The general iterative methods for equilibrium problems and fixed point problems of a countable family of nonexpansive mappings in Hilbert spaces

R E S E A R C H

Open Access

The general iterative methods for equilibrium

problems and fixed point problems of a

countable family of nonexpansive mappings

in Hilbert spaces

Kiattisak Rattanaseeha

*_{Correspondence:}

[email protected] Division of Mathematics, Department of Science, Faculty of Science and Technology, Loei Rajabhat University, Loei, 42000, Thailand

Abstract

In this paper, the researcher introduces the general iterative scheme for finding a common element of the set of equilibrium problems and fixed point problems of a countable family of nonexpansive mappings in Hilbert spaces. The results presented in this paper improve and extend the corresponding results announced by many others.

MSC: 47H10; 47H09

Keywords: equilibrium problem; fixed point; nonexpansive mapping; variational inequality; strongly positive operator; Hilbert spaces

1 Introduction

LetHbe a real Hilbert space with the inner product·,·and the norm · . LetCbe a nonempty closed and convex subset ofH, and letT:C→Cbe a nonlinear mapping. In this paper, we useF(T) to denote the fixed point set ofT.

Recall the following definitions.

() The mappingTis said to be nonexpansive if

Tx–Ty ≤ x–y, ∀x,y∈C. (.)

Further, letFbe a bifunction fromC×CintoR, whereRis the set of real numbers. The so-called equilibrium problem forF:C×C→Ris to findy∈Csuch that

F(y,u)≥, ∀u∈C. (.)

The set of solutions of (.) is denoted byEP(F). Given a mappingA:C→H, letF(y,u) =

Ay,u–yfor ally,u∈C. Thenz∈EP(F) if and only ifAz,u–z ≥ for allu∈C. Nu-merous problems in physics, optimization and economics reduce to finding a solution of (.).

() The mappings{Tn}n∈Nare said to be a family of nonexpansive mappings fromHinto itself if

Tnx–Tny ≤ x–y, ∀x,y∈H, (.)

and denoted byF(Tn) ={x∈H:Tnx=x}is the fixed point set ofTn. Finding an optimal point in_n∈NF(Tn) of the fixed point sets of each mapping is a matter of interest in various branches of science.

Recently, many authors considered the iterative methods for finding a common element of the set of solutions to problem (.) and of the set of fixed points of nonexpansive map-pings; see, for example, [, ] and the references therein.

Next, letA:C→Hbe a nonlinear mapping. We recall the following definitions. ()Ais said to bemonotoneif

Ax–Ay,x–y ≥, ∀x,y∈C.

()Ais said to bestrongly monotoneif there exists a constantα>  such that

Ax–Ay,x–y ≥αx–y, ∀x,y∈C.

In such a case,Ais said to beα-strongly monotone.

()Ais said to be inverse-strongly monotone if there exists a constantα>  such that

Ax–Ay,x–y ≥αAx–Ay, ∀x,y∈C.

In such a case,Ais said to beα-inverse-strongly monotone. The classical variational inequality is to findu∈Csuch that

Au,v–u ≥, ∀v∈C. (.)

In this paper, we useVI(C,A) to denote the set of solutions to problem (.). One can easily see that the variational inequality problem is equivalent to a fixed point problem.u∈C is a solution to problem (.) if and only ifuis a fixed point of the mappingPC(I–λ)T, whereλ>  is a constant.

The variational inequality has been widely studied in the literature; see, for example, the work of Plubtieng and Punpaeng [] and the references therein.

Recently, Cenget al.[] considered an iterative method for the system of variational inequalities (.). They got a strongly convergence theorem for problem (.) and a fixed point problem for a single nonexpansive mapping; see [] for more details.

On the other hand, Moudafi [] introduced the viscosity approximation method for nonexpansive mappings (see [] for further developments in both Hilbert and Banach spaces).

A mappingf :C→Cis calledα-contractive if there exists a constantα∈(, ) such that

f(x) –f(y)≤αx–y, ∀x,y∈C. (.)

Letf be a contraction onC. Starting with an arbitrary initialx∈C, define a sequence {xn}recursively by

where{σn}is a sequence in (, ). It is proved [, ] that under certain appropriate con-ditions imposed on{σn}, the sequence{xn}generated by (.) strongly converges to the unique solutionqinCof the variational inequality

(I–f)q,p–q≥, p∈C.

LetAbe a strongly positive linear bounded operator on a Hilbert spaceHwith a con-stantγ¯; that is, there existsγ¯>  such that

Ax,x ≥ ¯γx, ∀x∈H. (.)

Recently, Marino and Xu [] introduced the following general iterative method:

xn+= (I–αnA)Txn+αnγf(xn), n≥, (.)

whereAis a strongly positive bounded linear operator onH. They proved that if the se-quence{αn}of parameters satisfies appropriate conditions, then the sequence{xn} gener-ated by (.) converges strongly to the unique solution of the variational inequality

(A–γf)x∗,x–x∗≥, x∈C, (.)

which is the optimality condition for the minimization problem

min

x∈C 

Ax,x–h(x),

wherehis a potential function forγf (i.e.,h(x) =γf(x) forx∈H).

In , Takahashi and Takahashi [] introduced an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions (.) and the set of fixed points of a nonexpansive mapping in Hilbert spaces. LetS:C→Hbe a nonexpansive mapping. Starting with arbitrary initialx∈H, define sequences{xn}and

{un}recursively by

⎧ ⎨ ⎩

F(un,y) +_r_ny–un,un–xn ≥, ∀y∈C, xn+=αnγf(xn) + ( –αn)Sun, ∀n∈N.

(.)

They proved that under certain appropriate conditions imposed on{αn}and{rn}, the se-quences{xn}and{un}converge strongly toz∈F(S)∩EP(F), wherez=PF(S)∩EP(F)f(z).

Next, Plubtieng and Punpaeng, [] introduced an iterative scheme by the general itera-tive method for finding a common element of the set of solutions (.) and the set of fixed points of nonexpansive mappings in Hilbert spaces.

LetS:H→H be a nonexpansive mapping. Starting with an arbitraryx∈H, define

sequences{xn}and{un}by

⎧ ⎨ ⎩

F(un,y) +_r_ny–un,un–xn ≥, ∀y∈C, xn+=αnγf(xn) + (I–αnA)Sun, ∀n∈N.

They proved that if the sequences{αn}and{rn}of parameters satisfy appropriate condi-tions, then the sequence{xn}generated by (.) converges strongly to the unique solution of the variational inequality

(A–γf)z,x–z≥, ∀x∈F(S)∩EP(F), (.)

which is the optimality condition for the minimization problem

min

x∈F(S)∩EP(F)



Ax,x–h(x),

wherehis a potential function forγf (i.e.,h(x) =γf(x) forx∈H).

LetT,T, . . . be an infinite sequence of mappings ofCinto itself, and letλ,λ, . . . be

real numbers such that ≤λi≤ for everyi∈N. Then for anyn∈N, Takahashi [] (see []) defined a mappingWnofCinto itself as follows:

Un,n+=I,

Un,n=λnTnUn,n++ ( –λn)I,

Un,n–=λn–Tn–Un,n+ ( –λn–)I,

.. .

Un,k=λkTkUn,k++ ( –λk)I, (.)

Un,k–=λk–Tk–Un,k+ ( –λk–)I,

.. .

Un,=λTUn,+ ( –λ)I,

Wn=Un,=λTUn,+ ( –λ)I.

Such a mappingWnis called theW-mapping generated byTn,Tn–, . . . ,Tandλn,λn–,

. . . ,λ.

Recently, using process (.), Yaoet al.[] proved the following result.

Theorem . Let C be a nonempty closed convex subset of a real Hilbert space H. Let F:C×C→Rbe an equilibrium bifunction satisfying the conditions:

() Fis monotone,that is,F(x,y) +F(y,x)≤for allx,y∈C; () for eachx,y,z∈C,limt→F(tz+ ( –t)x,y)≤F(x,y);

() for eachx∈C,y→F(x,y)is convex and lower semicontinuous.

Let {Ti}∞i= be an infinite family of nonexpansive mappings of C into C such that ∞

i=F(Ti)∩EP(F)=∅.Suppose{αn},{βn}and{γn}are three sequences in(, )such that αn+βn+γn= and{rn} ⊂(,∞).Suppose the following conditions are satisfied:

Let f be a contraction of H into itself,and let x∈H be given arbitrarily.Then the sequences {xn}and{yn}generated iteratively by

⎧ ⎨ ⎩

F(yn,x) +_r_nx–yn,yn–xn ≥, ∀x∈C, xn+=αnf(xn) +βnxn+γnWnyn,

converge strongly to x∗∈∞_i=F(Ti)∩EP(F),the unique solution of the minimization prob-lem

min

x∈∞i=F(Ti)∩EP(F)

 x

_–h(x),

where h is a potential function for f.

Very recently, using process (.), Chen [] proved the following result.

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

{Tn}∞n=be a sequence of nonexpansive mappings from C to C such that the common fixed

point set=∞_n=F(Tn)=∅.Let f :C→H be anα-contraction,and let A:H→H be a self-adjoint,strongly positive bounded linear operator with a coefficientγ¯> .Letσ be a constant such that <γ α<γ¯.For an arbitrary initial point xbelonging to C,one defines

a sequence{xn}n≥iteratively

xn+=PC

αnγf(xn) + (I–αnA)Wnxn

, ∀n≥, (.)

where{αn}is a real sequence in[, ].Assume the sequence{αn}satisfies the following con-ditions:

(C) limn→∞αn= ; (C) ∞_n=αn=∞.

Then the sequence{xn}generated by(.)converges in norm to the unique solution x∗, which solves the following variational inequality:

x∗∈ such that(A–γf)x∗,x∗–xˆ≥,∀ˆx∈. (.)

Motivated by this result, we introduce the following explicit general iterative scheme:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

x∈H,

F(un,y) +_r_ny–un,un–xn ≥, ∀y∈H, xn+=PC[αnγf(xn) + (I–αnA)Wnun], ∀n∈N,

(.)

where {Tn}n∈N is a family of nonexpansive mappings from H into itself such that

n∈NF(Tn) is nonempty,F:C×C→Ris an equilibrium bifunction, Ais a strongly positive operator onH,f is a contraction ofH into itself withα∈(, ),{αn},{rn},{λn} suitable sequences inRand{Wn}is the sequence of aW-mapping generated by{Tn}n∈N and{λn}. LetUbe defined byUx=limn→∞Wnx=limn→∞Un,xfor everyx∈Cusing

a pointx∗∈∞_i=F(Ti)∩EP(F), which is the unique solution of the variational inequality

(A–γf)x∗,x∗–xˆ≥, ∀ˆx∈

∞

i=

F(Ti)∩EP(F), (.)

or, equivalently, the unique solution of the minimization problem

min

x∈∞i=F(Ti)∩EP(F)



Ax,ˆ xˆ–h(x),ˆ

wherehis a potential function forγf.

2 Preliminaries

LetHbe a real Hilbert space with the norm · and the inner product·,·, and letCbe a closed convex subset ofH. We callf:C→Hanα-contraction if there exists a constant α∈[, ) such that

f(x) –f(y)≤αx–y, ∀x,y∈C.

LetAbe a strongly positive linear bounded operator on a Hilbert spaceHwith a con-stantγ¯; that is, there existsγ¯>  such that

Ax,x ≥ ¯γx, ∀x∈H.

Next, we denote weak convergence and strong convergence by notationsand→,

re-spectively. A spaceXis said to satisfy Opial’s condition [] if for each sequence{xn}inX which converges weakly to a pointx∈X, we have

lim inf

n→∞ xn–x<lim infn→∞ xn–y, ∀y∈X,y=x.

For every pointx∈H, there exists a unique nearest point inC, denoted byPCx, such that

x–PCx ≤ x–y, ∀y∈C.

PCis called the (nearest point or metric) projection ofHontoC. In addition,PCxis char-acterized by the following properties:PCx∈Cand

x–PCx,y–PCx ≥, (.)

x–y≥ x–PCx+y–PCx, ∀x∈H,y∈C. (.)

Recall that a mappingT:H→His said to be firmly nonexpansive mapping if

Tx–Ty≤ Tx–Ty,x–y, ∀x,y∈H.

It is well known thatPCis a firmly nonexpansive mapping ofHontoCand satisfies

IfAis anα-inverse-strongly monotone mapping ofCintoH, then it is obvious thatAis



α-Lipschitzcontinuous. We also have that for allx,y∈Candλ> ,

(I–λA)x– (I–λA)y=x–y–λ(Ax–Ay)

=x–y– λAx–Ay,x–y+λAx–Ay

≤ x–y+λ(λ– α)Ax–Ay. (.)

So, ifλ≤α, thenI–λAis a nonexpansive mapping ofCintoH.

The following lemmas will be useful for proving the convergence result of this paper.

Lemma . Let H be a real Hilbert space.Then for all x,y∈H, () x+y≤ x+ y,x+y;

() x+y≥ x+ y,x.

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

{βn}be a sequence in[, ]with <lim infn→∞βn≤lim supn→∞βn< .Suppose that xn+=

( –βn)yn+βnxnfor all integers n≥andlim supn→∞(yn+–yn–xn+–xn)≤.Then

limn→∞yn–xn= .

Lemma .([]) Assume that F:C×C→R,let us assume that F satisfies 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 semicontinuous.

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

Tr(x) =

z∈C:F(z,y) +

ry–z,z–x ≥,∀y∈C

for all z∈H.Then the following hold: . Tris single-valued;

. Tris firmly nonexpansive,i.e.,for anyx,y∈H,

Trx–Try≤ Trx–Try,x–y;

. F(Tr) =EP(F);

. EP(F)is closed and convex.

Lemma .([]) Let H be a Hilbert space,C be a closed convex subset of H,and S:C→C be a nonexpansive mapping with F(S)=∅.If{xn}is a sequence in C weakly converging to x∈C and if{(I–S)xn}converges strongly to y,then(I–S)x=y.

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

where{γn}is a sequence in(, )and{δn}is a sequence inRsuch that () ∞_n=γn=∞;

() lim sup_n→∞δn

γn≤or ∞

n=|δn|<∞. Thenlimn→∞an= .

Lemma .([]) Let H be a Hilbert space,C be a nonempty closed convex subset of H, and f :H→H be a contraction with a coefficient <α< ,and let A be a strongly positive linear bounded operator with a coefficientγ¯> .Then,for <γ<γ_α¯,

x–y, (A–γf)x– (A–γf)y≥(γ¯–γ α)x–y, x,y∈H.

That is,A–γf is strongly monotone with a coefficientγ¯–γ α.

Lemma .([]) Assume A is a strongly positive linear bounded operator on a Hilbert space H with a coefficientγ¯> and <ρ≤ A–_{.ThenI–ρA ≤ –ργ¯.}

Lemma .([] and []) Let C be a nonempty closed convex subset of a Banach space E. Let{Ti}∞i=be a sequence of nonexpansive mappings of C into itself with

∞

i=F(Ti)=∅,and let{λi}∞i=be a real sequence such that <λi≤b< ,∀i≥.Then:

() Wnis nonexpansive andF(Wn) =

∞

i=F(Ti)for eachn≥;

() for eachx∈Cand for each positive integerk,thelimn→∞Un,kxexists; () the mappingU:C→Cdefined by

Ux= lim

n→∞Wnx=nlim→∞Un,x, x∈C

is a nonexpansive mapping satisfyingF(U) =∞_n=F(Ti)and it is called the W-mapping generated byT,T, . . .andλ,λ, . . .;

() limm,n→∞supx∈KWmx–Wnx= for any bounded subsetKofE.

3 Main results

In this section, we introduce our algorithm and prove its strong convergence.

Theorem . Let C be a closed convex subset of a real Hilbert space H.Let F be a bifunction from H×H intoRsatisfying(A)-(A).Let f be a contraction of H into itself withα∈ (, ),and let Tnbe a sequence of nonexpansive mappings of C into itself such that=

∞

n=F(Tn)∩EP(F)=∅.Let A:H→H be a strongly positive bounded linear operator with a coefficientγ¯ > with <γ < _αγ¯.Letλ,λ, . . .be a sequence of real numbers such that

 <λn≤b< for every n= , , . . . .Let Wnbe a W -mapping of C into itself generated by Tn,Tn–, . . . ,Tandλn,λn–, . . . ,λ.Let U be defined by Ux=limn→∞Wnx=limn→∞Un,x

for every x∈C.Let{xn}and{un}be sequences generated by x∈H and ⎧

⎨ ⎩

F(un,y) +_r_ny–un,un–xn ≥, ∀y∈H, xn+=PC[αnγf(xn) + (I–αnA)Wnun], ∀n∈N,

(.)

where{αn}is a sequence in(, )and{rn}is a sequence in[,∞).Suppose that{αn}and

(C) limn→∞αn= ; (C) ∞_n=αn=∞; (C) limn→∞rn=r> .

Then both{xn}and{un}converge strongly to x∗∈,which is the unique solution of the variational inequality

(A–γf)x∗,x∗–xˆ≥, ∀ˆx∈. (.)

Equivalently,one has x∗=P(I–A+γf)(x∗).

Proof We observe thatP(γf + (I–A)) is a contraction. Indeed, for allx,y∈H, we have

P

γf+ (I–A)(x) –P

γf + (I–A)(y)≤γf + (I–A)(x) –γf+ (I–A)(y)

≤γf(x) –f(y)+I–Ax–y

≤γ αx–y+ ( –γ)x–y

< – (γ –γ α)x–y.

Banach’s contraction mapping principle guarantees thatP(γf+ (I–A)) has a unique fixed point, sayx∗∈H. That is,x∗=P(γf+ (I–A))(x∗). Note that by Lemma ., we can write

xn+=PC

αnγf(xn) + (I–αnA)WnTrnxn

,

where

Trn(x) =

z∈H:F(z,y) +  rn

y–z,z–x ≥,∀y∈H

.

Moreover, sinceαn→ asn→ ∞by condition (C), we assume thatαn≤ A–for all n∈N. From Lemma ., we know that if  <ρ<A–_{, thenI–ρA ≤ –ργ. We divide}

the proof into seven steps as follows.

Step . Show that the sequences{xn}and{un}are bounded. Letxˆ∈. Thenxˆ∈EP(F). From Lemma ., we have

un–xˆ=Trnxn–Trnxˆ ≤ xn–xˆ.

Thus, we have

xn+–xˆ=PC

αnγf(xn) + (I–αnA)Wnun

–xˆ

≤αnγf(xn) + (I–αnA)Wnun–xˆ

≤αnγf(xn) –f(x)ˆ +I–αnAWnun–xˆ+αnγf(x) –ˆ Axˆ

≤αnγ αxn–xˆ+ ( –αnγ¯)un–xˆ+αnγf(x) –ˆ Axˆ

= –αn(γ¯–γ α)

xn–xˆ+αn(γ¯–γ α)

By induction, we have

xn–xˆ ≤max

x–xˆ,

γf(x) –ˆ Axˆ

¯

γ –γ α

, ∀n≥.

This shows that the sequence{xn}is bounded, so are{un},{f(xn)}and{Wnun}. Step . Show thatWn+un–Wnun → asn→ ∞.

Letxˆ∈. SinceTiandUn,iare nonexpansive andTixˆ=xˆ=Un,ixˆfor everyn∈Nand i≤n+ , it follows that

Wn+un–Wnun=λTUn+,un–λTUn,un

≤λUn+,un–Un,un

=λλTUn+,un–λTUn,un

≤λλUn+,un–Un,un .. . ≤ _n i= λi

Un+,n+un–xˆ+xˆ–Un,n+un

≤ _n i= λi

Un+,n+un–xˆ+xˆ–Un,n+un

≤ _n i= λi

un–xˆ.

Since{un}is bounded and  <λn≤b<  for anyn∈N, the following holds:

lim

n→∞Wn+un–Wnun= .

Step . Show thatxn+–xn → asn→ ∞.

SettingS= PC–I, we haveSis nonexpansive. Note thatWn= ( –λ)I+λTUn,. Then

we can write

xn+=

I+S 

αnγf(xn) + (I–αnA)Wnun

= –αn  Wnun+

αn 

γf(xn) –AWnun+Wnun

+ S

αnγf(xn) + (I–αnA)Wnun

= –αn 

( –λ)I+λTUn,

un+ αn



γf(xn) –AWnun+Wnun

+ S

αnγf(xn) + (I–αnA)Wnun

=( –λ)( –αn)

 un+

λ( –αn)

 TUn,un+ αn



γf(xn) –AWnun+Wnun

+ S

αnγf(xn) + (I–αnA)Wnun

Note that

 < lim

n→∞

( –λ)( –αn)

 =

 –λ

 < ,

and

λ( –αn)

 +

 =

 +λ

 –

λ

αn.

From (.), we have

xn+=

 –

 +λ

 +

 –λ

 αn

un+

 +λ

 +

 –λ

 αn

×

λ( –αn)

 TUn,un+ αn



γf(xn) –AWnun+Wnun

+ S

αnγf(xn) + (I–αnA)Wnun

 +λ

 +

 –λ

 αn

=

 –

 +λ

 +

 –λ

 αn

un+

 +λ

 +

 –λ

 αn

yn, (.)

where

yn=

λ( –αn)

 TUn,un+ αn



γf(xn) –AWnun+Wnun

+ S

αnγf(xn) + (I–αnA)Wnun

 +λ

 +

 –λ

 αn

=λ( –αn)TUn,un+αn

γf(xn) –AWnun+Wnun

+Sαnγf(xn) + (I–αnA)Wnun

/ +λ+ ( –λ)αn

.

Seten=γf(xn) –AWnun+Wnunanddn=αnγf(xn) + (I–αnA)Wnunfor alln. Then

yn=

λ( –αn)TUn,un+αnen+Sdn  +λ+ ( –λ)αn

, ∀n≥.

It follows that

yn+–yn=

λ( –αn+)TUn+,un++αn+en++Sdn+

 +λ+ ( –λ)αn+

–λ( –αn)TUn,un+αnen+Sdn  +λ+ ( –λ)αn

= λ( –αn+)  +λ+ ( –λ)αn+

(TUn+,un+–TUn,un)

+

λ( –αn+)

 +λ+ ( –λ)αn+

– λ( –αn)  +λ+ ( –λ)αn

TUn,un

+ αn+en+  +λ+ ( –λ)αn+

– αnen

 +λ+ ( –λ)αn

+ Sdn+–Sdn  +λ+ ( –λ)αn+

+



 +λ+ ( –λ)αn+

– 

 +λ+ ( –λ)αn

Thus,

yn+–yn ≤

λ( –αn+)

 +λ+ ( –λ)αn+

TUn+,un+–TUn,un

+ λ( –αn+)  +λ+ ( –λ)αn+

– λ( –αn)  +λ+ ( –λ)αn

TUn,un

+ αn+

 +λ+ ( –λ)αn+

en++

αn  +λ+ ( –λ)αn

en

+ 

 +λ+ ( –λ)αn+

Sdn+–Sdn

+ 

 +λ+ ( –λ)αn+

– 

 +λ+ ( –λ)αn

Sdn.

SinceSis nonexpansive, we obtain that

Sdn+–Sdn ≤ dn+–dn

=αn+γf(xn+) + (I–αn+A)Wn+un+–

αnγf(xn) + (I–αnA)Wnun

≤αn+γf(xn+) –AWn+un++αnγf(xn) –AWnun)

+Wn+un+–Wnun

≤αn+γf(xn+) –AWn+un++αnγf(xn) –AWnun)

+Wn+un+–Wn+un+Wn+un–Wnun

≤αn+γf(xn+) –AWn+un++αnγf(xn) –AWnun)+un+–un.

+Wn+un–Wnun.

SinceTiandUn,iare nonexpansive, we have

TUn+,un–TUn,un ≤ Un+,un–Un,un

=λTUn+,un–λTUn,un

≤λUn+,un–Un,un

≤ · · ·

≤λ· · ·λnUn+,n+un–Un,n+un

≤M

n

i=

λi,

whereM>  is a constant such thatUn+,n+un–Un,n+un ≤Mfor alln≥. So,

TUn+,un+–TUn,un ≤ TUn+,un+–TUn+,un+TUn+,un–TUn,un

≤ un+–un+M n

i=

Hence,

yn+–yn ≤

λ( –αn+)

 +λ+ ( –λ)αn+

un+–un+M n

i=

λi

+ λ( –αn+)  +λ+ ( –λ)αn+

– λ( –αn)  +λ+ ( –λ)αn

TUn,un

+ αn+

 +λ+ ( –λ)αn+

en++

αn  +λ+ ( –λ)αn

en

+ 

 +λ+ ( –λ)αn+

αn+γf(xn+) –AWn+un+

+αnγf(xn) –AWnun+un+–un

+Wn+un–Wnun

+ 

 +λ+ ( –λ)αn+

– 

 +λ+ ( –λ)αn

Sdn

= λ( –αn+)  +λ+ ( –λ)αn+

un+–un

+ λ( –αn+)  +λ+ ( –λ)αn+

– λ( –αn)  +λ+ ( –λ)αn

TUn,un

+ αn+

 +λ+ ( –λ)αn+

en++

αn  +λ+ ( –λ)αn

en

+ 

 +λ+ ( –λ)αn+

αn+γf(xn+) –AWn+un+

+αnγf(xn) –AWnun+un+–un

+Wn+un–Wnun

+ 

 +λ+ ( –λ)αn+

– 

 +λ+ ( –λ)αn

Sdn.

Note that:

() By condition (C), we have

λ( –αn+)

 +λ+ ( –λ)αn+

– λ( –αn)  +λ+ ( –λ)αn→



and



 +λ+ ( –λ)αn+

– 

 +λ+ ( –λ)αn

→.

() Wn+un–Wnun →asn→ ∞because of Step . Therefore,

lim sup

n→∞

yn+–yn–un+–un

≤.

By Lemma ., we get

lim

Hence, from (.), we deduce

lim

n→∞xn+–xn=nlim→∞

 +λ

 +

 –λ

 αn

yn–un= . (.)

Step . Show thatxn–Wnun → asn→ ∞. Indeed, we have

xn–Wnun ≤ xn–xn++xn+–Wnun

=xn–xn++PC

αnγf(xn) + (I–αnA)Wnun

–Wnun

≤ xn–xn++αnγf(xn) + (I–αnA)Wnun–Wnun

=xn–xn++–αnAWnun+αnγf(xn)

≤ xn–xn++αn

AWnun+γf(xn).

Then

lim

n→∞xn–Wnun ≤nlim→∞xn–xn++αn

AWnun+γf(xn)= . (.)

Thus, from (.), we obtain

lim

n→∞xn–Wnun= .

Step . Show thatxn–un → asn→ ∞.

Letxˆ∈. SinceTrnis firmly nonexpansive, it follows

ˆx–un =Trnxˆ–Trnxn 

≤ Trnxn–Trnx,ˆ xn–xˆ

≤ Trnxn–x,ˆ xn–xˆ

=un–x,ˆ xn–xˆ

= 



un–xˆ+xn–xˆ–xn–un

.

Then

un–xˆ≤ xn–xˆ–xn–un.

Since we have

xn+–xˆ =PC

αnγf(xn) + (I–αnA)Wnun

–PCxˆ



=αnγf(xn) + (I–αnA)Wnun–xˆ

=(I–αnA)(Wnun–x) +ˆ αn

γf(xn) –Axˆ



≤( –αnγ¯)Wnun–xˆ+ αn

γf(xn) –Ax,ˆ xn+–xˆ

≤( –αnγ¯)un–xˆ+ αnγ

+ αn

γf(x) –ˆ Ax,ˆ xn+–xˆ

≤( –αnγ¯)

xn–xˆ–xn–un

+ αnγ αxn–xˆxn+–xˆ

+ αnγf(x) –ˆ Axˆxn+–xˆ

= – αnγ¯+ (αnγ¯)

xn–xˆ– ( –αnγ¯)xn–un

+ αnγ αxn–xˆxn+–xˆ+ αnγf(x) –ˆ Axˆxn+–xˆ ≤ xn–xˆ+αnγ¯xn–xˆ– ( –αnγ¯)xn–un

+ αnγ αxn–xˆxn+–xˆ+ αnγf(x) –ˆ Axˆxn+–xˆ,

and hence

( –αnγ¯)xn–un≤ xn–xˆ–xn+–xˆ+αnγ¯xn–xˆ

+ αnγ αxn–xˆxn+–xˆ+ αnγf(x) –ˆ Axˆxn+–xˆ ≤ xn–xn+

xn–xˆxn+–xˆ

+αnγ¯xn–xˆ

+ αnγ αxn–xˆxn+–xˆ+ αnγf(x) –ˆ Axˆxn+–xˆ.

Therefore, we havexn–un → asn→ ∞.

Step . Show thatlim sup_n→∞γf(x∗) –Ax∗,xn–x∗ ≤. We can choose a subsequence{xni}of{xn}such that

lim

i→∞

γfx∗–Ax∗,xni–x

∗_{=lim sup}

n→∞

γfx∗–Ax∗,xn–x∗

.

Let

A(xni) =

x∈H:lim sup

i→∞

xni–x=inf

y∈Hlim sup_i→∞ xni–y

be the asymptotic center of{xni}. Since{xni}is bounded andHis a Hilbert space, it is well

known thatA(xni) is a singleton; sayA(xni) ={˜x}. Set

L=sup

i∈N

γf(xni) –AWniuni

and for everyx∈Hdefine

Wx=lim

i→∞Wnix (.)

and

Tr(x) =

z∈H:F(z,y) +

ry–z,z–x ≥,∀y∈H

.

Note that

xni–Wx˜ ≤ xni+–xni+xni+–Wx˜

=xni+–xni+PC

αniγf(xni) + (I–αniA)Wniuni

≤ xni+–xni+αniγf(xni) + (I–αniA)Wniuni–Wx˜

=xni+–xni+Wniuni–Wx˜+αni

γf(xni) –AWniuni

≤ xni+–xni+Wniuni–Wnixni+Wnixni–Wnix˜

+Wnix˜–Wx˜+αniL

≤ xni+–xni+uni–xni+xni–x˜+Wnix˜–Wx˜+αniL.

By Steps -, condition (C) and (.), we derive

lim sup

i→∞ xni–Wx˜ ≤lim supi→∞ uni–xni+xni–x˜+Wnix˜–Wx˜ ≤lim sup

i→∞

xni–x˜.

That is,Wx˜∈A(xni). ThereforeWx˜=x. Next, we show that˜ x˜=Trx.˜

Note that for anyx∈Handa,b> , we have

F(Tax,y) + 

ay–Tax,Tax–x ≥, ∀y∈H

and

F(Tbx,y) + 

by–Tbx,Tbx–x ≥, ∀y∈H,

then

F(Tax,Tbx) + 

aTbx–Tax,Tax–x ≥

and

F(Tbx,Tax) + 

bTax–Tbx,Tax–x ≥.

Summing up the last inequalities and using (A), we obtain

Tax–Tbx, Tbx–x

b – Tax–x

a

≥.

Hence we have

≤

Tax–Tbx,Tbx–x– b

a(Tax–x)

=

Tax–Tbx,Tbx–Tax+Tax–x– b

a(Tax–x)

=

Tax–Tbx, (Tbx–Tax) +

 –b a

(Tax–x)

≤–Tax–Tbx+

 –b

a

Tax–Tbx

Tax+x

We derive then

Tax–Tbx ≤| b–a|

a

Tax+x

.

It follows that

xni–Trx˜ ≤ xni+–xni+xni+–Trx˜

=xni+–xni+PC

αniγf(xni) + (I–αniA)Wniuni

–Trx˜

≤ xni+–xni+αniγf(xni) + (I–αniA)Wniuni–Trx˜

=xni+–xni+Wniuni–Trx˜+αni

γf(xni) –AWniuni

≤ xni+–xni+Wniuni–xni+Tr_nixni–Tr_nix˜+xni–Tr_nixni

+Tr_nix˜–Trx˜+αniL

≤ xni+–xni+uni–xni+xni–x˜+xni–uni

+Tr_nix˜–Trx˜+αniL

≤ xni+–xni+uni–xni+xni–x˜+xni–uni

+|rni–r|

r

Trx˜+˜x

+αniL.

By Steps -, conditions (C) and (C), we obtain

lim sup

i→∞

xni–Trx˜ ≤lim sup

i→∞

xni–x˜

andx˜=Trx. Thus˜ x˜∈F(W)∩F(Tr) =by Lemma . and .. Fixt∈(, ),x∈Hand sety=x˜+tx. Then

xni–x˜–tx _{≤ x}

ni–x˜

_{+ tx,x˜+tx–x}

ni.

By the minimizing property ofx˜and since · _{is continuous and increasing in [,∞), we}

have

lim sup

i→∞

xni–x˜

_{≤lim sup}

i→∞

xni–x˜–tx 

≤lim sup

i→∞ xni–x˜

_{+ tlim sup}

i→∞ x,x˜+tx–xni.

Thus,

lim sup

i→∞ x,x˜+tx–xni ≥.

On the other hand,

Hence we obtain

lim sup

i→∞

x,x˜–xni=lim

t→

lim sup

i→∞

x,x˜+tx–xni–tx  _≥.

Setx=γf(x∗) –Ax∗. Sincex˜∈, we obtain

≤lim sup

i→∞

γfx∗–Ax∗,x˜–xni

≤γfx∗–Ax∗,x˜–x∗+lim

i→∞

γfx∗–Ax∗,x∗–xni

≤ lim

i→∞

γfx∗–Ax∗,x∗–xni . So that lim sup n→∞

γfx∗–Ax∗,xn–x∗

= –lim

i→∞

γfx∗–Ax∗,xni–x ∗_≤.

Step . Show that both{xn}and{un}strongly converge tox∗∈, which is the unique solution of the variational inequality (.). Indeed, we note that

xn+–x∗ 

=xn+–dn,xn+–x∗

+dn–x∗,xn+–x∗

.

Sincexn+–dn,xn+–x∗ ≤, we get

xn+–x∗ 

≤dn–x∗,xn+–x∗

=αnγf(xn) + (I–αnA)Wnun–x∗,xn+–x∗

=αnγf(xn) –αnγf

x∗+Wnun–αnAWnun–x∗

+αnAx∗+αnγf

x∗–αnAx∗,xn+–x∗

=αnγ

f(xn) –f

x∗+ (I–αnA)

Wnun–x∗

,xn+–x∗

+αn

γfx∗–Ax∗,xn+–x∗

≤αnγf(xn) –f

x∗+I–αnAWnun–x∗xn+–x∗

+αn

γfx∗–Ax∗,xn+–x∗

≤ –αn(γ¯–γ α)xn–x∗xn+–x∗+αn

γfx∗–Ax∗,xn+–x∗

≤

[ –αn(γ¯–γ α)] 

xn–x∗



+

xn+–x

∗

+αn

γfx∗–Ax∗,xn+–x∗

.

It then follows that

xn+–x∗ 

≤ –αn(γ¯–γ α)xn–x∗



+ αn

γfx∗–Ax∗,xn+–x∗

. (.)

Then, we can write the last inequality as

an+≤( –γn)an+δn.

Note that in virtue of condition (C), ∞_n=γn=∞. Moreover,

lim sup

n→∞

δn γn

= 

¯

γ –γ αlim supn→∞ 

γfx∗–Ax∗,xn+–x∗

.

By Step , we obtain

lim sup

n→∞

δn γn ≤

. (.)

Now, applying Lemma . to (.), we conclude thatxn→x∗ asn→ ∞. Furthermore, sinceun–x∗=Trnxn–Trnx∗ ≤ xn–x∗, we then have thatun→x∗asn→ ∞. The

proof is now complete.

SettingA≡Iandγ =  in Theorem ., we have the following result.

Corollary . Let C be a nonempty closed convex subset of a real Hilbert space H.Let F:C×C→Rbe an equilibrium bifunction satisfying the conditions:

() Fis monotone,that is,F(x,y) +F(y,x)≤for allx,y∈C; () for eachx,y,z∈C,limt→F(tz+ ( –t)x,y)≤F(x,y);

() for eachx∈C,y→F(x,y)is convex and lower semicontinuous.

Let {Ti}∞i= be an infinite family of nonexpansive mappings of C into C such that ∞

i=F(Ti)∩EP(F)=∅.Suppose{αn} ⊂(, )and{rn} ⊂(,∞)satisfy the following condi-tions:

() limn→∞αn= and ∞n=αn=∞;

() lim inf_n→∞rn> andlimn→∞(rn+–rn) = .

Let f be a contraction of C into itself,and let x∈H be given arbitrarily.Then the sequences {xn}and{yn}generated iteratively by

⎧ ⎨ ⎩

F(yn,x) +_r_nx–yn,yn–xn ≥, ∀x∈C, xn+=αnf(xn) + ( –αn)Wnyn,

converge strongly to x∗∈∞_i=F(Ti)∩EP(F),the unique solution of the minimization prob-lem

min

x∈∞i=F(Ti)∩EP(F)

 x

_–h(x),

where h is a potential function for f.

SettingF=  in Theorem ., we have the following result.

fixed point set=∞_n=F(Tn)=∅.Let f :C→H be anα-contraction and A:H→H be a strongly positive bounded linear operator with a coefficientγ¯> .Letγ be a constant such that <γ α<γ¯.For an arbitrary initial point xbelonging to C,one defines a sequence {xn}n≥iteratively

xn+=PC

αnγf(xn) + (I–αnA)Wnxn

, ∀n≥, (.)

where{αn}is a real sequence in[, ].Assume that the sequence{αn}satisfies the following conditions:

(C) limn→∞αn= ; (C) ∞_n=αn=∞.

Then the sequence{xn}generated by(.)converges in norm to the unique solution x∗, which solves the following variational inequality:

x∗∈ such that(A–γf)x∗,x∗–xˆ≤,∀ˆx∈. (.)

Competing interests

The author declares that they have no competing interests.

Acknowledgements

The author would like to thank Asst. Prof. Dr. Rabian Wangkeeree for his useful suggestions and the referees for their valuable comments and suggestions.

Received: 2 October 2012 Accepted: 7 March 2013 Published: 3 April 2013

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Cite this article as:Rattanaseeha:The general iterative methods for equilibrium problems and fixed point problems of a countable family of nonexpansive mappings in Hilbert spaces.Journal of Inequalities and Applications2013