Strong convergence of a splitting algorithm for treating monotone operators

(1)

R E S E A R C H

Open Access

Strong convergence of a splitting algorithm

for treating monotone operators

Sun Young Cho

1

_{, Xiaolong Qin}

2,3

_{and Lin Wang}

4* *_{Correspondence:}

[email protected] 4_{College of Statistics and}

Mathematics, Yunnan University of Finance and Economics, Kunming, 650221, China

Full list of author information is available at the end of the article

Abstract

In this paper, we investigate a splitting algorithm for treating monotone operators. Strong convergence theorems are established in the framework of Hilbert spaces.

Keywords: maximal monotone operator; ﬁxed point; nonexpansive mapping;

proximal point algorithm; zero point

1 Introduction and preliminaries

In this article, we always assume thatHis a real Hilbert space with inner product·,·and norm · andCis a nonempty, closed, and convex subset ofH.

Let S:C→Cbe a mapping.F(S) stands for the ﬁxed point set ofS. Sis said to be contractiveiﬀ there exists a constantα∈(, ) such that

Sx–Sy ≤αx–y, ∀x,y∈C.

It is well known that every contractive mapping has a unique ﬁxed point in metric spaces. Sis 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 [] and the references therein.Sis said to bestrictly pseudocontractiveiﬀ there exists a constantκ∈[, ) such that

Sx–Sy≤ x–y+κx–y–Sx+Sy, ∀x,y∈C.

The class of strictly pseudocontractive mapping was introduced by Browder and Petryshyn []. It is clear that the class of strictly pseudocontractive mappings include the class of nonexpansive mappings as a special case. It is also not hard to see that strictly pseudocon-tractive mapping is continuous.

LetA:C→Hbe a mapping. Recall thatAis said to bemonotoneiﬀ

Ax–Ay,x–y ≥, ∀x,y∈C.

(2)

Ais said to bestrongly monotoneiﬀ there exists a constantκ>  such that

Ax–Ay,x–y ≥κx–y, ∀x,y∈C.

Ais said to beinverse-strongly monotoneiﬀ there exists a constantκ>  such that

Ax–Ay,x–y ≥κAx–Ay, ∀x,y∈C.

Ais inverse-strongly monotone iﬀ the inverse ofAis strongly monotone. It is not hard to see that every inverse-strongly monotone mapping is monotone and continuous. LetIbe the identity mapping onH. From [], we know thatI–Sis inverse-strongly monotone iﬀ Sis strictly pseudocontractive; for more details, see [] and the references therein.

The classical variational inequality problem is formulated as ﬁnding a pointx∈Csuch that

y–x,Ax ≥, ∀y∈C.

Such a pointx∈Cis called a solution of the variational inequality. In this paper, we use VI(C,A) to denote the solution set of the variational inequality. It is known thatx is a solution of the variational inequality iﬀxis a ﬁxed point of the mappingProj_C(I–rA), whereProj_Cis the metric projection fromHontoC,Iis the identity andris some posi-tive real number. Recently, many authors studied solutions of inverse-strongly monotone variational inequalities based on the equivalence; see [–].

Recall that a set-valued mappingB:H⇒His said to bemonotoneiﬀ, for allx,y∈H, f ∈Bxandg∈Byimplyx–y,f –g> . In this paper, we useB–_{() to stand for the zero}

point ofB. A monotone mappingB:H⇒H ismaximaliﬀ the graphGraph(B) ofBis not properly contained in the graph of any other monotone mapping. It is known that a monotone mappingBis maximal if and only if, for any (x,f)∈H×H,x–y,f –g ≥, for all (y,g)∈Graph(B) impliesf ∈Bx. For a maximal monotone operatorBonH, and r> , we may deﬁne the single-valued resolventJr:H→Dom(B), whereDom(B) denote

the domain ofB. It is known thatJris ﬁrmly nonexpansive, andB–() =F(Jr).

One of the most important techniques for solving zero point problem of monotone op-erators goes back to the work of Browder []. Many important problems have reformula-tions which require finding zero points, for instance, evolution equareformula-tions, complementar-ity problems, mini-max problems, variational inequalities and fixed point problems. It is well known that minimizing a convex functionf can be reduced to finding zero points of the subdifferential mappingA=∂f. One of the basic ideas in the case of a Hilbert spaceHis reducing the above inclusion problem to a fixed point problem of the operatorRAdefined

byRA= (I+A)–, which is called the classical resolvent ofA. IfAhas some monotonicity

conditions, the classical resolvent ofAis with full domain and ﬁrmly nonexpansive. The property of the resolvent ensures that the Picard iterative algorithmxn+=RAxnconverge

weakly to a ﬁxed point ofRA, which is necessarily a zero point ofA. Rockafellar introduced

this iteration method and call it the proximal point algorithm (PPA); for more details, see [] and [] and the references therein.

(3)

or of the same difficult as the original zero point problem. Therefore, one of the most inter-esting and important problems in the theory of monotone operators is to find an efficient iterative algorithm to compute their zero points. In many disciplines, including economics [], image recovery [], quantum physics [], and control theory [], problems arises in infinite dimension spaces. In such problems, strong convergence (norm convergence) is often much more desirable than weak convergence, for it translates the physically tangible property that the energyxn–xof the error between the iteratexnand the solutionx

eventually becomes arbitrarily small. The important of strong convergence is also under-lined in [], where a convex functionf is minimized via the proximal point algorithm: it is shown that the rate of convergence of the value sequence{f(xn)}is better when{xn}

converges strongly that it converges weakly. Such properties have a direct impact when the process is executed directly in the underlying inﬁnite dimensional space.

To improve the weak convergence of PPA, many authors considered lots of diﬀerent modiﬁcations; see [–] the references therein. One of the classic results was estab-lished by Solodov and Svaiter []. They obtained strong convergence theorems in Hilbert space without any compact assumption but with the aid of the metric projection.

In this paper, we are concerned with the problem of ﬁnding an element in the zero point set of the sum of two operators which are inverse-strongly monotone and a maxi-mal monotone and in the ﬁxed point set of a mapping which is strictly pseudocontractive. Strong convergence theorems are established without the aid of the metric projections. The organization of this paper is as follows. In Section , we provide an introduction and some necessary preliminaries. In Section , a regularization iterative algorithm is investi-gated. A strong convergence theorem is established without the aid of metric projections. In Section , applications of the main results are discussed.

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

Lemma .[] Let A:C→H be a mapping,and B:H⇒H a maximal monotone oper-ator.Then F(Jr(I–rB)) = (A+B)–().

Lemma .[] Let E be a Banach space and let A be an m-accretive operator.Forλ> , μ> ,and x∈E,we have Jλx=Jμ(μ_λx+ ( –μ_λ)Jλx),where Jλ= (I+λA)–and Jμ= (I+μA)–.

Lemma .[] Let{xn}and{yn}be bounded sequences in a Banach space E,and{βn}

be a sequence in (, )with <lim infn→∞βn≤lim supn→∞βn< .Suppose that xn+=

( –βn)yn+βnxn,∀n≥and

lim sup

n→∞

yn+–yn–xn+–xn

≤.

Thenlim_n_→∞yn–xn= .

Lemma .[] Let{an}be a sequence of nonnegative numbers satisfying the condition

an+≤( –tn)an+tnbn+cn,∀n≥,where{tn}is a number sequence in(, )such that

limn→∞tn= and

_∞

n=tn=∞,{bn}is a number sequence such thatlim supn→∞bn≤,

and{cn}is a positive number sequence such that∞n=cn<∞.Thenlimn→∞an= .

(4)

T:C→C by Tx:=ax+ ( –a)Sx for each x∈C.Then,as a∈[κ, ),T is nonexpansive such that F(S) =F(T).

2 Convergence analysis

Theorem . Let A:C→H be anα-inverse-strongly monotone mapping and let B be a maximal monotone operator on H.Let S:C→C be a strictly pseudocontractive mapping with the constantκ∈[, )and let f :C→C be a contractive mapping with the constant β∈[, ).Assume thatDom(B)⊂C and F(S)∩(A+B)–_()_{is not empty.}_{Let J}

rn= (I+rnB)–

and let{xn}be a sequence generated in the following process:x∈C and ⎧

⎪ ⎪ ⎨ ⎪ ⎪ ⎩

zn=αnf(xn) + ( –αn)xn,

yn=Jrn(zn–rnAzn+en),

xn+=βnxn+ ( –βn)(γnyn+ ( –γn)Syn), ∀n≥,

where{αn}, {βn}and{γn}are real number sequences in(, )and{rn}is a positive real

number sequence in(, α).Assume that the control sequences satisfy the following restric-tions:

(a) limn→∞αn= ,

_∞

n=αn=∞;

(b)  <lim inf_n_→∞βn≤lim supn→∞βn< ;

(c) κ≤γn≤a< ,limn→∞|γn+–γn|= ;

(d)  <b≤rn≤c< αand

∞

n=|rn–rn–|<∞; (e) ∞_n_=en<∞,

where a,b and c are three real numbers.Then{xn}converges strongly to a pointx¯∈F(S)∩

(A+B)–_(),_where_x_¯₌_Proj

F(S)∩(A+B)–_()f(¯x).

Proof First, we show that{xn}is bounded. Notice thatI–rnAis nonexpansive. Indeed, we

have

(I–rnA)x– (I–rnA)y



=x–y– rnx–y,Ax–Ay+rnAx–Ay

≤ x–y–rn(α–rn)Ax–Ay.

In view of the restriction (d), we ﬁnd thatI–rnAis nonexpansive. Fixingp∈F(S)∩(A+

B)–_{(), we ﬁnd that}

zn–p ≤αn f(xn) –p + ( –αn)xn–p

≤ –αn( –β)

xn–p+αn f(p) –p .

PuttingTnx:=γnx+ ( –γn)Sxfor eachx∈C, we see from Lemma . thatTnis

nonex-pansive withF(Tn) =F(S) for eachn≥. It follows that

xn+–p ≤βnxn–p+ ( –βn) TnJrn(zn–rnAzn+en) –p

≤βnxn–p+ ( –βn)zn–p+ ( –βn)en

≤βnxn–p+

 –αn( –β)

(5)

+αn( –βn) f(p) –p +en

≤ –αn( –β)( –βn)

xn–p+αn( –βn) f(p) –p +en

≤max

xn–p,

f(p) –p  –β

+en

≤max

xn––p,

f(p) –p  –β

+en–+en

.. .

≤max

x–p,f

(p) –p  –β

+ n i= ei ≤max

x–p,f

(p) –p  –β

+ ∞

i=

ei<∞.

This proves that the sequence{xn}is bounded, so are{yn}and{zn}. Notice that

zn–zn– ≤

 –αn( –β)

xn–xn–+|αn–αn–| f(xn–) –xn– .

Puttingρn=zn–rnAzn+en, we ﬁnd that

ρn–ρn– ≤ zn–zn–+rn–rn–Azn–+en+en–

≤ –αn( –β)

xn–xn–+|αn–αn–| f(xn–) –xn–

+|rn–rn–|Azn–+en+en–.

It follows from Lemma . that

yn–yn–= Jrn–

rn–

rn

ρn+

 –rn– rn

Jrnρn

–Jrn–ρn–

≤ rn–

rn

(ρn–ρn–) +

 –rn– rn

(Jrnρn–ρn–)

≤ (ρn–ρn–) +

 –rn– rn

(Jrnρn–ρn)

≤ ρn–ρn–+|rn

–rn–|

b Jrnρn–ρn

≤ –αn( –β)

xn–xn–+fn

≤ xn–xn–+fn,

where

fn=|αn–αn–| f(xn–) –xn– +|rn–rn–|

Azn–+

Jrnρn–ρn

b

+en+en–.

This implies that

Tnyn–Tn–yn– ≤ Tnyn–Tnyn–+Tnyn––Tn–yn–

(6)

In view of the restrictions (a), (c), (d), and (e), we ﬁnd that

lim sup

n→∞

Tnyn–Tn–yn––xn–xn–

≤.

It follows from Lemma . that

lim

n→∞Tnyn–xn= . (.)

This in turn implies that

lim

n→∞xn+–xn= . (.)

Notice that

xn+–p≤βnxn–p+ ( –βn)TnJrnρn–p

≤βnxn–p+ ( –βn) Jrn(zn–rnAzn+en) –p



≤βnxn–p+ ( –βn) (zn–rnAzn) – (I–rnA)p 

+en

en+  (zn–rnAzn) – (I–rnA)p

≤βnxn–p+ ( –βn)zn–p–rn( –βn)(α–rn)Azn–Ap

+en

en+  (zn–rnAzn) – (I–rnA)p

≤βnxn–p+αn( –βn) f(xn) –p



+ ( –βn)( –αn)xn–p

–rn( –βn)(α–rn)Azn–Ap+gn,

wheregn=en(en+ (zn–rnAzn) – (I–rnA)p). It follows that

rn( –βn)(α–rn)Azn–Ap

≤ xn–p–xn+–p+αn( –βn) f(xn) –p



+gn

≤xn–p+xn+–p

xn+–xn+αn f(xn) –p



+gn.

In view of the restrictions (a), (b), (c), (d), and (e), we ﬁnd from (.) that

lim

n→∞Azn–Ap= . (.)

SinceJrnis ﬁrmly nonexpansive, we ﬁnd that

Jrnρn–p

_≤_J

rnρn–p, (zn–rnAzn+en) – (p–rnAp)

= 

Jrnρn–p+ (zn–rnAzn+en) – (p–rnAp)



– (Jrnρn–p) –

(zn–rnAzn+en) – (p–rnAp)



≤  

Jrnρn–p

₊ _(I_–_r

nA)zn– (I–rnA)p



(7)

–Jrnρn–zn–en+rnAzn–rnAp



≤  

Jrnρn–p+zn–p+gn–Jrnρn–zn–en

–rnAzn–rnAp+ rnAzn–ApJrnρn–zn–en

.

It follows that

Jrnρn–p≤ zn–p+gn–Jrnρn–zn–en

–rnAzn–rnAp+ enJrnρn–zn+rnAzn–rnAp

≤αn f(xn) –p + ( –αn)xn–p+gn–Jrnρn–zn–en



+ rnAzn–ApJrnρn–zn–en.

This implies that

xn+–p≤βnxn–p+ ( –βn)TnJrnρn–p



≤βnxn–p+ ( –βn)Jrnρn–p



≤ xn–p+αn f(xn) –p



+gn– ( –βn)Jrnρn–zn–en



+ rnAzn–ApJrnρn–zn–en.

It follows that

( –βn)Jrnρn–zn–en

_≤_x

n–p+xn+–p

xn–xn++αn f(xn) –p



+gn+ rnAzn–ApJrnρn–zn–en.

In view of the restrictions (a), (b), and (e), we ﬁnd from (.) and (.) thatlimn→∞Jrnρn–

zn–en= . This in turn implies that

lim

n→∞Jrnρn–zn= . (.)

Notice that

yn–xn ≤ yn–zn+zn–xn

≤ yn–zn+αn f(xn) –xn .

It follows from (.) that

lim

n→∞yn–xn= . (.)

On the other hand, we have

Tnxn–xn ≤ γnxn+ ( –γn)Sxn

–γnyn+ ( –γn)Syn

+ γnyn+ ( –γn)Syn

–xn

(8)

SinceSis Lipschitz continuous, we ﬁnd from (.) and (.) that

lim

n→∞Tnxn–xn= . (.)

Notice that

Syn–yn ≤ Syn–Tnyn+Tnyn–Tnxn+Tnxn–xn+xn–yn

≤γnyn–Syn+ yn–xn+Tnxn–xn.

That is,

( –γn)Syn–yn ≤yn–xn+Tnxn–xn.

In view of (.) and (.), we ﬁnd from the restriction (c) that

lim

n→∞Syn–yn= . (.)

SinceProj_F₍_S₎_∩₍_A₊_B₎–_()f is contractive, we see that there exists a unique ﬁxed point, sayx.¯

Next, we show thatlim sup_n_→∞f(x) –¯ x,¯ zn–x¯ ≤. To show it, we can choose a

subse-quence{zni}of{zn}such that

lim sup

n→∞

f(x) –¯ x,¯ zn–x¯

= lim

i→∞

f(x) –¯ x,¯ zni–x¯

.

Since{zni}is bounded, we can choose a subsequence{zn_ij}of{zni}which converges weakly

to some pointx. We may assume, without loss of generality, thatzniconverges weakly tox.

In view of (.), we ﬁnd thatynialso converges weakly tox. It follows from Lemma . that

x∈F(S).

Now, we are in a position to show thatx∈(A+B)–_{(). Notice that}_y

n=Jrn(zn–rnAzn+

en). It follows that

zn–rnAzn+en∈(I+rnB)yn.

That is,

zn–yn

rn

–Azn+en∈Byn.

SinceBis monotone, we get, for any (μ,ν)∈B,

yn–μ,

zn–yn

rn

–Azn+en–ν

≥.

Replacingnbyniand lettingi→ ∞, we obtain from (.) that

(9)

This gives –Ax∈Bx, that is, ∈(A+B)(x). This proves thatx∈(A+B)–_{(). This complete}

the proof thatx∈F(S)∩(A+B)–_{(). It follows that}

lim sup

n→∞

f(¯x) –x,¯ zn–x¯

≤.

Finally, we show thatxn→ ¯x. Notice that

zn–x¯ ≤αn

f(xn) –x,¯ zn–x¯

+ ( –αn)xn–xz¯ n–x¯

≤αn f(xn) –f(¯x) zn–x¯ +αn

f(¯x) –x,¯ zn–x¯

+ ( –αn)xn–xz¯ n–x¯

≤ –αn( –β)



xn–x¯ +zn–x¯ 

+αn

f(¯x) –x,¯ zn–x¯

.

This implies that

zn–x¯≤

 –αn( –β)

xn–x¯+ αn

f(x) –¯ x,¯ zn–x¯

.

It follows that

yn–x¯ ≤ (zn–rnAzn) – (¯x–rnA¯x) +en



≤ (zn–rnAzn) – (¯x–rnA¯x)



+en

+ en (zn–rnAzn) – (¯x–rnA¯x)

≤ zn–x¯ +hn

≤ –αn( –β)

xn–x¯+ αn

f(¯x) –x,¯ zn–x¯

+hn, (.)

wherehn=en(en+ (zn–rnAzn) – (I–rnA)¯x). It follows from (.) that

xn+–x¯ ≤βnxn–x¯ + ( –βn)Tnyn–x¯ 

≤βnxn–x¯ + ( –βn)yn–x¯ 

≤ –αn( –βn)( –β)

xn–x¯ 

+ αn( –βn)

f(¯x) –x,¯ zn–x¯

+hn.

In view of the restriction (a), (b), and (e), we ﬁnd from Lemma . thatxn→ ¯x. This

com-pletes the proof.

IfSis nonexpansive andγn≡, then we have the following result immediately.

Corollary . Let A:C→H be anα-inverse-strongly monotone mapping and let B be a maximal monotone operator on H.Let S:C→C be a nonexpansive mapping and let f :C→C be a contractive mapping with the constantβ∈[, ).Assume thatDom(B)⊂C and F(S)∩(A+B)–_()_{is not empty.}_{Let J}

(10)

in the following process:x∈C and ⎧

⎨ ⎩

yn=αnf(xn) + ( –αn)xn,

xn+=βnxn+ ( –βn)SJrn(yn–rnAyn+en), ∀n≥,

where{αn}and{βn}are real number sequences in(, )and{rn}is a positive real number

sequence in(, α).Assume that the control sequences satisfy the following restrictions:

∞

n=αn=∞;

(b)  <lim infn→∞βn≤lim supn→∞βn< ;

(c)  <b≤rn≤c< αand

_∞

n=|rn–rn–|<∞; (d) ∞_n_=en<∞,

where b and c are two real numbers.Then{xn}converges strongly to a pointx¯∈F(S)∩(A+

B)–_(),_where_x_¯₌_Proj

F(S)∩(A+B)–_()f(x).¯

3 Applications

Many nonlinear problems arising in applied areas such as image recovery, signal process-ing, and machine learning are mathematically modeled as a nonlinear operator equation and this operator is decomposed as the sum of two nonlinear operators. The central prob-lem is to iteratively ﬁnd a zero point of the sum of two monotone operators, that is,

∈(A+B)(x).

Many real word problems can be formulated as a problem of the above form. For instance, a stationary solution to the initial value problem of the evolution equation

⎧ ⎨ ⎩

∈Fu+∂_∂u_t, u=u()

can be recast as the inclusion problem when the governing maximal monotoneFis of the formF=A+B; for more details, see [] and the references therein.

First, we give the following result.

Theorem . Let A:C→H be anα-inverse-strongly monotone mapping and let B be a maximal monotone operator on H.Let f :C→C be a contractive mapping with the constantβ∈[, ).Assume thatDom(B)⊂C and(A+B)–_()_{is not empty.}_{Let J}

rn= (I+

rnB)–and let{xn}be a sequence generated in the following process:x∈C and ⎧

⎨ ⎩

xn+=βnxn+ ( –βn)Jrn(yn–rnAyn+en), ∀n≥,

∞

n=αn=∞;

∞

(11)

where b and c are two real numbers.Then{xn}converges strongly to a pointx¯∈(A+B)–(),

wherex¯=Proj₍_A₊_B₎–_()f(¯x).

Proof PutS=I, the identity onH. The desired conclusion can be obtained immediately. LetHbe a Hilbert space andf :H→(–∞, +∞] 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.

From Rockafellar [], we know that ∂f is maximal monotone. It is easy to verify that ∈∂f(x) if and only iff(x) =miny∈Hf(y). 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

subdif-ferential∂ICofICis a maximal monotone operator. Theny= (I+λ∂IC)–x⇐⇒y=ProjCx,

∀x∈H,y∈C.

Theorem . Let A:C→H be anα-inverse-strongly monotone mapping.Let S:C→C be a strictly pseudocontractive mapping with the constantκ ∈[, )and let f:C→C be a contractive mapping with the constantβ∈[, ).Assume that F(S)∩VI(C,A)is not empty. Let{xn}be a sequence generated in the following process:x∈C and

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

yn=ProjC(zn–rnAzn+en),

where{αn}, {βn}and{γn}are real number sequences in(, )and{rn}is a positive real

∞

n=αn=∞;

(c) κ≤γn≤a< ,limn→∞|γn+–γn|= ;

∞

n=|rn–rn–|<∞; (e) ∞_n_=en<∞,

where a,b,and c are three real numbers.Then{xn}converges strongly to a pointx¯∈F(S)∩

(A+B)–_(),_where_x_¯₌_Proj

F(S)∩(A+B)–_()f(¯x).

Proof PutBx=∂IC. Next, we show thatVI(C,A) = (A+∂IC)–(). Notice that

x∈(A+∂IC)–() ⇐⇒ ∈Ax+∂ICx

⇐⇒ –Ax∈∂ICx

⇐⇒ Ax,y–x ≥ ⇐⇒ x∈VI(C,A).

(12)

IfS=I, the identity onH, then we ﬁnd from Theorem . the following result immedi-ately.

Corollary . Let A:C→H be anα-inverse-strongly monotone mapping.Let S:C→C be a strictly pseudocontractive mapping with the constantκ∈[, )and let f :C→C be a contractive mapping with the constantβ∈[, ).Assume that F(S)∩VI(C,A)is not empty. Let{xn}be a sequence generated in the following process:x∈C and

⎧ ⎨ ⎩

xn+=βnxn+ ( –βn)ProjC(yn–rnAyn+en), ∀n≥,

∞

n=αn=∞;

∞

n=|rn–rn–|<∞; (d) ∞_n_=en<∞,

where b and c are three real numbers.Then{xn}converges strongly to a pointx¯∈VI(C,A),

wherex¯=Proj_VI₍_C_,_A₎f(¯x).

LetFbe a bifunction ofC×CintoR, whereRdenotes the set of real numbers. Recall the following equilibrium problem:

Findx∈Csuch thatF(x,y)≥, ∀y∈C. (.)

In this paper, we useEP(F) to denote the solution set of the equilibrium problem (.). To study the equilibrium problems (.), we may assume thatFsatisﬁes the following conditions:

(A) F(x,x) = for allx∈C;

(A) Fis monotone,i.e.,F(x,y) +F(y,x)≤for allx,y∈C; (A) for eachx,y,z∈C,

lim sup

t↓

Ftz+ ( –t)x,y≤F(x,y);

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

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

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

F(z,y) +

ry–z,z–x ≥, ∀y∈C.

Further,deﬁne

Trx=

z∈C:F(z,y) +

ry–z,z–x ≥,∀y∈C

(13)

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

(a) Tris single-valued;

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

Trx–Try≤ Trx–Try,x–y;

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

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

Lemma .[] Let C be a nonempty closed convex subset of a real Hilbert space H,F a bifunction from C×C toRwhich satisﬁes(A)-(A)and AFa multivalued mapping of H

into itself deﬁned by

AFx=

⎧ ⎨ ⎩

{z∈H:F(x,y)≥ y–x,z,∀y∈C}, x∈C,

∅, x∈/C.

(.)

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

Trx= (I+rAF)–x, ∀x∈H,r> ,

where Tris deﬁned as in(.).

Theorem . Let A:C→H be anα-inverse-strongly monotone mapping and let FBbe a

bifunction from C×C toRwhich satisﬁes(A)-(A).Let S:C→C be a strictly pseudocon-tractive mapping with the constantκ∈[, )and let f :C→C be a contractive mapping with the constantβ∈[, ).Assume that F(S)∩EP(F)is not empty.Let{xn}be a sequence

generated in the following process:x∈C and ⎧

⎪ ⎪ ⎨ ⎪ ⎪ ⎩

yn=Trn(zn–rnAzn+en),

where{αn},{βn},and{γn}are real number sequences in(, )and{rn}is a positive real

∞

n=αn=∞;

(c) κ≤γn≤a< ,limn→∞|γn+–γn|= ;

∞

n=|rn–rn–|<∞; (e) ∞_n_=en<∞,

where a,b,and c are three real numbers.Then{xn}converges strongly to a pointx¯∈F(S)∩

EP(F),wherex¯=Proj_F₍_S₎_∩_EP₍_F₎f(¯x).

(14)

Corollary . Let A:C→H be anα-inverse-strongly monotone mapping and let FBbe

a bifunction from C×C toRwhich satisﬁes(A)-(A).Let f :C→C be a contractive mapping with the constantβ∈[, ).Assume that EP(F)is not empty.Let{xn}be a sequence

generated in the following process:x∈C and ⎧

⎨ ⎩

xn+=βnxn+ ( –βn)Trn(yn–rnAyn+en), ∀n≥,

_∞

n=αn=∞;

(c)  <b≤rn≤c< αandn∞=|rn–rn–|<∞; (d) ∞_n_=en<∞,

where b and c are two real numbers.Then{xn} converges strongly to a pointx¯∈EP(F),

wherex¯=Proj_EP₍_F₎f(x).¯

Competing interests

The authors declare that there is no competing interests.

Authors’ contributions

The main idea of this paper was proposed by SYC. XQ and LW participate the research and performed some steps of the proof in this research. All authors read and approved the ﬁnal manuscript.

Author details

1_{Department of Mathematics, Gyengsang National University, Jinju, 660-701, Korea.}2_{Department of Mathematics,}

Hangzhou Normal University, Hangzhou, 310036, China.3_{Department of Mathematics, Faculty of Science, King Abdulaziz} University, Jeddah, Saudi Arabia.4_{College of Statistics and Mathematics, Yunnan University of Finance and Economics,} Kunming, 650221, China.

Acknowledgements

The authors are grateful to the referees for useful suggestions which improved the contents of the paper.

Received: 5 January 2014 Accepted: 24 March 2014 Published:09 Apr 2014

References

1. Browder, FE: Nonlinear operators and nonlinear equations of evolution in Banach spaces. Proc. Symp. Pure Math.18, 78-81 (1976)

2. Browder, FE, Petryshyn, WV: Construction of ﬁxed points of nonlinear mappings in Hilbert spaces. J. Math. Anal. Appl.

20, 197-228 (1967)

3. Cho, SY, Kang, SM: Approximation of common solutions of variational inequalities via strict pseudocontractions. Acta Math. Sci.32, 1607-1618 (2011)

4. Cho, SY: Approximation of solutions of a generalized variational inequality problem based on iterative methods. Commun. Korean Math. Soc.25, 207-214 (2010)

5. Cho, SY, Qin, X, Kang, SM: Iterative processes for common ﬁxed points of two diﬀerent families of mappings with applications. J. Glob. Optim.57, 1429-1446 (2013)

6. Cho, SY, Kang, SM: Approximation of ﬁxed points of pseudocontraction semigroups based on a viscosity iterative process. Appl. Math. Lett.24, 224-228 (2011)

7. Luo, H, Wang, Y: Iterative approximation for the common solutions of a inﬁnite variational inequality system for inverse-strongly accretive mappings. J. Math. Comput. Sci.2, 1660-1670 (2012)

8. Cho, SY, Li, W, Kang, SM: Convergence analysis of an iterative algorithm for monotone operators. J. Inequal. Appl.

2013, Article ID 199 (2013)

9. Zegeye, H, Shahzad, N: Strong convergence theorem for a common point of solution of variational inequality and ﬁxed point problem. Adv. Fixed Point Theory2, 374-397 (2012)

10. Rodjanadid, B, Sompong, S: A new iterative method for solving a system of generalized equilibrium problems, generalized mixed equilibrium problems and common ﬁxed point problems in Hilbert spaces. Adv. Fixed Point Theory3, 675-705 (2013)

11. Qin, X, Cho, SY, Kang, SM: An extragradient-type method for generalized equilibrium problems involving strictly pseudocontractive mappings. J. Glob. Optim.49, 679-693 (2011)

(15)

13. Qin, X, Cho, SY, Kang, SM: Iterative algorithms for variational inequality and equilibrium problems with applications. J. Glob. Optim.48, 423-445 (2010)

14. Browder, FE: Existence and approximation of solutions of nonlinear variational inequalities. Proc. Natl. Acad. Sci. USA

56, 1080-1086 (1966)

15. Rockafellar, RT: Augmented Lagrangians and applications of the proximal point algorithm in convex programming. Math. Oper. Res.1, 97-116 (1976)

16. Rockafellar, RT: Monotone operators and proximal point algorithm. SIAM J. Control Optim.14, 877-898 (1976) 17. Eckstein, J: Approximate iterations in Bregman-function-based proximal algorithms. Math. Program.83, 113-123

(1998)

18. Khan, MA, Yannelis, NC: Equilibrium Theory in Inﬁnite Dimensional Spaces. Springer, New York (1991) 19. Combettes, PL: The convex feasibility problem in image recovery. In: Hawkes, P (ed.) Advanced in Imaging and

Electron Physcis, vol. 95, pp. 155-270. Academic Press, New York (1996)

20. Dautray, R, Lions, JL: Mathematical Analysis and Numerical Methods for Science and Technology, vol. 1. Springer, New York (1990)

21. Fattorini, HO: Inﬁnite-Dimensional Optimization and Control Theory. Cambridge University Press, Cambridge (1999) 22. Güler, O: On the convergence of the proximal point algorithm for convex minimization. SIAM J. Control Optim.29,

403-409 (1991)

23. Qin, X, Kang, SM, Cho, YJ: Approximating zeros of monotone operators by proximal point algorithms. J. Glob. Optim.

46, 75-87 (2010)

24. Song, J, Chen, M: A modiﬁed Mann iteration for zero points of accretive operators. Fixed Point Theory Appl.2013, Article ID 347 (2013)

25. Qing, Y, Cho, SY: Proximal point algorithms for zero points of nonlinear operators. Fixed Point Theory Appl.2014, Article ID 42 (2014)

26. Qing, Y, Cho, SY: A regularization algorithm for zero points of accretive operators. Fixed Point Theory Appl.2013, Article ID 341 (2013)

27. Cho, SY, Kang, SM: Zero point theorems form-accretive operators in a Banach space. Fixed Point Theory13, 49-58 (2012)

28. Wu, C, Lv, S: Bregman projection methods for zeros of monotone operators. J. Fixed Point Theory2013, Article ID 7 (2013)

29. Zhang, M: Iterative algorithms for common elements in ﬁxed point sets and zero point sets with applications. Fixed Point Theory Appl.2012, Article ID 21 (2012)

30. Qin, X, Su, Y: Approximation of a zero point of accretive operator in Banach spaces. J. Math. Anal. Appl.329, 415-424 (2007)

31. Kim, JK, Anh, PN, Nam, YM: Strong convergence of an extended extragradient method for equilibrium problems and ﬁxed point problems. J. Korean Math. Soc.49, 187-200 (2012)

32. Qin, X, Cho, SY, Wang, L: Iterative algorithms with errors for zero points ofm-accretive operators. Fixed Point Theory Appl.2013, Article ID 148 (2013)

33. Solodov, MV, Svaiter, BF: Forcing strong convergence of proximal point iterations in a Hilbert space. Math. Program.

87, 189-202 (2000)

34. Zhang, M: Strong convergence of a viscosity iterative algorithm in Hilbert spaces. J. Nonlinear Funct. Anal.2014, Article ID 1 (2014)

35. Qin, X, Shang, M, Su, Y: Strong convergence of a general iterative algorithm for equilibrium problems and variational inequality problems. Math. Comput. Model.48, 1033-1046 (2008)

36. Cho, SY: Strong convergence of an iterative algorithm for sums of two monotone operators. J. Fixed Point Theory

2013, Article ID 6 (2013)

37. Barbu, V: Nonlinear Semigroups and Diﬀerential Equations in Banach Space. Noordhoﬀ, Groningen (1976) 38. Suzuki, T: Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive

semigroups without Bochner integrals. J. Math. Anal. Appl.305, 227-239 (2005)

39. Xue, Z, Zhou, H, Cho, YJ: Iterative solutions of nonlinear equations form-accretive operators in Banach spaces. J. Nonlinear Convex Anal.1, 313-320 (2000)

40. Zhou, H: Convergence theorems of ﬁxed points forκ-strict pseudo-contractions in Hilbert spaces. Nonlinear Anal.

69, 456-462 (2008)

41. Lions, PL, Mercier, B: Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal.16, 964-979 (1979)

42. Blum, E, Oettli, W: From optimization and variational inequalities to equilibrium problems. Math. Stud.63, 123-145 (1994)

43. Fan, K: A minimax inequality and applications. In: Shisha, O (ed.) Inequality, vol. III, pp. 103-113. Academic Press, New York (1972)

44. Takahashi, S, Takahashi, W, Toyoda, M: Strong convergence theorems for maximal monotone operators with nonlinear mappings in Hilbert spaces. J. Optim. Theory Appl.147, 27-41 (2010)

10.1186/1687-1812-2014-94