On a finite family of variational inclusions with the constraints of generalized mixed equilibrium and fixed point problems

R E S E A R C H

Open Access

On a finite family of variational inclusions

with the constraints of generalized mixed

equilibrium and fixed point problems

Lu-Chuan Ceng

_{, Chi-Ming Chen}

_{and Chin-Tzong Pang}

*_{Correspondence:}

[email protected]

4_{Department of Information}

Management, Yuan Ze University, Chung-Li, 32003, Taiwan

5_{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 two iterative algorithms for finding common solutions of a finite family of variational inclusions for maximal monotone and inverse-strongly monotone mappings with the constraints of two problems: a generalized mixed equilibrium problem and a common fixed point problem of an infinite family of nonexpansive mappings and an asymptotically strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. We prove some strong and weak convergence theorems for the proposed iterative algorithms under suitable conditions.

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

Keywords: generalized mixed equilibrium; variational inclusion; nonexpansive mapping; asymptotically strict pseudocontractive mapping in the intermediate sense; maximal monotone mapping; inverse-strongly monotone mapping

1 Introduction

LetHbe a real Hilbert space with inner product·,·and norm · ,Cbe a nonempty closed convex subset ofHandPCbe the metric projection ofHontoC. LetS:C→H

be a nonlinear mapping onC. We denote byFix(S) the set of fixed points ofSand byR

the set of all real numbers. A mappingVis called strongly positive onHif there exists a constantγ¯∈(, ] such that

Vx,x ≥ ¯γx, ∀x∈H.

A mappingS:C→His calledL-Lipschitz-continuous if there exists a constantL>  such that

Sx–Sy ≤Lx–y, ∀x,y∈C.

In particular, ifL=  thenSis called a nonexpansive mapping; ifL∈(, ) thenAis called a contraction.

Letϕ:C→Rbe a real-valued function,A:H→Hbe a nonlinear mapping andΘ:C× C→Rbe a bifunction. We consider 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. The GMEP is further considered and studied in,e.g., [–].

Throughout this paper, it is assumed as in [] thatΘ:C×C→Ris a bifunction satis-fying conditions (H)-(H) andϕ:C→Ris a lower semicontinuous and convex function with restriction (H), where

(H) Θ(x,x) = for allx∈C;

(H) Θis monotone,i.e.,Θ(x,y) +Θ(y,x)≤for anyx,y∈C; (H) Θis upper-hemicontinuous,i.e., for eachx,y,z∈C,

lim sup t→+

Θtz+ ( –t)x,y≤Θ(x,y);

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

(H) 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< .

LetΘ,Θ:C×C→Rbe two bifunctions, andB,B:C→Hbe two nonlinear map-pings. Consider the system of generalized equilibrium problems (SGEP): find (x∗,y∗)∈ C×Csuch that

Θ(x∗,x) +By∗,x–x∗+_μ x

∗_–y∗_,x–x∗ _{≥, ∀x∈C,}

Θ(y∗,y) +Bx∗,y–y∗+_μ y

∗_–x∗_,y–y∗ _{≥, ∀y∈C,} (.)

whereμandμare two positive constants.

Let{Tn}∞n=be an infinite family of nonexpansive self-mappings onCand{λn}∞n=be a sequence of nonnegative numbers in [, ]. For anyn≥, define a self-mappingWnonH

as follows:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

Un,n+=I,

Un,n=λnTnUn,n++ ( –λn)I,

Un,n–=λn–Tn–Un,n+ ( –λn–)I,

.. .

Un,k=λkTkUn,k++ ( –λk)I, Un,k–=λk–Tk–Un,k+ ( –λk–)I, ..

.

Un,=λTUn,+ ( –λ)I, Wn=Un,=λTUn,+ ( –λ)I.

(.)

Such a mappingWnis called theW-mapping generated byTn,Tn–, . . . ,Tandλn,λn–,

Let f :H→H be a contraction and V be a strongly positive bounded linear oper-ator on H. Assume thatϕ:H→Ris a lower semicontinuous and convex functional, thatΘ,Θ,Θ:H×H→Rsatisfy conditions (H)-(H), and thatA,B,B:H→Hare inverse-strongly monotone. Very recently, motivated by Yaoet al.[], Cai and Bu [] in-troduced the following hybrid extragradient-like iterative algorithm:

⎧ ⎪ ⎨ ⎪ ⎩

zn=Sr(Θn,ϕ)(xn–rnAxn),

yn=TμΘ(I–μB)T

Θ

μ(I–μB)zn,

xn+=αn(u+γf(xn)) +βnxn+ (( –βn)I–αn(I+μV))Wnyn, ∀n≥,

(.)

for finding a common solution of GMEP (.), SGEP (.), and the fixed point problem of an infinite family of nonexpansive mappings{Ti}∞i=onH, where{rn} ⊂(,∞),{αn},{βn} ⊂

(, ), and x,u∈H are given. The authors proved the strong convergence of the se-quence generated by the hybrid iterative algorithm (.) to a pointx∗∈( ∞_i=Fix(Ti))∩ GMEP(Θ,ϕ,A)∩SGEP(G) under some suitable conditions, whereSGEP(G) is the fixed point set of the mappingG:=TΘ

μ(I–μB)T

Θ

μ(I–μB). This pointx

∗_{also solves the}

following optimization problem:

min

x∈( ∞n=Fix(Tn))∩GMEP(Θ,ϕ,A)∩SGEP(G)

μ

Vx,x+  x–u

_{–h(x), (OP)}

whereh:H→Ris the potential function ofγf.

Let Bbe a single-valued mapping ofC into H andRbe a set-valued mapping with D(R) =C. Consider the following variational inclusion: find a pointx∈Csuch that

∈Bx+Rx. (.)

We denote by I(B,R) the solution set of the variational inclusion (.). In particular, if B=R= , then I(B,R) =C. IfB= , then problem (.) becomes the inclusion problem introduced by Rockafellar []. It is known that problem (.) provides a convenient frame-work for the unified study of optimal solutions in many optimization related areas includ-ing mathematical programminclud-ing, complementarity problems, variational inequalities, op-timal control, mathematical economics, equilibria and game theory,etc. Let a set-valued mappingR:D(R)⊂H→H be maximal monotone. We define the resolvent operator JR,λ:H→D(R) associated withRandλas follows:

JR,λ= (I+λR)–, ∀x∈H,

whereλis a positive number.

In , for the case whereC=H, Yaoet al.[] introduced and analyzed an iterative algorithms for finding a common element of the set of solutions of the GMEP (.), the set of solutions of the variational inclusion (.) for maximal monotone and inverse-strongly monotone mappings and the set of fixed points of a countable family of nonexpansive mappings onH.

Recently, Kim and Xu [] introduced the concept of asymptoticallyκ-strict pseudocon-tractive mappings in a Hilbert space.

Definition . LetCbe a nonempty subset of a Hilbert spaceH. A mappingS:C→C

is said to be an asymptoticallyκ-strict pseudocontractive mapping with sequence{γn}if there exist a constantκ∈[, ) and a sequence{γn}in [,∞) withlimn→∞γn=  such that

_Sn_x–Sn_y_{≤( +γ}

n)x–y+κx–Snx–

y–Sny, ∀n≥,∀x,y∈C.

Subsequently, Sahuet al.[] considered the concept of asymptoticallyκ-strict pseudo-contractive mappings in the intermediate sense, which are not necessarily Lipschitzian.

Definition . LetCbe a nonempty subset of a Hilbert spaceH. A mappingS:C→Cis

said to be an asymptoticallyκ-strict pseudocontractive mapping in the intermediate sense with sequence{γn}if there exist a constantκ∈[, ) and a sequence{γn}in [,∞) with

limn→∞γn=  such that

lim sup n→∞

sup x,y∈C

_Sn_x–Sn_y_{– ( +γ}

n)x–y–κx–Snx–

y–Sny≤. (.)

Putcn:=max{,supx,y∈C(Snx–Sny– ( +γn)x–y–κx–Snx– (y–Sny))}. Then

cn≥ (∀n≥),cn→ (n→ ∞), and (.) reduce to the relation

Snx–Sny≤( +γn)x–y+κx–Snx–y–Sny+cn, ∀n≥,∀x,y∈C.

(.)

Whenevercn=  for alln≥ in (.), thenSis an asymptoticallyκ-strict

pseudocon-tractive mapping with sequence{γn}. The authors [] derived the weak and strong

con-vergence of the modified Mann iteration processes for an asymptoticallyκ-strict pseudo-contractive mapping in the intermediate sense with sequence{γn}. More precisely, they

first established one weak convergence theorem for the following iterative scheme:

x=x∈Cchosen arbitrarily, xn+= ( –αn)xn+αnSnxn, ∀n≥,

where  <δ≤αn≤ –κ–δ,

∞

n=αncn<∞, and

∞

n=γn<∞; and then obtained another

strong convergence theorem for the following iterative scheme:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

x=x∈Cchosen arbitrary, yn= ( –αn)xn+αnSnxn,

Cn={z∈C:yn–z≤ xn–z+θn},

Qn={z∈C:xn–z,x–xn ≥},

xn+=PCn∩Qnx, ∀n≥,

Inspired by the above facts, we in this paper introduce two iterative algorithms for find-ing common solutions of a finite family of variational inclusions for maximal monotone and inverse-strongly monotone mappings with the constraints of two problems: a gener-alized mixed equilibrium problem and a common fixed point problem of an infinite family of nonexpansive mappings and an asymptotically strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. We prove some strong and weak convergence theorems for the proposed iterative algorithms under suitable conditions. The results pre-sented in this paper are the supplement, extension, improvement, and generalization of the previously known results in this area.

2 Preliminaries

Throughout this paper, we assume thatH is a real Hilbert space whose inner product and norm are denoted by·,·and · , respectively. LetCbe a nonempty closed 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}

.

Definition . A mappingA:C→His called

(i) monotone if

Ax–Ay,x–y ≥, ∀x,y∈C;

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

Ax–Ay,x–y ≥ηx–y, ∀x,y∈C;

(iii) ζ-inverse-strongly monotone if there exists a constantζ> such that

Ax–Ay,x–y ≥ζAx–Ay, ∀x,y∈C.

It is easy to see that the projectionPCis -inverse-strongly monotone (in short, -ism).

Inverse-strongly monotone (also referred to as co-coercive) operators have been applied widely in solving practical problems in various fields.

Definition . A differentiable functionK:H→Ris called:

(i) convex, if

K(y) –K(x)≥K(x),y–x, ∀x,y∈H,

whereK(x)is the Frechet derivative ofKatx;

(ii) strongly convex, if there exists a constantσ> such that

K(y) –K(x) –K(x),y–x≥σ x–y

_{, ∀x,y∈H.}

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).

Some important properties of projections are gathered in the following proposition.

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

(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. (This implies thatPCis nonexpansive

and monotone.)

By using the technique of [], we can readily obtain the following elementary result.

Proposition .(see [, Lemma  and Proposition ]) Let C be a nonempty closed convex

subset of a real Hilbert space H and letϕ:C→Rbe a lower semicontinuous and convex function.LetΘ:C×C→Rbe a bifunction satisfying the conditions(H)-(H).Assume that

(i) K:H→Ris strongly convex with constantσ> and the functionx→ y–x,K(x)

is weakly upper semicontinuous for eachy∈H;

(ii) for eachx∈Handr> ,there exist a bounded subsetDx⊂Candyx∈Csuch that for anyz∈C\Dx,

Θ(z,yx) +ϕ(yx) –ϕ(z) +

 r

K(z) –K(x),yx–z

< .

Then the following hold:

(a) for eachx∈H,Sr(Θ,ϕ)(x)=∅;

(b) Sr(Θ,ϕ)is single-valued;

(c) S(rΘ,ϕ)is nonexpansive ifKis Lipschitz-continuous with constantν> and

K(x) –K(x),u–u≥K(u) –K(u),u–u, ∀(x,x)∈H×H,

whereui=Sr(Θ,ϕ)(xi)fori= , ;

(d) for alls,t> andx∈H

KS_s(Θ,ϕ)x–KS(tΘ,ϕ)x

,S(_sΘ,ϕ)x–S(tΘ,ϕ)x

≤s–t s

KS(Θ,ϕ)

s x

–K(x),S(Θ,ϕ)

s x–S

(Θ,ϕ)

t x

;

(e) Fix(Sr(Θ,ϕ)) =MEP(Θ,ϕ);

(f ) MEP(Θ,ϕ)is closed and convex.

In particular,wheneverΘ:C×C→Ris a bifunction satisfying the conditions(H)-(H) and K(x) =_x_{,∀x∈H,then,for any x,y∈H,}

(S(rΘ,ϕ)is firmly nonexpansive)and

S_s(Θ,ϕ)x–S(tΘ,ϕ)x≤

|s–t| s S

(Θ,ϕ)

s x–x, ∀s,t> ,x∈H.

In this case,S(rΘ,ϕ)is rewritten as T( Θ,ϕ)

r .If,in addition,ϕ≡,then T( Θ,ϕ)

r is rewritten as

TΘ

r (see[, Lemma .]for more details).

We need some facts and tools in a real Hilbert spaceHwhich 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.

Lemma .([, Lemma .]) Let H be a real Hilbert space.Given a nonempty closed

convex subset of H and points x,y,z∈H and given also a real number a∈R,the set

v∈C:y–v≤ x–v+z,v+a

is convex(and closed).

Lemma .([, Lemma .]) Let C be a nonempty subset of a Hilbert space H and S:

C→C be an asymptoticallyκ-strict pseudocontractive mapping in the intermediate sense with sequence{γn}.Then

Sn_x–Sn_y≤ 

 –κ

κx–y+ + ( –κ)γn

x–y_{+ ( –κ)c}

n

for all x,y∈C and n≥.

Lemma .([, Lemma .]) Let C be a nonempty subset of a Hilbert space H and S:

C→C be a uniformly continuous asymptoticallyκ-strict pseudocontractive mapping in the intermediate sense with sequence {γn}.Let{xn} be a sequence in C such thatxn–

xn+ →andxn–Snxn →as n→ ∞.Thenxn–Sxn →as n→ ∞.

Lemma .(Demiclosedness principle [, Proposition .]) Let C be a nonempty closed

convex subset of a Hilbert space H and S:C→C be a continuous asymptoticallyκ-strict pseudocontractive mapping in the intermediate sense with sequence {γn}. Then I–S is

Lemma .([, Proposition .]) Let C be a nonempty closed convex subset of a Hilbert space H and S:C→C be a continuous asymptoticallyκ-strict pseudocontractive mapping in the intermediate sense with sequence{γn}such thatFix(S)=∅.ThenFix(S)is closed and

convex.

Remark . Lemmas . and . give some basic properties of an asymptotically κ

-strict pseudocontractive mapping in the intermediate sense with sequence{γn}. Moreover,

Lemma . extends the demiclosedness principles studied for certain classes of nonlinear mappings; see [] for more details.

Lemma .([, p.]) Let{an}∞n=,{bn}∞n=,and{δn}∞n=be sequences of nonnegative real numbers satisfying the inequality

an+≤( +δn)an+bn, ∀n≥.

If∞_n=δn<∞and

∞

n=bn<∞,thenlimn→∞anexists.If,in addition,{an}∞n=has a sub-sequence which converges to zero,thenlim_n→∞an= .

Recall that a Banach spaceXis said to satisfy the Opial condition [] if, for any given sequence{xn} ⊂Xwhich converges weakly to an elementx∈X, we have the inequality

lim sup n→∞ xn

–x<lim sup n→∞ xn

–y, ∀y∈X,y=x.

It is well known in [] that every Hilbert spaceHsatisfies the Opial condition.

Lemma .(see [, Proposition .]) Let C be a nonempty closed convex subset of a real Hilbert space H and let{xn}be a sequence in H.Suppose that

xn+–p≤( +λn)xn–p+δn, ∀p∈C,n≥,

where{λn}and{δn}are sequences of nonnegative real numbers such that

∞

n=λn<∞and

∞

n=δn<∞.Then{PCxn}converges strongly in C.

Lemma .(see []) Let C be a closed convex subset of a real Hilbert space H.Let{xn}

be a sequence in H and u∈H.Let q=PCu.If{xn}is such thatωw(xn)⊂C and satisfies the

condition

xn–u ≤ u–q, for all n,

then xn→q as n→ ∞.

Lemma .(see [, Lemma .]) Let C be a nonempty closed convex subset of a real

Hilbert space H.Let{Tn}∞n=be a sequence of nonexpansive self-mappings on C such that

∞

n=Fix(Tn)=∅and let{λn}be a sequence in(,b]for some b∈(, ).Then,for every x∈C

Remark . (see [, Remark .]) It can be known from Lemma . that if D is a nonempty bounded subset ofC, then for>  there existsn≥ksuch that for alln>n

sup x∈DUn,k

x–Ukx ≤.

Remark .(see [, Remark .]) Utilizing Lemma ., we define a mappingW:C→C

as follows:

Wx= lim

n→∞Wnx=nlim→∞Un,x, ∀x∈C.

Such aWis called theW-mapping generated byT,T, . . . andλ,λ, . . . . SinceWnis

non-expansive,W:C→Cis also nonexpansive. Indeed, observe that for eachx,y∈C

Wx–Wy= lim

n→∞Wnx–Wny ≤ x–y.

If{xn}is a bounded sequence inC, then we putD={xn:n≥}. Hence, it is clear from

Remark . that for an arbitrary>  there existsN≥ such that for alln>N

Wnxn–Wxn=Un,xn–Uxn ≤sup x∈D

Un,x–Ux ≤.

This implies that

lim

n→∞Wnxn–Wxn= .

Lemma .(see [, Lemma .]) Let C be a nonempty closed convex subset of a real

Hilbert space H.Let{Tn}∞n=be a sequence of nonexpansive self-mappings on C such that

∞

n=Fix(Tn)=∅,and let{λn}be a sequence in(,b]for some b∈(, ).Then Fix(W) = ∞

n=Fix(Tn).

Lemma .(see [, Theorem . (Demiclosedness Principle)]) Let C be a nonempty

closed convex subset of a real Hilbert space H.Let T:C→C be nonexpansive.Then I–T is demiclosed on C.That is,whenever{xn}is a sequence in C weakly converging to some

x∈C and the sequence{(I–T)xn}strongly converges to some y,it follows that(I–T)x=y.

Here I is the identity operator of H.

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

D(R),f ∈R(x), andg∈R(y) imply

f–g,x–y ≥.

Let A:C→H be a monotone,k-Lipschitz-continuous mapping and let NCvbe the

normal cone toCatv∈C,i.e.,

NCv=

w∈H:v–u,w ≥,∀u∈C.

Define

Tv=

Av+NCv ifv∈C,

∅ ifv∈/C.

ThenTis maximal monotone and ∈Tvif and only ifAv,y–v ≥ for ally∈C(see []). Assume thatR:D(R)⊂H→H_{is a maximal monotone mapping. Letλ> . In terms}

of Huang [] (see also []), we have the following property for the resolvent operator JR,λ:H→D(R).

Lemma . JR,λis single-valued and firmly nonexpansive,i.e.,

JR,λx–JR,λy,x–y ≥ JR,λx–JR,λy, ∀x,y∈H.

Consequently,JR,λis nonexpansive and monotone.

Lemma .(see []) Let R be a maximal monotone mapping with D(R) =C.Then for

any givenλ> ,u∈C is a solution of problem(.)if and only if u∈C satisfies

u=JR,λ(u–λBu).

Lemma . (see []) Let R be a maximal monotone mapping with D(R) =C and let

B:C→H be a strongly monotone,continuous,and single-valued mapping.Then for each z∈H,the equation z∈(B+λR)x has a unique solution xλforλ> .

Lemma .(see []) Let R be a maximal monotone mapping with D(R) =C and B:C→

H be a monotone,continuous and single-valued mapping.Then(I+λ(R+B))C=H for each

λ> .In this case,R+B is maximal monotone.

Lemma .(see []) Let C be a nonempty closed convex subset of a real Hilbert space H, and g:C→R∪+∞be a proper lower semicontinuous differentiable convex function.If x∗ is a solution the minimization problem

gx∗=inf x∈Cg(x),

then

g(x),x–x∗≥, ∀x∈C.

In particular,if x∗solves(OP),then

3 Strong convergence theorems

In this section, we introduce and analyze an iterative algorithm for finding common so-lutions of a finite family of variational inclusions for maximal monotone and inverse-strongly monotone mappings with the constraints of two problems: a generalized mixed equilibrium problem and a common fixed point problem of an infinite family of nonexpan-sive mappings and an asymptotically strict pseudocontractive mapping in the intermedi-ate sense in a real Hilbert space. Under appropriintermedi-ate conditions imposed on the parameter sequences we will prove strong convergence of the proposed algorithm.

Theorem . Let C be a nonempty closed convex subset of a real Hilbert space H.Let N

be an integer.LetΘbe a bifunction from C×C toRsatisfying(H)-(H)andϕ:C→R

be a lower semicontinuous and convex functional.Let Ri:C→H be a maximal

mono-tone mapping and let A:H→H and Bi:C→H beζ-inverse-strongly monotone andηi

-inverse-strongly monotone,respectively,where i∈ {, , . . . ,N}.Let S:C→C be a uniformly continuous asymptoticallyκ-strict pseudocontractive mapping in the intermediate sense for some≤κ< with sequence{γn} ⊂[,∞)such thatlimn→∞γn= and{cn} ⊂[,∞)

such thatlimn→∞cn= .Let{Tn}∞n=be a sequence of nonexpansive self-mappings on C and {λn}be a sequence in(,b]for some b∈(, ).Let V be aγ¯-strongly positive bounded

lin-ear operator and f :H→H be an l-Lipschitzian mapping withγl< ( +μ)γ¯.Assume that

Ω:= ( ∞_n=Fix(Tn))∩GMEP(Θ,ϕ,A)∩( Ni=I(Bi,Ri))∩Fix(S)is nonempty and bounded.

Let Wnbe the W -mapping defined by(.)and{αn},{βn}and{δn}be three sequences in

(, )such thatlimn→∞αn= andκ≤δn≤d< .Assume that:

(i) K:H→Ris strongly convex with constantσ> and its derivativeKis

Lipschitz-continuous with constantν> such that the functionx→ y–x,K(x)is

weakly upper semicontinuous for eachy∈H;

(ii) for eachx∈H,there exist a bounded subsetDx⊂Candzx∈Csuch that for any

y∈/Dx,

Θ(y,zx) +ϕ(zx) –ϕ(y) +

 r

K(y) –K(x),zx–y

< ;

(iii)  <lim infn→∞βn≤lim sup_n→∞βn< ;

(iv) {λi,n} ⊂[ai,bi]⊂(, ηi),∀i∈ {, , . . . ,N},and{rn} ⊂[, ζ]satisfies

 <lim inf

n→∞ rn≤lim sup_n→∞ rn< ζ.

Pick any x∈H and set C=C,x=PCx.Let{xn}be a sequence generated by the following algorithm:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

un=Sr(Θn,ϕ)(I–rnA)xn,

zn=JRN,λN,n(I–λN,nBN)JRN–,λN–,n(I–λN–,nBN–)· · ·JR,λ,n(I–λ,nB)un,

kn=δnzn+ ( –δn)Snzn,

yn=αn(u+γf(xn)) +βnkn+ (( –βn)I–αn(I+μV))Wnzn,

Cn+={z∈Cn:yn–z≤ xn–z+θn},

xn+=PCn+x, ∀n≥,

(.)

whereθn= (αn+γn)Δn+cn,Δn=sup{xn–p+u+ (γf –I–μV)p:p∈Ω}<∞,

and=_–sup

n≥αn<∞.If S

(Θ,ϕ)

(I) {xn}converges strongly toPΩx;

(II) {xn}converges strongly toPΩx,which solves the optimization problem

min x∈Ω

μ

Vx,x+  x–u

_{–h(x), (OP)}

providedγn+cn+xn–yn=o(αn)additionally,whereh:H→Ris the potential function ofγf.

Proof Sincelimn→∞αn=  and  <lim infn→∞βn≤lim supn→∞βn< , we may assume,

without loss of generality, thatαn≤( –βn)( +μV)–. SinceVis aγ¯-strongly positive

bounded linear operator onH, we know that

V=supVu,u:u∈H,u= .

Observe that

( –βn)I–αn(I+μV)

u,u=  –βn–αn–αnμVu,u

≥ –βn–αn–αnμV ≥,

that is, ( –βn)I–αn(I+μV) is positive. It follows that

( –βn)I–αn(I+μV)=sup

( –βn)I–αn(I+μV)

u,u:u∈H,u= 

=sup –βn–αn–αnμVu,u:u∈H,u= 

≤ –βn–αn–αnμγ¯.

Put

Λi_n=JRi,λi,n(I–λi,nBi)JRi–,λi–,n(I–λi–,nBi–)· · ·JR,λ,n(I–λ,nB)

for alli∈ {, , . . . ,N}andn≥, andΛ

n=I, whereIis the identity mapping onH. Then

we have thatzn=ΛNnun. We divide the rest of the proof into several steps.

Step . We show that{xn}is well defined. It is obvious thatCnis closed and convex. As

the defining inequality inCnis equivalent to the inequality

(xn–zn),z

≤ xn–zn+θn,

by Lemma . we know thatCnis convex and closed for everyn≥.

First of all, we show thatΩ⊂Cnfor all n≥. Suppose thatΩ⊂Cnfor somen≥.

Takep∈Ωarbitrarily. Sincep=Sr(nΘ,ϕ)(p–rnAp),Aisζ-inverse strongly monotone and

≤rn≤ζ, we have

un–p=Sr(Θn,ϕ)(I–rnA)xn–S

(Θ,ϕ)

rn (I–rnA)p



≤(I–rnA)xn– (I–rnA)p

=(xn–p) –rn(Axn–Ap)



=xn–p– rnxn–p,Axn–Ap+rnAxn–Ap

≤ xn–p– rnζAxn–Ap+rnAxn–Ap

=xn–p+rn(rn– ζ)Axn–Ap

≤ xn–p. (.)

Sincep=JRi,λi,n(I–λi,nBi)p,Λ

i

np=p, andBiisηi-inverse-strongly monotone, whereηi∈

(, ηi),i∈ {, , . . . ,N}, by Lemma . we deduce that

zn–p=JRN,λN,n(I–λN,nBN)Λ

N–

n un–JRN,λN,n(I–λN,nBN)Λ

N–

n p



≤(I–λN,nBN)ΛNn–un– (I–λN,nBN)ΛNn–p



=ΛN_n–un–ΛNn–p

–λN,n

BNΛNn–un–BNΛNn–p



≤ΛN_n–un–ΛNn–p



+λN,n(λN,n– ηN)BNΛNn–un–BNΛNn–p



≤ΛN_n–un–ΛNn–p



.. .

≤Λ

nun–Λnp



=un–p. (.)

Combining (.) and (.), we have

zn–p ≤ xn–p. (.)

By Lemma .(b), we deduce from (.) and (.) that

kn–p=δn(zn–p) + ( –δn)

Snzn–p

=δnzn–p+ ( –δn)Snzn–p–δn( –δn)zn–Snzn

≤δnzn–p+ ( –δn)

( +γn)zn–p

+κzn–Snzn+cn

–δn( –δn)zn–Snzn

= +γn( –δn)

zn–p+ ( –δn)(κ–δn)zn–Snzn+ ( –δn)cn

≤( +γn)zn–p+ ( –δn)(κ–δn)zn–Snzn

 +cn

≤( +γn)zn–p+cn. (.)

SetV¯ =I+μV. Then, forγl≤( +μ)γ¯, by Lemma . we obtain from (.), (.), and (.)

yn–p

=αn

u+γf(xn)

+βnkn+

( –βn)I–αnV¯

Wnzn–p

=αn

u+γf(xn) –V p¯

+βn(kn–p) +

( –βn)I–αnV¯

(Wnzn–p)



=αn

u+γf(p) –V p¯ +αnγ

f(xn) –f(p)

+βn(kn–p) +

( –βn)I–αnV¯

(Wnzn–p)



≤αnγ

f(xn) –f(p)

+βn(kn–p) +

( –βn)I–αnV¯

(Wnzn–p)



+ αn

u+γf(p) –V p¯ ,yn–p

≤αnγf(xn) –f(p)+βnkn–p+( –βn)I–αnV¯

(Wnzn–p)

+ αn

u+γf(p) –V p¯ ,yn–p

≤αnγlxn–p+βnkn–p+ ( –βn–αn–αnμγ¯)Wnzn–p 

+ αnu+γf(p) –V p¯ yn–p

≤αn( +μ)γ¯xn–p+βnkn–p+

 –βn–αn( +μ)γ¯

zn–p 

+αnu+γf(p) –V p¯



+yn–p

≤αn( +μ)γ¯xn–p+βnkn–p+

 –βn–αn( +μ)γ¯zn–p

+αnu+γf(p) –V p¯ +yn–p

≤αn( +μ)γ¯xn–p+βn

( +γn)zn–p+cn

+ –βn–αn( +μ)γ¯

zn–p+αnu+γf(p) –V p¯



+yn–p

≤αn( +μ)γ¯xn–p+βn

( +γn)zn–p+cn

+ –βn–αn( +μ)γ¯

( +γn)zn–p+cn

+αnu+γf(p) –V p¯



+yn–p

=αn( +μ)γ¯xn–p+

 –αn( +μ)γ¯

( +γn)zn–p+cn

+αnu+γf(p) –V p¯ +yn–p

≤αn( +μ)γ¯

( +γn)xn–p+cn

+ –αn( +μ)γ¯

( +γn)xn–p+cn

+αnu+γf(p) –V p¯



+yn–p

= ( +γn)xn–p+cn+αnu+γf(p) –V p¯



+yn–p

,

which hence yields

yn–p≤

 +γn

 –αn

xn–p+ αn

 –αn

u+γf(p) –V p¯ +   –αn

cn

=

 +αn+γn  –αn

xn–p+ αn

 –αn

u+γf(p) –V p¯ +   –αn

cn

≤

 +αn+γn  –αn

xn–p+ αn+γn

 –αn

u+γf(p) –V p¯ +   –αn

cn

=xn–p+ αn+γn

 –αn

xn–p+u+γf(p) –V p¯



+ 

 –αn

≤ xn–p+ (αn+γn)

xn–p+u+γf(p) –V p¯

 +cn

≤ xn–p+ (αn+γn)Δn+cn

=xn–p+θn, (.)

whereθn= (αn+γn)Δn+cn,Δn=sup{xn–p+u+γf(p) –V p¯ :p∈Ω}<∞, and =_–sup

n≥αn<∞(due to{αn} ⊂(, ) andlimn→∞αn= ). Hencep∈Cn+. This implies

thatΩ⊂Cnfor alln≥. Therefore,{xn}is well defined.

Step . We prove thatxn–kn → asn→ ∞.

Indeed, letv=PΩx. Fromxn=PCnxandv∈Ω⊂Cn, we obtain

xn–x ≤ v–x. (.)

This implies that{xn}is bounded and hence{un},{zn},{kn}, and{yn}are also bounded.

Sincexn+∈Cn+⊂Cnandxn=PCnx, we have

xn–x ≤ xn+–x, ∀n≥.

Therefore limn→∞xn–x exists. From xn=PCnx, xn+ ∈Cn+ ⊂ Cn, by

Proposi-tion .(ii) we obtain

xn+–xn≤ x–xn+–x–xn,

which implies

lim

n→∞xn+–xn= . (.)

It follows fromxn+∈Cn+thatyn–xn+≤ xn–xn++θnand hence

xn–yn≤

xn–xn++xn+–yn

≤xn–xn++xn–xn++θn

= xn–xn++θn

.

From (.) andlim_n→∞θn= , we have

lim

n→∞xn–yn= . (.)

Also, utilizing Lemmas . and .(b) we obtain from (.), (.), and (.)

yn–p

=αn

u+γf(xn) –V W¯ nzn

+βn(kn–p) + ( –βn)(Wnzn–p)



≤βn(kn–p) + ( –βn)(Wnzn–p)

 + αn

u+γf(xn) –V W¯ nzn,yn–p

=βnkn–p+ ( –βn)Wnzn–p–βn( –βn)kn–Wnzn

≤βnkn–p+ ( –βn)zn–p–βn( –βn)kn–Wnzn

+ αnu+γf(xn) –V W¯ nznyn–p

≤βn

( +γn)zn–p+cn

+ ( –βn)zn–p–βn( –βn)kn–Wnzn

+ αnu+γf(xn) –V W¯ nznyn–p

≤βn

( +γn)zn–p+cn

+ ( –βn)

( +γn)zn–p+cn

–βn( –βn)kn–Wnzn+ αnu+γf(xn) –V W¯ nznyn–p

= ( +γn)zn–p+cn–βn( –βn)kn–Wnzn

+ αnu+γf(xn) –V W¯ nznyn–p

≤( +γn)xn–p+cn–βn( –βn)kn–Wnzn

+ αnu+γf(xn) –V W¯ nznyn–p,

which leads to

βn( –βn)kn–Wnzn

≤ xn–p–yn–p+γnxn–p+cn

+ αnu+γf(xn) –V W¯ nznyn–p

≤ xn–yn

xn–p+yn–p

+γnxn–p+cn

+ αnu+γf(xn) –V W¯ nznyn–p.

Sincelimn→∞αn= ,limn→∞γn= , andlimn→∞cn= , it follows from (.) and condition

(iii) that

lim

n→∞kn–Wnzn= . (.)

Note that

yn–kn=αn

u+γf(xn) –V W¯ nzn

+ ( –βn)(Wnzn–kn),

which yields

xn–kn ≤ xn–yn+yn–kn

≤ xn–yn+αn

u+γf(xn) –V W¯ nzn

+ ( –βn)(Wnzn–kn)

≤ xn–yn+αnu+γf(xn) –V W¯ nzn+ ( –βn)Wnzn–kn

≤ xn–yn+αnu+γf(xn) –V W¯ nzn+Wnzn–kn.

So, from (.), (.), andlimn→∞αn= , we get

lim

Step . We prove thatxn–un →,un–zn →,zn–Wzn →, andzn–Snzn →

 asn→ ∞.

Indeed, taking into consideration that  <lim inf_n→∞rn≤lim supn→∞rn< ζ, we may

assume, without loss of generality, that{rn} ⊂[c,d]⊂(, ζ). From (.) and (.) it follows

that

kn–p≤

 +γn( –δn)

zn–p+ ( –δn)(k–δn)zn–Snzn



+ ( –δn)cn

≤ zn–p+γnzn–p+cn

≤ zn–p+γnxn–p+cn. (.)

Next we prove that

lim

n→∞xn–un= . (.)

Forp∈Ω, we find that

un–p=Sr(Θn,ϕ)(I–rnA)xn–S

(Θ,ϕ)

rn (I–rnA)p



≤(I–rnA)xn– (I–rnA)p



=xn–p–rn(Axn–Ap)



≤ xn–p+rn(rn– ζ)Axn–Ap. (.)

By (.), (.), and (.), we obtain

kn–p≤ zn–p+γnxn–p+cn

≤ un–p+γnxn–p+cn

≤ xn–p+rn(rn– ζ)Axn–Ap+γnxn–p+cn,

which implies that

c(ζ –d)Axn–Ap≤rn(ζ–rn)Axn–Ap

≤ xn–p–kn–p+γnxn–p+cn

≤ xn–kn

xn–p+kn–p

+γnxn–p+cn.

Fromlimn→∞γn= ,limn→∞cn= , and (.), we have

lim

n→∞Axn–Ap= . (.)

By the firm nonexpansivity ofS(rn,ϕ)and Lemma .(a), we have

un–p

=S(_rΘ_n,ϕ)(I–rnA)xn–S(rΘn,ϕ)(I–rnA)p

≤(I–rnA)xn– (I–rnA)p,un–p

=

(I–rnA)xn– (I–rnA)p 

+un–p

–(I–rnA)xn– (I–rnA)p– (un–p)



≤ 

xn–p+un–p–xn–un–rn(Axn–Ap)



= 

xn–p+un–p–xn–un+ rnAxn–Ap,xn–un

–r_nAxn–Ap

,

which implies that

un–p≤ xn–p–xn–un+ rnAxn–Apxn–un. (.)

Combining (.) and (.), we have

kn–p≤ zn–p+γnxn–p+cn

≤ un–p+γnxn–p+cn

≤ xn–p–xn–un+ rnAxn–Apxn–un+γnxn–p+cn,

which implies

xn–un

≤ xn–p–kn–p+ rnAxn–Apxn–un+γnxn–p+cn

≤ xn–kn

xn–p+kn–p

+ rnAxn–Apxn–un

+γnxn–p+cn.

Fromlimn→∞γn= ,limn→∞cn= , (.), and (.), we know that (.) holds.

Next we show thatlimn→∞BiΛinun–Bip= ,i= , , . . . ,N. It follows from Lemma .

that

Λi nun–p



=JRi,λi,n(I–λi,nBi)Λ

i–

n un–JRi,λi,n(I–λi,nBi)p



≤(I–λi,nBi)Λni–un– (I–λi,nBi)p



≤Λi–

n un–p



+λi,n(λi,n– ηi)BiΛin–un–Bip



≤ un–p+λi,n(λi,n– ηi)BiΛin–un–Bip



≤ xn–p+λi,n(λi,n– ηi)BiΛni–un–Bip



. (.)

Combining (.) and (.), we have

kn–p≤ zn–p+γnxn–p+cn

≤Λi nun–p



≤ xn–p+λi,n(λi,n– ηi)BiΛin–un–Bip



+γnxn–p+cn,

together with{λi,n} ⊂[ai,bi]⊂(, ηi),i∈ {, , . . . ,N}, implies

ai(ηi–bi)BiΛin–un–Bip



≤λi,n(ηi–λi,n)BiΛin–un–Bip



≤ xn–p–kn–p+γnxn–p+cn

≤ xn–kn

xn–p+kn–p

+γnxn–p+cn.

Fromlimn→∞γn= ,limn→∞cn= , and (.), we obtain

lim n→∞BiΛ

i–

n un–Bip= , i= , , . . . ,N. (.)

By Lemma . and Lemma .(a), we obtain

Λi_nun–p



=JRi,λi,n(I–λi,nBi)Λ

i–

n un–JRi,λi,n(I–λi,nBi)p



≤(I–λi,nBi)Λin–un– (I–λi,nBi)p,Λinun–p

=

(I–λi,nBi)Λ

i–

n un– (I–λi,nBi)p+Λinun–p

–(I–λi,nBi)Λni–un– (I–λi,nBi)p–

Λi_nun–p

≤  Λ

i–

n un–p



+Λi_nun–p



–Λi_n–un–Λinun–λi,n

BiΛin–un–Bip



≤  

un–p+Λinun–p

–Λi_n–un–Λinun–λi,n

BiΛin–un–Bip

≤  

xn–p+Λinun–p



–Λi_n–un–Λinun–λi,n

BiΛin–un–Bip

 ,

which implies

Λi

nun–p

≤ xn–p–Λin–un–Λinun–λi,n

BiΛin–un–Bip



=xn–p–Λni–un–Λinun



–λ_i,nBiΛni–un–Bip



+ λi,n

Λi_n–un–Λinun,BiΛin–un–Bip

≤ xn–p–Λin–un–Λinun

Combining (.) and (.) we get

kn–p≤ zn–p+γnxn–p+cn

≤Λi_nun–p



+γnxn–p+cn

≤ xn–p–Λni–un–Λinun

+ λi,nΛin–un–ΛinunBiΛin–un–Bip

+γnxn–p+cn,

which implies

Λi_n–un–Λinun



≤ xn–p–kn–p+ λi,nΛin–un–ΛinunBiΛni–un–Bip

+γnxn–p+cn

≤ xn–kn

xn–p+kn–p

+ λi,nΛin–un–ΛinunBiΛin–un–Bip

+γnxn–p+cn.

From (.), (.),limn→∞γn= , andlimn→∞cn= , we have

lim n→∞Λ

i–

n un–Λinun= , i= , , . . . ,N. (.)

From (.) we get

un–zn=Λnun–ΛNnun

≤Λ

nun–Λnun+Λnun–Λnun+· · ·+ΛnN–un–ΛNnun

→ asn→ ∞. (.)

By (.) and (.), we have

xn–zn ≤ xn–un+un–zn

→ asn→ ∞. (.)

From (.) and (.), we have

zn+–zn ≤ zn+–xn++xn+–xn+xn–zn

→ asn→ ∞. (.)

By (.), (.), and (.), we get

kn–zn ≤ kn–xn+xn–un+un–zn

We observe that

kn–zn= ( –δn)

Snzn–zn

.

Fromδn≤d<  and (.), we have

lim n→∞S

n_z

n–zn= . (.)

We note that

Snzn–Sn+zn≤Snzn–zn+zn–zn++zn+–Sn+zn+

+Sn+zn+–Sn+zn.

From (.), (.), and Lemma ., we obtain

lim n→∞S

n_z

n–Sn+zn= . (.)

On the other hand, we note that

zn–Szn ≤zn–Snzn+Snzn–Sn+zn+Sn+zn–Szn.

From (.), (.), and the uniform continuity ofS, we have

lim

n→∞zn–Szn= . (.)

In addition, note that

zn–Wzn ≤ zn–kn+kn–Wnzn+Wnzn–Wzn.

So, from (.), (.), and Remark . it follows that

lim

n→∞zn–Wzn= . (.)

Step . we prove thatxn→v=PΩxasn→ ∞.

Indeed, since{xn}is bounded, there exists a subsequence{xni}which converges weakly

to somew. From (.) and (.)-(.), we see thatuniw,Λ

m

niuniw, andzniw,

wherem∈ {, , . . . ,N}. SinceSis uniformly continuous, by (.) we getlimn→∞zn–

Sm_z

n=  for anym≥. Hence from Lemma ., we obtainw∈Fix(S). In the meantime,

utilizing Lemma ., we deduce from (.) andzniwthatw∈Fix(W) =

∞

n=Fix(Tn)

(due to Lemma .). Next, we prove thatw∈ N_m=I(Bm,Rm). As a matter of fact, sinceBm

isηm-inverse-strongly monotone,Bmis a monotone and Lipschitz-continuous mapping. It

follows from Lemma . thatRm+Bmis maximal monotone. Let (v,g)∈G(Rm+Bm),i.e.,

g–Bmv∈Rmv. Again, sinceΛmnun=JRm,λm,n(I–λm,nBm)Λ

m–

n un,n≥,m∈ {, , . . . ,N},

we have

that is,



λm,n

Λm_n–un–Λmnun–λm,nBmΛmn–un

∈RmΛmnun.

In terms of the monotonicity ofRm, we get

v–Λm_nun,g–Bmv–



λm,n

Λm_n–un–Λmnun–λm,nBmΛmn–un

≥

and hence

v–Λm_nun,g

≥

v–Λm_nun,Bmv+



λm,n

Λm_n–un–Λmnun–λm,nBmΛmn–un

=

v–Λm_nun,Bmv–BmΛmnun+BmΛmnun–BmΛmn–un+



λm,n

Λm_n–un–Λmnun

≥v–Λm_nun,BmΛmnun–Bmmn–un

+

v–Λm_nun,



λm,n

Λm_n–un–Λmnun

.

In particular,

v–Λm_niuni,g

≥v–Λm_niuni,BmΛ

m

niuni–BmΛ

m–

ni uni

+

v–Λm_niuni,



λm,ni

Λm_ni–uni–Λ

m niuni

.

SinceΛm_nun–Λmn–un → (due to (.)) andBmΛmnun–BmΛmn–un → (due to the

Lipschitz-continuity ofBm), we conclude fromΛmniuniwand{λi,n} ⊂[ai,bi]⊂(, ηi),

i∈ {, , . . . ,N}that

lim i→∞

v–Λm_n

iuni,g

=v–w,g ≥.

It follows from the maximal monotonicity of Bm+Rm that ∈(Rm+Bm)w, i.e., w∈

I(Bm,Rm). Therefore,w∈ Nm=I(Bm,Rm).

Next, we show thatw∈GMEP(Θ,ϕ,A). In fact, fromzn=S( Θ,ϕ)

rn (I–rnA)xn, we know

that

Θ(un,y) +ϕ(y) –ϕ(un) +Axn,y–un+

 rn

K(un) –K(xn),y–un

≥, ∀y∈C.

From (H) it follows that

ϕ(y) –ϕ(un) +Axn,y–un+

 rn

K(un) –K(xn),y–un

≥Θ(y,un), ∀y∈C.

Replacingnbyni, we have

ϕ(y) –ϕ(uni) +Axni,y–uni+

K(uni) –K(xni)

rni

,y–uni

≥Θ(y,uni),

Putut=ty+ ( –t)wfor allt∈(, ] andy∈C. Then, from (.), we have

ut–uni,Aut

≥ ut–uni,Aut–ϕ(ut) +ϕ(uni) –ut–uni,Axni

–

K(uni) –K(xni)

rni

,ut–uni

+Θ(ut,uni)

≥ ut–uni,Aut–Auni+ut–uni,Auni–Axni–ϕ(ut) +ϕ(uni)

–

K(uni) –K(xni)

rni

,ut–uni

+Θ(ut,uni).

Sinceuni–xni → asi→ ∞, we deduce from the Lipschitz-continuity ofAandK

thatAuni–Axni → andK(uni) –K(xni) → asi→ ∞. Further, from the

mono-tonicity ofA, we haveut–uni,Aut–Auni ≥. So, from (H), we have the weakly lower

semicontinuity ofϕ,K(uni_r)–K(xni)

ni → anduniw, then we have

ut–w,Aut ≥–ϕ(ut) +ϕ(w) +Θ(ut,w), asi→ ∞. (.)

From (H), (H), and (.) we also have

 =Θ(ut,ut) +ϕ(ut) –ϕ(ut)

≤tΘ(ut,y) + ( –t)Θ(ut,w) +tϕ(y) + ( –t)ϕ(w) –ϕ(ut)

=tΘ(ut,y) +ϕ(y) –ϕ(ut)

+ ( –t)Θ(ut,w) +ϕ(w) –ϕ(w) –ϕ(ut)

≤tΘ(ut,y) +ϕ(y) –ϕ(ut)

+ ( –t)ut–w,Aut

=tΘ(ut,y) +ϕ(y) –ϕ(ut)

+ ( –t)ty–w,Aut,

and hence

≤Θ(ut,y) +ϕ(y) –ϕ(ut) + ( –t)y–w,Aut.

Lettingt→, we have, for eachy∈C,

≤Θ(w,y) +ϕ(y) –ϕ(w) +Aw,y–w.

This implies thatw∈GMEP(Θ,ϕ,A). Therefore,

w∈

∞

n=

Fix(Tn)∩GMEP(Θ,ϕ,A)∩ _N

i= I(Bi,Ri)

∩Fix(S) :=Ω.

This shows thatωw(xn)⊂Ω. From (.) and Lemma . we infer thatxn→v=PΩxas

n→ ∞.

Finally, assume additionally thatγn+cn+xn–yn=o(αn). Note thatVis aγ¯-strongly

positive bounded linear operator andf :H→His anl-Lipschitzian mapping withγl< ( +μ)γ¯. It is clear that

_¯