Iterative algorithm for a family of split equilibrium problems and fixed point problems in Hilbert spaces with applications

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R E S E A R C H

Open Access

Iterative algorithm for a family of split

equilibrium problems and ﬁxed point

problems in Hilbert spaces with applications

Shenghua Wang

1

_{, Xiaoying Gong}

2

_{, Afrah AN Abdou}

3

_{and Yeol Je Cho}

4,5*

*_{Correspondence: [email protected]} 4_{Department of Mathematics}

Education and RINS, Gyeongsang National University, Jinju, 660701, Korea

5_{Center for General Education,}

China Medical University, Taichung, 40402, Taiwan

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

Abstract

In this paper, we propose an iterative algorithm and, by using the proposed

algorithm, prove some strong convergence theorems for finding a common element of the set of solutions of a finite family of split equilibrium problems and the set of common fixed points of a countable family of nonexpansive mappings in Hilbert spaces. An example is given to illustrate the main result of this paper. As an application, we construct an algorithm to solve an optimization problem.

MSC: 47H10; 54H25; 65J15; 90C30

Keywords: split equilibrium problem; nonexpansive mapping; split feasible solution problem; Hilbert space

1 Introduction

Throughout this paper, letRdenote the set of all real numbers,Ndenote the set of all positive integer numbers,Hbe a real Hilbert space andCbe a nonempty closed convex subset ofH. A mappingS:C→Cis said tononexpansiveif

Sx–Sy ≤ x–y

for allx,y∈C. The set of ﬁxed points ofSis denoted byFix(S). It is known that the set Fix(S) is closed and convex.

LetF:C×C→Rbe a bifunction. Theequilibrium problemforFis to ﬁndz∈Csuch that

F(z,y)≥ (.)

for ally∈C. The set of all solutions of the problem (.) is denoted byEP(F),i.e.,

EP(F) =z∈C:F(z,y)≥,∀y∈C.

From the problem (.), we can consider some related problems, that is, variational in-equality problems, complementarity problems, ﬁxed point problems, game theory and

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other problems. Also, many problems in physics, optimization, and economics can be re-duced to ﬁnding a solution of the problem (.) (see [–]).

In , Combettes and Hirstoaga [] introduced an iterative scheme of ﬁnding a solu-tion of the problem (.) under the assumpsolu-tion thatEP(F) is nonempty. Later on, many iterative algorithms were considered to ﬁnd a common element of the set ofFix(S)∩EP(F) (see [–]).

Recently, some new problems calledsplit variational inequality problemswere consid-ered by some authors. Especially, Censoret al.[] initially studied this class of split vari-ational inequality problems.

LetHandHbe two real Hilbert spaces. Given the operatorsf:H→Handg:H→

H, bounded linear operatorA:H→H, and nonempty closed convex subsetsC⊂H andQ⊂H, the split variational inequality problem is formulated as follows:

Find a pointx∗∈Csuch that

fx∗,x–x∗≥ for allx∈Cand such that

y∗=Ax∗∈Q solves gy∗,y–y∗≥ for ally∈Q.

After investigating the algorithm of Censoret al., Moudaﬁ [] introduced a new itera-tive scheme to solve the following split monotone variational inclusion:

Findx∗∈Hsuch that

∈fx∗+B

x∗

and such that

y∗=Ax∗∈H solves ∈g

y∗+B

y∗,

whereB:Hi→Hiis a set-valued mappings fori= , .

In , Kazmi and Rizvi [] considered a new class of split equilibrium problems. Let

F:C×C→RandF:Q×Q→Rbe two bifunctions andA:H→Hbe a bounded linear operator. Thesplit equilibrium problemis as follows:

Findx∗∈Csuch that

F

x∗,x≥ (.)

for allx∈Cand such that

y∗=Ax∗∈Q solves F

y∗,y≥ (.)

for ally∈Q. The set of all solutions of the problems (.) and (.) is denoted by,i.e.,

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For more details as regards the split equilibrium problems, refer to [, ], in which the author gave an iterative algorithm to ﬁnd a common element of the sets of solutions of the split equilibrium problem and hierarchical ﬁxed point problem.

In this paper, inspired by the results in [] and [], we propose an iterative algorithm to ﬁnd a common element of the set of solutions for a family of split equilibrium problems and the set of common ﬁxed points of a countable family of nonexpansive mappings. In particular, we use some new methods to prove the main result of this paper. As an appli-cation, we propose an iterative algorithm to solve a split variational inequality problem.

2 Preliminaries

LetHbe a Hilbert space andCbe a nonempty closed subset ofH. For each pointx∈H, there exists a unique nearest point ofC, denoted byPCx, such that

x–PCx ≤ x–y

for ally∈C. Such aPCis called themetric projectionfromHontoC. It is well known that

PCis a ﬁrmly nonexpansive mapping fromHontoC,i.e.,

PCx–PCy≤ PCx–PCy,x–y

for allx,y∈H. Further, for anyx∈Handz∈C,z=PCxif and only if

x–z,z–y ≥

for ally∈C.

A mappingB:C→H is calledα-inverse strongly monotoneif there existsα>  such that

x–y,Bx–By ≥αBx–By

for allx,y∈H. For eachλ∈(, α],I–λBis a nonexpansive mapping ofCintoH (see []).

Consider the followingvariational inequalityfor an inverse strongly monotone map-pingB:

Findu∈Csuch that

v–u,Bu ≥

for allv∈C. The set of solutions of the variational inequality is denotedVI(C,B). It is well known that

u∈VI(C,B) ⇐⇒ u=PC(u–λBu)

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LetS:C→Cbe a mapping. It is well known thatSis nonexpansive if and only if the complementI–Sis_-inverse strongly monotone (see []). Assume thatFix(S)=∅. Then we have

Sx–x≤x–Sx,x–xˆ (.)

for allx∈Candxˆ∈Fix(S), which is obtained directly from

x–xˆ≥ Sx–Sxˆ=Sx–xˆ=Sx–x+ (x–xˆ) =Sx–x+x–xˆ+ Sx–x,x–xˆ.

LetFbe a bifunction ofC×CintoRsatisfying the following conditions: (A) F(x,x) = for allx∈C;

(A) Fis monotone,i.e.,F(x,y) +F(y,x)≤for allx,y∈C; (A) for eachx,y,z∈C,limt↓F(tz+ ( –t)x,y)≤F(x,y);

(A) for eachx∈C,y→F(x,y)is convex and lower semi-continuous.

Lemma .[] Let C be a nonempty closed convex subset of a Hilbert space H and F:

C×C→Rbe a bifunction which satisﬁes the conditions(A)-(A).For any x∈H and

r> ,deﬁne a mapping Tr:H→C by

T_rF(x) = z∈C:F(z,y) +

ry–z,z–x ≥,∀y∈C

. (.)

Then TF

r is well deﬁned and the following hold: () TF

r is single-valued; () TF

r is ﬁrmly nonexpansive,i.e.,for anyx,y∈H,

TF

rx–TrFy 

≤TF

rx–TrFy,x–y

;

() Fix(T_rF) =EP(F);

() EP(F)is closed and convex.

Lemma .[] Let F:C×C→Rbe a bifunction satisfying the conditions(A)-A().

Let TF

r and TsF be deﬁned as in Lemma.with r,s> .Then,for any x,y∈H,one has

T_rFx–T_sFy≤ |x–y|+ –s

r

T_rFx–x.

Remark . In [], some other conditions are required besides the conditions (A)-(A). In fact, the conditions (A)-(A) are enough for Lemma .. For the proof, refer to [, ].

Lemma .[] Let F:C×C→Rbe a functions satisfying the conditions(A)-(A)and TF

s,TtFbe deﬁned as in Lemma.with s,t> .Then the following holds:

T_sFx–T_tFx≤s–t

s Tsx–Ttx,Tsx–x

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Lemma .[] Let{an}be a sequence in[, ]such that

∞

n=an= .Then we have the

following:

∞

n=

anxn

 ≤

∞

n=

anxn

for any bounded sequence{xn}in a Hilbert space H.

Lemma .(Demiclosedness principle) Let T be a nonexpansive mapping on a closed

convex subset C of a real Hilbert space H.Then I–T is demiclosed at any point y∈H,that is,if xnx and xn–Txn→y∈H,then x–Tx=y.

Lemma .[] Assume that{an}is a sequence of nonnegative numbers such that

an+≤( –γn)an+δn

for each n≥,where{γn}is a sequence in(, )and{δn}is a sequence inRsuch that () ∞_n_=γn=∞;

() lim sup_n_→∞δn/γn≤or

_∞

=|δn|<∞.

Thenlimn→∞an= .

Lemma .[, ] Let U and V be nonexpansive mappings.Forσ ∈(, ),deﬁne S= σU+ ( –σ)V.Suppose thatFix(U)∩Fix(V)=∅.ThenFix(U)∩Fix(V) =Fix(S).

From [] we can see that Lemma . holds wheneverU andV are self or non-self mappings.

Lemma .[] Let C be a nonempty closed convex subset of a Hilbert space H and T:

C→H be a nonexpansive mapping withFix(T)=∅.Let PCbe the metric projection from

H onto C.ThenFix(PCT) =Fix(T) =Fix(TPC).

Remark . LetS,S:C→Hbe two nonexpansive mappings withFix(S)∩Fix(S)=∅. Letσ∈(, ) and deﬁne the mappingS:C→HbyS=σS+ ( –σ)S. By Lemmas . and ., it is easy to see thatFix(PCS) =Fix(PCS)∩Fix(PCS).

From Remark ., we get the following result.

Lemma . Let{Bi}Ni=be a ﬁnite family of inverse strongly monotone mappings from C

to H with the constants{βi}Ni=and assume that

N

i=VI(C,Bi)=∅.Let B=

N

i=αiBiwith {αi}Ni=⊂(, )and

N

i=αi= .Then B:C→H is aβ-inverse strongly monotone mapping

withβ=min{β, . . . ,βN}andVI(C,B) =

N

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Proof It is easy to show thatBis aβ-inverse strongly monotone mapping. In fact, for all

x,y∈C, by Lemma ., we have

βBx–By=β

N

i=

αi(Bix–Biy)



≤β N

i=

αiBix–Biy

≤ N

i=

αiβiBix–Biy

≤ N

i=

αix–y,Bix–Biy

=x–y,Bx–By,

which implies thatBis aβ-inverse strongly monotone mapping. Next, we prove thatVI(C,B) =N_i_=VI(C,Bi). Obviously, we have

N

i=

VI(C,Bi)⊂VI(C,B).

Now, for anyw∈VI(C,B), we show thatw∈_iN_=VI(C,Bi). Take a constantλ∈(, β]. ThenI–λBis nonexpansive. Note thatI–λB=N_i_=αi(I–λBi) and eachI–λBiis non-expansive. From Remark ., it follows that

FixPC(I–λB)

= N

i=

FixPC(I–λBi)

.

Thus we have

w∈VI(C,B) ⇐⇒ w=PC(I–λB)w=PC(I–λBi)w ⇐⇒ w∈VI(C,Bi)

for eachi= , . . . ,N. Therefore,w∈_iN_=VI(C,Bi). This completes the proof.

3 Main result

Now, we give the main results of this paper.

Theorem . Let H,H be two real Hilbert spaces and C⊂H,Q⊂H be nonempty

closed convex subsets.Let Ai:H→Hbe a bounded linear operator for each i= , . . . ,N

with N∈Nand Bi :C→H be aβi-inverse strongly monotone operator for each i= , . . . ,N with N∈N.Assume that F :C×C→Rsatisﬁes(A)-(A),Fi:Q×Q→R (i= , . . . ,N)satisﬁes(A)-(A).Let{Sn}be a countable family of nonexpansive mappings

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z∈EP(F)and Aiz∈EP(Fi),i= , . . . ,N}andVI=

N

i=VI(C,Bi).Let{γ, . . . ,γN} ⊂(, )

withN

i=γi= .Take v,x∈C arbitrarily and deﬁne an iterative scheme in the following

manner: ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

ui,n=TrFn(I–γA

∗

i(I–T Fi

rn)Ai)xn, i= , . . . ,N,

yn=PC(I–λn(

N

i=γiBi))(_N_

N

i=ui,n),

xn+=αnv+

n

i=(αi––αi)Siyn,

(.)

for each i= , . . . ,N and n∈N,where{rn} ⊂(r,∞)with r> ,{λn} ⊂(, β)withβ= min{β, . . . ,βN}andγ ⊂(, /L],L=max{L, . . . ,LN}and Liis the spectral radius of the

operator A∗_iAi and A∗i is the adjoint of Ai for each i∈ {, . . . ,N}, and{αn} ⊂(, )is a

strictly decreasing sequence.Letα= and assume that the control sequences{αn},{λn}, {rn}satisfy the following conditions:

() limn→∞αn= and∞n=αn=∞; () ∞_n_=|rn+–rn|<∞and

∞

n=|λn+–λn|<∞; () limn→∞λn=λ> .

Then the sequence{xn}deﬁned by(.)converges strongly to a point z=Pv.

Proof We ﬁrst show that, for eachi= , . . . ,N andn∈N,A∗i(I–T Fi

rn)Aiis a __L

i-inverse strongly monotone mapping. In fact, sinceTFi

rn is (ﬁrmly) nonexpansive andI–T Fi rn is

  -inverse strongly monotone, we have

A∗_iI–TFi rn

Aix–A∗i

I–TFi rn

Aiy 

=A∗_iI–TFi rn

(Aix–Aiy),A∗i

I–TFi rn

(Aix–Aiy)

=(I–Trn)(Aix–Aiy),AiA

∗

i

I–TFi rn

(Aix–Aiy)

≤L_iI–TFi rn

(Aix–Aiy),

I–TFi rn

(Aix–Aiy)

=L_iI–TFi rn

(Aix–Aiy) 

≤L_iAix–Aiy,

I–TFi rn

(Aix–Aiy)

= L_ix–y,A∗_iI–TFi rn

Aix–A∗i

I–TFi rn

Aiy

for allx,y∈H, which implies thatA∗_i(I–TFi rn)Aiis a



L_i-inverse strongly monotone map-ping. Note thatγ ∈(, 

L_i]. ThusI–γA

∗

i(I–T Fi

rn)Aiis nonexpansive for eachi= , . . . ,N andn∈N.

Now, we complete the proof by the next steps.

Step.{xn}is bounded. Letp∈. Thenp=TFi

rnpand (I–γA∗i(I–T Fi

rn)Ai)p=p. Thus we have

ui,n–p=TrFn

I–γA∗_iI–T_rF_nAi

xn–TrFni

I–γA∗_iI–TFi rn

Ai

p

≤I–γA∗_iI–TFi rn

Ai

xn–

I–γA∗I–TFi rn

Ai

p

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LetB=N

i=γiBi. ThenBis aβ-inverse strongly monotone mapping. Since{λn} ⊂(, β),

I–λnBis nonexpansive. Thus from (.), we have

yn–p=

PC(I–λnB) 

N N

i=

ui,n–PC(I–λnB)p

≤(I–λnB) 

N N

i=

ui,n– (I–λnB)p

≤  N N i=

ui,n–p

≤  N N i=

ui,n–p

≤ xn–p. (.)

Thus from (.), it follows that

xn+–p=

αn(v–p) + n

i=

(αi––αi)(Siyn–Sip)

≤αnv–p+ n

i=

(αi––αi)yn–p

≤αnv–p+ n

i=

(αi––αi)xn–p

=αnv–p+ ( –αn)xn–p ≤maxv–p,xn–p

for alln∈N, which implies that{xn}is bounded and so are{ui,n}(i= , . . . ,N) and{yn}.

Step.lim_n_→∞xn+–xn=  andlimn→∞ui,n+–ui,n=  for eachi= , . . . ,N. Since the mappings I–γA∗(I–TFi

rn)Aare nonexpansive, by Lemmas . and ., we have

ui,n+–ui,n

=T_rF_n_+I–γA∗_iI–TFi rn+

Ai

xn+–TrFn

Ai

xn ≤I–γA∗_iI–TFi

rn+

Ai

xn+–

Ai

xn +|rn+–rn|

rn+

T_rF_n_+I–γA∗_iI–TFi rn+

Ai

xn+–

I–γA∗_iI–TFi rn+

Ai

xn+ ≤ xn+–xn+I–γA∗i

I–TFi rn+

Ai

xn–

Ai

xn +|rn+–rn|

rn+ δn+

=xn+–xn+γA∗i

TFi

rn+Aixn–T

Fi

rnAixn+

|rn+–rn|

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≤ xn+–xn+γA∗i

_|

rn+–rn|

rn+

TFi

rn+Aixn–T

Fi rnAixn,T

Fi

rn+Aixn–Aixn

 

+|rn+–rn|

r δn+

≤ xn+–xn+γA∗i

_|

rn+–rn|

r σn+

 

+|rn+–rn|

r δn+

≤ xn+–xn+ηi,n+, (.)

where

σn+=sup n∈N

TFi

rn+Aixn–T

Fi rnAixn,T

Fi

rn+Aixn–Aixn,

δn+=sup n∈N

T_rF_n_+I–γA∗_iI–TFi rn+

Ai

xn+–

I–γA∗_iI–TFi rn+

Ai

xn+,

and

ηi,n+=γA∗i

_|

rn+–rn|

r σn+

 

+|rn+–rn|

r δn+.

Note that

(I–λn+B) 

N N

i=

ui,n+– (I–λnB) 

N N

i=

ui,n

=

(I–λn+B) 

N N

i=

ui,n+– (I–λn+B) 

N N

i=

ui,n+ (λn–λn+)Bwn

≤(I–λn+B) 

N N

i=

ui,n+– (I–λn+B) 

N N

i=

ui,n

+|λn–λn+|Bwn

≤ 

N N

i=

ui,n+–ui,n+|λn–λn+|Bwn, (.)

wherewn=_N_

N

i=ui,n. LetM=supn∈NBwn. By (.), (.), and (.), we have

yn+–yn=

PC(I–λn+B) 

N N

i=

ui,n+–PC(I–λnB) 

N N

i=

ui,n

≤(I–λn+B) 

N N

i=

ui,n+– (I–λnB) 

N N

i=

ui,n

≤  N N i=

ui,n+–ui,n+|λn–λn+|Bwn

≤ xn+–xn+ 

N N

i=

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Since{αn}is strictly decreasing, by using (.), we have

xn+–xn

=

(αn–αn–)v+ n–

i=

(αi––αi)(Siyn–Siyn–) + (αn––αn)Snyn

≤(αn––αn)v+ n–

i=

(αi––αi)Siyn–Siyn–+ (αn––αn)Snyn

≤(αn––αn)v+ n–

i=

(αi––αi)yn–yn–+ (αn––αn)Snyn

= (αn––αn)v+ ( –αn–)yn–yn–+ (αn––αn)Snyn

≤( –αn–)xn–xn–+ 

N N

i=

ηi,n+|λn––λn|M+ (αn––αn)M,

where M =sup{Snyn+v: n∈N}. By (i) and (ii) and Lemma ., we conclude that

lim

n→∞xn+–xn= . (.)

Further, by (.) and (.), we have

lim

n→∞yn+–yn= , nlim→∞ui,n+–ui,n= , i∈ {, . . . ,N}. (.)

Step.limn→∞Sixn–xn → for eachi∈N.

First, we show thatlimn→∞ui,n–xn=  for eachi∈ {, . . . ,N}. Since eachA∗i(I–T Fi rn)Ai is 

L_i-inverse strongly monotone, by (.), we have

ui,n–p =TrFn

Ai

xn–TrFn

Ai

p

≤I–γA∗_iI–TFi rn

Ai

xn–

Ai

p

=(xn–p) –γ

Aixn–A∗i

I–TFi rn

Aip =xn–p– γ

xn–p,A∗i

I–TFi rn

Aixn–A∗i

I–TFi rn

Aip

+γA∗_iI–TFi rn

Aixn–A∗i

I–TFi rn

Aip ≤ xn–p–

γ

L_iA

∗

i

I–TFi rn

Aixn–A∗i

I–TFi rn

Aip +γA∗_iI–TFi

rn

Aixn–A∗i

I–TFi rn

Aip 

=xn–p+γ

γ – 

L i

Aixn–A∗i

I–TFi rn

Aip 

=xn–p+γ

γ – 

L i

Aixn 

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From Lemma . and (.), it follows that

xn+–p =

αn(v–p) + n

i=

(αi––αi)(Siyn–p)



≤αnv–p+ n

i=

(αi––αi)Siyn–p

≤αnv–p+ n

i=

(αi––αi)yn–p

=αnv–p+ ( –αn)yn–p

≤αnv–p+ ( –αn) N

i= 

N

ui,n–p

≤αnv–p

+ ( –αn) N i=  N

xn–p+γ

γ– 

L i

Aixn 

=αnv–p+ ( –αn)xn–p

+ ( –αn) N i=  N γ

γ – 

L_i

Aixn ≤αnv–p+xn–p

+ ( –αn) N i=  N γ

γ – 

L i

Aixn 

.

Sinceγ<_L =max{_L , . . . ,

 L_N



}, we have

( –αn) 

N γ



L_i –γ

Aixn

≤( –αn) N i=  N γ  L i

–γA∗_iI–TFi rn

Aixn 

≤αnv–p+xn–p–xn+–p ≤αnv–p+xn–xn+

xn–p+xn+–p

.

Sinceαn→, by (.), we have

lim n→∞A

∗

i

I–TFi rn

Aixn=  (.)

for eachi∈ {, . . . ,N}, which implies that

lim n→∞I–T

Fi rn

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for eachi∈ {, . . . ,N}. SinceTF

rnis ﬁrmly nonexpansive andI–γA

∗

i(I–T Fi

rn)Aiis nonex-pansive, by (.), we have

ui,n–p =TrFn

xn+γA∗i

TFi rn–I

Aixn

–T_rF_n(p) ≤ui,n–p,xn+γA∗i

T_ri_n–IAixn–p

=  

ui,n–p+xn+γA∗i

TFi rn–I

Aixn–p 

–ui,n–p–

xn+γA∗i

TFi rn–I

Aixn–p  = 



ui,n–p+I–γA∗i

I–TFi rn

Ai

xn–

I–γA∗_iI–TF

rn

Ai

p

–ui,n–xn–γA∗i

TFi rn–I

Aixn

≤  

ui,n–p+xn–p–ui,n–xn–γA∗i

TFi rn–I

Aixn 

=  

ui,n–p+xn–p–

ui,n–xn+γA∗i

TFi rn–I

Aixn 

– γui,n–xn,A∗i

TFi rn–I

Aixn

,

which implies that

ui,n–p≤ xn–p–ui,n–xn+ γui,n–xnA∗i

TFi rn–I

Aixn. (.) Now, from (.) and (.), it follows that

xn+–p≤αnv–p+ ( –αn)yn–p

≤αnv–p+ ( –αn) N

i= 

N

ui,n–p

≤αnv–p+ ( –αn) N i=  N

xn–p–ui,n–xn

+ γui,n–xnA∗i

TFi rn–I

Aixn ≤αnv–p+xn–p– ( –αn)

N

i= 

N

ui,n–xn

+ γ N i=  N

ui,n–xnA∗i

TFi rn–I

Aixn,

and so

( –αn) 

N

ui,n–xn≤( –αn) N

i= 

N

ui,n–xn

≤αnv–p+xn–xn+

xn–p+xn+–p

+ γ N i=  N

ui,n+xnA∗i

TFi rn–I

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Sinceαn→, both{ui,n}and{xn}are bounded, by (.) and (.), we have

lim

n→∞ui,n–xn=  (.)

for eachi∈ {, . . . ,N}.

Next, we show that lim_n_→∞yn–un= , where un= _N_

N

i=ui,n. Note that p=

PC(I–λnB)p. By (.), we have

xn+–p≤αnv–p+ ( –αn)yn–p

≤αnv–p+ ( –αn)un–p–λn(Bun–Bp) 

=αnv–p

+ ( –αn)

un–p– λnun–p,Bun–Bp+λnBun–Bp

≤αnv–p

+ ( –αn)

un–p– λnβBun–Bp+λnBun–Bp

≤αnv–p

+ ( –αn)

xn–p– λnβBun–Bp+λnBun–Bp

=αnv–p+ ( –αn)xn–p

+ ( –αn)λn(λn– β)Bun–Bp

and so

( –αn)λn(β–λn)Bun–Bp ≤αnv–p+xn–xn+

xn–p+xn+–p

.

Sinceαn→ and  <limn→∞λn=λ< β, by (.), we have

lim

n→∞Bun–Bp= . (.)

SincePCis ﬁrmly nonexpansive and (I–λnB) is nonexpansive, by (.), we have

yn–p=PC(un–λnBun) –PC(p–λnBp) 

≤yn–p,un–λnBun– (p–λnBp)

= 

yn–p+(I–λnB)un– (I–λnB)p 

–yn–un+λn(Bun–Bp) 

≤ 

yn–p+un–p–yn–un+λn(Bun–Bp) 

= 

yn–p+un–p–yn–un–λnBun–Bp

– λnyn–un,Bun–Bp

≤ 

yn–p+un–p–yn–un–λnBun–Bp

+ λnyn–unBun–Bp

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and so

yn–p≤ un–p–yn–un–λnBun–Bp

+ λnyn–unBun–Bp

≤ xn–p–yn–un+ λnyn–unBun–Bp. (.)

From (.) and (.), we have

xn+–p≤αnv–p+ ( –αn)yn–p ≤αnv–p

+ ( –αn)

xn–p–yn–un+ λnyn–unBun–Bp

≤αnv–p+xn–p– ( –αn)yn–un

+ ( –αn)λnyn–unBun–Bp.

Therefore, we have

( –αn)yn–un≤αnv–p+xn–xn+

xn+–p+xn–p

+ ( –αn)λn

yn+un

Bun–Bp.

Sincelimn→∞αn=  and both{yn}and{un}are bounded, by (.) and (.), we have

lim

n→∞yn–un= . (.)

Further, from (.), (.), (.), and

xn+–yn ≤ xn+–xn+xn–un+un–yn

≤ xn+–xn+ N

i= 

N

xn–ui,n+un–yn,

it follows that

lim

n→∞xn+–yn= . (.)

Now, from (.), it follows that

n

i=

(αi––αi)(Siyn–yn) =xn+–yn–αn(v–yn). (.)

Since{αn}is strictly decreasing, for eachi∈N, by (.) and (.), we have

(αi––αi)Siyn–yn≤ n

i=

(αi––αi)Siyn–yn

≤ n

i=

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= xn+–yn,yn–p– αnv–yn,p–yn ≤xn+–ynyn–p+ αnv–ynyn–p.

Sincelimn→∞αn=  and{yn}is bounded, by (.), one has

lim

n→∞Siyn–yn=  (.)

for alli∈N. Further, since

Sixn–xn ≤ Sixn–Siyn+Siyn–yn+yn–xn ≤yn–xn+Siyn–yn

≤yn–xn++ xn+–xn+Siyn–yn,

by (.), (.), and (.), we obtain

lim

n→∞Sixn–xn=  (.)

for alli∈N.

Step.lim sup_n_→∞v–z,xn–z ≤.

Letz=Pv. Since{xn}is bounded, we can choose a subsequence{xnj}of{xn}such that

lim sup n→∞

v–z,xn–z=lim

j→∞v–z,xnj–z.

Since{xnj}is bounded, there exists a subsequence{xn_ji}of{xnj}converging weakly to a pointw∈C. Without loss of generality, we can assume thatxnjw.

Now, we show thatw∈. First of all, we prove thatw∈=∞_i_=Fix(Si). In fact, since

xn–Sixn→ for eachi∈Nandxnjw, by Lemma ., we obtainw∈

∞

i=Fix(Si) =. Next, we show thatw∈,i.e.,w∈EP(F) andAiw∈EP(Fi) for eachi= , . . . ,N. Letwi,n= (I–A∗i(I–T

Fi

rn))Aixnfor eachi= , . . . ,N. By (.) and (.) we see thatwi,n–

xn→ andTrFnwi,n–wi,n→ asn→ ∞. By Lemma . we see thatT F rnwi,n–T

F rwi,n ≤ | – r

rn|T F

rnwi,n–wi,n → asn→ ∞. HenceT F

rwi,n–wi,n→ asn→ ∞for eachi= , . . . ,N. SinceTrF is non-expansive and{wi,n}converges weakly tow, by Lemma . we getw=TF

rw,i.e.,w∈EP(F). On the other hand, since (I–γA∗i(I–T Fi

rn)Ai)xn–xn→ (by (.)) andI–γA∗_i(I–TFi

rn)Aiis non-expansive, from Lemmas . and . it follows that

w= (I–γA∗_i(I–TFi

r )Ai)w,i.e.,w=TrFAiw. Therefore,w∈. Finally, we prove thatw∈VI=N

i=VI(C,Bi) by demiclosedness principle. Obviously, we only need to show thatw=PC(w–λBiw), whereλ=limn→∞λn. By (.) and (.), one hasun–PC(I–λnB)un →, whereun=_N_

N

i=ui,n. Then we have

un–PC(I–λB)un≤un–PC(I–λnB)un+PC(I–λnB)un–PC(I–λB)un ≤un–PC(I–λnB)un+(I–λnB)un– (I–λB)un ≤un–PC(I–λnB)un+|λ–λn|Bun.

Sinceλn→λ> ,{Bun}is bounded andun–PC(I–λB)un →, we have

lim

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On the other hand, since{λn} ⊂(, β), one hasλ∈(, β]. ThusI–λBis nonexpansive and, further,PC(I–λB) is nonexpansive. Noting thatunjwasj→ ∞, by Lemma ., we obtainw=PC(I–λB)w. By Lemma ., we getw∈VI=

N

i=VI(C,Bi). Therefore,w∈. By the property onPC, we have

lim sup n→∞

v–z,xn–z=lim

j→∞v–z,xnj–z=v–z,w–z ≤. (.)

Step.xn→z=Pvasn→ ∞.

By (.), we have

xn+–z=

αnv+

n

i=

(αi––αi)Siyn–z



=αnv–z,xn+–z+ n

i=

(αi––αi)Siyn–z,xn+–z

≤αnv–z,xn+–z+

n

i=(αi––αi) 

Siyn–z+xn+–z

≤αnv–z,xn+–z+

n

i=(αi––αi) 

xn–z+xn+–z

=αnv–z,xn+–z+  –αn



xn–z+xn+–z

≤αnv–z,xn+–z+  –αn

 xn–z ₊

xn+–z _,

which implies that

xn+–z≤( –αn)xn–z+ αnv–z,xn+–z.

By Lemma . and (.), we can conclude thatlimn→∞xn–z= . This completes the

proof.

The following results follow directly from Theorem ..

Corollary . Let H,H be two real Hilbert spaces and C⊂H,Q⊂Hbe nonempty

closed convex subsets.Let A:H→Hbe a bounded linear operator and B:C→Hbe a β-inverse strongly monotone operator.Assume that F:C×C→R,F:Q×Q→Rare

bi-functions satisfying the conditions(A)-(A).Let{Sn}be countable family of nonexpansive

mappings from C into C.Assume that=∩∩VI(C,B)=∅,where=∞_n_=Fix(Sn)

and={z∈C:z∈EP(F)and Az∈EP(F)}.Take v∈C arbitrarily and deﬁne an iterative scheme in the following manner:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

un=TrFn(I–γA

∗₍_I_–_TF

rn)A)xn,

yn=PC(un–λnBun),

xn+=αnv+

n

i=(αi––αi)Siyn,

(.)

for all n∈N, where{rn} ⊂(r,∞)with r> ,{λn} ⊂(, β),andγ ⊂(, /L],L is the

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strictly decreasing sequence.Assume that the control sequences{αn},{λn},and{rn}satisfy

the following conditions: () lim_n_→∞αn= and

∞

n=αn=∞; () ∞_n_=|rn+–rn|<∞and

∞

n=|λn+–λn|<∞; () limn→∞λn=λ∈(, β).

Then the sequence{xn}deﬁned by(.)converges strongly to a point z=Pv.

Corollary . Let H,H be two real Hilbert spaces and C⊂H,Q⊂Hbe nonempty

closed convex subsets.Let A:H→Hbe a bounded linear operator and B:C→Hbe a β-inverse strongly monotone operator.Assume that F:C×C→R,F:Q×Q→Rare the

bifunctions satisfying the conditions(A)-(A).Let S:C→C be a nonexpansive mapping.

Assume that=Fix(S)∩∩VI(C,B)=∅,where={z∈C:z∈EP(F)and Az∈EP(F)}. Take v∈C arbitrarily and deﬁne an iterative scheme in the following manner:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

un=TrFn(I–γA

∗₍_I_–_TF

rn)A)xn,

yn=PC(un–λnBun),

xn+=αnv+ ( –αn)Syn

(.)

for all n∈N, where{rn} ⊂(r,∞)with r> ,{λn} ⊂(, β),andγ ⊂(, /L],L is the

spectral radius of the operator A∗A and A∗is the adjoint of A,{αn} ⊂(, )is a sequence.

Assume that the control sequences{αn},{λn},and{rn}satisfy the following conditions: () lim_n_→∞αn= and

∞

n=αn=∞; () ∞_n_=|rn+–rn|<∞and

∞

n=|λn+–λn|<∞; () limn→∞λn=λ∈(, β).

Then the sequence{xn}deﬁned by(.)converges strongly to a point z=Pv.

Remark . Theorem . and Corollary . extend the corresponding one of Kazmi and Rizvi [] from a nonexpansive mapping to a ﬁnite of family of nonexpansive mappings and from a split equilibrium problem to a ﬁnite of family of split equilibrium problems. It is a little simple to prove thatw∈VIby the demiclosedness principle in Theorem ..

We give an example to illustrate Theorem . as follows.

Example . LetH=RandH=R,C= [, ], andQ= [, ]×[, ]. LetA:H→H andA:H→Hdeﬁned byAx= (x,x)TandAx= (x_,x_)Tfor eachx∈H. ThenA∗y=

y+yandA∗y= y+y

 for eachy= (y,y)T∈H. ThenL=  andL= 

, whereLandL are the spectral radius ofA∗_AandA∗A, respectively.

LetB= (x– ) andB= – for allx∈C. Then it is easy to see thatBandBare_and -inverse strongly monotone operators fromCintoH. Find thatVI=VI(C,B)∩VI(C,B) = {}. For eachn∈N, letSn:C→Cdeﬁned bySn(x) =x+__nfor eachx∈[,_] andSn(x) =x for eachx∈(_, ]. Then{Sn}is a countable family of nonexpansive mappings fromCinto

Cand it is easy to see that=∞_n_=Fix(Sn) = (_, ]. For eachx,y∈C, deﬁne the bifunction

F:C×C→RbyF(x,y) =x–yfor allx,y∈C. For eachu= (u,u)T_and_v_{= (}_v_,_v_)T_∈_Q_, deﬁneF:Q×Q→RandF:Q×Q→Rby

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and

F(u,v) =

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

, ifu=v,

, ifu= (, ) or (_,_) andv= (, ) or (_,_),

–, ifv= (, ) or (_,_) andu= (, ) or (_,_),

u_+u_–v–v, otherwise.

It is easy to check that the bifunctionsF,F, andFsatisfy the conditions (A)-(A) andF. Moreover,={}, where={z∈C:z∈EP(F),Az∈EP(F) andAz∈EP(F)}. There-fore,=∩VI∩={}.

Letα= ,γ=γ=_, andγ =_. For eachn∈N, letrn= ,λn=_,αn=_n. Then the sequences{αn},{λn},{rn}satisfy the conditions ()-() in Theorem ..

For eachx∈Cand eachn∈N, we computeTF

rnAx,i.e.,TrFn(x,x). Findz= (, ) such that

F(z,y) + 

rn

y–z,z–Ax=  – (y+y) +  

(y– )( –x) + (y– )( –x)

=  – (y+y) + 

( –x)(y+y– )

= – (y+y)

 – ( –x)

≥

for ally= (y,y)∈Q. Thus, from Lemma .(), it follows thatTF

rnAx= (, ) for each

x∈C. Similarly, for eachx∈[, ], we can ﬁndz= (, ) such that, fory= (_,_),

F(z,y) + 

rn

y–z,z–Ax=  – 

( –x) =  +

x

≥;

fory= (, ),

F(z,y) + 

rn

y–z,z–Ax= ;

fory∈Q\ {(, ), ( ,

 )},

F(z,y) + 

rn

y–z,z–Ax=  +  

(y– )

 –x 

+ (y– )

 –x 

≥.

Thusz= (, ) =TF

rnAxfor allx∈Cby Lemma .().

Now, takev=_ andx=_ and deﬁne the sequence{xn}deﬁned by (.). Since each

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can get

I–γA∗_I–TF

rn

A

xn=

xn–γA∗

Axn–TrFnAxn

=xn–γA∗

(xn,xn) – (, )

=xn– γ(xn– )

= +xn  .

Note that

F(,y) + 

rn

y–z,z– +xn 

=  –y+ (y– )

 – +xn 

= ( –y)

 – 

 – +xn 

= ( –y)

 +

 +xn 

≥

for ally∈C. Thusu,n=  by Lemma .() for eachn∈N. Similarly, we can conclude that

u,n=  for eachn∈N.

Next, we compute the sequence{yn}. By the deﬁnition of{yn}, we see that

yn=PC

I–λn

B+B 

u,n+u,n 

=PC

 + 

= 

for alln∈N.

Finally, we compute the sequence{xn}by the following iteration:

xn+=αnv+ n

i=

(αi––αi)Siyn

=αnv+  –αn

=  –  n

→ =Pv=P{}



asn→ ∞as shown by Theorem ..

4 Applications

In this section, letH,H be two real Hilbert spaces andC,Qbe two nonempty closed convex subsets ofHandH, respectively. Letf :C→R,g:Q→Rbe two operators and

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We consider the following optimization problem:

ﬁndx∗∈Csuch thatfx∗≤f(x), ∀x∈C,

andy∗=Ax∗such thatgy∗≤g(y), ∀y∈Q.

(.)

We denote the set of solutions of (.) byand assume that=∅. LetF(x,y) =f(y) –f(x) for allx,y∈CandF(x,y) =g(y) –g(x) for allx,y∈Q. ThenF(x,y) andG(x,y) satisfy the conditions (A)-(A) in Section  provided thatf is convex and lower semicontinuous on

Candgis convex and lower semicontinuous onQ. Let={z∈C:z∈EP(F) andAz∈

EP(F)}. Obviously,=.

By Corollary . withB=IandS=I, we have the following iterative algorithm, which strongly converges to a pointz=Pv, which solves the optimization problem (.):

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

un=TrFn(I–γA

∗₍_I_–_TF

rn)A)xn,

yn=PC(un–λnun),

xn+=αnv+ ( –αn)yn,

(.)

where{rn} ⊂(r,∞) withr> ,{λn} ⊂(, ), andγ ⊂(, /L],Lis the spectral radius of the operatorA∗AandA∗is the adjoint ofA,{αn} ⊂(, ) is a sequence. Assume that the control sequences{αn},{λn}, and{rn}satisfy the following conditions:

() lim_n_→∞αn= and

∞

n=αn=∞; () ∞_n_=|rn+–rn|<∞,

∞

n=|αn+–αn|<∞, and

∞

n=|λn+–λn|<∞; () limn→∞λn=λ∈(, ).

For the special case withH=HandC=Q, we consider the following multi-objective optimization problem:

⎧ ⎨ ⎩

min{f(x),g(x)},

x∈C. (.)

We denote the set of solution of (.) byand assume that=∅. In (.), settingA=Iwe get the following algorithm, which strongly converges to the solution of multi-objective optimization problem (.):

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

un=TrFn(I–γ(I–T F

rn))xn,

yn=PC(un–λnun),

xn+=αnv+ ( –αn)yn,

whereγ ⊂(, /L_],_L_{is the spectral radius of the operator}_I∗_I_and_I∗_{is the adjoint of}_I_, other parameters such as{αn},{λn}, and{rn}satisfy the same conditions ()-().

5 Conclusion

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the proof is simple and is diﬀerent from the ones given in [–]. Also, in the results of this paper, we do not assume that eachFiis upper semi-continuous in the ﬁrst argument for eachi= , . . . ,N, which is required in the result in [–]. As an application, we solve an optimization problem by the result of this paper.

Competing interests

The authors declare that they have no competing interests. Authors’ contributions

All authors read and approved the ﬁnal manuscript. Author details

1_{Department of Mathematics and Physics, North China Electric Power University, Baoding, 071003, China.}2_Department

of Mathematics and Sciences, Shijiazhuang University of Economics, Shijiazhuang, 050031, China. 3_{Department of}

Mathematics, King Abdulaziz University, Jeddah, 21589, Saudi Arabia.4_{Department of Mathematics Education and RINS,}

Gyeongsang National University, Jinju, 660701, Korea.5_{Center for General Education, China Medical University, Taichung,}

40402, Taiwan. Acknowledgements

This work is supported by the Natural Science Funds of Hebei (Grant Number: A2015502021), the Fundamental Research Funds for the Central Universities (Grant Number: 2014ZD44) and the Project-sponsored by SRF for ROCS, SEM. The fourth author thanks the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and future Planning (Grant Number: 2014R1A2A2A01002100).

Received: 7 July 2015 Accepted: 29 November 2015 References

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