Iterative algorithms for monotone inclusion problems, fixed point problems and minimization problems

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R E S E A R C H

Open Access

Iterative algorithms for monotone inclusion

problems, ﬁxed point problems and

minimization problems

Jong Soo Jung

*

*_{Correspondence: [email protected];}

[email protected] Department of Mathematics, Dong-A University, Busan, 604-714, Korea

Abstract

We introduce new implicit and explicit iterative schemes for finding a common element of the set of fixed points ofk-strictly pseudocontractive mapping and the set of zeros of the sum of two monotone operators in a Hilbert space. Then we establish strong convergence of the sequences generated by the proposed schemes to a common point of two sets, which is a solution of a certain variational inequality. Further, we find the unique solution of the quadratic minimization problem, where the constraint is the common set of two sets mentioned above. As applications, we consider iterative schemes for the Hartmann-Stampacchia variational inequality problem and the equilibrium problem coupled with fixed point problem.

MSC: 47H05; 47H09; 47H10; 47J05; 47J07; 47J25; 47J20; 49M05

Keywords: maximal monotone operator; inverse-strongly monotone operator; strictly pseudocontractive mapping; ﬁxed points; variational inequality; zeros; minimum norm problem

1 Introduction

LetHbe a real Hilbert space with the inner product·,·and the induced norm · . Let Cbe a nonempty closed convex subset ofH, and letT :C→Cbe a self-mapping onC. We denote byF(T) the set of ﬁxed points ofT, that is,F(T) :={x∈C:Tx=x}.

LetA:C→Hbe a single-valued nonlinear mapping, and letB:H→H_{be a}

multival-ued mapping. Then we consider the monotone inclusion problem (MIP) of ﬁndingx∈H such that

∈Ax+Bx. (.)

The set of solutions of the MIP (.) is denoted by (A+B)–. That is, (A+B)– is the set of zeros ofA+B. The MIP (.) provides a convenient framework for studying a number of problems arising in structural analysis, mechanics, economics and others; see, for instance [, ]. Also, various types of inclusion problems have been extended and generalized, and there are many algorithms for solving variational inclusions. For more details, see [–] and the references therein.

The class of pseudocontractive mappings is one of the most important classes of map-pings among nonlinear mapmap-pings. We recall that a mappingT :C→H is said to be

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k-strictly pseudocontractiveif there exists a constantk∈[, ) such that

Tx–Ty≤ x–y+k(I–T)x– (I–T)y, ∀x,y∈C.

Note that the class ofk-strictly pseudocontractive mappings includes the class of non-expansive mappings as a subclass. That is,T is nonexpansive (i.e.,Tx–Ty ≤ x–y,

∀x,y∈C) if and only ifTis -strictly pseudocontractive. The mappingTis also said to be pseudocontractive ifk= , andT is said to be strongly pseudocontractive if there exists a constantλ∈(, ) such thatT–λIis pseudocontractive. Clearly, the class ofk-strictly pseudocontractive mappings falls into the one between classes of nonexpansive mappings and pseudocontractive mappings. Also, we remark that the class of strongly pseudocon-tractive mappings is independent of the class ofk-strictly pseudocontractive mappings (see []). Recently, many authors have been devoting the studies on the problems of ﬁnding ﬁxed points for pseudocontractive mappings (see, for example, [–] and the references therein).

Recently, in order to study the MIP (.) coupled with the fixed point problem, many au-thors have introduced some iterative schemes for finding a common element of the set of solutions of the MIP (.) and the set of fixed points of a countable family of nonexpansive mappings (see [, , ] and the references therein).

Inspired and motivated by the above-mentioned recent works, in this paper, we intro-duce new implicit and explicit iterative schemes for finding a common element of the set of the solutions of the MIP (.) with a set-valued maximal monotone operatorBand an inverse-strongly monotone mappingAand the set of fixed points of ak-strictly pseudo-contractive mappingT. Then we establish results of the strong convergence of the se-quences generated by the proposed schemes to a common point of two sets, which is a solution of a certain variational inequality. As a direct consequence, we find the unique solution of the quadratic minimization problem:

˜x=minx:x∈F(T)∩(A+B)–.

Moreover, as applications, we consider iterative algorithms for the Hartmann-Stampacchia variational inequality problem and the equilibrium problem coupled with ﬁxed point problem of nonexpansive mappings.

2 Preliminaries and lemmas

LetH be a real Hilbert space, and letC be a nonempty closed convex subset ofH. In the following, we writexnxto indicate that the sequence{xn}converges weakly tox.

xn→ximplies that{xn}converges strongly tox.

Recall that a mappingf:C→Cis said to becontractiveif there existsl∈[, ) such that

f(x) –f(y)≤lx–y, ∀x,y∈C.

A mappingAofCintoH is calledinverse-strongly monotoneif there exists a positive real numberαsuch that

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for all x,y∈C. For such a case, A is called α-inverse-strongly monotone. If A is an α-inverse-strongly monotone mapping of C into H, then it is obvious that A is _α -Lipschitzian and continuous. LetBbe a mapping ofHinto H_{. The eﬀective domain of}_B

is denoted bydom(B), that is,dom(B) ={x∈H:Bx=∅}. A multi-valued mappingBis said to be amonotone operatoronHifx–y,u–v ≥ for allx,y∈dom(B),u∈Bx, andv∈By. A monotone operatorBonHis said to bemaximalif its graph is not properly contained in the graph of any other monotone operator onH. For a maximal monotone operator BonH andr> , we may deﬁne a single-valued operatorJr= (I+rB)–:H→dom(B),

which is called the resolvent ofBforr. LetBbe a maximal monotone operator onH, and letB–_{ =}_{_x_∈_H_{: }_∈_Bx_}_{. It is well known that}_B–_{ =}_F(J

r) for allr>  and the resolvent

Jris ﬁrmly nonexpansive,i.e.,

Jrx–Jry≤ x–y,Jrx–Jry, ∀x,y∈H, (.)

and that the resolvent identity

Jλx=Jμ

μ λx+

 –μ λ

Jλx

(.)

holds for allλ,μ>  andx∈H. It is worth mentioning that the resolvent operatorJλis

nonexpansive and -inverse-strongly monotone, and that a solution of the MIP (.) is a ﬁxed point of the operatorJλ(I–λA) for allλ>  (see []).

In a real Hilbert spaceH, we have

x–y=x+y– x,y (.)

for allx,y∈Handλ∈R. For every pointx∈H, there exists a unique nearest point inC, denoted byPCx, such that

x–PCx=inf

x–y:y∈C.

PCis called themetric projectionofHontoC. It is well known thatPCis nonexpansive,

andPCis characterized by the property

u=PCx ⇐⇒ x–u,u–y ≥, ∀x∈H,y∈C. (.)

It is also well known thatHsatisﬁes theOpial condition, that is, for any sequence{xn}with

xnx, the inequality

lim inf

n→∞ xn–x<lim infn→∞ xn–y

holds for everyy∈Hwithy=x. For these facts, see [].

We need the following lemmas for the proof of our main results.

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

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Lemma .[] For all x,y,z∈H andα,β,γ ∈[, ]withα+β+γ = ,the following equality holds:

αx+βy+γz=αx+βy+γz–αβx–y–βγy–z–γ αz–x.

Lemma .[] Let H be a Hilbert space,let C be a closed convex subset of H.If T is a k-strictly pseudocontractive mapping on C,then the ﬁxed point set F(T)is closed convex, so that the projection PF(T)is well deﬁned,and F(PCT) =F(T).

Lemma .[] Let H be a real Hilbert space,let C be a closed convex subset of H,and let T:C→H be a k-strictly pseudocontractive mapping.Deﬁne a mapping T:C→H by Sx=λx+ ( –λ)Tx for all x∈C.Then,asλ∈[k, ),S is a nonexpansive mapping such that F(S) =F(T).

Lemma .[] Let C be a nonempty closed convex subset of a real Hilbert space H.Let the mapping A:C→H beα-inverse strongly monotone,and let r> be a constant.Then we have

(I–rA)x– (I–rA)y≤ x–y+r(r– α)Ax–Ay, ∀x,y∈C.

In particular,if≤r≤α,then I–rA is nonexpansive.

Lemma .[] Let B:H→H_{be a maximal monotone operator,}_{and let A}_:_H_→_{H be a}

Lipschitz continuous mapping.Then the mapping B+A:H→H_{is a maximal monotone}

operator.

Remark . Lemma . implies that (A+B)–_{ is closed and convex if}_B_:_H_→_H_{is a}

maximal monotone operator andA:H→His an inverse-strongly monotone mapping.

The following lemma is a variant of a Minty lemma (see []).

Lemma . Let C be a nonempty closed convex subset of a real Hilbert space H.Assume that the mapping G:C→H is monotone and weakly continuous along segments,that is, G(x+ty)→G(x)weakly as t→.Then the variational inequality

˜

x∈C, Gx,˜ p–x˜ ≥, ∀p∈C,

is equivalent to the dual variational inequality

˜

x∈C, Gp,p–x˜ ≥, ∀p∈C.

Lemma .[] Let{xn}and{zn}be bounded sequences in a real Banach space E,and let {γn}be a sequence in[, ],which satisﬁes the following condition:

 <lim inf

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Suppose that xn+=γnxn+ ( –γn)znfor all n≥,and

lim sup

n→∞

zn+–zn–xn+–xn

≤.

Thenlim_n→∞zn–xn= .

Lemma .[] Let{sn}be a sequence of non-negative real numbers satisfying

sn+≤( –ξn)sn+ξnδn, ∀n≥,

where{ξ}and{δn}satisfy the following conditions:

(i) {ξn} ⊂[, ]and ∞n=ξn=∞;

(ii) lim sup_n→∞δn≤or ∞n=ξnδn<∞.

Thenlim_n→∞sn= .

3 Iterative schemes

Throughout the rest of this paper, we always assume as follows: LetH be a real Hilbert space, and letCbe a nonempty closed convex subset ofH. LetA:C→Hbe anα-inverse strongly monotone mapping, and letBbe a maximal monotone operator onHsuch that the domain ofBis included inC. LetJλ= (I+λB)–be the resolvent ofBforλ> . Letf :

C→Cbe a contractive mapping with constantl∈(, ), and letT:C→Cbe ak-strictly pseudocontractive mapping. Deﬁne a mappingS:C→CbySx=λx+ ( –λ)Tx,∀x∈C, whereλ∈[k, ). Then, by Lemma .,Sis nonexpansive.

In this section, we introduce the following iterative scheme that generates a net{xt}in

an implicit way:

xt=tf(xt) + ( –t)SJλt(xt–λtAxt), t∈(, ), (.)

where  <a≤λt≤b< α. We prove strong convergence of{xt}, ast→, to a pointx˜in

F(T)∩(A+B)–_{, which is a solution of the following variational inequality:}

(I–f)x,˜ p–x˜≥, ∀p∈F(T)∩(A+B)–. (.)

Equivalently,x˜=P_F(T)∩(A+B)–_(I–f)x.˜ If we takef≡ in (.), then we have

xt= ( –t)SJλt(xt–λtAxt), t∈(, ). (.)

We also propose the following iterative scheme which generates a sequence{xn}in an

explicit way:

xn+=αnf(xn) +βnxn+ ( –αn–βn)SJλn(xn–λnAxn), ∀n≥, (.)

where{αn},{βn} ⊂(, ),{λn} ⊂(, α) andx∈Cis an arbitrary initial guess, and

estab-lish the strong convergence of this sequence to a ﬁxed pointx˜ofT, which is also a solution of the variational inequality (.). If we takef ≡ in (.), then we have

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3.1 Strong convergence of the implicit algorithm

Fort∈(, ), consider the following mappingQtonCdeﬁned by

Qtx=tf(x) + ( –t)SJλt(x–λtAx), ∀x∈C.

By Lemma ., we have

Qtx–Qty

=tf(x) + ( –t)SJλt(x–λtAx) –

tf(y) + ( –t)SJλt(y–λtAy)

≤tf(x) –f(y)+ ( –t)SJλt(x–λtAx) –SJλt(y–λtAy) ≤tlx–y+ ( –t)(I–λtA)x– (I–λtA)y

≤tlx–y+ ( –t)x–y

= – ( –l)tx–y.

Since  <  – ( –l)t< ,Qtis a contractive mapping. Therefore, by the Banach

contrac-tion principle,Qt has a unique ﬁxed pointxt∈C, which uniquely solves the ﬁxed point

equation

xt=tf(xt) + ( –t)SJλt(xt–λtAxt), t∈(, ).

Now, we prove strong convergence of the sequence{xt}, and show the existence ofx˜∈

F(T)∩(A+B)–_{, which solves the variational inequality (.).}

Theorem . Suppose that F(T)∩(A+B)–_._{Then the net}_{_x

t}deﬁned by the implicit

method(.)converges strongly,as t→,to a pointx˜∈F(T)∩(A+B)–_,_{which is the}

unique solution of the variational inequality(.).

Proof First, we can show easily the uniqueness of a solution of the variational inequality (.). In fact, ifx˜∈F(T)∩(A+B)–_{ and}_x_ˆ_∈_F(T)_∩_(A₊_B)–_{ both are solutions to (.).}

Then we have

(I–f)x,˜ xˆ–x˜≥, (.)

(I–f)x,ˆ x˜–xˆ≥. (.)

Adding up (.) and (.) yields

(I–f)x˜– (I–f)x,ˆ x˜–xˆ≤.

This implies that ( –l)˜x–xˆ_≤_{. So}_x_˜₌_{x, and the uniqueness is proved. Below, we use}_ˆ ˜

x∈F(T)∩(A+B)– to denote the unique solution of the variational inequality (.). Now, we prove that {xt} is bounded. Set yt =Jλt(xt –λtAxt) for all t∈(, ). Take

p∈F(T)∩(A+B)–_{. It is clear that}_p₌_J

λt(p–λtAp) =SJλt(p–λtAp) andp=Sp(by

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from Lemma . that

yt–p=Jλt(xt–λtAxt) –Jλt(p–λtAp) 

≤xt–λtAxt– (p–λtAp)

≤ xt–p+λt(λt– α)Axt–Ap

≤ xt–p. (.)

So, we have that

yt–p ≤ xt–p. (.)

Moreover, from (.), it follows that

xt–p=tf(xt) + (I–t)SJλt(xt–λtAxt) –p

≤tf(xt) –f(p)+tf(p) –p+ ( –t)Jλt(xt–λtAxt) –p ≤tlxt–p+tf(p) –p+ ( –t)yt–p

≤tlxt–p+tf(p) –p+ ( –t)xt–p

≤ –t( –l)xt–p+tf(p) –p, (.)

that is,

xt–p ≤

f(p) –p  –l .

Hence,{xt}is bounded, and so are{yt},{f(xt)},{Axt}and{Syt}.

From (.) and (.), we have

( –tl)xt–p≤

( –t)yt–p+tf(p) –p

= ( –t)yt–p+tf(p) –p

+ ( –t)tf(p) –pyt–p

≤ yt–p+tM

≤ xt–p+λt(λt– α)Axt–Ap+tM, (.)

whereM>  is an appropriate constant. This means that

a(α–b)Axt–Ap≤λt(α–λt)Axt–Ap

≤tl–tlxt–p+tM→ ast→.

Sincea(α–b) > , we deduce that

lim

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From (.) and (.), we also obtain

yt–p

=Jλt(xt–λtAxt) –Jλt(p–λtAp) 

≤(xt–λtAxt) – (p–λtAp),yt–p

=

(xt–λtAxt) – (p–λtAp)



+yt–p

–(xt–p) –λt(Axt–Ap) – (yt–p) 

≤



xt–p+yt–p–(xt–yt) –λt(Axt–Ap) 

= 

xt–p+yt–p–xt–yt+ λtxt–yt,Axt–Ap–λtAxt–Ap

.

So, we get

yt–p≤ xt–p–xt–yt

+ λtxt–yt,Axt–Ap–λtAxt–Ap. (.)

Since · is a convex function, by (.), we have

xt–p=t

f(xt) –p

+ ( –t)SJλt(xt–λtAxt) –p 

≤tf(xt) –f(p)+f(p) –p 

+ ( –t)Syt–Sp

≤tlxt–p+f(p) –p 

+ ( –t)yt–p

≤tlxt–p+f(p) –p 

+ ( –t)xt–p–xt–yt+ λtxt–yt,Axt–Ap

. (.)

We deduce from (.) that

( –t)xt–yt≤

t+Axt–Ap

M, (.)

whereM>  is an appropriate constant. Sincet→ andAxt–Ap →, we have

lim

t→∞xt–yt= . (.)

Observing that

Syt–xt=Syt–

tf(xt) + ( –t)Syt

=tSyt–f(xt)→ ast→,

by (.), we obtain

Sxt–xt ≤ Sxt–Syt+Syt–xt

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Let{tn} ⊂(, ) be a sequence such thattn→ asn→ ∞. Putxn:=xtn,yn:=ytn and

λn:=λtn. Since{xn}is bounded, there exists a subsequence{xni}of{xn}, which converges

weakly tox.˜

Next, we show thatx˜∈F(T)∩(A+B)–_{. Since}_C_{is closed and convex,} _C_{is weakly}

closed. So, we havex˜∈C. Let us showx˜∈F(T). Assume thatx˜∈/F(T) (=F(S)). Since xnix˜andx˜=Sx, it follows from the Opial condition and (.) that˜

lim inf

i→∞ xni–x˜<lim infi→∞ xni–Sx˜

≤lim inf

i→∞

xni–Sxni+Sxni–Sx˜

≤lim inf

i→∞ xni–x˜,

which is a contradiction. So we getx˜∈F(T). We shall show that x˜∈(A+B)–_{. Since}_x

t–yt → ast→, it follows that{yni}

converges weakly tox. We choose a subsequence˜ {λni}of{λn}such thatλni→λ. Letv∈

Bu. Noting that

yt=Jλt(xt–λtAxt) = (I+λtB)–(xt–λtAxt),

we have that

xt–λtAxt∈yt+λtByt,

and so,

xt–yt

λt

–Axt∈Byt.

SinceBis monotone, we have for (u,v)∈B,

xt–yt

λt

–Axt–v,yt–u

≥. (.)

Sincext–x,˜ Axt–Ax˜ ≥αAxt–Ax˜andxnix, we have˜ Axni→Ax. Then by (.)˜

and (.), we obtain

–Ax˜–v,x˜–u ≥.

SinceBis maximal monotone, this implies that –Ax˜∈Bx, that is, ˜ ∈(A+B)x. Hence, we˜ havex˜∈(A+B)–_{. Thus, we conclude that}_x_˜_∈_F(T)_∩_(A₊_B)–_.

On the one hand, we note that forp∈F(T)∩(A+B)–_,

xt–p=t

f(xt) –f(p)

+tf(p) –p+ ( –t)SJλt(xt–λtAxt) –p

.

Then it follows that

xt–p=xt–p,xt–p

=tf(xt) –f(p)

,xt–p

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+ ( –t)SJλt(xt–λtAxt) –p,xt–p

≤tlxt–p+t

f(p) –p,xt–p

+ ( –t)xt–p

= – ( –l)txt–p+t

f(p) –p,xt–p

.

Hence, we have

xt–p≤

  –l

f(p) –p,xt–p

. (.)

In particular,

xni–p _≤ 

 –l

f(p) –p,xni–p

. (.)

Sincex˜∈F(T)∩(A+B)–_{, by (.), we obtain}

xni–x˜

  –l

f(x) –˜ x,˜ xni–x˜

. (.)

Sincexnix, it follows from (.) that˜ xni→ ˜xasi→ ∞.

Now, we return to (.) and take the limit asi→ ∞to get

˜x–p≤   –l

(I–f)p,p–x˜. (.)

In particular,x˜solves the following variational inequality

˜

x∈F(T)∩(A+B)–, (I–f)p,p–x˜≥, p∈F(T)∩(A+B)–,

or the equivalent dual variational inequality (see Lemma .)

˜

x∈F(T)∩(A+B)–, (I–f)x,˜ p–x˜≥, p∈F(T)∩(A+B)–. (.)

Finally, we show that the net {xt}converges strongly, as t→, tox. To this end, let˜ {sk} ⊂(, ) be another sequence such thatsk→ ask→ ∞. Putxk:=xsk,yk:=ysk and

λk:=λsk. Let{xkj}be a subsequence of{xk}, and assume thatxkjx. By the same proofˆ

as the one above, we havexˆ∈F(T)∩(A+B)–_{. Moreover, it follows from (.) that}

(I–f)x,˜ x˜–xˆ≤. (.)

Interchangingx˜andx, we obtainˆ

(I–f)x,ˆ xˆ–x˜≤. (.)

Adding up (.) and (.) yields

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Hence,

˜x–xˆ≤f(x) –˜ f(x),ˆ x˜–xˆ≤l˜x–xˆ,

that is, ( –l)˜x–xˆ_≤_{. Since}_l_∈_{(, ), we have}_x_˜₌_{x. Therefore, we conclude that}_ˆ _x t→ ˜x

ast→.

Note thatP_F(T)∩(A+B)–_is well deﬁned by Lemma . and Remark .. The variational inequality (.) can be rewritten as

(I–f)x˜–x,˜ p–x˜≥, ∀p∈F(T)∩(A+B)–.

By (.), this is equivalent to the ﬁxed point equation

˜

x=P_F_(T)∩(A+B)–_(I–f)x.˜

This completes the proof.

From Theorem ., we can deduce the following result.

Corollary . Suppose that F(T)∩(A+B)–_₌_∅_._{Then the net}_{_x

t}deﬁned by the implicit

method(.)converges strongly,as t→,tox,˜ which solves the following minimum norm problem:ﬁndx˜∈F(T)∩(A+B)–such that

˜x= min

x∈F(T)∩(A+B)–_x. (.)

Proof From (.) withf ≡ andl= , we have

˜x–p≤ p,p–x˜, ∀p∈F(T)∩(A+B)–.

Equivalently,

˜x,p–x˜ ≥, ∀p∈F(T)∩(A+B)–.

This obviously implies that

˜x≤ p,x˜ ≤ p˜x, ∀p∈F(T)∩(A+B)–.

It turns out that˜x ≤ pfor allp∈F(T)∩(A+B)–_{. Therefore,}_x_˜_{is the minimum-norm}

point ofF(T)∩(A+B)–_.

3.2 Strong convergence of the explicit algorithm

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Theorem . Suppose that F(T)∩(A+B)–_₌_∅_._Let_{_α

n},{βn} ⊂(, )and{λn} ⊂(, α)

satisfy the following conditions:

(C) limn→∞αn= ;

(C) ∞_n=αn=∞;

(C)  <c≤βn≤d< ;

(C)  <a≤λn≤b< αandlimn→∞(λn–λn+) = .

Let the sequence{xn}be generated iteratively by(.):

xn+=αnf(xn) +βnxn+ ( –αn–βn)SJλn(xn–λnAxn), ∀n≥, (.)

where x∈C is an arbitrary initial guess.Then the sequence{xn}converges strongly to a

pointx in F(T˜ )∩(A+B)–_,_{which is the unique solution of the variational inequality}_(.).

Proof First, from condition (C), without loss of generality, we assume that(–l)αn

–αnl < , and

we note thatF(T) =F(S). From now, we putyn=Jλn(xn–λnAxn).

We divide the proof several steps as follows.

Step . We show thatxn–p ≤max{x–p,f(p)–p_–l }for alln≥ and allp∈F(T)∩

(A+B)– (=F(S)∩(A+B)–). Indeed, letp∈F(T)∩(A+B)–. Fromp=Jλn(p–λnAp) =

SJλn(p–λnAp),Sp=pand Lemma ., we get

yn–p=Jλn(xn–λnAxn) –Jλn(p–rAp) 

≤(xn–λnAxn) – (p–λnAp) 

=(xn–p) –λn(Axn–Ap) 

=xn–p– λnxn–p,Axn–Ap+λnAxn–Ap

≤ xn–p– λnαAxn–Ap+λnAxn–Ap

=xn–p+λn(λn– α)Axn–Ap

≤ xn–p. (.)

Using (.), we have

xn+–p=αnf(xn) +βnxn+ ( –αn–βn)SJλn(xn–λnAxn) –p

=αn

f(xn) –p

+βn(xn–p) + ( –αn–βn)(Syn–Sp)

≤αnf(xn) –f(p)+αnf(p) –p+βnxn–p

+ ( –αn–βn)yn–p

≤αnlxn–p+αnf(p) –p+βnxn–p+ ( –αn–βn)xn–p

= – ( –l)αn

xn–p+ ( –l)αn

f(p) –p  –l .

Using an induction, we have

xn–p ≤max

x–p,

f(p) –p  –l

.

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Step . We show thatlim_n→∞xn+–xn= . Putun=xn–λnAxn, and deﬁne

xn+=βnxn+ ( –βn)zn, ∀n≥. (.)

Then we have

zn+–zn

=xn+–βn+xn+  –βn+

–xn+–βnxn  –βn

=αn+f(xn+) +βn+xn++ ( –αn+–βn+)SJλn+un+–βn+xn+  –βn+

–αnf(xn) +βnxn+ ( –αn–βn)SJλnun–βnxn  –βn

=αn+f(xn+) + ( –αn+–βn+)SJλn+un+  –βn+

–αnf(xn) + ( –αn–βn)SJλnun  –βn

= αn+  –βn+

f(xn+) –

αn

 –βn

f(xn) +SJλn+un+–SJλnun

– αn+  –βn+

SJλn+un++ αn

 –βn

SJλnun

= αn+  –βn+

f(xn+) –SJλn+un+

+ αn  –βn

SJλnun–f(xn)

+SJλn+un+–SJλnun. (.)

SinceI–λn+Ais nonexpansive forλn+∈(, α) (by Lemma .), we have

(I–λn+A)xn+– (I–λn+A)xn≤ xn+–xn. (.)

By the resolvent identity (.) and (.), we get

Jλn+un+–Jλnun

=Jλn

λn

λn+

un++

 – λn λn+

Jλn+un+

–Jλnun

≤ λn

λn+

un++

 – λn λn+

Jλn+un+–un

≤ λn

λn+

un+–un+  – λn

λn+

Jλn+un+–un

≤ un+–un+  – λn

λn+

un+–un+Jλn+un+–un

≤(xn+–λn+Axn+) – (xn–λnAxn)

+λn+–λn a

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+|λn+–λn|

 a

≤ xn+–xn+|λn+–λn|M, (.)

whereM>  is an appropriate constant. Hence, from (.) and (.), we obtain

zn+–zn

≤ αn+

 –βn+

f(xn+)+SJλn+un+

+ αn  –βn

SJλnun+f(xn)

+SJλn+un+–SJλnun

≤ αn+

 –βn+

f(xn+)+SJλn+un+

+ αn  –βn

SJλnun+f(xn)

+Jλn+un+–Jλnun

≤ αn+

 –βn+

_f_(x_n+₎+SJλn+un+

+ αn  –βn

SJλnun+f(xn)

+xn+–xn+|λn+–λn|M. (.)

It follows from conditions (C) and (C) that

lim sup

n→∞

zn+–zn–xn+–xn

≤.

Thus, by Lemma ., we have

lim

n→∞zn–xn= . (.)

Consequently, we obtain

lim

n→∞xn+–xn=n→∞lim( –βn)zn–xn= .

Step . We show thatlim_n→∞Axn–Ap=  forp∈F(T)∩(A+B)–. From (.), (.)

and Lemma ., we have

xn+–p

=αnf(xn) +βnxn+ ( –αn–βn)SJλn(xn–λnAxn) –p 

=αn

f(xn) –p

+βn(xn–p) + ( –αn–βn)(Syn–p)

=αnf(xn) –p+βnxn–p+ ( –αn–βn)Syn–p

–αnβnf(xn) –xn–βn( –αn–βn)xn–Syn

–αn( –αn–βn)Syn–f(xn) 

≤αnf(xn) –f(p)+f(p) –p 

+βnxn–p+ ( –αn–βn)yn–p

≤αn

lxn–p+ lxn–pf(p) –p+f(p) –p 

+βnxn–p

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≤ – ( –l)αn

xn–p+ ( –αn–βn)λn(λn– α)Axn–Ap

+αn

lxn–pf(p) –p+f(p) –p 

≤ xn–p+ ( –αn–βn)λn(λn– α)Axn–Ap+αnM, (.)

whereM>  is an appropriate constant. From (.) and conditions (C) and (C), we

deduce that

( –αn–d)a(α–b)Axn–Ap≤( –αn–βn)λn(α–λn)Axn–Ap

≤ xn–xn+

xn–p+xn+–p

+αnM.

Sinceαn→ (by condition (C)) andxn+–xn → (by Step ), we conclude that

lim

n→∞Axn–Ap= .

Step . We show thatlim_n→∞xn–yn= . First, from (.) and (.), we get forp∈

F(T)∩(A+B)–_,

yn–p=Jλn(xn–λnAxn) –p 

=Jλn(xn–λnAxn) –Jλn(p–λnAp) 

≤(xn–λnAxn) – (p–λnAp),yn–p

=

(xn–λnAxn) – (p–λnAp)



+yn–p

–(xn–λnAxn) – (p–λnAp) – (yn–p) 

≤



xn–p+yn–p–xn–yn–λn(Axn–Ap) 

= 

xn–p+yn–p–xn–yn

+ λnxn–yn,Axn–Ap–λnAxn–Ap

.

So, we have

yn–p≤ xn–p–xn–yn+ λnxn–yn,Axn–Ap

–λ_nAxn–Ap

≤ xn–p–xn–yn+ λnxn–yn,Axn–Ap. (.)

Using (.) and (.), we obtain

xn+–p

≤αn

lxn–p+ lxn–pf(p) –p+f(p) –p 

+βnxn–p

+ ( –αn–βn)yn–p

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+ ( –αn–βn)

xn–p–xn–yn+ λnxn–yn,Axn–Ap

= – ( –l)αn

xn–p– ( –αn–βn)xn–yn

+ λnxn–yn,Axn–Ap+αnM

≤ xn–p– ( –αn–βn)xn–yn+ bMAxn–Ap+αnM, (.)

whereM,M>  are appropriate constants. This implies that

( –αn–d)xn–yn

≤( –αn–βn)xn–yn ≤ xn–xn+

xn+–p+xn–p

+ bMAxn–Ap+αnM.

Thus, from condition (C), Step  and Step , we deduce that

lim

n→∞xn–yn= .

Step . We show thatlim_n→∞Sxn–xn= . First, by (.), we have

Syn–xn ≤ Syn–xn++xn+–xn

=Syn–

αnf(xn) +βnxn+ ( –αn–βn)Syn+xn+–xn

≤αnSyn–f(xn)+βnxn–Syn+xn+–xn,

and so,

Syn–xn ≤

  –βn

αnSyn–f(xn)+xn+–xn

.

By conditions (C) and (C) and Step , we obtain

lim

n→∞Syn–xn= .

This together with Step  yields that

Sxn–xn ≤ Sxn–Syn+Syn–xn

≤ xn–yn+Syn–xn → asn→ ∞.

Step . We show that

lim sup

n→∞

f(x) –˜ x,˜ xn–x˜

≤,

wherex˜=lim_t→xt withxt being deﬁned by (.). We note that from Theorem .,x˜∈

Fix(T)∩(A+B)–_{, and}_x_˜ _{is the unique solution of the variational inequality (.). To}

show this, we can choose a subsequence{xni}of{xn}such that

lim

i→∞

f(x) –˜ x,˜ xni–x˜

=lim sup

n→∞

f(x) –˜ x,˜ xn–x˜

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Since{xni}is bounded, there exists a subsequence{xn_ij}of{xni}, which converges weakly

tow. Without loss of generality, we can assume thatxniw. By the same argument as in

the proof of Theorem . together with Step , we havew∈F(T)∩(A+B)–_{. Since}_x_˜₌

P_F(T)∩(A+B)–_(I–f)x˜is the unique solution of the variational inequality (.), we deduce that

lim sup

n→∞

f(x) –˜ x,˜ xn–x˜

= lim

i→∞

f(x) –˜ x,˜ xni–x˜

=f(x) –˜ x,˜ w–x˜≤.

Step . We show thatlimn→∞xn–x˜= , wherex˜=limt→xtwithxtbeing deﬁned by

(.), andx˜is the unique solution of the variational inequality (.). Indeed, from (.), we note that

xn+–x˜=αnf(xn) +βnxn+ ( –αn–βn)SJλn(xn–λnAxn) –x˜

=αn

f(xn) –x˜

+βn(xn–x) + ( –˜ αn–βn)(SJλnyn–x).˜

Applying Lemma ., we have

xn+–x˜≤βn(xn–x) + ( –˜ αn–βn)(SJλnyn–x)˜ 

+ αn

f(xn) –f(x),˜ xn+–x˜

+ αn

f(x) –˜ x,˜ xn+–x˜

≤βnxn–x˜+ ( –αn–βn)yn–x˜ 

+ αnlxn–x˜xn+–x˜+ αn

f(x) –˜ x,˜ xn+–x˜

≤( –αn)xn–x˜+αnl

xn–x˜+xn+–x˜

+ αn

f(x) –˜ x,˜ xn+–x˜

.

This implies that

xn+–x˜≤

 – ( –l)αn+αn

 –αnl

xn–x˜+

αn

 –αnl

f(x) –˜ x,˜ xn+–x˜

= – ( –l)αn  –αnl

xn–x˜+

α n

 –αnl

xn–x˜

+ αn  –αnl

f(x) –˜ x,˜ xn+–x˜

=

 –( –l)αn  –αnl

xn–x˜+

α_n  –αnl

xn–x˜

+ αn  –αnl

f(x) –˜ x,˜ xn+–x˜

≤

 –( –l)αn  –αnl

xn–x˜

+( –l)αn  –αnl

αnM

( –l)+   –l

f(x) –˜ x,˜ xn+–x˜

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whereM>  is an appropriate constant,ξn=(–l)_–αnlαn and

δn=

αnM

( –l)+   –l

f(x) –˜ x,˜ xn+–x˜

.

From conditions (C) and (C) and Step , it is easy to see that ξn→, ∞n=ξn=∞

andlim sup_n→∞δn≤. Hence, by Lemma ., we conclude thatxn→ ˜xasn→ ∞. This

completes the proof.

From Theorem ., we deduce immediately the following result.

Corollary . Suppose that F(T)∩(A+B)–=∅.Let{αn},{βn} ⊂(, )and{λn} ⊂(, α)

satisfy the following conditions:

(C) ∞_n=αn=∞;

(C)  <c≤βn≤d< ;

(C)  <a≤λn≤b< α.

Let the sequence{xn}be generated iteratively by(.):

xn+=βnxn+ ( –αn–βn)SJλn(xn–λnAxn), ∀n≥, (.)

pointx in F(T˜ )∩(A+B)–_,_{which is the unique solution of the minimum norm}

prob-lem(.).

Proof The variational inequality (.) is reduced to the inequality

˜x,p–x˜ ≥, ∀p∈F(T)∩(A+B)–.

This is equivalent to ˜x _≤_p,_x_˜_≤_p_˜_x_{for all}_p_∈_F(T₎_∩_(A₊_B)–_{. It turns out}

that˜x ≤ pfor allp∈F(T)∩(A+B)–_{ and}_x_˜_{is the minimum-norm point of}_F(T)_∩

(A+B)–_.

Remark . It is worth pointing out that our iterative schemes (.) and (.) are new ones diﬀerent from those in the literature. The iterative schemes (.) and (.) are also new ones diﬀerent from those in the literature (see [, ] and the references therein).

4 Applications

LetHbe a real Hilbert space, and letgbe a proper lower semicontinuous convex function ofHinto (–∞,∞]. Then the subdiﬀerential∂gofgis deﬁned as follows:

∂g(x) =z∈H|g(x) +z,y–x ≤g(y),y∈H

for allx∈H. From Rockafellar [], we know that∂g is maximal monotone. LetCbe a closed convex subset ofH, and letiCbe the indicator function ofC,i.e.,

iC(x) = ⎧ ⎨ ⎩

, x∈C,

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SinceiC is a proper lower semicontinuous convex function onH, the subdiﬀerential∂iC

ofiCis a maximal monotone operator. It is well known that ifB=∂iC, then the MIP (.)

is equivalent to ﬁndu∈Csuch that

Au,v–u ≥, ∀v∈C. (.)

This problem is called Hartman-Stampacchia variational inequality (see []). The set of solutions of the variational inequality (.) is denoted byVI(C,A).

The following result is proved by Takahashiet al.[].

Lemma .[] Let C be a nonempty closed convex subset of a real Hilbert space H,let PC

be the metric projection from H onto C,let∂iCbe the subdiﬀerential of iC,and let Jλbe the

resolvent of∂iCforλ> ,where iCis deﬁned by(.)and Jλ= (I+λ∂iC)–.Then

u=Jλx ⇐⇒ u=PCx, ∀x∈H,y∈C.

Applying Theorem ., we can obtain a strong convergence theorem for ﬁnding a com-mon element of the set of solutions to the variational inequality (.) and the set of ﬁxed points of a nonexpansive mapping.

Theorem . Let C be a nonempty closed convex subset of a real Hilbert space H.Let A be anα-inverse strongly monotone mapping of C into H,and let S be a nonexpansive mapping of C into itself such that F(S)∩VI(C,A)=∅.Let{αn},{βn} ⊂(, )and{λn} ⊂(, α)satisfy

the following conditions:

(C) ∞_n=αn=∞;

(C)  <c≤βn≤d< ;

Let the sequence{xn}be generated iteratively by

xn+=αnf(xn) +βnxn+ ( –αn–βn)SPC(xn–λnAxn), ∀n≥,

pointx in F(S)˜ ∩VI(C,A).

Proof PutB=∂iC. It is easy to show thatVI(C,A) = (A+∂iC)–. In fact,

x∈(A+∂iC)– ⇐⇒ ∈Ax+∂iCx

⇐⇒ –Ax∈∂iCx

⇐⇒ Ax,u–x ≥ (∀u∈C)

⇐⇒ x∈VI(C,A).

From Lemma ., we getJλn=PCfor allλn. Hence, the desired result follows from

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As in [, ], we consider the problem for ﬁnding a common element of the set of solu-tions of a mathematical model related to equilibrium problems and the set of ﬁxed points of a nonexpansive mapping in a Hilbert space.

LetCbe a nonempty closed convex subset of a Hilbert spaceH, and let us assume that a bifunction:C×C→Rsatisﬁes the following conditions:

(A) (x,x) = for allx∈C;

(A) is monotone, that is,(x,y) +(y,x)≤for allx,y∈C; (A) for eachx,y,z∈C,

lim

t↓

tz+ ( –t)x,y≤(x,y);

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

Then the mathematical model related to the equilibrium problem (with respect toC) is ﬁndxˆ∈Csuch that

(x,ˆ y)≥ (.)

for ally∈C. The set of such solutionsxˆis denoted byEP(). The following lemma was given in [, ].

Lemma .[, ] Let C be a nonempty closed convex subset of H,and letbe a bifunc-tion of C×C intoRsatisfying(A)-(A).Then for any r> and x∈H,there exists z∈C such that

(z,y) +

ry–z,z–x ≥, ∀y∈C.

Moreover,if we deﬁne Tr:H→C as follows:

Trx=

z∈C:(z,y) +

ry–z,z–x ≥,∀y∈C

for all x∈H,then the following hold:

() Tris single-valued;

() Tris ﬁrmly nonexpansive,that is,for anyx,y∈H,

Trx–Try≤ Trx–Try,x–y;

() F(Tr) =EP();

() EP()is closed and convex.

We call suchTrthe resolvent offorr> . The following lemma was given in Takahashi

et al.[].

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of H into itself deﬁned by

Ax= ⎧ ⎨ ⎩

{z∈H:(x,y)≥ y–x,z}, x∈C,

∅, x∈/C.

ThenEP() =A–

,and Ais a maximal monotone operator withdom(A)⊂C.

More-over,for any x∈H and r> ,the resolvent Trofcoincides with the resolvent of A;i.e.,

Trx= (I+rA)–x.

Applying Lemma . and Theorem ., we can obtain the following results.

Theorem . Let C be a nonempty closed convex subset of a real Hilbert space H,and let

be a bifunction of C×C intoRsatisfying(A)-(A).Let Abe a maximal monotone

operator with dom(A)⊂C deﬁned as in Lemma.,and let Tλ be the resolvent of

forλ> .Let A be anα-inverse strongly monotone mapping of C into H,and let S be a nonexpansive mapping from C into itself such that F(S)∩(A+A)–=∅.Let{αn},{βn} ⊂

(, )and{λn} ⊂(, α)satisfy the following conditions:

(C) lim_n→∞αn= ;

(C) ∞_n=αn=∞;

(C)  <c≤βn≤d< ;

xn+=αnf(xn) +βnxn+ ( –αn–βn)STλn(xn–λnAxn), ∀n≥,

pointx in F(S)˜ ∩(A+A)–.

Theorem . Let C be a nonempty closed convex subset of a real Hilbert space H,and let be a bifunction of C×C intoRsatisfying(A)-(A).Let Abe a maximal monotone

operator withdom(A)⊂C deﬁned as in Lemma.,and let Tλbe the resolvent offor

λ> ,and let S be a nonexpansive mapping from C into itself such that F(S)∩EP()=∅. Let{αn},{βn} ⊂(, )and{λn} ⊂(, α)satisfy the following conditions:

(C) lim_n→∞αn= ;

(C) ∞_n=αn=∞;

(C)  <c≤βn≤d< ;

xn+=αnf(xn) +βnxn+ ( –αn–βn)STλn(xn), ∀n≥,

pointx in F(S)˜ ∩EP().

Proof PutA=  in Theorem .. From Lemma ., we also know thatJλn=Tλn for all

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Remark . () As in Corollary ., if we takef ≡ in Theorems ., . and ., then we can obtain the minimum-norm point ofF(S)∩VI(C,A),F(S)∩(A+A)– andF(S)∩

EP(), respectively.

() For several iterative schemes for zeros of monotone operators, variational inequality problems, generalized equilibrium problems, convex minimization problems, and ﬁxed point problems, we can also refer to [–] and the references therein. By combining our methods in this paper and methods in [–], we will consider new iterative schemes for the above-mentioned problems coupled with the ﬁxed point problems of nonlinear operators.

Competing interests

The author declares that they have no competing interests.

Acknowledgements

The author would like to thank the anonymous referees for their careful reading and valuable suggestions, which improved the presentation of this manuscript, and the editor for his valuable comments along with providing some recent related papers. This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2013021600).

Received: 19 June 2013 Accepted: 3 September 2013 Published:08 Nov 2013

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