m_ki–_,_xki_. _(.)

According to the algorithm, we havezki,mki–_{= (–}ηmki–_)xki₊_ηm_ki–

yki_{. Since}_ηki,m_ki–_→

, xki_{converges weakly to}_{p, and}_yki_{converges weakly to}_{y, this implies that}_¯ _zki,m_ki–_p_as

i→ ∞. Beside that, { 

ρ_kiy

ki_–_xki_} _{is bounded, so without loss of generality we may}

assume thatlimi→+∞ρ_kiy

ki_–_xki_{exists. Hence, in the limit, from (.) we get that}

f(p,y)¯ ≤– lim i→+∞



ρki

yki–xki≤

θf(p,y).¯

Therefore,f(p,y) =  and¯ lim_i_→₊_∞yki_–_xki_{= . By}_Case_{ we get}_p_∈_Sol_(C,_f_).

Besides that, (.) implies thatSukj_–_ukj_→_{ as}_j_{→ ∞}_{; together with}_ukj _p_and

the demiclosedness ofI–S, we getp∈Fix(S). Therefore,

p∈Sol(C,f)∩Fix(S). (.)

Next, we need to show thatAp∈Sol(Q,g)∩Fix(T). Indeed, we haveSol(Q,g) =Fix(Tβg). So, ifT

g

βAp=Ap, then, using Opial’s condition, we have

lim inf j→+∞

Avkj–Ap<lim inf j→+∞

Avkj–T_βgAp

=lim inf j→+∞Av

kj_–_wkj₊_wkj_–_Tg

βAp ≤lim inf

j→+∞Av

kj_–_wkj₊_Tg

βAp–wkj. So it follows from (.) and Lemma  that

lim inf j→+∞Av

kj_–_Ap_<_{lim inf} j→+∞T

g

βAp–wkj =lim inf

j→+∞T

g

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≤lim inf j→+∞

Avkj–Ap+|αkj–β|

αkj

Tαg_kjAvkj–Avkj

=lim inf j→+∞

Avkj_–_Ap₊|αkj–β| αkj

wkj_–_Avkj

=lim inf j→+∞

Avkj_–_Ap_,

a contradiction. Thus,Ap∈Fix(Tαg) =Sol(Q,g).

Moreover, (.) shows thatlimj→∞Twkj–wkj= . Combining this withwkjApand the fact thatI–Tis demiclosed at , it is immediate thatAp∈Fix(T). Therefore,

Ap∈Sol(Q,g)∩Fix(T). (.)

From (.) and (.) we obtain thatp∈Ω.

To complete the proof, we must show that the whole sequence{xk}converges weakly top. Indeed, if there exists a subsequence{xli_}_of_{_xk_}_{such that}_xli_q_with_q₌_{p, then}

we haveq∈Ω. By Opial’s condition this yields

lim inf i→+∞

xli–q<lim inf i→+∞

xli–p

=lim inf j→+∞x

k_–_p

=lim inf j→+∞

_xkj_–_p

<lim inf j→+∞

xkj–q

=lim inf i→+∞

xli–q,

a contradiction. Hence,{xk}converges weakly top.

Combining this with (.), it is immediate that{uk_}_,_{_vk_}_{also converge weakly to}_p_and

wk_Ap_∈_Sol_(Q,_g)_∩_Fix_(T_).

A particular case of the problem SEPNM is the split equilibrium problem SEP, that is, S =IH and T =IH. In this case, we have the following linesearch algorithm for SEP.

Algorithm 

Initialization. Pickx_∈_C_{and choose the parameters}_η_,_θ_∈_{(, )}_,_{ <}_ρ_{≤ ¯}_ρ_,

CPxk min

fxk,y+  ρk

y–xk:y∈C

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such that

⎧ ⎨ ⎩

zk,m_{= ( –}_ηm_)xk₊_ηm_ym_,

f(zk,m_,_xk_{) –}_f_(zk,m_,_yk₎_≥ θ

ρkx k_–_yk_.

Step . Selectξk∈∂f(zk,xk)and computeσk=f(z k_,xk₎

ξk ,uk=PC(xk–γkσkξk).

Step . wk=TαgkAu k_.

Step . Takexk+₌_P

C(uk+μA∗(wk–Auk))and go toiterationkwithkis replaced

byk+ .

The following corollary is an immediate consequence of Theorem .

Corollary  Suppose that g,f are bifunctions satisfying AssumptionsAandB,respectively. Let A:H→Hbe a bounded linear operator with its adjoint A∗.IfΩ={x∗∈Sol(C,f) : Ax∗∈Sol(Q,g)} =∅,then the sequences{xk_}_and_{_uk_}_{converge weakly to an element p}_∈_Ω_,

and{wk}converges weakly to Ap∈Sol(Q,g).

4 A strong convergence algorithm

Algorithm 

Initialization. Pickxg∈C=Cand choose the parametersβ,η,θ∈(, ), <ρ≤ ¯ρ,

{ρk} ⊂[ρ,ρ¯], <γ≤ ¯γ < ,{γk} ⊂[γ,γ¯], <α,{αk} ⊂[α, +∞),μ∈(,_A). Iterationk(k= , , , . . .). Havingxk, do the following steps:

CPxk min

fxk,y+  ρk

y–xk:y∈C

such that

⎧ ⎨ ⎩

zk,m= ( –ηm)xk+ηmym,

f(zk,m_,_xk_{) –}_f_(zk,m_,_yk₎_≥ θ

ρkx

k_–_yk_. (.)

Step .

⎧ ⎨ ⎩

vk_{= ( –}_β_)uk₊_β_Suk_,

wk=TβgkAv k_.

Step . tk₌_P

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Step . DeﬁneCk+={x∈Ck:x–tk ≤ x–vk ≤ x–xk}. Compute

xk+₌_P Ck+(x

g₎_{and go to}_iteration_k_with_k_{is replaced by}_k_{+ }_.

Theorem  Let C and Q be two nonempty closed convex subsets inHandH,respectively. Let S:C→C;T:Q→Q be nonexpansive mappings,and let bifunctions g and f satisfy AssumptionsAandB,respectively.Let A:H→Hbe a bounded linear operator with its adjoint A∗.IfΩ={x∗∈Sol(C,f)∩Fix(S) :Ax∗∈Sol(Q,g)∩Fix(T)} =∅,then the sequences {xk},{uk},{vk}converge strongly to an element p∈Ω,and{wk}converges strongly to Ap∈

Sol(Q,g)∩Fix(T).

Proof First, we observe that the linesearch rule (.) is well deﬁned by Lemma . Let x∗∈Ω. From (.), (.), and (.) we have

tk–x∗≤vk–x∗–μ –μATwk–Avk–μwk–Avk

≤uk–x∗–β( –β)Suk–uk

–μ –μATwk–Avk–μwk–Avk

≤xk–x∗–β( –β)Suk–uk

–μ –μATwk–Avk–μwk–Avk. (.)

Sinceμ∈(,_A), (.) implies that

tk–x∗≤vk–x∗≤uk–x∗≤xk–x∗, ∀k. (.)

Sincex∗∈C, from(.) we get by induction thatx∗∈Ckfor allk∈N∗and, consequently, Ω⊂Ckfor allk.

By setting

Dk= x∈H:x–tk≤x–vk≤x–xk, k∈N,

it is clear thatDk is closed and convex for allk. In addition, C=Cis also closed and

convex, andCk+=Ck∩Dk. Hence,Ckis closed and convex for allk.

From the deﬁnition ofxk+_{we have}_xk+_∈_C

k+⊂Ckandxk=PCk(xg), so

xk–xg≤xk+–xg for allk.

Sincex∗∈Ck+, this implies that

xk+–xg≤x∗–xg.

Thus,

xk–xg≤xk+–xg≤x∗–xg, ∀k.

Consequently,{xk–xg}is nondecreasing and bounded, solimk→+∞xk–xgdoes exist.

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For all m>n, we have thatxm_∈_C

m⊂Cnandxn=PCn(xg). Combining this fact with

Lemma , we get

xm–xn≤xm–xg–xn–xg

=xm–xg–xn–xgxm–xg+xn–xg.

Since limk→+∞xk–xg exists, this implies thatlimm,n→∞xm–xn= ,i.e., {xk} is a

Cauchy sequence, so

lim k→∞x

k₌_p. _(.)

By Step  we get

_tk_–_xk+_≤_vk_–_xk+_≤_xk_–_xk+_.

Therefore,

tk–xk≤tk–xk++xk+–xk

≤xk–xk++xk–xk+

= xk–xk+ (.)

and

vk–xk≤vk–xk++xk+–xk

≤xk–xk++xk–xk+

= xk–xk+. (.)

So, from (.), (.), and (.) we get that

lim k→∞

tk–xk= lim k→∞

vk–xk= . (.)

In view of (.) and (.), we have

β( –β)Suk–uk+μ –μATwk–Avk+μwk–Avk

≤xk–x∗–tk–x∗

=xk–x∗+tk–x∗xk–x∗–tk–x∗

≤xk–tkxk–x∗+tk–x∗→ ask→ ∞. (.)

Sinceβ∈(, ) andμ∈(,_A), we deduce from (.) that

lim k→+∞

Suk–uk= , lim k→+∞

Twk–Avk= , and

lim k→+∞w

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In addition, from the inequality

Twk–wk≤Twk–Avk+wk–Avk,

combined with (.), we get

lim k→+∞

Twk–wk= . (.)

Besides, (.), (.), andlimk→+∞xk=pit imply

lim k→+∞u

k₌_p, _lim k→+∞v

k₌_p. _(.)

Since

Sp–p ≤Sp–Suk+Suk–uk+uk–p

≤p–uk+Suk–uk+uk–p

= uk–p+Suk–uk,

from (.) and (.) we get thatSp–p= , that is,p∈Fix(S). From (.) we have

lim k→∞ηkx

k_–_yk_{= .} _(.)

We now consider two distinct cases. Case.lim sup_k_→∞ηk> .

Then there existη¯>  and a subsequence{ηki} ⊂ {ηk}such thatηki>η¯for alli. So we

get from (.) that

lim i→∞x

ki_–_yki_{= .} _(.)

Sincexk_→_{p, (.) implies that}_yki_→_p_as_i_{→ ∞}_.

For eachy∈C, we get from (.) that

ρki

yki–xkiy–yki≥. (.)

Lettingi→ ∞, by the continuity off, sincexki_→_p_and_yki_→_{p, in the limit, from (.)}

we obtain that

f(p,y) –f(p,p)≥.

Hence,

f(p,y)≥, ∀y∈C,

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Case.limk→∞ηk= .

From the boundedness of{yk_}_{we deduce that there exists}_{_yki_{} ⊂ {}_yk_}_{such that}_yki_y_¯

asi→ ∞.

Replacingybyyki_{in (.), we get}

fxki,yki+ 

ρki

yki–xki≤. (.)

In the other hand, by the Armijo linesearch rule (.), formki– , there existszki,mki–such

that

fzki,mki–_,_xki_–_f_zki,m_ki–

,yki< θ ρki

yki–xki.

Combining this with (.), we get

fzki,mki–_,_yki_–_f_zki,m_ki–

,xki> – θ ρki

yki–xki≥

θf

xki,yki. (.)

According to the algorithm, we havezki,m_ki–_{= (–}_ηm_ki–_)xki₊_ηm_ki–

yki_{. Since}_ηki,m_ki–→_,

xki _{converges strongly to}_{p, and}_yki_{converges weakly to}_{y, this implies that}_¯ _zki,m_ki–_→

p asi→ ∞. Besides that,{_ρ

kiy

ki_–_xki_}_{is bounded, so, without loss of generality, we may}

assume thatlim_i_→₊_∞ 

ρ_kiy

ki_–_xki_{exists. Hence, we get in the limit (.) that}

f(p,y)¯ ≥– lim i→+∞



ρki

yki–xki≥θf(p,y).¯

Therefore, f(p,y) =  and¯ limi→+∞yki–xki = . ByCase it is immediate thatp∈

Sol(C,f). So

p∈Sol(C,f)∩Fix(S). (.)

We obtain from (.) thatlimk→+∞Avk=Ap. Combining this with (.) yields

lim k→+∞w

k₌_Ap. _(.)

Moreover,

TAp–Ap ≤TAp–Twk+Twk–wk+wk–Ap

≤Ap–wk+Twk–wk+wk–Ap

= wk–Ap+Twk–wk.

In view of (.) and (.), we obtainTAp–Ap= . Hence,Ap∈Fix(T). In addition,

TβgAp–Ap≤T

g

βAp–TαgkAv k₊_Tg

αkAv

k_–_Avk₊_Avk_–_Ap

=TβgAp–TαgkAv

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≤Avk–Ap+|αk–β|

αk

TαgkAv

k_–_Avk₊_wk_–_Avk₊_Avk_–_Ap

= Avk–Ap+|αk–β|

αk

wk–Avk+wk–Avk,

where the last inequality comes from Lemma . Letting k → ∞ and recalling that

limk→+∞Avk=Ap, from (.) we get

TαgAp–Ap= .

Therefore,Ap∈Fix(Tαg) =Sol(Q,g). Hence,

Ap∈Sol(Q,g)∩Fix(T).

Combining this with (.), we conclude thatp∈Ω. The proof is completed.

WhenS=IHandT=IH, Algorithm  becomes as follows.

Algorithm 

Initialization. Pickxg∈C=Cand choose the parametersη,θ∈(, ), <ρ≤ ¯ρ,

CPxk min

fxk,y+  ρk

y–xk:y∈C

to obtain its unique solutionyk.

such that

⎧ ⎨ ⎩

zk,m= ( –ηm)xk+ηmym,

f(zk,m_,_xk_{) –}_f_(zk,m_,_yk₎_≥ θ

ρkx k_–_yk_.

Step . wk₌_Tg

βkAu k_.

Step . tk₌_P

C(uk+μA∗(wk–Auk)).

Step . DeﬁneCk+={x∈Ck:x–tk ≤ x–uk ≤ x–xk}. Compute

xk+₌_P Ck+(x

g₎_{and go to}_iteration_k_with_k_{is replaced by}_k_{+ }_.

The following result is an immediate consequence of Theorem .

Corollary  Let g:Q×Q→Rbe a bifunction satisfying AssumptionA,and f :C×C→R be a bifunction satisfying AssumptionB.Let A:H→Hbe a bounded linear operator with its adjoint A∗.IfΩ={x∗∈Sol(C,f) :Ax∗∈Sol(Q,g)} =∅,then the sequences{xk_}_and_{_uk_}

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5 Conclusion

Two linesearch algorithms for solving a split equilibrium problem and nonexpansive map-pingSEPNM(C,Q,A,f,g,S,T) in Hilbert spaces have been proposed, in which the bifunc-tionf is pseudomonotone onCwith respect to its solution set, the bifunctiongis mono-tone onQ, andSandTare nonexpansive mappings. The weak and strong convergence of iteration sequences generated by the algorithms to a solution of this problem are obtained.

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_{Faculty of Information Technology, Le Quy Don Technical University, Hanoi, Vietnam.}2_{Department of Applied}

Mathematics, Pukyong National University, Busan, Korea.3_{Technical University, Hanoi, Vietnam.}

Acknowledgements

This work was supported by a Research Grant of Pukyong National University (2016).

Received: 12 November 2015 Accepted: 29 February 2016 References

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