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:

simplex_liou@hotmail.com

2Department of Information

Management, Cheng Shiu University, Kaohsiung, 833, Taiwan

3Research 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≥, ∀yC, (.)

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 findingxCsuch that

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

whereF:C×C−→His the set-valued mapping with the conditionF(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,zC,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∗,yx∗≥, ∀yC, (.)

whereF:C×C−→Ris a given bi-mapping withF(x,x) = , for allxCandT: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∗,yx∗+φ(y) –φx∗≥, ∀yC, (.)

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∗≥, ∀yC. (.)

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

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

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

Fλx∗+ ( –λ)b,y+Tx∗,yx∗+φ(y) –φx∗≥, ∀y,bC,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(xnγnA∗(AxnByn));

yn+=T(yn+γnB∗(AxnByn)),

(.)

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(xnγnA∗(AxnByn));

yn+= ( –αn)yn+αnT(ynγnB∗(AxnByn)),

(.)

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≥, ∀xC, (.)

and such that

y∗=Ax∗∈Q solves Gy∗,y≥, ∀yQ. (.)

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

algorithm: ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

F(un,u) +rnuun,unxn ≥, ∀uC;

G(vn,v) +rnvvn,vnAun ≥, ∀vQ;

xn+=PC(un) +μA∗(vnAun),

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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∗≥, ∀xC,

Gy∗,y+ϕ(y) –ϕy∗≥, ∀yQ, 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) +rnuun,unxn ≥, ∀uC;

G(vn,v) +ϕ(v) –ϕ(vn) +rnvvn,vnAun ≥, ∀vQ;

xn+=αnun+ ( –αn)T(unγnA∗(AunBvn));

yn+=αnvn+ ( –αn)S(vn+γnB∗(AunBvn)),

(.)

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

x∗+ ( –λ)b,x

+Tx∗,xx∗+φ(x) –φx∗≥,

x,bC,λ∈(, ],

y∗+ ( –λ)c,y

+Sy∗,yy∗+ϕ(y) –ϕy∗≥, (.)

y,cQ,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≥, ∀xC, Gy∗,y≥, ∀yQ, 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∗, ∀xC, ϕ(y)ϕy∗, ∀yQ, 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∗, ∀xC, and

y∗=Ax∗∈Q such that ϕ(y)ϕy∗, ∀yQ.

(.)

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 pointxH 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)xy, (.)

for allx,yHandλ∈[, ].

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

Txx∗≤xx∗+kxTx, ∀xC,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

Txx∗≤xx∗+√kxTx

≤( +√k)xx∗+√kTxx

=

 +√k  –√k

xx

=Lxx∗,

whereL=+

k

–√k, so that

Txx∗≤Lxx∗.

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

tozimplies thatTp=z.

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

subsequence{xnj}and a pointpCsuch thatxnjpasj→ ∞and (I–T)p=z.

Definition . A multi-valued mappingF:C−→His 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−→Hbe a KKM-mapping.If for each xC,F(x)is closed and if for

at least one point xC,F(x)is compact,thenxCF(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 allxC;

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

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(A) Tis weakly upper semicontinuous;

(A) for eachxC,λ∈(, ]andr> , there exist a bounded setDCandaC such that for anyzC\D,

–Fλa+ ( –λ)b,z+T(z),az+φ(a) –φ(z) +

raz,zx< , ∀bC.

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 xH,let JrF,T:H−→C be the resolvent operator of F and T,defined by

JrF,T(x) =

zC:z+ ( –λ)b,y

+T(z),yz

+φ(y) –φ(z) +

ryz,zx ≥, ∀y,bC

. (.)

Then:

(i) for eachxH,JrF,T(x)=∅;

(ii) JF,T

r is single-valued;

(iii) JF,T

r is firmly nonexpansive,i.e.,for anyx,yH, JrF,T(x) –JrF,T(y)≤JrF,T(x) –JrF,T(y),xy;

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

M(y) =

zC: –Fλy+ ( –λ)b,z

+T(z),yz

+φ(y) –φ(z) +

ryz,zx¯ ≥

. (.)

SinceyM(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

–Fλyi+ ( –λ)b,z¯

+T(z),¯ yiz¯

+φ(yi) –φ(z) +¯

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

 = 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

Tz),yiz¯

+

n

i=

tiφ(yi) – n

i= tiφ(¯z)

+ r

n

i=

tiyiz,¯ z¯–x¯

≤–

n

i= tiF

λyi+ ( –λ)b,z¯ + n i= ti

T(z),¯ yiz¯

+φ(yi) –φ(z) +¯

ryiz,¯ z¯–x¯ = n i= ti

–Fλyi+ ( –λ)b,z¯

+T(z),¯ yi–¯z

+φ(yi) –φ(z) +¯

ryiz,¯ z¯–x¯

< ,

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

Now, we prove thatM(y) =M(f)w for eachyC,i.e.,M(y) is weakly closed. LetzM(f)wand{zn}be a sequence inM(y) such thatznz. SinceznM(y), we have

–Fλy+ ( –λ)b,zn

+T(zn),yzn

+φ(y) –φ(zn) +

ryzn,znx¯ ≥. It follows from (A) and (A), and the weak lower semicontinuity ofφand · that

≤lim sup

n→∞

–Fλy+ ( –λ)b,zn

+T(zn),yzn

+φ(y) –φ(zn) +

ryzn,znx¯

≤lim sup

n→∞

–Fλy+ ( –λ)b,zn

+lim sup

n→∞

T(zn),yzn

+φ(y)

–lim inf

n→∞ φ(zn) +

rlim supn→∞ yzn,znx¯ ≤–lim inf

n→∞

y+ ( –λ)b,zn

+lim sup

n→∞

T(zn),yzn

+φ(y)

–lim inf

n→∞ φ(zn) +

rlim supn→∞

yzn,znx¯

≤–Fλy+ ( –λ)b,z

+T(z),yz+φ(y) –φ(z) +

ryz,zx¯.

This implies thatzM(y), and henceM(y) is weakly closed.

In order to show thatM(y) is weakly compact for at least a pointyC, from (A), we can see that there exist a bounded setDCandaDsuch that for anyzC\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 .,yCM(y)=∅. Hence, forzyCM(y), we have

–Fλy+ ( –λ)b,z

+T(z),yz+φ(y) –φ(z) +

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LetyCbe arbitrary, and letzt=ty+ ( –t)z,  <t≤. ThenztC, and hence we have

–Fλzt+ ( –λ)b,z

+T(z),ztz

+φ(zt) –φ(z) +

rztz,zx¯ ≥. (.) Applying (.), using (A), (A), and convexity ofφ, we have

 = zt+ ( –λ)b,zt

=zt+ ( –λ)b,ty+ ( –t)z

tFλzt+ ( –λ)b,y

+ ( –t)Fλzt+ ( –λ)b,z

tFλzt+ ( –λ)b,y

+ ( –t)T(z),ztz

+φ(zt) –φ(z) +

rztz,zx¯

tFλzt+ ( –λ)b,y

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

+t( –t)

r yz,zx¯

zt+ ( –λ)b,y

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

+( –t)

r yz,zx¯.

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

z+ ( –λ)b,y

+T(z),yz+φ(y) –φ(z) +

ryz,zx¯ ≥,

i.e.,zJF,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, forxH andr> , letz,z∈JrF,T(x).

Then

z+ ( –λ)b,z

+T(z),z–z

+φ(z) –φ(z) + 

rz–z,z–x ≥

and

z+ ( –λ)b,z

+T(z),z–z

+φ(z) –φ(z) + 

rz–z,z–x ≥.

Adding the above two inequalities, we obtain

z+ ( –λ)b,z

+z+ ( –λ)b,z

T(z) –T(z),z–z

+

rz–z,z–z ≥.

From (A)-(A) andr> , we have

z–z,z–z ≥,

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(iii) To prove thatJF,T

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

+TJrF,T(x),JrF,T(y) –JrF,T(x) +φJrF,T(y)–φJrF,T(x)+

r

JrF,T(y) –JrF,T(x),JrF,T(x) –x≥ and

JrF,T(y) + ( –λ)b,JrF,T(x)

+TJrF,T(y),JrF,T(x) –JrF,T(y) +φJrF,T(x)–φJrF,T(y)+

r

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

JrF,T(x) + ( –λ)b,JrF,T(y)

+JrF,T(y) + ( –λ)b,JrF,T(x)

TJrF,T(y)–TJrF,T(x),JrF,T(y) –JrF,T(x) +

r

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

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

JrF,T(x) –JrF,T(y)≤JrF,T(x) –JrF,T(y),xy.

(iv) TakexC. Then

x∈FixJrF,T

x=JrF,T(x)

x+ ( –λ)b,y

+T(x),yx+φ(y) –φ(x) +

ryx,xx ≥

x+ ( –λ)b,y

+T(x),yx+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∈(, ], anduH, letJsG,S:H−→Qbe the resolvent operator ofGandS, defined by

JsG,S(u) =

vQ:v+ ( –λ)c,w

+S(v),wv+ϕ(w) –ϕ(v)

+

swv,vu ≥,∀w,cQ

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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 WH satisfying the following conditions: (i) for everyμW,limn→∞μnμexists;

(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,yH,we have

xy=x–y– xy,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 Hbe real Hilbert spaces, CHand QHbe the nonempty closed convex subsets of Hand 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−→Hand Q:H−→Hbe the two demi-contractive mappings with constants kand 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−→Hand B:H−→Hbe bounded linear mappings.Assume that(x,y)∈C×Q and the iteration scheme{(xn,yn)}is defined as follows:

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

F(λun+ ( –λ)b,u) +T(un),uun

+φ(u) –φ(un) +rnuun,unxn ≥;

G(λvn+ ( –λ)c,v) +S(vn),vvn(v)

+ϕϕ(vn) +rnvvn,vnyn ≥;

xn+= ( –αn)(unγnA∗(AunBvn)) +αnP(unγnA∗(AunBvn));

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

(.)

for every u,bC,v,cQ,and n≥whereλAandλBdenote the spectral radii of AA and

BB,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∈(, ), <ααnβ< ;

(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

unx=JrFn,T(xn) –J

F,T

rn (x)≤ xnx, (.)

vny=JrGn,S(yn) –J

G,S

rn (y)≤ yny. (.)

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

xn+–x =( –αn)

unγnA∗(AunBvn)

+αnP

unγnA∗(AunBvn)

x

=( –αn)

unγnA∗(AunBvn)

x+αn

PunγnA∗(AunBvn)

x

= ( –αn)unγnA∗(AunBvn)

x+αnP

unγnA∗(AunBvn)

x

αn( –αn)unγnA∗(AunBvn)

PunγnA∗(AunBvn)

≤( –αn)unγnA∗(AunBvn)

x+αnunγnA∗(AunBvn)

x

+kunγnA∗(AunBvn)

PunγnA∗(AunBvn)

αn( –αn)unγnA∗(AunBvn)

PunγnA∗(AunBvn)

=unγnA∗(AunBvn)

x

+kαnunγnA∗(AunBvn)

PunγnA∗(AunBvn)

αn( –αn)unγnA∗(AunBvn)

PunγnA∗(AunBvn)

=unγnA∗(AunBvn)

x

αn( –k–αn)unγnA∗(AunBvn)

PunγnA∗(AunBvn)

=unx+γnA∗(AunBvn)– γn

A∗(AunBvn),unx

αn( –k–αn)unγnA∗(AunBvn)

PunγnA∗(AunBvn)

xnx+γnA∗(AunBvn)

 – γn

A∗(AunBvn),unx

αn( –k–αn)unγnA∗(AunBvn)

PunγnA∗(AunBvn)

. (.)

From the definition of the spectral radiusλAofAA, we have

γnA∗(AunBvn)=γn

AunBvn,AA∗(AunBvn)

λAγnAunBvn,AunBvn

=λAγnAunBvn. (.)

Combining (.) and (.), we have

xn+–x

xnx+λAγnAunBvn– γnAunBvn,AunAx

αn( –k–αn)unγnA∗(AunBvn)

PunγnA∗(AunBvn)

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

yn+–y

yny+λBγnAunBvn+ γnAunBvn,BvnBy

αn( –k–αn)vn+γnB∗(AunBvn)

Qvn+γnB∗(AunBvn)

. (.)

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

xn+–x+yn+–y

xnx+yny+

λAγn+λBγn

AunBvn– γnAunBvn

αn( –kαn)unγnA∗(AunBvn)

PunγnA∗(AunBvn)

+vn+γnB∗(AunBvn)

Qvn+γnB∗(AunBvn)

=xnx+yny–γn

 –γnA+λB)

AunBvn

αn( –kαn)unγnA∗(AunBvn)

PunγnA∗(AunBvn)

+vn+γnB∗(AunBvn)

Qvn+γnB∗(AunBvn)

. (.)

Now, putn(x,y) =xnx+yny. Therefore from (.), we have

n+(x,y)n(x,y) –γn

 –γnA+λB)

AunBvn

αn( –kαn)unγnA∗(AunBvn)

PunγnA∗(AunBvn)

+vn+γnB∗(AunBvn)

Qvn+γnB∗(AunBvn)

. (.)

Asαn∈(k, ) andγn∈(,λA+λB), we have  –γnA+λB) >  and ( –kαn) > . 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→∞AunBvn= , (.)

lim

n→∞unγnA(Au

nBvn)

PunγnA∗(AunBvn)= , (.)

and

lim

n→∞vn+γnB(Au

nBvn)

Qvn+γnB∗(AunBvn)= . (.)

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

y

(14)

lim supn→∞yny exist. From (.) and (.), we get lim supn→∞unx and

lim supn→∞vny also exist. Let x∗ and y∗ be weak limit points of the sequences

{xn} and {yn}, respectively. Also, {unγnA∗(AunBvn)} weakly converges to x∗ and

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

xn+–xn=xn+–xxn+x

=xn+–x–xnx– xn+–xn,xnx

=xn+–x–xnx– 

xn+–x∗,xnx

+ xnx∗,xnx

.

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 alluC, we

have

un+ ( –λ)b,u

+T(un),uun

+φ(u) –φ(un) +

rn

uun,unxn ≥ (.)

and

un++ ( –λ)b,u

+T(un+),uun+

+φ(u) –φ(un+) +  rn+

uun+,un+–xn+ ≥. (.)

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

≤un++ ( –λ)b,un

+un+ ( –λ)b,un+

+T(un+),unun+

+T(un),un+–un

+ 

rn+

unun+,un+–xn++  rn

(15)

By using (A)-(A), we have

≤  rn+

unun+,un+–xn++  rn

un+–un,unxn

un+–un,

unxn

rn

un+–xn+ rn+

=

un+–un,unxn

rn

rn+

(un+–xn+)

=

un+–un,unun++un+–xn

rn

rn+

(un+–xn+)

=un+–un,unun++

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+–unxn+–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

unx+λAγnAunBvn– γnAunBvn,AunAx

αn( –k–αn)unγnA∗(AunBvn)

PunγnA∗(AunBvn) (.)

and

yn+–y

vny+λBγnAunBvn+ γnAunBvn,BvnBy

αn( –k–αn)vn+γnB∗(AunBvn)

Qvn+γnB∗(AunBvn)

(16)

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

xn+–x+yn+–y

unx+vny–γn

 –γnA+λB)

AunBvn

αn( –kαn)unγnA∗(AunBvn)

PunγnA∗(AunBvn)

+vn+γnB∗(AunBvn)

Qvn+γnB∗(AunBvn)

, (.)

where

unx=JrFn,T(xn) –J

F,T rn (x)

xnx,unx

=  

xnx+unx–xnun

(.)

and

vny=JrGn,S(yn) –J

G,S rn (y)

yny,vny

=  

yny+vny–ynvn

. (.)

From (.)-(.), we conclude that

xnun+ynvn

xnx–xn+–x+yny–yn+–y –γn

 –γnA+λB)

AunBvn

αn( –kαn)unγnA∗(AunBvn)

PunγnA∗(AunBvn)

+vn+γnB∗(AunBvn)

Qvn+γnB∗(AunBvn)

. (.)

By using (.)-(.), we have

lim

n→∞xnun= , (.)

lim

n→∞ynvn= . (.)

Hence,unx∗andvny∗, respectively.

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

unPun

=unxn++xnn+  –Pun

unxn++xnn+  –Pun

=unun++un+–xn+ +( –αn)

unγnA∗(AunBvn)

+αnP

unγnA∗(AunBvn)

Pun

unun++un+–xn++unγnA∗(AunBvn)

(17)

+αnunγnA∗(AunBvn)

PunγnA∗(AunBvn)

unun++un+–xn++  +√k  –√k

|γn|AAunBvn

+αnunγnA∗(AunBvn)

PunγnA∗(AunBvn).

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

lim

n→∞unPun= . (.)

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

lim

n→∞vnQvn= . (.)

Since

xnPxn=xnun+unPun+PunPxn

xnun+unPun+PunPxn

xnun+unPun+

 +√k

 –√kunxn

= 

 –√kxnun+unPun, it follows from (.) and (.) that

lim

n→∞xnPxn= . (.)

Similarly, we have

lim

n→∞ynQyn= . (.)

As{xn}and{yn}weakly converge tox∗ andy∗, respectively, and (I–P) and (IQ) 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,uCandλ∈(, ],

un+ ( –λ)b,u

+T(un),uun

+φ(u) –φ(un) +

rn

uun,unxn ≥.

Using (A) and (A), we get

φ(u) –φ(un) +

rn

uun,unxn ≥–F

λun+ ( –λ)b,u

T(un),uun

u+ ( –λ)b,un

+T(u),unu

(18)

and hence

φ(u) –φ(unk) +  rnk

uunk,unkxnkF

λu+ ( –λ)b,unk

+T(u),unku

.

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

unkxnk

rnk = , and from the lower

semicontinuity ofφ, we have

u+ ( –λ)b,x

+T(u),x∗–u+φx∗–φ(u)≤, ∀b,uC. (.)

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

from (.), we have

ut+ ( –λ)b,x

+T(ut),x∗–ut

+φx∗–φ(ut)≤. (.)

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

 = ut+ ( –λ)b,ut

+ ( –t)T(ut),utut

+φ(ut) –φ(ut)

tFλut+ ( –λ)b,u

+ ( –t)Fλut+ ( –λ)b,x

+tφ(u) + ( –t)φx

φ(ut) + ( –t)

T(ut),utx

+ ( –t)T(ut),x∗–ut

=tFλut+ ( –λ)b,u

+ ( –t)T(ut),ux

+φ(u) –φ(ut)

×( –t)Fλut+ ( –λ)b,x

+T(ut),x∗–ut

+φx∗–φ(ut)

tFλut+ ( –λ)b,u

+ ( –t)T(ut),ux

+φ(u) –φ(ut)

,

which implies that

ut+ ( –λ)b,u

+ ( –t)T(ut),ux

+φ(u) –φ(ut)≥, ∀u,bC.

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

semi-continuity ofφ, we have

x∗+ ( –λ)b,u

+Tx∗,ux∗+φ(u) –φx∗≥, ∀u,bC,

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 arbitraryfH∗, 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

AunBvnAx∗–By∗,

which implies that

Ax∗–By∗≤lim inf

n→∞ AunBvn= ,

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→∞(xnx∗+yny∗)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= ,

limn→∞ynQyn= , 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→∞AxnkBynk= .

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

x+yny, 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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Hbe two demi-contractive mappings with constants kand 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−→Hand B:H−→Hbe bounded linear mappings.Assume that (x,y)∈C×Q and the iteration scheme{(xn,yn)}is defined as

follows:

⎧ ⎨ ⎩

xn+= ( –αn)(xnγnA∗(AxnByn)) +αnP(xnγnA∗(AxnByn));

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

whereλAandλBdenote the spectral radii of AA and BB,respectively,{γn}is a positive

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

If:=Fix(P)∩Fix(Q)∩SEP=∅,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

solution of problem(.).

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

Corollary . Let Hand Hbe real Hilbert spaces,CHand QHbe nonempty closed convex subsets of Hand H,respectively.Let P:H−→Hand Q:H−→Hbe two demi-contractive mappings with constants kand 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−→Hbe a bounded linear mapping.Assume that

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

⎧ ⎨ ⎩

xn+= ( –αn)(xnγnA∗(Axnyn)) +αnP(xnγnA∗(Axnyn));

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

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

(,λ

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

If:=Fix(P)∩Fix(Q)∩SFP=∅,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

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∗,yx∗≥, ∀yC. (.)

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∗,xx∗+φ(x) –φx∗≥, ∀xC,

ζy∗,yy∗+ϕ(y) –ϕy∗≥, ∀yQ, 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∗,xx∗+Tx∗,xx∗+φ(x) –φx∗≥, ∀xC,

ζy∗,yy∗+Sy∗,yy∗+ϕ(y) –ϕy∗≥, ∀yQ, 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∗,xx∗+Tx∗,xx∗+φ(x) –φx∗≥, ∀xC,

and such thaty∗=Ax∗∈Qsolves

ζy∗,yy∗+Sy∗,yy∗+ϕ(y) –ϕy∗≥, ∀yQ. (.)

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∗),xx∗andG(y∗,y) =ζ(y∗),yy∗, 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−→Hand Q:H −→Hbe two demi-contractive mappings with constants kand 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−→Hand B:H−→Hbe bounded linear mappings.Assume that(x,y)∈C×Q and the iteration scheme{(xn,yn)}is defined as follows:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

ψ(un),uun+T(un),uun+φ(u) –φ(un) +rnuun,unxn ≥;

ζ(vn),vvn+S(vn),vvn+ϕ(v) –ϕ(vn) +rnvvn,vnyn ≥;

xn+= ( –αn)(unγnA∗(AunBvn)) +αnP(unγnA∗(AunBvn));

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

for every uC,vQ and n≥whereλA andλB denote the spectral radii of AA and

BB,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∈(, ), <ααnβ< ;

(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(.).

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

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 Hand Hbe real Hilbert spaces,CHand QHbe nonempty closed convex subsets of Hand H,respectively.Suppose that the mappingsψ:C−→Handζ:Q−→Hare 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−→Hand Q:H −→Hbe two demi-contractive mappings with constants kand 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−→Hbe a bounded linear mapping.Assume that(x,y)∈ C×Q and the iteration scheme{(xn,yn)}is defined as follows:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

ψ(un),uun+T(un),uun+φ(u) –φ(un) +rnuun,unxn ≥;

ζ(vn),vvn+S(vn),vvn+ϕ(v) –ϕ(vn) +rnvvn,vnyn ≥;

xn+= ( –αn)(unγnA∗(Aunvn)) +αnP(unγnA∗(Aunvn));

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

for every uC,vQ,and n≥whereλAdenotes the spectral radius of AA,{γn}is a

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