Regularized hybrid iterative algorithms for triple hierarchical variational inequalities

R E S E A R C H

Open Access

Regularized hybrid iterative algorithms for

triple hierarchical variational inequalities

Lu-Chuan Ceng

1,2

_{, Ngai-Ching Wong}

3,4*

_{and Jen-Chih Yao}

5,6

*_{Correspondence:}

[email protected]

3_{Department of Applied}

Mathematics, National Sun Yat-sen University, Kaohsiung, 804, Taiwan

4_{Center for General Education,}

Kaohsiung Medical University, Kaohsiung, 807, Taiwan Full list of author information is available at the end of the article

Abstract

In this paper, we introduce and study a triple hierarchical variational inequality (THVI) with constraints of minimization and equilibrium problems. More precisely, let Fix(T) be the fixed point set of a nonexpansive mapping, let MEP(

Θ

,

ϕ

) be the solution set of a mixed equilibrium problem (MEP), and let

Γ

be the solution set of a

minimization problem (MP) for a convex and continuously Frechet differential functional in Hilbert spaces. We want to find a solutionx∗∈Fix(T)∩MEP(

Θ

,

ϕ

)∩

Γ

of a variational inequality with a variational inequality constraint over the intersection of Fix(T), MEP(

Θ

,

ϕ

), and

Γ

. We propose a hybrid iterative algorithm with regularization to compute approximate solutions of the THVI, and we present the convergence analysis of the proposed iterative algorithm.

MSC: 49J40; 47J20; 47H10; 65K05; 47H09

Keywords: triple hierarchical variational inequality; minimization problem; mixed equilibrium problem; nonexpansive mapping; maximal monotone mapping; regularization

1 Introduction

LetHbe a Hilbert space with inner product·,·and norm · over the real scalar fieldR. LetCbe a nonempty closed convex subset ofH, andPC be the metric projection ofH

ontoC. LetT:C→Cbe a self-mapping onC. Denote byFix(T) the set of fixed points ofT. We say thatTisL-Lipschitzianif there exists a constantL≥ such that

Tx–Ty ≤Lx–y, ∀x,y∈C.

WhenL=  or ≤L< , we callTanonexpansiveor acontractivemapping, respectively. We say that a mappingA:C→Hisα-inverse strongly monotoneif there exists a constant

α>  such that

Ax–Ay,x–y ≥αAx–Ay, ∀x,y∈C,

and thatAisη-strongly monotone(resp.monotone) if there exists a constantη>  (resp.

η= ) such that

Ax–Ay,x–y ≥ηx–y, ∀x,y∈C.

It is known thatT is nonexpansive if and only ifI–T is _-inverse strongly monotone. Moreover,L-Lipschitz continuous mappings are _L-inversely strong monotone (see,e.g., []).

Letf :C→Rbe a convex and continuously Frechet differentiable functional. Consider the minimization problem (MP):

min

x∈Cf(x) (.)

(assuming the existence of minimizers). We denote byΓ =∅the set of minimizers of prob-lem (.). The gradient-projection algorithm (GPA) generates a sequence{xn}determined by the gradient∇f and the metric projectionPC:

xn+:=PC

xn–λ∇f(xn)

, ∀n≥. (.)

The convergence of algorithm (.) depends on∇f. It is known that if∇f isη-strongly monotone andL-Lipschitz continuous, then for  <λ<_Lη, the operator

S:=PC(I–λ∇f)

is a contraction. Hence, the sequence{xn}defined by the GPA (.) converges in norm to the unique solution of (.). If the gradient∇f is only assumed to be Lipschitz continuous, then{xn}can only be weakly convergent whenHis infinite-dimensional (a counterexam-ple to the norm convergence of{xn}is given by Xu [, Section ]).

The regularization, in particular the traditional Tikhonov regularization, is usually used to solve ill-posed optimization problems. Consider the regularized minimization problem

min

x∈Cfα(x) :=f(x) + α

x _,

whereα>  is the regularization parameter, and againf is convex with Lipschitz continu-ous gradient∇f. While a regularization method provides the possible strong convergence to the minimum-norm solution, its disadvantage is the implicity. Hence explicit iterative methods seem to be attractive. See,e.g., Xu [, ].

On the other hand, for a given mappingA:C→H, we consider the variational inequal-ity problem (VIP) of findingx∗∈Csuch that

Ax∗,x–x∗≥, ∀x∈C. (.)

The solution set of VIP (.) is denoted byVI(C,A). It is well known that whenAis mono-tone,

x∈VI(C,A) ⇔ x=PC(x–λAx), ∀λ> .

WhenCis the fixed point setFix(T) of a nonexpansive mappingT andA=I–S, VIP (.) becomes the variational inequality problem of findingx∗∈Fix(T) such that

(I–S)x∗,x–x∗≥, ∀x∈Fix(T). (.)

This problem, introduced by Moudafi and Maingé [, ], is called the hierarchical fixed point problem. It is clear that ifShas fixed points, then they are solutions of VIP (.). IfS

is contractive, the solution set of VIP (.) is a singleton and it is well known as a viscosity problem. This was previously introduced by Moudafi [] and also developed by Xu []. In this case, solving VIP (.) is equivalent to finding a fixed point of the nonexpansive mappingPFix(T)S, wherePFix(T) is the metric projection onto the closed and convex set

Fix(T). Yaoet al.[] introduced a two-step algorithm to solve VIP (.).

LetΘ:C×C→Rbe a bifunction andϕ:C→Rbe a function. Consider the mixed equilibrium problem (MEP) of findingx∈Csuch that

Θ(x,y) +ϕ(y) –ϕ(x)≥, ∀y∈C, (.)

which was studied by Ceng and Yao []. The solution set of MEP (.) is denoted by

MEP(Θ,ϕ). The MEP (.) is very general in the sense that it includes, as special cases, fixed point problems, optimization problems, variational inequality problems, minimax problems, Nash equilibrium problems in noncooperative games and others; see, e.g., [–].

Recently, Iiduka [, ] considered a variational inequality with a variational inequal-ity constraint over the set of fixed points of a nonexpansive mapping. Since this problem has a triple structure in contrast with bilevel programming problems or hierarchical con-strained optimization problems or hierarchical fixed point problems, it is referred to as a triple hierarchical constrained optimization problem (THCOP). He presented some ex-amples of THCOP and developed iterative algorithms to find the solution of such a prob-lem. Since the original problem is a variational inequality, in this paper, we call it atriple hierarchical variational inequality(THVI). Cenget al.introduced and considered some THVI in []. A nice survey article on THVI is []. See also [–].

Extending the works done in [], we introduce and study in this paper the following triple hierarchical variational inequality with constraints of minimization and equilibrium problems.

The problem to study

LetCbe a nonempty closed convex subset of a real Hilbert spaceH. Letf :C→Rbe convex and continuously Frechet differentiable withΓ being the set of its minimizers. LetT:C→CandS:H→H be both nonexpansive. LetV :H→H beρ-contractive, andF:C→Hbeκ-Lipschitzian andη-strongly monotone with constantsρ∈[, ) and

κ,η> . Suppose  <μ< η/κand  <γ ≤τwhereτ=  – –μ(η–μκ_).

LetΞdenote the solution set of the following hierarchical variational inequality (HVI): findz∗∈Fix(T)∩MEP(Θ,ϕ)∩Γ such that

where the solution setΞ is assumed to be nonempty. Consider the following triple hier-archical variational inequality (THVI).

Findx∗∈Ξsuch that

(μF–γV)x∗,x–x∗≥, ∀x∈Ξ. (.)

Based on the iterative schemes provided by Xu [] and the two-step iterative scheme provided by Yaoet al.[], by virtue of the viscosity approximation method, hybrid steepest-descent method and the regularization method, we propose the following hybrid iterative algorithm with regularization:

⎧ ⎪ ⎨ ⎪ ⎩

Θ(un,y) +ϕ(y) –ϕ(un) +_r_ny–un,un–xn ≥, ∀y∈C,

yn=θnγSxn+ (I–θnμF)PC(un–λ∇fαn(un)),

xn+=βnγVyn+ (I–βnμF)TPC(un–λ∇fαn(un)), ∀n≥.

Here,  <λ< /L,{rn},{αn} ⊂(, +∞), and{βn},{θn} ⊂(, ). It is shown that under ap-propriate assumptions, the two iterative sequences{xn}and{yn}converge strongly to the unique solution of the THVI (.).

2 Preliminaries

LetKbe a nonempty closed convex subset of a real Hilbert spaceH. We writexnxand xn→xto indicate that the sequence{xn}converges weakly and strongly tox, respectively.

The weakω-limit set of the sequence{xn}is denoted by

ωw(xn) :=

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

.

The metric (or nearest point) projection fromH ontoK is the mappingPK :H→K

which assigns to each pointx∈Hthe unique pointPKx∈Ksatisfying the property

x–PKx=inf

y∈Kx–y=:d(x,K).

Proposition . For given x∈H and z∈K:

(i) z=PKx⇔ x–z,y–z ≤,∀y∈K;

(ii) z=PKx⇔ x–z≤ x–y–y–z,∀y∈K;

(iii) PKx–PKy,x–y ≥ PKx–PKy,∀y∈H. Hence,PKis nonexpansive and monotone.

Definition . A mappingT:H→His said to befirmly nonexpansiveif T–Iis non-expansive, or equivalently,

x–y,Tx–Ty ≥ Tx–Ty, ∀x,y∈H.

Alternatively,T is firmly nonexpansive if and only ifT can be expressed as

whereS:H→His nonexpansive. Projections are firmly nonexpansive. We callT an av-eraged mappingifTcan be expressed as a proper convex combination of the identity map

Iand a nonexpansive mapping. In particular, firmly nonexpansive mappings are averaged.

Proposition .(see []) Let T:H→H be a given mapping.

(i) Tis nonexpansive if and only if the complementI–T is _-inverse strongly monotone.

(ii) IfT isν-inverse strongly monotone,then so isγTfor allγ > .

(iii) Tis averaged if and only if the complementI–T isν-inverse strongly monotone for someν> /.Indeed,forα∈(, ),Tisα-averaged if and only ifI–T is 

α-inverse

strongly monotone.

Proposition .(see [, ]) Let S,T,V:H→H.

(i) IfT= ( –α)S+αVfor someα∈(, )and ifSis averaged andVis nonexpansive, thenTis averaged.

(ii) Tis firmly nonexpansive if and only if the complementI–T is firmly nonexpansive. (iii) IfT= ( –α)S+αVfor someα∈(, )and ifSis firmly nonexpansive andVis

nonexpansive,thenTis averaged.

(iv) The composition of finitely many averaged mappings is averaged.In particular,ifT isα-averaged andTisα-averaged,whereα,α∈(, ),thenTTis

(α+α–αα)-averaged.

(v) If the mappingsT, . . . ,TN are averaged and have a common fixed point,then

N

i=

Fix(Ti) =Fix(T· · ·TN).

For solving the equilibrium problem for a bifunctionΘ:C×C→R, let us consider the following conditions:

(A) Θ(x,x) = for allx∈C;

(A) Θis monotone, that is,Θ(x,y) +Θ(y,x)≤for allx,y∈C; (A) for eachx,y,z∈C,limt↓Θ(tz+ ( –t)x,y)≤Θ(x,y);

(A) for eachx∈C,y→Θ(x,y)is convex and lower semicontinuous; (A) for eachy∈C,x→Θ(x,y)is weakly upper semicontinuous;

(B) for eachx∈Handr> , there exist 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.

Lemma .(see []) Let C be a nonempty closed convex subset of a real Hilbert space H andΘ:C×C→Rbe a bifunction satisfying(A)-(A).Let r> and x∈H.Then there exists z∈C such that

Θ(z,y) +

Lemma .(see []) Let C be a nonempty closed convex subset of a real Hilbert space H.

LetΘ:C×C→Rbe a bifunction satisfying(A)-(A)andϕ:C→Rbe a proper lower semicontinuous and convex function.For r> and x∈H,define a mapping Tr:H→C as follows:

Trx:=

z∈C:Θ(z,y) +ϕ(y) –ϕ(z) +

ry–z,z–x ≥,∀y∈C

for all x∈H.Assume that either(B)or(B)holds.Then Tris a single-valued firmly non-expansive map on H,andFix(Tr) =MEP(Θ,ϕ)is closed and convex.

Lemma .(see []) Let{an}be a sequence of nonnegative real numbers such that

an+≤( –sn)an+sntn+n, ∀n≥.

Here,  <sn≤, ≤n,and tn∈Rfor all n= , , , . . . ,such that

(i) ∞_n=sn= +∞;

(ii) eitherlim sup_n→∞tn≤or

∞

n=sn|tn|< +∞; (iii) ∞_n=n< +∞.

Thenlim_n→∞an= .

Lemma .(Demiclosedness principle; see []) Let C be a nonempty closed convex subset of a real Hilbert space H and let T:C→C be a nonexpansive mapping withFix(T)=∅.

If{xn}is a sequence in C converging weakly to x and if{(I–T)xn}converges strongly to y,

then(I–T)x=y;in particular,if y= ,then x∈Fix(T).

Lemma .(see []) Let S:H→H be a nonexpansive mapping and V :H→H be a

ρ-contraction withρ∈[, ),respectively. (i) I–Sismonotone,i.e.,

(I–S)x– (I–S)y,x–y≥, ∀x,y∈H.

(ii) I–Vis( –ρ)-strongly monotone,i.e.,

(I–V)x– (I–V)y,x–y≥( –ρ)x–y, ∀x,y∈H.

Lemma .([]) Let H be a real Hilbert space.Then,for all x,y∈H andλ∈[, ],

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

Lemma . We have the following inequality in an inner product space X:

x+y≤ x+ y,x+y, ∀x,y∈X.

Notations Letλbe a number in (, ] and letμ,κ,η> . LetF:C→Hbeκ-Lipschitzian andη-strongly monotone. Associated with a nonexpansive mappingT:C→C, we define the mappingTλ_:C→Hby

Lemma .(see [, Lemma .]) The map Tλ_{is a contraction providedμ< η/κ}_,that

is,

Tλx–Tλy≤( –λτ)x–y, ∀x,y∈C,

whereτ =  – –μ(η–μκ₎∈_{(, ].In particular,if T=I the identity mapping,then}

(I–λμF)x– (I–λμF)y≤( –λτ)x–y, ∀x,y∈C.

A set-valued mappingT:H→His calledmonotoneifx–y,f –g ≥ for allx,y∈H,

f ∈Txandg∈Ty. A monotone set-valued mappingT:H→H_{is calledmaximalif its}

graphGph(T) is not properly contained in the graph of any other monotone set-valued mapping. It is known that a monotone set-valued mappingT:H→H_{is maximal if and}

only if for (x,f)∈H×H,x–y,f –g ≥ for every (y,g)∈Gph(T) implies thatf∈Tx. LetA:C→Hbe a monotone and Lipschitz continuous mapping and letNCvbe the

normal cone toCatv∈C, namely

NCv=

w∈H:v–u,w ≥,∀u∈C.

Define

Tv=

Av+NCv, ifv∈C,

∅, ifv∈/C.

Lemma .(see []) Let A:C→H be a monotone mapping. (i) T is maximal monotone;

(ii) v∈T–_{⇔v∈VI(C,A).}

3 Main results

Let us consider the following three-step iterative scheme with regularization:

⎧ ⎪ ⎨ ⎪ ⎩

Θ(un,y) +ϕ(y) –ϕ(un) +_r_ny–un,un–xn ≥, ∀y∈C,

yn=θnγSxn+ (I–θnμF)PC(un–λ∇fαn(un)),

xn+=βnγVyn+ (I–βnμF)TPC(un–λ∇fαn(un)), ∀n= , , , . . . .

(.)

Here,

• V:H→His aρ-contraction;

• S:H→HandT:C→Care nonexpansive mappings;

• F:C→His aκ-Lipschitzian andη-strongly monotone mapping; • Θ:C×C→Randϕ:C→Rare real-valued functions;

• ∇f :C→HisL-Lipschitz continuous with <λ<_L;

• {rn}and{αn}are sequences in(, +∞)with∞_n=αn< +∞andlim infn→∞rn> ;

• {βn}and{θn}are sequences in(, );

•  <μ< η/κand <γ ≤τ, whereτ=  – –μ(η–μκ_).

Theorem . Suppose thatΘ:C×C→Rsatisfies(A)-(A)and that(B)or(B)holds.

(H) ∞_n=βn= +∞,limn→∞βn| –

θn–

θn |= ; (H) limn→∞βn|



θn– 

θn–|= andlimn→∞



θn| –

βn–

βn |= ; (H) lim_n→∞θn= ,limn→∞αnθ+nβn= ,andlimn→∞

θ_n βn= ; (H) limn→∞|αnβ–nαθnn–|= andlimn→∞

|rn–rn–|

βnθn = .

Then we have the following: (i) limn→∞xn+θn–xn= ;

(ii) ωw(xn)⊂Fix(T)∩MEP(Θ,ϕ)∩Γ;

(iii) ωw(xn)⊂Ξ ifxn–yn=o(θn)held in addition,i.e.,limn→∞xnθ–nyn= .

Proof First, let us show thatPC(I–λ∇fα) isξ-averaged for eachλ∈(,α+L), where

ξ= +λ(α+L)  ∈(, ).

The Lipschitz condition implies that the gradient ∇f is _L-inverse strongly monotone [], that is,

∇f(x) –∇f(y),x–y≥

L∇f(x) –∇f(y)

 .

Observe that

(α+L)∇fα(x) –∇fα(y),x–y

= (α+L)αx–y+∇f(x) –∇f(y),x–y

=αx–y+α∇f(x) –∇f(y),x–y+αLx–y+L∇f(x) –∇f(y),x–y

≥αx–y+ α∇f(x) –∇f(y),x–y+∇f(x) –∇f(y)

=α(x–y) +∇f(x) –∇f(y)

=∇fα(x) –∇fα(y).

Hence, ∇fα =αI+∇f is _α_+L-inverse strongly monotone. Thus, λ∇fα is _λ(α_+L)-inverse strongly monotone by Proposition .(ii). By Proposition .(iii) the complementI–λ∇fα

isλ(α_+L)-averaged. Noting thatPCis_-averaged and utilizing Proposition .(iv), we know

that for eachλ∈(,_α_+L), the mapPC(I–λ∇fα) isξ-averaged with

ξ= +

λ(α+L)

 –

 ·

λ(α+L)

 =

 +λ(α+L)  ∈(, ).

In particular,PC(I–λ∇fα) is nonexpansive. Furthermore, forλ∈(,_L), utilizing the fact

thatlim_n→∞ 

αn+L=



L, we may assume

 <λ< 

αn+L

, ∀n≥.

Consequently, for each integern≥,PC(I–λ∇fαn) isξn-averaged with

ξn=

  +

λ(αn+L)

 –

 ·

λ(αn+L)

 =

 +λ(αn+L)

 ∈(, ).

We divide the proof into several steps. Step .limn→∞xn+θn–xn= .

For simplicity, putu˜n=PC(un–λ∇fαn(un)). Thenyn=θnγSxn+ (I–θnμF)u˜nandxn+=

βnγVyn+ (I–βnμF)Tu˜nfor everyn≥. We observe that

˜un–u˜n– ≤PC(I–λ∇fαn)un–PC(I–λ∇fαn)un–

+PC(I–λ∇fαn)un––PC(I–λ∇fαn–)un–

≤ un–un–+PC(I–λ∇fαn)un––PC(I–λ∇fαn–)un–

≤ un–un–+(I–λ∇fαn)un–– (I–λ∇fαn–)un–

=un–un–+λ∇fαn(un–) –λ∇fαn–(un–)

=un–un–+λ|αn–αn–|un–, ∀n≥. (.)

Moreover, from (.) we have

yn=θnγSxn+ (I–θnμF)u˜n,

yn–=θn–γSxn–+ (I–θn–μF)u˜n–, ∀n≥.

Thus

yn–yn–=θn(γSxn–γSxn–) + (θn–θn–)(γSxn––μFu˜n–)

+ (I–θnμF)u˜n– (I–θnμF)u˜n–.

Utilizing Lemma . from (.) we deduce that

yn–yn– ≤θnγSxn–γSxn–+|θn–θn–|γSxn––μFu˜n–

+(I–θnμF)u˜n– (I–θnμF)u˜n–

≤θnγxn–xn–+|θn–θn–|γSxn––μFu˜n–

+ ( –θnτ)˜un–u˜n–

≤θnγxn–xn–+|θn–θn–|γSxn––μFu˜n–

+ ( –θnτ)

un–un–+λ|αn–αn–|un–

, (.)

where τ =  – –μ(η–μκ_{). Taking into consideration that u}

n=Trnxnand un–=

Trn–xn–, we have

Θ(un,y) +ϕ(y) –ϕ(un) +



rn

y–un,un–xn ≥, ∀y∈C (.)

and

Θ(un–,y) +ϕ(y) –ϕ(un–) + 

rn–

y–un–,un––xn– ≥, ∀y∈C. (.)

Puttingy=un–in (.) andy=unin (.), we obtain

Θ(un,un–) +ϕ(un–) –ϕ(un) +



rn

and

Θ(un–,un) +ϕ(un) –ϕ(un–) + 

rn–

un–un–,un––xn– ≥, ∀y∈C.

Adding the last two inequalities, by (A) we get

un–un–,

un––xn–

rn–

–un–xn

rn

≥,

and hence

un–un–,un––un+un–xn––

rn–

rn

(un–xn)

≥.

Sincelim infn→∞rn> , we may assume, without loss of generality, that there exists a

pos-itive numbercsuch thatrn≥c>  for alln≥. Thus we have

un–un–≤

un–un–,xn–xn–+

 –rn–

rn

(un–xn)

≤ un–un–

xn–xn–+

 –rn–

rn

un–xn

,

and hence

un–un– ≤ xn–xn–+

 –rn–

rn

un–xn

≤ xn–xn–+

M

c |rn–rn–|. (.)

Here,M=sup{un–xn:n≥}< +∞.

Substituting (.) into (.) we derive

yn–yn– ≤θnγxn–xn–+|θn–θn–|γSxn––μFu˜n–

+ ( –θnτ)

xn–xn–+

M

c |rn–rn–|+λ|αn–αn–|un–

≤ –θn(τ–γ)

xn–xn–+|θn–θn–|γSxn––μFu˜n–

+M

c |rn–rn–|+λ|αn–αn–|un– ≤ xn–xn–+M

|θn–θn–|+|rn–rn–|+|αn–αn–|

. (.)

Here,γSxn–μFu˜n+M_c +λun ≤Mfor someM≥. On the other hand, from (.) we have

xn+=βnγVyn+ (I–βnμF)Tu˜n,

xn=βn–γVyn–+ (I–βn–μF)Tu˜n–, ∀n≥.

Simple calculations show that

xn+–xn= (I–βnμF)Tu˜n– (I–βnμF)Tu˜n–

Utilizing Lemma . from (.), (.), and (.) we deduce that

xn+–xn

≤(I–βnμF)Tu˜n– (I–βnμF)Tu˜n–+|βn–βn–|γVyn––μFTu˜n–

+βnγVyn–γVyn–

≤( –βnτ)˜un–u˜n–+|βn–βn–|γVyn––μFTu˜n–+βnγρyn–yn–

≤( –βnτ)

un–un–+λ|αn–αn–|un–

+|βn–βn–|γVyn––μFTu˜n–

+βnγρyn–yn–

≤( –βnτ)

xn–xn–+

M

c |rn–rn–|+λ|αn–αn–|un–

+|βn–βn–|γVyn––μFTu˜n–+βnγρ

xn–xn–+M

|θn–θn–|

+|rn–rn–|+|αn–αn–|

≤( –βnτ)

xn–xn–+M

|rn–rn–|+|αn–αn–|

+|βn–βn–|γVyn––μFTu˜n–

+βnγρ

xn–xn–+M

|θn–θn–|+|rn–rn–|+|αn–αn–|

≤( –βnτ)xn–xn–+ ( –βnτ)M

|rn–rn–|+|αn–αn–|

+|βn–βn–|γVyn––μFTu˜n–

+βnγρxn–xn–+βnτM

|θn–θn–|+|rn–rn–|+|αn–αn–|

≤ –βn(τ–γρ)

xn–xn–+|βn–βn–|γVyn––μFTu˜n–

+M

|θn–θn–|+|rn–rn–|+|αn–αn–|

≤ –βn(τ–γρ)

xn–xn–+M

|αn–αn–|+|βn–βn–|

+|θn–θn–|+|rn–rn–|

,

whereM+γVyn–μFTu˜n ≤M,∀n≥ for someM≥. Therefore,

xn+–xn

θn

≤ –βn(τ–γρ)

xn–xn–

θn

+M _|

αn–αn–|

θn

+|βn–βn–|

θn

+|θn–θn–|

θn

+|rn–rn–|

θn

= – (τ–γρ)βn

xn–xn–

θn–

+ – (τ–γρ)βn

xn–xn–



θn

– 

θn–

+M _|

αn–αn–|

θn

+|βn–βn–|

θn

+|θn–θn–|

θn

+|rn–rn–|

θn

≤ – (τ–γρ)βn

xn–xn–

θn–

+ (τ–γρ)βn·



τ–γρ

xn–xn– 

βn _θ_n– 

θn–

+M _|

αn–αn–|

βnθn

+|βn–βn–|

βnθn

+|θn–θn–|

βnθn

+|rn–rn–|

≤ – (τ–γρ)βn

xn–xn–

θn–

+ (τ–γρ)βn· M

τ–γρ



βn _θ_n– 

θn–

+|αn–αn–|

βnθn

+ 

θn  –βn–

βn

+ 

βn  –θn–

θn

+|rn–rn–|

βnθn

, (.)

whereM+xn–xn– ≤M,∀n≥ for someM≥. From (H), (H), and (H), it follows that∞_n=(τ–γρ)βn= +∞and

lim

n→∞

M

τ–γρ



βn _θ_n– 

θn–

+|αn–αn–|

βnθn

+ 

θn  –βn–

βn +  βn  –θn–

θn

+|rn–rn–|

βnθn

= .

Applying Lemma . to (.), we immediately conclude that

lim

n→∞

xn+–xn

θn

= .

In particular, from (H) it follows that

lim

n→∞xn+–xn= .

Step .limn→∞xn–un=  andlimn→∞yn–u˜n= .

By the firm nonexpansivity ofTrn, ifv∈MEP(Θ,ϕ) = Fix(Trn), we have

un–v=Trnxn–Trnv 

≤ Trnxn–Trnv,xn–v

= 

un–v+xn–v–Trnxn–Trnv– (xn–v)

 .

This immediately yields

un–v≤ xn–v–xn–un. (.)

Letp∈Fix(T)∩MEP(Θ,ϕ)∩Γ. We have

˜un–p=PC(I–λ∇fαn)un–PC(I–λ∇f)p ≤PC(I–λ∇fαn)un–PC(I–λ∇fαn)p

+PC(I–λ∇fαn)p–PC(I–λ∇f)p

≤ un–p+PC(I–λ∇fαn)p–PC(I–λ∇f)p

≤ un–p+λαnp. (.)

Note that

yn–p=θnγSxn–θnμFp+ (I–θnμF)u˜n– (I–θnμF)p

=θn(γSxn–μFp) + ( –θn)(u˜n–p) +θn

(I–μF)u˜n– (I–μF)p

=θn

γSxn+ (I–μF)u˜n–p

Hence we have

yn–u˜n=θn

γSxn+ (I–μF)u˜n–u˜n

.

By Lemmas . and ., we have from (.) and (.) that

yn–p

=θn

γSxn+ (I–μF)u˜n–p

+ ( –θn)(u˜n–p)

=θnγSxn+ (I–μF)u˜n–p



+ ( –θn)˜un–p

–θn( –θn)γSxn+ (I–μF)u˜n–u˜n



=θnγSxn+ (I–μF)u˜n–p



+ ( –θn)˜un–p–

 –θn θn

yn–u˜n

≤θnγSxn+ (I–μF)u˜n–p



+˜un–p–

 –θn θn

yn–u˜n. (.)

Furthermore, utilizing Lemmas . and . we have from (.) and (.) that

xn+–p

=βnγVyn–βnμFTp+ (I–βnμF)Tu˜n– (I–βnμF)Tp



≤βnγVyn–βnμFTp+(I–βnμF)Tu˜n– (I–βnμF)Tp



≤βnγVyn–μFp+ ( –βnτ)˜un–p 

≤βn



τγVyn–μFp

_{+ ( –β}

nτ)˜un–p

≤βn



τ

γVyn–γVp+ γVp–μFp,γVyn–μFp

+ ( –βnτ)˜un–p

≤βn γρ

τ yn–p

_+β

n



τγVp–μFpγVyn–μFp+ ( –βnτ)˜un–p



≤βn γρ

τ

θnγSxn+ (I–μF)u˜n–p



+˜un–p–

 –θn θn

yn–u˜n

+βn



τγVp–μFpγVyn–μFp+ ( –βnτ)˜un–p



=

 –βn

τ–γ _ρ

τ

˜un–p+βnθn γρ

τ γSxn+ (I–μF)u˜n–p



–βnγ

_ρ_{( –θ}

n)

τ θn

yn–u˜n+βn



τγVp–μFpγVyn–μFp

≤ ˜un–p+βnθn γρ

τ γSxn+ (I–μF)u˜n–p



–βnγ

_ρ_{( –θ}

n) τ θn

yn–u˜n+βn



τγVp–μFpγVyn–μFp

≤un–p+λαnp 

+βnθn γ_ρ

τ γSxn+ (I–μF)u˜n–p



–βnγ

_ρ_{( –θ}

n)

τ θn

yn–u˜n+βn



≤ un–p+λαnp

un–p+λαnp

+βnθn γρ

τ γSxn+ (I–μF)u˜n–p

 –βnγ

_ρ_{( –θ}

n) τ θn

yn–u˜n

+βn



τγVp–μFpγVyn–μFp

≤ xn–p–xn–un+λαnp

un–p+λαnp

+βnθn γρ

τ γSxn+ (I–μF)u˜n–p

 –βnγ

_ρ_{( –θ}

n)

τ θn

yn–u˜n

+βn



τγVp–μFpγVyn–μFp. (.)

It turns out therefore that

xn–un+

βnγρ( –θn)

τ θn

yn–u˜n

≤ xn–p–xn+–p+λαnp

un–p+λαnp

+βnθn γ_ρ

τ γSxn+ (I–μF)u˜n–p



+βn



τγVp–μFpγVyn–μFp ≤xn–p+xn+–p

xn–xn++λαnp

un–p+λαnp

+βnθn γρ

τ γSxn+ (I–μF)u˜n–p



+βn



τγVp–μFpγVyn–μFp.

Then it is clear that

xn–un≤

xn–p+xn+–p

xn–xn++λαnp

un–p+λαnp

+βnθn γρ

τ γSxn+ (I–μF)u˜n–p



+βn



τγVp–μFpγVyn–μFp.

Sinceαn→,βn→,θn→, andxn+–xn →, we conclude that

lim

n→∞xn–un= .

Furthermore,

βnγρ( –θn)

τ θn

yn–u˜n

≤xn–p+xn+–p

xn–xn++λαnp

un–p+λαnp

+βnθn γ_ρ

τ γSxn+ (I–μF)u˜n–p

 +βn



This yields

γ_ρ_{( –θ}

n)

τ yn–u˜n



≤xn–p+xn+–p

θn βn

xn–xn++

αnθn βn

λpun–p+λαnp

+θ_nγ

_ρ

τ γSxn+ (I–μF)u˜n–p

 +θn



τγVp–μFpγVyn–μFp.

Sincexn+–xn

θn →,

αn+βn

θn →, and

θ_n

βn → asn→ ∞, we have

lim

n→∞ θn βn

xn–xn+= lim

n→∞

xn–xn+

θn ·

θ_n βn

= 

and

lim

n→∞

αnθn βn

= lim

n→∞

αn θn ·

θ

n βn

= .

Therefore, from the last inequality we have

lim

n→∞yn–u˜n= .

Step .limn→∞un–u˜n=  andlimn→∞xn–yn= .

Letp∈Fix(T)∩Fix(Γ)∩Ξ. Utilizing Lemmas . and . we have from (.) that

xn+–p

≤ ˜un–p+βnθn γ_ρ

τ γSxn+ (I–μF)u˜n–p



–βnγ

_ρ_{( –θ}

n)

τ θn

yn–u˜n+βn



τγVp–μFpγVyn–μFp

≤PC(I–λ∇fαn)un–PC(I–λ∇f)p 

+βnθn γρ

τ γSxn+ (I–μF)u˜n–p



+βn



τγVp–μFpγVyn–μFp

≤(I–λ∇f)un– (I–λ∇f)p–λαnun

 +βnθn

γρ

τ γSxn+ (I–μF)u˜n–p



+βn



τγVp–μFpγVyn–μFp

≤(I–λ∇f)un– (I–λ∇f)p

 – λαn

un, (I–λ∇fαn)un– (I–λ∇f)p

+βnθn γ_ρ

τ γSxn+ (I–μF)u˜n–p

 +βn



τγVp–μFpγVyn–μFp

≤ un–p+λ

λ–

L

∇f(un) –∇f(p)



+ λαnun(I–λ∇fαn)un– (I–λ∇f)p

+βnθn γρ

τ γSxn+ (I–μF)u˜n–p

 +βn



≤ xn–p+λ

λ–

L

∇f(un) –∇f(p)

+ λαnun(I–λ∇fαn)un– (I–λ∇f)p

+βnθn γ_ρ

τ γSxn+ (I–μF)u˜n–p

 +βn



τγVp–μFpγVyn–μFp.

Hence,

λ



L–λ

∇f(un) –∇f(p)



≤ xn–p–xn+–p+ λαnun(I–λ∇fαn)un– (I–λ∇f)p

+βnθn γρ

τ γSxn+ (I–μF)u˜n–p

 +βn



τγVp–μFpγVyn–μFp ≤xn–p+xn+–p

xn–xn++ λαnun(I–λ∇fαn)un– (I–λ∇f)p

+βnθn γρ

τ γSxn+ (I–μF)u˜n–p

 +βn



τγVp–μFpγVyn–μFp.

Since αn→,βn→, θn→, and xn+ –xn →, it follows from  <λ< _L that

limn→∞∇f(un) –∇f(p)= , and hence

lim

n→∞∇fαn(un) –∇f(p)= .

Furthermore, from the firm nonexpansiveness ofPCwe obtain

˜un–p=PC(I–λ∇fαn)un–PC(I–λ∇f)p 

≤(I–λ∇fαn)un– (I–λ∇f)p,u˜n–p

= 

(I–λ∇fαn)un– (I–λ∇f)p 

+˜un–p

–(I–λ∇fαn)un– (I–λ∇f)p– (u˜n–p)



≤ 



un–p+ λ∇fαn(un) –∇f(p)(I–λ∇fαn)un– (I–λ∇f)p

+˜un–p–un–u˜n+ λ

un–u˜n,∇fαn(un) –∇f(p)

–λ∇fαn(un) –∇f(p) 

.

Consequently,

˜un–p

≤ un–p–un–u˜n+ λ∇fαn(un) –∇f(p)(I–λ∇fαn)un– (I–λ∇f)p

+ λun–u˜n,∇fαn(un) –∇f(p)

–λ∇fαn(un) –∇f(p) 

.

Thus, from (.) we have

xn+–p

≤ ˜un–p+βnθn γρ

τ γSxn+ (I–μF)u˜n–p

–βnγ

_ρ_{( –θ}

n) τ θn

yn–u˜n+βn



τγVp–μFpγVyn–μFp

≤ un–p–un–u˜n+ λ∇fαn(un) –∇f(p)(I–λ∇fαn)un– (I–λ∇f)p

+ λun–u˜n,∇fαn(un) –∇f(p)

–λ∇fαn(un) –∇f(p) 

+βnθn γ_ρ

τ γSxn+ (I–μF)u˜n–p

 –βnγ

_ρ_{( –θ}

n) τ θn

yn–u˜n

+βn



τγVp–μFpγVyn–μFp

≤ xn–p–un–u˜n+ λ∇fαn(un) –∇f(p)(I–λ∇fαn)un– (I–λ∇f)p

+ λun–u˜n∇fαn(un) –∇f(p)+βnθn

γρ

τ γSxn+ (I–μF)u˜n–p



+βn



τγVp–μFpγVyn–μFp.

This implies that

un–u˜n

≤ xn–p–xn+–p+ λ∇fαn(un) –∇f(p)(I–λ∇fαn)un– (I–λ∇f)p

+ λun–u˜n∇fαn(un) –∇f(p)+βnθn γρ

τ γSxn+ (I–μF)u˜n–p



+βn



τγVp–μFpγVyn–μFp ≤xn–p+xn+–p

xn–xn+

+ λ∇fαn(un) –∇f(p)(I–λ∇fαn)un– (I–λ∇f)p

+ λun–u˜n∇fαn(un) –∇f(p)+βnθn γρ

τ γSxn+ (I–μF)u˜n–p



+βn



τγVp–μFpγVyn–μFp.

Sinceθn→,βn→,xn–xn+ →, and∇fαn(un) –∇f(p) →, it follows that

lim

n→∞un–u˜n= .

This, together withxn–un → andyn–u˜n → (due to Step ), implies that

xn–yn ≤ xn–un+un–u˜n+˜un–yn → asn→ ∞,

and thus

lim

n→∞xn–yn= .

Letp∗∈ωw(xn). Then there exists a subsequence{xni}of{xn}such thatxnip∗. Since

xn+–u˜n=βnγSyn+ (I–βnμF)Tu˜n–u˜n

=βn(γSyn–μFTu˜n) +Tu˜n–u˜n,

we have

Tu˜n–u˜n=xn+–u˜n–βn(γSyn–μFTu˜n) ≤ xn+–u˜n+βnγSyn–μFTu˜n

≤ xn+–xn+xn–un+un–u˜n+βnγSyn–μFTu˜n.

Hence fromxn+–xn →,xn–un →,un–u˜n →, andβn→, we get

lim

n→∞Tu˜n–u˜n= .

Sincexn–un → andun–u˜n →, we haveu˜nip∗. Utilizing Lemma . we derive

p∗∈Fix(T).

Let us show thatp∗∈MEP(Θ,ϕ). As a matter of fact, sinceun=Trnxn, for anyy∈Cwe have

Θ(un,y) +ϕ(y) –ϕ(un) +



rn

y–un,un–xn ≥.

It follows from (A) that

ϕ(y) –ϕ(un) +



rn

y–un,un–xn ≥Θ(y,un).

Replacingnbyni, we have

ϕ(y) –ϕ(uni) + 

rni

y–uni,uni–xni ≥Θ(y,uni).

Sinceuni_r–xni

ni → andunip

∗_{, it follows from (A) that}

≥–ϕ(y) +ϕp∗+Θy,p∗, ∀y∈C.

Putzt=ty+ ( –t)p∗for allt∈(, ] andy∈C. We havezt∈Cand

≥–ϕ(zt) +ϕ

p∗+Θzt,p∗

. (.)

Utilizing (A), (A), and (.), we have

 =Θ(zt,zt) +ϕ(zt) –ϕ(zt)

≤tΘ(zt,y) + ( –t)Θ

zt,p∗

+tϕ(y) + ( –t)ϕp∗–ϕ(zt)

≤tΘ(zt,y) +ϕ(y) –ϕ(zt)

+ ( –t)Θzt,p∗

+ϕp∗–ϕ(zt)

≤tΘ(zt,y) +ϕ(y) –ϕ(zt)

and hence

≤Θ(zt,y) +ϕ(y) –ϕ(zt). (.)

Lettingt→ in (.) and utilizing (A), we get, for eachy∈C,

≤Θp∗,y+ϕ(y) –ϕp∗.

Hence,p∗∈MEP(Θ,ϕ).

Let us show thatp∗∈Γ. Fromxn–un → andun–u˜n →, we know thatunip∗ andu˜nip∗. Define

Tv=

∇f(v) +NCv, ifv∈C,

∅, ifv∈/C,

where

NCv=

w∈H:v–u,w ≥,∀u∈C.

ThenT is maximal monotone and ∈Tvif and only ifv∈VI(C,∇f); see [] for more details. Let (v,w)∈Gph(T). Then we have

w∈Tv=∇f(v) +NCv

and hence,

w–∇f(v)∈NCv.

Therefore,

v–u,w–∇f(v)≥, ∀u∈C.

On the other hand, from

˜ un=PC

un–λ∇fαn(un)

and v∈C,

we have

un–λ∇fαn(un) –u˜n,u˜n–v

≥,

and hence

v–u˜n, ˜

un–un

λ +∇fαn(un)

≥.

Therefore, from

we have

v–u˜ni,w ≥

v–u˜ni,∇f(v)

≥v–u˜ni,∇f(v)

–

v–u˜ni,

˜ uni–uni

λ +∇fαni(uni)

=v–u˜ni,∇f(v)

–

v–u˜ni,

˜ uni–uni

λ +∇f(uni)

–αniv–u˜ni,uni

=v–u˜ni,∇f(v) –∇f(u˜ni)

+v–u˜ni,∇f(u˜ni) –∇f(uni)

–

v–u˜ni,

˜ uni–uni

λ

–αniv–u˜ni,uni

≥v–u˜ni,∇f(u˜ni) –∇f(uni)

–

v–u˜ni,

˜ uni–uni

λ

–αniv–u˜ni,uni.

Hence, we obtain

v–p∗,w≥ asi→ ∞.

SinceTis maximal monotone, we havep∗∈T–_{, and hence,p}∗_{∈VI(C,∇f), which leads} top∗∈Γ. Consequently,p∗∈Fix(T)∩MEP(Θ,ϕ)∩Γ. This shows thatωw(xn)⊂Fix(T)∩

MEP(Θ,ϕ)∩Γ.

Utilizing Lemmas . and ., we have for everyp∈Fix(T)∩MEP(Θ,ϕ)∩Γ,

yn–p

=θnγSxn+ (I–μF)u˜n–p



=θn(γSp–μFp) +θn(γSxn–γSp) + (I–θnμF)u˜n– (I–θnμF)p



≤θn(γSxn–γSp) + (I–θnμF)u˜n– (I–θnμF)p



+ θnγSp–μFp,yn–p

≤θn(γSxn–γSp)+(I–θnμF)u˜n– (I–θnμF)p



+ θnγSp–μFp,yn–p

≤θnγxn–p+ ( –θnτ)˜un–p 

+ θnγSp–μFp,yn–p

≤θnγxn–p+ ( –θnτ)

un–p+λαnp 

+ θnγSp–μFp,yn–p

≤θnγxn–p+ ( –θnτ)

xn–p+λαnp 

+ θnγSp–μFp,yn–p

≤ –θn(τ–γ)

xn–p+λαnp 

+ θnγSp–μFp,yn–p

≤xn–p+λαnp 

+ θnγSp–μFp,yn–p. (.)

Suppose now thatxn–yn=o(θn) in addition. It follows from (.) that

γSp–μFp,yn–p

≤ 

θn

xn–p+λαnp 

–yn–p

≤ 

θn

xn–p+λαnp+yn–p

xn–p+λαnp–yn–p

≤ 

θn

xn–p+λαnp+yn–p

xn–yn+λαnp

This, together withαn

θn → and xn–yn

θn →, leads to

lim sup

n→∞ γSp–μFp,yn–p ≤.

Observe that

lim sup

n→∞ γSp–μFp,xn–p

=lim sup

n→∞

γSp–μFp,xn–yn+γSp–μFp,yn–p

=lim sup

n→∞ γSp–μFp,yn–p ≤.

So, it follows fromxnip∗that

γSp–μFp,p∗–p≤, ∀p∈Fix(T)∩MEP(Θ,ϕ)∩Γ.

Note also that  <γ≤τ and

μη≥τ ⇔ μη≥ –

 –μη–μκ

⇔  –μη–μκ≥_{ –μη}

⇔  – μη+μκ≥ – μη+μη

⇔ κ≥η

⇔ κ≥η.

It is clear that

(μF–γS)x– (μF–γS)y,x–y≥(μη–γ)x–y, ∀x,y∈H.

Hence, it follows from  <γ ≤τ ≤μηthatμF–γSis monotone. Sincep∗∈ωw(xn)⊂

Fix(T)∩MEP(Θ,ϕ)∩Γ, by Minty’s lemma [] we have

γSp∗–μFp∗,p–p∗≤, ∀p∈Fix(T)∩MEP(Θ,ϕ)∩Γ;

that is,p∗∈Ξ. This shows thatωw(xn)⊂Ξ.

Theorem . Assuming the conditions in Theorem..We have:

(i) {xn}and{yn}both converge strongly to an elementx∗∈Fix(T)∩MEP(Θ,ϕ)∩Γ, which is a unique solution of the variational inequality

γVx∗–μFx∗,x–x∗≤, ∀x∈Fix(T)∩MEP(Θ,ϕ)∩Γ.

(ii) {xn}and{yn}both converge strongly to a unique solution of THVI(.)if

Proof Utilizing Lemmas . and . we get from (.)

xn+–p

=βnγVyn+ (I–μF)Tu˜n–p



=βn(γVp–μFp) +βn(γVyn–γVp) + (I–βnμF)Tu˜n– (I–βnμF)Tp



≤βn(γVyn–γVp) + (I–θnμF)Tu˜n– (I–θnμF)Tp



+ βnγVp–μFp,xn+–p

≤βn(γVyn–γVp)+(I–βnμF)Tu˜n– (I–θnμF)Tp



+ βnγVp–μFp,xn+–p

≤βnγρyn–p+ ( –βnτ)˜un–p 

+ βnγVp–μFp,xn+–p

≤βn γ_ρ

τ yn–p

_{+ ( –β}

nτ)˜un–p+ βnγVp–μFp,xn+–p

≤βn γ_ρ

τ

xn–p+λαnp 

+ θnγSp–μFp,yn–p

+ ( –βnτ)

xn–p+λαnp 

+ βnγVp–μFp,xn+–p

=

 –βn

τ_–γ_ρ

τ

xn–p+λαnp 

+ βnθn γ_ρ

τ γSp–μFp,yn–p

+ βnγVp–μFp,xn+–p, (.)

whereτ=  – –μ(η–μκ_).

Note thatμF–γV:H→His (μκ+γρ)-Lipschitzian and (μη–γρ)-strongly monotone, namely

(μF–γV)x– (μF–γV)y≤(μκ+γρ)x–y, ∀x,y∈H

and

(μF–γV)x– (μF–γV)y,x–y≥(μη–γρ)x–y, ∀x,y∈H.

Hence there exists a unique solutionx∗∈Fix(T)∩MEP(Θ,ϕ)∩Γ of the variational in-equality problem

γVx∗–μFx∗,x–x∗≤, ∀x∈Fix(T)∩MEP(Θ,ϕ)∩Γ. (.)

Since the sequence{xn}is bounded, there exists a subsequence{xni}of{xn}such that

lim sup

n→∞

γVx∗–μFx∗,xn–x∗

=lim

i→∞

γVx∗–μFx∗,xni–x

∗_{. (.)}