Strong convergence theorems for equilibrium problems and fixed point problem of multivalued nonexpansive mappings via hybrid projection method

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R E S E A R C H

Open Access

Strong convergence theorems for equilibrium

problems and ﬁxed point problem of

multivalued nonexpansive mappings via

hybrid projection method

Ali Abkar

1

_{and Mohammad Eslamian}

2*

*_{Correspondence:}

[email protected]

2_{Young Researchers Club, Babol}

Branch, Islamic Azad University, Babol, Iran

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

Abstract

In this paper, a new iterative process by the hybrid projection method is constructed. Strong convergence of the iterative process to a common element of the set of common ﬁxed points of a ﬁnite family of generalized nonexpansive multivalued mappings and the solution set of two equilibrium problems in a Hilbert space is proved. Our results extend some important recent results.

MSC: 47H10; 47H09

Keywords: equilibrium problem; hybrid projection method; strong convergence; common ﬁxed point; generalized nonexpansive multivalued mapping

1 Introduction

LetCbe a nonempty closed convex subset of a Hilbert spaceH. A subsetC⊂His called proximal if for eachx∈Hthere exists an elementy∈Csuch that

x–y=dist(x,C) =infx–z:z∈C.

We denote byCB(C) andP(C) the collection of all nonempty closed bounded subsets and nonempty proximal bounded subsets ofC, respectively. The Hausdorﬀ metricHonCB(H) is deﬁned by

H(A,B) :=max

sup

x∈A

dist(x,B),sup

y∈B

dist(y,A)

,

for allA,B∈CB(H).

LetT:H→H_{be a multivalued mapping. An element}_x_∈_H_{is said to be a ﬁxed point}

ofT, ifx∈Tx. The set of ﬁxed points ofTwill be denoted byF(T).

Deﬁnition . A multivalued mappingT:H→CB(H) is called (i) nonexpansive if

H(Tx,Ty)≤ x–y, x,y∈H.

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(ii) quasi-nonexpansive ifF(T)=∅andH(Tx,Tp)≤ x–pfor allx∈Hand all

p∈F(T).

Recently, J. Garcia-Falset, E. Llorens-Fuster and T. Suzuki [] introduced a new condition on singlevalued mappings, called condition (E), which is weaker than nonexpansiveness.

Deﬁnition . A mappingT:H→His said to satisfy the condition (Eμ) provided that

x–Ty ≤μx–Tx+x–y, x,y∈H.

We say thatT satisﬁes the condition (E) wheneverTsatisﬁes (Eμ) for someμ≥.

Now we modify this condition for multivalued mappings as follows (see also []):

Deﬁnition . A multivalued mappingT:H→CB(H) is said to satisfy the condition (P) provided that

H(Tx,Ty)≤μdist(x,Tx) +ηx–y, x,y∈H,

for someμ,η≥.

It is obvious that every nonexpansive multivalued mapping satisfies the condition (P). The theory of multivalued mappings has applications in control theory, convex optimiza-tion, differential equations and economics. Theory of nonexpansive multivalued map-pings is harder than the corresponding theory of nonexpansive single valued mapmap-pings. Different iterative processes have been used to approximate fixed points of multivalued nonexpansive mappings (see [–]). Letbe a bifunction fromC×CintoR, whereRis the set of real numbers. The equilibrium problem for:C×C→Ris to findx∈Csuch that

(x,y)≥, ∀y∈C.

The set of solutions is denoted byEP(). It is well known that this problem is closely re-lated to minimax inequalities (see [] and []). The equilibrium problem includes ﬁxed point problems, optimization problems and variational inequality problems as special cases. Some methods have been proposed to solve the equilibrium problem, see, for ex-ample, [–].

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2 Preliminaries

Let us recall the following deﬁnitions and results which will be used in the sequel.

Lemma .([]) Let H be a real Hilbert space. Then for i,j= , , . . . ,k we have

ax+ax+· · ·+akxk≤ax+ax+· · ·+akxk–aiajxi–xj

for all xi,xj∈H and ai,aj∈[, ]with

k i=ai= .

LetCbe a closed convex subset ofH. For every pointx∈H, there exists a unique nearest point inC, denoted byPCxsuch that

x–PCx ≤ x–y, ∀y∈C.

PCis called the metric projection ofHontoC. It is well known thatPCis a nonexpansive

mapping.

Lemma .([]) Let C be a closed convex subset of H. Given x∈H and a point z∈C. Then z=PCx if and only if

x–z,z–y ≥, ∀y∈C.

Lemma .([]) Let C be a closed convex subset of H. Then for all x∈H and y∈C we have

y–PCx+x–PCx≤ x–y.

For solving the equilibrium problem, we assume that the bifunctionsatisﬁes the fol-lowing conditions:

(A) (x,x) = for anyx∈C,

(A) is monotone,i.e.,(x,y) +(y,x)≤for anyx,y∈C, (A) is upper-hemicontinuous,i.e., for eachx,y,z∈C,

lim sup

t→+

tz+ ( –t)x,y≤(x,y),

(A) (x,·)is convex and lower semicontinuous for eachx∈C. The following lemma was proved in [].

Lemma . Let C be a nonempty closed convex subset of H and letbe a bifunction of C×C intoRsatisfying (A)-(A). Let r> and x∈H. Then, there exists z∈C such that

(z,y) +

ry–z,z–x ≥ ∀y∈C.

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Lemma . Assume that:C×C→Rsatisﬁes (A)-(A). For r> and x∈H, deﬁne a mapping Tr:H→C as follows:

Trx= z∈C:(z,y) +



ry–z,z–x ≥,∀y∈C

.

Then, the following hold:

(i) Tris single valued;

(ii) Tris ﬁrmly nonexpansive, i.e., for anyx,y∈H,

Trx–Try≤ Trx–Try,x–y;

(iii) F(Tr) =EP();

(iv) EP()is closed and convex.

The following lemma was proved in [] for nonexpansive multivalued mappings. The statement is true for quasi-nonexpansive multivalued mappings as well. To avoid repeti-tion, we omit the details of the proof.

Lemma . Let C be a closed convex subset of a real Hilbert space H. Let T:C→CB(C)

be a quasi-nonexpansive multivalued mapping such that T(p) ={p}for all p∈F(T). Then F(T)is closed and convex.

Now, following Shahzad and Zegeye [], we remove the restriction T(p) ={p}for all

p∈F(T). LetT:C→P(C) be a multivalued mapping and

PT(x) =

y∈Tx:x–y=dist(x,Tx).

We use a similar argument as in the proof of Lemma . in [] to obtain the following lemma.

Lemma . Let C be a closed convex subset of a real Hilbert space H. Let T:C→P(C)

be a multivalued mapping such that PT is quasi-nonexpansive. Then F(T)is closed and convex.

Note that for allp∈F(T),PT(p) ={p}. We remark that there exist some examples of

multivalued mappings for whichPT is nonexpansive (see [] for details), so that the

as-sumption onTis not artiﬁcial.

3 The main result

Theorem . Let C be a nonempty closed convex subset of a real Hilbert space H,

andbe two bifunctions of C×C intoRsatisfying (A)-(A). Let Ti:C→CB(C), (i=

, , . . . ,m), be a ﬁnite family of quasi-nonexpansive multivalued mappings, each satisfying the condition (P). Assume further thatF=m_i_=F(Ti)∩EP()∩EP()=∅and Ti(p) =

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the following algorithm:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

x∈C,

un∈C such that(un,y) +_r_ny–un,un–xn ≥; ∀y∈C, u_n∈C such that(un,y) +sny–u

n,un–xn ≥; ∀y∈C, vn=δnun+ ( –δn)un,

wn=an,vn+an,zn,+· · ·+an,mzn,m, yn=bn,vn+bn,zn,+· · ·+bn,mzn,m, Cn+={v∈Cn:yn–v ≤ xn–v}, xn+=PCn+x: ∀n≥,

where zn,i∈Tivn, zn,i∈Tiwnfor i= , , . . . ,m. Assume that{an,j},{bn,j},{δn}and{rn},{sn} satisfy the following conditions:

(i) {an,j},{bn,j},{δn} ⊂[a,b]⊂(, )(j= , , , . . . ,m),

(ii) {rn},{sn} ⊂(,∞), andlim infn→∞rn> andlim infn→∞sn> . Then, the sequences{xn}and{vn}converge strongly to PFx.

Proof First, we show thatF⊂Cnfor alln≥. Fixq∈F. We set

T

r (x) = z∈C:(z,y) +



ry–z,z–x ≥∀y∈C

.

Hence we haveun=Trnxnandun=T



sn xn. By Lemma ., we have

un–q=Trnxn–T



rn q≤ xn–q and

u_n–q=T sn xn–T



sn q≤ xn–q, which implies that

vn–q ≤δnun–q+ ( –δn)un–q≤ xn–q.

SinceTiis quasi-nonexpansive, fori= , , . . . ,m, we have

wn–q=an,vn+an,zn,+· · ·+an,mzn,m–q

≤an,vn–q+an,zn,–q+· · ·+an,mzn,m–q

≤an,xn–q+an,dist(zn,,Tq) +· · ·+an,mdist(zn,m,Tmq)

≤an,xn–q+an,H(Tvn,Tq) +· · ·+an,mH(Tmvn,Tmq)

≤an,xn–q+an,vn–q+· · ·+an,mvn–q

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and also

yn–q =bn,vn+bn,zn,+· · ·+bn,mzn,m–q

≤bn,vn–q+bn,zn,–q+· · ·+bn,mzn,m–q

≤bn,xn–q+bn,dist

z_n_,,Tq

+· · ·+bn,mdist

z_n_,_m,Tmq

≤bn,xn–q+bn,H(Twn,Tq) +· · ·+bn,mH(Tmwn,Tmq)

≤bn,xn–q+bn,wn–q+· · ·+bn,mwn–q

≤bn,xn–q+bn,xn–q+· · ·+bn,mxn–q ≤ xn–q.

Thereforeq∈Cn, which implies that

F=

m

i=

F(Ti)∩EP()∩EP()⊂Cn, for alln≥.

We observe thatCnis closed and convex (see []). Now we show thatlimn→∞xn–x

exists. By Lemma . we haveFis closed and convex. Putw=PFx. Fromxn=PCnxand

xn+∈Cn+⊂Cnwe have

xn–x ≤ xn+–x.

Also fromw∈F⊂Cnandxn=PCnxfor alln≥, we get that

xn–x ≤ w–x.

It follows that the sequence {xn} is bounded and nondecreasing. Hence the limit

limn→∞xn–xexists. We show thatlimn→∞xn=u∈C. Fork>nwe havexk=PCkx∈

Ck⊂Cn. Now by applying Lemma . we have

xk–xn≤ xk–x–xn–x.

Sincelimn→∞xn–xexists, it follows that{xn}is a Cauchy sequence, and hence there

existsu∈Csuch thatlimn→∞xn=u. Puttingk=n+ , in the above inequality we have

lim

n→∞xn+–xn= .

Fromxn+∈Cn+, we have

yn–xn+ ≤ xn–xn+,

so thatlimn→∞yn–xn+= . This implies thatlimn→∞yn=u. Takeq∈F. By Lemma .,

for each ≤i≤m, we have

wn–q =an,vn+an,zn,+· · ·+an,mzn,m–q

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+· · ·+an,mzn,m–q–an,ian,ovn–zn,i

≤an,vn–q+an,dist(zn,,Tq)

+· · ·+an,mdist(zn,m,Tmq)–an,ian,ovn–zn,i

≤an,vn–q+an,H(Tvn,Tq)

+· · ·+an,mH(Tmvn,Tmq)–an,ian,ovn–zn,i

≤an,vn–q+an,vn–q

+· · ·+an,mvn–q–an,ian,ovn–zn,i

≤ vn–q–an,ian,ovn–zn,i,

and also

yn–q =bn,vn+bn,zn,+· · ·+bn,mzn,m–q



≤bn,vn–q+bn,zn,–q 

+· · ·+bn,mzn,m–q



≤bn,vn–q+bn,dist

z_n_,,Tq



+· · ·+bn,mdist

z_n_,_m,Tmq



≤bn,vn–q+bn,H(Twn,Tq)+· · ·+bn,mH(Tmwn,Tmq)

≤bn,vn–q+bn,wn–q+· · ·+bn,mwn–q

≤bn,vn–q+bn,vn–q+· · ·+bn,mvn–q

–bn,ian,ian,ovn–zn,i

≤ xn–q–bn,ian,ian,ovn–zn,i. ()

So we have that

abvn–zn,i≤bn,ian,ian,ovn–zn,i

≤ vn–q–yn–q

≤ xn–q–yn–q,

which implies that

lim

n→∞vn–zn,i= , fori= , , . . . ,m.

Hence

lim

n→∞dist(vn,Tivn)≤nlim→∞vn–zn,i=  (i= , , . . . ,m).

As aboveun=Trnxnso that

un–q =Trnxn–T

 rn q



≤T rn xn–T

 rn q,xn–q

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=un–q,xn–q

=  

un–q+xn–q–xn–un

and hence

un–q≤ xn–q–xn–un. ()

And also byu_n=T

sn xnwe have

u_n–q =T sn xn–T

 sn q



≤T sn xn–T

 sn q,xn–q

=u_n–q,xn–q

=  u

n–q



+xn–q–xn–un



and hence

u_n–q≤ xn–q–xn–un



. ()

Now we use () and () to obtain

vn–q≤δnun–q+ ( –δn)un–q



≤ xn–q–δnxn–un– ( –δn)xn–un



.

It follows from () and the last inequality that

yn–q≤ vn–q≤ xn–q–δnxn–un– ( –δn)xn–un



.

So we have

axn–un≤δnxn–un≤ xn–q–yn–q,

and

( –b)xn–un

_≤

( –δn)xn–un

_≤

xn–q–yn–q.

Sincelimn→∞xn=limn→∞yn=u, we obtain that

lim

n→∞un–xn=nlim→∞u

n–xn= ,

which implies that

lim

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Sincelimn→∞vn–xn= , fori= , , . . . ,m, we obtain that

dist(xn,Tixn)≤ xn–vn+dist(vn,Tivn) +H(Tivn,Tixn)

≤(η+ )xn–vn+ (μ+ )dist(vn,Tivn)→ asn→ ∞. ()

We observe thatu∈m_i_=F(Ti). Indeed,

dist(u,Tiu)≤ u–xn+dist(xn,Tixn) +H(Tixn,Tiu)

≤(η+ )u–xn+ (μ+ )dist(xn,Tixn)→ asn→ ∞,

which implies that u ∈ m_i_=F(Ti). Let us show that u ∈ EP() ∩ EP(). From

limn→∞xn=uandlimn→∞xn–un= , we haveun→uasn→ ∞. Sinceun=Trnxnwe obtain

(un,y) +



rn

y–un,un–xn ≥ ∀y∈C.

From (A), we have



rn

y–un,un–xn ≥(y,un),

and hence

y–un, un–xn

rn

≥(y,un).

Since

un–xn rn

→,

andun→u, from (A) we have

≥(y,u), ∀y∈C.

Fort∈(, ] andy∈C, letyt=ty+ ( –t)u. Sincey,u∈C, andCis convex we haveyt∈C

and hence(yt,u)≤. So, from (A) and (A), we have

 =(yt,yt)≤t(yt,y) + ( –t)(yt,u)≤t(yt,y),

which gives(yt,y)≥. From (A) we have ≤(u,y),∀y∈Cand henceu∈EP().

Similarly, we have u ∈ EP(). Now we show that u= PFx. Since xn =PCnx, by Lemma . we have

z–xn,x–xn ≤, ∀z∈Cn.

Sinceu∈F⊂Cnwe get

z–u,x–u ≤, ∀z∈F.

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By substitutingPTibyTiand using a similar argument as in Theorem ., we obtain the following result.

Theorem . Let C be a nonempty closed convex subset of a real Hilbert space H,

and  be two bifunctions of C ×C into R satisfying (A)-(A). Let Ti :C →P(C), (i= , , . . . ,m), be a ﬁnite family of multivalued mappings such that each PTi is

quasi-nonexpansive and satisﬁes the condition (P). Assume further that F = m_i_=F(Ti)∩ EP()∩EP()=∅. For C=C, let{xn}and{vn}be sequences generated by the following algorithm: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

x∈C,

wn=an,vn+an,zn,+· · ·+an,mzn,m, yn=bn,vn+bn,zn,+· · ·+bn,mzn,m, Cn+={v∈Cn:yn–v ≤ xn–v}, xn+=PCn+x: ∀n≥,

where zn,i∈PTi(vn), zn,i∈PTi(wn)for i= , , . . . ,m. Assume that{an,j},{bn,j},{δn}and{rn}, {sn}satisfy the following conditions:

(i) {an,j},{bn,j},{δn} ⊂[a,b]⊂(, )(j= , , , . . . ,m),

As a result, for single valued mappings we obtain the following theorem.

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

 and be two bifunctions of C×C intoRsatisfying (A)-(A). Let Ti:C→C (i=

, , . . . ,m), be a ﬁnite family of quasi-nonexpansive mappings, each satisfying the condition (P). Assume further thatF=m_i_=F(Ti)∩EP()∩EP()=∅. For C=C, let{xn}and

{vn}be sequences generated by the following algorithm:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

x∈C,

wn=an,vn+an,Tvn+· · ·+an,mTmvn, yn=bn,vn+bn,Twn+· · ·+bn,mTmwn, Cn+={v∈Cn:yn–v ≤ xn–v}, xn+=PCn+x: ∀n≥.

Assume that{an,j},{bn,j},{δn}and{rn},{sn}satisfy the following conditions:

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Theorem . Let C be a nonempty closed convex subset of a real Hilbert space H. Let Ti:C→P(C)(i= , , . . . ,m), be a ﬁnite family of multivalued mappings such that PTiis

quasi-nonexpansive and satisﬁes the condition (P). Assume further thatF=m_i_=F(Ti)=∅. For C=C, let{xn}be the sequence generated by the following algorithm:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

x∈C,

wn=an,xn+an,zn,+· · ·+an,mzn,m, yn=bn,xn+bn,zn,+· · ·+bn,mzn,m, Cn+={v∈Cn:yn–v ≤ xn–v}, xn+=PCn+x: ∀n≥,

where zn,i∈PTi(xn), zn,i∈PTi(wn)for i= , , . . . ,m. Assume that{an,j},{bn,j} ⊂[a,b]⊂(, )

(j= , , , . . . ,m), Then, the sequence{xn}converges strongly to PFx.

Proof Putting(x,y) =(x,y) =  for all x,y∈C andrn=sn=  in Theorem ., we

haveun=un=xnand hencevn=xn. Now, the desired conclusion follows directly from

Theorem ..

Now, we supply an example to illustrate the main result of this paper.

Example . We consider the nonempty closed convex subsetC= [, ] of the Hilbert

spaceR. Deﬁne two mappingsTandTonCas follows:

T(x) =

x

,

x



, T(x) =

⎧ ⎨ ⎩

[,x_], x= , {}, x= .

We note thatTandTare quasi-nonexpansive mappings satisfying the condition (P), (for

details, see []). Also we deﬁne two bifunctionsandas follows:

⎧ ⎨ ⎩

:C×C→R,

(x,y) =y–x,

⎧ ⎨ ⎩

:C×C→R,

(x,y) =y+xy– x.

It is easy to see thatandsatisfy the conditions (A)-(A). If we putrn=  andsn= ,

thenun=Trnxn=  andun=T



sn xn=__sx_nn_+=x_n (for details, see []). Putan,i=bn,i=_

fori= , ,  andδn=_. For any arbitraryx∈Cwe have

C=

v∈C:|y–v| ≤ |x–v|

=

,x+y 

.

Sincex+y

 ≤x, we obtain that

x=PCx= x+y

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By continuing this process we obtain

Cn+=

v∈Cn:|yn–v| ≤ |xn–v|

=

,xn+yn 

,

and hence

xn+=PCn+x= xn+yn

 .

Now, we have the following algorithm:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

x∈C,

vn=x_n,

zn,i∈Ti(vn), i= , , wn=_vn+_zn,+_zn,,

z_n_,_i∈Tiwn, i= , , yn=_vn+_zn,+zn,,

xn+=xn+_yn : ∀n≥.

Puttingzn,=zn,=v_n andzn,=zn,=wn we get that

xn+=

 , xn=

 , 

n+

x, ∀n≥.

We observe that for an arbitraryx∈C,xnis convergent to zero. We note thatF=F(T)∩

F(T)∩EP()∩EP() ={}.

Remark . Since every nonexpansive mapping is quasi-nonexpansive and satisﬁes the condition (P), our results hold for nonexpansive mappings.

Remark . Our results generalize the results of Tada and Takahashi [], of a nonex-pansive single valued mapping to a ﬁnite family of generalized nonexnonex-pansive multivalued mappings.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All the authors contributed equally to the writing of the present article. And they also read and approved the ﬁnal manuscript.

Author details

1_{Department of Mathematics, Imam Khomeini International University, Qazvin, 34149, Iran.}2_{Young Researchers Club,}

Babol Branch, Islamic Azad University, Babol, Iran.

Acknowledgements

Research of the ﬁrst author was supported in part by a grant from Imam Khomeini International University, under the grant number 751164-91.

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Cite this article as:Abkar and Eslamian:Strong convergence theorems for equilibrium problems and ﬁxed point problem of multivalued nonexpansive mappings via hybrid projection method.Journal of Inequalities and Applications