Fixed point solutions of generalized mixed equilibrium problems and variational inclusion problems for nonexpansive semigroups

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R E S E A R C H

Open Access

Fixed point solutions of generalized mixed

equilibrium problems and variational

inclusion problems for nonexpansive

semigroups

Jae Ug Jeong

*

*_{Correspondence:} [email protected] Department of Mathematics, Dongeui University, Busan, 614-714, South Korea

Abstract

In this paper, we introduce a composite iterative method for solving a common element of the set of solutions of ﬁxed points for nonexpansive semigroups, 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 sets under some mild conditions. Our results improve and extend the corresponding results of Kumam and Wattanawitoon (Math. Comput. Model.

53:998-1006, 2011), Shehu (Math. Comput. Model. 55:1301-1314, 2012), Plubtieng and Punpaeng (Math. Comput. Model. 48:279-286, 2008), Liet al.(Nonlinear Anal.

70:3065-3071, 2009), Plubtieng and Wangkeeree (Bull. Korean Math. Soc. 45:717-728, 2008) and some authors.

MSC: 46C05; 47H09; 47H10

Keywords: variational inequality; inverse strongly monotone mapping; generalized mixed equilibrium problem; nonexpansive semigroup

1 Introduction

LetHbe a real Hilbert space with inner product·,·and norm · . LetCbe a nonempty closed convex subset ofH. Recall that a mappingT:C→Cis said to be nonexpansive if

Tx–Ty ≤ x–y, ∀x,y∈C.

We denote byF(T) the set of ﬁxed points ofT. LetPCbe the projection ofH onto the

convex subsetC. Moreover, we also denote byRthe set of all real numbers.

A one-parameter family ={T(t) : ≤t<∞}is said to be a nonexpansive semigroup onCif it satisﬁes the following conditions:

() T()x=xfor allx∈C;

() T(s+t) =T(s)T(t)for alls,t≥;

() T(t)x–T(t)y ≤ x–yfor allx,y∈Candt≥; () for allx∈C,t→T(t)xis continuous.

We denote byF( ) the set of all common ﬁxed points of , that is,F( ) ={x∈C:T(t)x= x, ≤t<∞}. It is well known thatF( ) is closed and convex.

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A mappingAofCintoHis called monotone ifAx–Ay,x–y ≥ for allx,y∈C. A map-pingAis calledα-inverse strongly monotone if there exists a positive real numberαsuch thatAx–Ay,x–y ≥αAx–Ay_{for all}_x,_y_∈_{C. A mapping}_A_{is called}_α_-strongly

mono-tone if there exists a positive real numberαsuch thatAx–Ay,x–y ≥αx–y_{for all}

x,y∈C. It is obvious that anyα-inverse strongly monotone mappingsAis a monotone and



α-Lipschitz continuous mapping. A linear bounded operatorAis called strongly positive if there exists a constantγ >  with the propertyAx,x ≥γx_{for all}_x_∈_H.

LetB:H→H be a single-valued nonlinear mapping andM:H→H_{be a set-valued}

mapping. The variational inclusion problem is to ﬁndx∈Hsuch that

∈B(x) +M(x), (.)

where  is the zero vector inH.

The set of solutions of (.) is denoted byI(B,M) (see [–] and the reference therein). A set-valued mapping M:H→H _{is called monotone if for all} _x,_y_∈_H, _f _∈_M(x)

andg∈M(y) imply x–y,f –g ≥. A monotone mapping Mis maximal if its graph G(M) ={(f,x)∈H×H:f ∈M(x)}ofMis not properly contained in the graph of any other monotone mapping. It is well 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).

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 the resolvent operator associated withM, whereλis any positive number andI is the identity mapping. It is well known that the resolvent operatorJM,λis nonexpansive, -inverse strongly monotone and that a solution of problem (.) is a ﬁxed point of the operatorJM,λ(I–λB) for allλ> , whereIdenotes the identity operator onH(see []).

Peng and Yao [] considered the following generalized mixed equilibrium problem of ﬁndingx∈Csuch that

θ(x,y) +ϕ(y) –ϕ(x) +Fx,y–x ≥, ∀y∈C, (.)

whereF:C→His a nonlinear mapping,ϕ:C→Ris a function andθ:C×C→Ris a bifunction. The set of solutions of problem (.) is denoted byGMEP.

In the case ofF= , problem (.) reduces to the following mixed equilibrium problem of ﬁndingx∈Csuch that

θ(x,y) +ϕ(y) –ϕ(x)≥, ∀y∈C,

which was considered by Ceng and Yao [].GMEPis denoted byMEP.

In the case ofϕ= , problem (.) reduces to the following generalized equilibrium prob-lem of ﬁndingx∈Csuch that

θ(x,y) +Fx,y–x ≥, ∀y∈C,

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In the case ofϕ=  andF= , problem (.) reduces to the equilibrium problem of ﬁndingx∈Csuch that

θ(x,y)≥, ∀y∈C. (.)

The set of solution of (.) is denoted byEP(θ).

In the caseθ=  andϕ= , problem (.) reduces to the classical variational inequality of ﬁndingx∈Csuch that

Fx,y–x ≥, ∀y∈C. (.)

The set of solutions of problem (.) is denoted byVI(F,C).

The problem (.) is very general in the sense that it includes, as special cases, optimiza-tion problem, variaoptimiza-tional inequalities, minimax problems, the Nash equilibrium problem in noncooperative games and others (see [, –]).

Peng and Yao [] considered iterative methods for ﬁnding a common element of the set of solutions of problem (.), the set of solutions of problem (.), and the set of ﬁxed points of a nonexpansive mapping.

LetG,G:C×C→Rbe two bifunctions,φ,ψ:C→Rbe two functions andB,B:

C→Hbe two nonlinear mappings. We consider the generalized mixed equilibria prob-lem of ﬁnding (x,y)∈C×Csuch that

⎧ ⎨ ⎩

G(x,x) +φ(x) –φ(x) +By,x–x+μx–y,x–x ≥, ∀x∈C, G(y,y) +ψ(y) –ψ(y) +Bx,y–y+_μ_y–x,y–y ≥, ∀y∈C,

(.)

whereμ>  andμ>  are two constants.

In the caseφ=ψ = , problem (.) reduces to the following problem of the general system of generalized equilibria of ﬁnding (x,y)∈C×Csuch that

⎧ ⎨ ⎩

G(x,x) +By,x–x+μx–y,x–x ≥, ∀x∈C,

G(y,y) +Bx,y–y+_μ_y–x,y–y ≥, ∀y∈C,

(.)

which was considered by Ceng and Yao [].

In the caseG=G= , problem (.) reduces to the following problem of the general

system of variational inequalities of ﬁnding (x,y)∈C×Csuch that

⎧ ⎨ ⎩

μBy+x–y,x–x ≥, ∀x∈C,

μBx+y–x,y–y ≥, ∀y∈C,

(.)

which was considered by Ceng, Wang and Yao [].

In particular, ifB=B=D:C→His a nonlinear mapping, then problem (.) reduces

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such that ⎧ ⎨ ⎩

μDy+x–y,x–x ≥, ∀x∈C,

μDx+y–x,y–y ≥, ∀y∈C,

(.)

which was studied by Verma [].

Ifx=yin (.), then (.) reduces to the classical variational inequality (.).

For solving the mixed equilibrium problem, let us give the following assumptions for the bifunctionθ,ϕand the setC:

(H) θ(x,y) = for allx∈C;

(H) θis monotone,i.e.,θ(x,y) +θ(y,x)≤for allx,y∈C; (H) for eachx,y,z∈C

lim sup t→

θtz+ ( –t)x,y≤θ(x,y);

(H) for eachx∈C,y→θ(x,y)is convex and lower semicontinuous;

(A) for eachx∈Handr> , there exist a bounded subsetD⊂Candy∈Csuch that for anyz∈C\D

θ(z,y) +ϕ(y) –ϕ(z) +

ry–z,z–x< ;

(A) Cis a bounded set.

Recently, Shehu [] studied the problem of ﬁnding a common element of the set of common ﬁxed points of a one-parameter nonexpansive semigroup, the set of solutions to a variational inclusion, and the set of solutions to a generalized equilibrium problem. More precisely, the author proved the following theorem.

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

θ be a bifunction from C×C toRsatisfying(H)-(H),ψaμ-inverse strongly monotone mapping of C into H,B anα-inverse strongly monotone mapping of C into H and M: H→H_{a maximal monotone mapping.}_Let ₌_{_T_{(u) : }_≤_u_<_∞}_{be a one-parameter}

nonexpansive semigroup on H such thatF=F( )∩I(B,M)∩EP(θ)=φ and suppose f : H→H is a contraction mapping with a constantγ ∈(, ).Let{tn} ⊂(,∞)be a real

sequence such thatlimn→∞tn=∞.Suppose{xn}∞n=and{un}∞n=are generated by x∈H,

θ(un,y) +ψxn,y–un+

 rn

y–un,un–xn ≥, ∀y∈C,

xn+=βnxn+ ( –βn)

 tn

tn



T(u)αnf(xn) + ( –αn)JM,λ(un–λAun)

du

for all n≥,where{αn}∞n=and{βn}n∞=are sequences in(, )and{rn}∞n=⊂(,∞)

satisfy-ing:

(i) limn→∞βn= ,

∞

n=|βn+–βn|<∞,

(ii) lim_n_→∞αn= ,

∞

n=αn=∞,

∞

n=|αn+–αn|<∞,

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(iv)  <a≤rn≤b< μ,

∞

n=|rn+–rn|<∞,

(v) lim_n_→∞|tn–tn–|

tn 

αn(–βn)= .

Then{xn}∞n=converges strongly to z,where z=PFf(z).

In this paper, motivated by Shehu [], Kumam and Wattanawitoon [], Liet al.[], Plubtieng and Punpaeng [], Plubtieng and Wangkeeree [], we introduce the following general iterative scheme for ﬁnding a common element of the set of common ﬁxed points of a one-parameter nonexpansive semigroup, the set of solutions of the generalized mixed equilibrium problem (.), the set of solutions to a variational inclusion (.), and the set of solutions of the generalized mixed equilibria problem (.), which solves the variational inequality

(A–γf)x∗,x–x∗≥, ∀x∈F,

whereF=F( )∩GMEP∩∩I(B,M) andis the set of solutions of the generalized equilibria problem (.).

The results obtained in this paper improve and extend the recent results announced by [, –] and many others.

2 Preliminaries

LetCbe a nonempty closed convex subset of a real Hilbert spaceH. For every pointx∈H there exists a unique nearest point ofC, denoted byPCxsuch thatx–PCx ≤ x–y

for ally∈C. Such aPCis called the metric projection ofHontoC. We know thatPCis a

ﬁrmly nonexpansive mapping ofHontoC,i.e.,

x–y,PCx–PCy ≥ PCx–PCy, ∀x,y∈H.

Further, for anyx∈Handz∈C,z=PCxif and only if

x–z,z–y ≥, ∀y∈C.

It is also known thatHsatisﬁes Opial’s condition [], that is, if for each sequence{xn}∞n=

inHwhich converges weakly to a pointx∈H, we have

lim inf

n→∞ xn–x<lim infn→∞ xn–y, ∀y∈H,y=x.

In order to prove our main results in the next section, we need the following lemmas.

Lemma .([]) Let C be a nonempty closed convex subset of H.Letθ:C×C→Rbe a bifunction satisfying conditions(H)-(H)and letϕ:C→Rbe a lower semicontinuous and convex function.For r> and x∈H deﬁne a mapping

T_r(θ,ϕ)(x) =

z∈C:θ(z,y) +ϕ(y) –ϕ(z) +

ry–z,z–x ≥,∀y∈C

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Then the following results hold:

(i) Tr(θ,ϕ)(x)=φfor eachx∈HandTr(θ,ϕ)is single-valued;

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

T_r(θ,ϕ)x–T_r(θ,ϕ)y≤T_r(θ,ϕ)x–T_r(θ,ϕ)y,x–y;

(iii) F(Tr(θ,ϕ)) =MEP(θ,ϕ);

(iv) MEP(θ,ϕ)is closed and convex.

By similar argument as in the proof of Lemma . in [], we have the following result.

Lemma . Let C be a nonempty closed convex subset of H.Let G,G:C×C→Rbe two

bifunctions satisfying conditions(H)-(H).Letφ,ψ:C→Rbe two lower semicontinuous and convex functions with restriction(A)or(A)and let the mappings B,B:C→H

beβ-inverse strongly monotone andβ-inverse strongly monotone,respectively.Letμ∈

(, β)andμ∈(, β).Then for given x,y∈C, (x,y)is a solution of problem(.)if and

only if x is a ﬁxed point of the mapping:C→C deﬁned by

(x) =T(G,φ)

μ T

(G,ψ)

μ (x–μBx) –μBT

(G,ψ)

μ (x–μBx)

, ∀x∈C,

where y=T(G,ψ)

μ (x–μBx).

Remark . Under the conditions of Lemma ., the set of ﬁxed points of the mapping

is denoted by.

Proposition .([]) Let C,H,θ,ϕ and Tr(θ,ϕ) be as in Lemma..Then the following

holds:

T(θ,ϕ) s x–T

(θ,ϕ) t x



≤s–t s

T(θ,ϕ)

s x–T (θ,ϕ)

t x,Ts(θ,ϕ)x–x

for all s,t> and x∈H.

Lemma .([]) Assume that T is a nonexpansive self-mapping of a nonempty closed convex subset of C of a real Hilbert space H.If T has a ﬁxed point,then I–T is demiclosed, that is,when{xn}∞n=is a sequence in C converging weakly to some x∈C and the sequence

{(I–T)xn}n∞=converges strongly to some y,it follows that(I–T)x=y.

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

be a monotone and Lipschitz continuous mapping.Then the mapping M+B:H→H_{is a}

maximal monotone mapping.

Lemma .([]) Let C be a nonempty bounded closed convex subset of a Hilbert space H and let ={T(s) : ≤s<∞}be a nonexpansive semigroup on C.Then for any h≥,

lim t→∞sup_x_∈_C

_t t



T(s)x ds–T(h)

 t

t



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Lemma .([]) Let{an}∞n=be a sequence of nonnegative real numbers such that

an+≤( –δn)an+bn+σn, n∈N,

where{δn}∞n=⊂(, )and{bn}∞n=,{σn}∞n=are sequences inRsuch that

(i) limn→∞δn= ,

∞

n=δn=∞;

(ii) lim sup_n_→∞bn

δn ≤;

(iii) σn≥,

∞

n=σn<∞.

Thenlimn→∞an= .

Lemma .([]) Assume that A is a strongly positive linear bounded operator on a Hilbert space H with coeﬃcientγ> and <ρ≤ A–.ThenI–ρA ≤ –ργ.

3 Main results

Now we state and prove our main results.

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

θ,G,G :C× C→R be three bifunctions which satisfy assumptions (H)-(H) and ϕ,φ,ψ:C→Rbe three lower semicontinuous and convex functions with restriction(A) or(A).Let F,B,B,B:C→H beζ-inverse strongly monotone,β-inverse strongly

mono-tone, β-inverse strongly monotone and β-inverse strongly monotone, respectively and

M:H →H _{be a maximal monotone mapping.}_Let ₌_{_T_{(u) : }_≤_u_<_∞} _{be a}

one-parameter nonexpansive semigroup on H such thatF=F( )∩GMEP∩∩I(B,M)=φ. Let{tn} ⊂(,∞)be a real sequence such thatlimn→∞tn=∞.Let f be a contraction from

C into itself with a constantα( <α< )and let A be a strongly positive linear bounded operator with coeﬃcientγ > such thatA ≤.Assume that <γ <_α_(–γ_α

nγ).Let x∈C

and let{xn}be a sequence deﬁned by

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

zn=T(

θ,ϕ)

δn (xn–δnFxn),

yn=Tμ(G,φ)[T

(G,ψ)

μ (zn–μBzn) –μBT

(G,ψ)

μ (zn–μBzn)], xn+=αnγf(xn) +βnxn

+ (( –βn)I–αnA)[_t_n

tn

 T(u)(JM,λ(yn–λByn))du], n≥,

(.)

whereαn∈(, ),μ∈(, β],μ∈(, β],λ∈(, β],δn∈(, ζ]satisfy the following

conditions:

(C) lim_n_→∞αn= ,

∞

n=αn=∞and

∞

n=|αn+–αn|<∞;

(C) ∞_n_=βn<∞and

∞

n=|βn+–βn|<∞;

(C) lim infn→∞δn> and∞_n_=|δn+–δn|<∞;

(C) limn→∞|tn–_tt_nn–|αn= .

Then{xn}converges strongly to x∗=PF(γf+ (I–A))(x∗),which solves the following varia-tional inequality:

(A–γf)x∗,x–x∗≥, ∀x∈F,

and(x∗,y∗)is a solution of problem(.),where y∗=T(G,ψ)

μ (x ∗_–_μ

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Proof SinceFis aζ-inverse strongly monotone mapping, we have (I–δnF)x– (I–δnF)y



=x–y– δnx–y,Fx–Fy+δnFx–Fy

≤ x–y+δn(δn– ζ)Fx–Fy. (.)

In similar way, we can obtain

(I–λB)x– (I–λB)y≤ x–y+λ(λ– β)Bx–By,

(I–μB)x– (I–μB)y≤ x–y+μ(μ– β)Bx–By, (.)

(I–μB)x– (I–μB)y≤ x–y+μ(μ– β)Bx–By.

Noticing thatlimn→∞αn=  =limn→∞βn, we may assume, with no loss of generality, that

αn –βn <A

–_{for all}_n_≥_{. From Lemma . we know that if }_≤_ρ_≤_A–_{, then}_I_–_ρ_A_≤

 –ργ. SinceAis a strongly positive bounded linear operator onH, we have

A=supAx,x:x∈H,x= .

Observe that

( –βn)I–αnA

x,x=  –βn–αnAx,x ≥ –βn–αnA ≥.

This shows that ( –βn)I–αnAis positive. It follows that

( –βn)I–αnA=sup

x,x:x∈H,x= 

=sup –βn–αnAx,x:x∈H,x= 

≤ –βn–αnγ.

We divide the proof into several steps. Step .{xn}is bounded.

Indeed, takep∈Farbitrarily. Sincep=Tδ(nθ,ϕ)(p–δnFp),Fisζ-inverse strongly

mono-tone and  <δn≤ζ, we obtain for anyn≥

zn–p=Tδ(θn,ϕ)(xn–δnFxn) –T (θ,ϕ)

δn (p–δnFp) 

≤(xn–p) –δn(Fxn–Fp) 

=xn–pp– δnxn–p,Fxn–Fp+δnFxn–Fp

≤ xn–p+δn(δn– ζ)Fxn–Fp

≤ xn–p. (.)

Puttingun=Tμ(G,ψ)(zn–μBzn),wn=JM,λ(yn–λByn), andu=T

(G,ψ)

μ (p–μBp), we have

un–u=Tμ(G,ψ)(zn–μBzn) –T

(G,μ)

μ (p–μBp)



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=zn–p– μzn–p,Bzn–Bp+μBzn–Bp

≤ zn–p+μ(μ– β)Bzn–Bp

≤ zn–p (.)

and

wn–p=JM,λ(yn–λByn) –JM,λ(p–λBp) 

≤(yn–p) –λ(Byn–Bp) 

=yn–p– λyn–p,Byn–Bp+λByn–Bp

≤ yn–p+λ(λ– β)Byn–Bp

≤ yn–p. (.)

And since

p=T(G,φ)

μ T

(G,ψ)

μ (p–μBp) –μBT

(G,ψ)

=T(G,φ)

μ (u–μBu),

we know that for anyn≥

yn–p=Tμ(G,φ)(un–μBun) –T

(G,φ)

μ (u–μBu)



≤(un–u) –μ(Bun–Bu) 

=un–u– μun–u,Bun–Bu+μBun–Bu

≤ un–u+μ(μ– β)Bun–Bu

≤ un–u

≤ zn–p. (.)

SinceAis a strongly positive linear bounded operator with coeﬃcientγ > , we have

xn+–p

=αn

γf(xn) –Ap

+βn(xn–p) +

 tn

tn



T(u)wn–T(u)p

du

≤αnγf(xn) –Ap+βnxn–p+ ( –βn–αnγ)wn–p

≤αnγf(xn) –f(p)+αnγf(p) –Ap+βnxn–p+ ( –βn–αnγ)xn–p

≤  –αn(γ –γ α)

xn–p+αnγf(p) –Ap.

By induction, we obtain for alln≥

xn–p ≤max

x–p,



γ –γ αγf(p) –Ap

.

(10)

Step . We show thatlimn→∞xn+–xn= .

We estimateyn+–yn. From (.), we have

yn+–yn=Tμ(G,φ)(un+–μBun+) –T

(G,φ)

μ (un–μBun)



≤(un+–un) –μ(Bun+–Bun)

≤ un+–un+μ(μ– β)Bun+–Bun

≤ un+–un

=T(G,ψ)

μ (zn+–μBzn+) –T

(G,ψ)

μ (zn–μBzn)



≤(zn+–zn) –μ(Bzn+–Bzn)

≤ zn+–zn+μ(μ– β)Bzn+–Bzn

≤ zn+–zn, (.)

(xn+–δn+Fxn+) – (xn–δnFxn)

=xn+–xn–δn+(Fxn+–Fxn) + (δn–δn+)Fxn

≤xn+–xn–δn+(Fxn+–Fxn)+|δn–δn+|Fxn

≤ xn+–xn+|δn–δn+|Fxn

and

zn+–zn=Tδ(nθ+,ϕ)(xn+–δn+Fxn+) –T

(θ,ϕ)

δn (xn–δnFxn)

≤Tδ(nθ+,ϕ)(xn+–δn+Fxn+) –T

(θ,ϕ)

δn+(xn–δnFxn) +Tδ(θn+,ϕ)(xn–δnFxn) –T

(θ,ϕ)

δn (xn–δnFxn)

≤(xn+–δn+Fxn+) – (xn–δnFxn)

+T_δ(θ_n_+,ϕ)(xn–δnFxn) –Tδ(θn,ϕ)(xn–δnFxn)

≤ xn+–xn+|δn+–δn|Fxn

+Tδ(θn+,ϕ)(xn–δnFxn) –T

(θ,ϕ)

δn (xn–δnFxn). (.)

Without loss of generality, let us assume that there exists a real numberasuch thatδn>

a>  for alln. Utilizing Proposition . we have Tδ(θn+,ϕ)(xn–δnFxn) –T

(θ,ϕ)

δn (xn–δnFxn)

≤|δn+–δn| δn+

Tδ(θn+,ϕ)(I–δnF)xn–xn

≤|δn+–δn|

a T

(θ,ϕ)

δn+(I–δnF)xn–xn. (.)

It follows from (.)-(.) that

wn+–wn=JM,λ(yn+–λBun+) –JM,λ(yn–λByn)

≤(yn+–λByn+) – (yn–λByn)

(11)

and

yn+–yn ≤ zn+–zn

≤ xn+–xn+|δn+–δn|Fxn

+|δn+–δn|

a T

(θ,ϕ)

δn+(I–δnF)xn–xn. (.)

Puthn=_t_n

tn

 T(u)wndufor alln≥. We note that

hn–hn–=

_t_n_tnT(u)wndu–

 tn–

tn–



T(u)wn–du

≤ tn

tn



T(u)wn–T(u)wn–

du

+ tn

– 

tn–



T(u)wn––T(u)p

du

+ 

tn

tn–

T(u)wn––T(u)p

du

≤ yn–yn–+

|tn–tn–|

tn

yn––p. (.)

Using (.) and (.) we get

xn+–xn

= αnγf(xn) +βnxn+

hn

– αn–γf(xn–) +βn–xn–+

( –βn–)I–αn–A

hn–

≤αnγ αxn–xn–+γ|αn–αn–|f(xn–)+βnxn–xn–

+|βn–βn–|xn–+ ( –αnγ)hn–hn–+|αn–αn–|Ahn–

+|βn–βn–|hn–hn–

+|βn–βn–|xn–+ ( –αnγ)

yn–yn–+

|tn–tn–|

tn

yn––p

+|αn–αn–|Ahn–+|βn–βn–|

yn–yn–+

|tn–tn–|

tn

yn––p

+|βn–βn–|xn–+ ( –αnγ)

xn–xn–+|δn–δn–|Fxn–

+|δn–δn–|

a T

(θ,ϕ)

δn (I–δn–F)xn––xn–

+ ( –αnγ)

|tn–tn–|

tn

yn––p+|αn–αn–|Ahn–

+|βn–βn–|

(12)

+|δn–δn–|

a T

(θ,ϕ)

+|βn–βn–|

|tn–tn–|

tn

yn––p

≤  – (γ –γ α)αn

xn–xn–+

|tn–tn–|

tn

yn––p

+|αn–αn–|

γf(xn–)+Ahn–

+βn xn+xn–

+|βn–βn–| xn+ xn–

+ |δn–δn–|

Fxn–+

 aT

(θ,ϕ)

=  – (γ–γ α)αn

xn–xn–+D

|tn–tn–|

tn

+|αn–αn–|+βn

+|βn–βn–|+ |δn–δn–|

,

whereD=max{sup_n_≥_xn,supn≥(γf(xn)+Ahn),supn≥(Fxn+_aTδ(θn,ϕ)(I–δnF)×

xn–xn),supn≥yn–p}. From Lemma ., takingδn= (γ –γ α)αn,bn=|tn–_t_ntn–|D,σn=

D(|αn–αn–|+βn+|βn–βn–|+ |δn–δn–|), it follows thatlimn→∞xn+–xn= .

Step .limn→∞Fxn–Fp= ,limn→∞Bun–Bu= ,limn→∞Bzn–Bp=  and limn→∞Byn–Bp= .

Indeed, from (.), (.), (.), and (.) we get

xn+–p

=αn

γf(xn) –Ap

+βn(xn–p) +

( –βn)I

–αnA

 tn

tn



T(u)JM,λ(yn–λByn)

–T(u)JM,λ(p–λBp)

du



≤αnγf(xn) –Ap+βnxn–p+ ( –βn–αnγ)yn–p

≤αnγf(xn) –Ap+βnxn–p

+ ( –βn–αnγ) un–u+μ(μ– β)Bun–Bu

≤αnγf(xn) –Ap 

+βnxn–p+ ( –βn–αnγ) zn–p

+μ(μ– β)Bzn–Bp+μ(μ– β)Bun–Bu

+βnxn–p+ ( –βn–αnγ) xn–p

+δn(δn– ζ)Fxn–Fp+μ(μ– β)Bzn–Bp

+μ(μ– β)Bun–Bu

.

Therefore

δn(ζ–δn)Fxn–Fp+μ(β–μ)Bzn–Bp

(13)

+xn–p–xn+–p

+xn–p+xn+–p

xn–xn+.

Since αn →  and xn –xn+ →  as n → ∞, we have limn→∞Fxn –Fp = ,

limn→∞Bzn–Bp=  andlimn→∞Bun–Bu= . Similarly, from (.), (.), and

(.) we have

xn+–p≤αnγf(xn) –Ap 

+βnxn–p+ ( –βn–αnγ)wn–p

+ ( –αnγ)xn–p+λ(λ– β)Byn–Bp,

which implies that

λ(β–λ)Byn–Bp≤αnγf(xn) –Ap+

xn–p+xn+–p

xn–xn+.

We also havelim_n_→∞Byn–Bp= .

Step . We claim thatlimn→∞xn–yn= ,limn→∞xn–zn=  andlimn→∞wn–

yn= .

Indeed, from Lemma ., (.), (.), and (.) we have

un–u=Tμ(G,ψ)(zn–μBzn) –T

(G,ψ)



≤(zn–μBzn) – (p–μBp),un–u

=

 (zn–μBzn) – (p–μBp)



+un–u

–(zn–μBzn) – (p–μBzn) – (un–u)

≤

 xn–p

₊_u

n–u–(zn–un) – (p–u) 

+ μ

(zn–un) – (p–u),Bzn–Bp

–μ_Bzn–Bp

and

yn–p=Tμ(G,φ)(un–μBun) –T

(G,φ)

μ (u–μBu)

≤(un–μBun) – (u–μBu),yn–p

=

 (un–μBun) – (u–μBu)



+yn–p

–(un–μBun) – (u–μBu) – (yn–p)

≤

 un–u

₊_y

n–p–(un–yn) + (p–u) 

+ μ

Bun–Bu, (un–yn) + (p–u)

–μ_Bun–Bu

≤

 xn–p

₊_y

n–p–(un–yn) + (p–u) 

+ μ

Bun–Bu, (un–yn) + (p–u)

(14)

which imply that

un–u≤ xn–p–(zn–un) – (p–u)

+ μ

–μ_Bzn–Bp (.)

and

yn–p≤ xn–p–(un–yn) + (p–u) 

+ μBun–Bu(un–yn) + (p–u). (.)

It follows from (.) that

xn+–p≤αnγf(xn) –Ap+βnxn–p+ ( –βn–αnγ)yn–p

≤αnγf(xn) –Ap+ ( –αnγ)xn–p

– ( –βn–αnγ)(un–yn) + (p–u)

+ μ( –βn–αnγ)Bun–Bu(un–yn) + (p–u),

which gives

( –βn–αnγ)(un–yn) + (p–u) 

+xn–p+xn+–p

xn–xn+

+ μ( –βn–αnγ)Bun–Bu(un–yn) + (p–u).

Sinceαn→,xn+–xn → andBun–Bu → asn→ ∞, we have

lim

n→∞(un–yn) + (p–u)= . (.)

Also, from (.) we have

+βnxn–p+ ( –βn–αnγ)yn–p

≤αnγf(xn) –Ap+βnxn–p+ ( –βn–αnγ)un–u

≤αnγf(xn) –Ap+βnxn–p+ ( –βn–αnγ) xn–p

–(zn–un) – (p–u)+ μ

.

So, we have

( –βn–αnγ)(zn–un) – (p–u)

≤αnγf(xn) –Ap+

xn–p+xn+–p

xn–xn+

(15)

Note thatBzn–Bp → asn→ ∞. Then we have

lim

n→∞(zn–un) – (p–u)= . (.)

In addition, from the ﬁrm nonexpansivity ofT_δ(θ_n,ϕ), we obtain

zn–p=Tδ(θn,ϕ)(xn–δnFxn) –T (θ,ϕ)

δn (p–δnFp) 

≤(xn–δnFxn) – (p–δnFp),zn–p

=

 (xn–δnFxn) – (p–δnFp)



+zn–p

–(xn–δnFxn) – (p–δnFp) – (zn–p) 

≤ 

 xn–p

₊_z

n–p–xn–zn–δn(Fxn–Fp) 

=

 xn–p

₊_z

n–p–xn–zn

+ δnFxn–Fp,xn–zn–δnFxn–Fp

,

which implies that

zn–p≤ xn–p–xn–zn+ δnFxn–Fp,xn–zn

≤ xn–p–xn–zn+ δnFxn–Fpxn–zn. (.)

From (.) and (.), we have

+βnxn–p+ ( –βn–αnγ)yn–p

+βnxn–p+ ( –βn–αnγ)zn–p

+βnxn–p+ ( –βn–αnγ) xn–p

–xn–zn+ δnFxn–Fpxn–zn

.

It follows that

( –βn–αnγ)xn–zn≤αnγf(xn) –Ap 

+xn–p+xn+–p

xn–xn+

+ ( –βn–αnγ)δnFxn–Fpxn–zn.

SinceFxn–Fp → asn→ ∞, we obtain

lim

n→∞xn–zn= . (.)

Thus, from (.), (.), and (.) we obtain

lim

n→∞zn–yn=nlim→∞(zn–un) – (p–u) + (un–yn) + (p–u)

≤ lim

n→∞(zn–un) – (p–u)+nlim→∞(un–yn) + (p–u)

(16)

and

lim

n→∞xn–yn ≤nlim→∞xn–zn+nlim→∞zn–yn

= .

SinceJM,λis -inverse strongly monotone, we have

wn–p=JM,λ(yn–λByn) –JM,λ(p–λBp)

≤(yn–λByn) – (p–λBp),wn–p

= 

 (yn–λByn) – (p–λBp)



+wn–p

–(yn–λByn) – (p–λBp) – (wn–p) 

≤ 



yn–p+wn–p–(yn–wn) –λ(Byn–Bp)

≤ 



yn–p+wn–p–yn–wn+ λyn–wn,Byn–Bp

–λByn–Bp

,

which implies that

wn–p≤ yn–p–yn–wn+ λyn–wnByn–Bp. (.)

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

wn–p≤ xn–p–(un–yn) + (p–u)

+ μBun–Bu(un–yn) + (p–u)

–yn–wn+ λyn–wnByn–Bp. (.)

It follows from (.) that

xn+–p

≤αnγf(xn) –Ap+βnxn–p+ ( –βn–αnγ)wn–p

≤αnγf(xn) –Ap+ ( –αnγ)xn–p

– ( –βn–αnγ)(un–yn) + (p–u) 

+ μ( –βn–αnγ)Bun–Bu(un–yn) + (p–u)

– ( –βn–αnγ)yn–wn+ λ( –βn–αnγ)yn–wnByn–Bp,

which gives

( –βn–αnγ)yn–wn

+xn–xn+

xn–p+xn+–p

(17)

– ( –βn–αnγ)(un–yn) + (p–u) 

+ μ( –βn–αnγ)Bun–Bu(un–yn) + (p–u)

+ λ( –βn–αnγ)yn–wnByn–Bp.

Sinceαn→,xn+–xn →,(un–yn) + (p–u) →,Byn–Bp → asn→ ∞, we

havelim_n_→∞yn–wn= .

Step . We showlimn→∞T(u)yn–yn= .

Denotehn=_t_n

tn

 T(u)wndu. From (.),limn→∞αn= , andlimn→∞βn=  we have

xn+–hn=αnγf(xn) +βnxn+

hn–hn

≤αnγf(xn) –Ahn+βnxn–hn

→ asn→ ∞. (.)

LetK={w∈C:w–p ≤_γ_–_{γ α}γf(p) –Ap}. ThenKis a nonempty bounded closed convex subset ofCwhich isT(u)-invariant for eachu∈[,∞) and contains{xn}. It follows

from Lemma . that

lim

n→∞hn–T(u)hn= , u≥ (.)

and from (.) and (.), we have

_x_n_+_–_T_(u)x_n_+_≤_x_n_+_–_h_n₊_h_n_–_T(u)h_n₊_T_(u)h_n_–_T_(u)x_n_+ ≤xn+–hn+hn–T(u)hn

→ asn→ ∞.

Hence

lim

n→∞xn–T(u)xn= . (.)

Furthermore, from Step  we have for everyu∈[,∞) that

T(u)yn–T(u)xn≤ yn–xn → asn→. (.)

So, we obtain from (.)

T(u)yn–xn≤T(u)yn–T(u)xn+T(u)xn–xn

→ asn→ ∞. (.)

Hence, we have for everyu∈[,∞) that

T(u)yn–yn≤T(u)yn–xn+xn–yn

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Step . We show thatlim sup_n_→∞γf(x∗) –Ax∗,xn–x∗ ≤ andlim supn→∞γf(x∗) –

Ax∗,_t

n

tn

 T(u)wndu–x∗ ≤, wherex∗=PF(γf(x∗) + (I–A)(x∗)).

Indeed, take a subsequence{xni}of{xn}such that

lim sup n→∞

γfx∗–Ax∗,xn–x∗

=lim

i→∞

γfx∗–Ax∗,xni–x

∗_. _(.)

Since{yni}is bounded, we can assume thatyniw. First, we prove thatw∈F( ).

Assume the contrary thatw=T(u)wfor someu∈[,∞). Then by Opial’s condition, we obtain from Step  that

lim inf

j→∞ ynj–w<lim infj→∞

_y_n

j–T(u)w

≤lim inf j→∞

ynj–T(u)ynj+T(u)ynj–T(u)w

≤lim inf

j→∞ ynj–w.

This is a contraction. Hence,w∈F( ). Next, let us show thatw∈GMEP. Fromzn=Tδ(nθ,ϕ)(xn–δnFxn), we obtain

θ(zn,y) +ϕ(y) –ϕ(zn) +



δn

y–zn,zn– (xn–δnFxn)

≥, ∀y∈C.

It follows from (H) that

ϕ(y) –ϕ(zn) +



δn

y–zn,zn– (xn–δnFxn)

≥θ(y,zn), ∀y∈C.

Replacingnbyni, we have

ϕ(y) –ϕ(zni) +y–zni,Fxni+

y–zni,

zni–xni δni

≥θ(y,zni), ∀y∈C. (.)

Letzt=ty+ ( –t)wfor allt∈[, ] andy∈C. Then we havezt∈C. It follows from (.)

that

zt–zni,Fzt ≥ zt–zni,Fzt–ϕ(zt) +ϕ(zni) –zt–zni,Fxni

–

zt–zni,

+θ(zt,zni)

≤ zt–zni,Fzt–Fzni+zt–zni,Fzni–Fxni

–ϕ(zt) +ϕ(zni) –

zt–zni,

+θ(zt,zni).

Sincezni–xni →, we haveFzni–Fxni → asni→ ∞. From the monotonity ofF,

we have

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From (H),zni_δ–xni

ni → andzniw, we have

zt–w,Fzt ≥–ϕ(zt) +ϕ(w) +θ(zt,w) (.)

asni→ ∞. By (H), (H), and (.), we obtain

 =θ(zt,zt) +ϕ(zt) –ϕ(zt)

≤tθ(zt,y) + ( –t)θ(zt,w) +tϕ(y) + ( –t)ϕ(w) –ϕ(zt)

≤t θ(zt,y) +ϕ(y) –ϕ(zt)

+ ( –t)zt–w,Fzt

=t θ(zt,y) +ϕ(y) –ϕ(zt)

+ ( –t)ty–w,Fzt.

Hence we obtain

≤θ(zt,y) +ϕ(y) –ϕ(zt) + ( –t)y–w,Fzt.

Puttingt→, we have

≤θ(w,y) +ϕ(y) –ϕ(w) +y–w,Fw, ∀y∈C.

This implies thatw∈GMEP. Next, we prove thatw∈.

Utilizing Lemma ., we have for allx,y∈C (x) –(y)

=T(G,φ)

μ T

(G,ψ)

μ (x–μBx) –μBT

(G,ψ)

μ (x–μBx)

–T(G,φ)

μ T

(G,ψ)

μ (y–μBy) –μBT

(G,ψ)

μ (y–μBy)



≤T(G,ψ)

μ (x–μBx) –T

(G,ψ)

μ (y–μBy) –μ

BTμ(G,ψ)(x–μBx) –BT

(G,ψ)



≤T(G,ψ)

μ (x–μBx) –T

(G,ψ)



+μ(μ– β)BTμ(G,ψ)(x–μBx) –BT

(G,ψ)



≤T(G,ψ)

μ (x–μBx) –T

(G,ψ)



≤x–y–μ(Bx–By) 

≤ x–y+μ(μ– β)Bx–By

≤ x–y.

This shows that:C→Cis nonexpansive. Note that yn–(yn)=(zn) –(yn)

≤ zn–yn

→ asn→ ∞.

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Lastly, we show thatw∈I(B,M).

In fact, since Bis aβ-inverse strongly monotone,Bis monotone and Lipschitz

con-tinuous mapping. It follows from Lemma . thatM+Bis a maximal monotone

map-ping. Let (v,g)∈G(M+B). Theng–B(v)∈M(v). Sincewni=JM,λ(yni–λByni), we have

yni–λByni∈(I+λM)(wni),i.e., 

λ(yni–wni–λAyni)∈M(wni). By virtue of the maximal

monotonicity ofM+B, we have

v–wni,g–B(v) –



λ(yni–wni–λByni)

≥

and hence

v–wni,g ≥

v–wni,B(v) +



λ(yni–wni–λByni)

=v–wni,Bv–Bwni+v–wni,Bwni–Byni

+

v–wni,



λ(yni–wni)

.

It follows fromlimn→∞wn–yn=  that we have

lim

n→∞Bwn–Byn= 

and

wniw,

lim sup n→∞

v–wni,g=v–w,g ≥.

It follows from the maximal monotonicity ofM+Bthatθ∈(M+B)(w), that is,w∈I(B,M).

Thereforeω∈F=F( )∩GMEP∩∩I(B,M).

Byx∗=P_F(γf + (I–A))(x∗), we obtain

lim sup n→∞

(γf –A)x∗,xn–x∗

=lim

i→∞

(γf–A)x∗,xni–x

∗

=(γf–A)x∗,w–x∗

≤

and

lim sup n→∞

(γf–A)x∗,  tn

tn



T(u)wndu–x∗

=lim i→∞

(γf –A)x∗,  tni

t_ni



T(u)wnidu–x

∗

=(γf–A)x∗,w–x∗ ≤,

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Step . We provexn→x∗asn→ ∞.

By using (.), we have

xn+–x∗

=αn

γf(xn) –Ax∗

+βn

xn–x∗

+( –βn)I–αnA



tn



T(u)wndu–x∗

 ≤α_nγf(xn) –Ax∗



+βn

xn–x∗



tn



T(u)wndu–x∗



+ αn

γf(xn) –Ax∗,βn

xn–x∗



tn



T(u)wndu–x∗

≤α_nγf(xn) –Ax∗+βnxn–x∗+ ( –βn–αnγ)wn–x∗

+ βn

xn–x∗,



tn



T(u)wndu–x∗

+ αnβn

γf(xn) –γf

x∗,xn–x∗

+ αnβn

+ αn( –βn–αnγ)

γf(xn) –γf

x∗, 

tn



T(u)wndu–x∗

+ αn( –βn–αnγ)

γfx∗–Ax∗,  tn

tn



T(u)wndu–x∗

.

It follows from (.), (.), and (.) that

xn+–x∗

≤α_nγf(xn) –Ax∗ 

+β_nxn–x∗ 

+ ( –βn–αnγ)xn–x∗ 

+ βn( –βn–αnγ)xn–x∗ 

+ αnβnγ αxn–x∗ 

+ αnβn

+ αn( –βn–αnγ)γ αxn–x∗ 

+ αn( –βn–αnγ)

tn



T(u)wndu–x∗

≤  – αn(γ–γ α+αnγ γ α)xn–x∗ 

+α_nγf(xn) –Ax∗ 

+ αnβn

+ αn( –βn–αnγ)

tn



T(u)wndu–x∗

+α_nγxn–x∗ 

= ( –αn)xn–x∗ 

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where

αn= αn(γ –γ α+αnγ γ α),

β_n=α_nγf(xn) –Ax∗+ αnβn

+ αn( –βn–αnγ)

tn



T(u)wndu–x∗

+α_nγxn–x∗ 

.

It is easily seen thatαn→ asn→ ∞,

∞

n=αn= +∞andlim supn→∞βαnn ≤. Hence,

applying Lemma . we immediately obtainxn→x∗asn→ ∞. This completes the proof.

Remark . Let us consider the following sequences:

αn=



n

, βn=



n, δn=



n +

 and tn=n, ∀n≥.

It is easy to see that all hypotheses (C)-(C) of Theorem . are satisﬁed.

By Theorem ., we can obtain the following results immediately.

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

θ,G,G :C× C→R be three bifunctions which satisfy assumptions (H)-(H) and ϕ,φ,ψ:C→Rbe three lower semicontinuous and convex functions with restriction(A) or(A).Let B,B,B:C→H beβ-inverse strongly monotone,β-inverse strongly monotone

andβ-inverse strongly monotone,respectively and M:H→Hbe a maximal monotone

mapping.Let ={T(u) : ≤u<∞}be a one-parameter nonexpansive semigroup on H such thatF=F( )∩MEP∩∩I(B,M)=φ.Let{tn} ⊂(,∞)be a real sequence such

thatlimn→∞tn=∞.Let f be a contraction from C into itself with a constantα( <α< )

and let A be a strongly positive linear bounded operator with coeﬃcientγ > such that A ≤.Assume that <γ <_α_(–γ_α

nγ).Let x∈C and let{xn}be a sequence deﬁned by

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

θ(zn,y) +ϕ(y) –ϕ(zn) +δny–zn,zn–xn ≥, ∀y∈C,

yn=Tμ(G,φ)[T

(G,ψ)

μ (zn–μBzn) –μBT

(G,ψ)

μ (zn–μBzn)], xn+=αnγf(xn) +βnxn

+ (( –βn)I–αnA)[_t_n

tn

 T(u)(JM,λ(yn–λByn))]du, ∀n≥,

whereαn∈(, ),μ∈(, β],μ∈(, β],λ∈(, β],δn∈(, ζ]satisfy the following

conditions:

(i) limn→∞αn= ,

∞

n=αn=∞and

∞

n=|αn+–αn|<∞;

(ii) ∞_n_=βn<∞and

∞

n=|βn+–βn|<∞;

(iii) lim infn→∞δn> and

_∞

n=|δn+–δn|<∞;

(iv) limn→∞|tn–ttnn–| 