Convergence theorems for split equality generalized mixed equilibrium problems for demi contractive mappings

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R E S E A R C H

Open Access

Convergence theorems for split equality

generalized mixed equilibrium problems for

demi-contractive mappings

Mijanur Rahaman

1

_{, Yeong-Cheng Liou}

2,3*

_{, Rais Ahmad}

1

_{and Iqbal Ahmad}

1

*_{Correspondence:}

[email protected]

2_{Department of Information}

Management, Cheng Shiu University, Kaohsiung, 833, Taiwan

3_{Research Center of Nonlinear}

Analysis and Optimization and Center for Fundamental Science, Kaohsiung Medical University, Kaohsiung, 807, Taiwan Full list of author information is available at the end of the article

Abstract

In this paper, we introduce a new iterative algorithm for solving the split equality generalized mixed equilibrium problems. The weak and strong convergence theorems are proved for demi-contractive mappings in real Hilbert spaces. Several special cases are also discussed. As applications, we employ our results to get the convergence results for the split equality convex diﬀerentiable optimization problem, the split equality convex minimization problem, and the split equality mixed

equilibrium problem. The results in this paper generalize, extend, and unify some recent results in the literature.

MSC: 47H09; 47J25; 49M37; 90C25

Keywords: split equality generalized mixed equilibrium problem; demi-contractive mappings; ﬁxed point; iterative scheme

1 Introduction

The equilibrium problem has been extensively studied, beginning with Blum and Oettli [] where they proposed it as a generalization of optimization and variational inequality problem. The classical, scalar-valued equilibrium problem deals with the existence ofx∗∈ Csuch that

Fx∗,y≥, ∀y∈C, (.)

whereCis the nonempty closed convex subset of a real Hilbert spaceHandF:C×C−→ Ris a bi-mapping.

Very recently, Ahmad and Rahaman [] introduced the generalized vector equilibrium problem of ﬁndingx∈Csuch that

Fλx+ ( –λ)z,y–C\ {}, ∀y,z∈C,λ∈(, ],

whereF:C×C−→H_{is the set-valued mapping with the condition}_F(λx_{+ ( –}_λ)z,_x)_⊇

{}, and [·,z) denotes the line-segment excluding the pointz. In the scalar case, the gen-eralized equilibrium problem takes the form to ﬁndx∗∈Csuch that

Fλx∗+ ( –λ)z,y≥, ∀y,z∈C,λ∈(, ], (.)

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with the conditionF(λx+ ( –λ)z,x) = . Ifλ= , then the generalized equilibrium problem (.) reduces to the classical equilibrium problem (.).

In , Moudaﬁ and Thèra [] introduced the mixed equilibrium problem of ﬁnding x∗∈Csuch that

Fx∗,y+Tx∗,y–x∗≥, ∀y∈C, (.)

whereF:C×C−→Ris a given bi-mapping withF(x,x) = , for allx∈CandT:C−→ C is a continuous mapping. Problem (.) has useful applications in nonlinear analysis, including optimization problems, variational inequalities, ﬁxed-point problems, and the problems of Nash equilibria as special cases.

In , Peng and Yao [] considered the following extended mixed equilibrium prob-lem (EMEP): Findx∗∈Csuch that

Fx∗,y+Tx∗,y–x∗+φ(y) –φx∗≥, ∀y∈C, (.)

whereφ:C−→R∪ {+∞}is a mapping.

IfT= , then the extended mixed equilibrium problem (.) becomes the mixed equi-librium problem to ﬁndx∗∈Csuch that

Fx∗,y+φ(y) –φx∗≥, ∀y∈C. (.)

Problem (.) was studied by Ceng and Yao []. IfT=  andF(x,y) = , for allx,y∈C, the extended mixed equilibrium problem (.) becomes the following convex minimization problem:

Findingx∗∈C such that φ(y) –φx∗≥, ∀y∈C. (.)

Now, we mention the following generalized mixed equilibrium problem of ﬁnding x∗∈C:

Fλx∗+ ( –λ)b,y+Tx∗,y–x∗+φ(y) –φx∗≥, ∀y,b∈C,λ∈(, ]. (.)

The solution set of problem (.) is denoted byGMEP(F,T,φ).

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LetH, H, andH be the real Hilbert spaces. LetCandQbe two nonempty closed convex subsets of real Hilbert spacesHandH, respectively,A:H−→Hbe a bounded linear mapping. The (SFP) is to ﬁnd a pointx∗such that

x∗∈C and Ax∗∈Q. (.)

Recently, Moudaﬁ and Al-Shemas [] introduced the split equality problem (SEP) of ﬁndingx∗andy∗with the property

x∗∈C,y∗∈Q such that Ax∗=By∗, (.)

whereA:H−→H andB:H−→H are bounded linear mappings, which allows for asymmetric and partial relations between the variablesx∗ andy∗. IfH=HandB=I, then the (SEP) (.) reduces to the (SFP) (.).

In order to study the weak convergence properties of (SEP) (.), Moudaﬁ and Al-Shemas [] introduced the following simultaneous iterative method:

⎧ ⎨ ⎩

xn+=U(xn–γnA∗(Axn–Byn));

yn+=T(yn+γnB∗(Axn–Byn)),

(.)

whereU:H−→H,T:H−→Hare ﬁrmly quasi-nonexpansive mappings,A∗andB∗ are the adjoint ofAandB, respectively. Under some suitable conditions, they proved the weak convergence result for the SEP (.).

To get the strong and weak convergence theorems for the (SEP) (.), Maet al.[] gen-eralized the corresponding results of Moudaﬁ and Al-Shemas [], and they introduced the following iterative algorithm under some mild control conditions in Hilbert spaces:

⎧ ⎨ ⎩

xn+= ( –αn)xn+αnU(xn–γnA∗(Axn–Byn));

yn+= ( –αn)yn+αnT(yn–γnB∗(Axn–Byn)),

(.)

where{αn} ⊂[α, ], for someα> .

Recently, He [] introduced the following split equilibrium problem (SEqP). LetF:C×

C−→RandG:Q×Q−→Rbe two bi-mappings,A:H−→H be a bounded linear mapping. The split equilibrium problem is to ﬁnd an elementx∗∈Csuch that

Fx∗,x≥, ∀x∈C, (.)

and such that

y∗=Ax∗∈Q solves Gy∗,y≥, ∀y∈Q. (.)

To solve the split equilibrium problem (SEqP), He [] proposed the following iterative

algorithm: ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

F(un,u) +_r_nu–un,un–xn ≥, ∀u∈C;

G(vn,v) +_r_nv–vn,vn–Aun ≥, ∀v∈Q;

xn+=PC(un) +μA∗(vn–Aun),

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where PC is the metric projection operator from H onto C, rn ⊂ (,∞) with

limn→∞infrn> , andμ∈(,_A∗) is a constant.

Very recently, Maet al.[] considered the following split equality mixed equilibrium problem (SEMEP). Let φ :C−→R∪ {+∞} andϕ :Q−→R∪ {+∞} be proper lower semicontinuous and convex mappings such thatC∩domφ=∅andQ∩domϕ=∅, and A:H−→HandB:H−→Hbe bounded linear mappings. Then the SEMEP is to ﬁnd x∗∈Candy∗∈Qsuch that

Fx∗,x+φ(x) –φx∗≥, ∀x∈C,

Gy∗,y+ϕ(y) –ϕy∗≥, ∀y∈Q, and (.)

Ax∗=By∗.

The set of solutions of (.) is denoted bySEMEP(F,G,φ,ϕ).

In order to obtain the weak and strong convergence results of (SEMEP) (.), Maet al. [] presented the following simultaneous iterative algorithm:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

F(un,u) +φ(u) –φ(un) +_r_nu–un,un–xn ≥, ∀u∈C;

G(vn,v) +ϕ(v) –ϕ(vn) +_r_nv–vn,vn–Aun ≥, ∀v∈Q;

xn+=αnun+ ( –αn)T(un–γnA∗(Aun–Bvn));

yn+=αnvn+ ( –αn)S(vn+γnB∗(Aun–Bvn)),

(.)

whereT:H−→H,S:H−→Hare nonexpansive mappings.

In this paper, we consider the following split equality generalized mixed equilibrium problem (SEGMEP).

LetF:C×C−→RandG:Q×Q−→Rbe two nonlinear bi-mappings,T:C−→C andS:Q−→Qbe two nonlinear mappings, andφ:C−→R∪ {+∞}andϕ:Q−→R∪

{+∞}be proper lower semicontinuous and convex mappings such thatC∩domφ=∅and Q∩domϕ=∅. LetA:H−→H andB:H−→Hbe bounded linear mappings. Then the split equality generalized mixed equilibrium problem (SEGMEP) is to ﬁndx∗∈Cand y∗∈Qsuch that

Fλx∗+ ( –λ)b,x

+Tx∗,x–x∗+φ(x) –φx∗≥,

∀x,b∈C,λ∈(, ],

Gλy∗+ ( –λ)c,y

+Sy∗,y–y∗+ϕ(y) –ϕy∗≥, (.)

∀y,c∈Q,λ∈(, ], and

Ax∗=By∗.

The solution set of problem (.) is denoted bySEMEP(F,G,T,S,φ,ϕ). Special cases:

() Ifλ=λ= andT=S= , then the split equality generalized mixed equilibrium problem (.) becomes the split equality mixed equilibrium problem (.). () Ifλ=λ= ,T=S= , andφ=ϕ= , then the split equality generalized mixed

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problem: ﬁndx∗∈Candy∗∈Qsuch that

Fx∗,x≥, ∀x∈C, Gy∗,y≥, ∀y∈Q, and Ax∗=By∗. (.)

The set of solutions of (.) is denoted bySEEP(F,G).

() IfF=G= andT=S= , then the split equality generalized mixed equilibrium problem (.) becomes the split equality convex minimization problem to ﬁnd x∗∈Candy∗∈Qsuch that

φ(x)≥φx∗, ∀x∈C, ϕ(y)≥ϕy∗, ∀y∈Q, and Ax∗=By∗. (.)

The set of solutions of (.) is denoted bySECMP(φ,ϕ).

() IfB=I, then the split equality convex minimization problem (.) reduces to the following split convex minimization problem. Findx∗∈Csuch that

φ(x)≥φx∗, ∀x∈C, and

y∗=Ax∗∈Q such that ϕ(y)≥ϕy∗, ∀y∈Q.

(.)

The set of solutions of (.) is denoted bySCMP(φ,ϕ).

In this paper, by using the well-known KKM technique, we derive an important lemma which is a foundation for studying the generalized mixed equilibrium problem (.). Mo-tivated by the recent above work of Moudaﬁet al.[], Maet al.[], Maet al.[] and Chidumeet al.[], in this paper, we introduce a new iterative algorithm for solving the split equality generalized mixed equilibrium problem (.) for demi-contractive map-pings. We obtain weak and strong convergence results for the sequences generated by these processes. As applications, we employ our results to study the convergence results for the split equality convex diﬀerentiable optimization problem, the split equality convex minimization problem, and the split equality mixed equilibrium problem. The results of this paper generalize, extend, and unify some well-known weak and strong convergence results in the literature mentioned above.

2 Preliminaries

We ﬁrst recall some deﬁnitions and known results which are needed to prove our main results.

In the sequel, letHbe a real Hilbert space with inner product·,·, and norm · . LetC be a nonempty closed convex subset ofH. Let the symbols→anddenote strong and weak convergence, respectively. A pointx∈H is said to be a ﬁxed point ofT provided Tx=x, whereT:H−→His a mapping. We denote the set of ﬁxed points of the mapping T byFix(T). It is well known that

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

for allx,y∈Handλ∈[, ].

Deﬁnition . A mappingT:H−→His said to be ﬁrmly quasi-nonexpansive ifFix(T)=

∅and

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Deﬁnition . LetCbe a nonempty subset ofH. The mappingT:C−→Cis said to be k-demi-contractive ifFix(T)=∅and there exists a constantk∈(, ) such that

Tx–x∗≤x–x∗+kx–Tx, ∀x∈C,x∗∈Fix(T). (.)

Remark . Clearly, the class of ﬁrmly quasi-nonexpansive mappings is a subclass of demi-contractive mappings. Note also that the mappingT satisfying (.) withk=  is usually called hemicontractive. It is easy to observe from (.) that

Tx–x∗≤x–x∗+√kx–Tx

≤( +√k)x–x∗+√kTx–x∗

=

 +√k  –√k

x–x∗

=Lx–x∗,

whereL=+

√

k

–√k, so that

Tx–x∗≤Lx–x∗.

Deﬁnition . A mappingT:C−→H is said to be demi-closed at a pointz∈Hif, the weak convergence of{xn}inCto some pointp∈Cand the strong convergence of{Txn}

tozimplies thatTp=z.

Deﬁnition . A mappingT :C−→H is said to be demi-compact at a pointz∈Hif, for any bounded sequence{xn}inCsuch that (I–T)xn→zasn→ ∞, then there exist a

subsequence{xnj}and a pointp∈Csuch thatxnj→pasj→ ∞and (I–T)p=z.

Deﬁnition . A multi-valued mappingF:C−→H_{is said to be KKM-mapping if, for}

each ﬁnite subset{x, . . . ,xn}ofC,Co{x, . . . ,xn} ⊆

n

i=F(xi), whereCo{x, . . . ,xn}denotes

the convex hull of{x, . . . ,xn}.

Theorem . (KKM theorem []) Let C be a subset of a Hausdorﬀ topological vector space H and let F:C−→H_{be a KKM-mapping.}_{If for each x}_∈_C,_F(x)_{is closed and if for}

at least one point x∈C,F(x)is compact,then_x_∈_CF(x)=∅.

Lemma .([]) Let C be a nonempty closed convex subset of a strictly convex Banach space X,and T :C−→C be a nonexpansive mapping withFix(T)=∅. ThenFix(T)is closed and convex.

For solving the generalized mixed equilibrium problem (.), let us give the following assumptions for the bi-mappingF:C×C−→R, and the mappingT:C−→C:

(A) F(λx+ ( –λ)b,x) = , for allx∈C;

(A) Fis monotone,i.e.,F(λx+ ( –λ)b,y) +F(λy+ ( –λ)b,x)≤, for allx,y∈C; (A) Tis monotone,i.e.,T(x) –T(y),x–y ≥, for allx,y∈C;

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(A) Tis weakly upper semicontinuous;

(A) for eachx∈C,λ∈(, ]andr> , there exist a bounded setD⊂Canda∈C such that for anyz∈C\D,

–Fλa+ ( –λ)b,z+T(z),a–z+φ(a) –φ(z) +

ra–z,z–x< , ∀b∈C.

Lemma . Let C be a nonempty closed convex subset of a Hilbert space H.Suppose that the bi-mapping F:C×C−→Rand the mapping T:C−→C satisfy the conditions (A)-(A).Letφ:C−→R∪ {+∞}be a proper lower semicontinuous and convex mapping such that C∩domφ=∅.For r> ,λ∈(, ]and x∈H,let JrF,T:H−→C be the resolvent operator of F and T,deﬁned by

J_rF,T(x) =

z∈C:Fλz+ ( –λ)b,y

+T(z),y–z

+φ(y) –φ(z) +

ry–z,z–x ≥, ∀y,b∈C

. (.)

Then:

(i) for eachx∈H,JrF,T(x)=∅;

(ii) JF,T

r is single-valued;

(iii) JF,T

r is ﬁrmly nonexpansive,i.e.,for anyx,y∈H, J_rF,T(x) –J_rF,T(y)≤J_rF,T(x) –J_rF,T(y),x–y;

(iv) Fix(J_rF,T) =GMEP(F,T,φ)and it is closed and convex. Proof (i) Letx¯be any given point inH. For eachy∈C, we deﬁne

M(y) =

z∈C: –Fλy+ ( –λ)b,z

+T(z),y–z

+φ(y) –φ(z) +

ry–z,z–x¯ ≥

. (.)

Sincey∈M(y),M(y) is nonempty. Now, we show thatMis a KKM-mapping. To the con-trary, suppose thatMis not a KKM-mapping. Then there exist a ﬁnite subset{y, . . . ,yn}

ofCandti≥ for alli= , . . . ,nwith

n

i=ti=  such thatz¯=

n i=tiyi∈/

n

i=M(yi), for

eachi. Then we have

–Fλyi+ ( –λ)b,z¯

+T(z),¯ yi–z¯

+φ(yi) –φ(z) +¯



ryi–¯z,z¯–x¯< , ∀i. By using (A)-(A) and convexity ofφ, we have

 = Fλz¯+ ( –λ)b,z¯

+T(z),¯ z¯–z¯+φ(z) –¯ φ(z) +¯ 

r¯z–¯z,z¯–x¯

=F

λ¯z+ ( –λ)b,

n

i= tiyi

+

T(z),¯

n

i= tiyi–

n

i= tiz¯

+φ

_n

i= tiyi

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–φ(z) +¯  r

_n

i=

tiyi–¯z,z¯–x¯

≤

n

i= tiF

λ¯z+ ( –λ)b,yi

+ n i= ti

T(¯z),yi–z¯

+

n

i=

tiφ(yi) – n

i= tiφ(¯z)

+ r

n

i=

tiyi–z,¯ z¯–x¯

≤–

n

i= tiF

λyi+ ( –λ)b,z¯ + n i= ti

T(z),¯ yi–z¯

+φ(yi) –φ(z) +¯



ryi–z,¯ z¯–x¯ = n i= ti

–Fλyi+ ( –λ)b,z¯

+T(z),¯ yi–¯z

+φ(yi) –φ(z) +¯



ryi–z,¯ z¯–x¯

< ,

which is not possible, and henceMis a KKM-mapping.

Now, we prove thatM(y) =M(f)w for eachy∈C,i.e.,M(y) is weakly closed. Letz∈ M(f)wand{zn}be a sequence inM(y) such thatznz. Sincezn∈M(y), we have

–Fλy+ ( –λ)b,zn

+T(zn),y–zn

+φ(y) –φ(zn) +



ry–zn,zn–x¯ ≥. It follows from (A) and (A), and the weak lower semicontinuity ofφand · _that

≤lim sup

n→∞

–Fλy+ ( –λ)b,zn

+T(zn),y–zn

+φ(y) –φ(zn) +



ry–zn,zn–x¯

≤lim sup

n→∞

–Fλy+ ( –λ)b,zn

+lim sup

n→∞

T(zn),y–zn

+φ(y)

–lim inf

n→∞ φ(zn) +



rlim supn→∞ y–zn,zn–x¯ ≤–lim inf

n→∞

Fλy+ ( –λ)b,zn

+lim sup

n→∞

T(zn),y–zn

+φ(y)

–lim inf

n→∞ φ(zn) +



rlim supn→∞

y–zn,zn–x¯

≤–Fλy+ ( –λ)b,z

+T(z),y–z+φ(y) –φ(z) +

ry–z,z–x¯.

This implies thatz∈M(y), and henceM(y) is weakly closed.

In order to show thatM(y) is weakly compact for at least a pointy∈C, from (A), we can see that there exist a bounded setD⊂Canda∈Dsuch that for anyz∈C\D, we have z∈/M(a). ThenM(a)⊂D,i.e., it is bounded, which shows thatM(y) is weakly compact. Then by KKM theorem .,_y_∈_CM(y)=∅. Hence, forz∈_y_∈_CM(y), we have

–Fλy+ ( –λ)b,z

+T(z),y–z+φ(y) –φ(z) +

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Lety∈Cbe arbitrary, and letzt=ty+ ( –t)z,  <t≤. Thenzt∈C, and hence we have

–Fλzt+ ( –λ)b,z

+T(z),zt–z

+φ(zt) –φ(z) +



rzt–z,z–x¯ ≥. (.) Applying (.), using (A), (A), and convexity ofφ, we have

 = Fλzt+ ( –λ)b,zt

=Fλzt+ ( –λ)b,ty+ ( –t)z

≤tFλzt+ ( –λ)b,y

+ ( –t)Fλzt+ ( –λ)b,z

≤tFλzt+ ( –λ)b,y

+ ( –t)T(z),zt–z

+φ(zt) –φ(z) +



rzt–z,z–x¯

≤tFλzt+ ( –λ)b,y

+t( –t)T(z),y–z+t( –t)φ(y) –t( –t)φ(z)

+t( –t)

r y–z,z–x¯

≤Fλzt+ ( –λ)b,y

+ ( –t)T(z),y–z+ ( –t)φ(y) – ( –t)φ(z)

+( –t)

r y–z,z–x¯.

Lettingt→ and thereforezt→z, and by (A), we get

Fλz+ ( –λ)b,y

+T(z),y–z+φ(y) –φ(z) +

ry–z,z–x¯ ≥,

i.e.,z∈JF,T

r (x). Hence, from the arbitrariness of¯ x, we see that¯ JrF,T(x) is nonempty.

(ii) We claim thatJF,T

r is single-valued. Indeed, forx∈H andr> , letz,z∈JrF,T(x).

Then

Fλz+ ( –λ)b,z

+T(z),z–z

+φ(z) –φ(z) + 

rz–z,z–x ≥

and

Fλz+ ( –λ)b,z

+T(z),z–z

+φ(z) –φ(z) + 

rz–z,z–x ≥.

Adding the above two inequalities, we obtain

Fλz+ ( –λ)b,z

+Fλz+ ( –λ)b,z

–T(z) –T(z),z–z

+

rz–z,z–z ≥.

From (A)-(A) andr> , we have

z–z,z–z ≥,

(10)

(iii) To prove thatJF,T

r is ﬁrmly nonexpansive, for anyx,y∈H, we have FλJrF,T(x) + ( –λ)b,JrF,T(y)

+TJ_rF,T(x),J_rF,T(y) –J_rF,T(x) +φJ_rF,T(y)–φJ_rF,T(x)+

r

J_rF,T(y) –J_rF,T(x),J_rF,T(x) –x≥ and

FλJrF,T(y) + ( –λ)b,JrF,T(x)

+TJ_rF,T(y),J_rF,T(x) –J_rF,T(y) +φJ_rF,T(x)–φJ_rF,T(y)+

r

J_rF,T(x) –J_rF,T(y),J_rF,T(y) –y≥. Adding the above two inequalities, we get

FλJrF,T(x) + ( –λ)b,JrF,T(y)

+FλJrF,T(y) + ( –λ)b,JrF,T(x)

–TJ_rF,T(y)–TJ_rF,T(x),J_rF,T(y) –J_rF,T(x) +

r

J_rF,T(y) –J_rF,T(x),J_rF,T(x) –J_rF,T(y) –x+y≥. Using (A)-(A) andr> , we have

J_rF,T(y) –J_rF,T(x),J_rF,T(x) –J_rF,T(y) – (x–y)≥, which implies that

J_rF,T(x) –J_rF,T(y)≤J_rF,T(x) –J_rF,T(y),x–y.

(iv) Takex∈C. Then

x∈FixJ_rF,T

⇔ x=J_rF,T(x)

⇔ Fλx+ ( –λ)b,y

+T(x),y–x+φ(y) –φ(x) +

ry–x,x–x ≥

⇔ Fλx+ ( –λ)b,y

+T(x),y–x+φ(y) –φ(x)≥

⇔ x∈GMEP(F,T,φ).

SinceJF,T

r is ﬁrmly nonexpansive, thereforeJrF,Tis also nonexpansive. By Lemma ., we

see thatFix(JF,T

r ) =GMEP(F,T,φ) is closed and convex.

Let the mappingsG:Q×Q−→RandS:Q−→Qsatisfy (A)-(A). Letϕ:Q−→R∪

{+∞}be a proper lower semicontinuous and convex mapping such thatQ∩domϕ=∅. Fors> ,λ∈(, ], andu∈H, letJsG,S:H−→Qbe the resolvent operator ofGandS, deﬁned by

J_sG,S(u) =

v∈Q:Gλv+ ( –λ)c,w

+S(v),w–v+ϕ(w) –ϕ(v)

+

sw–v,v–u ≥,∀w,c∈Q

(11)

Then clearlyJG,S

s satisﬁes (i)-(iv) of Lemma ., andFix(JsG,S) =GMEP(G,S,ϕ).

Lemma .(Opial’s lemma []) Let H be a real Hilbert space and{μn}be a sequence in

H such that there exists a nonempty set W⊂H satisfying the following conditions: (i) for everyμ∈W,limn→∞μn–μexists;

(ii) any weak cluster point of the sequence{μn}belongs toW.

Then there exists w∗∈W such that{μn}converges weakly to w∗.

Lemma .([]) Let H be a real Hilbert space.Then for all x,y∈H,we have

x–y=x–y– x–y,y.

3 Convergence results

In this section, we prove the weak and strong convergence result for split equality gener-alized mixed equilibrium problem (.).

Theorem . Let H, H, and H be real Hilbert spaces, C⊆H and Q⊆H be the nonempty closed convex subsets of Hand H,respectively.Suppose that the bi-mappings F:C×C−→Rand G:Q×Q−→R,and the mappings T:C−→C and S:Q−→Q sat-isfy the conditions(A)-(A).Letφ:C−→R∪{+∞}andϕ:Q−→R∪{+∞}be the proper lower semicontinuous and convex mappings such that C∩domφ=∅and Q∩domϕ=∅. Let P:H−→Hand Q:H−→Hbe the two demi-contractive mappings with constants kand k,respectively,with the condition k∈(, ),where k=max{k,k}such that(I–P) and(I–Q)are demi-closed at zero,andFix(P)=∅andFix(Q)=∅.Let A:H−→Hand B:H−→Hbe bounded linear mappings.Assume that(x,y)∈C×Q and the iteration scheme{(xn,yn)}is deﬁned as follows:

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

F(λun+ ( –λ)b,u) +T(un),u–un

+φ(u) –φ(un) +_r_nu–un,un–xn ≥;

G(λvn+ ( –λ)c,v) +S(vn),v–vn(v)

+ϕ–ϕ(vn) +_r_nv–vn,vn–yn ≥;

xn+= ( –αn)(un–γnA∗(Aun–Bvn)) +αnP(un–γnA∗(Aun–Bvn));

yn+= ( –αn)(vn+γnB∗(Aun–Bvn)) +αnQ(vn+γnB∗(Aun–Bvn)),

(.)

for every u,b∈C,v,c∈Q,and n≥whereλAandλBdenote the spectral radii of A∗A and

B∗B,respectively,{γn}is a positive real sequence such thatγn∈(,λA+λB–),for suﬃciently small,{αn}is a sequence in(k, ),and{rn} ⊂(,∞)such that the following conditions are

satisﬁed:

(i) for someα,β∈(, ), <α≤αn≤β< ;

(ii) lim infn→∞rn> andlimn→∞|rn+–rn|= .

If:=Fix(P)∩Fix(Q)∩SEMEP(F,G,T,S,φ,ϕ)=∅,then

(I) The sequence{(xn,yn)}weakly converges to a solution of problem(.).

(II) In addition,ifPandQare demi-compact,then{(xn,yn)}strongly converges to a

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Proof To prove (I), let (x,y)∈. From Lemma ., we have

un–x=JrFn,T(xn) –J

F,T

rn (x)≤ xn–x, (.)

vn–y=JrGn,S(yn) –J

G,S

rn (y)≤ yn–y. (.)

SincePis a demi-contractive mapping, using the well-known identity (.) and Lemma ., we obtain the following estimates:

xn+–x =( –αn)

un–γnA∗(Aun–Bvn)

+αnP

–x

=( –αn)

–x+αn

Pun–γnA∗(Aun–Bvn)

–x

= ( –αn)un–γnA∗(Aun–Bvn)

–x+αnP

–x

–αn( –αn)un–γnA∗(Aun–Bvn)

–Pun–γnA∗(Aun–Bvn)

≤( –αn)un–γnA∗(Aun–Bvn)

–x+αnun–γnA∗(Aun–Bvn)

–x

+kun–γnA∗(Aun–Bvn)

–Pun–γnA∗(Aun–Bvn)



=un–γnA∗(Aun–Bvn)

–x

+kαnun–γnA∗(Aun–Bvn)



=un–γnA∗(Aun–Bvn)

–x

–αn( –k–αn)un–γnA∗(Aun–Bvn)



=un–x+γnA∗(Aun–Bvn)– γn

A∗(Aun–Bvn),un–x

≤ xn–x+γnA∗(Aun–Bvn)

 – γn

A∗(Aun–Bvn),un–x



. (.)

From the deﬁnition of the spectral radiusλAofA∗A, we have

γ_nA∗(Aun–Bvn)=γn

Aun–Bvn,AA∗(Aun–Bvn)

≤λAγnAun–Bvn,Aun–Bvn

=λAγnAun–Bvn. (.)

Combining (.) and (.), we have

xn+–x

≤ xn–x+λAγnAun–Bvn– γnAun–Bvn,Aun–Ax



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Similarly, the last equality of the iterative scheme (.) leads to

yn+–y

≤ yn–y+λBγnAun–Bvn+ γnAun–Bvn,Bvn–By

–αn( –k–αn)vn+γnB∗(Aun–Bvn)

–Qvn+γnB∗(Aun–Bvn)



. (.)

Adding the inequalities (.) and (.), usingk=max{k,k}andAx=By, we get

xn+–x+yn+–y

≤ xn–x+yn–y+

λAγn+λBγn

Aun–Bvn– γnAun–Bvn

–αn( –k–αn)un–γnA∗(Aun–Bvn)

+vn+γnB∗(Aun–Bvn)

–Qvn+γnB∗(Aun–Bvn)

=xn–x+yn–y–γn

 –γn(λA+λB)

Aun–Bvn



. (.)

Now, putn(x,y) =xn–x+yn–y. Therefore from (.), we have

n+(x,y)≤n(x,y) –γn

 –γn(λA+λB)

Aun–Bvn



. (.)

Asαn∈(k, ) andγn∈(,λ_A+λB –), we have  –γn(λA+λB) >  and ( –k–αn) > . It follows from (.) that

n+(x,y)≤n(x,y).

Hence, the sequence{n(x,y)}is non-increasing and lower bounded by . Therefore, it

converges to some ﬁnite limit, sayσ(x,y). So, condition (i) of Lemma . is satisﬁed with μn= (xn,yn),μ∗= (x,y), andW=. It follows from inequality (.) and the convergence

of the sequence{n(x,y)}that

lim

n→∞Aun–Bvn= , (.)

lim

n→∞un–γnA ∗_(Au

n–Bvn)

–Pun–γnA∗(Aun–Bvn)= , (.)

and

lim

n→∞vn+γnB ∗_(Au

n–Bvn)

–Qvn+γnB∗(Aun–Bvn)= . (.)

Moreover, as {n(x,y)} converges to a ﬁnite limit and xn –x ≤ n(x,y), yn –

y _≤

(14)

lim sup_n_→∞yn – y exist. From (.) and (.), we get lim supn→∞un – x and

lim sup_n_→∞vn –y also exist. Let x∗ and y∗ be weak limit points of the sequences

{xn} and {yn}, respectively. Also, {un–γnA∗(Aun–Bvn)} weakly converges to x∗ and

{vn+γnB∗(Aun–Bvn)}weakly converges toy∗. Using Lemma ., we have

xn+–xn=xn+–x–xn+x

=xn+–x–xn–x– xn+–xn,xn–x

=xn+–x–xn–x– 

xn+–x∗,xn–x

+ xn–x∗,xn–x

.

Therefore,

lim sup

n→∞

xn+–xn= .

Similarly, we obtain

lim sup

n→∞

yn+–yn= .

We conclude that

lim

n→∞xn+–xn=  (.)

and

lim

n→∞yn+–yn= . (.)

From Lemma ., we haveun=JrFn,T(xn) andun+=J

F,T

rn+(xn+). Therefore, for allu∈C, we

have

Fλun+ ( –λ)b,u

+T(un),u–un

+φ(u) –φ(un) +

 rn

u–un,un–xn ≥ (.)

and

Fλun++ ( –λ)b,u

+T(un+),u–un+

+φ(u) –φ(un+) +  rn+

u–un+,un+–xn+ ≥. (.)

Puttingu=unin (.) andu=un+in (.), and adding together the resulting inequali-ties, we have

≤Fλun++ ( –λ)b,un

+Fλun+ ( –λ)b,un+

+T(un+),un–un+

+T(un),un+–un

+ 

rn+

un–un+,un+–xn++  rn

(15)

By using (A)-(A), we have

≤  rn+

un–un+,un+–xn++  rn

un+–un,un–xn

≤

un+–un,

un–xn

rn

–un+–xn+ rn+

=

un+–un,un–xn–

rn

rn+

(un+–xn+)

=

un+–un,un–un++un+–xn–

rn

rn+

(un+–xn+)

=un+–un,un–un++

un+–un,xn+–xn+

 – rn

rn+

(un+–xn+)

= –un+–un+

un+–un,xn+–xn+

 – rn

rn+

(un+–xn+)

,

which implies that

un+–un≤ un+–un

xn+–xn+

 – rn

rn+

un+–xn+

.

Thus,

un+–un ≤ xn+–xn+

 – rn

rn+

un+–xn+. (.)

Using (.) and condition (ii) of the hypothesis, (.) implies that

lim

n→∞un+–un= . (.)

Similarly, using the same arguments as above, we have

lim

n→∞vn+–vn= . (.)

From (.) and (.), we have

xn+–x

≤ un–x+λAγnAun–Bvn– γnAun–Bvn,Aun–Ax

–Pun–γnA∗(Aun–Bvn) (.)

and

yn+–y

≤ vn–y+λBγnAun–Bvn+ γnAun–Bvn,Bvn–By

–αn( –k–αn)vn+γnB∗(Aun–Bvn)



(16)

Adding the inequalities (.) and (.), usingk=max{k,k}andAx=By, we obtain

xn+–x+yn+–y

≤ un–x+vn–y–γn

 –γn(λA+λB)

Aun–Bvn



, (.)

where

un–x=JrFn,T(xn) –J

F,T rn (x)



≤ xn–x,un–x

=  

xn–x+un–x–xn–un

(.)

and

vn–y=JrGn,S(yn) –J

G,S rn (y)



≤ yn–y,vn–y

=  

yn–y+vn–y–yn–vn

. (.)

From (.)-(.), we conclude that

xn–un+yn–vn

≤ xn–x–xn+–x+yn–y–yn+–y –γn

 –γn(λA+λB)

Aun–Bvn



. (.)

By using (.)-(.), we have

lim

n→∞xn–un= , (.)

lim

n→∞yn–vn= . (.)

Hence,unx∗andvny∗, respectively.

SincePisk-demi-contractive mapping and (I–P) is demi-closed at , we have

un–Pun

=un–xn++xnn+  –Pun

≤ un–xn++xnn+  –Pun

=un–un++un+–xn+ +( –αn)

+αnP

–Pun

≤ un–un++un+–xn++un–γnA∗(Aun–Bvn)

(17)

+αnun–γnA∗(Aun–Bvn)

≤ un–un++un+–xn++  +√k  –√k

|γn|A∗Aun–Bvn

+αnun–γnA∗(Aun–Bvn)

–Pun–γnA∗(Aun–Bvn).

Using (.), (.), (.), and (.), we have

lim

n→∞un–Pun= . (.)

Similarly, using the same steps as above forQ, we have

lim

n→∞vn–Qvn= . (.)

Since

xn–Pxn=xn–un+un–Pun+Pun–Pxn

≤ xn–un+un–Pun+Pun–Pxn

≤ xn–un+un–Pun+

 +√k

 –√kun–xn

= 

 –√kxn–un+un–Pun, it follows from (.) and (.) that

lim

n→∞xn–Pxn= . (.)

Similarly, we have

lim

n→∞yn–Qyn= . (.)

As{xn}and{yn}weakly converge tox∗ andy∗, respectively, and (I–P) and (I–Q) are

demi-closed at , it follows from (.) and (.) thatx∗∈Fix(P) andy∗∈Fix(Q). Every Hilbert space satisﬁes Opial’s condition, which shows that the weakly subsequential limit of{(xn,yn)}is unique.

Now, we show thatx∗∈GMEP(F,T,φ) andy∗∈GMEP(G,S,ϕ). Sinceun=JrFn,T(xn), we have, for allb,u∈Candλ∈(, ],

Fλun+ ( –λ)b,u

+T(un),u–un

+φ(u) –φ(un) +

 rn

u–un,un–xn ≥.

Using (A) and (A), we get

φ(u) –φ(un) +

 rn

u–un,un–xn ≥–F

λun+ ( –λ)b,u

–T(un),u–un

≥Fλu+ ( –λ)b,un

+T(u),un–u

(18)

and hence

φ(u) –φ(unk) +  rnk

u–unk,unk–xnk ≥F

λu+ ( –λ)b,unk

+T(u),unk–u

.

From (.), we haveunk x∗. It shows that limk→∞

u_nk–x_nk

r_nk = , and from the lower

semicontinuity ofφ, we have

Fλu+ ( –λ)b,x∗

+T(u),x∗–u+φx∗–φ(u)≤, ∀b,u∈C. (.)

Setut=tu+ ( –t)x∗, for allt∈(, ] andu∈C. SinceCis a convex set,ut∈C. Hence

from (.), we have

Fλut+ ( –λ)b,x∗

+T(ut),x∗–ut

+φx∗–φ(ut)≤. (.)

Using the conditions (A)-(A), convexity ofφ, and (.), we get

 = Fλut+ ( –λ)b,ut

+ ( –t)T(ut),ut–ut

+φ(ut) –φ(ut)

≤tFλut+ ( –λ)b,u

+ ( –t)Fλut+ ( –λ)b,x∗

+tφ(u) + ( –t)φx∗

–φ(ut) + ( –t)

T(ut),ut–x∗

+ ( –t)T(ut),x∗–ut

=tFλut+ ( –λ)b,u

+ ( –t)T(ut),u–x∗

+φ(u) –φ(ut)

×( –t)Fλut+ ( –λ)b,x∗

+T(ut),x∗–ut

+φx∗–φ(ut)

≤tFλut+ ( –λ)b,u

+ ( –t)T(ut),u–x∗

+φ(u) –φ(ut)

,

which implies that

Fλut+ ( –λ)b,u

+ ( –t)T(ut),u–x∗

+φ(u) –φ(ut)≥, ∀u,b∈C.

Lett→ and thereforeut→x∗. Using the conditions (A)-(A) and proper lower

semi-continuity ofφ, we have

Fλx∗+ ( –λ)b,u

+Tx∗,u–x∗+φ(u) –φx∗≥, ∀u,b∈C,

which shows thatx∗∈GMEP(F,T,φ). Using the equivalent assertions to the above, we obtainy∗∈GMEP(G,S,ϕ).

SinceA:H−→HandB:H−→Hare bounded linear mappings, and{un}and{vn}

converges weakly tox∗andy∗, respectively, for arbitraryf ∈H_∗, we have

f(Aun) = (f ◦A)(un) −→ (f ◦A)

x∗=fAx∗.

Likewise,

f(Bvn) = (f◦B)(vn) −→ (f ◦B)

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Therefore, we have

Aun–BvnAx∗–By∗,

which implies that

Ax∗–By∗≤lim inf

n→∞ Aun–Bvn= ,

so that Ax∗ = By∗. This implies that (x∗,y∗) ∈ SEMEP(F,G,T,S,φ,ϕ). Therefore, (x∗,y∗)∈.

Finally, we conclude that

. for each(x∗,y∗)∈,limn→∞(xn–x∗+yn–y∗)exists;

. each weak cluster point of the sequence(x∗,y∗)belongs to. On takingH=H×H with the norm(x,y)=

(x₊_y_),_W ₌_,_μ

n= (xn,yn),

andμ= (x∗,y∗) in Lemma ., we see that there exists (x,¯ y)¯ ∈such that xnx¯ and

yny. Therefore, the sequence¯ {(xn,yn)}generated by the iterative scheme (.) weakly

converges to a solution of problem (.) in. This completes the proof of conjecture (I). We now prove the strong convergence conjecture (II).

SincePandQare demi-compact,{xn}and{yn}are bounded, andlimn→∞xn–Pxn= ,

lim_n_→∞yn–Qyn= , there exist (without loss of generality) subsequences{xnk}of{xn} and{ynk} of{yn}such that {xnk}and{ynk}converge strongly to some pointsu∗ andv∗, respectively. Since{xnk}and{ynk}converge weakly tox∗andy∗, respectively, this implies thatx∗=u∗andy∗=v∗. It follows from the demi-closedness ofPandQthatx∗∈Fix(P) andy∗∈Fix(Q). Using similar steps to the previous ones, we getx∗∈GMEP(F,T,φ) and y∗∈GMEP(G,S,ϕ). Thus, we have

Ax∗–By∗= lim

k→∞Axnk–Bynk= .

This implies thatAx∗=By∗. Hence (x∗,y∗)∈. On the other hand, sincen(x,y) =xn–

x+yn–y, for any (x,y)∈, we know thatlimk→∞n(x∗,y∗) = . From conjecture

(I), we see thatlimn→∞n(x∗,y∗) exists, thereforelimn→∞n(x∗,y∗) = . So, the iterative

scheme (.) converges strongly to a solution of problem (.). This completes the proof

of the conjecture (II).

Remark . The convergence theorem, Theorem ., generalizes, extends, and unifies some well-known weak and strong convergence results of Moudafiet al.[], Maet al. [], Maet al.[] and Chidumeet al.[] as we considered the class of demi-contractive mappings, which is much larger than the class of nonexpansive mappings, firmly quasi-nonexpansive mappings. Also, we studied the split equality generalized equilibrium prob-lem (.), which is a more general probprob-lem than the split equality probprob-lem (.), the split equality mixed equilibrium problem (.),etc.

On takingF=G= ,T =S= , andφ=ϕ=  in Theorem ., we get the following convergence theorem for the split equality problem (.).

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Hbe two demi-contractive mappings with constants kand k,respectively,with the con-dition k∈(, ),where k=max{k,k}such that(I–P)and(I–Q)are demi-closed at zero, andFix(P)=∅andFix(Q)=∅.Let A:H−→Hand B:H−→H be bounded linear mappings.Assume that (x,y)∈C×Q and the iteration scheme{(xn,yn)}is deﬁned as

follows:

⎧ ⎨ ⎩

xn+= ( –αn)(xn–γnA∗(Axn–Byn)) +αnP(xn–γnA∗(Axn–Byn));

yn+= ( –αn)(yn+γnB∗(Axn–Byn)) +αnQ(yn+γnB∗(Axn–Byn)),

whereλAandλBdenote the spectral radii of A∗A and B∗B,respectively,{γn}is a positive

real sequence such thatγn∈(,λA+λB –) (forsmall enough),{αn}is a sequence in(k, ) such that for someα,β∈(, ),  <α≤αn≤β< .

If:=Fix(P)∩Fix(Q)∩SEP=∅,then:

(I) The sequence{(xn,yn)}weakly converges to a solution of problem(.).

solution of problem(.).

On takingB=IandH=Hin Corollary ., we obtain the following convergence the-orem for the split feasibility problem (.).

Corollary . Let H and H be real Hilbert spaces,C⊆H and Q⊆H be nonempty closed convex subsets of Hand H,respectively.Let P:H−→Hand Q:H−→Hbe two demi-contractive mappings with constants kand k,respectively,with the condition k∈(, ),where k=max{k,k}such that(I–P)and(I–Q)are demi-closed at zero,and

Fix(P)=∅andFix(Q)=∅.Let A:H−→Hbe a bounded linear mapping.Assume that

(x,y)∈C×Q and the iteration scheme{(xn,yn)}is deﬁned as follows:

⎧ ⎨ ⎩

xn+= ( –αn)(xn–γnA∗(Axn–yn)) +αnP(xn–γnA∗(Axn–yn));

yn+= ( –αn)(yn+γn(Axn–yn)) +αnQ(yn+γn(Axn–yn)),

whereλAdenotes the spectral radii of A∗A,{γn}is a positive real sequence such thatγn∈

(,_λ

A –) (forsmall enough),{αn}is a sequence in(k, )such that for someα,β∈(, ),  <α≤αn≤β< .

If:=Fix(P)∩Fix(Q)∩SFP=∅,then:

(I) The sequence{(xn,yn)}weakly converges to a solution of problem(.).

solution of problem(.).

4 Applications

4.1 Application to the split equality convex differentiable optimization problem

The familiar problem

⎧ ⎨ ⎩

minimizeψ(x)

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in whichψ:C−→Cis convex and diﬀerentiable has a special optimality criterion. A vec-torx∗solves (.) if and only if it solves the following variational inequality problem: ﬁnd x∗∈Csuch that

∇ψx∗,y–x∗≥, ∀y∈C. (.)

By puttingF(x∗,y) =∇ψ(x∗),y–x∗in (.), we see that the variational inequality problem (.) and the equilibrium problem (.) have the same set of solutions.

In , Maet al.[] introduced the so-called split equality mixed variational inequality problem which is ﬁndingx∗∈Candy∗∈Qsuch that

ψx∗,x–x∗+φ(x) –φx∗≥, ∀x∈C,

ζy∗,y–y∗+ϕ(y) –ϕy∗≥, ∀y∈Q, and

Ax∗=By∗,

whereψ:C−→Handζ :Q−→Hare the mappings.

The split equality mixed convex diﬀerentiable optimization problem can be viewed as analogous to the problem of ﬁndingx∗∈Candy∗∈Qsuch that

∇ψx∗,x–x∗+Tx∗,x–x∗+φ(x) –φx∗≥, ∀x∈C,

∇ζy∗,y–y∗+Sy∗,y–y∗+ϕ(y) –ϕy∗≥, ∀y∈Q, and (.)

Ax∗=By∗,

whereψ:C−→H andζ :Q−→Hare convex and diﬀerentiable mappings. The set of solutions of the split equality mixed convex diﬀerentiable optimization problem (.) is denoted bySEMCDOP(ψ,ζ,T,S,φ,ϕ).

IfB=IandH=H, then the split mixed convex diﬀerentiable optimization problem can be viewed as analogous to the problem of ﬁndingx∗∈Csuch that

∇ψx∗,x–x∗+Tx∗,x–x∗+φ(x) –φx∗≥, ∀x∈C,

and such thaty∗=Ax∗∈Qsolves

∇ζy∗,y–y∗+Sy∗,y–y∗+ϕ(y) –ϕy∗≥, ∀y∈Q. (.)

The set of solutions of the split mixed convex diﬀerentiable optimization problem (.) is denoted bySMCDOP(ψ,ζ,T,S,φ,ϕ).

By settingF(x∗,x) =∇ψ(x∗),x–x∗andG(y∗,y) =∇ζ(y∗),y–y∗, forλ=λ= , it is easy to see thatFandGsatisfy all the conditions (A)-(A) since the gradients∇ψ and

∇ζare monotone mappings due to convexity and diﬀerentiability ofψandζ, respectively. Then from Theorem ., we have the following result.

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T:C−→C and S:Q−→Q satisfy the conditions(A), (A), (A).Letφ:C−→R∪ {+∞} andϕ:Q−→R∪ {+∞}be proper lower semicontinuous and convex mappings such that C∩domφ=∅and Q∩domϕ=∅.Let P:H−→H and Q:H −→H be two demi-contractive mappings with constants kand k,respectively,with the condition k∈(, ), where k=max{k,k}such that(I–P)and(I–Q)are demi-closed at zero,andFix(P)=∅ andFix(Q)=∅.Let A:H−→Hand B:H−→Hbe bounded linear mappings.Assume that(x,y)∈C×Q and the iteration scheme{(xn,yn)}is deﬁned as follows:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

∇ψ(un),u–un+T(un),u–un+φ(u) –φ(un) +_r_nu–un,un–xn ≥;

∇ζ(vn),v–vn+S(vn),v–vn+ϕ(v) –ϕ(vn) +_r_nv–vn,vn–yn ≥;

xn+= ( –αn)(un–γnA∗(Aun–Bvn)) +αnP(un–γnA∗(Aun–Bvn));

yn+= ( –αn)(vn+γnB∗(Aun–Bvn)) +αnQ(vn+γnB∗(Aun–Bvn)),

for every u∈C,v∈Q and n≥whereλA andλB denote the spectral radii of A∗A and

B∗B,respectively,{γn}is a positive real sequence such thatγn∈(,_λ 

A+λB –) (for small enough),{αn}is a sequence in(k, ),and{rn} ⊂(,∞)satisfying the following conditions:

(i) for someα,β∈(, ), <α≤αn≤β< ;

(ii) lim infn→∞rn> andlimn→∞|rn+–rn|= .

If:=Fix(P)∩Fix(Q)∩SEMCDOP(ψ,ζ,T,S,φ,ϕ)=∅,then:

(I) The sequence{(xn,yn)}weakly converges to a solution of problem(.).

solution of problem(.).

If we takeB=IandH=Hin Theorem ., then we have the following convergence result for the split mixed convex diﬀerentiable optimization problem (.).

Corollary . Let H and H be real Hilbert spaces,C⊆H and Q⊆H be nonempty closed convex subsets of Hand H,respectively.Suppose that the mappingsψ:C−→H andζ:Q−→Hare convex and diﬀerentiable with optimality criterion,and the mappings T:C−→C and S:Q−→Q satisfy the conditions(A), (A), (A).Letφ:C−→R∪ {+∞} andϕ:Q−→R∪ {+∞}be proper lower semicontinuous and convex mappings such that C∩domφ=∅and Q∩domϕ=∅.Let P:H−→H and Q:H −→H be two demi-contractive mappings with constants kand k,respectively,with the condition k∈(, ), where k=max{k,k}such that(I–P)and(I–Q)are demi-closed at zero,andFix(P)=∅ andFix(Q)=∅.Let A:H−→H be a bounded linear mapping.Assume that(x,y)∈ C×Q and the iteration scheme{(xn,yn)}is deﬁned as follows:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

∇ψ(un),u–un+T(un),u–un+φ(u) –φ(un) +_r_nu–un,un–xn ≥;

∇ζ(vn),v–vn+S(vn),v–vn+ϕ(v) –ϕ(vn) +_r_nv–vn,vn–yn ≥;

xn+= ( –αn)(un–γnA∗(Aun–vn)) +αnP(un–γnA∗(Aun–vn));

yn+= ( –αn)(vn+γn(Aun–vn)) +αnQ(vn+γn(Aun–vn)),

for every u∈C,v∈Q,and n≥whereλAdenotes the spectral radius of A∗A,{γn}is a