Strong convergence of a Halpern type algorithm for common solutions of fixed point and equilibrium problems

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R E S E A R C H

Open Access

Strong convergence of a Halpern-type

algorithm for common solutions of ﬁxed

point and equilibrium problems

Qingnian Zhang

*

*_{Correspondence:}

[email protected] School of Mathematics and Information Science, North China University of Water Resources and Electric Power, Zhengzhou, Henan 450011, China

Abstract

In this article, ﬁxed points of nonexpansive mappings and equilibrium problems based on a Halpern-type algorithm are investigated. Strong convergence theorems for common solutions of the two problems are obtained in the framework of real Hilbert spaces.

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

1 Introduction

The study of equilibrium problems is an important branch of optimization theory and nonlinear functional analysis. Numerous problems in physics, optimization, transporta-tion, signal processing, and economics are reduced to find a solution to equilibrium prob-lems, which cover fixed point probprob-lems, variational inequalities, saddle probprob-lems, inclu-sion problems, and so on. A closely related subject of current interest is the problem of finding common elements in the fixed point set of nonlinear operators and in the solution set of monotone variational inequalities; see [–] and the references therein. The moti-vation for this subject is mainly due to its possible applications to mathematical modeling of concrete complex problems. The aim of this paper is to investigate a common element problem based on a Halpern-type algorithm. Strong convergence of the algorithm is ob-tained in the framework of real Hilbert spaces. The organization of this paper is as follows. In Section , we provide some necessary preliminaries. In Section , a Halpern-type algo-rithm is proposed and analyzed. Strong convergence theorems for common solutions of two problems are established in the framework of Hilbert spaces. In Section , applica-tions of the main results are provided.

2 Preliminaries

LetHbe a real Hilbert space with inner product·,·and norm · . LetCbe a nonempty, closed, and convex subset ofHand letProj_Cbe the metric projection fromHontoC.

LetT:C→Cbe a mapping. In this paper, we useF(T) to denote the ﬁxed point set ofT. Recall thatTis said to becontractiveiﬀ there exists a constantα∈(, ) such that

Tx–Ty ≤αx–y, ∀x,y∈C.

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For such a case,T is also said to beα-contractive. Recall thatT is said to benonexpansive

iﬀ

Tx–Ty ≤ x–y, ∀x,y∈C.

It is well known that the ﬁxed point set of nonexpansive mappings is nonempty provided that the subsetCis bounded, convex, and closed.

LetA:C→Hbe a mapping. Recall thatAis said to bemonotoneiﬀ

Ax–Ay,x–y ≥, ∀x,y∈C.

Recall thatAis said to beinverse-strongly monotoneiﬀ there exists a constantα>  such that

Ax–Ay,x–y ≥αAx–Ay, ∀x,y∈C.

For such a case,Ais also said to beα-inverse-strongly monotone.

Recall that the classical variational inequality is to ﬁnd anx∈Csuch that

Ax,y–x ≥, ∀y∈C. (.)

In this paper, we useVI(C,A) to denote the solution set of (.). It is well known thatx∈C

is a solution of the variational inequality (.) iﬀxis a solution of the ﬁxed point equation

PC(I–rA)x=x, wherer>  is a constant.

Recall that a set-valued mappingM:H⇒His said to bemonotoneiﬀ, for allx,y∈H,

f ∈Mxandg∈Myimplyx–y,f–g> .Mismaximaliﬀ the graphGraph(M) ofRis not properly contained in the graph of any other monotone mapping. It is well known that a monotone mappingMis maximal if and only if, for any (x,f)∈H×H,x–y,f–g ≥, for all (y,g)∈Graph(M) impliesf ∈Rx.

For a maximal monotone operatorMonH, andr> , we may deﬁne the single-valued resolventJr:H→D(M), whereD(M) denotes the domain ofM. It is known that Jr is

ﬁrmly nonexpansive, andM–() =F(Jr), whereF(Jr) :={x∈D(M) :x=Jrx}, andM–() :=

{x∈H: ∈Mx}.

LetA:C→Hbe a monotone mapping, and letFbe a bifunction ofC×CintoR, where

Rdenotes the set of real numbers. We consider the following generalized equilibrium problem:

Findx∈Csuch thatF(x,y) +Ax,y–x ≥, ∀y∈C. (.)

In this paper, we useEP(F,A) to denote the solution set of the generalized equilibrium problem (.).

Next, we give some special cases of the generalized equilibrium problem (.).

(I) IfF≡, then problem (.) is reduced to the classical variational inequality (.). (II) IfA≡, the zero mapping, then problem (.) is reduced to the following

equilibrium problem:

Findx∈Csuch thatF(x,y)≥, ∀y∈C. (.)

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To study the equilibrium problems, we may assume thatFsatisﬁes the following condi-tions:

(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 weakly lower semi-continuous.

Recently, many authors have studied ﬁxed point problems of nonexpansive mappings and solution problems of the equilibrium problems (.) and (.); for more details, see [–] and the references therein. In this paper, motivated and inspired by the research going on in this direction, we consider common element problems based on a mean it-erative process. Strong convergence of the itit-erative process is obtained in the framework of real Hilbert spaces. The results presented in this paper improve and extend the corre-sponding results in Hao [], Qinet al.[], Changet al.[].

In order to prove our main results, we need the following lemmas.

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

() ∞_n_=γn=∞;

() lim sup_n_→∞δn/γn≤or

∞

n=|δn|<∞.

Thenlimn→∞αn= .

Lemma .[] 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) +

ry–z,z–x ≥, ∀y∈C.

Deﬁne a mapping Tr:H→C as follows:

Trx=

z∈C:F(z,y) +

ry–z,z–x ≥,∀y∈C

, x∈H,

then the following conclusions 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.

Let{Si:C→C}be a family of inﬁnitely nonexpansive mappings and{γi}be a

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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,

Wn=Un,=γSUn,+ ( –γ)I.

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

bySn,Sn–, . . . ,Sandγn,γn–, . . . ,γ; see [] and the references therein.

Lemma .[] Let{Si:C→C}be a family of inﬁnitely nonexpansive mappings with a

nonempty common ﬁxed point set and let{γi}be a real sequence such that <γi≤l< ,

where l is some real number,∀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 B:C→H be a mapping and let M:H⇒H be a maximal monotone operator.Then F(Jr(I–rB)) = (B+M)–(),where r is some positive real number.

Lemma .[] Let{Si:C→C}be a family of inﬁnitely nonexpansive mappings with a

nonempty common ﬁxed point set and let{γi}be a real sequence 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 A:C→H a Lipschitz monotone mapping and let NCx be the normal

cone to C at x∈C;that is,NCx={y∈H:x–u,y,∀u∈C}.Deﬁne

Wx= ⎧ ⎨ ⎩

Ax+NCx, x∈C,

∅ x∈/C.

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

in(, )with <lim infn→∞βn≤lim supn→∞βn< .Suppose that xn+= ( –βn)yn+βnxn

for all n≥and

lim sup

n→∞

yn+–yn–xn+–xn

≤.

Thenlim_n_→∞yn–xn= .

3 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 B:C→H be aβ-inverse-strongly monotone mapping.

Let M:H⇒H be a maximal monotone operator.Let{Si:C→C}be a family of inﬁnitely

nonexpansive mappings.Assume that F:=∞_i_=F(Si)∩EP(F,B)∩(A+M)–()=∅.Let

f :C→C be aκ-contraction.Let{xn}be a sequence generated by the process:x∈C and ⎧

⎨ ⎩

F(un,y) +Bxn,y–un+_s_ny–un,un–xn ≥, ∀y∈C,

xn+=αnf(xn) +βnxn+γnWnProjCJrn(un–rnAun), ∀n≥,

where{Wn:C→C}is the sequence generated in(.),{αn},{βn},and{γn}are sequences

in(, )such thatαn+βn+γn= and{rn}and{sn}are positive number sequences.Assume

that the above control sequences satisfy the following restrictions:

(a)  <r≤rn≤r< α, <r≤sn≤r< β;

(b) limn→∞αn= and

_∞

n=αn=∞;

(c)  <lim infn→∞βn≤lim supn→∞βn< ;

(d) limn→∞|sn–sn+|=limn→∞|rn–rn+|= .

Then the sequence{xn}converges strongly tox¯=ProjFf(¯x).

Proof First, we show thatI–rnAis nonexpansive. For∀x,y∈C, we have

(I–rnA)x– (I–rnA)y



=x–y– rnx–y,Ax–Ay+rnAx–Ay

≤ x–y– rnαAx–Ay+rnAx–Ay

=x–y+rn(rn– α)Ax–Ay.

Using restriction (a), we haveI–rnAis nonexpansive, so isI–snB. Fixx∗∈F. It follows

thatun–x∗ ≤ (I–snB)xn– (I–snB)x∗ ≤ xn–x∗. Puttingyn=Jrn(un–rnAun), one ﬁnds thatyn–x∗ ≤ un–x∗ ≤ xn–x∗. It follows that

xn+–x∗≤αnf(xn) –x∗+βnxn–x∗+γnWnProjCyn–x∗

≤αnf(xn) –f

x∗+αnf

x∗–x∗+βnxn–x∗+γnyn–x∗

≤ –αn( –κ)xn–x∗+αnf

x∗–x∗.

Hence, we havexn–x∗ ≤max{x–x∗,f(x ∗_)–_x∗

–α }. This yields the result that{xn}is

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we assume that there exists a bounded setK ⊂C such thatxn,yn,un∈K. Notice that

F(un+,y) +_s_n_+y–un+,un+– (I–sn+B)xn+ ≥,∀y∈C, andF(un,y) +_s_ny–un,un–

(I–snB)xn ≥,∀y∈C. It follows that

un+–un,

un– (I–snB)xn

sn

–un+– (I–sn+B)xn+

sn+

≥.

Hence, we have

un+–un≤

un+–un, (I–sn+B)xn+– (I–snB)xn

+

 – sn

sn+

un+– (I–sn+B)xn+

≤ un+–un

(I–sn+B)xn+– (I–snB)xn

+ – sn

sn+

.

This yields the result that

un+–un ≤(I–sn+B)xn+– (I–snB)xn

+|sn+–sn|

sn+

=(I–sn+B)xn+– (I–sn+B)xn+ (I–sn+B)xn– (I–snB)xn

+|sn+–sn|

sn+

≤ xn+–xn+|sn+–sn|M, (.)

whereMis an appropriate constant such that

M=sup

n≥

Bxn+

un+– (I–sn+B)xn+ ¯

a

.

SinceJrnis ﬁrmly nonexpansive, one sees that yn+–yn

=Jrn(un+–rn+Aun+) –Jrn(un–rnAun) ≤un+–rn+Aun+– (un–rnAun)

=(I–rn+A)un+– (I–rn+A)un+ (rn–rn+)Aun

≤ un+–un+|rn–rn+|Aun. (.)

Substituting (.) into (.), one ﬁnds that

yn+–yn ≤ xn+–xn+

|sn+–sn|+|rn–rn+|

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where M is an appropriate constant such that M=max{supn≥{Aun},M}. On the other hand, one has

Wn+ProjCyn+–WnProjCyn

≤ Wn+yn+–Wnyn

≤ Wn+yn+–Wyn++Wyn+–Wyn+Wyn–Wnyn

≤sup

x∈K

Wn+x–Wx+Wx–Wnx

+yn+–yn, (.)

whereKis the bounded subset ofCdeﬁned above. Combining (.) with (.), one ﬁnds

Wn+ProjCyn+–WnProjCyn

≤sup

x∈K

+xn+–xn

+|rn+–rn|+|sn–sn+|

M. (.)

Lettingxn+= ( –βn)zn+βnxnwe see that

zn+–zn ≤

αn+  –βn+

f(xn+) –Wn+yn++ αn

 –βn

f(xn) –Wnyn

+Wn+ProjCyn+–WnProjCyn. (.)

Substituting (.) into (.), we see that

zn+–zn–xn+–xn

≤ αn+  –βn+

f(xn+) –Wn+yn++ αn

 –βn

f(xn) –Wnyn

+sup

x∈K

+|rn+–rn|+|sn–sn+|

M.

It follows from restrictions (a), (c), and (d) thatlim sup_n_→∞(zn+–zn–xn+–xn)≤.

Using Lemma ., we ﬁnd thatlimn→∞zn–xn= . It follows that

lim

n→∞xn+–xn= . (.)

For anyx∗∈F, we see that xn+–x∗



≤αnf(xn) –x∗



+βnxn–x∗



+γnWnProjCyn–x∗



+βnxn–x∗



+γnyn–x∗



. (.)

Since

yn–x∗



=Jrn(un–rnAun) –x ∗

≤(I–rnA)un– (I–rnA)x∗

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=un–x∗

 – rn

un–x∗,Aun–Ax∗

+r_nAun–Ax∗



≤xn–x∗



+rn(rn– α)Aun–Ax∗

 ,

we ﬁnd from (.) that

lim

n→∞Aun–Ax

∗_{= .} _(.)

It also follows from (.) that

xn+–x∗≤αnf(xn) –x∗



+βnxn–x∗



+γnxn–x∗–sn

Bxn–Bx∗



+βnxn–x∗



+γnxn–x∗+snBxn–Bx∗– sn

Bxn–Bx∗,xn–x∗



+βnxn–x∗



+γnxn–x∗



–snγn(β–sn)Bxn–Bx∗

 .

Using (.), one arrives at

lim

n→∞Bxn–Bx

∗_{= .} _(.)

SinceTsnis ﬁrmly nonexpansive, we ﬁnd that un–x∗



≤(I–snB)xn– (I–snB)x∗,un–x∗

≤ 

xn–x ∗

+un–x∗–xn–un–snBxn–Bx∗

+ sn

Bxn–Bx∗,xn–un

,

which implies thatun–x∗≤ xn–x∗–xn–un+ snBxn–Bx∗xn–un. Hence

γnxn–un≤αnf(xn) –x∗



+xn–x∗+xn+–x∗xn–xn+ + snBxn–Bx∗xn–un.

Using (.) and (.), one has

lim

n→∞xn–un= . (.)

Similarly, one also has

lim

n→∞yn–un= . (.)

Since

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we ﬁnd from (.) and (.) that

lim

n→∞Wnyn–yn= . (.)

Now, we are in a position to showlim sup_n_→∞f(¯x) –x¯,xn–z ≤, wherex¯=ProjFf(¯x).

To prove this, we choose a subsequence{xni}of{xn}such that

lim sup

n→∞

f(¯x) –x¯,xn–x¯

= lim

i→∞

f(¯x) –x¯,xni–x¯

. (.)

Since{xni}is bounded, without loss of generality, we may assume thatxniq. Using (.) and (.), we havelimn→∞xn–yn= . Therefore, we see thatyniq. Now, we are in a position to proveq∈(A+M)–_{(). Notice that}un–yn

rn –Aun∈Myn. Letμ∈Mν. SinceMis monotone, we ﬁnd thatun–yn

rn –Aun–μ,yn–ν ≥. This implies that–Aq–μ,q–ν ≥. This implies that –Aq∈Mq, that is,q∈(A+M)–(). Next, we show thatq∈EP(F,B). Sinceun=Tsn(I–snB)xn, we ﬁnd from (A) that

Bxni,y–uni+

y–uni,

uni–xni

sni

≥F(y,uni), ∀y∈C. (.)

Puttingyt=ty+ ( –t)qfor anyt∈(, ] andy∈C, we see thatyt∈C. Using (.), we

ﬁnd that

yt–uni,Byt

≥ yt–uni,Byt–Bxni,yt–uni–

yt–uni,

uni–xni

sni

+F(yt,uni)

=yt–uni,Byt–Buni+yt–uni,Buni–Bxni–

yt–uni,

uni–xni

sni

+F(yt,uni).

SinceBis monotone, we obtain from (A) thatyt–w,Byt ≥F(yt,w). Using (A) and (A),

we ﬁnd that

 = F(yt,yt)≤tF(yt,y) + ( –t)F(yt,w)

≤tF(yt,y) + ( –t)yt–w,Byt

=tF(yt,y) + ( –t)ty–w,Byt.

Hence, ≤F(yt,y) + ( –t)y–w,Byt, ∀y∈C. It follows from (A) thatw∈EP(F,B).

Next, we prove thatq∈∞_i_=F(Si). Suppose to the contrary,q∈/

∞

i=F(Si),i.e.,Wq=q.

Sinceyniqand the space satisﬁes Opial’s condition, one has

lim inf

i→∞ yni–q<lim inf

i→∞ yni–Wq ≤lim inf

i→∞

yni–Wyni+Wyni–Wq

≤lim inf

i→∞

yni–Wyni+yni–q

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SinceWyn–yn ≤supx∈KWx–Wnx+Wnyn–yn, we ﬁnd from Lemma . that

limn→∞Wyn–yn= . It follows thatlim infi→∞yni –q<lim infi→∞yni–q. This leads to a contradiction. Thus, we haveq∈∞_i_=F(Si). This proves thatq∈F. Therefore,

one has

lim sup

n→∞

f(x¯) –x¯,xn–x¯

≤.

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

xn+–x¯ ≤αn

f(xn) –f(¯x),xn+–x¯

+αn

f(¯x) –x¯,xn+–x¯

+βnxn–x¯xn+–x¯+γnyn–x¯xn+–x¯ ≤κ

αn

xn–x¯+xn+–x¯

+αn

f(¯x) –x¯,xn+–x¯

+ ( –αn)xn–x¯xn+–x¯ ≤ –αn( –κ)

 xn–x¯ ₊

xn+–x¯ ₊_α

n

f(¯x) –x¯,xn+–x¯

,

which implies that

xn+–x¯≤

 –αn( –κ)

xn–x¯+ αn

f(¯x) –x¯,xn+–x¯

.

Using Lemma ., we ﬁnd thatlimn→∞xn–x¯= . This completes the proof.

4 Applications

Recall that a mappingT:C→Cis said to be ak-strict pseudo-contraction if there exists a constantk∈[, ) such that

Tx–Ty≤ x–y+k(I–T)x– (I–T)y

for allx,y∈C. Note that the class ofk-strict pseudo-contractions strictly includes the class of nonexpansive mappings. PutA=I–T, where T:C→C is ak-strict pseudo-contraction. ThenAis–_k-inverse-strongly monotone. Now, we are in a position to state a results on ﬁxed points of strict pseudo-contractions.

Theorem . Let C be a nonempty closed convex subset of a Hilbert space H and let F be a bifunction from C × C to R which satisﬁes (A)-(A). Let T : C → H be a k-strict pseudo-contraction, B: C→ H be a β-inverse-strongly monotone map-ping, and{Si:C →C} be a family of inﬁnitely nonexpansive mappings. Assume that

F:=∞_i_=F(Si)∩EP(F,B)∩F(T)=∅.Let f :C→C be aκ-contraction.Let{xn}be a

se-quence generated by x∈C and

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

yn= ( –rn)un+rnTun,

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in(, )such thatαn+βn+γn= and{rn},and{sn}are positive number sequences.Assume

(a)  <r≤sn≤r< β, <r≤rn≤r<  –k;

∞

n=αn=∞;

(d) limn→∞|sn–sn+|=limn→∞|rn–rn+|= .

Proof TakingA=I–T, wee see that A:C→H is aα-strict pseudo-contraction with α=–k

 andF(T) =VI(C,A). Using Theorem ., we ﬁnd the desired conclusion

immedi-ately.

Letg:H→(–∞, +∞] be a proper convex lower semi-continuous function. Then the

subdiﬀerential∂gofgis deﬁned as follows:

∂fg(x) =y∈H:g(z)≥g(x) +z–x,y,z∈H, ∀x∈H.

From Rockafellar [], we know that∂gis maximal monotone. It is not hard to verify that ∈∂g(x) if and only ifg(x) =miny∈Hg(y).

LetICbe the indicator function ofC,i.e.,

IC(x) =

⎧ ⎨ ⎩

, x∈C,

+∞, x∈/C.

SinceICis a proper lower semi-continuous convex function onH, we see that the

subdif-ferential∂ICofICis a maximal monotone operator. It is clear thatJrx=PCx,∀x∈H. Notice

that (A+∂IC)–() =VI(C,A). Now, we are in a position to state the result on variational inequalities.

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,B:C→H be aβ-inverse-strongly monotone mapping,and

{Si:C→C}be a family of inﬁnitely nonexpansive mappings.Assume that F:=∞i=F(Si)∩

EP(F,B)∩VI(C,A)=∅.Let f :C→C be aκ-contraction.Let{xn}be a sequence generated

by x∈C and

⎧ ⎨ ⎩

xn+=αnf(xn) +βnxn+γnWnPC(un–snAun), ∀n≥,

in(, )such thatαn+βn+γn= and{rn},and{sn}are positive number sequences.Assume

(a)  <r≤sn≤r< β, <r≤rn≤r< α;

∞

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(d) lim_n_→∞|sn–sn+|=limn→∞|rn–rn+|= .

Competing interests

The author declares to have no competing interests. Acknowledgements

The author is grateful to Prof. Tian Cheng and the reviewers’ suggestions which improved the contents of the article. Received: 5 May 2014 Accepted: 16 July 2014 Published: 21 August 2014

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doi:10.1186/1029-242X-2014-313