Constructed nets with perturbations for equilibrium and fixed point problems

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R E S E A R C H

Open Access

Constructed nets with perturbations for

equilibrium and ﬁxed point problems

Yonghong Yao

1

_{, Yeol Je Cho}

2,3*

_{, Yeong-Cheng Liou}

4

_{and Ravi P Agarwal}

3,5

This paper is dedicated to Professor Shih-sen Chang for his 80th birthday.

*_{Correspondence: [email protected]} 2_{Department of Mathematics}

Education and RINS, Gyeongsang National University, Chinju, 660-701, Korea

3_{Department of Mathematics, King}

Abdulaziz University, P.O. Box 80203, Jeddah, Saudi Arabia

Full list of author information is available at the end of the article

Abstract

In this paper, an implicit net with perturbations for solving the mixed equilibrium problems and ﬁxed point problems has been constructed and it is shown that the proposed net converges strongly to a common solution of the mixed equilibrium problems and ﬁxed point problems. Also, as applications, some corollaries for solving the minimum-norm problems are also included.

MSC: 47J05; 47J25; 47H09

Keywords: equilibrium problem; ﬁxed point problem; minimization problem; nonexpansive mapping

1 Introduction

The present paper is devoted to solving the following mixed equilibrium problem: Find

u∈Csuch that

F(u,v) +Au,v–u ≥ (.)

for allv∈C, whereCis a nonempty closed convex subset of a real Hilbert spaceH,F:

C×C→Ris a bifunction andA:C→His a nonlinear operator. The solution set of (.) is denoted by S(MEP).

This problem (.) includes optimization problems, variational inequalities, minimax problems, and Nash equilibrium problems in noncooperative games as special cases.

Case. IfA=  in (.), then (.) reduces to the following equilibrium problem: Find

u∈Csuch that

F(u,v)≥ (.)

for allv∈C. The solution of (.) is denoted by S(EP).

Case. IfF=  in (.), then (.) reduces to the variational inequality problem: Find

z∈Csuch that

Au,v–u ≥ (.)

for allv∈C. The solution of (.) is denoted by S(VI).

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In the literature, there are a large number of references associated with some equilibrium problems and variational inequality problems (see, for instance, [–]).

The main purpose of the present paper is to construct the following implicit net with perturbations for solving the mixed equilibrium (.) and the ﬁxed point problem:

F(zt,y) +  λ

y–zt,zt–

tut+ ( –t)Tzt–λATzt

≥

for ally∈C. Also, it is shown that the proposed net{zt}converges strongly to a common solution of the mixed equilibrium problems and ﬁxed point problems. As applications, some corollaries for solving the minimum-norm problems are also included.

2 Preliminaries

LetHbe a real Hilbert space with an inner product·,·and a norm · , respectively, and

Cbe a nonempty closed convex subset of a real Hilbert spaceH. () A mappingT:C→Cis said to benonexpansiveif

Tu–Tv ≤ u–v

for allu,v∈C.F(T) denotes the set of ﬁxed points ofT.

() A mapping A:C→H is said to be α-inverse-strongly monotone if there exists a positive real numberα>  such that

Au–Av,u–v ≥αAu–Av

for allu,v∈C. It is clear that anyα-inverse-strongly monotone mapping is monotone and 

α-Lipschitz continuous.

Throughout this paper, we assume that a bifunctionF:C×C→Rsatisﬁes the following conditions:

(C) F(u,u) = for allu∈C;

(C) Fis monotone,i.e.,F(u,v) +F(v,u)≤for allu,v∈C; (C) for eachu,v,w∈C,limt↓F(tw+ ( –t)u,v)≤F(u,v);

(C) for eachu∈C,v→F(u,v)is convex and lower semicontinuous.

In fact, some eﬀorts to construct the algorithms for solving the equilibrium problems have been carried out. For instance, Moudaﬁ [] presented an iterative algorithm and proved a weak convergence theorem for solving the mixed equilibrium problem (.). Takahashi and Takahashi [] constructed the following iterative algorithm:

⎧ ⎨ ⎩

F(zn,y) +Axn,y–zn+λny–zn,zn–xn ≥,

xn+=βnxn+T(αnu+ ( –βn)zn)

(.)

for ally∈Candn≥, whereT :C→Cis a nonexpansive mapping. They proved that the sequence{xn}generated by (.) converges strongly toz=ProjF(T)∩S(MEP)(u).

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Implicit iterative algorithm{xn}:

⎧ ⎨ ⎩

F(un,y) +_rny–un,un–xn ≥,

xn=αnγf(xn) + (I–αnA)Tun

(.)

for ally∈Handn≥.

Explicit iterative algorithm{xn}:

⎧ ⎨ ⎩

F(un,y) +_rny–un,un–xn ≥,

xn+=αnγf(xn) + (I–αnA)Tun

(.)

for ally∈Handn≥.

They proved that, under certain conditions, the sequences{xn}generated by (.) and (.) converge strongly to the unique solution of the variational inequality:

(A–γf)z,x–z≥

for allx∈F(T)∩S(EP), which is the optimality condition for the minimization problem:

min

x∈F(T)∩S(EP) 

Ax,x–h(x),

wherehis a potential function forγf.

We know that there are perturbations always occurring in the iterative processes be-cause the manipulations are inaccurate. Recently, Chuanget al.([]) constructed the fol-lowing iteration process with perturbations for ﬁnding a common element of the set of solutions of the equilibrium problem and the set of ﬁxed points for a quasi-nonexpansive mapping with perturbation:q∈Hand

⎧ ⎨ ⎩

xn∈C such that F(xn,y) +λny–xn,xn–qn ≥,

qn+=αnun+ ( –αn)(βnxn+ ( –βn)Txn)

for all y∈C and n ≥. They shown that the sequence {qn} converges strongly to

Proj_F(T)_∩_S(EP).

Now, we need the following useful lemmas for our main results.

Lemma .[] Let C be a nonempty closed convex subset of a real Hilbert space H.Let

F:C×C→R be a bifunction which satisﬁes the conditions(C)-(C).Let r> and x∈H.

Then there exists z∈C such that

F(z,y) +

ry–z,z–x ≥

for all y∈C.Further,if

Tr(x) = z∈C:F(z,y) + 

ry–z,z–x ≥,∀y∈C

,

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()Tris single-valued and Tris ﬁrmly nonexpansive,i.e.,for any x,y∈H,

Trx–Try≤ Trx–Try,x–y;

()S(EP)is closed and convex and S(EP) =F(Tr).

Lemma .[] Let C,H,F,and Trx be as in Lemma..Then the following holds:

Tsx–Ttx≤

s–t

s Tsx–Ttx,Tsx–x

for all s,t> and x∈H.

Lemma .[] Let C be a nonempty closed convex subset of a real Hilbert space H.Let a

mapping A:C→H beα-inverse-strongly monotone and r> be a constant.Then we have

(I–rA)x– (I–rA)y≤ x–y+r(r– α)Ax–Ay

for all x,y∈C.In particular,if≤r≤α,then I–rA is nonexpansive.

Lemma .[] Let C be a closed convex subset of a real Hilbert space H and let T:C→C be a nonexpansive mapping. Then the mapping I–T is demiclosed,that is,if{xn} is a

sequence in C such that xn→u weakly and(I–T)xn→v strongly,then(I–T)u=v.

3 Convergence results

In this section, ﬁrst, we give our main results.

Part I: Assumptions on the setting ofC,F,A, andT:

(A) Cis a nonempty closed convex subset of a real Hilbert spaceH; (A) F:C×C→Ris a bifunction satisfying the conditions (C)-(C); (A) A:C→His anα-inverse-strongly monotone mapping;

(A) T:C→Cis a nonexpansive mapping.

Part II: Parameter restrict:

λis a constant satisfyinga≤λ≤b, where [a,b]⊂(, α).

Part III: Perturbations:

{ut} ⊂His a net satisfyinglimt→+ut=u∈H.

Algorithm . For anyt∈(,  –_λ_α), deﬁne a net{zt} ⊂Cby the implicit manner:

F(zt,y) +  λ

y–zt,zt–

≥ (.)

for ally∈C.

Remark . We show that the net{zt}is well deﬁned. Next, we prove that (.) can be

rewritten as

zt=Tλ

(.)

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In fact, for anyt∈(,  –_λ_α),ut∈H, andx∈H, we ﬁndzsuch that, for ally∈C,

F(z,y) +  λ

y–z,z–tut+ ( –t)Tx–λATx

≥.

From Lemma ., we get immediately

z=Tλtut+ ( –t)Tx–λATx

.

Now, we can deﬁne a mapping

ϑt:=Tλ

tut+ ( –t)T–λAT

for allt∈(,  – λ

α). Again, from Lemma ., we know thatTλis nonexpansive. Thus, for

anyx,y∈C, we have

ϑtx–ϑty

=Tλtut+ ( –t)Tx–λATx

–Tλtut+ ( –t)Ty–λATy

≤tut+ ( –t)Tx–λATx

–tut+ ( –t)Ty–λATy

= ( –t)

I– λ

 –tA

Tx–

I– λ

 –tA

Ty. (.)

From Lemma .,I– λ

–tAis nonexpansive for allt∈(,  –

λ

α). Note thatTis also

non-expansive. By (.), we deduce

ϑtx–ϑty ≤( –t)x–y

for allx,y∈C. This indicates thatϑtis a contraction onCand so it has a unique ﬁxed point, denoted byzt, inC. That is,zt=Tλ(tut+ ( –t)Tzt–λATzt). Hence{zt}is well deﬁned.

Theorem . Suppose that F(T)∩S(MEP)= ∅.Then the net{zt}deﬁned by(.)converges

strongly as t→+to PF(T)∩S(MEP)(u).

Proof Pick upz∈F(T)∩S(MEP). It is obvious thatz=Tλ(z–λAz) for allλ> . So, we have

z=Tz=Tλ(z–λAz) =Tλ(Tz–λATz) =Tλ

tTz+ ( –t)

Tz– λ

 –tATz

for allt∈(,  –_λ_α). Then we have

zt–z

=Tλtut+ ( –t)Tzt–λATzt

–z

=Tλ

tut+ ( –t)

Tzt–

λ  –tATzt

–Tλ

tz+ ( –t)

Tz– λ

 –tATz

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≤

tut+ ( –t)

Tzt–

λ  –tATzt

–

tz+ ( –t)

Tz– λ

 –tATz



=( –t)

Tzt–

λ  –tATzt

–

Tz– λ

 –tATz

+t(ut–z)

. (.)

Using the convexity of · and the inverse-strong monotonicity ofA, we derive

( –t)

Tzt–

λ  –tATzt

–

Tz– λ

 –tATz

+t(ut–z)



≤( –t)

Tzt–

λ  –tATzt

–

Tz– λ

 –tATz

+tut–z

= ( –t)(Tzt–Tz) –λ(ATzt–ATz)/( –t) 

+tut–z

= ( –t)

Tzt–Tz– λ

 –tATzt–ATz,Tzt–Tz

+ λ 

( –t)ATzt–ATz 

+tut–z

≤( –t)

Tzt–Tz– αλ

 –tATzt–ATz

₊ λ

( –t)ATzt–ATz 

+tut–z

= ( –t)

Tzt–Tz+ λ ( –t)

λ– ( –t)αATzt–ATz

+tut–z

≤( –t)

zt–z+ λ ( –t)

+tut–z. (.)

By the assumption, we haveλ– ( –t)α≤ for allt∈(,  – λ

α). Then, from (.) and

(.), it follows that

zt–z

≤( –t)

zt–z+ λ ( –t)

+tut–z

≤( –t)zt–z+tut–z (.)

and so

zt–z ≤ ut–z. (.)

Sincelimt→+ut=u, there exists a positive constantM>  such that supt{ut} ≤M. Then, from (.), we deduce that{zt}is bounded. Hence{Tzt}and{ATzt}are also bounded.

From (.) and (.), we obtain

zt–z≤( –t)zt–z+ λ ( –t)

λ– ( –t)αATzt–Az+tut–z

and so

λ ( –t)

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This implies that

lim

t→+ATzt–Az= . (.)

Next, we showzt–Tzt →. SinceTλis ﬁrmly nonexpansive (see Lemma .), we have

zt–z=Tλ

–z

=Tλtut+ ( –t)Tzt–λATzt

–Tλ(Tz–λATz)

≤tut+ ( –t)Tzt–λATzt– (Tz–λATz),zt–z

=

tut+ ( –t)Tzt–λATzt– (Tz–λATz) 

+zt–z

–tut+ ( –t)Tzt–λ(ATzt–λATz) –zt 

.

SinceI–λA/( –t) is nonexpansive, we have

tut+ ( –t)Tzt–λATzt– (Tz–λATz)

=( –t)Tzt–λATzt/( –t)

–Tz–λATz/( –t)+t(ut–z)

≤( –t)Tzt–λATzt/( –t)

–Tz–λATz/( –t)+tut–z

≤( –t)Tzt–Tz+tut–z

≤( –t)zt–z+tut–z.

Thus we have

zt–z≤  

( –t)zt–z+tut–z+zt–z

–tut+ ( –t)Tzt–zt–λ(ATzt–ATz) 

.

It follows that

≤tut–z–tut+ ( –t)Tzt–zt–λ(ATzt–ATz)

=tut–z–tut+ ( –t)Tzt–zt

+ λtut+ ( –t)Tzt–zt,ATzt–ATz

–λATzt–ATz

≤tut–z–tut+ ( –t)Tzt–zt

+ λtut+ ( –t)Tzt–ztATzt–ATz

and so

tut+ ( –t)Tzt–zt 

≤tut–z+ λtut+ ( –t)Tzt–ztATzt–Az.

SinceATzt–Az → by (.), we deduce

lim

t→+

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Therefore, we have

lim

t→+zt–Tzt= . (.)

From (.), it follows that

zt–z

≤( –t)

Tzt–

λ  –tATzt

–

Tz– λ

 –tATz

+t(ut–z)



= ( –t)

Tzt–

λ  –tATzt

–

Tz– λ

 –tATz



+ t( –t)

ut–z,

Tzt–

λ  –tATzt

–

Tz– λ

 –tATz

+tut–z

≤( –t)zt–z+ t( –t)

ut–z,Tzt– λ

 –t(ATzt–Az) –z

+tut–z

= ( – t)zt–z+ t ( –t)

ut–z,Tzt–z– λ

 –t(ATzt–Az)

+tut–z+zt–z

,

which implies that

zt–z≤

ut–z,Tzt–z– λ

+ t 

ut–z+zt–z

+tut–z

Tzt–z– λ

≤ z–u,z–Tzt+ λ

 –tut–zATzt–Az+

t+ut–u

M, (.)

whereMis a constant such that

sup ut–z+zt–z+ut–z

Tzt–z–

λ

 –t(ATzt–Az) :t∈

,  – λ α

≤M.

Next, we show that{zt}is relatively norm-compact ast→+. Assume that{tn} ⊂(, ) is a sequence such thattn→+ asn→ ∞. Putzn:=ztn. From (.), it follows that

zn–z

≤ z–u,z–Tzn+ λ  –tn

un–zATzn–Az+

tn+un–u

M (.)

for allz∈F(T)∩S(MEP). Since{zn}is bounded, without loss of generality, we may assume thatznx˜∈C. From (.), we have

lim

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We can use Lemma . to (.) to deducex˜∈F(T). Further, we show thatx˜ is also in

S(MEP). Sincezn=Tλ(tnun+ ( –tn)Tzn–λATzn) for anyy∈C, we have

F(zn,y) +ATzn,y–zn+  λ

y–zn,zn–

tnun+ ( –tn)Tzn

≥.

From (C), it follows that

ATzn,y–zn+  λ

y–zn,zn–

≥F(y,zn). (.)

Putxt=ty+ ( –t)x˜ for allt∈(,  –_λ_α) andy∈C. Then we havext∈Cand so, from (.), it follows that

xt–zn,Axt ≥ xt–zn,Axt–xt–zn,ATzn

– λ

xt–zn,zn–

+F(xt,zn)

=xt–zn,Axt–Azn+xt–zn,Azn–ATzn

– λ

xt–zn,zn–

+F(xt,zn).

Sincezn–Tzn →, we haveAzn–ATzn →. Further, sinceAis monotone, we have

xt–zn,Axt–Azn ≥. So, from (C), it follows that, asn→ ∞,

xt–x˜,Axt ≥F(xt,x˜). (.)

Also, it follows from (C), (C), and (.) that

 =F(xt,xt)

≤tF(xt,y) + ( –t)F(xt,x˜)

≤tF(xt,y) + ( –t)xt–x˜,Axt

=tF(xt,y) + ( –t)ty–x˜,Axt

and hence

≤F(xt,y) + ( –t)y–x˜,Axt.

Lettingt→, we have

≤F(x˜,y) +y–x˜,Ax˜

for ally∈C. This impliesx˜∈EP. Therefore, we can substitutex˜forzin (.) to get

zn–x˜≤ ˜x–u,x˜–Tzn+ λ  –tn

un–x˜ATzn–Ax˜

+tn+un–x˜

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for all x˜∈F(T)∩S(MEP). By (.), we know thatATzn–Az → for anyz∈F(T)∩

S(MEP). Then we getATzn–Ax˜ →. Consequently, the weak convergence of{zn}(and

{Tzn}) tox˜actually implies thatzn→ ˜x. This proves the relative norm-compactness of the net{zt}ast→+.

Now, we return to (.) and take the limit asn→ ∞to get

˜x–z≤ z–u,z–x˜

for allz∈F(T)∩S(MEP). Equivalently, we have

u–x˜,z–x˜ ≤

for allz∈F(T)∩S(MEP). This clearly implies that

˜

x=PF(T)∩S(MEP)(u).

Therefore,x˜is the unique cluster point of the net{zt}. Hence the whole net{zt}converges strongly tox˜=PF(T)∩S(MEP)(u). This completes the proof.

4 Induced algorithms and corollaries (I) TakingT=Iin (.), we get the following.

Algorithm . For anyt∈(,  –_λ_α), deﬁne a net{zt} ⊂Cby the implicit manner:

F(zt,y) +  λ

y–zt,zt–

tut+ ( –t)zt–λAzt

≥ (.)

for ally∈C.

Corollary . Suppose that S(MEP)=∅. Then the net{zt} deﬁned by (.) converges

strongly as t→+to PS(MEP)(u).

(II) TakingF=  in (.), we get the following.

Algorithm . For anyt∈(,  –_λ_α), deﬁne a net{zt} ⊂Cby the implicit manner:

y–zt,zt–

tut+ ( –t)zt–λAzt

≥ (.)

for ally∈C.

Corollary . Suppose that S(VI)=∅.Then the net{zt}deﬁned by(.)converges strongly

as t→+to PS(VI)(u).

(III) TakingA=  in (.), we get the following.

Algorithm . For anyt∈(, ), deﬁne a net{zt} ⊂Cby the implicit manner:

F(zt,y) +

t

λy–zt,zt–ut ≥ (.)

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Corollary . Suppose that S(EP)=∅.Then the net{zt}deﬁned by(.)converges strongly

as t→+to PS(EP)(u).

5 Minimum-norm solutions

In many problems, one needs to ﬁnd a solution with the minimum norm. In an abstract way, we may formulate such problems as ﬁnding a pointx†with the property:

x†∈C, x†=min

x∈Cx _,

whereCis a nonempty closed convex subset of a real Hilbert spaceH. In other words,x†

is the (nearest point or metric) projection of the origin ontoC, that is,

x†=PC(),

wherePCis the metric (or nearest point) projection fromHontoC.

A typical example is the least-squares solution of the constrained linear inverse problem:

⎧ ⎨ ⎩

Ax=b,

x∈C,

whereAis a bounded linear operator fromH to another real Hilbert spaceH andbis a given point inH. Theleast-squares solutionis the least-norm minimizer of the mini-mization problem:

min

x∈CAx–b _.

Motivated by the above least-squares solution of the constrained linear inverse prob-lems, we study the general case of ﬁnding the minimum-norm solutions for the mixed equilibrium problem (.), the equilibrium problem (.), the variational inequality (.), and the ﬁxed point problem.

Now, we state our algorithms which can be inducted from the above section. (I) Takingut=  for alltin (.), we get the following.

Algorithm . For anyt∈(,  – λ

α), deﬁne a net{zt} ⊂Cby the implicit manner:

F(zt,y) +  λ

y–zt,zt–

( –t)Tzt–λATzt

≥ (.)

for ally∈C.

Corollary . Suppose that F(T)∩S(MEP)= ∅.Then the net {zt}deﬁned by(.)

con-verges strongly as t →+ to PF(T)∩S(MEP)(), which is the minimum-norm element in

F(T)∩S(MEP).

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Algorithm . For anyt∈(,  –_λ_α), deﬁne a net{zt} ⊂Cby the implicit manner:

F(zt,y) +  λ

y–zt,zt–

( –t)zt–λAzt

≥ (.)

for ally∈C.

Corollary . Suppose that S(MEP)=∅. Then the net {zt} deﬁned by (.) converges

strongly as t→+to PS(MEP)(),which is the minimum-norm element in S(MEP).

(III) Takingut=  for alltin (.), we get the following.

Algorithm . For anyt∈(,  –_λ_α), deﬁne a net{zt} ⊂Cby the implicit manner:

y–zt,zt–

( –t)zt–λAzt

≥ (.)

for ally∈C.

Corollary . Suppose that S(VI)=∅.Then the net{zt}deﬁned by(.)converges strongly

as t→+to PS(VI)(),which is the minimum-norm element in S(VI).

(IV) TakingA=  in (.), we get the following.

Algorithm . For anyt∈(, ), deﬁne a net{zt} ⊂Cby the implicit manner:

F(zt,y) +  λ

y–zt,zt– ( –t)Tzt

≥ (.)

for ally∈C.

Corollary . Suppose that F(T)∩S(EP)= ∅.Then the net{zt}deﬁned by(.)converges

strongly as t→+to PF(T)∩S(EP)(),which is the minimum-norm element in F(T)∩S(EP).

(V) TakingA=  in (.), we get the following.

Algorithm . For anyt∈(, ), deﬁne a net{zt} ⊂Cby the implicit manner:

F(zt,y) +

t

λy–zt,zt ≥ (.)

for ally∈C.

Corollary . Suppose that S(EP)=∅. Then the net {zt} deﬁned by (.) converges

strongly as t→+to PS(EP)(),which is the minimum-norm element in S(EP).

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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Author details

1_{Department of Mathematics, Tianjin Polytechnic University, Tianjin, 300387, China.}2_{Department of Mathematics}

Education and RINS, Gyeongsang National University, Chinju, 660-701, Korea.3_{Department of Mathematics, King}

Abdulaziz University, P.O. Box 80203, Jeddah, Saudi Arabia.4_{Department of Information Management, Cheng Shiu}

University, Kaohsiung, 833, Taiwan.5_{Department of Mathematics, Texas A&M University, Kingsville, USA.}

Acknowledgements

The ﬁrst author was supported in part by NSFC 11071279 and NSFC 71161001-G0105, the second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant Number: NRF-2013053358) and the third author was supported in part by NSC 101-2628-E-230-001-MY3 and NSC 101-2622-E-230-005-CC3.

Received: 25 February 2014 Accepted: 15 August 2014 Published: 2 September 2014 References

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doi:10.1186/1029-242X-2014-334