Projection methods of iterative solutions in Hilbert spaces

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R E S E A R C H

Open Access

Projection methods of iterative solutions in

Hilbert spaces

Feng Gu and Jing Lu

* *_{Correspondence:}

[email protected]; [email protected]

Department of Mathematics, Hangzhou Normal University, Hangzhou, 310036, China

Abstract

In this paper, zero points of the sum of two monotone mappings, solutions of a monotone variational inequality, and ﬁxed points of a nonexpansive mapping are investigated based on a hybrid projection iterative algorithm. Strong convergence of the purposed iterative algorithm is obtained in the framework of real Hilbert spaces without any compact assumptions.

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

Keywords: ﬁxed point; monotone operator; nonexpansive mapping; variational inequality; zero point

1 Introduction and preliminaries

Throughout this paper, we always assume thatH is a real Hilbert space with an inner product·,· and a norm · . LetC be a nonempty, closed, and convex subset of H. LetS:C→C be a nonlinear mapping.F(S) stands for the ﬁxed point set ofS; that is, F(S) :={x∈C:x=Tx}.

Recall thatSis said to benonexpansiveiﬀ

Sx–Sy ≤ x–y, ∀x,y∈C.

IfCis a bounded, closed, and convex subset ofH, thenF(S) is not empty, closed, and convex; see [].

LetA:C→H be a mapping. 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. Ais said to bemonotoneiﬀ

Ax–Ay,x–y ≥, ∀x,y∈C.

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

Ax,y–x ≥, ∀y∈C. (.)

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In this paper, we useVI(C,A) to denote the solution set of (.). It is known thatx∈Cis a solution of (.) if and only ifxis a ﬁxed point of the mappingProj_C(I–rA), wherer>  is a constant,Istands for the identity mapping, andProj_C stands for the metric projection fromH ontoC. IfAisα-inverse-strongly monotone andr∈(, α], then the mapping Proj_C(I–rA) is nonexpansive; see [] for more details. It follows thatVI(C,A) is closed and convex.

Monotone variational inequality theory has emerged as a fascinating branch of math-ematical and engineering sciences with a wide range of applications in industry, finance, economics, ecology, social, regional, pure, and applied sciences. In recent years, much at-tention has been given to developing efficient numerical methods for treating solution problems of monotone variational inequality. The gradient-projection method is a pow-erful tool for solving constrained convex optimization problems and has extensively been studied; see [–] and the references therein. It has recently been applied to solving split feasibility problems which find applications in image reconstructions and the intensity modulated radiation theory; see [–] and the references therein. However, the gradient-projection method requires the operator to be strongly monotone and Lipschitz con-tinuous. These strong conditions rule out many applications. Extra gradient-projection method which was first introduce by Korpelevich [] in the finite dimensional Euclidean space has been studied recently for relaxing the strong monotonicity of operators; see [– ] and the references therein.

Recall that a set-valued mappingM:H⇒H is said to bemonotoneiﬀ, for allx,y∈H, f ∈Mxandg∈Myimplyx–y,f –g> . A monotone mappingM:H⇒Hismaximal iﬀ the graphGraph(M) ofRis not properly contained in the graph of any other monotone mapping. It is 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 operatorMonHandr> , 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(J

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

{x∈H: ∈Mx}.

Recently, variational inequalities, ﬁxed point problems, and zero point problems have been investigated by many authors based on iterative methods; see, for example, [–] and the references therein. In this paper, zero point problems of the sum of a maximal monotone operator and an inverse-strongly monotone mapping, solution problems of a monotone variational inequality, and ﬁxed point problems of a nonexpansive mapping are investigated. A hybrid iterative algorithm is considered for analyzing the convergence of iterative sequences. Strong convergence theorems are established in the framework of real Hilbert spaces without any compact assumptions.

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

Lemma . Let C be a nonempty,closed,and convex subset of H.Then the following in-equality holds:

x–Proj_Cx+y–Proj_C≤ x–y, ∀x∈H,y∈C.

Lemma .[] Let C be a nonempty,closed,and convex subset of H.Let S:C→C be a nonexpansive mapping.Then the mapping I–S is demiclosed at zero,that is,if{xn}is a

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Lemma . Let C be a nonempty,closed,and convex subset of H,B:C→H be a mapping, and M:H⇒H be a maximal monotone operator.Then F(Jr(I–sB)) = (B+M)–().

Proof Notice that

p∈FJr(I–sB)

⇐⇒ p=Jr(I–sB)p ⇐⇒ p–sBp∈p+sMp

⇐⇒ ∈(B+M)–() ⇐⇒ p∈(B+M)–().

This completes the proof.

Lemma .[] Let C be a nonempty,closed,and convex subset of H,A:C→H be a Lipschitz monotone mapping,and 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.

Then W is maximal monotone and∈Wx if and only if x∈VI(C,A).

2 Main results

Now, we are in a position to give our main results.

Theorem . Let C be a nonempty,closed,and convex subset of H.Let S:C→C be a nonexpansive mapping with a nonempty ﬁxed point set,A:C→H be anα-Lipschitz con-tinuous and monotone mapping,and B:C→H be aβ-inverse-strongly monotone map-ping.Let M:H⇒H be a maximal monotone operator such that D(M)⊂C.Assume that

F:=F(S)∩(B+M)–_()_∩_VI_(C,_A)_{is not empty.}_Let_{_x

n}be a sequence generated by the

following iterative process:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

x∈C,

C=C,

zn=ProjC(Jsn(xn–snBxn) –rnAJsn(xn–snBxn)),

yn=αnxn+ ( –αn)SProjC(Jsn(xn–snBxn) –rnAzn),

Cn+={v∈Cn:yn–v ≤ xn–v},

xn+=ProjCn+x, n≥,

(.)

where Jsn= (I+snM)–,{rn}is a sequence in(,α),{sn}is a sequence in(, β),and{αn}is a sequence in(, ).Assume that the following restrictions are satisﬁed:

(a)  <a≤rn≤b<α; (b)  <c≤sn≤d< β;

(c) ≤αn≤e< ,

where a, b, c, d, and e are real constants.Then the sequence{xn} converges strongly to

Proj_Fx.

Proof First, we show thatCnis closed and convex for eachn≥. From the assumption,

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m≥. We show thatCm+ is closed and convex for the samem. Letv,v∈Cm+ and v=tv+ ( –t)v, wheret∈(, ). Notice that

ym–v ≤ xm–v

is equivalent to

ym–xm– v,ym–xm ≥.

It is clear thatv∈Cm+. This shows thatCnis closed and convex for eachn≥.

Next, we show thatF⊂Cnfor eachn≥. Putun=ProjC(vn–rnAzn), wherevn=Jsn(xn–

snBxn). From the assumption, we see thatF⊂C=C. Suppose thatF⊂Cm for some

m≥. For anyp∈F⊂Cm, we see from Lemma . that

um–p≤ vm–rmAzm–p–vm–rmAzm–um

=vm–p–vm–um+ rmAzm,p–um

=vm–p–vm–um+ rm

Azm–Ap,p–zm+Ap,p–zm

+Azm,zm–um

≤ vm–p–vm–zm+zm–um+ rmAzm,zm–um

=vm–p–vm–zm–zm–um

+ vm–zm–rmAzm,um–zm. (.)

Notice thatAis Lipschitz continuous. In view ofzm=ProjC(vm–rmAvm), we ﬁnd that

vm–zm–rmAzm,um–zm

=vm–zm–rmAvm,um–zm+rmAvm–rmAzm,um–zm

≤rmαvm–zmum–zm. (.)

Substituting (.) into (.), we obtain that

um–p≤ vm–p–vm–zm–zm–um+ rmαvm–zmum–zm

≤ vm–p–

 –r_mαvm–zm. (.)

This in turn implies from restriction (a) that

ym–p≤αmxm–p+ ( –αm)Sum–p

≤αmxm–p+ ( –αm)um–p

≤αmxm–p+ ( –αm)

vm–p–

 –r_mαvm–zm

≤ xm–p– ( –αm)

 –r_mαvm–zm

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This shows thatp∈Cm+. This proves thatF⊂Cnfor eachn≥. Notexn=ProjCnx. For

eachp∈F⊂Cn, we havex–xn ≤ x–p. SinceBis inverse-strongly monotone, we

see from Lemma . that (B+M)–() is closed and convex. SinceAis Lipschitz continu-ous, we ﬁnd thatVI(C,A) is close and convex. This proves thatFis closed and convex. It follows that

x–xn ≤ x–Proj_Fx. (.)

This implies that{xn}is bounded. Sincexn=ProjCnxandxn+=ProjCn+x∈Cn+⊂Cn,

we have

≤ x–xn,xn–xn+

=x–xn,xn–x+x–xn+

≤–x–xn+x–xnx–xn+.

It follows that

xn–x ≤ xn+–x.

This proves thatlim_n_→∞xn–xexists. Notice that

xn–xn+

=xn–x+ xn–x,x–xn++x–xn+

=xn–x– xn–x+ xn–x,xn–xn++x–xn+

≤ x–xn+–xn–x.

It follows that

lim

n→∞xn–xn+= . (.)

In view ofxn+=ProjCn+x∈Cn+, we see that

yn–xn+ ≤ xn–xn+.

This implies that

yn–xn ≤ yn–xn++xn–xn+ ≤xn–xn+.

From (.), we ﬁnd that

lim

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SinceBisβ-inverse-strongly monotone, we see from restriction (b) that

(I–snB)x– (I–snB)y



=x–y– snx–y,Bx–By+snBx–By

≤ x–y–sn(β–sn)Bx–By

≤ x–y, ∀x,y∈C.

This implies from (.) that

yn–p≤αnxn–p+ ( –αn)vn–p

=αnxn–p+ ( –αn)Jsn(xn–snBxn) –Jsn(p–snBp)



≤ xn–p– ( –αn)sn(β–sn)Bxn–Bp.

It follows that

( –αn)sn(β–sn)Bxn–Bp≤ xn–p–yn–p

≤ xn–yn

xn–p+yn–p

.

In view of restrictions (b) and (c), we ﬁnd from (.) that

lim

n→∞Bxn–Bp= . (.)

SinceJsnis ﬁrmly nonexpansive, we ﬁnd that

vn–p=Jsn(xn–snBxn) –Jsn(p–snBp)



≤ vn–p, (xn–snBxn) – (p–snBp)

= 

vn–p+(xn–snBxn) – (p–snBp)

–(vn–p) –

(xn–snBxn) – (p–snBp)

≤



vn–p+xn–p–vn–xn+sn(Bxn–Bp)



= 

vn–p+xn–p–vn–xn–snBxn–Bp

– snvn–xn,Bxn–Bp

≤



vn–p+xn–p–vn–xn+ snvn–xnBxn–Bp

.

This in turn implies that

vn–p≤ xn–p–vn–xn+ snvn–xnBxn–Bp. (.)

Combining (.) with (.), we arrive at

yn–p≤αnxn–p+ ( –αn)vn–p

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It follows that

( –αn)vn–xn≤ xn–p–yn–p+ snvn–xnBxn–Bp

≤ xn–yn

xn–p+yn–p

+ snvn–xnBxn–Bp.

In view of (.) and (.), we see from restriction (c) that

lim

n→∞vn–xn= . (.)

On the other hand, we ﬁnd from (.) that

( –αn)

 –r_nαvn–zn≤ xn–p–yn–p

≤ xn–yn

xn–p+yn–p

.

In view of restrictions (a) and (c), we obtain from (.) that

lim

n→∞vn–zn= . (.)

Notice that

un–zn=PC(vn–rnAzn) –PC(vn–rnAvn)



≤(vn–rnAzn) – (vn–rnAvn)



≤r_nαzn–vn.

Thanks to (.), we arrive at

lim

n→∞un–zn= . (.)

Notice that

xn–Sxn ≤ xn–Sun+Sun–Sxn

≤xn–yn

 –αn

+un–xn

≤xn–yn

 –αn

+un–zn+zn–vn+vn–xn.

In view of (.), (.), (.), and (.), we ﬁnd from restriction (c) that

lim

n→∞xn–Sxn= .

Since{xn}is bounded, we ﬁnd that there exists a subsequence{xni}of{xn}such thatxni

q∈C. From Lemma ., we easily conclude thatq∈F(S). Now, we are in a position to show thatx∈VI(C,A). Deﬁne

Wx=

⎧ ⎨ ⎩

Ax+NCx, x∈C,

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For any given (x,y)∈G(W), we havey–Ax∈NCx. Sinceun∈C, we see from the deﬁnition

ofNC

x–un,y–Ax ≥. (.)

In view ofun=PC(vn–rnAzn), we obtain that

x–un,un+rnAzn–vn ≥

and hence

x–un,

un–vn

rn

+Azn

≥. (.)

In view of (.) and (.), we ﬁnd that

x–uni,y ≥ x–uni,Ax

≥ x–uni,Ax–

x–uni,

uni–vni

rni

+Azni

=x–uni,Ax–Auni+x–uni,Auni–Azni

–

x–uni,

uni–vni

rni

≥ x–uni,Auni–Azni–

x–uni,

uni–vni

rni

. (.)

Notice that

xn–un ≤ xn–vn+vn–zn+zn–un.

It follows from (.), (.), and (.) that

lim

n→∞xn–un= .

SinceAis Lipschitz continuous, we ﬁnd from (.) that

x–q,y ≥.

Since W is maximal monotone, we conclude that q∈ W–(). This proves that q ∈ VI(C,A).

Finally, we prove thatq∈(B+M)–_{(). Notice that}

xn–snBxn∈vn+snMvn;

that is,

xn–vn

sn

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Letμ∈Mν. SinceMis monotone, we ﬁnd from (.)

xn–vn

sn

–Bxn–μ,vn–ν

≥.

In view of the restriction (b), we see that

–Bq–μ,q–ν ≥.

This implies that –Bq∈Mq, that is,q∈(B+M)–_{(). This completes}_q_∈_F_{. Assume that} there exists another subsequence{xnj}of{xn}which converges weakly toq∈F. We can

easily conclude from Opial’s condition thatq=q.

Finally, we show thatq=Proj_Fx and{xn}converges strongly toq. This completes the

proof of Theorem .. In view of the weak lower semicontinuity of the norm, we obtain from (.) that

x–Proj_Fx ≤ x–q ≤lim inf

n→∞ x–xn

≤lim sup

n→∞ x–xn ≤ x–ProjFx,

which yields thatlimn→∞x–xn=x–Proj_Fx=x–q. It follows that{xn}

con-verges strongly toProj_Fx. This completes the proof.

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

Corollary . Let C be a nonempty,closed,and convex subset of H.Let S:C→C be a nonexpansive mapping with a nonempty ﬁxed point set and A:C→H be anα-Lipschitz continuous and monotone mapping.Let M:H⇒H be a maximal monotone operator such that D(M)⊂C.Assume thatF :=F(S)∩M–_()_∩_VI_(C,_A)_{is not empty.}_Let_{_x

n}be a

sequence generated by the following iterative process:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

x∈C,

C=C,

zn=ProjC(Jsnxn–rnAJsnxn),

yn=αnxn+ ( –αn)SProjC(Jsnxn–rnAzn),

where Jsn= (I+snM)–,{rn}is a sequence in(,_α),{sn}is a sequence in(, +∞),and{αn}

is a sequence in(, ).Assume that the following restrictions are satisﬁed: (a)  <a≤rn≤b<α;

(b)  <c≤sn<∞;

(c) ≤αn≤d< ,

where a,b,c,and d are real constants.Then the sequence{xn}converges strongly toProjFx.

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Corollary . Let C be a nonempty,closed,and convex subset of H.Let S:C→C be a nonexpansive mapping with a nonempty ﬁxed point set and A:C→H be anα-Lipschitz continuous and monotone mapping.Assume thatF:=F(S)∩VI(C,A)is not empty.Let

{xn}be a sequence generated by the following iterative process: ⎧

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

x∈C,

C=C,

zn=ProjC(xn–rnAxn),

yn=αnxn+ ( –αn)SProjC(xn–rnAzn),

xn+=ProjCn+x, n≥,

where{rn}is a sequence in(,α),and{αn}is a sequence in(, ).Assume that the following restrictions are satisﬁed:

(a)  <a≤rn≤b<α; (b) ≤αn≤c< ,

where a,b,and c are real constants.Then the sequence{xn}converges strongly toProj_Fx.

IfA= , then Theorem . is reduced to the following.

Corollary . Let C be a nonempty,closed,and convex subset of H.Let S:C→C be a non-expansive mapping with a nonempty ﬁxed point set and B:C→H be aβ-inverse-strongly monotone mapping.Let M:H⇒H be a maximal monotone operator such that D(M)⊂C. Assume thatF:=F(S)∩(B+M)–_()_{is not empty.}_Let_{_x

n}be a sequence generated by the

following iterative process:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

x∈C,

C=C,

yn=αnxn+ ( –αn)SJsn(xn–snBxn),

where Jsn= (I+snM)–,{sn}is a sequence in(, β),and{αn}is a sequence in(, ).Assume

that the following restrictions are satisﬁed: (a)  <a≤sn≤b< β;

(b) ≤αn≤c< ,

Letf :H→(–∞, +∞] be a proper convex lower semicontinuous function. Then the subdiﬀerential∂f off is deﬁned as follows:

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

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LetICbe the indicator function ofC,i.e.,

IC(x) = ⎧ ⎨ ⎩

, x∈C,

+∞, x∈/C.

SinceICis a proper lower semicontinuous convex function onH, we see that the

subdiﬀer-ential∂ICofICis a maximal monotone operator. It is clear thatJsx=PCx,∀x∈H. Notice

that (B+∂IC)–() =VI(C,B). Indeed,

x∈(B+∂IC)–() ⇐⇒ ∈Bx+∂ICx

⇐⇒ –Bx∈∂ICx

⇐⇒ Bx,y–x ≥

⇐⇒ x∈VI(C,B). (.)

In the light of the above, the following is not hard to derive from Corollary ..

Corollary . Let C be a nonempty,closed,and convex subset of H.Let S:C→C be a nonexpansive mapping with a nonempty ﬁxed point set and B:C→H be aβ -inverse-strongly monotone mapping.Assume thatF:=F(S)∩VI(C,B)is not empty.Let{xn}be a

sequence generated by the following iterative process:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

x∈C,

C=C,

yn=αnxn+ ( –αn)SPC(xn–snBxn),

where{sn}is a sequence in(, β),and{αn}is a sequence in(, ).Assume that the following

restrictions are satisﬁed: (a)  <a≤sn≤b< β;

(b) ≤αn≤c< ,

3 Applications

First, we consider the problem of ﬁnding a minimizer of a proper convex lower semicon-tinuous function.

Theorem . Let f :H→(–∞, +∞]be a proper convex lower semicontinuous function

such that(∂f)–_()_{is not empty.}_Let_{_x

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process:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

x∈H,

C=H,

yn=arg mins∈H{f(z) +z–xn 

sn },

where{sn}is a positive sequence such that <a≤sn,where a is a real constant.Then the

sequence{xn}converges strongly toProj(∂f)–_()x.

Proof PuttingA=B= ,S=I, andαn≡, we can immediately draw the desired

conclu-sion from Theorem ..

Second, we consider the problem of approximating a common ﬁxed point of a pair of nonexpansive mappings.

Theorem . Let C be a nonempty,closed,and convex subset of H.Let S:C→C and T:C→C be a pair of nonexpansive mappings with a nonempty common ﬁxed point set. Let{xn}be a sequence generated by the following iterative process:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

x∈C,

C=C,

zn= ( –sn)xn+snTxn,

yn=αnxn+ ( –αn)Szn,

where{sn},and{αn}are sequences in(, ).Assume that the following restrictions are

sat-isﬁed:

(a)  <a≤sn≤b< ;

(b) ≤αn≤c< ,

where a, b, and c are real constants. Then the sequence {xn} converges strongly to

Proj_F₍_S₎_∩_F₍_T₎x.

Proof PuttingA= ,M=∂IC, andB=I–T, we see thatBis_-inverse-strongly monotone.

We also haveF(T)=VI(C,B) andPC(xn–snBxn) = ( –sn)xn+snTxn. In view of (.), we

can immediately obtain the desired result.

LetFbe a bifunction ofC×CintoR, whereRdenotes the set of real numbers. Recall the following equilibrium problem in the terminology of Blum and Oettli [] (see also Fan []):

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To study the equilibrium problem (.), we may assume thatF satisﬁes 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.

PuttingF(x,y) =Ax,y–xfor everyx,y∈C, we see that the equilibrium problem (.) is reduced to the variational inequality (.).

The following lemma can be found in [] and [].

Lemma . Let C be a nonempty,closed,and convex subset of H and F:C×C→Rbe a bifunction satisfying(A)-(A).Then,for any s> and x∈H,there exists z∈C such that

F(z,y) +

sy–z,z–x ≥, ∀y∈C.

Further,deﬁne

Tsx=

z∈C:F(z,y) +

sy–z,z–x ≥,∀y∈C

(.)

for all s> and x∈H.Then,the following hold: (a) Tsis single-valued;

(b) Tsis ﬁrmly nonexpansive;that is,

Tsx–Tsy≤ Tsx–Tsy,x–y, ∀x,y∈H;

(c) F(Ts) =EP(F);

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

Lemma .[] Let C be a nonempty,closed,and convex subset of H,F be a bifunction from C×C toRwhich satisﬁes(A)-(A),and AF be a multivalued mapping of H into

itself deﬁned by

AFx= ⎧ ⎨ ⎩

{z∈H:F(x,y)≥ y–x,z,∀y∈C}, x∈C,

∅, x∈/C.

(.)

Then AF is a maximal monotone operator with the domain D(AF)⊂C,EP(F) =A–F (),

where FP(F)stands for the solution set of(.),and

Tsx= (I+sAF)–x, ∀x∈H,r> ,

where Tsis deﬁned as in(.).

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Theorem . Let C be a nonempty,closed,and convex subset of H.Let F:C×C→Rbe a bifunction satisfying(A)-(A)such that EP(F)=∅.Let{xn}be a sequence generated by

the following iterative process:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

x∈C,

C=C,

yn= (I+snAF)–xn,

where AFis deﬁned as(.),and{sn}is a positive sequence such that <a≤sn<∞,where

a is a real constant Then the sequence{xn}converges strongly toProjFP(F)x.

Proof PuttingA=B= ,S=Iandαn≡, we immediately reach the desired conclusion

from Lemma ..

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors contributed equally to this work. Both authors read and approved the ﬁnal manuscript.

Author details

Department of Mathematics, Hangzhou Normal University, Hangzhou, 310036, China.

Authors’ information

Author’s information

Acknowledgements

The ﬁrst author was supported by the National Natural Science Foundation of China (11071169, 11271105), the Natural Science Foundation of Zhejiang Province (Y6110287, Y12A010095).

Received: 27 June 2012 Accepted: 6 September 2012 Published: 25 September 2012 References

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doi:10.1186/1687-1812-2012-162