Affine algorithms for the split variational inequality and equilibrium problems

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R E S E A R C H

Open Access

Afﬁne algorithms for the split variational

inequality and equilibrium problems

Yonghong Yao

1*

_{, Rudong Chen}

1

_{and Yeong-Cheng Liou}

2

*_{Correspondence:}

[email protected] 1_{Department of Mathematics,}

Tianjin Polytechnic University, Tianjin, 300387, China Full list of author information is available at the end of the article

Abstract

An aﬃne algorithm for the split variational inequality and equilibrium problems is presented. Strong convergence result is given.

Keywords: aﬃne algorithm; split method; variational inequality; equilibrium problem

1 Introduction

In the present manuscript, we focus on the following split variational inequality and equi-librium problem: Finding a pointx∗such that

x∗∈GVI(B,ψ,C) and ψx∗∈EP(F,A), (.)

whereGVI(B,ψ,C) is the solution set of the generalized variational inequality of ﬁnding u∈C,ψ(u)∈Csuch that

Bu,ψ(v) –ψ(u)≥, ∀ψ(v)∈C, (.)

andEP(F,A) is the solution set of the equilibrium problem, which is to ﬁndx†_∈_C_such

that

Fx†,y+Ax†,y–x†≥, ∀y∈C. (.)

Our main motivations are inspired by the following reasons.

Reason  Recently, the split problems have been considered by some authors. Especially, the split feasibility problem which can mathematically be formulated as the problem of ﬁnding a pointx˜with the property

˜

x∈C and g(x)˜ ∈Q

has received much attention due to its applications in signal processing and image recon-struction with particular progress in intensity modulated radiation therapy [–]. Note that the involved operatorgis a bounded linear operator. However, in the present paper, the involved mappingψin (.) is a nonlinear mapping.

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Reason  The variational inequality problem [–] and equilibrium problem [–], which include the ﬁxed point problems and optimization problems [–], have been studied by many authors. It is an interesting topic associated with the analytical and algo-rithmic approach to the variational inequality and equilibrium problems.

Motivated and inspired by the results in the literature, we present an aﬃne algorithm for solving the split problem (.). Strong convergence theorem is given under some mild assumptions.

2 Preliminaries

LetHbe a real Hilbert space with the inner product·,·and the norm · , respectively. LetCbe a nonempty closed convex subset ofH.

2.1 Monotonicity and convexity

An operator A:C→H is said to be monotone ifx–y,Ax–Ay ≥ for allx,y∈C. A:C→His said to be strongly monotone if there exists a constantγ >  such thatx– y,Ax–Ay ≥γx–y_{for all}_x,_y_∈_C._A_:_C_→_H_{is called an inverse-strongly-monotone} operator if there existsα>  such thatx–y,Ax–Ay ≥αAx–Ay_{for all}_x,_y_∈_{C. Let} g:C→Cbe a nonlinear operator.A:C→His said to beα-inverse stronglyg-monotone iﬀ g(x) –g(y),Ax–Ay ≥αAx–Ay _{for all}_x,_y_∈_C _{and for some}_α_{> . Let}_B_{be a} mapping ofHinto H_{. The eﬀective domain of}_B_{is denoted by}_dom_{(B), that is,}_dom_{(B) =}

{x∈H:Bx=∅}. A multi-valued mappingBis said to be a monotone operator onHiﬀ

x–y,u–v ≥ for allx,y∈dom(B),u∈Bx, andv∈By. A monotone operatorBonHis said to be maximal iﬀ its graph is not strictly contained in the graph of any other monotone operator onH.

A functionF:H→Ris said to be convex if for anyx,y∈Hand for anyλ∈[, ],F(λx+ ( –λ)y)≤λF(x) + ( –λ)F(y).

2.2 Nonexpansivity and continuity

A mappingT:C→Cis said to be nonexpansive [–] ifTx–Ty ≤ x–yfor all x,y∈C. We useFix(T) to denote the set of fixed points ofT.T:C→Cis called a firmly nonexpansive mapping if, for allx,y∈C,Tx–Ty_≤_x_–_y,_Tx_–_Ty_{. It is known that} T is firmly nonexpansive if and only if a mapping T–Iis nonexpansive, whereIis the identity mapping onH.T:C→His said to beL-Lipschitz continuous if there exists a constantL>  such thatTx–Ty ≤Lx–yfor allx,y∈C. In such a case,T is said to beL-Lipschitz continuous. Given a nonempty, closed convex subsetCofH, the mapping that assigns every pointx∈Hto its unique nearest point inCis called a metric projection ontoCand denoted byPC, that is,PCx∈Candx–PCx=inf{x–y:y∈C}. The metric projectionPCis a typical firmly nonexpansive mapping. The characteristic inequality of the projection isx–PCx,y–PCx ≤ for allx∈H,y∈C.

2.3 Equilibrium problem

In this paper, we consider the split problem (.). In the sequel, we assume that the solution setSof (.) is nonempty.

Problem . Assume that

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(A) ψ:C→Cis a weakly continuous andγ-strongly monotone mapping such that

R(ψ) =C;

(A) F:C×C→Ris a bifunction;

(A) A:C→His aβ-inverse-strongly monotone mapping.

Our objective is to

ﬁndx∗∈GVI(B,ψ,C) such thatψx∗∈EP(F,A),

whereFsatisﬁes the following conditions:

(F) F(x,x) = for allx∈C;

(F) Fis monotone,i.e.,F(x,y) +F(y,x)≤for allx,y∈C; (F) for eachx,y,z∈C,limt↓F(tz+ ( –t)x,y)≤F(x,y);

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

In order to solve Problem ., we need the following useful lemmas.

2.4 Useful lemmas

The following three lemmas are important tools for our main results in the next section. Note that these lemmas are used extensively in the literature.

Lemma .(Combettes and Hirstoaga’s lemma []) Let C be a nonempty closed convex subset of a real Hilbert space H.Let F:C×C→R be a bifunction which satisﬁes conditions (F)-(F).Letλ> and x∈C.Then there exists z∈C such that

F(z,y) + 

λy–z,z–x ≥, ∀y∈C.

Further,if Tλ(x) ={z∈C:F(z,y) +_λy–z,z–x ≥for all y∈C},then the following hold: (a) Tλis single-valued andTλis ﬁrmly nonexpansive;

(b) EP(F)is closed and convex andEP(F) =Fix(Tλ).

Lemma .(Suzuki’s 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 xn+= ( –βn)yn+βnxnfor all n≥andlim supn→∞(yn+–yn–xn+–xn)≤. Thenlimn→∞yn–xn= .

Lemma .(Xu’s lemma []) Assume that{an}is a sequence of nonnegative real numbers such that an+≤( –γn)an+δnγn,where{γn}is a sequence in(, )and{δn}is a sequence such that∞_n_=γn=∞andlim supn→∞δn≤ (or

∞

n=|δnγn|<∞).Thenlimn→∞an= .

3 Algorithms and convergence analysis

In this section, we ﬁrst present our algorithm for solving Problem .. Assume that the conditions in Problem . are all satisﬁed.

Algorithm . LetCbe a nonempty closed and convex subset of a real Hilbert spaceH. Step . (Initialization)

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Step . (Projection step) For{xn}, let the sequence{un}be generated iteratively by

un=PC

αnδϕ(xn) + ( –αn)

ψ(xn) –μnBxn

, n≥,

wherePCis the metric projection,{αn} ⊂[.] is a real number sequence,ϕ:C→His an L-Lipschitz continuous mapping andδ>  is a constant.

Step . (Proximal step) Find{zn}such that

F(zn,y) +Aun,y–zn+  λn

y–zn,zn–un ≥, ∀y∈C,

where{λn} ⊂(,∞) is a real number sequence.

Step . (Aﬃne step) For the above sequences{xn}and{zn}, let the (n+ )th sequence

{xn+}be generated by

ψ(xn+) =βnψ(xn) + ( –βn)zn, n≥,

where{βn} ⊂[, ] is a real number sequence.

Theorem . Suppose S=∅.Assume that the following restrictions are satisﬁed:

(C) λn∈(a,b)⊂(, β),μn∈(c,d)⊂(, α)andγ ∈(Lδ, α);

(C) limn→∞(μn+–μn) = andlimn→∞(λn+–λn) = ;

(C) limn→∞αn= and

nαn=∞;

(C) βn∈[ξ,ξ]⊂(, ).

Then the sequence{xn}generated by Algorithm.converges strongly to x∗∈S,which solves the following variational inequality:

δϕx∗–ψx∗,ψ(x) –ψx∗≤, ∀x∈. (.)

Remark . The solution of variational inequality (.) is unique. As a matter of fact, if

˜

x∈Salso solves (.), we have

δϕx∗–ψx∗,ψ(x) –˜ ψx∗≤ and δϕ(x) –˜ ψ(x),˜ ψx∗–ψ(x)˜ ≤.

Adding up the above two inequalities, we deduce

δϕ(x) –˜ ψ(x) –˜ δϕx∗+ψx∗,ψx∗–ψ(x)˜ ≤.

It follows that

ψx∗–ψ(x)˜ ≤δϕx∗–ϕ(x),˜ ψx∗–ψ(x)˜

≤δ ϕx∗–ϕ(x)˜ ψx∗–ψ(x)˜ ,

which implies that

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Sinceψisγ-strongly monotone, we have

γ x∗–x˜ ≤ψx∗–ψ(x),˜ x∗–x˜≤ ψx∗–ψ(x)˜ x∗–x˜ .

Hence,

γ x∗–x˜ ≤ ψx∗–ψ(x)˜ ≤δ ϕx∗–ϕ(x)˜ ≤δL x∗–x˜ .

This deduces the contraction because ofδL<γ by the assumption. Therefore,x∗=x. So,˜ the solution of variational inequality (.) is unique.

Remark . Using the characteristic inequality of the projection, we have

˘

x∈GVI(B,ψ,C) ⇐⇒ ψ(x) =˘ PC

ψ(x) –˘ νBx˘, ∀ν> .

Remark .

ψ(x) –μBx–ψ(y) –μBy ≤ ψ(x) –ψ(y) +μ(μ– α)Bx–By.

In fact,

ψ(x) –μBx–ψ(y) –μBy 

= ψ(x) –ψ(y) – μBx–By,ψ(x) –ψ(y)+μBx–By

≤ ψ(x) –ψ(y) – μαBx–By+μBx–By

≤ ψ(x) –ψ(y) +μ(μ– α)Bx–By.

Next, we prove Theorem ..

Proof Letx‡_∈_{. Hence}_x‡_∈_GVI_(B,_ψ_,_{C) and}_ψ_(x‡₎_∈_EP_(F,_{A). Since}_μ

n> , from Re-mark . we haveψ(x‡) =PC[ψ(x‡) –μnBx‡] for alln≥. Thus,

un–ψ

x‡ = PC

ψ(xn) –μnBxn

–PC

ψx‡–μnBx‡

≤ αn

δϕ(xn) –ψ

x‡+μnBx‡

+ ( –αn)

ψ(xn) –μnBxn

–ψx‡_–_μ

nBx‡

≤αn δϕ(xn) –δϕ

x‡ +αn δϕ

x‡–ψx‡+μnBx‡

+ ( –αn) ψ(xn) –μnBxn

–ψx‡–μnBx‡

≤αnδL xn–x‡ +αn δϕ

+ ( –αn) ψ(xn) –ψ

x‡

≤αnδL/γ ψ(xn) –ψ

x‡ +αn δϕ

+ ( –αn) ψ(xn) –ψ

x‡

= – ( –δL/γ)αn ψ(xn) –ψ

x‡ +αn δϕ

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≤ – ( –δL/γ)αn ψ(xn) –ψ

x‡

+αn δϕ

x‡–ψx‡ + α Bx‡ . (.)

By Algorithm ., we havezn=Tλn(I–λnA)unfor alln≥. Noting thatψ(x‡)∈EP(F,A), we deduceψ(x‡_{) =}_T

λn(I–λnA)ψ(x‡) for alln≥. It follows that ψ(xn+) –ψ

x‡ ≤βn ψ(xn) –ψ

x‡

+ ( –βn) Tλn(I–λnA)un–Tλn(I–λnA)ψ

x‡

≤βn ψ(xn) –ψ

x‡ + ( –βn) un–ψ

x‡

≤βn ψ(xn) –ψ

x‡

+ ( –βn)

 – ( –δL/γ)αn ψ(xn) –ψ

x‡

+ ( –βn)αn δϕ

x‡–ψx‡ + α Bx‡

= – ( –δL/γ)( –βn)αn ψ(xn) –ψ

x‡

+ ( –δL/γ)( –βn)αn

δϕ(x‡_{) –}_ψ_(x‡₎_{+ α}_Bx‡

 –δL/γ .

By induction

ψ(xn) –ψ

x‡ ≤max ψ(x) –ψ

x‡ ,δϕ(x

‡_{) –}_ψ_(x‡₎_{+ α}_Bx‡

 –δL/γ

.

Hence, {ψ(xn)} is bounded. Since ψ is γ-strongly monotone, we can get (by a simi-lar technique as that in Remark .) γxn–x‡ ≤ ψ(xn) –ψ(x‡). So, xn–x‡ ≤



γψ(xn) –ψ(x

‡₎_≤ 

γ max{ψ(x) –ψ(x

‡₎_,δϕ(x‡)–ψ(x‡)+αBx‡

–δL/γ }. This implies that{xn}

is bounded. Next, we showxn+–xn →. From Step  in Algorithm ., we have

F(zn,y) +  λn

y–zn,zn– (un–λnAun)

≥, ∀y∈C.

Takingy=zn+, we get

F(zn,zn+) +  λn

zn+–zn,zn– (un–λnAun)

≥.

Similarly, we also have

F(zn+,zn) +  λn+

zn–zn+,zn+– (un+–λn+Aun+)

≥.

Adding up the above two inequalities, we get

F(zn,zn+) +F(zn+,zn) +Aun–Aun+,zn+–zn+

zn+–zn, zn–un

λn

–zn+–un+ λn+

≥.

By the monotonicity ofF, we have

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So,

Aun–Aun+,zn+–zn+

zn+–zn, zn–un

λn

–zn+–un+ λn+

≥.

Thus,

λnAun–Aun+,zn+–zn+

zn+–zn,zn–zn++zn+–un– λn λn+

(zn+–un+)

≥.

It follows that

zn+–zn≤λnAun–Aun+,zn+–zn

+

zn+–zn,un+–un+

 – λn λn+

(zn+–un+)

=(I–λnA)un+– (I–λnA)un,zn+–zn

+

zn+–zn,

 – λn λn+

(zn+–un+)

≤ (I–λnA)un+– (I–λnA)un zn+–zn

+ – λn λn+

zn+–znzn+–un+

≤ zn+–zn

un+–un+  – λn

λn+

zn+–un+

and hence

zn+–zn ≤ un+–un+  – λn

λn+

zn+–un+

≤ un+–un+ 

a|λn+–λn|zn+–un+.

By Algorithm ., we have

un+–un= PC

αn+δϕ(xn+) + ( –αn+)

ψ(xn+) –μn+Bxn+

–PC

≤ αn+δϕ(xn+) + ( –αn+)

ψ(xn+) –μn+Bxn+

–αnδϕ(xn) + ( –αn)

ψ(xn) –μnBxn

≤αn+δ ϕ(xn+) –ϕ(xn) +δ|αn+–αn| ϕ(xn)

+ ( –αn+) ψ(xn+) –μn+Bxn+–

ψ(xn) –μn+Bxn

+|αn+–αn| ψ(xn) +|μn+–μn| B(xn) +|αn+μn+–αnμn| B(xn)

≤αn+δLxn+–xn+ ( –αn+) ψ(xn+) –ψ(xn)

+|αn+–αn|

δ ϕ(xn) + ψ(xn) +|μn+–μn| B(xn)

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≤αn+(δL/γ) ψ(xn+) –ψ(xn) + ( –αn+) ψ(xn+) –ψ(xn)

+|αn+–αn|

δ ϕ(xn) + ψ(xn) +|μn+–μn| B(xn)

+|αn+μn+–αnμn| B(xn)

= – ( –δL/γ)αn+ ψ(xn+) –ψ(xn)

+|αn+–αn|

δ ϕ(xn) + ψ(xn)

+|μn+–μn| B(xn) +|αn+μn+–αnμn| B(xn) .

Therefore,

zn+–zn ≤

 – ( –δL/γ)αn+ ψ(xn+) –ψ(xn)

+|αn+–αn|

δ ϕ(xn) + ψ(xn)

+|μn+–μn| B(xn) +|αn+μn+–αnμn| B(xn)

+

It follows that

zn+–zn– ψ(xn+) –ψ(xn) ≤ |αn+–αn|

δ ϕ(xn) + ψ(xn)

+|μn+–μn| B(xn) +|αn+μn+–αnμn| B(xn)

+

Sincelim_n_→∞αn= ,limn→∞(μn+–μn) = ,limn→∞(λn+–λn) =  and the sequences

{ϕ(xn)},{ψ(xn)},{zn},{un}and{Bxn}are bounded, we have

lim sup

n→∞

zn+–zn– ψ(xn+) –ψ(xn) ≤.

By Lemma ., we obtain

lim

n→∞ zn–ψ(xn) = .

Hence,

lim

n→∞ ψ(xn+) –ψ(xn) =nlim→∞( –βn) zn–ψ(xn) = .

This together with theγ-strong monotonicity ofψimplies that

lim

n→∞xn+–xn= .

By the convexity of the norm, we have

ψ(xn+) –ψ

x‡ 

= βn

ψ(xn) –ψ

x‡+ ( –βn)

zn–ψ

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≤βn ψ(xn) –ψ

x‡ + ( –βn) zn–ψ

x‡ 

≤βn ψ(xn) –ψ

x‡ + ( –βn) αn

δϕ(xn) –ψ

x‡+μnBx‡

+ ( –αn)

–ψx‡–μnBx‡ 

≤βn ψ(xn) –ψ

x‡ + ( –βn)

αn δϕ(xn) –ψ

x‡+μnBx‡ 

+ ( –αn) ψ(xn) –μnBxn

+ αn( –αn) δϕ(xn) –ψ

x‡₊_μ

nBx‡ ψ(xn) –μnBxn

–ψx‡_–_μ

nBx‡

≤βn ψ(xn) –ψ

x‡ 

+ ( –βn)( –αn) ψ(xn) –μnBxn

+αnM, (.)

whereM>  is some constant. From Remark ., we derive ψ(xn)–μnBxn

–ψx‡–μnBx‡ ≤ ψ(xn)–ψ

x‡ +μn(μn–α) Bxn–Bx‡ .

Thus,

ψ(xn+) –ψ

x‡ ≤βn ψ(xn) –ψ

x‡ + ( –βn)( –αn) ψ(xn) –ψ

x‡ 

+μn(μn– α) Bxn–Bx‡ 

+αnM

≤ ψ(xn) –ψ

x‡ 

+ ( –βn)( –αn)μn(μn– α) Bxn–Bx‡ 

+αnM.

So,

( –βn)( –αn)μn(α–μn) Bxn–Bx‡ 

≤ ψ(xn) –ψ

x‡ – ψ(xn+) –ψ

x‡ +αnM

≤ ψ(xn) –ψ

x‡ + ψ(xn+) –ψ

x‡ ψ(xn+) –ψ(xn) +αnM.

Sinceαn→,ψ(xn+) –ψ(xn) → andlim infn→∞( –βn)( –αn)μn(α–μn) > , we obtain

lim

n→∞ Bxn–Bx

‡

= .

Setyn=ψ(xn) –μnBxn– (ψ(x‡) –μnBx‡) for alln. By using the property of projection, we get

un–ψ

x‡  = PC

ψ(xn) –μnBxn

–PC

ψx‡–μnBx‡ 

≤αn

δϕ(xn) –ψ

x‡+μnBx‡

+ ( –αn)yn,un–ψ

x‡

=   αn

δϕ(xn) –ψ

x‡+μnBx‡

+ ( –αn)yn 

+ un–ψ

x‡ 

– αn

δϕ(xn) –ψ

x‡+μnBx‡

+ ( –αn)yn–un+ψ

x‡ 

≤ 



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+ ( –αn) ψ(xn) –ψ

x‡ + un–ψ

x‡ 

– αn

δϕ(xn) –ψ

x‡+μnBx‡–yn

+ψ(xn) –un–μn

Bxn–Bx‡ 

=  

x‡+μnBx‡ 

+ ( –αn) ψ(xn) –ψ

x‡ + un–ψ

x‡ 

– ψ(xn) –un 

–μ_n Bxn–Bx‡

–α_n δϕ(xn) –ψ

x‡+μnBx‡–yn 

+ μnαn

Bxn–Bx‡,δϕ(xn) –ψ

+ μn

ψ(xn) –un,Bxn–Bx‡

– αn

ψ(xn) –un,δϕ(xn) –ψ

. (.)

It follows that

un–ψ

x‡ ≤αn δϕ(xn) –ψ

x‡+μnBx‡ 

+ ( –αn) ψ(xn) –ψ

x‡ – ψ(xn) –un 

+ μnαn Bxn–Bx‡ δϕ(xn) –ψ

x‡₊_μ

nBx‡–yn

+ μn ψ(xn) –un Bxn–Bx‡

+ αn ψ(xn) –un δϕ(xn) –ψ

x‡+μnBx‡–yn . (.)

From (.) and (.), we have

ψ(xn+) –ψ

x‡ 

≤βn ψ(xn) –ψ

x‡ + ( –βn) un–ψ

x‡ 

≤βn ψ(xn) –ψ

x‡ + ( –βn)αn δϕ(xn) –ψ

x‡+μnBx‡ 

+ ( –αn)( –βn) ψ(xn) –ψ

x‡ – ( –βn) ψ(xn) –un 

+ μn( –βn)αn Bxn–Bx‡ δϕ(xn) –ψ

+ μn( –βn) ψ(xn) –un Bxn–Bx‡

+ ( –βn)αn ψ(xn) –un δϕ(xn) –ψ

≤ ψ(xn) –ψ

x‡ +αn δϕ(xn) –ψ

x‡+μnBx‡ 

– ( –βn) ψ(xn) –un 

(11)

Then we obtain

( –βn) ψ(xn) –un 

≤ ψ(xn) –ψ

x‡ + ψ(xn+) –ψ

x‡ ψ(xn+) –ψ(xn)

+αn δϕ(xn) –ψ

x‡+μnBx‡ 

x‡+μnBx‡–yn .

Sincelimn→∞αn= ,limn→∞ψ(xn+) –ψ(xn)=  andlimn→∞Bxn–Bx‡= , we have

lim

n→∞ ψ(xn) –un = . (.)

Next, we provelim sup_n_→∞δϕ(x∗) –ψ(x∗),un–ψ(x∗) ≤, wherex∗is the unique solution of (.). We take a subsequence{uni}of{un}such that

lim sup

n→∞

δϕx∗–ψx∗,un–ψ

x∗

=lim

i→∞

δϕx∗–ψx∗,uni–ψ

x∗

=lim

i→∞

δϕx∗–ψx∗,ψ(xni) –ψ

x∗. (.)

Since{xni}is bounded, there exists a subsequence{xn_ij}of{xni}which converges weakly to some pointz∈C. Without loss of generality, we may assume thatxniz. This implies

thatψ(xni) ψ(z) due to the weak continuity ofψ. Now, we showz∈S. We ﬁrstly show

z∈EP(F,A). Sincezn=Tλn(un–λnAun), for anyy∈C, we have

F(zn,y) +  λn

≥.

From the monotonicity ofF, we have

 λn

≥F(y,zn), ∀y∈C.

Hence,

y–zni,

zni–uni

λni

+Auni

≥F(y,zni), ∀y∈C. (.)

Putvt=ty+ ( –t)zfor allt∈(, ] andy∈C. Then we havevt∈C. So, from (.) we have

vt–zni,Avt ≥ vt–zni,Avt–

vt–zni,

zni–uni

λni

+Auni

+F(vt,zni)

=vt–zni,Avt–Azni+vt–zni,Azni–Auni

–

vt–zni,

zni–uni

λni

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Note thatAzni–Auni ≤



βzni–uni →. Further, from the monotonicity ofA, we

havevt–zni,Avt–Azni ≥. Lettingi→ ∞in (.), we havevt–z,Avt ≥F(vt,z). This

together with (F), (F) implies that

 = F(vt,vt)≤tF(vt,y) + ( –t)F(vt,z)

≤tF(vt,y) + ( –t)vt–z,Avt

=tF(vt,y) + ( –t)ty–z,Avt,

and hence ≤F(vt,y) + ( –t)Avt,y–z. Lettingt→, we have ≤F(z,y) +y–z,Az. This implies thatz∈EP(F,A). Next, we only need to provez∈GVI(B,ψ,C). Set

Rv= ⎧ ⎨ ⎩

Bv+NC(v), v∈C,

∅, v∈/C.

By [], we know thatRis maximalψ-monotone. Let (v,w)∈G(R). Sincew–Bv∈NC(v) and xn∈C, we haveψ(v) –ψ(xn),w–Bv ≥. Noting thatun=PC[αnδϕ(xn) + ( – αn)(ψ(xn) –μnBxn)], we get

ψ(v) –un,un–

ψ(xn) –μnBxn

≥.

It follows that

ψ(v) –un,

un–ψ(xn) μn

+Bxn– αn μn

δϕ(xn) –ψ(xn) +μnBxn

≥.

Then

ψ(v) –ψ(xni),w

≥ψ(v) –ψ(xni),Bv

–

ψ(v) –uni,

uni–ψ(xni)

μni

–ψ(v) –uni,Bxni

+αni

μni

ψ(v) –uni,δϕ(xni) –ψ(xni) +μniBxni

=ψ(v) –ψ(xni),Bv–Bxni

+ψ(v) –ψ(xni),Bxni

–

ψ(v) –uni,

uni–ψ(xni)

μni

–ψ(v) –uni,Bxni

+αni

μni

ψ(v) –uni,δϕ(xni) –ψ(xni) +μniBxni

≥–

ψ(v) –uni,

uni–ψ(xni)

μni

–ψ(xni) –uni,Bxni

+αni

μni

ψ(v) –uni,δϕ(xni) –ψ(xni) +μniBxni

. (.)

Since ψ(xni) –uni → and ψ(xni) ψ(z), we deduce thatψ(v) –ψ(z),w ≥ by

(13)

z∈GVI(B,ψ,C). Therefore,z∈S. From (.), we obtain

lim sup

n→∞

x∗

=lim

i→∞

δϕx∗–ψx∗,ψ(xni) –ψ

x∗

=δϕx∗–ψx∗,ψ(z) –ψx∗≤.

Note that

un–ψ

x∗ ≤αn

δϕ(xn) –ψ

x∗+ ( –αn)yn,un–ψ

x∗

≤αnδ

ϕ(xn) –ϕ

x∗,un–ψ

x∗+αn

x∗

+ ( –αn) ψ(xn) –μnBxn–

ψx∗–μnBx∗ un–ψ

x∗

≤αnLδ xn–x∗ un–ψ

x∗ +αn

x∗

+ ( –αn) ψ(xn) –ψ

x∗ un–ψ

x∗

≤αn(δL/γ) ψ(xn) –ψ

x∗ un–ψ

x∗

+αn

x∗

+ ( –αn) ψ(xn) –ψ

x∗ un–ψ

x∗

= – ( –Lδ/γ)αn ψ(xn) –ψ

x∗ un–ψ

x∗

+αn

x∗

=  – ( –Lδ/γ)αn

 ψ(xn) –ψ

x∗ +  un–ψ

x∗ 

+αn

x‡.

It follows that

un–ψ

x∗ ≤ – ( –Lδ/γ)αn ψ(xn) –ψ

x∗ 

+ αn

x∗.

Therefore,

ψ(xn+) –ψ

x∗ ≤βn ψ(xn) –ψ

x∗ + ( –βn) un–ψ

x∗ 

≤βn ψ(xn) –ψ

x∗ 

+ ( –βn)

 – ( –δL/γ)αn ψ(xn) –ψ

x∗ 

+ ( –βn)αn

x∗

= – ( –δL/γ)( –βn)αn ψ(xn) –ψ

x∗ 

+ ( –βn)αn

x∗

= – ( –δL/γ)( –βn)αn ψ(xn) –ψ

x∗ 

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×αn

  –δL/γ

x∗

= ( –γn) ψ(xn) –ψ

x∗ +δnγn,

whereγn= ( –δL/γ)( –βn)αnandδn=_–δL/γδϕ(x∗) –ψ(x∗),un–ψ(x∗). It is easily seen

that_nγn=∞andlim supn→∞δn≤. We can therefore apply Lemma . to conclude thatψ(xn)→ψ(x∗) andxn→x∗. This completes the proof.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the ﬁnal manuscript.

Author details

1_{Department of Mathematics, Tianjin Polytechnic University, Tianjin, 300387, China.}2_{Department of Information} Management, Cheng Shiu University, Kaohsiung, 833, Taiwan.

Acknowledgements

Yonghong Yao was supported in part by NSFC 11071279 and NSFC 71161001-G0105. Rudong Chen was supported in part by NSFC 11071279. Yeong-Cheng Liou was supported in part by NSC 101-2628-E-230-001-MY3 and NSC 101-2622-E-230-005-CC3.

Received: 11 October 2012 Accepted: 14 May 2013 Published: 29 May 2013 References

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