A new iterative algorithm for solving common solutions of generalized mixed equilibrium problems, fixed point problems and variational inclusion problems with minimization problems

R E S E A R C H

Open Access

A new iterative algorithm for solving

common solutions of generalized mixed

equilibrium problems, fixed point problems

and variational inclusion problems with

minimization problems

Thanyarat Jitpeera

1

_{and Poom Kumam}

1,2*

*_{Correspondence:}

[email protected]

1_{Department of Mathematics,}

Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangmod, Thrungkru, Bangkok, 10140, Thailand

2_{Computational Science and}

Engineering Research Cluster (CSEC), King Mongkut’s University of Technology Thonburi (KMUTT), Bangmod, Thrungkru, Bangkok, 10140, Thailand

Abstract

In this article, we introduce a new general iterative method for solving a common element of the set of solutions of fixed point for nonexpansive mappings, the set of solutions of generalized mixed equilibrium problems and the set of solutions of the variational inclusion for a

β

-inverse-strongly monotone mapping in a real Hilbert space. We prove that the sequence converges strongly to a common element of the above three sets under some mild conditions. Our results improve and extend the corresponding results of Marino and Xu (J. Math. Anal. Appl. 318:43-52, 2006), Suet al. (Nonlinear Anal. 69:2709-2719, 2008), Tan and Chang (Fixed Point Theory

Appl. 2011:915629, 2011) and some authors. MSC: 46C05; 47H09; 47H10

Keywords: nonexpansive mapping; inverse-strongly monotone mapping; generalized mixed equilibrium problem; variational inclusion

1 Introduction

LetCbe a nonempty closed convex subset of a real Hilbert spaceHwith the inner prod-uct·,·and the norm · , respectively. A mappingS:C→Cis said to benonexpansive

ifSx–Sy ≤ x–y, ∀x,y∈C. IfC is bounded closed convex andS is a nonexpan-sive mapping ofCinto itself, then F(S) :={x∈C:Sx=x} is nonempty []. A mapping

S:C→Cis said to be ak-strictly pseudo-contraction[] if there exists ≤k<  such that

Sx–Sy≤ x–y+k(I–S)x– (I–S)y,∀x,y∈C, whereIdenotes the identity oper-ator onC. We denote weak convergence and strong convergence by notationsand→, respectively. A mappingAofCintoHis calledmonotoneifAx–Ay,x–y ≥,∀x,y∈C. A mappingAis calledα-inverse-strongly monotoneif there exists a positive real number αsuch thatAx–Ay,x–y ≥αAx–Ay_{,∀x,y∈C. A mappingAis calledα-strongly}

monotoneif there exists a positive real numberαsuch thatAx–Ay,x–y ≥αx–y_,

∀x,y∈C. It is obvious that anyα-inverse-strongly monotone mappingsAis a monotone and_α-Lipschitz continuous mapping. A linear bounded operatorAis calledstrongly pos-itiveif there exists a constantγ¯ >  with the propertyAx,x ≥ ¯γx_{, ∀x∈H. A self}

mappingf :C→Cis calledcontractiononCif there exists a constantα∈(, ) such that

f(x) –f(y) ≤αx–y,∀x,y∈C.

LetB:H→H be a single-valued nonlinear mapping andM:H→Hbe a set-valued mapping. Thevariational inclusion problemis to findx∈Hsuch that

θ∈B(x) +M(x), (.)

whereθ is the zero vector inH. The set of solutions of (.) is denoted byI(B,M). The variational inclusion has been extensively studied in the literature. See,e.g.[–] and the reference therein.

A set-valued mappingM:H→H_{is calledmonotoneif∀x,y∈H,f ∈M(x) andg∈M(y)} implyx–y,f–g ≥. A monotone mappingMismaximalif its graphG(M) :={(f,x)∈

H×H:f ∈M(x)} ofMis not properly contained in the graph of any other monotone mapping. It is known that a monotone mappingMis maximal if and only if for (x,f)∈

H×H,x–y,f –g ≥ for all (y,g)∈G(M) implyf ∈M(x).

LetBbe an inverse-strongly monotone mapping ofCintoHand letNCvbe normal cone toCatv∈C,i.e.,NCv={w∈H:v–u,w ≥,∀u∈C}, and define

Mv= ⎧ ⎨ ⎩

Bv+NCv, ifv∈C,

∅, ifv∈/C.

ThenMis a maximal monotone andθ∈Mvif and only ifv∈VI(C,B) (see []).

LetM:H→H_{be a set-valued maximal monotone mapping, then the single-valued} mappingJM,λ:H→Hdefined by

JM,λ(x) = (I+λM)–(x), x∈H (.)

is called theresolvent operatorassociated withM, whereλis any positive number andIis the identity mapping. In the worth mentioning that the resolvent operator is nonexpan-sive, -inverse-strongly monotone and that a solution of problem (.) is a fixed point of the operatorJM,λ(I–λB) for allλ>  (see []).

LetFbe a bifunction ofC×CintoR, whereRis the set of real numbers,:C→Hbe a mapping andψ:C→Rbe a real-valued function. Thegeneralized mixed equilibrium problemfor findingx∈Csuch that

F(x,y) +x,y–x+ψ(y) –ψ(x)≥, ∀y∈C. (.)

The set of solutions of (.) is denoted byGMEP(F,ψ,), that is

GMEP(F,ψ,) =x∈C:F(x,y) +x,y–x+ψ(y) –ψ(x)≥,∀y∈C.

If≡ andψ≡, the problem (.) is reduced into theequilibrium problem(see also []) for findingx∈Csuch that

The set of solutions of (.) is denoted byEP(F), that is

EP(F) =x∈C:F(x,y)≥,∀y∈C.

This problem contains fixed point problems, includes as special cases numerous problems in physics, optimization and economics. Some methods have been proposed to solve the equilibrium problem, please consult [–].

IfF≡ andψ≡, the problem (.) is reduced into theHartmann-Stampacchia vari-ational inequality[] for findingx∈Csuch that

x,y–x ≥, ∀y∈C. (.)

The set of solutions of (.) is denoted byVI(C,). The variational inequality has been extensively studied in the literature [].

IfF≡ and≡, the problem (.) is reduced into theminimize problemfor finding

x∈Csuch that

ψ(y) –ψ(x)≥, ∀y∈C. (.)

The set of solutions of (.) is denoted byArgmin(ψ). Iterative methods for nonexpan-sive mappings have recently been applied to solve convex minimization problems. Convex minimization problems have a great impact and influence in the development of almost all branches of pure and applied sciences. A typical problem is to minimize a quadratic func-tion over the set of the fixed points of a nonexpansive mapping on a real Hilbert spaceH:

θ(x) = 

Ax,x–x,y, ∀x∈F(S), (.)

whereAis a linear bounded operator,F(S) is the fixed point set of a nonexpansive mapping

Sandyis a given point inH[].

In , Moudafi [] introduced the viscosity approximation method for nonexpan-sive mapping and prove that ifHis a real Hilbert space, the sequence{xn}defined by the iterative method below, with the initial guessx∈Cis chosen arbitrarily,

xn+=αnf(xn) + ( –αn)Sxn, n≥, (.)

where{αn} ⊂(, ) satisfies certain conditions, converge strongly to a fixed point ofS(say

¯

x∈C) which is the unique solution of the following variational inequality:

(I–f)x¯,x–x¯≥, ∀x∈F(S). (.)

In , Iiduka and Takahashi [] introduced following iterative processx∈C,

xn+=αnu+ ( –αn)SPC(xn–λnAxn), ∀n≥, (.)

generated by (.) converges strongly to a common element of the set of fixed points of nonexpansive mapping and the set of solutions of the variational inequality for an inverse-strongly monotone mapping (sayx¯∈C) which solve some variational inequality

¯x–u,x–x¯ ≥, ∀x∈F(S). (.)

In , Marino and Xu [] introduced a general iterative method for nonexpansive mapping. They defined the sequence{xn}generated by the algorithmx∈C,

xn+=αnγf(xn) + (I–αnA)Sxn, n≥, (.)

where{αn} ⊂(, ) andAis a strongly positive linear bounded operator. They proved that ifC=Hand the sequence{αn}satisfies appropriate conditions, then the sequence{xn} generated by (.) converge strongly to a fixed point ofS(sayx¯∈H) which is the unique solution of the following variational inequality:

(A–γf)x¯,x–x¯≥, ∀x∈F(S), (.)

which is the optimality condition for the minimization problem

min

x∈F(S) 

Ax,x–h(x), (.)

wherehis a potential function forγf (i.e.,h(x) =γf(x) forx∈H).

In , Suet al.[] introduced the following iterative scheme by the viscosity approx-imation method in a real Hilbert space:x,un∈H

⎧ ⎨ ⎩

F(un,y) +_r_ny–un,un–xn ≥, ∀y∈C,

xn+=αnf(xn) + ( –αn)SPC(un–λnAun),

(.)

for alln∈N, where{αn} ⊂[, ) and{rn} ⊂(,∞) satisfy some appropriate conditions. Furthermore, they proved{xn}and{un}converge strongly to the same pointz∈F(S)∩

VI(C,A)∩EP(F) wherez=PF(S)∩VI(C,A)∩EP(F)f(z).

In , Tan and Chang [] introduced following iterative process for{Tn:C→C}is a sequence of nonexpansive mappings. Let{xn}be the sequence defined by

xn+=αnxn+ ( –αn)SPC ( –tn)JM,λ(I–λA)Tn(I–μB)

xn

, ∀n≥, (.)

where{αn} ⊂(, ),λ∈(, α] andμ∈(, β]. The sequence{xn}defined by (.) con-verges strongly to a common element of the set of fixed points of nonexpansive mappings, the set of solutions of the variational inequality and the generalized equilibrium problem. In this article, we mixed and modified the iterative methods (.), (.) and (.) by purposing the following new general viscosity iterative method:x,un∈Cand

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

un=Tr(nF,ψ)(xn–rnBxn),

vn=Ts(nF,ψ)(xn–snBxn),

where{αn},{ξn} ⊂(, ),λ∈(, β) such that  <a≤λ≤b< β,{rn} ∈(, η) with  <

c≤d≤ –ηand{sn} ∈(, ρ) with  <e≤f ≤ –ρsatisfy some appropriate conditions. The purpose of this article, we show that under some control conditions the sequence

{xn}converges strongly to a common element of the set of fixed points of nonexpansive mappings, the common solutions of the generalized mixed equilibrium problem and the set of solutions of the variational inclusion in a real Hilbert space.

2 Preliminaries

LetHbe a real Hilbert space with the inner product·,·and the norm · , respectively. LetCbe a nonempty closed convex subset ofH. Recall that the metric (nearest point) projectionPCfromHontoCassigns to eachx∈H, the unique point inPCx∈Csatisfying the property

x–PCx=min y∈Cx–y.

The following characterizes the projection PC. We recall some lemmas which will be needed in the rest of this article.

Lemma . The function u∈C is a solution of the variational inequality (.) if and only if u∈C satisfies the relation u=PC(u–λu)for allλ> .

Lemma . For a given z∈H, u∈C, u=PCz⇔ u–z,v–u ≥,∀v∈C.

It is well known that PCis a firmly nonexpansive mapping of H onto C and satisfies

PCx–PCy≤ PCx–PCy,x–y, ∀x,y∈H. (.)

Moreover, PCx is characterized by the following properties: PCx∈C and for all x∈H, y∈C,

x–PCx,y–PCx ≤. (.)

Lemma .([]) Let M:H→H_{be a maximal monotone mapping and let B:H→H}

be a monotone and Lipshitz continuous mapping. Then the mapping L=M+B:H→H

is a maximal monotone mapping.

Lemma .([]) Each Hilbert space H satisfies Opial’s condition, that is, for any sequence

{xn} ⊂H with xnx, the inequalitylim infn→∞xn–x<lim infn→∞xn–y, hold for

each y∈H with y=x.

Lemma .([]) Assume{an}is a sequence of nonnegative real numbers such that

an+≤( –γn)an+δn, ∀n≥,

where{γn} ⊂(, )and{δn}is a sequence inRsuch that

(i) ∞_n=γn=∞;

(ii) lim sup_n→∞δn

γn≤or

∞

n=|δn|<∞.

Lemma .([]) Let C be a closed convex subset of a real Hilbert space H and let T :

C→C be a nonexpansive mapping. Then I–T is demiclosed at zero, that is, xnx, xn–Txn→

implies x=Tx.

For solving the generalized mixed equilibrium problem, let us assume that the bifunction

F:C×C→R, the nonlinear mapping:C→His continuous monotone andψ:C→

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

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

(A) for each fixedy∈C,x→F(x,y)is weakly upper semicontinuous; (A) for each fixedx∈C,y→F(x,y)is convex and lower semicontinuous;

(B) for eachx∈Candr> , there exist a bounded subsetDx⊆Candyx∈Csuch that for anyz∈C\Dx,

F(z,yx) +ψ(yx) –ψ(z) + 

ryx–z,z–x< , (.)

(B) Cis a bounded set.

Lemma .([]) Let C be a nonempty closed convex subset of a real Hilbert space H. Let F:C×C→Rbe a bifunction mapping satisfies (A)-(A) and letψ:C→Ris convex and lower semicontinuous such that C∩domψ=∅. Assume that either (B) or (B) holds. For r> and x∈H, then there exists u∈C such that

F(u,y) +ψ(y) –ψ(u) +

ry–u,u–x ≥. Define a mapping Tr(F,ψ):H→C as follows:

T_r(F,ψ)(x) =

u∈C:F(u,y) +ψ(y) –ψ(u) +

ry–u,u–x ≥,∀y∈C

(.)

for all x∈H. Then, the following hold: (i) Tr(F,ψ)is single-valued;

(ii) Tr(F,ψ)is firmly nonexpansive, i.e., for anyx,y∈H,

T_r(F,ψ)x–T_r(F,ψ)y≤T_r(F,ψ)x–T_r(F,ψ)y,x–y;

(iii) F(Tr(F,ψ)) =MEP(F,ψ);

(iv) MEP(F,ψ)is closed and convex.

Lemma .([]) Assume A is a strongly positive linear bounded operator on a Hilbert space H with coefficientγ¯> and <ρ≤ A–_{, thenI–ρA ≤ –ργ¯.}

Lemma .([]) Let H be a real Hilbert space and A:H→H a mapping.

(ii) IfAisδ-strongly monotone andμ-strictly pseudo-contraction withδ+μ> , then for any fixed numberτ∈(, ),I–τAis contraction with constant –τ( –( –δ)/μ). 3 Strong convergence theorems

In this section, we show a strong convergence theorem which solves the problem of finding a common element ofF(S),GMEP(F,ψ,B),GMEP(F,ψ,B) andI(B,M) of an inverse-strongly monotone mapping in a real Hilbert space.

Theorem . Let H be a real Hilbert space, C be a closed convex subset of H. Let F, F

be two bifunctions of C×C intoRsatisfying (A)-(A) and B,B,B:C→H beβ,η,ρ

-inverse-strongly monotone mappings,ψ,ψ:C→Rbe convex and lower semicontinuous

function, f :C→C be a contraction with coefficientα( <α< ), M:H→H_{be a}

maxi-mal monotone mapping and A be aδ-strongly monotone andμ-strictly pseudo-contraction mapping withδ+μ> ,γ is a positive real number such thatγ<_α( –

–δ

μ ). Assume that

either (B) or (B) holds. Let S be a nonexpansive mapping of H into itself such that

:=F(S)∩GMEP(F,ψ,B)∩GMEP(F,ψ,B)∩I(B,M)=∅.

Suppose{xn}is a sequence generated by the following algorithm x∈C arbitrarily: ⎧

⎪ ⎪ ⎨ ⎪ ⎪ ⎩

un=Tr(nF,ψ)(xn–rnBxn),

vn=Ts(nF,ψ)(xn–snBxn),

xn+=ξnPC[αnγf(xn) + (I–αnA)SJM,λ(I–λB)un] + ( –ξn)vn, n≥,

(.)

where{αn},{ξn} ⊂(, ),λ∈(, β)such that <a≤λ≤b< β,{rn} ∈(, η)with <c≤

d≤ –ηand{sn} ∈(, ρ)with <e≤f ≤ –ρsatisfy the following conditions:

(C): limn→∞αn= ,n∞=αn=∞,n∞=|αn+–αn|<∞,

(C):  <lim inf_n→∞ξn<lim supn→∞ξn< ,n∞=|ξn+–ξn|<∞,

(C): lim infn→∞rn> and limn→∞|rn+–rn|= ,

(C): lim infn→∞sn> and limn→∞|sn+–sn|= .

Then{xn}converges strongly to q∈, where q=P(γf +I–A)(q)which solves the

fol-lowing variational inequality:

(γf–A)q,p–q≤, ∀p∈

which is the optimality condition for the minimization problem

min

q∈ 

Aq,q–h(q), (.)

where h is a potential function forγf (i.e., h(q) =γf(q)for q∈H). Proof SinceBisβ-inverse-strongly monotone mappings, we have

(I–λB)x– (I–λB)y=(x–y) –λ(Bx–By)

≤ x–y+λ(λ– β)Bx–By

≤ x–y. (.)

AndB,Bareη,ρ-inverse-strongly monotone mappings, we have

(I–rnB)x– (I–rnB)y 

=(x–y) –rn(Bx–By) 

=x–y– rnx–y,Bx–By+rnBx–By

≤ x–y+rn(rn– η)Bx–By

≤ x–y. (.)

In similar way, we can obtain

(I–snB)x– (I–snB)y≤ x–y. (.)

It is clear that if  <λ< β,  <rn< η,  <sn≤ρ thenI–λB,I–rnB,I–snB are all nonexpansive. We will divide the proof into six steps.

Step . We will show{xn}is bounded. Putyn=JM,λ(un–λBun),n≥. It follows that

yn–q =JM,λ(un–λBun) –JM,λ(q–λBq)

≤ un–q. (.)

By Lemma ., we haveun=T(F, ψ)

rn (xn–rnBxn) for alln≥. Then, we note that

un–q =Tr(nF,ψ)(xn–rnBxn) –T

(F,ψ)

rn (q–rnBq)



≤(xn–rnBxn) – (q–rnBq)

≤ xn–q+rn(rn– η)Bxn–Bq

≤ xn–q. (.)

In similar way, we can obtain

vn–q =Ts(nF,ψ)(xn–snBxn) –T

(F,ψ)

sn (q–snBq)



≤(xn–snBxn) – (q–snBq) 

≤ xn–q+sn(sn– ρ)Bxn–Bq

≤ xn–q. (.)

Putzn=PC[αnγf(xn) + (I–αnA)Syn] for alln≥. From (.) and Lemma .(ii), we deduce that

xn+–q

=ξn(zn–q) + ( –ξn)(vn–q)

≤ξnPC

αnγf(xn) + (I–αnA)Syn

≤ξnαnγf(xn) + (I–αnA)Syn–q+ ( –ξn)vn–q =ξnαn γf(xn) –Aq

+ (I–αnA)(Syn–q)+ ( –ξn)vn–q

≤ξnαnγf(xn) –Aq+ξn

 –αn

 –

 –δ

μ

yn–q+ ( –ξn)vn–q

≤ξnαnγ αxn–q+ξnαnγf(q) –Aq+ξn

 –αn

 –

 –δ

μ

xn–q + ( –ξn)xn–q

=

 –

 –

 –δ μ –γ α

ξnαn

xn–q+ξnαnγf(q) –Aq

≤

 –

 –

 –δ

μ –γ α

ξnαn

xn–q

+

 –

 –δ μ –γ α

ξnαn

γf(q) –Aq

( ––δ μ –γ α)

≤max

xn–q,

γf(q) –Aq

 –

–δ μ –γ α

. (.)

It follows from induction that

xn–q ≤max

x–q,

γf(q) –Aq

 ––δ μ –γ α

, n≥.

Therefore{xn}is bounded, so are{vn},{yn},{zn},{Syn},{f(xn)}and{ASyn}. Step . We claim thatlimn→∞xn+–xn+= . From (.), we have

xn+–xn+=ξn+zn++ ( –ξn+)vn+–ξnzn– ( –ξn)vn =ξn+(zn+–zn) + (ξn+–ξn)zn

+ ( –ξn+)(vn+–vn) + (ξn+–ξn)vn

≤ξn+zn+–zn+ ( –ξn+)vn+–vn +|ξn+–ξn| zn+vn

. (.)

We will estimatevn+–vn. On the other hand, fromvn–=Ts(nF–,ψ)(xn––sn–Bxn–) and vn=Ts(nF,ψ)(xn–snBxn), it follows that

F(vn–,y) +Bxn–,y–vn–+ψ(y) –ψ(vn–)

+ 

sn–

y–vn–,vn––xn– ≥, ∀y∈C (.)

and

F(vn,y) +Bxn,y–vn+ψ(y) –ψ(vn) + 

sn

Substitutingy=vnin (.) andy=vn–in (.), we get

F(vn–,vn) +Bxn–,vn–vn–+ψ(vn) –ψ(vn–) + 

sn–

vn–vn–,vn––xn– ≥

and

F(vn,vn–) +Bxn,vn––vn+ψ(vn–) –ψ(vn) + 

sn

vn––vn,vn–xn ≥.

From (A), we obtain

vn–vn–,Bxn––Bxn+

vn––xn–

sn–

–vn–xn

sn

≥,

and then

vn–vn–,sn–(Bxn––Bxn) +vn––xn––

sn–

sn

(vn–xn)

≥,

so

vn–vn–,sn–Bxn––sn–Bxn+vn––vn+vn–xn––

sn–

sn

(vn–xn)

≥.

It follows that

vn–vn–, (I–sn–B)xn– (I–sn–B)xn–+vn––vn+vn–xn–

sn–

sn

(vn–xn)

≥,

vn–vn–,vn––vn+

vn–vn–,xn–xn–+

 –sn–

sn

(vn–xn)

≥.

Without loss of generality, let us assume that there exists a real numberesuch thatsn–>

e> , for alln∈N. Then, we have

vn–vn–≤

vn–vn–,xn–xn–+

 –sn–

sn

(vn–xn)

≤ vn–vn–

xn–xn–+  –sn–

sn

vn–xn

and hence

vn–vn– ≤ xn–xn–+ 

sn|

sn–sn–|vn–xn

≤ xn–xn–+

M

e |sn–sn–|, (.)

whereM=sup{vn–xn:n∈N}. Substituting (.) into (.) that

xn+–xn+ ≤ξn+zn+–zn+ ( –ξn+)

xn+–xn+

M

e |sn–sn–|

+|ξn+–ξn| zn+vn

We note that

zn+–zn=PC

αn+γf(xn+) + (I–αn+A)Syn+

–PC

αnγf(xn) – (I–αnA)Syn

≤αn+γf(xn+) + (I–αn+A)Syn+– αnγf(xn) – (I–αnA)Syn

≤αn+γ f(xn+) –f(xn)

+ (αn+–αn)γf(xn) + (I–αn+A)(Syn+–Syn)

+ (αn–αn+)ASyn

≤αn+γ αxn+–xn+|αn+–αn|γf(xn)

+

 –αn+

 –

 –δ μ

yn+–yn +|αn+–αn|ASyn

≤αn+γ αxn+–xn+|αn+–αn| γf(xn)+ASyn

+

 –αn+

 –

 –δ μ

yn+–yn. (.)

SinceI–λBbe nonexpansive, we have

yn+–yn=JM,λ(un+–λBun+) –JM,λ(un–λBun)

≤(un+–λBun+) – (un–λBun)

≤ un+–un. (.)

On the other hand, fromun–=Tr(nF–,ψ)(xn––rn–Bxn–) andun=T

(F,ψ)

rn (xn–rnBxn), it follows that

F(un–,y) +Bxn–,y–un–+ψ(y) –ψ(un–)

+ 

rn–

y–un–,un––xn– ≥, ∀y∈C (.)

and

F(un,y) +Bxn,y–un+ψ(y) –ψ(un) + 

rn

y–un,un–xn ≥, ∀y∈C. (.)

Substitutingy=unin (.) andy=un–in (.), we get

F(un–,un) +Bxn–,un–un–+ψ(un) –ψ(un–) + 

rn–

un–un–,un––xn– ≥

and

F(un,un–) +Bxn,un––un+ψ(un–) –ψ(un) + 

rn

un––un,un–xn ≥.

From (A), we obtain

un–un–,Bxn––Bxn+

un––xn–

rn–

–un–xn

rn

and then

un–un–,rn–(Bxn––Bxn) +un––xn––

rn–

rn

(un–xn)

≥,

so

un–un–,rn–Bxn––rn–Bxn+un––un+un–xn––

rn–

rn

(un–xn)

≥.

It follows that

un–un–, (I–rn–B)xn– (I–rn–B)xn–+un––un+un–xn–

rn–

rn

(un–xn)

≥,

un–un–,un––un+

un–un–,xn–xn–+

 –rn–

rn

(un–xn)

≥.

Without loss of generality, let us assume that there exists a real numbercsuch thatrn–>

c> , for alln∈N. Then, 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–rn–|un–xn

≤ xn–xn–+

M

c |rn–rn–|, (.)

whereM=sup{un–xn:n∈N}. Substituting (.) into (.), we have

yn–yn– ≤ xn–xn–+

M

c |rn–rn–|. (.)

Substituting (.) into (.), we obtain that

zn+–zn ≤αn+γ αxn+–xn+|αn+–αn| γf(xn)+ASyn

+

 –αn+

 –

 –δ μ

xn–xn–+

M

c |rn–rn–|

. (.)

And substituting (.), (.) into (.), we get

xn+–xn+ ≤ξn+

αn+γ αxn+–xn+|αn+–αn| γf(xn)+ASyn

+

 –αn+

 –

 –δ μ

xn–xn–+

M

c |rn–rn–|

+ ( –ξn+)

xn–xn–+

M

e |sn–sn–|

+|ξn+–ξn| zn+vn

≤

 –

 –

 –δ

μ

–γ α

ξn+αn+

xn+–xn+ |αn+–αn|

+|ξn+–ξn|

M+

M

e |sn–sn–|+ M

c |rn–rn–|, (.)

whereM>  is a constant satisfying

sup

n

γf(xn)+ASyn,zn+vn

≤M.

This together with (C)-(C) and Lemma ., imply that

lim

n→∞xn+–xn+= . (.)

From (.), we also haveyn+–yn → asn→ ∞. Step . We show the followings:

(i) limn→∞Bun–Bq= ; (ii) limn→∞Bxn–Bq= ; (iii) lim_n→∞Bxn–Bq= .

Forq∈andq=JM,λ(q–λBq), then we get

yn–q =JM,λ(un–λBun) –JM,λ(q–λBq) 

≤(un–λBun) – (q–λBq)

≤ un–q+λ(λ– β)Bun–Bq

≤ xn–q+λ(λ– β)Bun–Bq. (.)

It follows that

zn–q =PC αnγf(xn) + (I–αnA)Syn

–PC(q)



≤αn γf(xn) –Aq

+ (I–αnA)(Syn–q)



≤αnγf(xn) –Aq

 +

 –αn

 –

 –δ

μ

yn–q

+ αn

 –αn

 –

 –δ

μ

γf(xn) –Aqyn–q

≤αnγf(xn) –Aq+ αn

 –αn

 –

 –δ

μ

γf(xn) –Aqyn–q

+

 –αn

 –

 –δ

μ

xn–q+λ(λ– β)Bun–Bq

≤αnγf(xn) –Aq+ αn

 –αn

 –

 –δ

μ

γf(xn) –Aqyn–q

+xn–q+

 –αn

 –

 –δ

μ

By the convexity of the norm · , we have

xn+–q=ξnzn+ ( –ξn)vn–q



≤ξn(zn–q) + ( –ξn)(vn–q)

≤ξnzn–q+ ( –ξn)vn–q. (.)

Substituting (.), (.) into (.), we obtain

xn+–q

≤ξn

αnγf(xn) –Aq

 + αn

 –αn

 –

 –δ

μ

γf(xn) –Aqyn–q

+xn–q+

 –αn

 –

 –δ

μ

λ(λ– β)Bun–Bq

+ ( –ξn)xn–q

≤ξnαnγf(xn) –Aq

 + ξnαn

 –αn

 –

 –δ

μ

γf(xn) –Aqyn–q

+ξnxn–q+ξn

 –αn

 –

 –δ

μ

λ(λ– β)Bun–Bq + ( –ξn)xn–q.

So, we obtain

ξn

 –αn

 –

 –δ

μ

λ(β–λ)Bun–Bq

≤ξnαnγf(xn) –Aq

 +n

+xn–xn+ xn–q+xn+–q

,

wheren= ξnαn( –αn( –

–δ

μ ))γf(xn) –Aqyn–q. Since conditions (C)-(C) and

limn→∞xn+–xn= , then we obtain thatBun–Bq → asn→ ∞. We consider this inequality in (.) that

zn–q≤αnγf(xn) –Aq+

 –αn

 –

 –δ

μ

yn–q

+ αn

 –αn

 –

 –δ

μ

γf(xn) –Aqyn–q. (.)

Substituting (.) and (.) into (.), we have

zn–q≤αnγf(xn) –Aq

 +

 –αn

 –

 –δ

μ

+ αn

 –αn

 –

 –δ

μ

γf(xn) –Aqyn–q

=αnγf(xn) –Aq

 +

 –αn

 –

 –δ

μ

xn–q

+

 –αn

 –

 –δ

μ

rn(rn– η)Bxn–Bq

+ αn

 –αn

 –

 –δ

μ

γf(xn) –Aqyn–q

≤αnγf(xn) –Aq



+xn–q

+

 –αn

 –

 –δ

μ

rn(rn– η)Bxn–Bq

+ αn

 –αn

 –

 –δ

μ

γf(xn) –Aqyn–q. (.)

Substituting (.) and (.) into (.), we obtain

xn+–q≤ξn

αnγf(xn) –Aq



+xn–q

+

 –αn

 –

 –δ

μ

rn(rn– η)Bxn–Bq

+ αn

 –αn

 –

 –δ

μ

γf(xn) –Aqyn–q

+ ( –ξn)xn–q =ξnαnγf(xn) –Aq



+ξnxn–q+ξn

 –αn

 –

 –δ

μ

rn(rn– η)Bxn–Bq

+ ξnαn

 –αn

 –

 –δ

μ

γf(xn) –Aqyn–q

+ ( –ξn)xn–q. (.)

So, we also have

ξn

 –αn

 –

 –δ

μ

rn(η–rn)Bxn–Bq

≤ξnαnγf(xn) –Aq

 +n

+xn–xn+ xn–q+xn+–q

,

where n= ξnαn( –αn( –

–δ

μ ))γf(xn) –Aqyn–q. Since conditions (C)-(C),

(.) into (.), we have

zn–q

≤αnγf(xn) –Aq



+

 –αn

 –

 –δ

μ

xn–q+λ(λ– β)Bun–Bq

+ αn

 –αn

 –

 –δ

μ

γf(xn) –Aqyn–q

≤αnγf(xn) –Aq



+xn–q

+

 –αn

 –

 –δ

μ

λ(λ– β)Bun–Bq

+ αn

 –αn

 –

 –δ

μ

γf(xn) –Aqyn–q. (.)

Substituting (.) and (.) into (.), we obtain

xn+–q

≤ξn

αnγf(xn) –Aq+xn–q

+

 –αn

 –

 –δ

μ

λ(λ– β)Bun–Bq

+ αn

 –αn

 –

 –δ

μ

γf(xn) –Aqyn–q

+ ( –ξn)

xn–q+sn(sn– ρ)Bxn–Bq

=ξnαnγf(xn) –Aq



+ξnxn–q+ξn

 –αn

 –

 –δ

μ

λ(λ– β)Bun–Bq

+ ξnαn

 –αn

 –

 –δ

μ

γf(xn) –Aqyn–q + ( –ξn)xn–q+ ( –ξn)sn(sn– ρ)Bxn–Bq =ξnαnγf(xn) –Aq



+xn–q+ξn

 –αn

 –

 –δ

μ

λ(λ– β)Bun–Bq

+ ξnαn

 –αn

 –

 –δ

μ

γf(xn) –Aqyn–q

So, we also have

( –ξn)sn(ρ–sn)Bxn–Bq

≤ξnαnγf(xn) –Aq



+n+xn–xn+ xn–q+xn+–q

+ξn

 –αn

 –

 –δ

μ

λ(λ– β)Bun–Bq,

wheren= ξnαn( –αn( –

–δ

μ ))γf(xn) –Aqyn–q. Since conditions (C), (C), (C),

limn→∞xn+–xn=  andlimn→∞Bun–Bq= , then we obtain thatBxn–Bq → asn→ ∞.

Step . We show the followings: (i) limn→∞xn–un= ; (ii) limn→∞un–yn= ; (iii) limn→∞yn–Syn= . SinceT(F,ψ)

rn is firmly nonexpansive, we observe that

un–q =Tr(nF,ψ)(xn–rnBxn) –T

(F,ψ)

rn (q–rnBq)



≤(xn–rnBxn) – (q–rnBq),un–q

= 

 (xn–rnBxn) – (q–rnBq) 

+un–q –(xn–rnBxn) – (q–rnBq) – (un–q)



≤ 

 xn–q _+u

n–q–(xn–un) –rn(Bxn–Bq) 

= 

 xn–q _+u

n–q–xn–un + rnBxn–Bq,xn–un–rnBxn–Bq

.

Hence, we have

un–q≤ xn–q–xn–un+ rnBxn–Bqxn–un. (.)

SinceJM,λis -inverse-strongly monotone, we compute

yn–q =JM,λ(un–λBun) –JM,λ(q–λBq)

≤(un–λBun) – (q–λBq),yn–q

= 

 (un–λBun) – (q–λBq) 

+yn–q –(un–λBun) – (q–λBq) – (yn–q)



= 

 un–q _+y

n–q–(un–yn) –λ(Bun–Bq)



≤ 

 un–q _+y

n–q–un–yn + λun–yn,Bun–Bq–λBun–Bq

which implies that

yn–q≤ un–q–un–yn+ λun–ynBun–Bq. (.)

Substitute (.) into (.), we have

yn–q≤

xn–q–xn–un+ rnBxn–Bqxn–un

–un–yn+ λun–ynBun–Bq. (.)

Substitute (.) into (.), we have

zn–q≤αnγf(xn) –Aq

 +

 –αn

 –

 –δ

μ

xn–q–xn–un

+ rnBxn–Bqxn–un–un–yn+ λun–ynBun–Bq

+ αn

 –αn

 –

 –δ

μ

γf(xn) –Aqyn–q

≤αnγf(xn) –Aq



+xn–q–xn–un

+ 

 –αn

 –

 –δ

μ

rnBxn–Bqxn–un–un–yn

+ 

 –αn

 –

 –δ

μ

λun–ynBun–Bq

+ αn

 –αn

 –

 –δ

μ

γf(xn) –Aqyn–q. (.)

SinceT(F,ψ)

sn is firmly nonexpansive, we observe that

vn–q =Ts(nF,ψ)(xn–snBxn) –T

(F,ψ)

sn (q–snBq)



≤(xn–snBxn) – (q–snBq),vn–q

= 

 (xn–snBxn) – (q–snBq) 

+vn–q –(xn–snBxn) – (q–snBq) – (vn–q)



≤ 

 xn–q _+v

n–q–(xn–vn) –sn(Bxn–Bq) 

= 

 xn–q _+v

n–q–xn–vn + snBxn–Bq,xn–vn–snBxn–Bq

.

Hence, we have

Substitute (.) and (.) into (.), we obtain

xn+–q

≤ξnzn–q+ ( –ξn)vn–q

≤ξn

αnγf(xn) –Aq



+xn–q–xn–un–un–yn

+ 

 –αn

 –

 –δ

μ

rnBxn–Bqxn–un

+ 

 –αn

 –

 –δ

μ

λun–ynBun–Bq

+ αn

 –αn

 –

 –δ

μ

γf(xn) –Aqyn–q

+ ( –ξn)

xn–q–xn–vn+ snBxn–Bqxn–vn

≤ξnαnγf(xn) –Aq



+ξnxn–q–xn–un–un–yn

+ ξn

 –αn

 –

 –δ

μ

rnBxn–Bqxn–un

+ ξn

 –αn

 –

 –δ

μ

λun–ynBun–Bq

+ ξnαn

 –αn

 –

 –δ

μ

γf(xn) –Aqyn–q

+ ( –ξn)xn–q–xn–vn+ ( –ξn)snBxn–Bqxn–vn. (.)

Then, we derive

xn–un+un–yn+xn–vn

≤ξnαnγf(xn) –Aq



+xn–q–xn+–q

+ ξn

 –αn

 –

 –δ

μ

rnBxn–Bqxn–un

+ ξn

 –αn

 –

 –δ

μ

λun–ynBun–Bq

+ ξnαn

 –αn

 –

 –δ

μ

γf(xn) –Aqyn–q + ( –ξn)snBxn–Bqxn–vn

≤ξnαnγf(xn) –Aq+xn+–xn xn–q+xn+–q

+ ξn

 –αn

 –

 –δ

μ

+ ξn

 –αn

 –

 –δ

μ

λun–ynBun–Bq

+ ξnαn

 –αn

 –

 –δ

μ

γf(xn) –Aqyn–q

+ ( –ξn)snBxn–Bqxn–vn. (.)

By conditions (C)-(C),limn→∞xn+–xn= ,limn→∞Bun–Bq= ,limn→∞Bxn–

Bq=  and limn→∞Bxn–Bq= . So, we have xn–un →, un–yn →,

xn–vn → asn→ ∞. We note thatxn+–xn=ξn(zn–xn) + ( –ξn)(vn–xn). From

limn→∞xn–vn= ,limn→∞xn+–xn= , and hence

lim

n→∞zn–xn= . (.)

It follows that

xn–yn ≤ xn–un+un–yn →, asn→ ∞. (.)

Since

zn–yn ≤ zn–xn+xn–yn.

So, by (.) andlimn→∞xn–yn= , we obtain

lim

n→∞zn–yn= . (.)

Therefore, we observe that

Syn–zn=PCSyn–PC αnγf(xn) + (I–αnA)Syn

≤Syn–αnγf(xn) – (I–αnA)Syn

=αnγf(xn) –ASyn. (.)

By condition (C), we haveSyn–zn → asn→ ∞. Next, we observe that

Syn–yn ≤ Syn–zn+zn–yn.

By (.) and (.), we haveSyn–yn → asn→ ∞.

Step . We show thatq∈:=F(S)∩GMEP(F,ψ,B)∩GMEP(F,ψ,B)∩I(B,M) and

lim sup_n→∞(γf –A)q,Syn–q ≤. It is easy to see thatP(γf + (I–A)) is a contraction ofHinto itself. In fact, from Lemma ., we have

P γf+ (I–A)

x–P γf + (I–A)

y

≤ γf + (I–A)x– γf + (I–A)y

≤γ αx–y+

 –

 –

 –δ μ

x–y

=

 –δ μ +γ α

x–y.

HenceHis complete, there exists a unique fixed pointq∈Hsuch thatq=P(γf + (I–

A))(q). By Lemma . we obtain that(γf–A)q,w–q ≤ for allw∈.

Next, we show thatlim sup_n→∞(γf –A)q,Syn–q ≤, whereq=P(γf+I–A)(q) is the unique solution of the variational inequality(γf –A)q,p–qr ≥,∀p∈. We can choose a subsequence{yni}of{yn}such that

lim sup

n→∞

(γf –A)q,Syn–q

=lim

i→∞

(γf–A)q,Syni–q

.

As{yni}is bounded, there exists a subsequence{yn_ij}of{yni}which converges weakly tow.

We may assume without loss of generality thatyniw. We claim thatw∈. Sinceyn–

Syn → and by Lemma ., we havew∈F(S).

Next, we show thatw∈GMEP(F,ψ,B). Sinceun=Tr(nF,ψ)(xn–rnBxn), we know that

F(un,y) +ψ(y) –ψ(un) +Bxn,y–un+ 

rn

y–un,un–xn ≥, ∀y∈C.

It follows by (A) that

ψ(y) –ψ(un) +Bxn,y–un+ 

rn

y–un,un–xn ≥F(y,un), ∀y∈C.

Hence,

ψ(y) –ψ(uni) +Bxni,y–uni+



rni

y–uni,uni–xni ≥F(y,uni), ∀y∈C. (.)

Fort∈(, ] andy∈H, letyt=ty+ ( –t)w. From (.), we have

yt–uni,Byt

≥ yt–uni,Byt–ψ(yt) +ψ(uni) –Bxni,yt–uni

– 

rni

yt–uni,uni–xni+F(yt,uni)

=yt–uni,Byt–Buni+yt–uni,Buni–Bxni–ψ(yt) +ψ(uni)

– 

rni

yt–uni,uni–xni+F(yt,uni).

Fromuni–xni →, we haveBuni–Bxni →. Further, from (A) and the weakly

lower semicontinuity ofψ,

u_ni–x_ni

r_ni → anduniw, we have

From (A), (A) and (.), we have

 = F(yt,yt) –ψ(yt) +ψ(yt)

≤tF(yt,y) + ( –t)F(yt,w) +tψ(y) + ( –t)ψ(w) –ψ(yt) =tF(yt,y) +ψ(y) –ψ(yt)

+ ( –t)F(yt,w) +ψ(w) –ψ(yt)

≤tF(yt,y) +ψ(y) –ψ(yt)

+ ( –t)yt–w,Byt =tF(yt,y) +ψ(y) –ψ(yt)

+ ( –t)ty–w,Byt,

and hence

≤F(yt,y) +ψ(y) –ψ(yt) + ( –t)y–w,Byt.

Lettingt→, we have, for eachy∈C,

F(w,y) +ψ(y) –ψ(w) +y–w,Bw ≥.

This implies thatw∈GMEP(F,ψ,B). By following the same arguments, we can show thatw∈GMEP(F,ψ,B).

Lastly, we show thatw∈I(B,M). In fact, sinceBis aβ-inverse-strongly monotone,Bis monotone and Lipschitz continuous mapping. It follows from Lemma . thatM+Bis a maximal monotone. Let (v,g)∈G(M+B), sinceg–Bv∈M(v). Again sinceyni=JM,λ(uni–

λBuni), we haveuni–λBuni∈(I+λM)(yni), that is,



λ(uni–yni–λBuni)∈M(yni). By virtue

of the maximal monotonicity ofM+B, we have

v–yni,g–Bv–



λ(uni–yni–λBuni)

≥,

and hence

v–yni,g ≥

v–yni,Bv+



λ(uni–yni–λBuni)

=v–yni,Bv–Byni+v–yni,Byni–Buni

+

v–yni,



λ(uni–yni)

.

It follows fromlimn→∞un–yn= , we havelimn→∞Bun–Byn=  andyniwthat

lim sup

n→∞ v–yn,g=v–w,g ≥.

It follows from the maximal monotonicity ofB+Mthatθ∈(M+B)(w), that is,w∈I(B,M). Therefore,w∈. It follows that

lim sup

n→∞

(γf –A)q,Syn–q

=lim

i→∞

(γf–A)q,Syni–q

Step . We prove xn→q. By using (.) and together with Schwarz inequality, we have

xn+–q=ξnPC αnγf(xn) + (I–αnA)Syn

–q+ ( –ξn)(vn–q)



≤ξnPC αnγf(xn) + (I–αnA)Syn

–PC(q)



+ ( –ξn)vn–q

≤ξnαn γf(xn) –Aq

+ (I–αnA)(Syn–q)



+ ( –ξn)xn–q

≤ξn(I–αnA)Syn–q+ξnαnγf(xn) –Aq



+ ξnαn

(I–αnA)(Syn–q),γf(xn) –Aq

+ ( –ξn)xn–q

≤ξn

 –αn

 –

 –δ

μ 

yn–q+ξnαnγf(xn) –Aq

+ ξnαn

Syn–q,γf(xn) –Aq

– ξnαn

A(Syn–q),γf(xn) –Aq

+ ( –ξn)xn–q

≤ξn

 –αn

 –

 –δ

μ 

xn–q+ξnαnγf(xn) –Aq



+ ξnαn

Syn–q,γf(xn) –γf(q)

+ ξnαn

Syn–q,γf(q) –Aq

– ξnαn

A(Syn–q),γf(xn) –Aq

+ ( –ξn)xn–q

≤ξn

 –αn

 –

 –δ

μ 

xn–q+ξnαnγf(xn) –Aq



+ ξnαnSyn–qγf(xn) –γf(q)+ ξnαn

Syn–q,γf(q) –Aq

– ξnαn

A(Syn–q),γf(xn) –Aq

+ ( –ξn)xn–q

≤ξn

 –αn

 –

 –δ

μ 

xn–q+ξnαnγf(xn) –Aq



+ ξnγ ααnyn–qxn–q+ ξnαn

Syn–q,γf(q) –Aq

– ξnαn

A(Syn–q),γf(xn) –Aq

+ ( –ξn)xn–q

≤

ξn– ξnαn

 –

 –δ

μ

+ξnαn

 –

 –δ

μ 

xn–q

+ξnαnγf(xn) –Aq



+ ξnγ ααnxn–q + ξnαn

Syn–q,γf(q) –Aq

– ξnαn

A(Syn–q),γf(xn) –Aq

+ ( –ξn)xn–q

≤

 – ξnαn

 –

 –δ

μ

+ ξnγ ααn

xn–q

+αn

ξnαnγf(xn) –Aq



+ ξn

Syn–q,γf(q) –Aq

– ξnαnA(Syn–q)γf(xn) –Aq

+ξnαn

 –

 –δ

μ 

xn–q