A hybrid approximation algorithm for finding common solutions of equilibrium problems, a finite family of variational inclusions, and fixed point problems in Hilbert spaces

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R E S E A R C H

Open Access

A hybrid approximation algorithm for ﬁnding

common solutions of equilibrium problems,

a ﬁnite family of variational inclusions, and

ﬁxed point problems in Hilbert spaces

Bin-Chao Deng

*

_{and Tong Chen}

*_{Correspondence:}

[email protected] School of Management, Tianjin University, Tianjin, 300072, China

Abstract

In this paper, we introduce an iterative method for finding a common element of the set of fixed points of nonexpansive mappings, the set of solutions of a finite family of variational inclusions with set-valued maximal monotone mappings and inverse strongly monotone mappings, and the set of solutions of an equilibrium problem in Hilbert spaces. Furthermore, using our new iterative scheme, under suitable conditions, we prove some strong convergence theorems for approximating these common elements. The results presented in the paper improve and extend many recent important results.

Keywords: variational inclusions; inverse strongly monotone; maximal monotone mapping; ﬁxed point; equilibrium problem

1 Introduction

LetHbe a real Hilbert space whose inner product and norm are denoted by·,·and · , respectively. LetCbe a nonempty closed convex subset ofH, and letFbe a bifunction of

C×CintoRwhich is the set of real numbers. The equilibrium problem forF:C×C→R

is to ﬁndx∈Csuch that

F(x,y)≥, ∀y∈C. (.)

The set of solutions of (.) is denoted byEP(F). Recently, Combettes and Hirstoaga [] introduced an iterative scheme of ﬁnding the best approximation to the initial data when

EP(F) was nonempty and proved a strong convergence theorem. LetA:C→Hbe a non-linear mapping. The classical variational inequality which is denoted byVI(A,C) is to ﬁnd

x∈Csuch that

Ax,y–x ≥, ∀y∈C. (.)

The variational inequality has been extensively studied in literature; see, for example, [, ] and the references therein. Recall that the mappingTofCinto itself is called nonexpansive

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if

Tx–Ty ≤ x–y, ∀x,y∈C.

A mappingf:C→Cis called contractive if there exists a constantβ∈(, ) such that

fx–fy ≤βx–y, ∀x,y∈C.

We denote byFix(T) the set of ﬁxed points ofT.

Some methods have been proposed to solve the equilibrium problem and the ﬁxed point problem of nonexpansive mappings; see, for instance, [, –] and the references therein. Recently, Plubtieng and Punpaeng [] introduced the following iterative scheme. Letx∈

H, and let{xn}and{un}be sequences generated by

⎧ ⎨ ⎩

F(un,y) +_r_ny–un,un–xn ≥, ∀y∈H, xn+=αnγf(xn) + (I–αnA)Tun, ∀n∈N.

They proved that if the sequences{αn}and{rn}of parameters satisﬁed appropriate

con-ditions, then the sequences{xn}and{un}both converged strongly to the unique solution

of the variational inequality

(A–γf)z,z–x≥, ∀x∈Fix(T)∩EP(F),

which was the optimality condition for the minimization problem

min x∈Fix(T)∩EP(F)



Ax,x–h(x),

wherehis a potential function forγf.

LetA:H→H be a single-valued nonlinear mapping, and letM:H→H be a set-valued mapping. We consider the following variational inclusion, which is to ﬁnd a point

u∈Hsuch that

θ∈A(u) +M(u), (.)

whereθis the zero vector inH. The set of solutions of problem (.) is denoted byI(A,M). LetAi:H→H,i= , , . . . ,N, be single-valued nonlinear mappings, and letMi:H→H, i= , , . . . ,N, be set-valued mappings. IfA≡, then problem (.) becomes the inclusion problem introduced by Rockafellar []. IfM=∂δC, whereCis a nonempty closed convex

subset ofHandδC:H→[,∞] is the indicator function ofC, that is,

δC(x) =

⎧ ⎨ ⎩

, x∈C,

+∞, x∈/C, (.)

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of optimal solutions in many optimization-related areas including mathematical pro-gramming, complementarity, variational inequalities, optimal control, mathematical eco-nomics, equilibria, and game theory. Also, various types of variational inclusions problems have been extended and generalized (see [] and the references therein). We introduce the following ﬁnite family of variational inclusions, which are to ﬁnd a pointu∈Hsuch that

θ∈Ai(u) +Mi(u), i= , , . . . ,N, (.)

where θ is the zero vector in H. The set of solutions of problem (.) is denoted by

N

i=I(Ai,Mi). The formulation (.) extends this formalism to a ﬁnite family of variational

inclusions covering, in particular, various forms of feasibility problems (see,e.g., []). In , Plubtemg and Sripard [] introduced the following iterative scheme for finding a common element of the set of solutions to problem (.) with a multi-valued maximal monotone mapping and an inverse-strongly monotone mapping, the set solutions of an equilibrium problem, and the set of fixed points of a nonexpansive mapping in a Hilbert space. Starting with an arbitraryx∈H, define sequences{xn},{yn}, and{un}by

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

F(un,y) +_r_ny–un,un–xn ≥, ∀y∈H, yn=JM,λ(un–λAun), ∀n> ,

xn+=αnγf(xn) + (I–αnB)Snyn,

(.)

for alln∈N, whereλ∈(, α],{αn} ⊂[, ], and{rn} ⊂(,∞);Bis a strongly positive

bounded linear operator onH and{Sn}is a sequence of nonexpansive mappings onH.

They proved that under certain appropriate conditions imposed on {αn} and{rn}, the

sequences{xn},{yn}, and{un}generated by (.) converge strongly toz∈

_∞

i=Fix(Si)∩ I(A,M)∩EP(F), wherez=P∞

i=Fix(Si)∩I(A,M)∩EP(F)f(z).

In , Tian [] introduced the following general iterative scheme for finding an el-ement of the set of solutions to the fixed point of a nonexpansive mapping in a Hilbert space: Define the sequence{xn}by

xn+=αnγf(xn) + (I–μαnB)Txn, n≥, (.)

where Bis ak-Lipschitzian andη-strongly monotone operator. Then he proved that if the sequence{αn}satisﬁes appropriate conditions, the sequence{xn}generated by (.)

converges strongly to the unique solutionx∗∈Cof the variational inequality

(γf–μB)x∗,x–x∗≤, ∀x∈C,

whereC=Fix(T).

In , Denget al.[] considered the following hybrid approximation scheme for find-ing common solutions of mixed equilibrium problems, a finite family of variational inclu-sions, and fixed point problems in Hilbert spaces. Starting with an arbitraryx∈H, define sequences{xn},{yn}, and{un}by

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

F(un,y) +F(un,y) +_r_ny–un,un–xn ≥, ∀y∈H, yn=JMN,λN,n(I–λN,nAN)· · ·JM,λ,n(I–λ,nA)un,

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for alln∈N, whereλi,n∈(, αi],i∈ {, , . . . ,N},{n} ⊂[, ], and{rn} ⊂(,∞),Bis a

strongly positive bounded linear operator onH, and{Sn}is a sequence of nonexpansive

mappings onH. Under suitable conditions and from this iterative scheme, they proved that{xn},{yn}, and{un}converge strongly toz, wherez=P(I–B+γf)(z) is a unique

solution of the variational inequality

(B–γf)z,z–x≤, x∈,

where:= (∞_n_=Fix(Sn))∩MEP(F,F)∩(

N

i=I(Ai,Mi))=∅.

Motivated and inspired by Aoyamaet al.[], Plubieng and Punpaeng [], Plubtemg and Sripard [], Peng et al.[], Tian [], and Denget al.[], we introduce an iter-ative scheme for finding a common element of the set of solutions of a finite family of variational inclusion problems (.) with multi-valued maximal monotone mappings and inverse-strongly monotone mappings, the set of solutions of an equilibrium problem, and the set of fixed points of nonexpansive mappings in a Hilbert space. Starting with an arbi-traryx∈H, define sequences{xn},{yn}, and{un}by

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

F(un,y) +_r_ny–un,un–xn ≥, ∀y∈H, yn=JMN,λN,n(I–λN,nAN)· · ·JM,λ,n(I–λ,nA)un,

xn+=nγf(xn) + (I–μnB)Snyn,

for all n∈N, whereλi,n∈(, αi],i∈ {, , . . . ,N}, {n} ⊂[, ], and{rn} ⊂(,∞),f is

anL-Lipschitz mapping onH,Bis ak-Lipschitzian andη-strongly monotone operator onHwith coeﬃcientsk>  andη> , and{Sn}is a sequence of nonexpansive mappings

onH. Under suitable conditions, some strong convergence theorems for approximating to these common elements are proved. Our results extend and improve some corresponding results in [, ] and the references therein.

2 Preliminaries

This section collects some lemmas which are used in the proofs of the main results in the next section.

LetHbe a real Hilbert space with the inner product·,·and the norm·, respectively. It is well known that for allx,y∈Handλ∈[, ], the following holds:

λx+ ( –λ)y =λx+ ( –λ)y–λ( –λ)x–y.

LetC be a nonempty closed convex subset ofH. Then, for anyx∈H, there exists a unique nearest point ofC, denoted byPCx, such thatx–PCx ≤ x–yfor ally∈C. Such

aPCis called the metric projection fromHintoC. We know thatPCis nonexpansive. It is

also known thatPCx∈Cand

x–PCx,PCx–z ≥, ∀x∈Handz∈C. (.)

It is easy to see that (.) is equivalent to

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For solving the equilibrium problem for a bifunctionF:C×C→R, let us assume that

Fsatisﬁes the following conditions: (A) F(x,x) = for allx∈C;

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

lim t→F

tz+ ( –t)x,y≤F(x,y);

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

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

F(z,y) +

ry–z,z–x ≥, ∀y∈C.

Deﬁne a mapping Tr:H→C as follows:

Tr(x) =

z∈C:F(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;

() Fix(Tr) =EP(F);

() EP(F)is closed and convex.

By the proof of Lemma  in [], we have the following lemma.

Lemma . Let C be a nonempty closed convex subset of a Hilbert space H,and let F:

C×C→Rbe a bifunction.Let x∈C and r,r∈(,∞).Then

Trx–Trx ≤  –r

r

Trx+x

. (.)

Lemma .[] Let H be a Hilbert space,and let f :H→H be a Lipschitz mapping with coeﬃcient <L.B:H→H is a k-Lipschitzian and η-strongly monotone operator with k> andη> .Then for <γ <μη/α,

x–y, (μB–γf)x– (μB–γf)y≥(μη–γL)x–y, x,y∈H.

That is,μB–γf is strongly monotone with coeﬃcientμη–γL.

Lemma .[] Assume that{αn}is a sequence of nonnegative real numbers such that

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where{αn}is a sequence in(, )and{δn}is a sequence in R such that

(i) ∞_n_=γn=∞,

(ii) lim sup_n_→∞δn

γn ≤or

∞

n=|δn|<∞. Thenlimn→∞αn= .

Deﬁnition . LetA:C→Hbe a nonlinear mapping.Ais said to be: (i) Monotone if

Ax–Ay,x–y ≥, ∀x,y∈C.

(ii) Strongly monotone if there exists a constantα> such that

Ax–Ay,x–y ≥αx–y, ∀x,y∈C.

For such a case,Ais said to beα-strongly-monotone.

(iii) Inverse-strongly monotone if there exists a constantα> such that

Ax–Ay,x–y ≥αAx–Ay, ∀x,y∈C.

For such a case,Ais said to beα-inverse-strongly-monotone. (iv) k-Lipschitz continuous if there exists a constantk≥such that

Ax–Ay ≤kx–y, ∀x,y∈C.

LetI be the identity mapping onH. It is well known that ifA:H→H isα -inverse-strongly monotone, thenAis a_α-Lipschitz continuous and monotone mapping. In addi-tion, if  <λ≤α, thenI–λAis a nonexpansive mapping.

A set-valued mappingM:H→H is called monotone if for allx,y∈H,f ∈Mxand

g∈Myimplyx–y,f –g ≥. A monotone mappingM:H→H_{is maximal if its graph} G(M) :{(x,f)∈H×H|f∈M(x)}ofMis not properly contained in the graph of any other monotone mapping. It is known that a monotone mappingMis maximal if and only if for (x,f)∈H×H,x–y,f –g ≥ for every (y,g)∈G(H) impliesf ∈Mx.

Let the set-valued mappingM:H→H_{be maximal monotone. We deﬁne the resolvent}

operatorJM,λassociated withMandλas follows:

JM,λ(u) = (I+λM)–(u), ∀u∈H,

whereλis a positive number. It is worth mentioning that the resolvent operatorJM,λis

single-valued, nonexpansive, and -inverse-strongly monotone (see, for example, []) and that a solution of problem (.) is a ﬁxed point of the operatorJM,λ(I–λA) for allλ> ;

see, for instance, []. Furthermore, a solution of a ﬁnite family of variational inclusion problems (.) is a common ﬁxed point ofJMk,λ(I–λAk),k∈ {, . . . ,N},λ> .

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Lemma . For all x,y∈H,the following inequality holds:

x+y≤ x+ y,x+y.

Lemma .(The resolvent identity) Let E be a Banach space,forλ> ,μ> ,and x∈E,

Jλx=Jμ

μ λx+

 –μ

λ

Jλx

.

Lemma .[] Let H be a Hilbert space.Let Ai:H→H,i= , , . . . ,N beαi -inverse-strongly monotone mappings,let Mi:H→H,i= , , . . . ,N be maximal monotone map-pings,and let{ωn}be a bounded sequence in H.Assume thatλj,n> ,j= , , . . . ,N,satisfy the following:

(H) limn→∞∞n=|λj,n–λj,n+|<∞,

(H) lim infn→∞λj,n> .

Setk_n=JMk,λk,n(I–λk,nAk)· · ·JM,λ,n(I–λ,nA)for k∈ {, , . . . ,N}and 

n=I for all n. Then,for k∈ {, , . . . ,N},

∞

i=

k_i_+ωi–kiωi <∞. (.)

Lemma .[] Let H be a real Hilbert space and B be a k-Lipschitzian andη-strongly monotone operator with k > ,η> .Let  <μ< _kη andτ =μ(η–

ηk

 ).Then for t∈

min{,

τ},I–tμB is a contraction with a constant –tτ.

3 Main results

LetHbe a real Hilbert space andTbe a nonexpansive mapping onH. Assume that the set

Fix(Sn) is nonempty, that is,Fix(Sn) :={x∈H:Snx=x} =∅. SinceFix(Sn) is closed convex,

the nearest point projection fromHontoFix(Sn) is well defined. Recall also thatf is an L-Lipschitz mapping onHwith coefficientL> . LetBis ak-Lipschitzian andη-strongly monotone operator onHwith coefficientsk>  andη> .

Now givef is anL-Lipschitz mapping onHwith coeﬃcientL> ,t∈(, ). Let  <μ< η/k_{,  <}_γ _<_μ₍_η_–μk

 )/L=τ/L. Consider a mappingWtonHdeﬁned by

Wtx=tγf(x) + (I–μtB)Snx, n> .

According to Lemma ., we can easily see that

Wtx–Wty ≤tγ f(x) –f(y) + (I–μtB)Tx– (I–μtB)Sny

≤ –t(τ–γL)x–y. (.)

Theorem . Let H be a real Hilbert space,let F be a bifunction H×H→Rsatisfying

(A)-(A),and let{Sn}be a sequence of nonexpansive mappings on H.Let Ai:H→H, i= , , . . . ,N,beαi-inverse-strongly monotone mappings,let Mi:H→H,i= , , . . . ,N,be maximal monotone mappings such that:= (∞_n_=Fix(Sn))∩EP(F)∩(

N

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andη-strongly monotone operator on H with coeﬃcients k> andη> .Let{xn},{yn},and

{un}be sequences generated by x∈H and

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

F(un,y) +_r_ny–un,un–xn ≥, ∀y∈H, yn=JMN,λN,n(I–λN,nAN)· · ·JM,λ,n(I–λ,nA)un,

xn+=nγf(xn) + (I–μnB)Snyn,

(.)

for all n∈N,whereλi,n∈(, αi],i∈ {, , . . . ,N},satisfy(H)-(H),{n} ⊂[, ]and{rn} ⊂

(,∞)satisfy

(C) lim_n_→∞n= ;

(C) ∞_n_=n=∞;

(C) ∞_n_=|n+–n|<∞;

(C) lim infn→∞rn> ;

(C) ∞_n_=|rn+–rn|<∞.

Suppose that∞_n_=sup{Sn+z–Snz:z∈K}<∞for any bounded subset K of H.Let S be a mapping of H into itself deﬁned by Sx=limn→∞Snx for all x∈H,and suppose that Fix(S) =

_∞

n=Fix(Sn).Then{xn},{yn},and{un}converge strongly to z,where z=P(I–μB+γf)(z) is a unique solution of the variational inequality

(μB–γf)z,z–x≤, x∈. (.)

Proof Using the deﬁnition ofk_nin Lemma ., we haveyn=Nnun. We divide the proof

into several steps.

Step . The sequence{xn}is bounded.

Letp∈. Using the fact thatJMk,λk,n(I–λk,nAk),k∈ {, , . . . ,N}, is nonexpansive and

p=JMk,λk,n(I–λk,nAk)p, we have

yn–p= Nnun–Nnp ≤ un–p ≤ Trxn–Trp ≤ xn–p

for alln≥. Then we have

xn+–p= nγf(xn) + (I–μnB)Snyn–p

≤n γf(xn) –μBp +I–μnByn–p

≤n γf(xn) –μBp + ( –nτ)xn–p

≤n γ

f(xn) –f(p)

+γf(p) –μBp + ( –nτ)xn–p

≤nγLxn–p+n γf(p) –μBp + ( –nτ)xn–p

= –n(γ¯–γL)

xn–p+n γf(p) –μBp

= –n(τ–γL)

xn–p+n(τ–γL)

γf(p) –μBp

(τ–γL) . (.)

It follows from (.) and induction that

xn–p ≤max

x–p,

γf(p) –μBp τ–γL

, n> .

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Step . We show thatxn+–xn →.

SinceI–λAis nonexpansive,yn=Nnunandyn+=Nn+un+, it follows that

yn+–yn= nN+un+–Nnun

≤ N_n_+un–Nnun + Nn+un–Nn+un+

≤ N_nun–Nn+un +un–un+. (.)

Then we have

xn+–xn+

= n+γf(xn+) + (I–μn+B)Sn+yn+–nγf(xn) – (I–μnB)Snyn

= (I–μn+B)(Sn+yn+–Sn+yn) + (n–n+)μBSn+yn

+ ( –μnB)(Sn+yn–Snyn) + (n+–n)γf(xn) +n+γ

f(xn+) –f(xn)

≤( –n+τ)yn+–yn+|n–n+|μBSn+yn+ ( –nτ)Sn+yn–Snyn

+|n+–n| γf(xn) +n+γLxn+–xn

≤( –n+τ)yn+–yn+n+γLxn+–xn+|n–n+|

μBSn+yn+ γf(xn)

+Sn+yn–Snyn

≤( –n+τ) Nnun–Nn+un +un–un+

+n+γLxn+–xn

+|n–n+|M+sup

Sn+z–Snz:z∈ {yn}

, (.)

where M =sup{max{μBSn+yn,γf(xn)}:n ≥} <∞. On the other hand, using

Lemma ., we have

un+–un=Tn+xn+–Tnxn ≤ Tn+xn+–Tn+xn+Tn+xn–Tnxn

≤ xn+–xn+

 –rn+

rn

Tnxn+xn

. (.)

Combining (.) and (.), we have

xn+–xn+

≤ –n+(γ¯–γ β)

xn+–xn+

 –rn+

rn

Tnxn+xn

+|n–n+|M

+ N_n_+un–Nnun +sup

Sn+z–Snz:z∈ {un}

. (.)

From boundedness of{un}and Lemma ., using the condition of (H)-(H), we obtain

∞

n=

N_n_+un–Nnun <∞. (.)

Since{yn}is bounded, it follows that

∞

n=sup{Sn+z–Snz:z∈K}<∞. Hence, using

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Step . We now show that

lim n→∞

k

nun–kn–un = , k= , , . . . ,N. (.)

Indeed, letp∈. It follows from the ﬁrmly nonexpansiveness ofJMk,λk,n(I–λk,nAk) that

k_nun–p



= JMk,λk,n(I–λk,nAk)

k–

n un–JMk,λk,n(I–λk,nAk)p 

≤k_nun–p,kn–un–p

=  

k

nun–p + kn–un–p – knun–kn–un 

, (.)

for eachk∈ {, , . . . ,N}. Thus we get k_nun–p



≤ k_n–un–p



– k_nun–kn–un

 ,

which implies that for eachk∈ {, , . . . ,N},

yn–p = Nnun–p



≤ _nun–p

 –

N

k=

k_nun–kn–un



≤ un–p– nkun–kn–un



≤ xn–p– nkun–kn–un



. (.)

Using Lemma . and noting that · _{is convex, we derive from (.)}

xn+–p = nγf(xn) + (I–μnB)Snyn–p



= ( –μnB)(Snyn–p) +n

γf(xn) –Bp



≤( –nτ)Snyn–p+ n

γf(xn) –μBp,xn+–p

≤( –nτ)yn–p+ n

γf(xn) –μBp,xn+–p

≤( –nτ)

xn–p– nkun–kn–un 

+ nγ

f(xn) –f(p),xn+–p

+ n

γf(p) –μBp,xn+–p

≤( –nτ)

xn–p– nkun–kn–un



+ nγLxn–pxn+–p+ n f(p) –μBp xn+–p. (.)

PutM=supn≥{f(p) –μBpxn+–p}. It follows from (.) that

( –nτ) knun–kn–un



≤ xn–p–xn+–p+

(nτ)– nτ

xn–p

+nγLxn–pxn+–p+ nM ≤ xn–xn+

xn–p+xn+–p

+n

nτ– τ

xn–p

+nγLxn–pxn+–p+ nM.

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Step . We provelimn→∞un–xn= .

We note from (.),

xn–Snyn ≤ xn–Sn–yn–+Sn–yn––Sn–yn+Sn–yn–Snyn

≤n– γf(xn–) –μBSn–yn– +yn––yn

+supSn+z–Snz:z∈ yn

. (.)

Sincen→,limn→∞yn+–yn= , andsup{Sn+z–Snz:z∈ {yn}} →, we get

xn–Snyn →. (.)

Letp∈. Sinceun=Trnxn, it follows from Lemma . that

un–p=Trnxn–Trnp≤ Trnxn–Trnp,xn–p=un–p,xn–p

≤  

un–p+xn–p–un–xn

,

and henceun–p≤ xn–p–un–xn. Therefore, using Lemma . and (.), we

have

xn+–p ≤ –nτ

yn–p+ nγ

+ n

≤ –nτ

un–p+ nγ

+ n

≤( –nτ)

xn–p–un–xn

+ nγLxn–pxn+–p

+ n γf(p) –μBp xn+–p ≤ xn–p+n

τ– τxn–p– ( –nτ)un–xn

+ nγLxn–pxn+–p

+ n γf(p) –μBp xn+–p,

and hence

(I–nτ)un–xn

≤ xn–p–xn+–p+n

τ– τxn–p

+ nγLxn–pxn+–p+ n γf(p) –μBp xn+–p ≤ xn–xn+

xn–p+xn+–p

+n

τ– τxn–p

+ nγLxn–pxn+–p+ n γf(p) –μBp xn+–p. (.)

Since{xn}is bounded,n→ andlimn→∞xn–xn+= , it follows that

lim

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Next we provelimn→∞un–yn= .

un–yn= Nnun–un

≤ N_nun–Nn–un + Nn–un–Nn–un

+· · ·+ _nun–nun + nun–nun +un–un.

From (.), we obtain

lim

n→∞un–yn= . (.)

In addition, according toxn–yn ≤ xn–un+un–yn, we have

lim

n→∞xn–yn= . (.)

It follows from (.), (.) and the inequalityyn–Snyn ≤ yn–xn+xn–Snynthat limn→∞yn–Snyn= . Since

Syn–yn ≤ Syn–Snyn+Snyn–yn

≤supSz–Snz:z∈ {yn}

+Snyn–yn,

for alln∈N, it follows that

lim

n→∞Syn–yn= . (.)

Step . We showω∈(∞_n_=Fix(Sn))∩EP(F)∩(

N

i=I(Ai,Mi)).

Since{xn}is bounded, there exists a subsequence{xni}of{xn}which converges weakly to ω. From (.), we obtain{uni}which converges weakly toω. From (.), it follows

yni ω. We showω∈EP(F). According to (.) and (A),



rn

y–un,un–xn ≥F(y,un),

and hence

y–uni,

uni–xni

rni

≥F(y,uni).

Since uni_r–xni

ni → anduni ω, from (A) it follows that ≥F(y,ω) for ally∈H. Fort with  <t≤ andy∈H, letyt=ty+ ( –t)ω, then we get ≥F(yt,ω). So, from (A) and

(A), we have

 =F(yt,yt)≤tF(yt,y) + ( –t)F(yt,ω)≤tF(yt,y),

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We showω∈∞_n_=Fix(Sn). Assume thatω∈/

∞

n=Fix(Sn), then we haveω=Sω. It

fol-lows, by Opial’s condition and (.), that

lim inf

n→∞ yn–ω<lim infn→∞ yn–Sω

≤lim inf n→∞

yn–Syn+Syn–Sω

≤lim inf

n→∞ yn–ω.

This is a contradiction. Henceω∈∞_n_=Fix(Sn).

We now show thatω∈N_i_=I(Ai,Mi). In fact, sinceAiisαi-inverse-strongly monotone,

thenAi,i= , , . . . ,N, is an αi-Lipschitz continuous monotone mapping andD(Ai) =H,

i= , , . . . ,N. It follows from Lemma . thatMi+Ai,i= , , . . . ,N, is maximal monotone.

Let (p,g)∈G(Mi+Ai),i= , , . . . ,N, that is,g–Aip∈(Mip),i= , , . . . ,N. Sinceknun= JMk,λk,n(I–λk,nAk)

k–

n un, we havekn–un–λk,nAknk–un∈(I+λk,nMk)(knun), that is,



λn,k

k_n–un–knun–λN,nAkkn–un

∈Mk

k_nun

.

By the maximal monotonicity ofMi+Ai,i= , , . . . ,N, we have

p–k_nun,g–Akp–



λk,n

k_n–un–knun–λk,nAkkn–un

≥,

which implies

p–k_nun,g

≥

p–k_nun,Akp+



λn,k

k_n–un–knun–λk,nAkkn–un

=

p–k_nun,Akp–Akknun+Akknun–Akkn–un+



λk,n

k_n–un–knun

≥ +p–k_nun,Akknun–Akkn–un

+

p–k_nun,



λk,n

k_n–un–knun

(.)

for k ∈ {, , . . . ,N}. From (.), it follows limn→∞nkun–kn–un = , especially, k_n_iuni ω. Since Ak, k = , . . . ,N, are Lipschitz continuous operators, we have Akkn–un–Akknun →. So, from (.), we have

lim i→∞

p–k_n

iuni,g

=p–ω,g ≥.

SinceAk+Mk,k∈ {, , . . . ,N}is maximal monotone, this implies that ∈(Mk+Ak)(ω), k∈ {, , . . . ,N},i.e.,ω∈N_i_=I(Ai,Mi). So, we obtain the result.

Step . We show that

lim sup n→∞

(μB–γf)z,z–xn

≤,

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

lim i→∞

(μB–γf)z,z–xni

=lim sup n→∞

(μB–γf)z,z–xn

.

By the proof of Step , we obtain that

lim sup n→∞

(μB–γf)z,z–xn

= lim i→∞

(μB–γf)z,z–xni

=(μB–γf)z,z–ω≤.

Step . We prove thatxn→ω.

Using Lemma . and (.), we obtain

xn+–ω

= nγf(xn) + (I–μnB)Snyn–ω 

= n

γf(xn) –μBω

+ (I–μnB)(Snyn–ω) 

≤ (I–μnB)(Snyn–ω) + n

γf(xn) –μBω,xn+–ω

≤( –nτ)yn–ω+ n

γf(xn) –f(ω),xn+–ω

+ n

γf(ω) –μBω,xn+–ω

≤( –nτ)xn–ω+ nγLxn–ωxn+–ω+ n

≤( –nτ)xn–ω+nγL

xn–ω+xn+–ω

+ n

.

This implies that

xn+–ω≤

 – nτ+ (nτ)+nγL

 –nγL

xn–ω+

n

 –nγL

=

 –(n(τ) _–_τ₎  –nγL

xn–ω+

(nτ)

 –nγL

xn–ω

+ n  –nγL

≤( –γn)xn–ω+δn,

whereγn=_–n(τ_n–_γLγL) andδn=_–n_n_γ_L(nτxn–ω+ γf(ω) –μBω,xn+–ω). It is easily veriﬁed thatγn→,

∞

n=γn=∞, andlim supn→∞δn/γn≤. Hence, by Lemma ., the

sequence{xn}converges strongly toω. Furthermore, from (.) and (.), we obtain that

the sequences{yn}and{un}converge strongly toω.

LetB≡Iandγ =  in Theorem .; we obtain the following corollary.

Corollary . Let H be a real Hilbert space,let F be a bifunction H×H→R satisfying

(A)-(A),and let Snbe a sequence of nonexpansive mappings on H.Let A:H→H be anα-inverse-strongly monotone mapping,and let M:H→H_{be a maximal monotone} mapping such that=∞_n_=Fix(Sn)∩EP(F)∩(

N

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x∈H and

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

F(un,y) +_r_ny–un,un–xn ≥, ∀y∈H,

yn=JMN,λN,n(I–λN,nAN)· · ·JM,λ,n(I–λ,nA)un, ∀n> ,

xn+=nf(xn) + (I–n)Snyn,

(.)

for all n∈N,whereλi,n∈(, αi],i∈ {, , . . . ,N},satisfy(H)-(H),{n} ⊂[, ]and{rn} ⊂

(,∞)satisfy:

(C) lim_n_→∞n= ;

(C) ∞_n_=n=∞;

(C) ∞_n_=|n+–n|<∞;

(C) lim infn→∞rn> ;

(C) ∞_n_=|rn+–rn|<∞.

Suppose that∞_n_=sup{Sn+z–Snz:z∈K}<∞for any bounded subset K of H.Let S be a mapping of H into itself deﬁned by Sx=limn→∞Snx for all x∈H,and suppose that Fix(S) =

∞

n=Fix(Sn).Then{xn},{yn},and{un}converge strongly to z,where z=P(I–γf)(z) is a unique solution of the variational inequality

(I–γf)z,z–x≤, x∈.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and signiﬁcantly in writing this paper. All authors read and approved the ﬁnal manuscript.

Acknowledgements

This work is supported in part by National Natural Science Foundation of China (71272148), the Ph.D. Programs Foundation of Ministry of Education of China (20120032110039) and China Postdoctoral Science Foundation (Grant No. 20100470783).

Received: 14 April 2012 Accepted: 13 March 2013 Published: 10 April 2013

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doi:10.1186/1029-242X-2013-165