Implicit and explicit iterative methods for mixed equilibria with constraints of system of generalized equilibria and hierarchical fixed point problem

R E S E A R C H

Open Access

Implicit and explicit iterative methods for

mixed equilibria with constraints of system of

generalized equilibria and hierarchical fixed

point problem

Lu-Chuan Ceng

_{, Chin-Tzong Pang}

_{and Ching-Feng Wen}

[email protected]

2_{Department of Information}

Management, and Innovation Center for Big Data and Digital Convergence, Yuan Ze University, Chung-Li, 32003, Taiwan Full list of author information is available at the end of the article

Abstract

In this paper, we introduce one composite implicit relaxed extragradient-like scheme and another composite explicit relaxed extragradient-like scheme for finding a common solution of a finite family of generalized mixed equilibrium problems (GMEPs) with the constraints of a system of generalized equilibrium problems (SGEP) and the hierarchical fixed point problem (HFPP) for a strictly pseudocontractive mapping in a real Hilbert space. We establish the strong convergence of these two composite relaxed extragradient-like schemes to the same common solution of finitely many GMEPs and the SGEP, which is the unique solution of the HFPP for a strictly pseudocontractive mapping. In particular, we make use of weaker control conditions than previous ones for the sake of proving strong convergence. Utilizing these results, we first propose the composite implicit and explicit relaxed

extragradient-like schemes for finding a common fixed point of a finite family of strictly pseudocontractive mappings, and then we derive their strong convergence to the unique common solution of the SGEP and some HFPP. Our results complement, develop, improve, and extend the corresponding ones given by some authors recently in this area.

MSC: Primary 49J30; 47H09; secondary 47J20; 49M05

Keywords: composite relaxed extragradient-like method; generalized mixed equilibrium problem; system of generalized equilibrium problems; inverse strongly monotone mapping; strictly pseudocontractive mapping; fixed point

1 Introduction

LetH be a real Hilbert space with inner product·,·and induced norm · ,C be a nonempty, closed, and convex subset ofH, andPCbe the metric projection ofHontoC.

LetT:C→Cbe a self-mapping onC. We denote byFix(T) the set of fixed points ofT and by R the set of all real numbers. A mappingA:H→His calledγ¯-strongly positive onHif there exists a constantγ¯>  such that

Ax,x ≥ ¯γx, ∀x∈H.

A mappingF:C→H is calledL-Lipschitz-continuous if there exists a constantL≥ such that

Fx–Fy ≤Lx–y, ∀x,y∈C.

In particular, ifL=  thenFis called a nonexpansive mapping; ifL∈[, ) thenFis called a contraction. A mappingT:C→Cis calledk-strictly pseudocontractive (or ak-strict pseudocontraction) if there exists a constantk∈[, ) such that

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

In particular, ifk= , thenT is a nonexpansive mapping. The mappingTis pseudocon-tractive if and only if

Tx–Ty,x–y ≤ x–y, ∀x,y∈C.

Tis strongly pseudocontractive if and only if there exists a constantλ∈(, ) such that

Tx–Ty,x–y ≤λx–y, ∀x,y∈C.

Note that the class of strictly pseudocontractive mappings includes the class of nonex-pansive mappings as a subclass. That is,T is nonexpansive if and only ifT is -strictly pseudocontractive. The mappingT is also said to be pseudocontractive ifk=  andT is said to be strongly pseudocontractive if there exists a positive constantλ∈(, ) such that T+ ( –λ)Iis pseudocontractive. Clearly, the class of strictly pseudocontractive mappings falls into the one between the classes of nonexpansive mappings and of pseudocontractive mappings. Also it is clear that the class of strongly pseudocontractive mappings is inde-pendent of the class of strictly pseudocontractive mappings (see []). The class of pseu-docontractive mappings is one of the most important classes of mappings among non-linear mappings. Recently, many authors have been devoting to the study of the problem of finding fixed points of pseudocontractive mappings; seee.g., [–] and the references therein.

LetA:C→Hbe a nonlinear mapping onC. The variational inequality problem (VIP) associated with the setCand the mappingAis stated as follows: findx∗∈Csuch that

Ax∗,x–x∗≥, ∀x∈C. (.)

The solution set of VIP (.) is denoted byVI(C,A).

The VIP (.) was first discussed by Lions []. There are many applications of VIP (.) in various fields; see,e.g., [, , , ]. It is well known that, ifAis a strongly monotone and Lipschitz-continuous mapping onC, then VIP (.) has a unique solution. In , Kor-pelevich [] proposed an iterative algorithm for solving VIP (.) in Euclidean space Rn_:

yn=PC(xn–τAxn),

xn+=PC(xn–τAyn), ∀n≥,

withτ>  a given number, which is known as the extragradient method. The literature on the VIP is vast and Korpelevich’s extragradient method has received great attention given by many authors, who improved it in various ways; see,e.g., [, , –] and references therein, to name but a few.

In , Cenget al.[] also introduced the following iterative method:

xn+=PC

αnγVxn+ (I–αnμF)Txn

, ∀n≥, (.)

where T :C→C is a nonexpansive mapping such that Fix(T)=∅,F :C→H is a κ -Lipschitzian andη-strongly monotone operator with positive constantsκ,η> ,V:C→ H is an l-Lipschitzian mapping with constantl≥ and  <μ< _κη. They proved that, under mild conditions, the sequence{xn}generated by (.) converges strongly to a point

˜

x∈Fix(T) which is the unique solution to the VIP

(μF–γV)x,˜ p–x˜≥, ∀p∈Fix(T). (.)

Their results also improve Tian’s results [] from the contractive mappingf to the Lips-chitzian mappingV.

In , Cenget al.[] introduced one general composite implicit scheme that gener-ates a net{xt}_{t∈(,min{,}–γ¯

τ–γ α})in an implicit way

xt= (I–θtA)Txt+θt

Txt–t

μFTxt–γf(xt) , (.)

and also proposed another general composite explicit scheme that generates a sequence {xn}in an explicit way

yn= (I–αnμF)Txn+αnγf(xn),

xn+= (I–βnA)Txn+βnyn, ∀n≥,

(.)

wherex∈His an arbitrary initial guess,F:H→His aκ-Lipschitzian andη-strongly

monotone operator with positive constantsκ,η> ,T:H→His a nonexpansive map-ping,A:H→His aγ¯-strongly positive bounded linear operator, andf :H→His anα -contractive mapping withα∈(, ). They proved that, under appropriate conditions, the net{xt}and the sequence{xn}generated by (.) and (.), respectively, converge strongly

to the same pointx˜∈Fix(T), which is the unique solution to the VIP

(A–I)x,˜ p–x˜≥, ∀p∈Fix(T). (.)

Their results supplement and develop the corresponding ones of Marino and Xu [], Yamada [] and Tian [].

Very recently, inspired by Cenget al.[], Jung [] introduced one general composite implicit scheme that generates a net{xt}_{t∈(,min{,}–γ¯

τ–γl})in an implicit way

xt= (I–θtA)Ttxt+θt

tγVxt+ (I–tμF)Ttxt

and also proposed another general composite explicit scheme that generates a sequence {xn}in an explicit way,

yn=αnγVxn+ (I–αnμF)Tnxn,

xn+= (I–βnA)Tnxn+βnyn, ∀n≥,

(.)

wherex∈His an arbitrary initial guess and the following conditions are satisfied:

T:H→His ak-strictly pseudocontractive mapping withFix(T)=∅;

Ais aγ¯-strongly positive bounded linear operator onHwithγ¯∈(, );

F:H→His aκ-Lipschitzian andη-strongly monotone operator with <μ<_κη;

V:H→His anl-Lipschitzian mapping with≤γl<τ and

τ =  – –μ(η–μκ_);

Tt:H→His a mapping defined byTtx=λtx+ ( –λt)Tx,t∈(, ), for

≤k≤λt≤λ< andlimt→λt=λ;

Tn:H→His a mapping defined byTnx=λnx+ ( –λn)Txfor≤k≤λn≤λ< and

lim_n→∞λn=λ;

{αn} ⊂[, ],{βn} ⊂(, ]and{θt}_{t∈(,min{,}–γ¯

τ–γl})⊂(, ).

The author of [] proved that, under weaker control conditions than the previous ones, the net {xt} and the sequence {xn} generated by (.) and (.), respectively, converge

strongly to the same pointx˜∈Fix(T), which is the unique solution to the VIP

(A–I)˜x,p–x˜≥, ∀p∈Fix(T). (.)

His results extend and improve Ceng et al.’s corresponding ones [] from the nonex-pansive mappingTto the strictly pseudocontractive mappingTand from the contractive mappingf to the Lipschitzian mappingV.

On the other hand, letϕ:C→Rbe a real-valued function,A:C→Hbe a nonlinear mapping andΘ:C×C→Rbe a bifunction. In , Peng and Yao [] introduced the generalized mixed equilibrium problem (GMEP) of findingx∈Csuch that

Θ(x,y) +ϕ(y) –ϕ(x) +Ax,y–x ≥, ∀y∈C. (.) We denote the set of solutions of GMEP (.) byGMEP(Θ,ϕ,A). The GMEP (.) is very general in the sense that it includes, as special cases, optimization problems, variational in-equalities, minimax problems, Nash equilibrium problems in noncooperative games and others. Recently, many authors have been devoting to the study of the GMEP (.) and its special cases,e.g., generalized equilibrium problem (GEP), mixed equilibrium problem (MEP),equilibrium problem (EP),etc.; see,e.g., [, , –, , –] and the refer-ences therein.

It was assumed in [] thatΘ:C×C→Ris a bifunction satisfying conditions (A)-(A) andϕ:C→Ris a lower semicontinuous and convex function with restriction (B) or (B), where

(A) Θ(x,x) = for allx∈C;

(A) Θis monotone,i.e.,Θ(x,y) +Θ(y,x)≤for anyx,y∈C;

(A) Θis upper-hemicontinuous,i.e., for eachx,y,z∈C,

lim sup t→+

(A) Θ(x,·)is convex and lower semicontinuous for eachx∈C;

(B) for eachx∈Handr> , there exists a bounded subsetDx⊂Candyx∈Csuch

that for anyz∈C\Dx,

Θ(z,yx) +ϕ(yx) –ϕ(z) +



ryx–z,z–x< ;

(B) Cis a bounded set.

Given a positive numberr> . LetTr(Θ,ϕ):H→Cbe the solution set of the auxiliary

mixed equilibrium problem, that is, for eachx∈H,

T_r(Θ,ϕ)(x) :=

y∈C:Θ(y,z) +ϕ(z) –ϕ(y) +

ry–x,z–y ≥,∀z∈C

.

In particular, ifϕ≡ thenTr(Θ,ϕ)is rewritten asTrΘ:H→C,i.e.,

TΘ

r (x) :=

y∈C:Θ(y,z) +

ry–x,z–y ≥,∀z∈C

.

LetΦ,Φ:C×C→Rbe two bifunctions andF,F:C→Hbe two mappings.

Con-sider the problem of finding (x∗,y∗)∈C×Csuch that

Φ(x∗,x) +Fy∗,x–x∗+νx

∗_–y∗_,x–x∗ _{≥, ∀x∈C,}

Φ(y∗,y) +Fx∗,y–y∗+νy

∗_–x∗_,y–y∗ _{≥, ∀y∈C,} (.)

which is called a system of generalized equilibrium problems (SGEP) whereν>  and ν>  are two constants. In , Ceng and Yao [] transformed the SGEP (.) into

the fixed point problem of the mappingG=TΦ

ν (I–νF)T Φ

ν (I–νF), that is,Gx

∗_=x∗_,

wherey∗=TΦ

ν (I–νF)x

∗_{. Throughout this paper, the fixed point set of the mappingG}

is denoted byΞ.

In particular, ifΦ≡Φ≡, then problem (.) reduces to the system of variational

inequalities (SVI) of finding (x∗,y∗)∈C×Csuch that

νFy∗+x∗–y∗,x–x∗ ≥, ∀x∈C,

νFx∗+y∗–x∗,y–y∗ ≥, ∀y∈C,

(.)

where ν >  and ν >  are two constants. Recently, many authors have addressed

the study of the SVI (.); see, e.g., [, , , –, –] and the references therein.

LetT :C→Cbe ak-strictly pseudocontractive mapping. In , Ceng and Yao [] proposed and analyzed the following relaxed extragradient-like iterative scheme for find-ing a common solutionx∗∈Ω:=Fix(T)∩GMEP(Θ,ϕ,A)∩Ξ of the GMEP (.), the SGEP (.), and the fixed point problem ofT:

⎧ ⎪ ⎨ ⎪ ⎩

zn=T(

Θ,ϕ)

λn (I–λnA)xn, yn=TνΦ(I–νF)T

Φ

ν (I–νF)zn,

xn+=αnu+βnxn+γnyn+δnTyn, ∀n≥,

where  <νj< ζjforj= , , and{λn} ⊂[, η],{αn},{βn},{γn},{δn} ⊂[, ] such thatαn+ βn+γn+δn=  and (γn+δn)k≤γn,∀n≥. Under some mild assumptions, the authors

[] proved that{xn}converges strongly tox∗=PΩuand (x∗,y∗) is a solution of the SGEP (.), wherey∗=TΦ

ν (I–νF)x

∗_.

In this paper, we introduce one composite implicit relaxed extragradient-like scheme and another composite explicit relaxed extragradient-like scheme for finding a common solution of a finite family of generalized mixed equilibrium problems (GMEP) with the constraints of the SGEP (.) and the hierarchical fixed point problem (HFPP) for a strictly pseudocontractive mapping in a real Hilbert space. We establish the strong con-vergence of these two composite relaxed extragradient-like schemes to the same common solution of finitely many GMEPs and the SGEP (.), which is the unique solution of the HFPP for a strictly pseudocontractive mapping. In particular, we make use of weaker con-trol conditions than the previous ones for the sake of proving strong convergence. Utilizing these results, we first propose the composite implicit and explicit relaxed extragradient-like schemes for finding a common fixed point of a finite family of strictly pseudocontrac-tive mappings, and then derive their strong convergence to the unique common solution of the SGEP (.) and some HFPP. Our results complement, develop, improve, and ex-tend the corresponding ones given by some authors recently in this area. See,e.g., Cenget al.[], Jung [], and Ceng and Yao [].

2 Preliminaries

Throughout this paper, we assume thatHis a real Hilbert space whose inner product and norm are denoted by·,·and · , respectively. LetCbe a nonempty, closed, and convex subset ofH. We writexnxto indicate that the sequence{xn}converges weakly tox

andxn→xto indicate that the sequence{xn}converges strongly tox. Moreover, we use ωw(xn) to denote the weakω-limit set of the sequence{xn},i.e.,

ωw(xn) :=

x∈H:xnixfor some subsequence{xni}of{xn}

.

The metric (or nearest point) projection fromH ontoCis the mapping PC:H→C

which assigns to each pointx∈Hthe unique pointPCx∈Csatisfying the property

x–PCx=inf

y∈Cx–y=:d(x,C).

The following properties of projections are useful and pertinent to our purpose.

Proposition . Given any x∈H and z∈C.One has

(i) z=PCx⇔ x–z,y–z ≤,∀y∈C;

(ii) z=PCx⇔ x–z≤ x–y–y–z,∀y∈C;

(iii) PCx–PCy,x–y ≥ PCx–PCy,∀y∈H,which hence implies thatPCis

nonexpansive and monotone.

Definition . A mappingT:H→His said to be firmly nonexpansive if T–Iis non-expansive, or equivalently, ifTis -inverse strongly monotone (-ism),

alternatively,Tis firmly nonexpansive if and only ifTcan be expressed as

T= (I+S),

whereS:H→His nonexpansive; projections are firmly nonexpansive.

Definition . A mappingF:C→His said to be

(i) monotone if

Fx–Fy,x–y ≥, ∀x,y∈C;

(ii) η-strongly monotone if there exists a constantη> such that

Fx–Fy,x–y ≥ηx–y, ∀x,y∈C;

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

Fx–Fy,x–y ≥αFx–Fy, ∀x,y∈C.

It can easily be seen that ifTis nonexpansive, thenI–Tis monotone. It is also easy to see that the projectionPCis -ism. Inverse strongly monotone (also referred to as co-coercive)

operators have been applied widely in solving practical problems in various fields. On the other hand, it is obvious that ifF:C→Hisα-inverse strongly monotone, then F is monotone and_α-Lipschitz-continuous. Moreover, we also have, for allu,v∈Cand

λ> ,

(I–λF)u– (I–λF)v≤ u–v+λ(λ– α)Fu–Fv. (.)

Consequently, ifλ≤α, thenI–λFis a nonexpansive mapping fromCtoH. Next we list some elementary conclusions for the MEP.

Proposition .(see []) Assume thatΘ:C×C→Rsatisfies(A)-(A)and letϕ:C→

Rbe a proper lower semicontinuous and convex function.Assume that either(B)or(B) holds.For r> and x∈H,define a mapping Tr(Θ,ϕ):H→C as follows:

T(Θ,ϕ) r (x) =

z∈C:Θ(z,y) +ϕ(y) –ϕ(z) +

ry–z,z–x ≥,∀y∈C

for all x∈H.Then the following hold:

(i) for eachx∈H,Tr(Θ,ϕ)(x)is nonempty and single-valued; (ii) Tr(Θ,ϕ)is firmly nonexpansive,that is,for anyx,y∈H,

T(Θ,ϕ)

r x–Tr(Θ,ϕ)y 

≤T(Θ,ϕ)

r x–Tr(Θ,ϕ)y,x–y

;

(iii) Fix(Tr(Θ,ϕ)) =MEP(Θ,ϕ); (iv) MEP(Θ,ϕ)is closed and convex;

(v) Ts(Θ,ϕ)x–Tt(Θ,ϕ)x≤s–ts T (Θ,ϕ)

s x–Tt(Θ,ϕ)x,T(

Θ,ϕ)

In , Ceng and Yao [] transformed the SGEP (.) into a fixed point problem in the following way:

Proposition .(see []) LetΦ,Φ:C×C→Rbe two bifunctions satisfying conditions

(A)-(A).Then(x∗,y∗)∈C×C is a solution of the SGEP(.)if and only if x∗is a fixed point of the mapping G:C→C defined by

Gx=TΦ

ν (I–νF)T Φ

ν (I–νF)x, ∀x∈C,

where y∗=TΦ

ν (I–νF)x

∗_.

In particular,if the mapping Fj:C→H isζj-inverse strongly monotone for j= , ,then

the mapping G is nonexpansive providedνj∈(, ζj]for j= , .We denote byΞ the fixed

point set of the mapping G.

In Proposition ., puttingΦ≡Φ≡, we get the following.

Corollary .(see [], Lemma .) For given x∗,y∗∈C, (x∗,y∗)is a solution of the SVI (.)if and only if x∗ is a fixed point of the mapping G:C→C defined by Gx=PC(I– νF)PC(I–νF)x for all x∈C,where y∗=PC(I–νF)x∗.

In particular,if the mapping Fj:C→H isζj-inverse strongly monotone for j= , ,then

the mapping G is nonexpansive providedνj∈(, ζj]for j= , .We denote byΞ the fixed

point set of the mapping G.

We need some facts and tools in a real Hilbert spaceH; these are listed as lemmas below.

Lemma . Let X be a real inner product space.Then we have the following inequality:

x+y≤ x+ y,x+y, ∀x,y∈X.

Lemma . Let H be a real Hilbert space.Then the following hold:

(a) x–y=x–y– x–y,yfor allx,y∈H;

(b) λx+μy=λx+μy–λμx–yfor allx,y∈Handλ,μ∈[, ]with λ+μ= ;

(c) if{xn}is a sequence inHsuch thatxnx,it follows that

lim sup n→∞ xn–y

_{=lim sup}

n→∞ xn–x

_+x–y_{, ∀y∈H.}

It is clear that, in a real Hilbert spaceH,T:C→Cisk-strictly pseudocontractive if and only if the following inequality holds:

Tx–Ty,x–y ≤ x–y– –k

 (I–T)x– (I–T)y



, ∀x,y∈C.

This immediately implies that ifT is ak-strictly pseudocontractive mapping, thenI–T is –k

 -inverse strongly monotone; for further detail, we refer to [] and the references

Lemma .(see [], Proposition .) Let C be a nonempty,closed,and convex subset of a real Hilbert space H and T:C→C be a mapping.

(i) IfT is ak-strictly pseudocontractive mapping,thenTsatisfies the Lipschitzian condition

Tx–Ty ≤ +k

 –kx–y, ∀x,y∈C.

(ii) IfT is ak-strictly pseudocontractive mapping,then the mappingI–Tis semiclosed at,that is,if{xn}is a sequence inCsuch thatxnx˜and(I–T)xn→,then

(I–T)˜x= .

(iii) IfT isk-(quasi-)strict pseudocontraction,then the fixed point setFix(T)ofTis

closed and convex so that the projectionPFix(T)is well defined.

Lemma .(see []) Let C be a nonempty,closed, and convex subset of a real Hilbert space H.Let T :C→C be a k-strictly pseudocontractive mapping.Letγ andδ be two nonnegative real numbers such that(γ +δ)k≤γ.Then

γ(x–y) +δ(Tx–Ty)≤(γ +δ)x–y, ∀x,y∈C.

Lemma .(see [], Demiclosedness principle) Let C be a nonempty,closed,and convex subset of a real Hilbert space H.Let S be a nonexpansive self-mapping on C.Then I–S is demiclosed.That is,whenever{xn}is a sequence in C weakly converging to some x∈C and

the sequence{(I–S)xn}strongly converges to some y,it follows that(I–S)x=y.Here I is

the identity operator of H.

Lemma . Let F:C→H be a monotone mapping.In the context of the variational in-equality problem the characterization of the projection(see Proposition.(i))implies

u∈VI(C,F) ⇔ u=PC(u–λFu), λ> .

Let C be a nonempty,closed,and convex subset of a real Hilbert space H.We introduce some notations.Letλbe a number in(, ]and letμ> .Associating with a nonexpansive mapping T:C→C,we define the mapping Tλ_{:C→H by}

Tλx:=Tx–λμF(Tx), ∀x∈C,

where F:C→H is an operator such that, for some positive constantsκ,η> ,F isκ -Lipschitzian andη-strongly monotone on C;that is,F satisfies the conditions:

Fx–Fy ≤κx–y and Fx–Fy,x–y ≥ηx–y

for all x,y∈C.

Lemma .(see [], Lemma .) Tλ_{is a contraction provided <μ<}η

κ;that is, _Tλ

x–Tλy≤( –λτ)x–y, ∀x,y∈C,

Lemma .(see [], Lemma .) Let{an}be a sequence of nonnegative real numbers

satisfying

an+≤( –ωn)an+ωnδn+rn, ∀n≥,

where{ωn},{δn},and{rn}satisfy the following conditions: (i) {ωn} ⊂[, ]and

∞

n=ωn=∞; (ii) eitherlim sup_n→∞δn≤or

∞

n=ωn|δn|<∞;

(iii) rn≥for alln≥,and

∞

n=rn<∞.

Thenlimn→∞an= .

Lemma .(see []) Assume that A is aγ¯-strongly positive bounded linear operator on H with <ρ≤ A–_{.ThenI–ρA ≤ –ργ¯.}

LetLIMbe a Banach limit. According to time and circumstances, we useLIMnaninstead

ofLIMafor everya={an} ∈l∞. The following properties are well known:

(i) for alln≥,an≤cnimpliesLIMnan≤LIMncn;

(ii) LIMnan+N=LIMnanfor any fixed positive integerN;

(iii) lim infn→∞an≤LIMnan≤lim supn→∞anfor all{an} ∈l∞.

The following lemma was given in [], Proposition .

Lemma . Let a∈ R be a real number and let a sequence {an} ∈l∞ satisfy the

condition LIM_nan ≤a for all Banach limit LIM. If lim supn→∞(an+ –an)≤, then lim sup_n→∞an≤a.

Recall that a set-valued mappingT:D(T)⊂H→H_{is called monotone if for allx,y∈}

D(T),f ∈Tx, andg∈Tyimply

f–g,x–y ≥.

A set-valued mappingTis called maximal monotone ifTis monotone and (I+λT)D(T) = Hfor eachλ> , whereIis the identity mapping ofH. We denote byG(T) the graph ofT. It is well known that a monotone mappingT is maximal if and only if, for (x,f)∈H×H, f –g,x–y ≥ for every (y,g)∈G(T) impliesf ∈Tx. Next we provide an example to illustrate the concept of a maximal monotone mapping.

LetΓ :C→Hbe a monotone and Lipschitz-continuous mapping and letNCvbe the

normal cone toCatv∈C,i.e.,

NCv=

u∈H:v–p,u ≥,∀p∈C.

Define

Tv=

Γv+NCv, ifv∈C,

∅, ifv∈/C.

Then it is well known [] that T is maximal monotone and ∈Tvif and only ifv∈

3 Main results

LetCbe a nonempty, closed, and convex subset of a real Hilbert spaceH. Throughout this section, we always assume the following:

F:C→His aκ-Lipschitzian andη-strongly monotone operator with positive

constantsκ,η> , andFj:C→Hisζj-inverse strongly monotone forj= , ;

T:C→Cis ak-strictly pseudocontractive mapping andAi:C→Hisηi-inverse

strongly monotone for eachi= , . . . ,N;

Ais aγ¯-strongly positive bounded linear operator onHwithγ¯∈(, )and

V:C→His anl-Lipschitzian mapping withl≥;

Θi,Φj:C×C→Rare the bifunctions satisfying conditions (A)-(A) and

ϕi:C→R∪ {+∞}be a proper lower semicontinuous and convex function with

restrictions (B) or (B) for eachi= , . . . ,Nandj= , ;

 <μ<_κη and≤γl<τwithτ=  –

 –μ(η–μκ_);

S:C→Cis a mapping defined bySx=λx+ ( –λ)Txfor≤k≤λ< ;

G:C→Cis a mapping defined byGx=TΦ

ν(I–νF)T Φ

ν (I–νF)xwith <νj< ζj

forj= , ; ΔN

t :C→Cis a mapping defined by

ΔN_t x=T(ΘN,ϕN)

rN,t (I–rN,tAN)· · ·T

(Θ,ϕ)

r,t (I–r,tA)x,t∈(, ), for {ri,t} ⊂[ai,bi]⊂(, ηi),i= , . . . ,N;

ΔN_n :C→Cis a mapping defined by

ΔN_nx=T(ΘN,ϕN)

rN,n (I–rN,nAN)· · ·T

(Θ,ϕ)

r,n (I–r,nA)xwith{ri,n} ⊂[ai,bi]⊂(, ηi)and

limn→∞ri,n=ri, for eachi= , . . . ,N;

Ω:= (N_i=GMEP(Θi,ϕi,Ai))∩Fix(T)∩Ξ=∅andPΩis the metric projection ofH

ontoΩ;

{αn} ⊂[, ],{βn} ⊂(, ]and{θt}_{t∈(,min{,}–γ¯

τ–γl})⊂(, ). Next, put

Δi_t=T(Θi,ϕi)

ri,t (I–ri,tAi)T

(Θi–,ϕi–)

ri–,t (I–ri–,tAi–)· · ·T

(Θ,ϕ)

r,t (I–r,tA), ∀t∈(, ), and

Δi_n=T(Θi,ϕi)

ri,n (I–ri,nAi)T

(Θi–,ϕi–)

ri–,n (I–ri–,nAi–)· · ·T

(Θ,ϕ)

r,n (I–r,nA), ∀n≥, for alli∈ {, . . . ,N}, andΔ_t=Δ_n=I, whereIis the identity mapping onH.

By Lemma ., we know thatSis nonexpansive. It is clear thatFix(S) =Fix(T). Since {λi,t} ⊂[ai,bi]⊂(, ηi), utilizing (.) and Proposition .(ii) we have for allx,y∈C

ΔN_t x–ΔN_t y=T(ΘN,ϕN)

rN,t (I–rN,tAN)Δ

N–

t x–Tr(NΘ,Nt,ϕN)(I–rN,tAN)Δ

N–

t y

≤(I–rN,tAN)ΔN–t x– (I–rN,tAN)ΔN–t y

≤ΔN_t –x–ΔN_t –y ≤ · · ·

≤Δi_tx–Δi_ty ≤ · · ·

≤Δ_tx–Δ_ty

which implies thatΔi

t :C→Cis a nonexpansive mapping for allt∈(, ). Also, since

{ri,n} ⊂[ai,bi]⊂(, ηi), utilizing (.) and Proposition .(ii) we have for allx,y∈C

ΔN_nx–ΔN_ny=T(ΘN,ϕN)

rN,n (I–rN,nAN)Δ

N– n x–T(

ΘN,ϕN)

rN,n (I–rN,nAN)Δ

N–

n y

≤(I–rN,nAN)ΔN–n x– (I–rN,nAN)ΔNn–y

≤ΔN_n–x–ΔN–_n y ≤ · · ·

≤Δi_nx–Δi_ny ≤ · · ·

≤Δ_nx–Δ_ny

=x–y,

which implies thatΔi_n:C→Cis a nonexpansive mapping for alln≥.

In this section, we introduce the first composite relaxed extragradient-like scheme that generates a net{xt}_{t∈(,min{,}–γ¯

τ–γl})in an implicit manner:

xt=PC

(I–θtA)SΔNt Gxt+θt

tγVxt+ (I–tμF)SΔNt Gxt . (.)

We prove the strong convergence of {xt}ast→ to a pointx˜ ∈Ω which is a unique

solution to the VIP

(A–I)˜x,p–x˜≥, ∀p∈Ω. (.)

For arbitrarily givenx∈C, we also propose the second composite relaxed

extragra-dient-like scheme, which generates a sequence{xn}in an explicit way:

yn=αnγVxn+ (I–αnμF)SΔNnGxn,

xn+=PC[(I–βnA)SΔnNGxn+βnyn], ∀n≥,

(.)

and establish the strong convergence of{xn}asn→ ∞to the same pointx˜∈Ω, which is

also the unique solution to VIP (.).

Now, fort∈(,min{,_τ–_–γγ¯_l}), andθt∈(,A–], consider a mappingQt:C→Cdefined

by

Qtx=PC

(I–θtA)SΔNt Gx+θt

tγVx+ (I–tμF)SΔN_t Gx , ∀x∈C.

It is easy to see thatQt is a contractive mapping with constant  –θt(γ¯–  +t(τ–γl)).

Indeed, by Proposition . and Lemmas . and ., we have

Qtx–Qty≤(I–θtA)SΔNt Gx+θt

tγVx+ (I–tμF)SΔN_t Gx

– (I–θtA)SΔNt Gy–θt

tγVx+ (I–tμF)SΔN_t Gy

+θttγVx+ (I–tμF)SΔNt Gx –

tγVy+ (I–tμF)SΔN_t Gy

≤( –θtγ¯)SΔtNGx–SΔNt Gy+θt

tγVx–Vy

+(I–tμF)SΔN_t Gx– (I–tμF)SΔN_t Gy ≤( –θtγ¯)x–y+θt

tγlx–y+ ( –tτ)x–y

= –θt

¯

γ –  +t(τ–γl) x–y.

Sinceγ¯∈(, ),τ–γl> , and

 <t<min

,  –γ¯

τ–γl

≤  –γ¯

τ–γl,

it follows that

 <γ¯–  +t(τ–γl) < ,

which together with  <θt≤ A–<  yields

 <  –θt

¯

γ –  +t(τ–γl) < .

HenceQt:C→Cis a contractive mapping. By the Banach contraction principle,Qthas

a unique fixed point, denoted byxt, which uniquely solves the fixed point equation (.).

We summarize the basic properties of{xt}. The argument techniques in [, , ] extend

to developing the new argument ones for these basic properties. We include the argument process for the sake of completeness.

Proposition . Let{xt}be defined via(.).Then (i) {xt}is bounded fort∈(,min{,τ––γγ¯l});

(ii) lim_t→xt–Sxt= ,limt→xt–Gxt= andlimt→xt–ΔNt xt= provided limt→θt= ;

(iii) xt: (,min{,τ––γγ¯l})→His locally Lipschitzian provided

θt: (,min{,τ––γγ¯l})→(,A

–_{]is locally Lipschitzian,and}

λi,t: (,min{,τ––γγ¯l})→[ai,bi]is locally Lipschitzian for eachi= , . . . ,N; (iv) xtdefines a continuous path from(,min{,_τ–_–γγ¯_l})intoHprovided

θt: (,min{,_τ–_–γγ¯_l})→(,A–]is continuous,andλi,t: (,min{,_τ–_–γγ¯_l})→[ai,bi]

is continuous for eachi= , . . . ,N.

Proof (i) Letp∈Ω. Noting thatFix(S) =Fix(T),Sp=p, Gp=p, andΔi

tp=pfor each

i= , . . . ,N, by the nonexpansivity ofS,G, andΔi

t, and Lemmas . and . we get

xt–p

≤(I–θtA)SΔNt Gxt+θt

tγVxt+ (I–tμF)SΔNt Gxt –p

=(I–θtA)SΔNt Gxt– (I–θtA)SΔNt Gp

+θt

tγVxt+ (I–tμF)SΔtNGxt–p +θt(I–A)p

+θttγVxt+ (I–tμF)SΔNt Gxt–p+θt(I–A)p

=(I–θtA)SΔNt Gxt– (I–θtA)SΔNt Gp

+θt(I–tμF)SΔtNGxt– (I–tμF)SΔNt Gp

+t(γVxt–μFp)+θt(I–A)p

≤( –θtγ¯)SΔNt Gxt–SΔNt Gp

+θt(I–tμF)SΔNt Gxt– (I–tμF)SΔNt Gp

+tγVxt–Vp+γVp–μFp +θt(I–A)p

≤( –θtγ¯)xt–p+θt

( –tτ)xt–p

+tγlxt–p+(γV–μF)p +θtI–Ap

= –θt

¯

γ –  +t(τ–γl) xt–p+θt

I–Ap+t(γV–μF)p.

So, it follows that

xt–p ≤

I–Ap+t(γV–μF)p ¯

γ –  +t(τ–γl)

≤I–Ap+_¯t(γV–μF)p

γ – 

≤I–Ap+(γV–μF)p ¯

γ –  .

Hence{xt}is bounded and so are{Vxt},{ΔNt xt},{SΔNt Gxt}, and{FSΔNt Gxt}.

(ii) By the definition of{xt}, we have

xt–SΔNt Gxt

=PC

(I–θtA)SΔNt Gxt+θt

tγVxt+ (I–tμF)SΔNt Gxt –PCSΔNt Gxt

≤(I–θtA)SΛNt Gxt+θt

tγVxt+ (I–tμF)SΛtNGxt –SΛNt Gxt

=θt

(I–A)SΔN_t Gxt+t

γVxt–μFSΔNt Gxt

=θt(I–A)SΔNt Gxt+t

γVxt–μFSΔNt Gxt

≤θtI–ASΔNt Gxt+tγVxt–μFSΔNt Gxt→ ast→,

by the boundedness of{Vxt},{SΔNt Gxt}, and{FSΔNt Gxt}in the assertion (i). That is,

lim

t→xt–SΔ N

t Gxt= . (.)

Sincep=Gp=TΦ

ν (I–νF)T Φ

ν (I–νF)pandFjisζj-inverse strongly monotone with  <νj< ζjforj= , , by Proposition .(ii) we deduce that

Gxt–p

=TΦ

ν(I–νF)T Φ

ν (I–νF)xt–T Φ

ν (I–νF)T Φ

ν (I–νF)p



≤(I–νF)TνΦ(I–νF)xt– (I–νF)T Φ

ν (I–νF)p

=TΦ

ν (I–νF)xt–T Φ

ν (I–νF)p

–ν

FTνΦ(I–νF)xt–FT Φ

ν (I–νF)p



≤TΦ

ν (I–νF)xt–T Φ

ν (I–νF)p



+ν(ν– ζ)FTνΦ(I–νF)xt–FT Φ

ν (I–νF)p



≤TΦ

ν (I–νF)xt–T Φ

ν (I–νF)p



≤(I–νF)xt– (I–νF)p 

=(xt–p) –ν(Fxt–Fp)

≤ xt–p+ν(ν– ζ)Fxt–Fp

≤ xt–p. (.)

In the meantime, utilizing theηi-inverse strong monotonicity ofAi, we obtain

Δi_tGxt–p=Tr(iΘ,ti,ϕi)(I–ri,tAi)Δ

i–

t Gxt–Tr(iΘ,ti,ϕi)(I–ri,tAi)p



≤(I–ri,tAi)Δti–Gxt– (I–ri,tAi)p 

=Δi–_t Gxt–p–ri,t

AiΔi–t Gxt–Aip 

≤Δi–_t Gxt–p 

+ri,t(ri,t– ηi)AiΔi–t Gxt–Aip 

≤ Gxt–p+ri,t(ri,t– ηi)AiΔi–t Gxt–Aip 

, (.)

for eachi∈ {, , . . . ,N}. Simple calculations show that

xt–p

=xt–wt+wt–p

=xt–wt+ (I–θtA)SΔNt Gxt+θt

tγVxt+ (I–tμF)SΔNt Gxt –p

=xt–wt+ (I–θtA)SΔNt Gxt– (I–θtA)SΔNt Gp+θt

tγVxt

+ (I–tμF)SΔN_t Gxt–p

+θt(I–A)p

=xt–wt+ (I–θtA)

SΔN_t Gxt–SΔNt Gp +θt

t(γVxt–μFp)

+ (I–tμF)SΔN_t Gxt– (I–tμF)p

+θt(I–A)p, (.)

wherewt= (I–θtA)SΔNt Gxt+θt(tγVxt+ (I–tμF)SΔNt Gxt).

For simplicity, we writex˜t=TνΦ(I–νF)xt,p˜=T Φ

ν (I–νF)pandyt=T Φ

ν (I–νF)x˜t. Then we have yt =TνΦ(I–νF)T

Φ

ν (I–νF)xt andp=Gp=T Φ

ν (I–νF)p. Then, by˜ Propositions .(i) and ., and Lemmas . and ., from (.)-(.) we obtain

xt–p

=xt–wt,xt–p+

(I–θtA)

SΔN_t Gxt–SΔNt Gp ,xt–p

+θt

tγVxt–μFp,xt–p+

(I–tμF)SΔN_t Gxt– (I–tμF)p,xt–p

+θt

(I–A)p,xt–p

≤(I–θtA)

SΔN_t Gxt–SΔNt Gp ,xt–p

+θt

tγVxt–μFp,xt–p

+(I–tμF)SΔN_t Gxt– (I–tμF)p,xt–p

+θt

(I–A)p,xt–p

=(I–θtA)

SΔN_t Gxt–SΔNt Gp ,xt–p

+θt

(I–tμF)SΔN_t Gxt– (I–tμF)p,xt–p

+tγVxt–Vp,xt–p+γVp–μFp,xt–p

+θt

(I–A)p,xt–p

≤(I–θtA)

SΔN_t Gxt–SΔNt Gp xt–p

+θt(I–tμF)SΔNt Gxt– (I–tμF)pxt–p

+tγVxt–Vpxt–p+γVp–μFpxt–p +θt(I–A)pxt–p

≤( –θtγ¯)SΔNt Gxt–SΔNt Gpxt–p+θt

( –tτ)ΔN_t Gxt–pxt–p

+tγlxt–p+γVp–μFpxt–p +θt(I–A)pxt–p

≤( –θtγ¯)ΔNt Gxt–pxt–p+θt

( –tτ)ΔN_t Gxt–pxt–p

+tγlxt–p+γVp–μFpxt–p +θt(I–A)pxt–p

= –θt(γ¯–  +tτ) ΔNt Gxt–pxt–p

+θtt

γlxt–p+γVp–μFpxt–p +θt(I–A)pxt–p

≤ –θt(γ¯–  +tτ)

 Δ

N

t Gxt–p+xt–p

+θtt

γlxt–p+γVp–μFpxt–p +θt(I–A)pxt–p

≤ –θt(γ¯–  +tτ)

 Δ

i

tGxt–p 

+xt–p

+θtt

γlxt–p+γVp–μFpxt–p +θt(I–A)pxt–p

≤ –θt(γ¯–  +tτ)

 

Gxt–p

+ri,t(ri,t– ηi)AiΔi–t Gxt–Aip 

+xt–p

+θtt

γlxt–p+γVp–μFpxt–p +θt(I–A)pxt–p

≤ –θt(γ¯–  +tτ)

 

xt–p+ν(ν– ζ)Fxt–Fp

+ν(ν– ζ)Fx˜t–Fp˜

+ri,t(ri,t– ηi)AiΔi–t Gxt–Aip 

+xt–p

+θtt

γlxt–p+γVp–μFpxt–p +θt(I–A)pxt–p

= –θt(γ¯–  +t(τ–γl)

xt–p–

 –θt(γ¯–  +tτ)



ν(ζ–ν)Fxt–Fp

+ν(ζ–ν)Fx˜t–Fp˜ +ri,t(ηi–ri,t)AiΔi–t Gxt–Aip 

+θt

tγVp–μFpxt–p+(I–A)pxt–p

≤ xt–p–

 –θt(γ¯–  +tτ)



+ν(ζ–ν)Fx˜t–Fp˜ +ri,t(ηi–ri,t)AiΔi–t Gxt–Aip 

+θt

tγVp–μFpxt–p+(I–A)pxt–p , (.)

which together withνj∈(, ζj),j= , , and{ri,t} ⊂[ai,bi]⊂(, ηi),i= , . . . ,N, implies

that

 –θt(γ¯–  +tτ)



ν(ζ–ν)Fxt–Fp

+ν(ζ–ν)Fx˜t–Fp˜ +ai(ηi–bi)AiΔi–t Gxt–Aip 

≤ –θt(γ¯–  +tτ)



ν(ζ–ν)Fxt–Fp

+ν(ζ–ν)Fx˜t–Fp˜+ri,t(ηi–ri,t)AiΔi–t Gxt–Aip 

≤θt

tγVp–μFpxt–p+(I–A)pxt–p .

Sincelimt→θt=  and{xt}is bounded, we have

lim

t→Fxt–Fp= , tlim→Fx˜t–Fp˜ =  and

lim t→AiΔ

i–

t Gxt–Aip= 

(.)

for eachi= , . . . ,N.

On the other hand, in terms of the firm nonexpansivity ofTνΦjjand theζj-inverse strong

monotonicity ofFjforj= , , we obtain fromνj∈(, ζj),j= , , and (.)

˜xt–p˜ =TνΦ(I–νF)xt–T Φ

ν (I–νF)p



≤(I–νF)xt– (I–νF)p,x˜t–p˜

= 

(I–νF)xt– (I–νF)p



+˜xt–p˜

–(I–νF)xt– (I–νF)p– (x˜t–p)˜ 

≤  

xt–p+˜xt–p˜–(xt–x˜t) –ν(Fxt–Fp) – (p–p)˜ 

=  

xt–p+˜xt–p˜ –(xt–x˜t) – (p–p)˜ 

+ ν

(xt–x˜t) – (p–p),˜ Fxt–Fp

–ν_Fxt–Fp

and

yt–p =TνΦ(I–νF)˜xt–T Φ

ν(I–νF)(I–νF)˜p



≤(I–νF)x˜t– (I–νF)p,˜ yt–p

= 

(I–νF)˜xt– (I–νF)˜p



+yt–p

–(I–νF)˜xt– (I–νF)˜p– (yt–p) 

≤  

+ ν

Fx˜t–Fp, (˜˜ xt–yt) + (p–p)˜

–ν_Fx˜t–Fp˜ 

≤  

xt–p+yt–p–(˜xt–yt) + (p–p)˜ 

+ ν

Fx˜t–Fp, (˜ x˜t–yt) + (p–p)˜

.

Thus, we have

˜xt–p˜ 

≤ xt–p–(xt–x˜t) – (p–p)˜ 

+ ν

(xt–x˜t) – (p–p),˜ Fxt–Fp

–ν_Fxt–Fp (.)

and

yt–p≤ xt–p–(˜xt–yt) + (p–p)˜ 

+ νFx˜t–Fp˜ (˜xt–yt) + (p–p)˜ . (.)

Consequently, from (.), (.), and (.) it follows that

xt–p

≤ –θt(γ¯–  +tτ)

 

Gxt–p

+ri,t(ri,t– ηi)AiΔi–t Gxt–Aip 

+xt–p

+θtt

γlxt–p+γVp–μFpxt–p +θt(I–A)pxt–p

≤ –θt(γ¯–  +tτ)

 

Gxt–p+xt–p

+θtt

γlxt–p+γVp–μFpxt–p +θt(I–A)pxt–p

≤ –θt(γ¯–  +tτ)

 

˜xt–p˜ +ν(ν– ζ)Fx˜t–Fp˜ +xt–p

+θtt

γlxt–p+γVp–μFpxt–p +θt(I–A)pxt–p

≤ –θt(γ¯–  +tτ)

 

˜xt–p˜ +xt–p

+θtt

γlxt–p+γVp–μFpxt–p +θt(I–A)pxt–p

≤ –θt(γ¯–  +tτ)

 

xt–p–(xt–x˜t) – (p–p)˜ 

+ ν

(xt–x˜t) – (p–p),˜ Fxt–Fp

–ν_Fxt–Fp+xt–p

+θtt

γlxt–p+γVp–μFpxt–p +θt(I–A)pxt–p

≤ –θt

¯

γ –  +t(τ–γl) xt–p

– –θt(γ¯–  +tτ)



(xt–x˜t) – (p–p)˜



+ν(xt–x˜t) – (p–p)˜ Fxt–Fp

+θt

≤ xt–p–

 –θt(γ¯–  +tτ)



(xt–x˜t) – (p–p)˜



+ν(xt–x˜t) – (p–p)˜ Fxt–Fp

+θt

tγVp–μFpxt–p+(I–A)pxt–p ,

which hence leads to

 –θt(γ¯–  +tτ)



(xt–x˜t) – (p–p)˜



≤ν(xt–x˜t) – (p–p)˜ Fxt–Fp

+θt

tγVp–μFpxt–p+(I–A)pxt–p .

Sincelimt→θt=  andlimt→Fxt–Fp=  (due to (.)), we deduce from the

bound-edness of{xt}and{˜xt}that

lim

t→(xt–x˜t) – (p–p)˜ = . (.)

Furthermore, from (.), (.), and (.) it follows that

xt–p

≤ –θt(γ¯–  +tτ)

 

Gxt–p

+ri,t(ri,t– ηi)AiΔi–t Gxt–Aip 

+xt–p

+θtt

γlxt–p+γVp–μFpxt–p +θt(I–A)pxt–p

≤ –θt(γ¯–  +tτ)

 

Gxt–p+xt–p

+θtt

γlxt–p+γVp–μFpxt–p +θt(I–A)pxt–p

= –θt(γ¯–  +tτ)

 

yt–p+xt–p

+θtt

γlxt–p+γVp–μFpxt–p +θt(I–A)pxt–p

≤ –θt(γ¯–  +tτ)

 

xt–p–(x˜t–yt) + (p–p)˜ 

+ νFx˜t–Fp˜(x˜t–yt) + (p–p)˜ +xt–p

+θtt

γlxt–p+γVp–μFpxt–p +θt(I–A)pxt–p

≤ –θt

¯

γ –  +t(τ–γl) xt–p

– –θt(γ¯–  +tτ)



(˜xt–yt) + (p–p)˜



+νFx˜t–Fp˜ (˜xt–yt) + (p–p)˜

+θt

tγVp–μFpxt–p+(I–A)pxt–p

≤ xt–p–

 –θt(γ¯–  +tτ)



(˜xt–yt) + (p–p)˜



+νFx˜t–Fp˜ (˜xt–yt) + (p–p)˜

+θt

which hence yields

 –θt(γ¯–  +tτ)



(x˜t–yt) + (p–p)˜



≤νFx˜t–Fp˜ (˜xt–yt) + (p–p)˜

+θt

tγVp–μFpxt–p+(I–A)pxt–p .

Sincelimt→θt=  andlimt→Fx˜t–Fp˜ =  (due to (.)), we deduce from the

bound-edness of{xt},{yt}, and{˜xt}that

lim

t→(˜xt–yt) + (p–p)˜ = . (.)

Note that

xt–yt ≤(xt–x˜t) – (p–p)˜ +(˜xt–yt) + (p–p)˜ .

Hence from (.) and (.) we get

lim

t→xt–Gxt=tlim→xt–yt= . (.)

Utilizing Proposition .(ii) and Lemma .(a), we obtain for eachi∈ {, . . . ,N}

Δi_tGxt–p 

=T(Θi,ϕi)

ri,t (I–ri,tAi)Δ

i–

t Gxt–Tr(iΘ,ti,ϕi)(I–ri,tAi)p



≤(I–ri,tAi)Δi–t Gxt– (I–ri,tAi)p,ΔitGxt–p

=

(I–ri,tAi)Δ

i–

t Gxt– (I–ri,tAi)p 

+Δi_tGxt–p 

–(I–ri,tAi)Δti–Gxt– (I–ri,tAi)p–

Δi_tGxt–p 

≤ Δ

i–

t Gxt–p+ΔitGxt–p–Δi–t Gxt–ΔitGxt

–ri,t

AiΔi–t Gxt–Aip 

≤ 

xt–p+ΔitGxt–p 

–Δi–_t Gxt–ΔitGxt–ri,t

AiΔi–t Gxt–Aip 

,

which immediately leads to

Δi_tGxt–p

≤ xt–p–Δi–t Gxt–ΔitGxt–ri,t

AiΔi–t Gxt–Aip 

=xt–p–Δti–Gxt–ΔitGxt 

–r_i,tAiΔi–t Gxt–Aip 

+ ri,t

Δi–_t Gxt–ΔitGxt,AiΔi–t Gxt–Aip

≤ xt–p–Δti–Gxt–ΔitGxt 

Combining (.) and (.) we conclude that

xt–p

≤ –θt(γ¯–  +tτ)

 Δ

i

tGxt–p 

+xt–p

+θtt

γlxt–p+γVp–μFpxt–p +θt(I–A)pxt–p

≤ –θt(γ¯–  +tτ)

 

xt–p–Δti–Gxt–ΔitGxt 

+ ri,tΔi–t Gxt–ΔitGxtAiΔi–t Gxt–Aip+xt–p

+θtt

γlxt–p+γVp–μFpxt–p +θt(I–A)pxt–p

≤ –θt

¯

γ –  +t(τ–γl) xt–p

– –θt(γ¯–  +tτ)

 Δ

i–

t Gxt–ΔitGxt 

+ri,tΔi–t Gxt–ΔitGxtAiΔi–t Gxt–Aip

+θt

tγVp–μFpxt–p+(I–A)pxt–p

≤ xt–p–

 –θt(γ¯–  +tτ)

 Δ

i–

t Gxt–ΔitGxt 

+ri,tΔi–t Gxt–ΔitGxtAiΔi–t Gxt–Aip

+θt

tγVp–μFpxt–p+(I–A)pxt–p ,

which hence yields

 –θt(γ¯–  +tτ)

 Δ

i–

t Gxt–ΔitGxt

≤ri,tΔi–t Gxt–ΔitGxtAiΔi–t Gxt–Aip

+θt

tγVp–μFpxt–p+(I–A)pxt–p .

Since{ri,t} ⊂[ai,bi]⊂(, ηi),limt→θt =  andlimt→AiΔi–t Gxt –Aip=  (due to

(.)), we deduce from the boundedness of{xt}and{ΔitGxt}that

lim t→Δ

i–

t Gxt–ΔitGxt= , ∀i∈ {, . . . ,N}. (.)

Note that

Gxt–ΔNt Gxt=ΔtGxt–ΔNt Gxt

≤Δ_tGxt–ΔtGxt+ΔtGxt–ΔtGxt+· · ·

+ΔN–_t Gxt–ΔNt Gxt.

Hence, from (.) we get

lim t→