Weak convergence theorems of a hybrid algorithm in Hilbert spaces

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R E S E A R C H

Open Access

Weak convergence theorems of a hybrid

algorithm in Hilbert spaces

Yan Hao

* *_{Correspondence:}

[email protected]

School of Mathematics, Physics and Information Science, Zhejiang Ocean University, Zhoushan, 316004, China

Abstract

In this paper, a hybrid algorithm is investigated for solving common solutions of a generalized equilibrium problem, a variational inequality, and ﬁxed point problems of an asymptotically strict pseudocontraction. Weak convergence theorems are

established in the framework of real Hilbert spaces.

Keywords: equilibrium problem; variational inequality; nonexpansive mapping; common solution

1 Introduction

Monotone variational inequalities recently have been investigated as an effective and pow-erful tool for studying a wide class of real world problems which arise in economics, finance, image reconstruction, ecology, transportation, and network; see [–] and the ref-erences therein. Monotone variational inequalities, which include many important prob-lems in nonlinear analysis and optimization, such as the Nash equilibrium problem, com-plementarity problems, fixed point problems, saddle point problems, and game theory recently have been extensively studied based on projection methods. Many well-known problems can be studied by using methods which are iterative in their nature. As an ex-ample, in computer tomography with limited data, each piece of information implies the existence of a convex set in which the required solution lies. The problem of finding a point in the intersection of these convex subsets is then of crucial interest, and it cannot be usually solved directly. Therefore, an iterative algorithm must be used to approximate such a point. Krasnoselskii-Mann iteration, which is also known as a one-step iteration, is a classic algorithm to study fixed points of nonlinear operators. However, Krasnoselskii-Mann iteration only enjoys weak convergence for nonexpansive mappings; see [] and the references therein.

The purposes of this paper is to study common solutions of a generalized equilibrium problem, a variational inequality, and ﬁxed point problems of an asymptotically strict pseudocontraction based on a hybrid algorithm. Weak convergence theorems are estab-lished 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 hybrid algorithm is introduced and the convergence analysis is given. Weak convergence theorems are estab-lished in a real Hilbert space.

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2 Preliminaries

From now on, we always assume thatHis a real Hilbert space with the inner product·,· and the norm · ,Cis a nonempty closed convex subset ofHandPCdenotes the metric

projection fromHontoC.

LetA:C→Hbe a mapping. Recall thatAis said to be monotone if

Ax–Ay,x–y ≥, ∀x,y∈C.

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

Ax–Ay,x–y ≥αAx–Ay, ∀x,y∈C.

For such a case, we also call it anα-inverse-strongly monotone mapping.

A set-valued mappingT:H→H_{is said to be monotone if for all}_x,_y_∈_H,_f _∈_Tx_and_g_∈

Tyimplyx–y,f–g> . A monotone mappingT:H→H_{is maximal if the graph}_G(T₎

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

the normal cone toCatv∈C,i.e.,

NCv=

w∈H:v–u,w ≥,∀u∈C and deﬁne a mappingTonCby

Tv=

⎧ ⎨ ⎩

Av+NCv, v∈C,

∅, v∈/C.

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

Recall that the classical variational inequality problem is to ﬁndx∈Csuch that

Ax,y–x ≥, ∀y∈C. (.)

It is known thatx∈Cis a solution to (.) if and only ifxis a ﬁxed point of the mapping PC(I–λA), whereλ>  is a constant andIis the identity mapping. Projection methods

recently have been studied for variational inequality (.); see [–] and the references therein.

LetS:C→Cbe a nonlinear mapping. In this paper, we useF(S) to denote the ﬁxed point set ofS. Recall thatSis said to be nonexpansive if

Sx–Sy ≤ x–y, ∀x,y∈C.

Sis said to be asymptotically nonexpansive if there exists a sequence{kn} ⊂[,∞) with limn→∞kn=  such that

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Sis said to beκ-strictly pseudocontractive if there exists a constantk∈[, ) such that

Sx–Sy≤ x–y+κ(x–Sx) – (y–Sy), ∀x,y∈C.

The class of strict pseudocontractions was introduced by Browder and Petryshyn []. It is clear that every nonexpansive mapping is a -strict pseudocontraction.

T is said to be an asymptoticallyκ-strict pseudocontraction if there exists a sequence {kn} ⊂[,∞) withlimn→∞kn=  and a constantκ∈[, ) such that

Tnx–Tny≤knx–y+κI–Tn x–

I–Tn y, ∀x,y∈C,n≥.

The class of asymptotically strict pseudocontractions was introduced by Qihou []. It is clear that every asymptotically nonexpansive mapping is an asymptotically -strict pseu-docontraction.

LetFbe a bifunction ofC×CintoR, whereRdenotes the set of real numbers andA: C→His an inverse-strongly monotone mapping. In this paper, we consider the following generalized equilibrium problem:

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

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

EP(F,A) =x∈C:F(x,y) +Ax,y–x ≥,∀y∈C.

To study the generalized equilibrium problem (.), we may assume thatFsatisﬁes 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.

If A≡, then the generalized equilibrium problem (.) is reduced to the following equilibrium problem:

Findx∈Csuch thatF(x,y)≥, ∀y∈C. (.)

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

EP(F) =x∈C:F(x,y)≥,∀y∈C.

IfF≡, then the generalized equilibrium problem (.) is reduced to the classical vari-ational inequality (.).

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(.), generalized equilibrium problem (.), and ﬁxed points of an asymptotically strict pseudocontraction. Possible computation errors are taken into account. Weak conver-gence theorems are established in the framework of real Hilbert spaces.

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

Lemma .[] Let C be a nonempty closed convex subset of H,and let F:C×C→R

be 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.

Further,deﬁne

Trx=

z∈C:F(z,y) +

ry–z,z–x ≥,∀y∈C

for all r> and x∈H.Then the following hold:

(a) Tris single-valued;

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

Trx–Try≤ Trx–Try,x–y;

(c) F(Tr) =EP(F);

(d) EP(F)is closed and convex.

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

C→C be an asymptotically strict pseudocontraction.Then I–S is demi-closed,that is,if {xn}is a sequence in C with xnx and xn–Sxn→,then x∈F(S).

Lemma .[] Let H be a Hilbert space and <p≤tn≤q< for all n≥.Suppose that

{xn}and{yn}are sequences in H such that

lim sup n→∞

xn ≤d, lim sup n→∞

yn ≤d

and

lim

n→∞tnxn+ ( –tn)yn=d

hold for some d≥.Thenlimn→∞xn–yn= .

Lemma .[] Let{an},{bn},and{cn}be three nonnegative sequences satisfying the

fol-lowing condition:

an+≤( +bn)an+cn, ∀n≥n,

where n is some nonnegative integer,

∞

n=bn<∞ and

∞

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3 Main results

Theorem . Let C be a nonempty closed convex subset of a real Hilbert space H.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 S:C→C be an asymptoticallyκ-strict pseudocontraction with the sequence{kn}such

that∞_n=(kn– ) <∞.Assume that=F(S)∩VI(C,B)∩EP(F,A)is not empty.Let{αn},

{α_n},{α_n},and{βn}be real number sequences in(, ).Let{rn}and{sn}be two positive real

number sequences.Let{xn}be a sequence generated in the following process:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

x∈C,

F(zn,z) +Axn,z–zn+_r_nz–zn,zn–xn ≥, ∀z∈C,

yn=PC(zn–snBzn),

xn+=αnxn+αn(βnyn+ ( –βn)Snyn) +αnen,

where{en}is a bounded sequence in C.Assume that the control sequences satisfy the

fol-lowing restrictions:

(a) αn+αn+αn= ; (b)  <p≤αn≤q< and

_∞

n=αn<∞; (c)  <κ<βn≤b< ;

(d)  <s≤sn≤s< βand <r≤rn≤r< α,

where p,q,b,s,s,r,rare real constants.Then{xn}converges weakly to some point in.

Proof First, we show that the sequences{xn},{yn}, and{zn}are bounded. Letp∈be

ﬁxed arbitrarily. For anyx,y∈C, we see that

(I–rnA)x– (I–rnA)y 

=(x–y) –rn(Ax–Ay) 

=x–y– rnx–y,Ax–Ay+rnAx–Ay 

≤ x–y–rn(α–rn)Ax–Ay. (.)

Using the restriction (d), we see that(I–rnA)x– (I–rnA)y ≤ x–y. This implies that

I–rnAis nonexpansive. In the same way, we ﬁnd thatI–snBis also nonexpansive. Using

the restriction (c), we obtain that

βnyn+ ( –βn)Snyn–p 

=βnyn–p+ ( –βn)Snyn–Snp 

–βn( –βn)(yn–p) –

Snyn–Snp 

≤βnyn–p+ ( –βn)

knyn–p+κ(yn–p) –

Snyn–Snp 

–βn( –βn)(yn–p) –

Snyn–Snp 

=knyn–p– ( –βn)(βn–κ)(yn–p) –

Snyn–Snp 

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It follows that

xn+–p≤αnxn–p+αnβnyn+ ( –βn)Snyn–p 

+α_nen–p

≤αnxn–p+αnknyn–p+αnen–p

=αnxn–p+αnknPC(I–snB)zn–p+αnen–p

≤αnxn–p+αnknTrn(I–rnA)xn–p



+α_nen–p

≤knxn–p+αnen–p.

This implies from Lemma . thatlimn→∞xn–pexists. This shows that{xn}is bounded,

so are{yn}and{zn}. From (.), we have

xn+–p≤αnxn–p+αnknyn–p+αnen–p

≤αnxn–p+αnkn(I–snB)zn–p+αnen–p

≤αnxn–p+αnkn

zn–p–sn(β–sn)Bzn–Bp +αnen–p

≤knxn–p–snknαn(β–sn)Bzn–Bp+αnen–p.

It follows that

snknαn(β–sn)Bzn–Bp≤knxn–p–xn+–p+αnen–p.

With the aid of the restrictions (b) and (d), we ﬁnd that

lim

n→∞Bzn–Bp= . (.)

SincePCis ﬁrmly nonexpansive, we have

yn–p =PC(I–snB)zn–PC(I–snB)p 

≤(I–snB)zn– (I–snB)p,yn–p

= 

(I–snB)zn– (I–snB)p



+yn–p

–(I–snB)zn– (I–snB)p– (yn–p) 

≤  

zn–p+yn–p–zn–yn–sn(Bzn–Bp)

=  

zn–p+yn–p–zn–yn

+ snzn–yn,Bzn–Bp–snBzn–Bp

≤  

xn–p+yn–p–zn–yn

+ snzn–yn,Bzn–Bp–snBzn–Bp

,

which implies that

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Hence, we ﬁnd from (.) that

xn+–p≤αnxn–p+αnknyn–p+αnen–p

≤knxn–p–αnknzn–yn+ αnsnknzn–ynBzn–Bp

+α_nen–p.

Therefore, we obtain that

α_nknzn–yn≤knxn–p–xn+–p+ snknzn–ynBzn–Bp

+α_nen–p.

From the restrictions (b) and (d), we ﬁnd from (.) that

lim

n→∞zn–yn= . (.)

It follows from (.) that

zn–p=Trn(I–rnA)xn–p



≤ xn–p–rn(α–rn)Axn–Ap.

Hence, we have

xn+–p≤αnxn–p+αnknyn–p+αnen–p

≤αnxn–p+αnknzn–p+αnen–p

≤knxn–p–αnrn(α–rn)knAxn–Ap+αnen–p.

This implies that

α_nrn(α–rn)knAxn–Ap≤knxn–p–xn+–p+αnen–p.

Using the restrictions (b) and (d), we obtain that

lim

n→∞Axn–Ap= . (.)

SinceTrnis ﬁrmly nonexpansive, we ﬁnd that

zn–p=Trn(I–rnA)xn–Trn(I–rnA)p



≤(I–rnA)xn– (I–rnA)p,zn–p

= 

(I–rnA)xn– (I–rnA)p



+zn–p

–(I–rnA)xn– (I–rnA)p– (zn–p) 

≤  

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=  

xn–p+zn–p–

xn–zn

– rnxn–zn,Axn–Ap–rnAxn–Ap

,

which implies that

zn–p≤ xn–p–xn–zn+ rnxn–znAxn–Ap.

It follows that

xn+–p≤αnxn–p+αnknyn–p+αnen–p

≤αnxn–p+αnknzn–p+αnen–p

≤knxn–p–αnknxn–zn+ rnαnknxn–znAxn–Ap

+α_nen–p,

which yields that

α_nknxn–zn≤knxn–p–xn+–p+ rnαnxn–znAxn–Ap

+α_nen–p.

Using the restrictions (b) and (d), we ﬁnd from (.) that

lim

n→∞xn–zn= . (.)

It follows from (.) and (.) that

lim

n→∞xn–yn= . (.)

Since{xn}is bounded, we see that there exists a subsequence{xni}of{xn}which converges

weakly toξ. LetTbe a maximal monotone mapping deﬁned by

Tx=

⎧ ⎨ ⎩

Bx+NCx, x∈C,

∅, x∈/C.

For any given (x,y)∈Graph(T), we havey–Bx∈NCx. Sinceyn∈C, by the deﬁnition ofNC,

we havex–yn,y–Bx ≥. Sinceyn=PC(I–snB)zn, we see thatx–yn,yn– (I–snB)zn ≥

and hence

x–yn,

yn–zn

sn

+Bzn

≥.

It follows that

x–yni,y ≥ x–yni,Bx

≥ x–yni,Bx–

x–yni, yni–zni

sni

+Bzni

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=x–yni,Bx–Byni+x–yni,Byni–Bzni–

sni

≥ x–yni,Byni–Bzni–

sni

.

Sinceyniconverges weakly toξandBis



β-Lipschitz continuous, we see thatx–ξ,y ≥. Notice thatTis maximal monotone and hence ∈Tξ. This shows thatξ∈VI(C,B). From (.), we see thatzniconverges weakly toξ. It follows that

F(zn,z) +Axn,z–zn+

 rnz

–zn,zn–xn ≥, ∀z∈C.

From condition (A), we see that

Axn,z–zn+

 rnz

–zn,zn–xn ≥F(z,zn), ∀z∈C.

Replacingnbyni, we arrive at

Axni,z–zni+

z–zni,

zni–xni rni

≥F(z,zni), ∀z∈C. (.)

Fortwith  <t≤ andz∈C, letut=tz+ ( –t)ξ. Sincez∈Candξ∈C, we haveut∈C.

In view of (.), we ﬁnd that

ut–zni,Aut ≥ ut–zni,Aut–Axni,ut–zni–

ut–zni,

zni–xni rni

+F(ut,zni)

=ut–zni,Aut–Azni+ut–zni,Azni–Axni

–

ut–zni, zni–xni

rni

+F(ut,zni).

Using (.), we have limi→∞Azni –Axni = . Since A is monotone, we see thatut – zni,Aut–Azni ≥. It follows from condition (A) that

ut–ξ,Aut ≥F(ut,ξ). (.)

Using conditions (A) and (A), we see from (.) that

 =F(ut,ut)≤tF(ut,z) + ( –t)F(ut,ξ)

≤tF(ut,z) + ( –t)ut–ξ,Aut

=tF(ut,u) + ( –t)tz–ξ,Aut,

which yields that

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Lettingt→, we ﬁnd

F(ξ,z) +z–ξ,Aξ ≥,

which implies thatξ∈EP(F,A).

Now, we are in a position to showξ∈F(S). Sincelimn→∞xn–pexists, we may

as-sume thatlimn→∞xn–p=d> . Putλn=βnyn+ ( –βn)Snyn. It follows from (.) that lim sup_n_→∞xn–p+αn(en–λn) ≤dandlim supn→∞λn–p+αn(en–λn) ≤d. On the

other hand, we have

lim

n→∞xn+–p=nlim→∞αn

(xn–p) +αn(en–λn) + ( –αn)

λn–p+αn(en–λn) =d.

Using Lemma ., we obtain thatlim_n_→∞λn–xn= . Note that

Snyn–xn= λn–xn

 –βn

+βn(xn–yn)  –βn

.

Hence, we havelim_n_→∞Sn_y

n–xn= . Note thatSnxn–xn ≤ Snxn–Snyn+Snyn–

xn. SinceSis Lipschitz continuous, we havelimn→∞Snxn–xn= . Further, we ﬁnd that lim_n_→∞Sxn–xn= . Using Lemma ., we see thatξ∈F(S). This proves thatη∈.

Finally, we show that the sequence{xn}converges weakly toξ. Assume that there exists

another subsequence{xnj}of{xn}such that{xnj}converges weakly toη. In the same way, we ﬁndη∈. Ifη=ξ, we see from the Opial condition [] that

lim

n→∞xn–ξ=lim infi→∞ xni–ξ<lim infi→∞ xni–η

=lim inf

n→∞ xn–η=lim infj→∞ xnj–η <lim inf

j→∞ xnj–ξ= lim

n→∞xn–ξ.

This derives a contradiction. Hence, we haveη=ξ. This implies thatxn ξ ∈. This

completes the proof.

Remark . The key of the weak convergence of the algorithm is due to the fact thatAis inverse-strongly monotone, which yields thatI–rnAis nonexpansive. The nonexpansivity

of the mappingI–rnAplays an important role in this theorem. Therefore, it is of interest

to relax the monotonicity ofAsuch that the algorithm is still weakly convergent.

Next, we give some subresults of Theorem .. IfSis asymptotically nonexpansive, we ﬁnd the following result.

Corollary . Let C be a nonempty closed convex subset of a real Hilbert space H.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 S:C→C be an asymptotically nonexpansive mapping with the sequence{kn}such that

∞

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and{α_n}be real number sequences in(, ).Let{rn}and{sn}be two positive real number

sequences.Let{xn}be a sequence generated in the following process:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

x∈C,

F(zn,z) +Axn,z–zn+_r_nz–zn,zn–xn ≥, ∀z∈C,

yn=PC(zn–snBzn),

xn+=αnxn+αnSnyn+αnen,

(a) αn+αn +αn= ;

(b)  <p≤αn≤p< and∞n=αn<∞; (c)  <s≤sn≤s< βand <r≤rn≤r< α,

where p,q,s,s,r,rare real constants.Then{xn}converges weakly to some point in.

Further, ifSis an identity mapping, we have the following result.

Corollary . Let C be a nonempty closed convex subset of a real Hilbert space H.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. Assume that=VI(C,B)∩EP(F,A)is not empty.Let{αn},{αn},and{αn}be real number

sequences in(, ).Let{rn}and{sn}be two positive real number sequences.Let{xn}be a

sequence generated in the following process:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

x∈C,

F(yn,z) +Axn,z–yn+_r_nz–yn,yn–xn ≥,

xn+=αnxn+αnPC(yn–snByn) +αnen,

(a) αn+αn +αn= ; (b)  <p≤αn≤q< and

∞

n=αn<∞; (c)  <s≤sn≤s< βand <r≤rn≤r< α,

Next, we give a result on variational inequality (.).

Corollary . Let C be a nonempty closed convex subset of a real Hilbert space H.Let

A:C→H be anα-inverse-strongly monotone mapping,and let B:C→H be aβ -inverse-strongly monotone mapping.Assume that=VI(C,B)∩VI(C,A)is not empty.Let{αn},

{α_n},and{α_n}be real number sequences in(, ).Let{rn} and{sn} be two positive real

number sequences.Let{xn}be a sequence generated in the following process:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

x∈C,

zn=PC(xn–snAxn),

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∞

n=αn<∞; (c)  <s≤sn≤s< βand <r≤rn≤r< α,

Proof PuttingF≡, we see that

Axn,z–zn+

 rn

z–zn,zn–xn ≥, ∀z∈C

is equivalent to

xn–rnAxn–zn,zn–z ≥, ∀z∈C.

This implies thatzn=PC(xn–rnAxn). Letβn=  andSbe the identity. Then we can obtain

from Theorem . the desired results immediately.

Finally, we consider solving common ﬁxed points of a pair of strict pseudocontractions.

Corollary . Let C be a nonempty closed convex subset of a real Hilbert space H.Let

T:C→C be anα-strict pseudocontraction,and let T:C→C be aβ-strict

pseudo-contraction.Assume that=F(T)∩F(T)is not empty.Let{αn},{αn},and{αn}be real

number sequences in(, ).Let{rn}and{sn}be two positive real number sequences.Let{xn}

be a sequence generated in the following process:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

x∈C,

zn= ( –rn)xn+rnTxn,

yn= ( –sn)xn+snTxn,

xn+=αnxn+αnyn+αnen,

∞

n=αn<∞;

(c)  <s≤sn≤s<  –αand <r≤rn≤r<  –β,

Proof Put F≡ ,A=I–T andB=I –T. It follows that A is –_α-inverse-strongly

monotone andBis –_β-inverse-strongly monotone. We also have F(T) =VI(C,B) and

F(T) =VI(C,A). In view of Theorem ., we ﬁnd the desired result immediately.

Competing interests

The author declares that they have no competing interests.

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