Convergence of an extragradient-like iterative algorithm for monotone mappings and nonexpansive mappings

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R E S E A R C H

Open Access

Convergence of an extragradient-like

iterative algorithm for monotone mappings

and nonexpansive mappings

Yuan Qing

1

_{and Meijuan Shang}

2,3*

*_{Correspondence:}

[email protected]

2_{Department of Mathematics,}

School of Science, Beijing Jiaotong University, Beijing, 100044, China

3_{Department of Mathematics,}

Shijiazhuang University, Shijiazhuang, 050035, China Full list of author information is available at the end of the article

Abstract

In this paper, we investigate the problem of finding some common element in the set of common fixed points of an infinite family of nonexpansive mappings and in the set of solutions of variational inequalities based on an extragradient-like iterative

algorithm. Strong convergence of the purposed iterative algorithm is obtained.

MSC: 47H05; 47H09; 47J25; 90C33

Keywords: inverse-strongly monotone mapping; ﬁxed point; nonexpansive mapping; projection; strong convergence

1 Introduction

Iterative algorithms have been playing an important role in the approximation solvability, especially of nonlinear variational inequalities as well as of nonlinear equations in several fields such as mechanics, traffic, economics, information, medicine, and many others. The well-known convex feasibility problem which captures applications in various disciplines 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; see, for example, [–]. The Mann iterative algorithm is an efficient method to study the class of nonexpansive mappings. Indeed, Picard cannot converge even that the fixed point set of nonexpansive mappings is nonempty.

It is known that Mann iterative algorithm only has weak convergence for nonexpansive mappings in inﬁnite-dimensional Hilbert spaces; see [] for more details and the refer-ences therein. In many disciplines, including economics [], image recovery [], quan-tum physics [–], and control theory [], problems arise in inﬁnite dimension spaces. To improve the weak convergence of the Mann iterative algorithm, many authors consid-ered using contractions to approximate nonexpansive mappings; for more details, see [] and [] and the references therein.

In this paper, we focus on the problem of finding some common element in the set of common fixed points of an infinite family of nonexpansive mappings and in the set of so-lutions of variational inequalities based on an extragradient-like iterative algorithm. Some deduced sub-results and applications are obtained.

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2 Preliminaries

Throughout this paper, we assume thatHis a real Hilbert space whose inner product and norm are denoted by·,·and · , respectively. LetKbe a nonempty, closed, and convex subset ofH. LetPKbe the metric projection fromHontoK.

Recall that a mappingB:K→His said to be inverse-strongly monotone iﬀ there exists a positive real numberμsuch that

Bx–By,x–y ≥μBx–By, ∀x,y∈K.

For such a case,Bis also said to beμ-inverse-strongly monotone. Recall that a mappingT:K→Kis said to be nonexpansive iﬀ

Tx–Ty ≤ x–y, ∀x,y∈K.

In this paper, we useF(T) to denote the ﬁxed point set of the mappingT.

Recall that a mappingf :K→Kis said to be a contraction iﬀ there exists a coeﬃcient

α∈(, ) such that

f(x) –f(y)≤αx–y, ∀x,y∈K.

For such a case,f is also said to be anα-contraction.

Recall that a linear bounded operatorA:K→K is strongly positive iﬀ there exists a constantγ¯>  such that

Ax,x ≥ ¯γx, ∀x∈K.

Recall that a set-valued mappingS:H→H_{is said to be monotone iﬀ}_f _∈_Sx_and_g_∈_Sy imply

x–y,f –g ≥, ∀x,y∈H.

A monotone mappingS:H→H _{is maximal iﬀ the graph of}_G₍_S_{) of}_S_{is not properly} contained in the graph of any other monotone mapping. It is known that a monotone mappingSis maximal iﬀ for (x,f)∈H×H,x–y,f –g ≥ for every (y,g)∈G(S) implies

f ∈Sx. LetQ:C→Hbe a monotone mapping andNKvbe the normal cone toKatv∈K,

i.e.,NKv={w∈H:v–u,w ≥, ∀u∈K}, and deﬁne

Sv=

⎧ ⎨ ⎩

Qv+NKv, v∈K,

∅, v∈/K.

ThenSis maximal monotone and ∈Sviﬀv∈VI(K,A); see [] for more details. Recall that the classical variational inequality is to ﬁnd au∈Ksuch that

Bu,v–u ≥, ∀v∈K, (.)

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for the identity mapping. In this paper, we useVI(K,B) to denote the solution set of the variational inequality (.).

Iterative algorithms for nonexpansive mappings have recently been applied to solve con-vex minimization problems. A typical problem is to minimize a quadratic function over the set of ﬁxed points of a nonexpansive mappingTon a real Hilbert spaceH,

min

x∈F(T) 

Ax,x–x,u, (.)

whereAis a linear bounded self-adjoint operator onHanduis a given point inH. In [], it is proved that the sequence{xn}deﬁned by the iterative algorithm

x∈H, xn+= (I–αnA)Txn+αnu, n≥,

strongly converges to the unique solution of the minimization problem (.) provided that the sequence{αn}satisﬁes certain restriction.

Recently, Marino and Xu [] reconsidered the problem by viscosity approximation method. They investigated the following iterative algorithm:

x∈H, xn+= (I–αnA)Txn+αnγf(xn), n≥,

where Ais a linear bounded self-adjoint operator onH,T :H→H is a nonexpansive mapping, andf :H→His a contraction. They proved that the sequence{xn}generated in the above iterative process converges strongly to the unique solution of the following variational inequality:

(A–γf)x∗,x–x∗≥, x∈K,

which is the optimality condition for the minimization problem

min

x∈F(T) 

Ax,x–h(x),

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

Recently, the problem of ﬁnding a common element in the ﬁxed point set of a nonex-pansive mapping and in the solution set of a variational inequality has been considered by many authors; see, for example, [–] and the references therein. In , Takahashi and Toyoda [] considered the following iterative algorithm:

x∈K, xn+=αnxn+ ( –αn)TPK(xn–λnBxn), n≥, (.) whereT:K→Kis a nonexpansive mapping,B:K→His aμ-inverse-strongly mono-tone mapping,{αn}is a sequence in (, ), and{λn}is a sequence in (, μ). They showed that the sequence {xn} generated in (.) weakly converges to some point z∈F(T)∩

VI(K,B).

Iiduka and Takahashi [] reconsidered the common element problem via the following iterative algorithm:

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whereT:K→Kis a nonexpansive mapping,B:K→His aμ-inverse-strongly mono-tone mapping,{αn}is a sequence in (, ), and{λn}is a sequence in (, μ). They proved that the sequence{xn}strongly converges to some pointz∈F(T)∩VI(K,B).

In this paper, we will consider an inﬁnite family of nonexpansive mappings. More pre-cisely, we consider the mappingWndeﬁned by

Un,n+=I,

Un,n=γnTnUn,n++ ( –γn)I,

Un,n–=γn–Tn–Un,n+ ( –γn–)I,

.. .

Un,k=γkTkUn,k++ ( –γk)I, (.)

Un,k–=γk–Tk–Un,k+ ( –γk–)I,

.. .

Un,=γTUn,+ ( –γ)I,

Wn=Un,=γTUn,+ ( –γ)I,

whereγ,γ, . . . are real numbers such that ≤γn≤,T,T, . . . is an inﬁnite family of mappings ofKinto itself. Nonexpansivity of eachTiensures the nonexpansivity ofWn.

RegardingWn, we have the following lemmas which are important to prove our main results.

Lemma .[] Let K be a nonempty,closed,and convex subset of a strictly convex Banach space E.Let T,T, . . .be nonexpansive mappings of K into itself such that

∞

n=F(Tn)is

nonempty,and letγ,γ, . . .be real numbers such that <γn≤b< for any n≥.Then,

for every x∈K and k∈N,the limitlimn→∞Un,kx exists. Using Lemma ., one can deﬁne the mappingWas follows:

Wx= lim

n→∞Wnx=nlim→∞Un,x, ∀x∈K. (.) Such a mappingWis calledW-mapping generated byT,T, . . . andγ,γ, . . . .

Throughout this paper, we will assume that  <γn≤b<  for eachn≥.

Lemma .[] Let K be a nonempty,closed,and convex subset of a strictly convex Banach space E.Let T,T, . . .be nonexpansive mappings of K into itself such that

∞

n=F(Tn)is

nonempty,and letγ,γ, . . .be real numbers such that <γn≤b< for each n≥.Then

F(W) =∞_n_=F(Tn).

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

Lemma . In a real Hilbert space H,the following inequality holds:

x+y≤ x+ y,x+y, ∀x,y∈H.

Lemma .[] Assume A is a strongly positive linear bounded self-adjoint operator on a Hilbert space H with the coeﬃcientγ¯> and <ρ≤ A–_._Then_I_–_ρA_≤_{ –}_ρ_γ_¯_.

Lemma .[] Let H be a Hilbert space. Let A be a strongly positive linear bounded self-adjoint operator with the coeﬃcientγ¯> .Assume that <γ <γ¯/α.Let T be a non-expansive mapping with a ﬁxed point xt∈H of the contraction x→tγf(x) + (I–tA)Tx.

Then{xt}converges strongly as t→to a ﬁxed pointx of T¯ ,which solves the variational

inequality

(A–γf)x¯,z–x¯≤, ∀z∈F(T).

Equivalently,we have PF(T)(I–A+γf)x¯=x¯.

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= .

Lemma .[] Let K be a nonempty closed convex subset of a Hilbert space H, {Ti:

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

sequence such that <γn≤b< for each n≥.If C is any bounded subset of K,then

lim_n_→∞sup_x_∈_CWx–Wnx= .

3 Main results

Theorem . Let K be a nonempty,closed,and convex subset of a real Hilbert space H.Let Bi:K→H beμi-inverse-strongly monotone mappings for each i= , ,and f:K→K be

anα-contraction.Let A:K→K be a strongly positive linear bounded self-adjoint operator with the coeﬃcientγ¯> .Let{xn}be a sequence generated in the following

extragradient-like iterative algorithm:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

x∈K,

yn=PK(xn–ηnBxn),

xn+=PK(αnγf(xn) + (I–αnA)WnPK(I–λnB)yn), n≥,

(.)

where PK is the metric projection from H onto K,Wnis a mapping deﬁned by(.),{αn}

is a real number sequence in(, ),and{λn},{ηn}are two positive real number sequences.

Assume that F =∞_i_=F(Ti)∩VI(K,B)∩VI(K,B)=∅,  <γ <γ¯/α and the following

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(a) limn→∞αn= , ∞n=αn=∞,and ∞n=|αn+–αn| ≤ ∞;

(b) ∞_n_=|ηn+–ηn|<∞, ∞n=|λn+–λn|<∞;

(c) {ηn},{λn} ⊂[u,v],where <u<v< min{μ,μ}.

Then the sequence{xn}strongly converges to x∗∈F,where x∗=PF(γf+ (I–A))x∗.

Proof First, we show thatI–λnBandI–ηnBare nonexpansive. Indeed, we see from the restriction (c) that

(I–λnB)x– (I–λnB)y 

=x–y–λn(Bx–By) 

=x–y– λnx–y,Bx–By+λnBx–By

≤ x–y+λn(λn– μ)Bx–By

≤ x–y, ∀x,y∈C.

This shows thatI–λnB is nonexpansive, so isI–ηnB. Noticing the condition (a), we may assume, with no loss of generality, thatαn≤ A–for eachn≥. It follows from Lemma . thatI–αnA ≤ –αnγ¯.

Next, we show that the sequence{xn}is bounded. Lettingp∈F, we see that

yn–p=PK(I–ηnB)xn–PK(I–ηnB)p≤ xn–p.

It follows that

xn+–p ≤αn

γf(xn) –Ap

+ (I–αnA)

WnPC(I–λnB)yn–p

≤αnγf(xn) –Ap+ ( –αnγ¯)WnPC(I–λnB)yn–p

≤αnγf(xn) –f(p)+αnγf(p) –Ap+ ( –αnγ¯)yn–p = –αn(γ¯–γ α)

xn–p+αnγf(p) –Ap.

By simple induction, we have

xn–p ≤max

x–p,

Ap–γf(p)

¯

γ–γ α

,

which yields that the sequence{xn}is bounded, so is{yn}. Notice that

yn+–yn

=PK(I–ηn+B)xn+–PK(I–ηnB)xn

≤(I–ηn+B)xn+– (I–ηn+B)xn+ (I–ηn+B)xn– (I–ηnB)xn

≤ xn+–xn+|ηn+–ηn|Bxn. (.)

Puttingρn=PK(I–λnB)yn, we have

ρn+–ρn

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≤(I–λn+B)yn+– (I–λn+B)yn+ (I–λn+B)yn– (I–λnB)yn

≤ yn+–yn+|λn+–λn|Byn. (.)

Substituting (.) into (.), we arrive at

ρn+–ρn ≤ xn+–xn+M

|ηn+–ηn|+|λn+–λn|

, (.)

whereMis an appropriate constant such that

M≥max

sup

n≥

Byn,sup n≥

Bxn

.

Notice that

xn+–xn+

≤(I–αn+A)(Wn+ρn+–Wnρn) – (αn+–αn)AWnρn +γαn+

f(xn+) –f(xn)

+f(xn)(αn+–αn)

≤( –αn+γ¯)

ρn+–ρn+Wn+ρn–Wnρn

+|αn+–αn|AWnρn +γαn+αxn+–xn+f(xn)|αn+–αn|

. (.)

SinceTiandUn,iare nonexpansive, we have from (.) that

Wn+ρn–Wnρn=γTUn+,ρn–γTUn,ρn

≤γUn+,ρn–Un,ρn

=γγTUu+,ρn–γTUn,ρn

≤γγUu+,ρn–Un,ρn

≤ · · ·

≤γγ· · ·γnUn+,n+ρn–Un,n+ρn

≤M

n

i=

γi, (.)

whereM≥ is an appropriate constant such thatUn+,n+ρn–Un,n+ρn ≤Mfor each

n≥. Substituting (.) and (.) into (.), we arrive at

xn+–xn+ ≤

 –αn+(γ¯–αγ)

xn+–xn

+M

_n

i=

γi+|αn+–αn|+|λn+–λn|+|ηn+–ηn|

, (.)

whereMis an appropriate constant such that

M=max

M,M,γsup

n≥

f(xn)+AWnρn

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From the restrictions (a) and (b), we obtain from Lemma . that

lim

n→∞xn+–xn= . (.)

Notice that

xn+–Wnρn=PK

αnγf(xn) + (I–αnA)WnPK(I–λnB)yn

–PK(Wnρn)

≤αnγf(xn) –AWnρn.

It follows from the restriction (a) that

lim

n→∞Wnρn–xn+= . (.)

Notice that

yn–p≤(xn–p) –ηn(Bxn–Bp)

≤ xn–p– ηnμBxn–Bp+ηnBxn–Bp =xn–p+

η_n– ηnμ

Bxn–Bp. (.)

In a similar way, we ﬁnd that

ρn–p≤ xn–p+

λ_n– λμ

Byn–Bp. (.)

On the other hand, we have

xn+–p≤αn

γf(xn) –Ap

+ (I–αnA)(Wnρn–p)

≤αnγf(xn) –Ap+ ( –αnγ¯)ρn–p



≤αnγf(xn) –Ap+ρn–p+ αnγf(xn) –Apρn–p. (.)

Substituting (.) into (.) gives

xn+–p≤αnγf(xn) –Ap



+xn–p+

λ_n– λnμ

Byn–Bp + αnγf(xn) –Apρn–p.

It follows from the restriction (c) that

u(μ–v)Byn–Bp

≤αnγf(xn) –Ap+xn–p–xn+–p+ αnγf(xn) –Apρn–p

≤αnγf(xn) –Bp+

xn–p+xn+–p

xn–xn+ + αnγf(xn) –Apρn–p.

In view of the restriction (a), we obtain from (.) that

lim

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From (.), we also have

xn+–p≤αnγf(xn) –Ap+yn–p

+ αnγf(xn) –Apρn–p. (.)

Combining (.) with (.), we arrive at

xn+–p≤αnγf(xn) –Ap+xn–p+

η_n– ηnμ

Bxn–Bp + αnγf(xn) –Apρn–p,

which implies from the restriction (c) that

u(μ–v)Bxn–Bp

≤αnγf(xn) –Ap+xn–p–xn+–p + αnγf(xn) –Apρn–p

≤αnγf(xn) –Ap



+xn–p+xn+–p

xn–xn+ + αnγf(xn) –Apρn–p.

In view of the restriction (a), we obtain from (.) that

lim

n→∞Bxn–Bp= . (.)

On the other hand, we see from the ﬁrm expansivity ofPKthat

yn–p =PK(I–ηnB)xn–PK(I–ηnB)p

≤(I–ηnB)xn– (I–ηnB)p,yn–p

= 

(I–ηnB)xn– (I–ηnB)p 

+yn–p –(I–ηnB)xn– (I–ηnB)p– (yn–p)



≤ 



xn–p+yn–p–(xn–yn) –ηn(Bxn–Bp) 

=  

xn–p+yn–p–xn–yn–ηnBxn–Bp + ηnxn–yn,Bxn–Bp

,

which yields

yn–p≤ xn–p–xn–yn+ ηnxn–ynBxn–Bp. (.)

In the same way, we can obtain that

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Substituting (.) into (.) yields

xn+–p≤αnγf(xn) –Ap



+xn–p–xn–yn

+ ηnxn–ynBxn–Bp+ αnγf(xn) –Apρn–p.

It follows that

xn–yn≤αnγf(xn) –Ap



+xn–p–xn+–p

+ ηnxn–ynBxn–Bp+ αnγf(xn) –Apρn–p

≤αnγf(xn) –Ap+

xn–p+xn+–p

xn+–xn + ηnxn–ynBxn–Bp+ αnγf(xn) –Apρn–p.

In view of (.) and (.), we see from the restriction (a) that

lim

n→∞xn–yn= . (.)

Similarly, we can obtain that

lim

n→∞ρn–yn= . (.)

Observe that

ρn–Wnρn ≤ yn–ρn+xn–yn+xn–xn++xn+–Wnρn.

It follows from (.), (.), (.) and (.) that

lim

n→∞Wnρn–ρn= . (.)

From Lemma ., we haveWρn–Wnρn → asn→ ∞. This in turn implies that

lim

n→∞Wρn–ρn= . (.)

Next, we show that

lim sup

n→∞

γfx∗–Ax∗,xn–x∗

≤. (.)

To show it, we choose a subsequence{xni}of{xn}such that

lim sup

n→∞

=lim

i→∞

γfx∗–Ax∗,xni–x

∗_.

As {xni} is bounded, we have that a subsequence{xn_ij} of{xni}converges weakly to p.

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haveynipandznip, respectively. Notice thatp∈F. Indeed, let us ﬁrst show that p∈VI(K,B). Put

Sw=

⎧ ⎨ ⎩

Bv+NKv, v∈K,

∅, v∈/K.

ThenSis maximal monotone. Let (v,w)∈G(S). Sincew–Bv∈NKvandρn∈K, we have

v–ρn,w–Bv ≥.

On the other hand, we have fromρn=PK(I–λnB)ynthat

v–ρn,ρn– (I–λnB)yn

≥

and hence

v–ρn,

ρn–yn

λn

+Byn

≥.

It follows that

v–ρni,w ≥ v–ρni,Bv

≥ v–ρni,Bv–

v–ρni,

ρni–yni λni

+Byni

≥

v–ρni,Bv–

–Byni

=v–ρni,Bv–Bρni+v–ρni,Bρni–Byni–

v–ρni,

≥ v–ρni,Bρni–Byni–

v–ρni,

,

which implies thatv–p,w ≥. We havep∈B–

  and hencep∈VI(K,B). In a similar way, we can showp∈VI(K,B). Next, let us showp∈

∞

i=F(Ti). Since Hilbert spaces satisfy Opial’s condition, we see from (.) that

lim inf

i→∞ ρni–p <lim infi→∞ ρni–Wp =lim inf

i→∞ ρni–Wρni+Wρni–Wp

≤lim inf

i→∞ Wρni–Wp

≤lim inf

i→∞ ρni–p,

which derives a contradiction. Thus, we havep∈F(W) =∞_i_=F(Ti). From Lemma ., we see that there exists a uniquex∗such thatx∗=PF(γf + (I–A))x∗. It follows that

lim sup

n→∞

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That is, (.) holds. It follows from Lemma . that

xn+–x∗ 

≤αn

γf(xn) –Ax∗

+ (I–αnA)

Wnρn–x∗



≤( –αnγ¯)Wnρn–x∗+ αn

γf(xn) –Ax∗,xn+–x∗

≤( –αnγ¯)xn–x∗+αγ αnxn–x∗+xn+–x∗

+ αn

γfx∗–Ax∗,xn+–x∗

.

It follows that

xn+–x∗≤

( –αnγ¯)+αnγ α  –αnγ α

xn–x∗+ αn  –αnγ α

= ( – αnγ¯+αnαγ)  –αnγ α

xn–x∗+

α_nγ¯

 –αnγ α

xn–x∗

+ αn  –αnγ α

≤

 –αn(γ¯–αγ)  –αnγ α

xn–x∗



+αn(γ¯–αγ)  –αnγ α



¯

γ–αγ

+ αnγ¯ 

(γ¯–αγ)M

,

whereMis an appropriate constant such thatM≥supn≥xn–x∗. Putbn=α_–n(_αγ¯_n–_αγαγ) andcn=_γ_¯_–_αγγf(x∗) –Ax∗,xn+–x∗+ αnγ¯



(γ¯–αγ)M. That is,

xn+–x∗ 

≤( –bn)xn–x∗+bncn. (.)

In view of the restrictions (a) and (b), we see from (.) that

lim

n→∞bn= , ∞

n=

bn=∞ and lim sup n→∞

cn≤.

Apply Lemma . to (.) to conclude that xn→x∗ as n→ ∞. This completes the

proof.

IfB= , the zero mapping, then Theorem . is reduced to the following.

Corollary . Let K be a nonempty,closed,and convex subset of a real Hilbert space H.Let B:K→H beμ-inverse-strongly monotone mappings and f :K→K be anα-contraction.

Let A:K→K be a strongly positive linear bounded self-adjoint operator with the coeﬃ-cientγ¯ > .Assume that <γ <γ¯/α.Let{xn} be a sequence generated in the following

iterative algorithm:

⎧ ⎨ ⎩

x∈K,

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where PKis the metric projection from H onto K,Wnis a mapping deﬁned by(.),{αn}is

a real number sequence in(, ),and{λn}is a positive real number sequence.Assume that

F=∞_i_=F(Ti)∩VI(K,B)=∅and the following restrictions are satisﬁed:

(a) limn→∞αn= , ∞n=αn=∞,and ∞n=|αn+–αn| ≤ ∞;

(b) ∞_n_=|λn+–λn|<∞;

(c) {λn} ⊂[u,v],where <u<v< μ.

Remark . Corollary . includes the corresponding results in Iiduka and Takahashi [] as a special case.

As an application of our main results, we consider another class of important nonlinear operators: strict pseudocontractions.

Recall that a mappingS:K→Kis said to be aκ-strict pseudocontraction if there exists a constantκ∈[, ) such that

Sx–Sy≤ x–y+κ(I–S)x– (I–S)y, ∀x,y∈K.

It is easy to see that the class ofκ-strict pseudocontractions strictly includes the class of nonexpansive mappings as a special case.

PuttingB=I–S, whereS:K→Kis aκ-strict pseudocontraction, we know thatBis –κ

 -inverse-strongly monotone; see [] and the references therein.

Corollary . Let H be a real Hilbert space and K be a nonempty closed convex subset of H.

Let Si:K→K beκi-inverse-strongly monotone mappings for each i= , and f:K→K be

anα-contraction.Let A:K→K be a strongly positive linear bounded self-adjoint operator with the coeﬃcientγ¯> .Assume that <γ <γ¯/α.Let{xn}be a sequence generated in the

following iterative process:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

x∈K,

yn= ( –ηn)xn+ηnSxn,

xn+=PK(αnγf(xn) + (I–αnA)Wn(( –λn)yn+λnSyn)), n≥,

where PK is the metric projection from H onto K,Wnis a mapping deﬁned by(.),{αn}

is a real number sequence in(, ),and{λn},{ηn}are two positive real number sequences.

Assume that F=∞_i_=F(Ti)∩F(S)∩F(S)=∅and the following restrictions are satisﬁed:

(a) lim_n_→∞αn= , ∞n=αn=∞,and ∞n=|αn+–αn| ≤ ∞;

(b) ∞_n_=|ηn+–ηn|<∞, ∞n=|λn+–λn|<∞;

(c) {ηn},{λn} ⊂[u,v],where <u<v< min{μ,μ}.

Proof PutB=I–SandB=I–S. ThenBis ( –κ)/-inverse-strongly monotone and

Bis (–κ)/-inverse-strongly monotone, respectively. We haveF(S) =VI(K,B),F(S) =

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Competing interests

The authors declare that they have no competing interests. Authors’ contributions

Both authors participated in the design of this work and performed equally. Both authors read and approved the ﬁnal manuscript.

Author details

1_{Department of Mathematics, Hangzhou Normal University, Hangzhou, 310036, China.}2_{Department of Mathematics,}

School of Science, Beijing Jiaotong University, Beijing, 100044, China.3_{Department of Mathematics, Shijiazhuang}

University, Shijiazhuang, 050035, China. Acknowledgements

This research was supported by the Natural Science Foundation of Hebei Province (A2010001943), the Science Foundation of Shijiazhuang Science and Technology Bureau (121130971) and the Science Foundation of Beijing Jiaotong University (2011YJS075).

Received: 31 December 2012 Accepted: 21 February 2013 Published: 25 March 2013 References

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doi:10.1186/1687-1812-2013-67