A general iterative algorithm for monotone operators with λ hybrid mappings in Hilbert spaces

R E S E A R C H

Open Access

A general iterative algorithm for monotone

operators with

λ-hybrid mappings in Hilbert

spaces

Chung-Chien Hong

*

*_{Correspondence:}

[email protected] Department of Industrial Management, National Pingtung University of Science and Technology, Pingtung, Taiwan

Abstract

LetCbe a nonempty closed convex subset of a Hilbert spaceH, letB,Gbe two set-valued maximal monotone operators onCintoH, and letg:H→Hbe a

k-contraction with 0 <k< 1.A:C→His an

α-inverse strongly monotone mapping,

V:H→His a

γ

¯-strongly monotone andL-Lipschitzian mapping with

γ

¯> 0 and

L> 0,T:C→Cis a

λ-hybrid mapping. In this paper, a general iterative scheme for

approximating a point ofF(T)∩(A+B)–1_0∩G–1_{0=∅is introduced, whereF(T) is the} set of fixed points ofT, and a strong convergence theorem of the sequence

generated by the iterative scheme is proved under suitable conditions. As applications of our strong convergence theorem, the related equilibrium and variational problems are also studied.

MSC: 47H05; 47H10; 58E35

Keywords: fixed point; maximal monotone operator; inverse-strongly monotone mapping; resolvent; equilibrium problem; variational inequality

1 Introduction

Throughout this paper,Hdenotes a real Hilbert space,Ca nonempty closed convex subset ofH,Nthe set of all natural numbers andRthe set of all real numbers. For a self-mapping

TonH,F(T) denotes the set of all fixed points ofT.

A set-valued mapB:H→Hwith domainD(B) :={x∈H:Bx=∅}is called monotone if

x–y,u–v ≥

for allx,y∈D(B) and for anyu∈Bx,v∈By;Bis said to be maximal monotone if its graph

{(x,u) :x∈H,u∈B(x)} is not properly contained in the graph of any other monotone operator. For a positive real numberr, the resolventJ_rBof a monotone operatorBforris a single-valued mappingJB

r :H→D(B) defined byz=JrB(x) if and only ifx∈z+rBz, that

is,JB

r(x) = (I+rB)–(x) for anyx∈H, whereIis the identity mapping onH. The Yosida

approximationArofBforr>  is defined asAr=_r(I–JrB). It is known [] thatArx∈B(JrBx)

for allx∈H.

The fixed point theory for nonexpansive mappings can be applied to the problem of finding a zero pointvof a maximal monotone operatorBonH, that is, finding a point

v∈Hsatisfying ∈B(v). In the sequel, we shall denote the set of all zero points ofBby

B–_.

A self-mappingVonHis calledγ¯-strongly monotone if there is a positive real number

¯

γ such that

x–y,Vx–Vy ≥ ¯γ x–y , ∀x,y∈H;

Vis calledL-Lipschitzian if there is a positive real numberLsuch that

Vx–Vy ≤L x–y , ∀x,y∈H.

A mappingA:C→His said to beα-inverse strongly monotone if there is a positive real numberαsuch that

x–y,Ax–Ay ≥α Ax–Ay , ∀x,y∈C.

As easily seen, anα-inverse strongly monotone mapping is_α-Lipschitzian onC. Forλ∈R, a mappingT:C→His said to beλ-hybrid if

Tx–Ty ≤ x–y +λx–Tx,y–Ty, ∀x,y∈C.

Whenλ= ,Tis called nonspreading. It is known thatF(T) is closed and convex provided

T is aλ-hybrid self-mapping onC,cf.[].

Recently, Lin and Takahashi [] introduced an algorithm for finding a pointp∈(A+

B)–_∩G–_{, whereAis anα-inverse strongly monotone mapping ofCintoH, andB,} Gare two set-valued maximal monotone operators withD(B)⊂CandD(G)⊂C. More precisely, let gbe ak-contraction and V be aγ¯-strongly monotone andL-Lipschitzian mapping. Chooseμ,γ ∈Rso that  <_Lγ¯ and  <γ <

¯

γ–L_μ

k . Then the algorithm starts

with anyx∈Hand generates a sequence{xn}iteratively by xn+=αnγg(xn) + (I–αnV)JσBn(I–σnA)J

G

rnxn, n∈N, ()

where{αn} ⊆(, ),{σn} ⊆(,∞), and{rn} ⊆(,∞) satisfy

lim n→∞αn= ,

∞

n=

αn=∞,

∞

n=

|αn–αn+|<∞,  <a≤σn≤α,

∞

n=

|σn–σn+|<∞, lim inf

n→∞ rn>  and ∞

n=

|rn–rn+|<∞.

They proved thatP(A+B)–_∩G–_(I–V+γg) has a unique fixed pointpin, and thisp

is also a unique solutionp∈(A+B)–_∩G–_{ to the hierarchical variational inequality}

(V–γg)p,q–p, ∀q∈.

On the other hand, Manaka and Takahashi [] used the algorithm

xn+=αnxn+ ( –αn)T

JσBn(I–σnA)xn

() to find a pointp∈F(T)∩(A+B)– for a nonspreading mappingT, anα-inverse strongly monotone mappingAand a maximal monotone operatorBunder the conditions

 <c≤αn≤d<  and  <a≤σn≤b< α,

wherea,b,c,d∈Rare fixed. They proved that the sequence{xn}constructed above

con-verges weakly to a pointp∈F(T)∩(A+B)–_.

Very recently, Liuet al.[] modified the iterative scheme () to approximate a point

p∈F(T)∩(A+B)– for a nonspreading mappingT, anα-inverse strongly monotone mapping and a maximal monotone operatorB. For anyu∈H, they putxto be any point

ofHand define recursively for alln∈N,

⎧ ⎪ ⎨ ⎪ ⎩

zn=JαBn(I–σnA)xn,

yn=_n

n–

i=Tizn, xn+=αnu+ ( –αn)yn,

()

where{αn}is a suitable sequence in [, ].

Motivated by the above works, in this paper we introduce a general iterative scheme for approximating a point ofF(T)∩(A+B)–∩G–=∅, whereTis aλ-hybrid self-mapping onC,A:C→His anα-inverse strongly monotone mapping andBandGare two maximal monotone operators. A strong convergence theorem of the sequence generated by our iterative scheme is proved under suitable conditions. Our result improves and generalizes the main theorem of Lin and Takahashi []. As applications of our strong convergence theorem, the related equilibrium and variational problems are also studied.

2 Preliminaries

In order to facilitate our investigation, in what follows we recall some basic facts. A map-pingT:C→His said to be

(i) nonexpansive if

Tx–Ty ≤ x–y , ∀x,y∈C;

(ii) firmly nonexpansive if

Tx–Ty ≤ x–y,Tx–Ty, ∀x,y∈C.

The metric projectionPCfromHontoCis the mapping that assigns eachx∈Hthe unique

pointPCxinCwith the property x–PCx =min

y∈C y–x .

It is known thatPCis nonexpansive and characterized by the inequality: for anyx∈H,

x–PCx,y–PCx ≤, ∀y∈C, ()

For anyx,y∈H, one has

x+y ≤ x + y,x+y. ()

Lemma .(Demiclosedness principle) [] Let T be a nonexpansive self-mapping on a nonempty closed convex subset C ofH,and suppose that{xn}is a sequence in C such that

{xn}converges weakly to some z∈C andlimn→∞ xn–Txn = .Then Tz=z.

Fors> , the resolventJ_sBof the maximal monotone operatorBonHhas the following properties,cf.[].

Lemma . Let B be a maximal monotone operator onH.Then,for any s> ,

(a) JB

s is single-valued and firmly nonexpansive;

(b) D(JB

s) =HandF(JsB) =B–.

The following lemma can be derived easily from the resolvent identity of a monotone operatorB:

J_sBx=J_tB

t sx+

 –t

s

J_sBx

for anys,t>  and anyx∈H.

Lemma . Let B be a monotone operator onH.Then,for any s,t∈Rwith s,t> and for any x∈H,

J_sBx–J_tBx≤|s–t| s x–J

B sx.

WhenBis maximal monotone, a different proof may be found in Takahashiet al.[].

Lemma .[] Let A:C→Hbe anα-inverse strongly monotone mapping,and let B be a maximal monotone operator onHwithD(B)⊆C.Then,for anyσ> ,one has(A+B)– =

F(JσB(I–σA)).

Lemma .[] Let A:C→Hbe anα-inverse strongly monotone mapping.Then,for any σ∈(, α], (I–σA)is nonexpansive.

Lemma .[] Let g :H→Hbe a k-contraction with  <k< ,let V :H→Hbe a

¯

γ-strongly monotone and L-Lipschitzian mapping withγ¯> and L> ,and letγ be a real number satisfying <γ <γ_k¯.Then V–γg is a(γ¯–γk)-strongly monotone and(L+γk) -Lipschitzian mapping.Furthermore,for any nonempty closed convex subsetofH,P(I– V+γg)has a unique fixed point p∈,which is also a unique solution of the variational inequality

(V–γg)z,q–z≥, ∀q∈.

Lemma .[] Let{sn}be a sequence of nonnegative real numbers satisfying sn+≤( –αn)sn+αnμn+νn, n∈N,

(i) {αn} ⊆[, ],

∞

n=αn=∞;

(ii) lim sup_n→∞μn≤;

(iii) {νn} ⊆[,∞)and

∞

n=νn<∞. Thenlimn→∞sn= .

3 Strong convergence theorems

We begin the proof of the main result of this paper. As the proof is rather lengthy, we divide the proof into many assertions.

Theorem . Suppose that

(..) G:C→HandB:C→Hare two maximal monotone operators with D(G)⊆CandD(B)⊆C;

(..) g:H→His ak-contraction,A:C→His anα-inverse strongly monotone mapping,andV:H→His aγ¯-strongly monotone andL-Lipschitzian mapping withγ¯> andL> ;

(..) T:C→Cis aλ-hybrid mapping; (..) :=F(T)∩(A+B)–_∩G–_=∅;

(..) μandγ are two real numbers satisfying <μ<_Lγ¯ and <γ <

¯

γ–Lμ

k .

Start with any x∈Hand define a sequence{xn}iteratively by

⎧ ⎪ ⎨ ⎪ ⎩

zn=JσBn(I–σnA)J

G

rnxn, n∈N,

yn=_n n–

i=Tizn, n∈N, xn+=αnγg(xn) + (I–αnV)yn,

()

where the sequences{αn},{σn}and{rn}verify the following conditions:

(..) {αn}is a sequence in[, ]withlimn→∞αn= and

∞

n=αn=∞;

(..) {σn}and{rn}are sequences in(,∞)so thatlim infn→∞rn> and there are a,b∈Rwith <a≤σn≤b< αfor alln∈N.

Then the sequence{xn}constructed by algorithm()converges strongly to a point p∈, where p is the unique fixed point of P(I–V +γg),and this point pis also a unique solution of the hierarchical inequality

(V–γg)p,q–p≥, ∀q∈. ()

Proof In what follows,pis a point in,τ=γ¯–L_μ, andun=JrGnxnfor alln∈N. •Assertion (A):There is N∈Nsuch that

(I–αnV)yn– (I–αnV)p≤( –αnτ) yn–p .

Sincelimn→∞αn= , there isN∈Nsuch that  –αnτ >  andμ–αn>  for alln≥N.

Then, asVisγ¯-strongly monotone andL-Lipschitzian, we have that for alln≥N,

(I–αnV)yn– (I–αnV)p 

=(yn–p) –αn(Vyn–Vp) 

≤ yn–p – αnγ¯ yn–p +αnL yn–p 

= – αnτ–αnLμ+αnL

yn–p 

≤ – αnτ+αnτ

yn–p –αnL(μ–αn) yn–p 

≤( –αnτ) yn–p , ()

and so the assertion holds.

•Assertion (B):The sequences{xn},{yn},{zn},{un},{Aun},{Vyn},{g(xn)}and{Tnzn}are bounded.

We firstly show that

yn–p ≤ zn–p ≤ xn–p , ∀n∈N. ()

On account ofp=JB

σn(I–σnA)pby Lemma .,p=J

G

rnpby assumption and the facts that a

resolvent is nonexpansive andAisα-inverse strongly monotone, we have

zn–p =JσBn(I–σnA)J

G rnxn–J

B

σn(I–σnA)J

G rnp



≤(I–σnA)JrGnxn– (I–σnA)J

G rnp



=J_rG_nxn–JrGnp–σn

AJ_rG_nxn–AJrGnp



=J_rG_nxn–JrGnp



– σnun–p,Aun–Ap+σn Aun–Ap  ()

≤ xn–p – σnα Aun–Ap +σn Aun–Ap 

= xn–p –σn(α–σn) Aun–Ap  ()

≤ xn–p ,

where the last inequality follows from the hypothesis that α≥σn. Therefore,

zn–p ≤ xn–p .

SinceTisλ-hybrid, we have, for anyn∈N,

Tzn–p = Tzn–Tp ≤ zn–p +λzn–Tzn,p–Tp= zn–p ,

and so, by induction, it comes easily that

Tizn–p≤ zn–p for alli,n∈N. ()

Consequently,

yn–p =



n n–

i=

Tizn–p

≤ 

n n–

i=

≤ 

n n–

i= zn–p

= zn–p ≤ xn–p .

Next, we show that{ xn–p }is bounded. Indeed, asp= (I–αnV)p+αnVp, we have xn+–p

=αnγg(xn) + (I–αnV)yn–αnVp– (I–αnV)p

≤αn

γg(xn) –Vp+(I–αnV)yn– (I–αnV)p

≤αnγg(xn) –γg(p)+αnγg(p) –Vp+(I–αnV)yn– (I–αnV)p

≤αnγk xn–p +αnγg(p) –Vp+(I–αnV)yn– (I–αnV)p,

which together with Assertion (A) implies that for alln≥N,

xn+–p ≤αnγk xn–p +αnγg(p) –Vp+ ( –αnτ) yn–p

≤ –αn(τ–γk)

xn–p +αnγg(p) –Vp

≤ xn–p +γg(p) –Vp,

from which we inductively deduce that

xn–p ≤ x–p +γg(p) –Vp.

Thus,{xn}is bounded, and so are{yn},{zn}by (). And then the boundedness of{Tnzn}

follows from (). The fact that{Vyn}and{g(xn)}are bounded is due to the fact thatVand gareL-Lipschitzian andk-contraction, respectively.

Finally, from un–p = JσGnxn–J

G

σnp ≤ xn–p , we deduce that{un}is bounded, and {Aun}is bounded comes fromAisα-Lipschitzian.

•Assertion (C):lim_n→∞ xn+–xn = .

We at first show thatlimn→∞ yn+–yn = . By Assertion (B), we can choose a positive

real numberMso that

max

sup n∈N

Tnzn,sup n∈N

yn

≤M.

Then, for alln∈N,

yn+–yn =



n+ 

n

i= Tizn–



n n–

i= Tizn

≤ 

n+ T

n_z n+



n–



n+ 

n– i= Tizn

= 

n+ T

n_z n+



Now, for alln≥N, we have from Assertion (A) that

xn+–xn+

=αn+γg(xn+) + (I–αn+V)yn+–αnγg(xn) – (I–αnV)yn

≤αn+γg(xn+) –αn+γg(xn)+αn+γg(xn) –αnγg(xn)

+(I–αn+V)yn+– (I–αn+V)yn+(I–αn+V)yn– (I–αnV)yn

≤αn+γk xn+–xn +γ|αn+–αn|g(xn)+ ( –αn+τ) yn+–yn

+|αn+–αn| Vyn .

Consequently, using condition (..), Assertion (B) andlimn→∞ yn+–yn = , we get limn→∞ xn+–xn = .

•Assertion (D):limn→∞ Aun–Ap = .

Using (), it follows from (), () and () that

xn+–p 

=(I–αnV)yn– (I–αnV)p+αnγg(xn) –Vp 

≤(I–αnV)yn– (I–αnV)p 

+ αn

γg(xn) –Vp,xn+–p

by ()

≤( –αnτ) yn–p + αn

γg(xn) –Vp,xn+–p

by ()

≤( –αnτ) zn–p + αn

γg(xn) –Vp,xn+–p

by () ()

≤( –αnτ)

xn–p –σn(α–σn) Aun–Ap 

+ αn

γg(xn) –Vp,xn+–p

by ()

≤( –αnτ) xn–p – ( –αnτ)σn(α–σn) Aun–Ap 

+ αn

γg(xn) –Vp,xn+–p

≤ xn–p +αnτ xn–p – ( –αnτ)σn(α–σn) Aun–Ap 

+ αn

γg(xn) –Vp,xn+–p

.

Hence, on account of  <a≤σn≤b< αfor alln∈N, we have

( –αnτ)a(α–b) Aun–Ap 

≤( –αnτ)σn(α–σn) Aun–Ap 

≤ xn–p – xn+–p +αnτ xn–p 

+ αn

γg(xn) –Vp,xn+–p

≤ xn–xn+

xn–p + xn+–p

+α_nτ xn–p 

+ αn

γg(xn) –Vp,xn+–p

,

which together with Assertions (B) and (C) implies thatlim_n→∞ Aun–Ap = .

From

xn–yn ≤ xn–xn+ + xn+–yn

= xn–xn+ +αnγg(xn) + (I–αnV)yn–yn

= xn–xn+ +αnγg(xn) –Vyn,

condition (..) and Assertions (B) and (C), we conclude thatlim_n→∞ xn–yn = .

AsJG

rnis firmly nonexpansive, one has

 un–p = JrGnxn–J

G rnp



≤xn–p,un–p

= xn–p + un–p – un–xn ,

and so

un–p ≤ xn–p – un–xn . ()

In addition, sinceAisα-inverse strongly monotone, we see from () that

zn–p ≤JrGnxn–J

G rnp



– σnun–p,Aun–Ap+σn Aun–Ap 

= un–p – σnun–p,Aun–Ap+σn Aun–Ap 

≤ un–p – σnα Aun–Ap +σn Aun–Ap 

= un–p –σn(α–σn) Aun–Ap . ()

Then (), () and () give us that

xn+–p 

≤( –αnτ) zn–p + αn

γg(xn) –Vp,xn+–p

by ()

≤( –αnτ)

un–p –σn(α–σn) Aun–Ap 

by () + αn

γg(xn) –Vp,xn+–p

≤( –αnτ)

xn–p – un–xn 

– ( –αnτ)σn(α–σn) Aun–Ap 

+ αn

γg(xn) –Vp,xn+–p

by ()

≤ xn–p +αnτ xn–p – ( –αnτ) un–xn 

– ( –αnτ)σn(α–σn) Aun–Ap + αn

γg(xn) –Vp,xn+–p

. Therefore

( –αnτ) un–xn 

≤ xn–p – xn+–p +αnτ xn–p 

– ( –αnτ)σn(α–σn) Aun–Ap + αn

≤ xn–xn+

xn–p + xn+–p

+α_nτ xn–p 

– ( –αnτ)σn(α–σn) Aun–Ap + αn

γg(xn) –Vp,xn+–p

.

By Assertions (C), (D) and condition (..), we obtain

lim

n→∞ xn–un = .

•Assertion (F):is a nonempty closed convex subset of H.

Sinceis nonempty by assumption, it suffices to show thatis closed and convex. From Lemma ., we have that for anyσ> ,

(A+B)– =FJ_σB(I–σA).

In case  <σ≤α, we have thatI–σAis nonexpansive by Lemma .. ThenJB

σ(I–σA) is

nonexpansive, and so (A+B)–_{ =F(J}B

σ(I–σA)) is closed and convex. In the like manner,

for anyr> ,G– =F(J_rG) is closed and convex. Besides, it is shown in [] thatF(T) is closed and convex. Hence comes the conclusion for.

•Assertion (G):P(I–V+γg)has a unique fixed point pin,and this pis also a unique solution p∈to the hierarchical variational inequality()

(V–γg)p,q–p≥, ∀q∈.

Taking into account thatis a nonempty closed convex subset, the conclusion follows from Lemma ..

•Assertion (H):Let pbe the unique solution to the hierarchical variational inequality

().Then lim sup

n→∞

(V–γg)p,xn–p

≤.

Choose a subsequence{xni}of{xn}so that

lim sup n→∞

(V–γg)p,xn–p

=lim i→∞

(V–γg)p,xni–p

and{xni}converges weakly to w∈H. In view of Assertion (E), we see that both of the

sequences{uni}and{yni}converge weakly tow. We at first show thatw∈F(T). SinceT

isλ-hybrid, one has Tx–Ty ≤ x–y +λx–Tx,y–Tyfor allx,y∈C. Expressλas

λ= ( –δ). Then, for allx,y∈C, we have

Tx–Ty 

≤ x–Ty+Ty–y + ( –δ)x–Tx,y–Ty

= x–Ty + x–Ty,Ty–y+ Ty–y + ( –δ)x–Tx,y–Ty

= x–Ty + Ty–y + x– ( –δ)x+ ( –δ)Tx–Ty,Ty–y

that is,

≤ x–Ty – Tx–Ty + Ty–y + δx+ ( –δ)Tx–Ty,Ty–y. In particular, for alli∈N, allznand ally∈C,

≤Tizn–Ty 

–Ti+zn–Ty 

+ Ty–y 

+ δTizn+ ( –δ)Ti+zn–Ty,Ty–y

.

Summing these inequalities fromi=  ton–  and dividing byn, we obtain

≤ 

n

zn–Ty – Tnzn–Ty 

+ Ty–y 

+ 

δyn+ ( –δ)

n+ 

n yn+– zn

n

–Ty,Ty–y

. ()

Replacingnwithniin () and lettingi→ ∞, we get via Assertion (B) that

≤ Ty–y + δw+ ( –δ)w–Ty,Ty–y. () Puttingy=win (), we arrive at ≤– Tw–w _{. This shows thatw∈F(T).}

Since  <a≤σni≤b< α,{σni}has a convergent subsequence. For simplicity, we

as-sume that{σni}converges to a numberσ∈[a,b]. Note that for alln∈N, JσB(I–σA)un–zn

≤JσB(I–σA)un–JσB(I–σnA)un+JσB(I–σnA)un–zn

≤(I–σA)un– (I–σnA)un+JσB(I–σnA)un–JσBn(I–σnA)un ≤ |σn–σ| Aun +|

σn–σ|

σ J

B

σvn–vn, ()

wherevn= (I–σnA)un. Furthermore, we have for alln∈N, JσBvn–vn≤JσBvn–p+ vn–p

=JσBvn–JσBp+ vn–p

≤ vn–p

and

vn–p =(I–σnA)un– (I–σnA)p–σnAp

≤ un–p +b Ap by Lemma ..

Hence,{JB

σvn–vn}is bounded. Now, replacenwithniin () and note that both{Auni}

and{JB

σvni–vni}are bounded. Lettingi→ ∞, we get that

Applying Lemma . to the nonexpansive mappingJB

σ(I–σA), we conclude thatw=JσB(I– σA)w, that is,w∈(A+B)–_.

We now show thatw∈G–_{. SinceGis a maximal monotone operator, the Yosida}

ap-proximationArx=x–J

G rx

r ofGforr>  is inF(J G

r(x)). So, for any (z,v)∈G, one has

z–uni,v–

xni–uni

rni

≥.

Sincelim inf_n→∞rn> ,{uni}converges weakly towandxni–uni→, we have z–w,v ≥,

and then the maximality ofTshows that ∈Gw, that is, ∈G–.

In summary, we have shown thatw∈, and so, by Assertion (G), we conclude that

lim sup n→∞

(V–γg)p,xn–p

=lim i→∞

(V–γg)p,xni–p

=(V–γg)p,w–p≥.

•Assertion (I):The sequence{xn}converges strongly to p.

Replacingpwithpin (), we have

xn+–p ≤( –αnτ) zn–p + αn

γg(xn) –Vp,xn+–p

≤( –αnτ) zn–p + αnγk xn–p xn+–p

+ αn

γg(p) –Vp,xn+–p

≤( –αnτ) xn–p +αnγk

xn–p + xn+–p 

+ αn

γg(p) –Vp,xn+–p

by () =( –αnτ)+αnγk

xn–p +αnγk xn+–p 

+ αn

γg(p) –Vp,xn+–p

.

Hence

xn+–p ≤

 – αnτ+ (αnτ)+αnγk

 –αnγk

xn–p 

+ αn  –αnγk

γg(p) –Vp,xn+–p

=

 –(τ–γk)αn  –αnγk

xn–p +

(αnτ)

 –αnγk

xn–p 

+ αn  –αnγk

γg(p) –Vp,xn+–p

= ( –βn) xn–p 

+βn

αnτ xn–p 

(τ–γk) + 

τ–γk

γg(p) –Vp,xn+–p

whereβn=(_–τ–αγnkγ)kαn. Since

∞

n=βn=∞, and since

lim sup n→∞

αnτ xn–p 

(τ–γk) + 

τ–γk

γg(p) –Vp,xn+–p

≤,

by Assertion (H), it follows from Lemma . that{xn}converges strongly top. This

com-pletes the proof.

WhenTis the identity mapping, the theorem reduces to the following corollary.

Corollary . Suppose that

(..) G:C→HandB:C→Hare two maximal monotone operators with D(G)⊆CandD(B)⊆C;

(..) g:H→His ak-contraction,A:C→His anα-inverse strongly monotone mapping,andV:H→His aγ¯-strongly monotone andL-Lipschitzian mapping withγ¯> andL> ;

(..) := (A+B)–∩G–=∅;

(..) μandγ are two real numbers satisfying <μ<_Lγ¯ and <γ <

¯

γ–L_μ

k .

Start with any x∈Hand define a sequence{xn}iteratively by

yn=JσBn(I–σnA)J

G

rnxn, n∈N,

xn+=αnγg(xn) + (I–αnV)yn,

where the sequences{αn},{σn}and{rn}verify the following conditions:

(..) {αn}is a sequence in[, ]withlimn→∞αn= and

∞

n=αn=∞;

(..) {σn}and{rn}are sequences in(,∞)so thatlim infn→∞rn> and there are a,b∈Rwith <a≤σn≤b< αfor alln∈N.

Then P(I–V+γg)has a unique fixed point pin,and this pis also a unique solution p∈to the hierarchical variational inequality

(V–γg)p,q–p, ∀q∈.

Here we would like to remark that Corollary . is related to Theorem  in Lin and Takahashi [], although our conditions (..) and (..) are different from the corre-sponding ones in [].

4 Applications

In this section, we shall apply Theorem . to study the related equilibrium problem. Let

f :C×C→RandA:C→H. Then a generalized equilibrium problem is the problem of findingxˆ∈Csuch that

f(xˆ,y) +Axˆ,y–xˆ ≥, ∀y∈C. ()

The solution set for Eq. () is denoted byEP(f,A), that is,

In caseA= , problem () reduces to the equilibrium problem of findingxˆ∈Csuch that

f(xˆ,y)≥, ∀y∈C,

whose solution set is denoted byEP(f). Whenf = , the generalized equilibrium problem becomes the variational problem of findingxˆ∈Csuch that

Axˆ,y–xˆ ≥, ∀y∈C,

whose solution set is denoted byVI(C,A). For solving an equilibrium problem, we assume that the functionf satisfies the following conditions:

(A) f(x,x) = ,∀x∈C;

(A) f is monotone,that is,f(x,y) +f(y,x)≤,∀x∈C; (A) for allx,y,z∈C,lim sup_t↓f(( –t)x+tz,y)≤f(x,y); (A) for allx∈C,f(x,·)is convex and lower semicontinuous.

The following Lemma . appears implicitly in Blum and Oettli [] and is proved in detail by Aoyamaet al.[], while Lemma . is Lemma . of Combettes and Hirstoaga [].

Lemma .[, ] Let f :C×C→Rbe a function satisfying conditions(A)-(A),and let r> and x∈H.Then there exists a unique z∈C such that

f(z,y) +

ry–z,z–x ≥, ∀y∈C.

Lemma .[] Let f :C×C→Rbe a function satisfying conditions(A)-(A).For r> ,

define Jrf:H→C by

J_rfx=

z∈C:f(z,y) +

ry–z,z–x ≥,∀y∈C

for all x∈H.Then the following hold:

(a) Jrf is single-valued;

(b) Jrf is firmly nonexpansive;

(c) F(Jrf) =EP(f);

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

We callJrf the resolvent off forr> . Using Lemmas . and ., Takahashiet al.[]

established the lemma below.

Lemma .[] Let f :C×C→Rbe a function satisfying conditions(A)-(A)and define a set-valued mapping ofHinto itself by

Gf(x) =

{z∈H:f(x,y)≥ y–x,z,∀y∈C}, ∀x∈C,

∅, ∀x∈/C.

Then the following hold:

(a) Gf is a maximal monotone operator withD(Gf)⊆C;

(b) EP(f) =G– f ;

Theorem . Suppose that

(..) f :C×C→Ris a function satisfying conditions(A)-(A)andB:C→His a maximal monotone operator withD(B)⊆C;

(..) g:H→His ak-contraction,A:C→His anα-inverse strongly monotone mapping,andV:H→His aγ¯-strongly monotone andL-Lipschitzian mapping withγ¯> andL> ;

(..) T:C→Cis aλ-hybrid mapping; (..) :=F(T)∩(A+B)–∩EP(f)=∅;

(..) μandγ are two real numbers satisfying <μ<_Lγ¯ and <γ<

¯

γ–L_μ

k .

Start with any x∈Hand define a sequence{xn}iteratively by

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

f(un,y) +_r_ny–un,un–xn ≥, ∀y∈C, zn=JσBn(I–σnA)un, n∈N,

yn=_n

n–

i=Tizn, n∈N, xn+=αnγg(xn) + (I–αnV)yn,

()

where the sequences{αn},{σn}and{rn}verify the following conditions:

(..) {αn}is a sequence in[, ]withlimn→∞αn= and

∞

n=αn=∞;

(..) {σn}and{rn}are sequences in(,∞)so thatlim infn→∞rn> and there are two a,b∈Rwith <a≤σn≤b< αfor alln∈N.

Then the sequence{xn}constructed by algorithm()converges strongly to a point p∈, where p is the unique fixed point of P(I–V +γg),and this point pis also a unique solution of the hierarchical inequality

(V–γg)p,q–p≥, ∀q∈.

Proof The set-valued mappingGf associated withf defined in Lemma . is a maximal

monotone operator withD(Gf)⊆C, and it follows from Lemmas . and . thatF(J Gf

r ) = F(Jrf) =EP(f) =Gf– for anyr> . PuttingG=Gf in Theorem ., we see thatun=J

Gf

rnxn,

and so the conclusion follows from Theorem .. SinceCis a nonempty closed convex subset ofH, the indicator functionιCdefined by

ιC(x) =

, x∈C,

∞, x∈/C

is a proper lower semicontinuous convex function, and its subdifferential∂ιCdefined by

∂ιC(x) =

z∈H:y–x,z ≤ιC(y) –ιC(x),∀y∈H

is a maximal monotone operator,cf.Rockafellar []. As shown in Lin and Takahashi [] the resolventJ∂ιC

r of∂ιCforr>  is the same as the metric projectionPC. Theorem . Suppose that

(..) g:H→His ak-contraction,A:C→His anα-inverse strongly monotone mapping,andV:H→His aγ¯-strongly monotone andL-Lipschitzian mapping withγ¯> andL> ;

(..) T:C→Cis aλ-hybrid mapping; (..) :=F(T)∩VI(C,A)=∅;

(..) μandγ are two real numbers satisfying <μ<_Lγ¯ and <γ <

¯

γ–L_μ

k .

Start with any x∈Hand define a sequence{xn}iteratively by

⎧ ⎪ ⎨ ⎪ ⎩

zn=PC(I–σnA)PCxn, n∈N, yn=_n

n–

i=Tizn, n∈N, xn+=αnγg(xn) + (I–αnV)yn,

()

where the sequences{αn},{σn}and{rn}verify the following conditions:

(..) {αn}is a sequence in[, ]withlimn→∞αn= and

∞

n=αn=∞;

(..) {σn}is a sequence in(,∞)so that there are twoa,b∈Rwith

 <a≤σn≤b< αfor alln∈N.

Then the sequence{xn}constructed by algorithm()converges strongly to a point p∈, where p is the unique fixed point of P(I–V +γg),and this point pis also a unique solution of the hierarchical inequality

(V–γg)p,q–p≥, ∀q∈.

Proof PutB=G=∂ιCin Theorem .. Then, forσn>  andrn> , we have thatJσBn=J

∂ιC

rn =

PC. Furthermore, as shown in Theorem  in Lin and Takahashi [], we have

(∂ιC)– =C and (A+∂ιC)– =VI(C,A).

Thus we obtain the desired results from Theorem ..

Competing interests

The author declares that they have no competing interests.

Received: 28 December 2013 Accepted: 1 July 2014 Published:22 Jul 2014 References

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