Split hierarchical variational inequality problems and related problems

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R E S E A R C H

Open Access

Split hierarchical variational inequality

problems and related problems

Qamrul Hasan Ansari

1,2

_{, Nimit Nimana}

3

_{and Narin Petrot}

3,4*

*_{Correspondence: narinp@nu.ac.th} 3_{Department of Mathematics,}

Faculty of Science, Naresuan University, Phitsanulok, 65000, Thailand

4_{Centre of Excellence in}

Mathematics, CHE, Si Ayutthaya Rd., Bangkok, 10400, Thailand Full list of author information is available at the end of the article

Abstract

The main objective of this paper is to introduce a split hierarchical variational inequality problem. Several related problems are also considered. We propose an iterative method for ﬁnding a solution of our problem. The weak convergence of the sequence generated by the proposed method is studied.

MSC: 47H09; 47H10; 47J25; 49J40; 65K10

Keywords: split hierarchical variational inequality problem; split common ﬁxed point problem; split convex minimization problem; strongly nonexpansive operators; cutter operators

1 Introduction

In , Censor and Elfving [] introduced the following split feasibility problem (in short, SFP): ﬁnd a point

x∗∈C such that Ax∗∈Q, (.)

whereCandQare nonempty closed convex subsets ofRn_and_Rm_{, respectively, and}_A_is

anm×nmatrix. They proposed an algorithm to ﬁnd the solution of SFP. Their algorithm did not become popular, since it concerns the complicated matrix inverse computations and, subsequently, is considered in the case whenn=m. Based on these observations, Byrne [] applied the forward-backward method, a type of projected gradient method, presenting the so-called CQ-iterative procedure, which is deﬁned by

xk+=PC

xk+γA(PQ–I)Axk

, ∀k≥,

where an initialx∈Rn_, _γ _∈_{(, /}_A_{), and}_P

C andPQdenote the metric projections

ontoCandQ, respectively. The convergence result of the sequence{xk}∞_k_=to a solution of the considered split feasibility problem was presented. Further, Byrne also proposed an application to dynamic emission tomographic image reconstruction. A few years later, Censoret al.[, ] proposed an incredible application of the split feasibility problem to the inverse problem of intensity-modulated radiation therapy treatment planning. Recently, Xu [] considered SFP in the setting of inﬁnite-dimensional Hilbert spaces and established the following CQ-algorithm: LetHandHbe real Hilbert spaces andA:H→Hbe a

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bounded linear operator. For a givenx∈H, generated a sequence{xk}∞_k_=by the following iterative scheme:

xk+=PC

xk+γA∗(PQ–I)Axk

, ∀k≥,

whereγ ∈(, /A_{), and}_A∗_{is the adjoint operator of}_{A. He also proved the weak} con-vergence of the sequence produced by the above procedure to a solution of the SFP. The details of the CQ-algorithm for SFP problem are given in []. A comprehensive literature, survey, and references on SFP can be found in [].

In , Censor and Segal [] presented an important form of the split feasibility prob-lem called the split common ﬁxed point probprob-lem, which is to ﬁnd a point

x∗∈Fix(T) such that Ax∗∈Fix(S), (.)

whereTandSare some nonlinear operators onRn_and_Rm_{, respectively, and}_A_{is a real}

m×nmatrix. Based on the properties of operatorsT andS, called cutter or directed operators, they presented the following algorithm for solving the split common ﬁxed point problem:

xk+=T

xk+γA(S–I)Axk

, ∀k≥, (.)

where an initial x∈Rn, γ ∈(, /A). They also presented a convergence result for this algorithm. Moudafi [] studied the split common fixed point problem in the context of the demicontractive operatorsT andS in the setting of infinite-dimensional Hilbert spaces. He established the weak convergence of the sequence generated by his scheme to a solution of the split common fixed point problem.

On the other hand, the theory of variational inequalities is well known and well devel-oped because of its applications in different areas of science, social science, engineering, and management. There are several monographs on variational inequalities, but we men-tion here a few [–]. LetCbe a nonempty closed convex subset of a real Hilbert space Handf :H→Hbe an operator. The variational inequality problem defined byCandf is to findx∗∈Csuch that

(VIP(C;f)) fx∗,x–x∗≥, for allx∈C.

Another problem closely related toVIP(C;f) is known as the Minty variational inequality problem: ﬁndx∗∈Csuch that

(MVIP(C;f)) f(x),x–x∗≥, for allx∈C.

The trivial unlikeness of two proposed problems is the linearity of variational inequalities. In fact, the Minty variational inequalityMVIP(C;f) is linear but the variational inequality

VIP(C;f) is not. However, under the (hemi)continuity and monotonicity off, the solution sets of these problems are the same (see [, Lemma ]).

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LetT:H→H be a nonlinear operator with the set of ﬁxed pointsFix(T)=∅andf : H→H be an operator. The hierarchical variational inequality problem is to ﬁndx∗ ∈

Fix(T) such that

fx∗,x–x∗≥, for allx∈Fix(T). (.)

A closely related problem to hierarchical variational inequality problem is the hierarchical Minty variational inequality problem: ﬁndx∗∈Fix(T) such that

f(x),x–x∗≥, for allx∈Fix(T). (.)

In the recent past, several methods for solving hierarchical variational inequalities have been investigated in the literature; see, for example, [–] and the references therein.

The goal of this paper is to introduce a split-type problem, by combining a split ﬁxed point problem and a hierarchical variational inequality problem. The considered problem can be applied to solve many existing problems. We present an iterative procedure for ﬁnd-ing a solution of the proposed problem and show that under some suitable assumptions, the sequence generated by our algorithm converges weakly to a solution of the considered problem.

The rest of this paper is divided into four sections. In Section , we recall and state preliminaries on numerous nonlinear operators and their useful properties. Section  we divide in two subsections, that is, ﬁrst, we present a split problem, called the split hierar-chical Minty variational inequality problem. We subsequently propose an algorithm for solving our problem and establish a convergence result under some assumptions. Second, we present the split hierarchical variational inequality problem. In the last section, we investigate some related problems, where we can apply the considered problem.

2 Preliminaries

LetH be a real Hilbert space whose inner product and norm are denoted by·,·and

· , respectively. The strong convergence and weak convergence of a sequence {xk}∞_k_= tox∈Hare denoted byxk→xandxkx, respectively. LetT :H→Hbe an operator.

We denote byR(T) the range of T, and byFix(T) the set of all ﬁxed points ofT, that is,Fix(T) ={x∈H:x=Tx}. The operatorTis said to benonexpansiveif for allx,y∈H,

Tx–Ty ≤ x–y;strongly nonexpansive[, ] ifTis nonexpansive and for all bounded sequences{xk}∞_k_=,{yk}∞_k_=inH, the conditionlim_k_→∞(xk–yk–Txk–Tyk) =  implies limk→∞(xk–yk) – (Txk–Tyk)= ;averaged nonexpansiveif for allx,y∈H,T = ( –

α)I+αSholds for a nonexpansive operatorS:H→Handα∈(, );ﬁrmly nonexpansive if T–Iis nonexpansive, or equivalently for allx,y∈H,Tx–Ty_≤_x_–y,_Tx_–Ty_;_cutter [] ifx–Tx,z–Tx ≤ for allx∈Hand allz∈Fix(T);monotoneifTx–Ty,x–y ≥, for allx,y∈H;α-inverse strongly monotone(orα-cocoercive) if there exists a positive real numberαsuch thatTx–Ty,x–y ≥αTx–Ty, for allx,y∈H.

LetB:H→H _{be a set-valued operator, we deﬁne a graph of}_B_by_{_(x,_y)_∈_H_×_H_:

y∈B(x)}and an inverse operator ofB, denotedB–_{, by}_{_(y,_x)_∈_H_×_H_:_x_∈_B(y)_}_{. A} set-valued operator B:H→H_{is called}_monotone_{if for all}_x,_y_∈_H,_u_∈_{B(x) and}_v_∈_B(y)

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ofB. For a maximal monotone operatorB, we know that for eachx∈Hand a positive real numberσ, there is a uniquez∈Hsuch thatx∈(I+σB)z. We deﬁne theresolventofB with parameterσbyJB

σ := (I+σB)–. It is well known that the resolvent is a single-valued and ﬁrmly nonexpansive operator.

Remark . It can be seen that the class of averaged nonexpansive operators is a proper subclass of the class of strongly nonexpansive operators. Since any firmly nonexpansive operator is an averaged nonexpansive operator, it is clear that a class of firmly nonex-pansive operators is contained in the class of strongly nonexnonex-pansive operators. Further, a firmly nonexpansive operator with fixed point is cutter. However, a nonexpansive cutter operator need not to be firmly nonexpansive; for further details, see [].

The following well-known lemma is due to Opial [].

Lemma .(Demiclosedness principle) [, Lemma ] Let C be a nonempty closed convex

subset of a real Hilbert space H and T:C→H be a nonexpansive operator.If the sequence

{xk}∞_k_=⊆C converges weakly to an element x∈C and the sequence{xk–Txk}∞k=converges strongly to,then x is a ﬁxed point of the operator T.

To prove the main theorem of this paper, we need the following lemma.

Lemma .[, Section .., Lemma ] Assume that{ak}∞_k_=and{bk}∞_k_=are nonnegative real sequences such that ak+≤ak+bk.If

∞

k=bk<∞,thenlimk→∞akexists.

The following lemma can be immediately obtained by the properties of an inner product.

Lemma . Let H be a real Hilbert space H.Then,for all x,y∈H,

x–y≤ x+ y,y–x.

3 Split hierarchical variational inequality problems and convergent results

In this section we introduce split hierarchical variational inequality problems and discuss some related problems. Further, we propose an iterative method for ﬁnding a solution of the hierarchical variational inequality problem and prove the convergence result for the sequence generated by the proposed iterative method.

3.1 Split hierarchical Minty variational inequality problem

LetHandHbe two real Hilbert spaces,f,T:H→Hbe operators such thatFix(T)=∅, andh,S:H→Hbe operators such thatFix(S)=∅. LetA:H→Hbe an operator with R(A)∩Fix(S)=∅. Thesplit hierarchical Minty variational inequality problem(in short, SHMVIP) is to ﬁndx∗∈Fix(T) such that

f(x),x–x∗≥, for allx∈Fix(T), (.)

and such thatAx∗∈Fix(S) satisﬁes

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The solution set of SHMVIP (.)-(.) is denoted by, that is,

:=xwhich solves (.) :Axsolves (.).

We note that the problems (.) and (.) are nothing but hierarchical Minty variational inequality problems. Iff andhare zero operators, that is,f ≡h≡, then SHMVIP (.) and (.) reduces to the split ﬁxed point problem (.). Moreover, SHMVIP (.) and (.) can be applied to several existing split-type problems, which we will discuss in Section . Inspired by the iterative scheme (.) for solving split common ﬁxed point problem (.) and the existing algorithms, a generalization of the projected gradient method, for solving the hierarchical variational inequality problem of Yamada [] and Iiduka [], we now present an iterative algorithm for solving HMVIP (.)-(.).

Algorithm .

Initialization: Choose{αk}∞_k_=,{βk}∞_k_=⊂(, +∞). Take arbitraryx∈H. Iterative Step: For a given current iteratexk∈H, compute

yk:=xk–γA∗

I–S(I–βkh)

Axk,

whereγ ∈(,_A) and deﬁnexk+∈Hby

xk+:=T(I–αkf)yk.

Updatek:=k+ .

Rest of the section, unless otherwise speciﬁed, we assume thatT is a strongly nonex-pansive operator onHwithFix(T)=∅andSis a strongly nonexpansive cutter operator onHwithFix(S)=∅,f (respectively,h) is a monotone and continuous operator onH (respectively,H) andA:H→His a bounded linear operator withR(A)∩Fix(S)=∅.

Remark . (i) The Algorithm . can be applied to the iterative scheme (.) by setting the operatorsf ≡h≡.

(ii) The iterative Algorithm . extends and develops the algorithms in [, Algo-rithm .] and [, AlgoAlgo-rithm ()] in many aspects. Indeed, the metric projectionsPC

andPQin [, Algorithm .] and the resolvent operatorsJλB andJ

B

λ in [, Algorithm ()] are ﬁrmly nonexpansive operators which are special cases of the assumption on the operators T andS. Second, its involves control sequences{αk}∞_k_= and{βk}∞_k_=, while in [, Algorithm .] and [, Algorithm ()] they are constant. Furthermore, we assume

γ ∈(,_A), while in [, Algorithm .] and [, Algorithm ()], it was assumed to be in

(,_A), which clearly is a more restrictive assumption.

(iii) By setting an operatorAto be zero operator, the iterative Algorithm . relates to existing iterative schemes for solving the hierarchical variational inequality problem in, for example, [, Algorithm .] and [, Algorithm ].

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Theorem . Let a sequence{xk}∞_k_=be generated by Algorithm.with x∈H,and let

{αk}∞_k_= and {βk}∞_k_=⊂(, ) be sequences such that∞_k_=αk<∞andlimk→∞βk= .If

=∅,then the following statements hold.

(i) If there exists a natural numberksuch that

⊂

∞

k=k

z∈H:

h(Axk),S(I–βkh)Axk–Az

≥,

then,for allk≥kandz∈,we have

yk–z≤ xk–z–γ

 –γAI–S(I–βkh)

Axk

 .

(ii) If the sequence{f(yk)}∞k=is bounded,thenlimk→∞xk–zexists for allz∈.

(iii) If the sequence{h(Axk)}∞_k_=is bounded,thenlimk→∞xk+–xk= ,

lim_k_→∞xk–Txk= ,andlimk→∞Axk–SAxk= .

(iv) Ifxk+–xk=o(αk),andαk=o(βk),then the sequence{xk}∞k=converges weakly to an element of.

Proof (i) Letz∈be given. Then

yk–z =xk–γA∗

I–S(I–βkh)

Axk–z

=xk–z+γA∗

I–S(I–βkh)

Axk

– γxk–z,A∗

I–S(I–βkh)

Axk

≤ xk–z+γAI–S(I–βkh)

Axk



– γxk–z,A∗

I–S(I–βkh)

Axk

, (.)

for allk≥. On the other hand, sinceSis a cutter operator, we have with

xk–z,A∗

S(I–βkh) –I

Axk

=Axk–Az,

S(I–βkh) –I

Axk

=S(I–βkh)(Axk) –Az,S(I–βkh)(Axk) –Axk

–S(I–βkh) –I

Axk

=S(I–βkh)(Axk) –Az,S(I–βkh)(Axk) – (I–βkh)Axk

–βk

S(I–βkh)(Axk) –Az,h(Axk)

–S(I–βkh) –I

Axk

≤–S(I–βkh) –I

Axk, (.)

for allk≥k. By using (.), the inequality (.) becomes

yk–z≤ xk–z+γAS(I–βkh) –I

Axk



– γS(I–βkh) –I

Axk



=xk–z–γ

 –γAS(I–βkh) –I

Axk

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for allk≥k, as required. Furthermore, sinceγ <_A, we observe thatγ( –γA) > ,

and hence

yk–z ≤ xk–z, ∀k≥k. (.)

(ii) LetM:=sup{f(yk):k≥}. Fork≥k, we havexk+–z ≤ xk–z+αkM. Since

_∞

k=αk<∞, by using Lemma ., we obtain the result thatlimk→∞xk–zexists.

(iii) We ﬁrst show thatlimk→∞xk–xk+= .

Letwk:=yk–αkf(yk) for allk≥. Note that{wk}∞_k_=is a bounded sequence, since for all

k≥kwe know thatwk–z ≤ yk–z+αkf(yk)and{yk}∞k=is a bounded sequence. By using the nonexpansiveness ofT and (.), we have

≤ wk–z–Twk–Tz ≤ xk–z+αkf(yk)–xk+–z, ∀k≥k.

Subsequently, by the existence oflim_k_→∞xk–zand the fact thatαk→, it follows that lim

k→∞

wk–z–Twk–Tz

= .

By the strong nonexpansiveness ofT and the boundedness of{wk}∞_k_=, we have

lim

k→∞wk–Twk= .

On the other hand, by (i), we observe that

xk+–z≤yk–αkf(yk) –z



≤ yk–z+αkf(yk)



+ αkf(yk)yk–z ≤ xk–z–γ

 –γAI–S(I–βkh)

Axk



+αkf(yk)



+ αkf(yk)yk–z,

which is equivalent to

γ –γAI–S(I–βkh)

Axk



≤ xk–z–xk+–z+αkf(yk)



+ αkf(yk)yk–z, (.)

for allk≥k. Taking the limit ask→ ∞, we get

lim

k→∞I–S(I–βkh)

Axk= , (.)

which implies, by the deﬁnition ofyk, that

lim

k→∞yk–xk= , (.)

and also

lim

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These together imply that

lim

k→∞xk–xk+ ≤klim→∞

xk–wk+wk–Twk

= , (.)

as desired.

We now show thatlimk→∞xk–Txk= .

Observe thatxk+–Tyk ≤αkf(yk) → ask→ ∞. Using this one together with (.),

(.) and the nonexpansiveness ofT, we obtain

lim

k→∞xk–Txk= . (.)

Next, we show thatlimk→∞Axk–SAxk= .

Setzk:=Axk–βkh(Axk) for allk≥. For everyk≥k, we observe that

≤ zk–Az–Szk–SAz ≤βkh(Axk)+Axk–Szk.

Thus, the boundedness of{h(Axk)}∞k=and (.) yield

lim k→∞

zk–Az–Szk–SAz

= ,

and hence, by the fact that the sequence{zk}∞_k_=is bounded andSis a strongly nonexpansive mapping, we have

lim

k→∞zk–Szk= . (.)

We note thatAxk–zk=βkh(Axk) → ask→ ∞. Also, we observe that zk–SAxk ≤ zk–Szk+S

Axk–βkh(Axk)

–SAxk≤ zk–Szk+βkh(Axk),

for allk≥k. By using (.) and taking the limit ask→ ∞, we get

lim

k→∞zk–SAxk= .

Hence, we get

lim

k→∞Axk–SAxk= , (.)

as required.

(iv) Since{xk}∞_k_=is a bounded sequence, there exist a subsequence{xkj}∞j=of{xk}∞k=and q∈Hsuch thatxkjq∈H. By (iii) and the demiclosed principle of the nonexpansive

operatorT, we obtainq∈Fix(T). We claim thatq∈Fix(T) solves (.). In fact, fork≥k, we have

xk+–z≤ yk–z+ αk

f(yk),z–wk

=xk–z+ αk

f(yk),z–yk

+ α_kf(yk)

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=xk–z+ αk

f(z),z–yk

– αk

f(yk) –f(z),yk–z

+ α_kf(yk)



≤ xk–z+ αk

f(z),z–yk

+ α_kf(yk)



, (.)

and then

f(z),yk–z

≤ 

αk

xk–z–xk+–z

+ αkf(yk)



≤ 

αk

xk–z+xk+–z

xk–z–xk+–z

+ αkf(yk)



≤ xk–xk+

αk

M+ αkf(yk),

whereM:=sup{xk–z+xk+–z:k≥}<∞. Sincexkjq,αk→, andxk+–xk= o(αk), by (.), we havef(z),q–z ≤ for allz∈Fix(T) which means thatq∈Fix(T)

solves (.).

Next, byAxkjAq∈Htogether with (iii) and the demiclosedness of the nonexpansive

operatorS, we know thatAq∈Fix(S). We show that suchAq∈Fix(S) solves (.). Sinceαk=o(βk), we may assume thatαk≤βkfor allk≥k. From (.), for allk≥k,

we have

γ –γAAxk–Szk≤ xk–xk+M+αkf(yk)



+ αkf(yk)yk–z,

whereM:=sup{xk–z+xk+–z:k≥k}. This implies that, fork≥k,

γ –γAAxk–Szk 

β_k ≤

xk–xk+

β_k M+ αk

β_kM≤

xk–xk+

αk

M+αk

β_kM,

where M:=max{M,sup_k_≥{f(yk)+ f(yk)yk–z}}. Subsequently, sincexk+– xk=o(αk),αk=o(βk), andγ( –γA) > , we have

lim k→∞

Axk–Szk

βk

= . (.)

Fork≥k, we compute

Szk–SAz≤Axk–βkh(Axk) –Az



≤ Axk–Az+ βk

h(Axk),Az–zk

=Axk–Az+ βk

h(Axk),Az–Axk

+ β_kh(Axk)



=Axk–Az– βk

Axk–Az,h(Axk) –h(Az)

+ βk

h(Az),Az–Axk

+ β_kh(Axk)



≤ Axk–Az+βkh(Axk) –h(Az)

 + βk

h(Az),Az–Axk

+ β_kh(Axk)



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This gives

h(Az),Axk–Az

≤ 

βk

Axk–Az–Szk–SAz

+βkh(Axk) –h(Az)

+ βkh(Axk)



≤Axk–Szk

βk

M+βkh(Axk) –h(Az)



+ βkh(Axk)

 ,

whereM:=sup{Axk–Az+Szk–SAz:k≥}<∞. Using this together withAxkj

Aq∈Fix(S) and (.), we obtainh(Az),Aq–Az ≤ for allAz∈Fix(S). Thus,Aq∈Fix(S) solves (.), and therefore,q∈.

Finally, it remains to show thatxkq∈. By the boundedness of{xk}∞k=, it suﬃces to show that there is no subsequence{xki}∞_i_=of{xk}∞_k_=such thatxkip∈Handp=q.

Indeed, if this is not true, the well-known Opial theorem would imply

lim

k→∞xk–p=jlim→∞xkj–q

< lim

j→∞xkj–p

= lim

k→∞xk–p

= lim

i→∞xki–p < lim

i→∞xki–q

= lim

k→∞xk–p,

which leads to a contradiction. Therefore, the sequence{xk}∞_k_=converges weakly to a point

q∈.

3.2 Split hierarchical variational inequality problems

We consider another kind of split hierarchical variational inequality problem (in short, SHVIP) in which we consider a variational inequality formulation instead of the Minty variational inequality. More precisely, we consider the followingsplit hierarchical varia-tional inequality problem:ﬁnd a pointx∗∈Fix(T) such that

fx∗,x–x∗≥, for allx∈Fix(T), (.)

such thatAx∗∈Fix(S) and it satisﬁes

hAx∗,y–Ax∗≥, for ally∈Fix(S). (.)

The solution set of the SHVIP is denoted by . SinceFix(T) andFix(S) are nonempty closed convex and f and h are monotone and continuous, by the Minty lemma [, Lemma ], SHVIP (.)-(.) and SHMVIP (.)-(.) are equivalent. Hence, Algo-rithm . and Theorem . are also applicable for SHVIP (.)-(.).

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Remark . It is worth to note that, in the context of SHVIP (.)-(.), if we assume either the ﬁnite dimensional settings ofH andH or the compactness of spacesHand H, then the monotonicity assumptions of operatorsf andhcan be omitted.

In fact, as in the statement and the proof of Theorem ., since we know thatxkjq∈ Fix(T), we also havexkj→q∈Fix(T). Recalling the inequality (.), we have, fork≥k,

xk+–z≤ xk–z+ αk

f(yk),z–yk

+ α_kf(yk)

 ,

f(yk),yk–z

≤xk–xk+

αk

M+ αkf(yk)

 ,

whereM:=sup{xk–z+xk+–z:k≥}<∞. Since we know thatykj→q∈Fix(T) and

f is continuous, by approachingjto inﬁnity, we obtainf(q),q–z ≤, for allz∈Fix(T), which means thatq∈Fix(T) solves (.). Similarly, from the inequality (.), we note that, fork≥k,

Szk–SAz≤ Axk–Az+ βk

h(Axk),Az–Axk

+ β_kh(Axk)

 ,

h(Axk),Axk–Az

≤Axk–Szk

βk

M+ βkh(Axk)

 ,

whereM:=sup{Axk–Az+Szk–SAz:k≥}<∞. SinceAxkj →Aq∈Fix(S) and

his continuous, we also geth(Aq),Aq–Az ≤, for allAz∈Fix(S), which means that Aq∈Fix(S) solves (.), and subsequently,q∈ .

Remark . The strong nonexpansiveness of operatorsT andSin Theorem . can be

applied to the cases when the operatorsT andSare not only ﬁrmly nonexpansive, but with relaxation of being ﬁrmly nonexpansive and averaged nonexpansive but also strictly nonexpansive, that is, Tx–Ty<x–yor x–y=Tx–Tyfor allx,y; regarding the compactness of the spacesHandH, for more details, see [, Remark ..].

4 Some related problems

In this section, we present some split-type problems which are special cases of SHVIP (.) and (.) and can be solved by using Algorithm ..

4.1 A split convex minimization problem

Letφ:H→Randϕ:H→Rbe convex continuously diﬀerentiable functions andA: H→Hbe a bounded linear operator such thatA(H)∩Fix(S)=∅. Consider the following split convex minimization problem(in short, SCMP): ﬁnd

x∗∈Fix(T) such that x∗=argmin x∈Fix(T)

φ(x), (.)

and such that

Ax∗∈Fix(S) such that Ax∗=argmin y∈Fix(S)

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We know that the SCMP (.)-(.) can be formulated as the following split hierarchical variational inequality problem: ﬁnd a point

x∗∈Fix(T) such that ∇φx∗,x–x∗≥, for allx∈Fix(T), and such that the point

Ax∗∈Fix(S) such that ∇ϕAx∗,y–Ax∗≥, for ally∈Fix(S),

where ∇φ and∇ϕ denote the gradient ofφ andϕ, respectively. Since∇φ and∇ϕ are monotone [, Proposition .] and continuous, we can apply Algorithm . to obtain the solution of SCMP (.)-(.), and Theorem . will provide the convergence of the sequence{xk}∞_k_=to a solution of SCMP (.)-(.). Furthermore, in the context of ﬁnite-dimensional cases, we can remove the convexity ofφandϕ.

4.2 A split variational inequality problem over the solution set of monotone variational inclusion problem

LetHandHbe two real Hilbert spaces,φ:H→H,ϕ:H→HbeL, (respectively, L)-inverse strongly monotone operators,B :H→H_, _B_:_H →_H _{are set-valued}

maximal monotone operators. Let us consider the followingmonotone variational inclu-sion problem(in short, MVIP): ﬁnd

x∗∈H such that ∈φx∗+Bx∗. We denote the solution set of MVIP bySOL(φ,B).

Forσ> ,λ∈(, L), we know that the operatorJσB(I–λφ) is an averaged nonexpansive operator and

FixJB

σ (I–λφ)

=SOL(φ,B), whereJB

σ represents the resolvent ofBwith parameterσ.

We consider the following split variational inequality problem: ﬁndx∗∈SOL(φ,B) such that

f(x),x–x∗≥, for allx∈SOL(φ,B), (.)

and such thatAx∗∈SOL(ϕ,B) satisﬁes

h(y),y–Ax∗≥, for ally∈SOL(ϕ,B). (.) By using these facts and adding the assumption thatJB

σ (I–λϕ) is cutter, we can apply Algorithm . and Theorem . to obtain the solution of SVIP (.)-(.).

4.3 A split variational inequality problem over the solution set of equilibrium problem

LetHandHbe real Hilbert spaces,C⊆HandQ⊂Hbe nonempty closed convex sets,

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andφis the problem of ﬁndingx∗∈Csuch that

φx∗,x≥, for allx∈C,

and we denote the solution set of equilibrium problem ofCandφbyEP(C,φ).

Recall that Blum and Oettli [] showed that the bifunctionφ satisﬁes the following conditions:

(A) φ(x,x) = for allx∈C;

(A) φis monotone, that is,φ(x,y) +φ(y,x)≤for allx,y∈C; (A) for allx,y,z∈C,

lim sup t↓ φ

tz+ ( –t)x,y≤φ(x,y);

(A) for allx∈C,φ(x,·)is a convex and lower semicontinuous function,

and letr>  andz∈H, then the set

Tr(z) :=

x∗∈C:φx∗,x+ r

x–x∗,x∗–x≥ for allx∈C

=∅.

Moreover, from [], we know that

(i) Tris single-valued;

(ii) Tris ﬁrmly nonexpansive;

(iii) Fix(Tr) =EP(C;φ).

We now consider the following split variational inequality problem: ﬁndx∗∈EP(C,φ) such that

f(x),x–x∗≥, for allx∈EP(C,φ), (.)

and such thatAx∗∈EP(Q,ϕ) satisﬁes

h(y),y–Ax∗≥, for ally∈EP(Q,ϕ). (.) Similarly, by assuming that conditions (A)-(A) in the context of the equilibrium problem deﬁned byQandϕhold, we see that, for allu∈Hands> , the set

Ts(u) :=

y∗∈Q:ϕy∗,y+ s

y–y∗,y∗–u≥ for ally∈Q

is a nonempty set. Moreover, we also see thatTsis single-valued and ﬁrmly nonexpansive;

andFix(Ts) =EP(Q;ϕ).

By using these facts, we obtain the result that the problem (.)-(.) forms a special case of the SHVIP (.)-(.).

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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Author details

1_{Department of Mathematics, Aligarh Muslim University, Aligarh, 202 002, India.}2_{Department of Mathematics and}

Statistics, King Fahd University of Petroleum & Minerals, Dhahran, Saudi Arabia.3_{Department of Mathematics, Faculty of}

Science, Naresuan University, Phitsanulok, 65000, Thailand. 4_{Centre of Excellence in Mathematics, CHE, Si Ayutthaya Rd.,}

Bangkok, 10400, Thailand.

Acknowledgements

The authors would like to thank the anonymous referees for their careful reading and suggestions, which allowed us to improve the ﬁrst version of this paper. This research is partially support by the Center of Excellence in Mathematics the Commission on Higher Education, Thailand. QH Ansari was partially supported by a KFUPM Funded Research Project No. IN121037. N Nimana was supported by the Thailand Research Fund through the Royal Golden Jubilee PhD Program (Grant No. PHD/0079/2554).

Received: 14 May 2014 Accepted: 16 September 2014 Published: 13 October 2014 References

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