On strong convergence of an iterative algorithm for common fixed point and generalized equilibrium problems

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R E S E A R C H

Open Access

On strong convergence of an iterative

algorithm for common ﬁxed point and

generalized equilibrium problems

Jian-Min Song

*

*_{Correspondence:}

[email protected]

Department of Mathematics and Sciences, Shijiazhuang University of Economics, Shijiazhuang, Hebei 050031, China

Abstract

In this article, an iterative algorithm for ﬁnding a common element in the solution set of generalized equilibrium problems and in the common ﬁxed point set of a family of nonexpansive mappings. Strong convergence of the algorithm is established in the framework of Hilbert spaces.

MSC: 47H05; 47H09; 47J25

Keywords: equilibrium problem; ﬁxed point; nonexpansive mapping; variational inequality

1 Introduction and preliminaries

Equilibrium problems which were introduced by Ky Fan [] and further studied by Blum and Oettli [] have intensively been investigated based on iterative methods. The equilib-rium problems have emerged as an eﬀective and powerful tool for studying a wide class of problems which arise in economics, ecology, transportation, network, elasticity, and optimization; see [–] and the references therein. It is well known that the equilibrium problems cover ﬁxed point problems, variational inequality problems, saddle problems, inclusion problems, complementarity problems, and minimization problems; see [–] and the references therein.

In this paper, an iterative algorithm is proposed for treating common ﬁxed point and generalized equilibrium problems. It is proved that the sequence generated in the algo-rithm converges strongly to a common element in the solution set of generalized equilib-rium problems and in the common ﬁxed point set of a family of nonexpansive mappings. From now on, we always assume thatHis a real Hilbert space, whose inner product and norm are denoted by·,·and·, respectively. LetCbe a nonempty closed convex subset ofHand letPCbe the projection ofHontoC.

LetS:C→Cbe a mapping. Throughout this paper, we useF(S) to denote the ﬁxed point set of the mappingS. Recall thatS:C→Cis said to be nonexpansive iﬀ

Sx–Sy ≤ x–y, ∀x,y∈C.

S:C→Cis said to be ﬁrmly nonexpansive iﬀ

Sx–Sy≤ Sx–Sy,x–y, ∀x,y∈C.

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It is easy to see that every ﬁrmly nonexpansive mapping is nonexpansive. LetA:C→Hbe a mapping. Recall thatAis said to be monotone iﬀ

Ax–Ay,x–y ≥, ∀x,y∈C.

Ais said to be inverse-strongly monotone iﬀ there exists a constantα>  such that

Ax–Ay,x–y ≥δAx–Ay, ∀x,y∈C.

For such a case,Ais also said to beα-inverse-strongly monotone. It is known that ifS: C→Cis nonexpansive, thenA=I–Sis_-inverse-strongly monotone. Recall that a set-valued mappingT:H→His called monotone if, for allx,y∈H,f ∈Txandg∈Tyimply x–y,f–g ≥. A monotone mappingT:H→H_{is maximal if the graph of}_G(T_{) of}_T

is not properly contained in the graph of any other monotone mapping. It is well known that a monotone mappingTis maximal if and only if for (x,f)∈H×H,x–y,f–g ≥ for every (y,g)∈G(T) impliesf ∈Tx. LetBbe a monotone map ofCintoHand letNCv

be the normal cone toCatv∈C,i.e.,NCv={w∈H:v–u,w ≥,∀u∈C}and deﬁne

Tv=

⎧ ⎨ ⎩

Bv+NCv, v∈C,

∅, v∈/C.

ThenTis maximal monotone and ∈Tvif and only ifAv,u–v ≥, for∀u∈C; see [] and the references therein

Recall that the classical variational inequality is to ﬁndu∈Csuch that

Au,v–u ≥, ∀v∈C. (.)

In this paper, we useVI(C,A) to denote the solution set of the variational inequality (.). One can see that the variational inequality (.) is equivalent to a ﬁxed point problem. The elementu∈Cis a solution of the variational inequality (.) if and only ifu∈Cis a ﬁxed point of the mappingPC(I–λA), whereλ>  is a constant andIdenotes the identity

mapping. IfAis anα-inverse strongly monotone, we remark here that the mappingPC(I–

λA) is nonexpansive iﬀ  <λ< α. Indeed,

PC(I–λA)x–PC(I–λA)y≤(I–λA)x– (I–λA)y

=x–y– λx–y,Ax–Ay+λAx–Ay ≤ x–y–λ(α–λ)Ax–Ay.

This alternative equivalent formulation has played a signiﬁcant role in the studies of the variational inequalities and related optimization problems.

LetAbe an inverse-strongly monotone mapping,Fa bifunction ofC×CintoR, where Ris the set of real numbers. We consider the following equilibrium problem:

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In this paper, the set of suchz∈Cis denoted byEP(F,A),i.e.,

EP(F,A) =z∈C:F(z,y) +Az,y–z ≥,∀y∈C.

If the case ofA≡, the zero mapping, the problem (.) is reduced to

Findz∈Csuch thatF(z,y)≥, ∀y∈C. (.)

In this paper, we useEP(F) to denote the solution set of the problem (.). The problem of (.) and (.) have been considered by many authors; see, for example, [–] and the references therein. In the case ofF≡, the problem (.) is reduced to the classical variational inequality (.).

To study the equilibrium problems, we assume that the bifunctionF:C×C→R satis-ﬁes 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,

lim sup t↓

Ftz+ ( –t)x,y ≤F(x,y);

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

The well-known convex feasibility problem which captures applications in various dis-ciplines such as image restoration, and radiation therapy treatment planning is to find a point in the intersection of common fixed point sets of a family of nonlinear mappings. In this paper, we propose an iterative algorithm for finding a common element in the solution set of the generalized equilibrium problem (.) and in the common fixed point set of a family of nonexpansive mappings. Strong convergence of the algorithm is established in the framework of Hilbert spaces.

In order to prove our main results, we need the following deﬁnitions and lemmas. A spaceXis said to satisfy Opial’s condition [] if for each sequence{xn}∞n=inXwhich converges weakly to pointx∈X, we have

lim inf

n→∞ xn–x<lim infn→∞ xn–y, ∀y∈X,y=x.

It is well known that the above inequality is equivalent to

lim sup

n→∞ xn–x<lim supn→∞ xn–y, ∀y∈X,y=x.

The following lemma can be found in [].

Lemma . Let C be a nonempty closed convex subset of H ad let F:C×C→Rbe a

bifunction satisfying(A)-(A).Then,for any r> and x∈H,there exists z∈C such that

F(z,y) +

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Further,if Trx={z∈C:F(z,y) +_ry–z,z–x ≥,∀y∈C},then the following hold:

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 C,H,F and Trbe as in Lemma..Then the following holds:

Tsx–Ttx≤

s–t

s Tsx–Ttx,Tsx–x

for all s,t> and x∈H.

Lemma .[] Assume that{αn}is a sequence of nonnegative real numbers such that

αn+≤( –γn)αn+δn,

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

(b) lim sup_n_→∞δn/γn≤or

∞

n=|δn|<∞.

Thenlimn→∞αn= .

Deﬁnition .[] Let{Si:C→C}be a family of inﬁnitely nonexpansive mappings and

{γi}be a nonnegative real sequence with ≤γi< ,∀i≥. Forn≥ deﬁne a mapping

Wn:C→Cas follows:

Un,n+=I,

Un,n=γnSnUn,n++ ( –γn)I,

Un,n–=γn–Sn–Un,n+ ( –γn–)I,

.. .

Un,k=γkSkUn,k++ ( –γk)I, (.)

un,k–=γk–Sk–Un,k+ ( –γk–)I,

.. .

Un,=γSUn,+ ( –γ)I,

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Such a mappingWnis nonexpansive fromCtoCand it is called aW-mapping generated

bySn,Sn–, . . . ,Sandγn,γn–, . . . ,γ.

Lemma .[] Let C be a nonempty closed convex subset of a Hilbert space H,{Si:C→

C}be a family of inﬁnitely nonexpansive mappings with∞_i_=F(Si)=∅,{γi}be a real

se-quence such that <γi≤l< ,∀i≥.Then () Wnis nonexpansive andF(Wn) =

_∞

i=F(Si),for eachn≥;

() for eachx∈Cand for each positive integerk,the limitlimn→∞Un,kexists; () the mappingW:C→Cdeﬁned by

Wx:= lim

n→∞Wnx=nlim→∞Un,x, x∈C, (.)

is a nonexpansive mapping satisfyingF(W) =∞_i_=F(Si)and it is called the

W-mapping generated byS,S, . . .andγ,γ, . . ..

Lemma .[] Let C be a nonempty closed convex subset of a Hilbert space H,{Si:C→

C}be a family of inﬁnitely nonexpansive mappings with∞_i_=F(Si)=∅,{γi}be a real

se-quence such that <γi≤l< ,∀i≥.If K is any bounded subset of C,then

lim

n→∞sup_x_∈_KWx–Wnx= .

Throughout this paper, we always assume that  <γi≤l< ,∀i≥.

Lemma .[] Let{xn}and{yn}be bounded sequences in a Hilbert space H and let{βn}

be a sequence in [, ]with <lim infn→∞βn≤lim supn→∞βn< .Suppose that xn+= ( –βn)yn+βnxnfor all n≥and

lim sup n→∞

yn+–yn–xn+–xn ≤.

Thenlim_n_→∞yn–xn= .

2 Main results

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

a bifunction from C×C toRwhich satisﬁes(A)-(A).Let A:C→H be anα -inverse-strongly monotone mapping and let {Si:C→C}be a family of inﬁnitely nonexpansive

mappings.Assume that:=∞_i_=F(Si)∩EP(F,A)=∅.Let f :C→C be aκ-contractive

mapping.Let{xn} be a sequence generated in the following process:let it be a sequence

generated in

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

x∈C, chosen arbitrarily,

F(yn,y) +Axn,y–yn+_r_ny–yn,yn–xn ≥, ∀y∈C,

xn+=βnxn+ ( –βn)Wn(αnf(Wnxn) + ( –αn)yn), ∀n≥,

where{Wn}is the mapping sequence deﬁned by(.),{αn},and{βn}are sequences in[, ]

and{rn}is a positive number sequence.Assume that the above control sequences satisfy the

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(a)  <a≤βn≤b< , <c≤rn≤d< α; (b) limn→∞αn= ,and∞n=αn=∞;

(c) limn→∞(rn–rn+) = .

Then{xn}converges strongly to a point x∈,where x=Pf(x).

Proof First, we show that the sequence{xn}and{yn}are bounded. Fixingx∗∈, we ﬁnd

that

yn–x∗



=Trn(xn–rnAxn) –Trn

x∗–rnAx∗



≤(xn–rnAxn) –

x∗–rnAx∗



=xn–x∗ –rn

Axn–Ax∗ 

=xn–x∗

 – rn

xn–x∗,Axn–Ax∗

+r_nAxn–Ax∗



≤xn–x∗



– rnαAxn–Ax∗



+r_nAxn–Ax∗



=xn–x∗+rn(rn– α)Axn–Ax∗. (.)

Using the restriction (a), we ﬁnd that

yn–x∗≤xn–x∗. (.)

From the above, we also ﬁnd that the mappings I–rnAis nonexpansive. Puttingzn=

αnf(Wnxn) + ( –αn)yn, we ﬁnd from (.) that zn–x∗=αnf(Wnxn) + ( –αn)yn–x∗

≤αnf(Wnxn) –x∗+ ( –αn)yn–x∗

≤αnκxn–x∗+αnf

x∗ –x∗+ ( –αn)yn–x∗

≤ –αn( –κ) xn–x∗+αnf

x∗ –x∗. (.)

It follows from (.) that

xn+–x∗≤βnxn–x∗+ ( –βn)Wnzn–x∗

≤βnxn–x∗+ ( –βn)zn–x∗

≤ –αn( –βn)( –κ) xn–x∗+αn( –βn)f

x∗ –x∗

≤ · · ·

≤maxx–x∗,f(x

∗_{) –}_x∗

 –κ

.

This shows that the sequence{xn}is bounded, and so are{yn}and{zn}. Without loss of

generality, we can assume that there exists a bounded setK⊂Csuch thatxn,yn,zn∈K;

yn+–yn=Trn+(xn+–rn+Axn+) –Trn+(xn–rnAxn)

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≤(xn+–rn+Axn+) – (xn–rnAxn)

+Trn+(xn–rnAxn) –Trn(xn–rnAxn)

≤ xn+–xn+|rn+–rn|Axn

+Trn+(xn–rnAxn) –Trn(xn–rnAxn). (.)

It follows that

zn+–zn

≤αn+f(Wn+xn+) –f(Wnxn)+|αn+–αn|f(Wn+xn+)+yn

+ ( –αn+)yn+–yn

≤αn+κWn+xn+–Wnxn+|αn+–αn|f(Wn+xn+)+yn

+xn+–xn+|rn+–rn|Axn+Trn+(xn–rnAxn) –Trn(xn–rnAxn). (.)

Note that

Wn+zn+–Wnzn

=Wn+zn+–Wzn++Wzn+–Wzn+Wzn–Wnzn

≤ Wn+zn+–Wzn++Wzn+–Wzn+Wzn–Wnzn

≤sup x∈K

Wn+x–Wx+Wx–Wnx

+zn+–zn. (.)

Combing (.) with (.) yields

Wn+zn+–Wnzn–xn+–xn

≤sup x∈K

Wn+x–Wx+Wx–Wnx

+αn+κWn+xn+–Wnxn

+|αn+–αn|f(Wn+xn+)+yn

+|rn+–rn|Axn+Trn+(xn–rnAxn) –Trn(xn–rnAxn).

From the restrictions (a), (b), and (c), we ﬁnd from Lemma . that

lim sup n→∞

Wn+zn+–Wnzn–xn+–xn

≤.

Using Lemma ., we obtain

lim

n→∞Wnzn–xn= . (.)

It follows that

lim

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Using (.), we ﬁnd that

xn+–x∗ =βnxn+ ( –βn)Wnzn–x∗

≤βnxn–x∗



+ ( –βn)zn–x∗



≤βnxn–x∗



+ ( –βn)

αnf(Wnxn) –x∗



+ ( –αn)yn–x∗



≤xn–x∗+αnf(Wnxn) –x∗

+rn(rn– α)( –αn)( –βn)Axn–Ax∗

 ,

which in turn yields

rn(α–rn)( –αn)( –βn)Axn–Ax∗

≤xn–x∗



–xn+–x∗+αnf(Wnxn) –x∗



≤xn–x∗+xn+–x∗ xn–xn++αnf(Wnxn) –x∗

 .

Using (.), we ﬁnd from the restrictions (a), (b), and (c) that

lim

n→∞Axn–Ax

∗_{= .} _(.)

On the other hand, we see that

yn–x∗



=Trn(I–rnA)xn–Trn(I–rnA)x

∗

≤(I–rnA)xn– (I–rnA)x∗,yn–x∗

≤ 

xn–x

∗

+yn–x∗



–(xn–yn) –rn

Axn–Ax∗



= 

xn–x

∗

+yn–x∗



–xn–yn

+ rn

xn–yn,Axn–Ax∗

–r_nAxn–Ax∗

 .

Hence, we have

yn–x∗



≤xn–x∗



–xn–yn+ rnxn–ynAxn–Ax∗.

It follows that

xn+–x∗ 

≤xn–x∗



+αnf(Wnxn) –x∗



– ( –αn)( –βn)xn–yn

+ rn( –αn)( –βn)xn–ynAxn–Ax∗

≤xn–x∗+αnf(Wnxn) –x∗– ( –αn)( –βn)xn–yn

+ rnxn–ynAxn–Ax∗.

This implies that

( –αn)( –βn)xn–yn≤xn–x∗+xn+–x∗ xn–xn+

+αnf(Wnxn) –x∗



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Using (.) and (.), we ﬁnd from the restrictions (a), (b), and (c) that

lim

n→∞xn–yn= . (.)

Sincezn=αnf(Wnxn) + ( –αn)yn, we ﬁnd that

lim

n→∞zn–yn= . (.)

Notice that

xn+–xn= ( –βn)Wnzn–xn.

This implies from (.)

lim

n→∞Wnzn–xn= . (.)

Note that

Wnzn–zn ≤ zn–yn+yn–xn+xn–Wnzn.

From (.), (.), and (.), we obtain

lim

n→∞Wnzn–zn= . (.)

Since the mappingPf is contractive, we denote the unique ﬁxed point byx. Next, we

prove thatlim sup_n_→∞f(x) –x,zn–x ≤. To see this, we choose a subsequence{zni}of

{zn}such that

lim sup n→∞

f(x) –x,zn–x

= lim i→∞

f(x) –x,zni–x

.

Since{zni}is bounded, there exists a subsequence{zn_ij}of{zni}which converges weakly

toz. Without loss of generality, we may assume thatzni z. Indeed, we also haveyni f. First, we show thatz∈∞_i_=F(Si). Suppose the contrary,Wz=z. Note that

zn–Wzn ≤ Wzn–Wnzn+Wnzn–zn

≤sup x∈K

Wx–Wnx

+Wnzn–zn.

Using Lemma ., we obtain from (.) thatlimn→∞zn–Wzn= . By Opial’s condition,

we see that

lim inf

i→∞ zni–z <lim inf

i→∞ zni–Wz ≤lim inf

i→∞

zni–Wzni+Wzni–Wz

≤lim inf i→∞

zni–Wzni+zni–z

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This implies thatlim infi→∞zni–z<lim infi→∞zni–z, which leads to a contradiction. Thus, we havez∈∞_i_=F(Si).

Next, we show thatf∈EP(F,A). Note thatyn z. Sinceyn=Trn(xn–rnAxn), we have

F(yn,y) +Axn,y–yn+

 rn

y–yn,yn–xn ≥, ∀y∈C.

From the condition (A), we see that

Axn,y–yn+

 rny

–yn,yn–xn ≥F(y,yn), ∀y∈C.

Replacingnbyni, we arrive at

Axni,y–yni+

y–yni,

yni–xni rni

≥F(y,yni), ∀y∈C. (.)

Fortwith  <t≤ andρ∈C, letρt=tρ+ ( –t)z. Sinceρ∈Candz∈C, we haveρt∈C.

It follows from (.) that

ρt–yni,Aρt ≥ ρt–yni,Aρt–Axni,ρt–yni–

ρt–yni,

yni–xni rni

+F(ρt,yni)

=ρt–yni,Aρt–Ayn,i+ρt–yni,Ayn,i–Axni

–

ρt–yni,

yni–xni rni

+F(ρt,yni). (.)

Using (.), we have Ayn,i–Axni→ asi→ ∞. On the other hand, we get from the monotonicity ofAthatρt–yni,Aρt–Ayn,i ≥. It follows from (A) and (.) that

ρt–z,Aρt ≥F(ρt,z). (.)

From (A) and (A), we see from (.) that

 = F(ρt,ρt)≤tF(ρt,ρ) + ( –t)F(ρt,z)

≤tF(ρt,ρ) + ( –t)ρt–z,Aρt

=tF(ρt,ρ) + ( –t)tρ–z,Aρt,

which yieldsF(ρt,ρ) + ( –t)ρ–f,Aρt ≥. Lettingt→ in the above inequality, we

arrive atF(z,ρ) +ρ–z,Az ≥. This shows thatf∈EP(F,A). It follows that

lim sup n→∞

f(x) –x,zn–x

≤. (.)

Finally, we show thatxn→x, asn→ ∞. Note that

zn–x =αn

f(Wnxn) –x,zn–x

+ ( –αn)yn–x,zn–x

≤ –αn( –κ)xn–xzn–x+αn

f(x) –x,zn–x

≤ –αn( –κ)



xn–x+zn–x +αn

f(x) –x,zn–x

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Hence, we have

zn–x≤

 –αn( –κ) xn–x+ αn

f(x) –x,zn–x

.

This implies that

xn+–x =βnxn+ ( –βn)Wnzn–x



≤βnxn–x+ ( –βn)zn–x

≤ –αn( –βn)( –κ)xn–x+ αn( –βn)

f(x) –x,zn–x

.

Using Lemma . and (.), we ﬁnd from the restrictions (a), (b), and (c) thatlimn→∞xn–

x= . This completes the proof.

3 Applications

For a single mapping, we ﬁnd from Theorem . the following result.

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

be a bifunction from C×C toRwhich satisﬁes(A)-(A).Let A:C→H be anα -inverse-strongly monotone mapping and let S be a nonexpansive mapping.Assume that:=F(S)∩ EP(F,A)=∅.Let f :C→C be aκ-contractive mapping.Let{xn}be a sequence generated

in the following process:let it be a sequence generated in

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

F(yn,y) +Axn,y–yn+_r_ny–yn,yn–xn ≥, ∀y∈C,

xn+=βnxn+ ( –βn)S(αnf(Sxn) + ( –αn)yn), ∀n≥,

where{αn}and{βn}are sequences in[, ]and{rn}is a positive number sequence.Assume

that the above control sequences satisfy the following restrictions:

(a)  <a≤βn≤b< , <c≤rn≤d< α; (b) limn→∞αn= ,and

∞

n=αn=∞; (c) limn→∞(rn–rn+) = .

Then{xn}converge strongly to a point x∈,where x=Pf(x).

IfSis the identity, we ﬁnd the following result on the generalized equilibrium problem.

Corollary . Let C be a nonempty closed convex subset of a Hilbert space H and let F

be a bifunction from C×C toRwhich satisﬁes(A)-(A).Let A:C→H be anα -inverse-strongly monotone mapping.Assume that EP(F,A)=∅.Let f :C→C be aκ-contractive mapping.Let{xn} be a sequence generated in the following process:let it be a sequence

generated in

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

F(yn,y) +Axn,y–yn+_r_ny–yn,yn–xn ≥, ∀y∈C,

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(a)  <a≤βn≤b< , <c≤rn≤d< α; (b) limn→∞αn= ,and

∞

n=αn=∞; (c) limn→∞(rn–rn+) = .

Then{xn}converge strongly to a point x∈EP(F,A),where x=PEP(F,A)f(x).

Next, we give a result on the equilibrium problem (.).

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

be a bifunction from C×C toRwhich satisﬁes(A)-(A).Let{Si:C→C}be a family of

inﬁnitely nonexpansive mappings.Assume that:=∞_i_=F(Si)∩EP(F)=∅.Let f:C→C

be aκ-contractive mapping.Let{xn}be a sequence generated in the following process:let it

be a sequence generated in

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

F(yn,y) +_r_ny–yn,yn–xn ≥, ∀y∈C,

where{Wn}is the mapping sequence deﬁned by(.),{αn}and{βn}are sequences in[, ]

following restrictions:

(a)  <a≤βn≤b< , <c≤rn≤d< +∞; (b) limn→∞αn= ,and

∞

n=αn=∞; (c) limn→∞(rn–rn+) = .

Proof By puttingA≡, the zero operator, we can easily get the desired conclusion. This

completes the proof.

Next, we give a result on the classical variational inequality.

Theorem . Let C be a nonempty closed convex subset of a Hilbert space H.Let A:C→

H be anα-inverse-strongly monotone mapping and let{Si:C→C}be a family of inﬁnitely

nonexpansive mappings.Assume that:=∞_i_=F(Si)∩VI(C,A)=∅.Let f :C→C be a

κ-contractive mapping.Let{xn}be a sequence generated in the following process:let it be a

sequence generated in

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

yn=PC(xn–rnAxn),

where{Wn}is the mapping sequence deﬁned by(.),{αn}and{βn}are sequences in[, ]

following restrictions:

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(b) limn→∞αn= ,and

∞

n=αn=∞; (c) limn→∞(rn–rn+) = .

Proof PuttingF≡, we see from Theorem . that

Axn,y–yn+

 rn

y–yn,yn–xn ≥, ∀y∈C,∀y∈C,∀n≥.

This implies that

y–yn,xn–rnAxn–yn ≤, ∀y∈C.

It follows that

yn=PC(xn–rnAxn).

This completes the proof.

Finally, we utilize the results presented in the paper to study the following optimization problem:

min

x∈Ch(x), (.)

whereCis a nonempty closed convex subset of a Hilbert space, andh:C→Ris a convex and lower semi-continuous functional. We useto denote the solution set of the prob-lem (.). LetF:C×C→Rbe a bifunction deﬁned byF(x,y) =h(y) –h(x). We consider the following equilibrium problem: to ﬁndx∈Csuch that

F(x,y)≥, ∀y∈C.

It is easy to see that the bifunctionFsatisﬁes conditions (A)-(A) andEP(F) =.

Theorem . Let C be a nonempty closed convex subset of a Hilbert space H and let h:

C→Rbe deﬁned as above.Assume that=∅.Let f :C→C be aκ-contractive mapping. Let{xn}be a sequence generated in the following process:let it be a sequence generated in

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

h(y) –h(un) +_r_ny–yn,yn–xn ≥, ∀y∈C,

xn+=βnxn+ ( –βn)(αnf(xn) + ( –αn)yn), ∀n≥,

(a)  <a≤βn≤b< , <c≤rn≤d< +∞; (b) limn→∞αn= ,and

∞

n=αn=∞; (c) limn→∞(rn–rn+) = .

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Competing interests

The author declares to have no competing interests.

Acknowledgements

The author is grateful to the anonymous referees for useful suggestions which improved the contents of the article.

Received: 19 April 2014 Accepted: 28 June 2014 Published:22 Jul 2014 References

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