Linesearch algorithms for split equilibrium problems and nonexpansive mappings

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 Son³, Liguo Jiao²and Do Sang Kim^2*

*Correspondence:

dskim@pknu.ac.kr

2Department 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 writex^k→xorx^kx iffx^kconverges strongly or weakly tox, respectively, ask→ ∞. LetC,Qbe 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:

x^k+=PC

x^k+γA^T(PQ–I)Ax^k ,

whereIis the identity mapping, andPCis projection mapping ontoC. Xu [] investigated the SFP setting in infinite-dimensional Hilbert spaces. In this case, the CQ-algorithm be-

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comes x^k+=PC

x^k+γA^∗(PQ–I)Ax^k , 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{x^k}by

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩

x^∈C,{rk} ⊂(, +∞), μ> , f(y^k,y) +_r^

ky–y^k,y^k–x^k ≥, ∀y∈C,

g(u^k,v) +_r^

kv–u^k,u^k–Ay^k ≥, ∀v∈Q,

x^k+=PC(y^k+μA^∗(u^k–Ay^k)), ∀k≥.

Under certain conditions on bifunctions and parameters, the author shows that{x^k}, {y^k}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), whereFix(S) andFix(T) are the fixed points of the mappingsSandT, respectively.

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 P_C 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) P_C(x) –P_C(y)^≤ P_C(x) –P_C(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{x^k} ⊂Hwith x^kx,we have the inequal- ity

lim inf

k→+∞x^k–x<lim inf

k→+∞x^k–y for all y∈Hsuch that y=x.

Definition  We say that an operatorT:H→His demiclosed at  if, for any sequence {x^k}such thatx^kxandTx^k→ 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↓ g

λw+ ( –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{x^k},{y^k} ⊂C converge weakly toxandy, respectively, thenf(x^k,y^k)→f(x,y)ask→+∞.

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{x^k},{y^k}be two sequences in Δconverging weakly tox,¯ ¯y,respectively.Then,for anyε> ,there existη> and kε∈N such that

∂_ϕ x^k,y^k

⊂∂_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{x^k} ⊂C,  <ρ≤ ¯ρ,{ρ_k} ⊂[ρ,ρ].¯ Consider the sequence{y^k}defined as

y^k=arg min

ϕ x^k,y

+ 

ρk

y–x^k:y∈C

. If{x^k}is bounded,then{y^k}is also bounded.

Proof First, we show that if{x^k}converges weakly tox^∗, then{y^k}is bounded. Indeed, y^k=arg min

ϕ

x^k,y + 

ρk

y–x^k:y∈C

and ϕ

x^k,x^k + 

ρ_kx^k–x^k= .

Therefore, ϕ

x^k,y^k + 

ρ_ky^k–x^k≤, ∀k.

In addition, for allξˆ^k∈∂_ϕ(x^k,x^k), we have ϕ

x^k,y^k + 

ρk

y^k–x^k≥ξˆ^k,y^k–x^k + 

ρk

y^k–x^k.

This implies

–ˆξ^ky^k–x^k+ 

ρk

y^k–x^k≤.

Hence,

y^k–x^k≤ρ_kˆξ^k, ∀k.

Because{ρ_k}is bounded,{x^k}converges weakly tox^∗andξˆ^k∈∂_ϕ(x^k,x^k). By Lemma  the sequence{ˆξ^k}is bounded; combining this with the boundedness of{x^k}, we get that {y^k}is also bounded.

Now let us prove Lemma . Suppose that{y^k}is unbounded, that is, there exists a subse- quence{y^ki} ⊆ {y^k}such thatlim_i→∞y^ki= +∞. By the boundedness of{x^k}this implies that{x^ki}is also bounded, and without loss of generality, we may assume that{x^ki}con- verges weakly to somex^∗. By the same argument as before, we get that{y^ki}is bounded,

a contradiction. Therefore,{y^k}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 Tα^gdefined onHas T_α^g(u) =

w∈Q:g(w,v) + 

αv–w,w–u ≥,∀v∈Q

has following properties:

(i) Tα^g is single-valued;

(ii) Tα^g is firmly nonexpansive,that is,for anyu,v∈H, T_α^g(u) –T_α^g(v)^≤

T_α^g(u) –T_α^g(v),u–v

; (iii) Fix(Tα^g) =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 T_α^g(u) –T_β^g(v)≤ v–u+|β–α|

β T_β^g(v) –v. 3 A weak convergence algorithm

Algorithm 

Initialization. Pickx^∈Cand choose the parametersβ,η,θ∈(, ), <ρ≤ ¯ρ, {ρ_k} ⊂[ρ,ρ],¯  <γ≤ ¯γ < ,{γ_k} ⊂[γ,γ¯], <α,{α_k} ⊂[α, +∞),μ∈(,_A^ ).

Iterationk(k= , , , . . .). Havingx^k, do the following steps:

Step . Solve the strongly convex program

CP x^k

min

f x^k,y

+ 

ρk

y–x^k:y∈C

to obtain its unique solutiony^k.

Ify^k=x^k, then setu^k=x^kand go to Step . Otherwise, go to Step .

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

⎧⎨

⎩

z^k,m= ( –η^m)x^k+η^my^m,

f(z^k,m,x^k) –f(z^k,m,y^k)≥_ρ^θ_kx^k–y^k. (.) Setη_k=η^mk,z^k=z^k,mk.

Step . Selectξ^k∈∂_f(z^k,x^k)and computeσ_k=^f_ξ^(zkk^,xk),u^k=PC(x^k–γ_kσ_kξ^k).

Step .

⎧⎨

⎩

v^k= ( –β)u^k+βSu^k, w^k=Tα^g_kAv^k.

Step . Takex^k+=P_C(v^k+μA^∗(Tw^k–Av^k))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(z^k,x^k) > ;

(c)  /∈∂_f(z^k,x^k);

(d)

u^k–p≤x^k–p^–γ_k( –γ_k)

σ_kξ^k.

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 {x^k},{u^k},{v^k}converge weakly to an element p∈Ω,and{w^k}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

u^k–x^∗≤x^k–x^∗–γ_k( –γ_k)

σ_kξ^k

≤x^k–x^∗. By Step  we get

v^k–x^∗=( –β)u^k+βSu^k–x^∗

=( –β) u^k–x^∗

+β

Su^k–Sx^∗

≤( –β)u^k–x^∗+βSu^k–Sx^∗

≤( –β)u^k–x^∗+βu^k–x^∗

=u^k–x^∗. Thus,

v^k–x^∗≤u^k–x^∗≤x^k–x^∗. (.)

Assertions (iii) and (ii) in Lemma  imply that T_α^g_kAv^k–Ax^∗=T_α^g_kAv^k–T_α^g_kAx^∗

≤ T_α^g

kAv^k–T_α^g

kAx^∗,Av^k–Ax^∗

= T_α^g

kAv^k–Ax^∗,Av^k–Ax^∗

= 

T_α^g_kAv^k–Ax^∗+Av^k–Ax^∗–T_α^g_kAv^k–Av^k . Hence,

T_α^g_kAv^k–Ax^∗≤Av^k–Ax^∗–T_α^g_kAv^k–Av^k.

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

Tw^k–Ax^∗=TT_α^g_kAv^k–TAx^∗

≤T_α^g_kAv^k–Ax^∗

≤Av^k–Ax^∗–T_α^g_kAv^k–Av^k. (.) Using (.), we have

A v^k–x^∗

,Tw^k–Av^k

= A

v^k–x^∗

+Tw^k–Av^k–

Tw^k–Av^k

,Tw^k–Av^k

=

Tw^k–Ax^∗,Tw^k–Av^k

–Tw^k–Av^k

= 

Tw^k–Ax^∗+Tw^k–Av^k–Av^k–Ax^∗ –Tw^k–Av^k

= 

Tw^k–Ax^∗–Av^k–Ax^∗

–Tw^k–Av^k

≤–

T_α^g

kAv^k–Av^k–

Tw^k–Av^k. (.) By the definition ofx^k+we have

x^k+–x^∗=PC

v^k+μA^∗

Tw^k–Av^k –PC

x^∗

≤v^k–x^∗ +μA^∗

Tw^k–Av^k

=v^k–x^∗+μA^∗

Tw^k–Av^k+ μ

v^k–x^∗,A^∗

Tw^k–Av^k

≤v^k–x^∗+μ^A^∗Tw^k–Av^k+ μ A

v^k–x^∗

,Tw^k–Av^k . In combination with (.) and (.), the last inequality becomes

x^k+–x^∗≤v^k–x^∗+μ^A^∗Tw^k–Av^k –μTw^k–Av^k–μT_α^g

kAv^k–Av^k

=v^k–x^∗–μ

 –μA^Tw^k–Av^k–μw^k–Av^k

≤x^k–x^∗–μ

 –μA^Tw^k–Av^k–μw^k–Av^k. (.) In view of (.), (.), andμ∈(,_A^), we get

x^k+–x^∗≤v^k–x^∗≤u^k–x^∗≤x^k–x^∗ (.)

and μ

 –μA^Tw^k–Av^k+μw^k–Av^k≤x^k–x^∗–x^k+–x^∗. (.)

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

k→+∞lim x^k–x^∗= lim

k→+∞v^k–x^∗= lim

k→+∞u^k–x^∗ and

k→+∞lim Tw^k–Av^k= lim

k→+∞w^k–Av^k= .

(.)

From (.) and the inequality

Tw^k–w^k≤Tw^k–Av^k+w^k–Av^k

we get

k→+∞lim Tw^k–w^k= . (.)

Besides, Lemma (d) implies

u^k–x^∗≤x^k–x^∗–γ_k( –γ_k)

σ_kξ^k. Hence,

γ_k( –γ_k)

σ_kξ^k≤x^k–x^∗–u^k–x^∗

=x^k–x^∗–u^k–x^∗x^k–x^∗+u^k–x^∗. In view of (.), we get

k→+∞lim σ_kξ^k= . (.)

In addition, by the definition ofu^k,u^k=P_C(x^k–γ_kσ_kξ^k). We have u^k–x^k≤γ_kσ_kξ^k.

So we get from (.) that

k→+∞lim u^k–x^k= . (.)

Usingv^k= ( –β)u^k+βSu^k, Lemma , and the nonexpansiveness ofS, we have v^k–x^∗=( –β)u^k+βSu^k–x^∗

=( –β) u^k–x^∗

+β

Su^k–x^∗

= ( –β)u^k–x^∗+βSu^k–x^∗–β( –β)Su^k–u^k

= ( –β)u^k–x^∗+βSu^k–Sx^∗–β( –β)Su^k–u^k

≤( –β)u^k–x^∗+βu^k–x^∗–β( –β)Su^k–u^k

=u^k–x^∗–β( –β)Su^k–u^k. (.)

Therefore,

β( –β)Su^k–u^k≤u^k–x^∗–v^k–x^∗. Combining the last inequality with (.), we obtain that

k→+∞lim Su^k–u^k= . (.)

In addition,

v^k–x^k≤v^k–u^k+u^k–x^k

=αSu^k–u^k+u^k–x^k. Therefore, we get from (.) and (.) that

k→+∞lim v^k–x^k= . (.)

Becauselim_k→+∞x^k–x^∗exists,{x^k}is bounded. By Lemma ,{y^k}is bounded, and consequently{z^k}is bounded. By Lemma ,{ξ^k}is bounded. Step  and (.) yield

k→∞limf z^k,x^k

= lim

k→∞

σ_kξ^kξ^k= . (.)

We have

 =f z^k,z^k

=f

z^k, ( –η_k)x^k+η_ky^k

≤( –η_k)f z^k,x^k

+η_kf z^k,y^k

, so, we get from (.) that

f z^k,x^k

≥η_k f

z^k,x^k –f

z^k,y^k

≥ θ

ρk

η_kx^k–y^k. Combining this with (.), we have

k→∞limη_kx^k–y^k= . (.)

Suppose thatpis a weak accumulation point of{x^k}, that is, there exists a subsequence {x^kj}of{x^k}such thatx^kjconverges weakly top∈Casj→+∞. Then, it follows from (.) and (.) thatu^kjp,v^kjp, andAv^kjAp.

Sincelim_k→+∞w^k–Av^k= , we deduce thatw^kj Ap. Because{w^k} ⊂QandQis closed and convex, we have thatAp∈Q.

From (.) we get

i→∞limη_kix^ki–y^ki= . (.)

We now consider two distinct cases.

Case.lim sup_i→∞η_ki> .

In this case, there existη¯>  and a subsequence of{η_ki}, denoted again by{η_ki}, such that, for somei> ,η_ki>η¯for alli≥i. Using this fact and (.), we have

i→∞limx^ki–y^ki= . (.)

Recall thatx^kp, together with (.), implies thaty^kipasi→ ∞.

By the definition ofy^ki, y^ki=arg min

f

x^ki,y + 

ρki

y–x^ki:y∈C

,

we have

∈∂_f x^ki,y^ki

+  ρk_i

y^ki–x^ki +NC

y^ki ,

so there existsξˆ^ki∈∂_f(x^ki,y^ki) such that ξˆ^ki,y–y^ki

+  ρ_ki

y^ki–x^ki,y–y^ki

≥, ∀y∈C.

Combining this with f

x^ki,y –f

x^ki,y^ki

≥ξˆ^ki,y–y^ki

, ∀y∈C, yields

f x^ki,y

–f x^ki,y^ki

+  ρ_ki

y^ki–x^ki,y–y^ki

≥, ∀y∈C. (.)

Since

y^ki–x^ki,y–y^ki

≤y^ki–x^kiy–y^ki, from (.) we get that

f x^ki,y

–f x^ki,y^ki

+ 

ρ_kiy^ki–x^kiy–y^ki≥. (.) 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,

which means thatpis a solution ofEP(C,f).

Case.lim_i→∞η_ki= .

From the boundedness of{y^ki}, without loss of generality, we may assume thaty^kiy¯ asi→ ∞.

Replacingybyx^kiin (.), we get f

x^ki,y^ki

≤– 

ρ_kiy^ki–x^ki. (.)

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

z^ki,mki–,x^ki –f

z^ki,mki–,y^ki

< θ

ρ_kiy^ki–x^ki. Combining this with (.), we get

f x^ki,y^ki

≤– 

ρ_kiy^ki–x^ki≤ θ

f

z^ki,mki–,y^ki –f

z^ki,mki–,x^ki

. (.)

According to the algorithm, we havez^ki,mki–= (–η^mki–)x^ki+η^mki–y^ki. Sinceη^ki,mki–→, x^kiconverges weakly top, andy^kiconverges weakly toy, this implies that¯ z^ki,mki–pas i→ ∞. Beside that, {_ρ^

kiy^ki–x^ki} is bounded, so without loss of generality we may assume thatlim_i→+∞ρ^

kiy^ki–x^kiexists. Hence, in the limit, from (.) we get that f(p,y)¯ ≤– lim

i→+∞

 ρk_i

y^ki–x^ki≤ θf(p,y).¯

Therefore,f(p,y) =  and¯ lim_i→+∞y^ki–x^ki= . ByCase we getp∈Sol(C,f).

Besides that, (.) implies thatSu^kj–u^kj → asj→ ∞; together withu^kj pand 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(T_β^g). So, ifT_β^gAp=Ap, then, using Opial’s condition, we have

lim inf

j→+∞Av^kj–Ap<lim inf

j→+∞Av^kj–T_β^gAp

=lim inf

j→+∞Av^kj–w^kj+w^kj–T_β^gAp

≤lim inf

j→+∞Av^kj–w^kj+T_β^gAp–w^kj. So it follows from (.) and Lemma  that

lim inf

j→+∞Av^kj–Ap<lim inf

j→+∞T_β^gAp–w^kj

=lim inf

j→+∞T_β^gAp–T_α^g

kjAv^kj

≤lim inf

j→+∞

Av^kj–Ap+|α_kj–β| α_kj T_α^g

kjAv^kj–Av^kj

=lim inf

j→+∞

Av^kj–Ap+|α_kj–β|

α_kj w^kj–Av^kj

=lim inf

j→+∞Av^kj–Ap, a contradiction. Thus,Ap∈Fix(Tα^g) =Sol(Q,g).

Moreover, (.) shows thatlim_j→∞Tw^kj–w^kj= . Combining this withw^kjApand 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{x^k}converges weakly top. Indeed, if there exists a subsequence{x^li}of{x^k}such thatx^liqwithq=p, then we haveq∈Ω. By Opial’s condition this yields

lim inf

i→+∞x^li–q<lim inf

i→+∞x^li–p

=lim inf

j→+∞x^k–p

=lim inf

j→+∞x^kj–p

<lim inf

j→+∞x^kj–q

=lim inf

i→+∞x^li–q,

a contradiction. Hence,{x^k}converges weakly top.

Combining this with (.), it is immediate that{u^k},{v^k}also converge weakly topand

w^kAp∈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^∈Cand choose the parametersη,θ∈(, ), <ρ≤ ¯ρ, {ρ_k} ⊂[ρ,ρ],¯  <γ≤ ¯γ < ,{γ_k} ⊂[γ,γ¯], <α,{α_k} ⊂[α, +∞),μ∈(,_A^ ).

Iterationk(k= , , , . . .). Havingx^k, do the following steps:

Step . Solve the strongly convex program

CP x^k

min

f x^k,y

+ 

ρ_ky–x^k:y∈C

to obtain its unique solutiony^k.

Ify^k=x^k, then setu^k=x^kand go to Step . Otherwise, go to Step .