Mathematical programming with multiple sets split monotone variational inclusion constraints

R E S E A R C H

Open Access

Mathematical programming with multiple

sets split monotone variational inclusion

constraints

Zenn-Tsun Yu

_{, Lai-Jiu Lin}

_{and Chih-Sheng Chuang}

*_{Correspondence:}

[email protected]

2_{Department of Mathematics,}

National Changhua University of Education, Changhua, 50058, Taiwan

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

Abstract

In this paper, we first study a hierarchical problem of Baillon’s type, and we study a strong convergence theorem of this problem. For the special case of this

convergence theorem, we obtain a strong convergence theorem for the ergodic theorem of Baillon’s type. Our result of the ergodic theorem of Baillon’s type improves and generalizes many existence theorems of this type of problem. Two numerical examples are given to demonstrate our results.

As applications of our convergence theorem of the hierarchical problem, we study the unique solution for the following problems: mathematical programming with multiply sets split variational inclusion and fixed point set constraints; mathematical programming with multiple sets split variational inequalities and fixed point set constraints; the variational inequality problem with a system of mixed type equilibria and fixed point set constraints; the variational inequality problem with multiple sets split system of mixed type equilibria and fixed point set constraints; mathematical programming with a system of mixed type equilibria and fixed point set constraints. We give iteration processes for these types of problems and establish the strong convergence for the unique solution of these problems. For our special case, our results can be reduced to the following problems: the unique minimal norm solution of the multiply sets split monotonic variational inclusion problems; the minimum norm solutions for the multiple sets split system of mixed type equilibria problem; the minimum norm solution of the system of mixed type equilibria problem. Our results will have many applications in diverse fields of science.

Keywords: hierarchical problems; split variational inclusion problems; fixed point problems; mathematical programming; minimum norm solution

1 Introduction

LetC,C, . . . ,Cm be nonempty closed convex subsets of a Hilbert spaceH. The

well-known convex feasiblity problem (CFP) is to findx∗∈Hsuch that

x∗∈C∩C∩ · · · ∩Cm.

The split feasibility problem (SFP) is to find a point

x∗∈Csuch thatAx∗∈Q,

whereCis a nonempty closed convex subset of a Hilbert spaceH,Qis a nonempty closed

convex subset of a Hilbert spaceH, andA:H→His an operation. The split feasibility

problem (SFP) in the finite dimensional Hilbert spaces was first introduced by Censoret al. [] for modeling inverse problems which arise from phase retrievals and in medical image reconstruction. Since then, the convex feasibility problem and the split feasibility prob-lem (SFP) has received much attention due to its applications in signal processing, image reconstruction, approximation theory, control theory, biomedical engineering, commu-nications, and geophysics. For example, one can refer to [–] and related literature.

Let C,C, . . . ,Cm be nonempty closed convex subsets of a Hilbert space H, let

Q,Q, . . . ,Qn be nonempty closed convex setsH and letA,A, . . . ,Am:H→H be

linear operator operators. The well-known multiple sets split feasibility problem studied by Censoret al.[]. Xu [] and Lopezet al.[] (MSSFP) is to findx∗∈Hsuch that

x∗∈Cisuch thatAix∗∈Qi for alli= , , . . . ,m.

In , Moudafi [] introduced and studied the following split monotone variational inclusion (SMVI):

Findx¯∈Hsuch thatx¯∈(B+G)–, ()

and

¯

y=Ax¯∈Hsuch that¯y∈(B+G)–, ()

whereHandH are real Hilbert spaces,A:H→His a bounded linear operator,B:

H→HandB:H→Hare given operators,G:HHandG:HHare given

multivalued mappings.

Moudafi [] proved a weakly convergence theorem for the solution of the split monotone variational inclusion (SMVI) with an iteration process.

In , Maruyamaet al.[] proved the following ergodic theorem of Baillon’s type [].

Theorem .[] Let C be a nonempty closed convex subset of a real Hilbert space H, T:C→C be a-generalized hybrid mapping withFix(T)=∅and let PFix(T)be the metric

projection of HontoFix(T).Then,for any x∈C,

Snx:=  n

n–

k=

Tkx

converges weakly to an element p ofFix(T),where p=limn→∞PFix(T)Tnx.

As applications of our convergence theorem of the hierarchical problem, we study the unique solution for the following problems: mathematical programming with multiply sets split variational inclusion and a fixed point set constraints; mathematical program-ming with multiple sets split variational inequalities and fixed point set constraints; the variational inequality problem with a system of mixed type equilibria and fixed point set constraints; the variational inequality problem with multiple sets split system of mixed type equilibria and a fixed point set constraints; mathematical programming with system of mixed type equilibrium and a fixed point set constraints. We give iteration processes for these types of problems and establish the strong convergence for the unique solution of these problems. For the special case of our results, our results can be reduced to the fol-lowing problems: the unique minimal norm solution of the multiply sets split monotonic variational inclusion problems; the minimum norm solutions for the multiple sets split system of mixed type equilibrium problem; the minimum norm solution of the system of the mixed type equilibria problem. Our results will have many applications in diverse fields of science.

2 Preliminaries

Throughout this paper, letNbe the set of positive integers and letRbe the set of real numbers,Hbe a (real) Hilbert space with inner product ·,·and norm · , respectively,

andCbe a nonempty closed convex subset ofH. We denote the strongly convergence

and the weak convergence of{xn}tox∈Hbyxn→xandxnx, respectively.

LetT :C→Hbe a mapping, and letFix(T) :={x∈C:Tx=x}denote the set of fixed

points ofT. A mappingT:C→His called

(i) a-generalized hybrid mapping [] if there existδ,δ,,∈Rsuch that

δTx–Ty+δTx–Ty+ ( –δ–δ)x–Ty

≤Tx–y+Tx–y+ ( ––)x–y

for allx,y∈C.

We know that the class of -generalized hybrid mapping contains the classes of nonexpan-sive mappings, nonspreading mappings, and a (α,β)-generalized hybrid [] in a Hilbert space. We give an example for a -generalized hybrid mapping.

Example .[] LetT: [, ]→Rbe defined as

Tx=

 ifx∈[, ),  ifx= .

ThenT is a -generalized hybrid mapping andFix(T) ={}. Proof Let==_,δ=δ=_.

Case : Ifx∈[, ),y= , thenTx=Tx= ,Ty=  andx–Ty ≤. We know that

δTx–Ty 

+δTx–Ty+ ( –δ–δ)x–Ty

=δ+δ+ ( –δ–δ)x–Ty

and

Tx–y+Tx–y+ ( ––)x–y

= + + ( ––)x– 

≥+ ≥ +  = .

Therefore,

δTx–Ty 

+δTx–Ty+ ( –δ–δ)x–Ty

≤Tx–y



+Tx–y+ ( ––)x–y.

Case : Ifx∈[, ),y∈[, ), thenTx=Tx= ,Ty=Ty= . We know that

δTx–Ty 

+δTx–Ty+ ( –δ–δ)x–Ty

= ( –δ–δ)x

= 

≤Tx–y



+Tx–y+ ( ––)x–y.

Case : Ifx=y= , thenTx= ,T_{x= ,Ty= ,T}_{y= . We know that}

δTx–Ty+δTx–Ty+ ( –δ–δ)x–Ty

=δ+ ( –δ–δ)

= ( –δ) =

 

and

Tx–y 

+Tx–y+ ( ––)x–y

= +≥

 .

Therefore,

δTx–Ty+δTx–Ty+ ( –δ–δ)x–Ty

≤Tx–y+Tx–y+ ( ––)x–y.

By the above case, we know thatT is a -generalized hybrid.

A mappingV:H→His called

(i) strongly monotone if there existsγ¯> such that x–y,Vx–Vy ≥ ¯γx–y_{for all}

x,y∈H;

(ii) α-inverse-strongly monotone if x–y,Vx–Vy ≥αVx–Vy_{for allx,y∈H} and

We also know that ifV is aα-inverse-strongly monotone mapping and  <λ≤α, then I–λV:C→His nonexpansive.

LetG:HHbe a multivalued mapping. The effective domain ofGis denoted by

D(G), that is,D(G) ={x∈H:Gx=∅}.

ThenG:HHis called

(i) a monotone operator onHif x–y,u–v ≥for allx,y∈D(G),u∈Gx, and

v∈Gy;

(ii) a maximal monotone operator onHifGis a monotone operator onHand its

graph is not properly contained in the graph of any other monotone operator onH.

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

T be a nonexpansive mapping of C into itself,and let{xn}be a sequence in C.If xnw andlim_n→∞xn–Txn= ,then Tw=w.

In , Hojoet al.[] also gave an example for a -generalized hybrid mapping which is not a generalized hybrid mapping withFix(T) ={(, )}. We shall prove that this ex-ample for a -generalized hybrid mapping does not satisfy the demiclosed property as in Lemma ..

Example . LetA={x∈R:x ≤}andTA→Rbe defined as

Tx=

(, ) ifx∈A, x

x ifx∈R/A.

Hojoet al.[] showed thatT is a -generalized hybrid mapping, butT is not a gen-eralized hybrid mapping. Note that T does not have the demiclosed property. Indeed, there exists a sequence{xn} ∈Asuch thatxnwandlimn→∞xn–Txn= , butwin

R_{/Fix(T) =R}_{/{(, )}.}

Proof Letrn=  + 

n,xn= (rncosθ,rnsinθ) for alln∈N, thenxn→(cosθ,sinθ) andTxn= (cosθ,sinθ). We also haveTxn–xn=((rn– )cosθ, (rn– )sinθ)=rn– →, but

(cosθ,sinθ)= (, ).

Lemma .[] Let V:H→Hbe aγ¯-strongly monotone and L-Lipschitzian continuous

operator withγ¯> and L> .Letθ∈H,and V:H→Hsuch that Vx=Vx–θ.Then

Vis aγ¯-strongly monotone and L-Lipschitzian continuous mapping.Furthermore,there

is a unique fixed point zin C satisfying z=PC(z–Vz+θ).This point z∈C is also a

unique solution of the hierarchical variational inequality Vz–θ,q–z ≥,for all q∈C.

LetC be a nonempty subset of a real Hilbert spaceH. ThenT:C→H is a firmly

nonexpansive mapping ifTx–Ty≤ x–y–(I–T)x– (I–T)yfor everyx,y∈C, that is,Tx–Ty_{≤ x–y,Tx–Tyfor everyx,y∈C.}

Lemma .[] Let Gbe a maximal monotone mapping on H.Let JrGbe the resolvent of Gdefined by JrG= (I+rG)–for each r> .Then the following holds:

|s–t| s

JG

s x–J G

t x,JsGx–x

≥JG

s x–J G

t x

for all s,t> and x∈H.In particular,

JG

s x–J G

t x≤

|s–t| s J

G

s x–x

for all s,t> and x∈H.

A mapping T :H→H is said to be averaged if T = ( –α)I+αS, whereα∈(, )

andS:H→His nonexpansive. In this case, we also say thatTisα-averaged. A firmly

nonexpansive mapping is_-averaged.

Lemma .([, ]) Let C be a nonempty closed convex subset of a real Hilbert space H, and let T:C→C be a mapping.Then the following are satisfied:

(i) Tis nonexpansive if and only if the complement(I–T)is/-ism. (ii) IfSisυ-ism,then forγ > ,γSisυ/γ-ism.

(iii) Sis averaged if and only if the complementI–Sisυ-ism for someυ> /. (iv) IfSandT are both averaged,then the product(composite)STis averaged.

(v) If the mappings{Ti}n

i=are averaged and have a common fixed point,then

n

i=Fix(Ti) =Fix(T· · ·Tn).

Lemma .[] Let{an}be a sequence of real numbers such that there exists a subsequence

{ni}of{n}such that ani<ani+for all i∈N.Then there exists a nondecreasing sequence

{mk} ⊆Nsuch that mk→ ∞and the following properties are satisfied by all(sufficiently large)numbers k∈N:

am_k ≤am_k+ and ak≤amk+.

In fact,mk=max{j≤k:aj<aj+}.

Lemma .[] Let{an}n∈Nbe a sequence of nonnegative real numbers,{αn}a sequence of real numbers in[, ]with∞_n=αn=∞,{un}a sequence of nonnegative real numbers with∞_n=un<∞,{tn}a sequence of real numbers withlim suptn≤.Suppose that an+≤

( –αn)an+αntn+unfor each n∈N.Thenlimn→∞an= .

3 Convergence theorems of hierarchical problems

LetHbe a real Hilbert space and letCbe a nonempty closed convex subset ofH. For

eachi= , , andκi> , letFibe aκi-inverse-strongly monotone mapping ofCintoH. For

eachi= , , letGibe a maximal monotone mapping onHsuch that the domain ofGiis included inCand define the setG–

i  asG–i  ={x∈H: ∈Gix}. LetJλGn= (I+λnG)

–

andJG

rn = (I+rnG)– for eachn∈N,λn>  andrn> . Let{θn} ⊂H be a sequence.

LetVbe aγ¯-strongly monotone andL-Lipschitzian continuous operator withγ¯>  and L> . Throughout this paper, we use these notations and assumptions unless specified otherwise.

Theorem . Let T :C→H be a-generalized hybrid mapping withFix(T)∩(F+

G)–∩(F+G)–=∅.Takeμ∈Ras follows:

 <μ<γ¯ L.

Let{xn} ⊂Hbe defined by ⎧

⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

x∈C chosen arbitrarily,

yn=JG

λn(I–λnF)J

G

rn (I–rnF)xn,

sn=_nn_k–_=Tk_yn,

xn+=αnxn+ ( –αn)(βnθn+ (I–βnV)sn)

(.)

for each n∈N,{λn} ⊂(,∞),{αn} ⊂(, ),{βn} ⊂(, ),and{rn} ⊂(,∞).Assume that: (i)  <lim infn→∞αn≤lim supn→∞αn< ;

(ii) limn→∞βn= ,and

∞

n=βn=∞;

(iii)  <a≤λn≤b< κ,and <a≤rn≤b< κ;

(iv) limn→∞θn=θfor someθ∈H.

Thenlimn→∞xn=x,¯ wherex¯=PFix(T)∩(F+G)–_{∩(F+G)}–_(x¯–Vx¯+θ).This pointx is also a¯

unique solution of the hierarchical variational inequality:

Vx¯–θ,q–x¯ ≥, ∀q∈Fix(T)∩(F+G)–∩(F+G)–.

Proof Take anyx¯∈Fix(T)∩(F+G)–∩(F+G)– and letx¯be fixed. Then we have ¯

x=JG

λn(I–λnF)x¯andx¯=J

G

rn (I–rnF)x. Let¯ un=JrGn(I–rnF)xn. For eachn∈N, by the same argument as the proof of Theorem . [], we have

un–x¯≤ xn–x¯–rn(κ–rn)Fxn–Fx¯≤ xn–x¯ ()

and

yn–x¯≤ un–x¯–λn(κ–λn)Fun–Fx¯≤ xn–x¯. ()

By equations () and (), we have

yn–x¯ ≤ un–x¯ ≤ xn–x¯.

SinceT is a -generalized hybrid mapping withFix(T)=∅, we know thatT is a quasi-nonexpansive, and

sn–x¯=

 n

n–

k=

Tkyn–x¯

≤

 n

n–

k=

Tkyn–x¯

≤ yn–x¯ ≤ un–x¯ ≤ xn–x¯. ()

By the same argument as in the proof of Theorem . [], we find that the sequence{xn} is bounded. Furthermore,{un},{zn},{yn}, and{sn}are bounded. We also have

xn+–xn≤( –αn)

β_nθn–Vsn+sn–xn+ βnθn–Vsnsn–xn

and

xn+–x¯–xn–x¯+ ( –αn)αnsn–xn

≤( –αn)βn θn,xn–x¯– ( –αn)βnVsn,xn–x¯

+ ( –αn)β_nθn–Vsn+ βnθn–Vsnsn–xn. ()

We will divide the proof into two cases.

Case : there exists a natural numberNsuch thatxn+–x¯ ≤ xn–x¯for eachn≥N. Therefore,limn→∞xn–x¯exists. Hence, it follows from equation (), (i), and (ii) that

lim

n→∞sn–xn= . ()

By equations (), (), (i), and (ii), we have

lim

n→∞xn+–xn= . ()

We also have

zn–sn ≤βnθn+ ( –βnV)sn–sn≤βnθn–Vsn. ()

By equation (), (iv), and (ii) we have

lim

n→∞zn–sn= . ()

By equations () and (),

lim

n→∞zn–xn= . ()

By the same argument as in the proof of Theorem . [], we have

lim

n→∞un–xn=  ()

and

lim

n→∞yn–un= . ()

SinceFix(T)∩(F+G)–∩(F+G)– is a nonempty closed convex subset ofH, by

Lemma ., we can takex¯∈Fix(T)∩(F+G)–∩(F+G)– such that ¯

x=PFix(T)∩(F+G)–_{∩(F+G)}–_(x¯_–Vx¯_+θ).

This pointx¯is also a unique solution of the hierarchical variational inequality:

We shall show that

lim sup

n→∞

Vx¯–θ,zn–x¯ ≥.

Without loss of generality, there exists a subsequence{zn_k}of{zn}such that

znkw ()

for somew∈Hand

lim sup

n→∞

Vx¯–θ,zn–x¯= lim

k→∞

Vx¯–θ,znk–x¯

. ()

By equations () and (), we have

lim

n→∞un–zn= 

andunkw. On the other hand, since  <a≤λn≤b< κ, there exists a subsequence

{λn_kj}of{λnk}such that{λn_kj}converges to a numberλ¯ ∈[a,b]. By equation () and

Lemma ., we have

_un

kj–J

G ¯

λ (I–λF¯ )unkj

≤un_kj–JG

λ

nkj(I–λnkjF)unkj+J

G

λ

nkj(I–λF¯ )unkj–J

G ¯

λ (I–λF¯ )unkj

+JG

λ_nkj(I–λn_kjF)un_kj –JλG_nkj (I–λF¯ )un_kj

≤ un_kj –yn_kj+|λn_kj–λ¯|Fun_kj

+| λn_kj –λ¯|

¯

λ J G ¯

λ (I–λF¯ )unkj– (I–λF¯ )unkj→. ()

By equation (),un_kj w, and Lemma .,w∈Fix(JG ¯

λ (I–λF)) = (F¯ +B)

–_.

Since  <a≤rn≤b< κ, there exists a subsequence{rn_kj}of{rnk}such that{rn_kj}

converges to a number r¯ ∈[a,b]. By the same argument as for equation (), we

have

xn_kj–JG ¯

r (I–rF¯ )xn_kj→. ()

By equation () andunkw, we havexnkw.

By equation (),xnkwand Lemma ., we havew∈Fix(J

G ¯

r (I–¯rF)) = (F+G)–.

SinceTis a -generalized hybrid mapping, there existδ,δ,,∈Rsuch that

δTx–Ty 

+δTx–Ty+ ( –δ–δ)x–Ty

≤Tx–y



for allx,y∈C. ReplacingxbyTk_{ynin equation (), we have, for ally∈Candk= , , . . . ,n,}

δTk+yn–Ty



+δTk+yn–Ty



+ ( –δ–δ)Tkyn–Ty



≤Tk+yn–y



+Tk+yn–y



+ ( ––)Tkyn–y



≤Tk+yn–Ty



+Ty–y+ Tk+yn–Ty,Ty–y

+Tk+yn–Ty+Ty–y+ 

Tk+yn–Ty,Ty–y

+ ( ––)Tkyn–Ty+Ty–y+ 

Tkyn–Ty,Ty–y.

This implies that

≤(–δ)Tk+yn–Ty+Ty–y+ 

Tk+yn–Ty,Ty–y

+ (–δ)Tk+yn–Ty+ 

Tk+yn–Ty,Ty–y

+ (δ–+δ–)Tkyn–Ty

+ ( ––)

Tkyn–Ty,Ty–y

≤(–δ)Tk+yn–Ty–Tkyn–Ty



+ (–δ)Tk+yn–Ty 

–Tkyn–Ty+Ty–y

+ Tkyn–Ty+

Tk+yn–Tkyn+

Tk+yn–Tkyn,Ty–y. ()

Summing up these inequalities () with respect tok=  tok=n–  and dividing byn, we have

≤(–δ) n T

n+_y

n–Ty



+Tnyn–Ty



–Tyn–Ty–yn–Ty

+(–δ) n T

n_yn–Ty

–yn–Ty

+Ty–y+  sn–Ty,Ty–y

+ n



Tn+yn+Tnyn–Tyn–yn

+

Tnyn–yn

,Ty–y. ()

Replacing nby nkj and let nkj → ∞. Then from equation (), (), and (), we have

sn_kj w, and

≤ Ty–y+ w–Ty,Ty–y.

Takingy=win the above inequality, we have

≤ Tw–w+ w–Tw,Tw–w=Tw–w– Tw–w= –Tw–w.

This implies thatw∈Fix(T). Hence,w∈Fix(T)∩(F+G)–∩(F+G)–. Therefore,

we have from equations () and ()

lim sup

n→∞

Vx¯–θ,zn–x¯

= lim

k→∞

Vx¯–θ,znk–x¯

By the same argument as the proof of Theorem . [], we have

xn+–x¯

≤ – ( –αn)βnτ

xn–x¯+ ( –αn)βnτ

βnτxn–x¯



+ θn–θ,zn–x¯

τ +

θ–Vx¯,zn–x¯

τ

. ()

By equations (), (), assumptions, and Lemma ., we know thatlimn→∞xn=x¯, where

¯

x=PFix(T)∩(F+G)–_{∩(F+G)}–_(x¯–Vx¯+θ).

Case : Suppose that there exists{ni}of{n}such thatxni–x¯ ≤ xni+–x¯for alli∈N.

By Lemma ., there exists a nondecreasing sequence{mj}inNsuch thatmj→ ∞and

xmj–x¯ ≤ xmj+–x¯ and xj–x¯ ≤ xmj+–x¯. ()

Hence, it follows from equations () and () that

( –αmj)αmjsmj–xmj



≤( –αmj)βmj θmj,xmj–x¯– ( –αmj)βmj Vsmj,xmj–x¯

+ ( –αmj)

_β

mjθmj–Vsmj

_{+ β}

mjθmj–Vsmjsmj–xmj

()

for eachj∈N. Hence, it follows from equation (), (i), and (ii) that

lim

j→∞smj–xmj= . ()

We show that

lim sup

j→∞ Vx¯–θ,zmj–x¯ ≥.

Without loss of generality, there exists a subsequence{zm_jk}of{zmj}such thatzm_jk w

for somew∈Hand

lim sup

j→∞

Vx¯–θ,zmj–x¯= lim

k→∞ Vx¯–θ,zmjk –x¯. ()

By a similar argument as in the proof of Case , we havew∈Fix(T)∩(F+G)–∩(F+

G)–. Therefore, we have from equations () and ()

lim sup

Following a similar argument as in the proof of Case , we have

xmj+–x¯



≤ – ( –αmj)βmjτ

xmj–x¯

_{+ ( –α}

mj)(βmjτ)

_xm

j–x¯



+ βmj( –αmj)θn–θ,zmj–x¯+ βmj( –αmj) θ–Vx¯,zmj–x¯. ()

Fromxmj–x¯ ≤ xmj+–x¯, we have

( –αmj)βmjτxmj–x¯



≤( –αmj)(βmjτ)

_xm

j–x¯

_{+ β}

mj( –αmj)θn–θ,zmj–x¯

+ βmj( –αmj)θ–Vx¯,zmj–x¯. ()

Since ( –αmj)βmj> , we have

τxmj–x¯

_≤β

mjτxmj–x¯

_{+  θ}

n–θ,zmj–x¯+  θ–Vx¯,zmj–x¯. ()

By equations (), (), and the assumptions, we know that

lim

j→∞xmj–x¯= .

By (), (), and the assumptions, we know that

lim

j→∞xmj+–xmj= .

Thus, we have

lim

j→∞xmj+–x¯= . ()

By equations () and (),

lim

j→∞xj–x¯ ≤j→∞limxmj+–x¯= .

Thus, the proof is completed.

Remark .

(i) The assumptions, method, conclusion, and applications of Theorem . are different from Theorem . in [] and []. In Theorem ., Lemma . is used to prove the result, but in [] and [] we did not use this lemma.

(ii) The assumptions, method, and conclusion of Theorem . are different from Theorem . []. In Theorem . [],Tis a quasi-nonexpansive with the

Figure 1 x-axis: index;y-axis:xn.

Example . LetT be the same as Example .. Letαn= /, βn= /n,θn= ,Vx=x,

Gx=Fx= x, Gx=Fx= x, rn= /, λn= /. Then V, G, G, F, F satisfy all

conditions of Theorem . andFix(T)∩(F+G)–∩(F+G)– ={}, and if we let

x= ., we see the following numerical results and graph (see Figure )

demon-strating Theorem .:

n=  –  . . . . . n=  –  . . . . . n=  –  . . . . . . . .

Besides, we know the following.

If|xn–xn–|< –, thenn= ; if|xn–xn–|< –, thenn= ; if|xn–xn–|< –,

thenn= ; if|xn–xn–|< –, thenn= ,.

Fori= , , letFi= ,Gi=∂iC, andλn=rn=  for alln∈Nin Theorem .. Furthermore, putθn=θ, andV(x) =xfor allx∈H; we obtain the following theorem which generalizes

Theorem . in [].

Theorem . Let T:C→Hbe a-generalized hybrid mapping such that F(T)=∅.Let

θ∈C,and{xn} ⊂C be defined by

⎧ ⎪ ⎨ ⎪ ⎩

x∈C chosen arbitrarily,

sn=_nn_k–_=Tk_xn,

xn+=αnxn+ ( –αn)(βnθ+ ( –βn)sn)

(.)

for each n ∈ N, {αn} ⊂ (, ), and {βn} ⊂ (, ). Assume that  < lim infn→∞αn ≤

lim sup_n→∞αn<  and limn→∞βn= , and

∞

n=βn =∞. Then limn→∞xn=x,¯ where

¯

Figure 2 x-axis: index;y-axis:xn.

Example . LetT be the same as Example .. Letαn= /,βn= /n,θ = . Then the following numerical results and graph (see Figure ) demonstrate Theorem .:

n=  –  . . . . . n=  –  . . . . . n=  –  . . . . . n=  –  . . . . .

Besides, we know the following.

If|xn–xn–|< –, thenn= ; if|xn–xn–|< –, thenn= ; if|xn–xn–|< –,

thenn= ; if|xn–xn–|< –, thenn= ,.

4 Mathematical programming with multiple sets split feasibility constraints

LetHbe a Hilbert space, letf be a proper lower semicontinuous convex function ofH

into (–∞,∞). The subdifferential∂f off is defined as follows:

∂f(x) =z∈H:f(x) + z,y–x ≤f(y),∀y∈H

for allx∈H. From Rockafellar [], we know that∂f is a maximal monotone operator. Let

Cbe a nonempty closed convex subset of a real Hilbert spaceH, andiCbe the indicator

function ofC,i.e.

iCx=

 ifx∈C,

∞ ifx∈/C.

Furthermore, we also define the normal coneNCuofCatuas follows:

NCu=

z∈H: z,v–u ≤,∀v∈C

.

TheniCis a proper lower semicontinuous convex function onH, and the subdifferential ∂iC ofiC is a maximal monotone operator. Thus, we can define the resolventJ∂iC

λ of∂iC forλ> ,i.e.

J∂iC

for allx∈H. Since

∂iCx=

z∈H:iCx+ z,y–x ≤iCy,∀y∈H

=z∈H: z,y–x ≤,∀y∈C

=NCx

for allx∈C, we have

u=J∂iC

λ x ⇔ x∈u+λ∂iCu ⇔ x–u∈λNCu

⇔ x–u,y–u ≤, ∀y∈C

⇔ u=PCx. ()

The equilibrium problem is to findz∈Csuch that

g(z,y)≥ for eachy∈C. (EP)

The solutions set of the equilibrium problem (EP) is denoted byEP(g). For solving the equilibrium problem, let us assume that the bifunctiong:C×C→Rsatisfies the follow-ing conditions:

(A) g(x,x) = for eachx∈C;

(A) gis monotone,i.e.,g(x,y) +g(y,x)≤for anyx,y∈C; (A) for eachx,y,z∈C,limt↓g(tz+ ( –t)x,y)≤g(x,y);

(A) for eachx∈C, the scalar functiony→g(x,y)is convex and lower semicontinuous.

Lemma .[, ] Let g:C×C→Rbe a bifunction which satisfies conditions(A)-(A). Let r> and x∈C.Then there exists z∈C such that

g(z,y) +

r y–z,z–x ≥ for all y∈C.

Furthermore,if

T_rg(x) :=

z∈C:g(z,y) +

r y–z,z–x ≥for all y∈C

,

then we have:

(i) Trg is single-valued;

(ii) Trg is a firmly nonexpansive mapping; (iii) EP(g)is a closed convex subset ofC; (iv) EP(g) =Fix(Trg).

We call such Trg the resolvent ofg forr> . Throughout these section, we use these notations and assumptions unless specified otherwise.

Lemma .[] Let g:C×C→Rbe a bifunction satisfying the conditions(A)-(A). Define Agas follows:

Agx=

{z∈H:g(x,y)≥ y–x,z,∀y∈C} if x∈C;

∅ if x∈/C. (L.)

Then,EP(g) =A–

g and Agis a maximal monotone operator with the domain of Ag⊂C. Furthermore,for any x∈H and r> ,the resolvent Trg of g coincides with the resolvent of Ag,i.e.,Trgx= (I+rAg)–x.

LetC,Q, andQbe nonempty closed convex subsets of real Hilbert spacesH,H, and

H, respectively, letGibe a maximal monotone mapping onHsuch that the domains of

Gi is included inCfor each i= , . LetJG

λ = (I+λG)–andJrG= (I+rG)–for each

λ>  andr> , letLbe aκ-inverse-strongly monotone mapping ofCintoH, let L

be aκ-inverse-strongly monotone mapping ofC intoH, letBbe aν-inverse-strongly

monotone mapping ofQintoH and letBbe aν-inverse-strongly monotone mapping

ofQintoH. LetGbe a maximal monotone mappings onHsuch that the domain ofG

is included inQand letGbe maximal a monotone mappings onHsuch that the domain

ofGis included inQ. LetJG

λ= (I+λG)

–_andJG

r = (I+rG)

–_{for eachλ>  andr> .}

LetA,A:H →H be bounded linear operators,A∗ andA∗ the adjoints of A andA

respectively,A:H→Ha bounded linear operator, andA∗the adjoint ofA. LetRibe

the spectral radius of the operatorA∗_iAifori= , , respectively, andRthe spectral radius of the operatorA∗A. LetI,I,Ibe the identity mappings ofH,H,H, respectively. We

use these notations throughout this section unless specified otherwise.

In order to study the convergence theorems for the solutions set of the multiple sets split monotone variational inclusion problem, we study the following essential problem (SFP-):

Findx¯∈Hsuch thatAx¯∈(G+B)–().

Theorem . Given anyx¯∈Hwe have the following.

(i) Ifx¯is a solution of(SFP-),then(I–λA∗(I–U)A)x¯=x¯,whereλ> ,

U=JG

σ(I–σB)andσ> .

(ii) Suppose thatU=JG

σ(I–σB), <λ<_R, <σ< ν.ThenA∗(I–U)Ais aκ_R-ism mapping,JG

σ(I–σB),andI–λA∗(I–U)Aare averaged for someκ>_.Suppose further that solution set of(SFP-)is nonempty and(I–λA∗(I–U)A)x¯=x¯.Thenx¯ is a solution of(SFP-).

Proof (i) Suppose thatx¯∈His a solution of (SFP-). Thenx¯∈H,Ax¯∈Fix(U). It is easy

to see that (I–λA∗(I–U)A)x¯=x.¯

(ii) Since the solutions set of (SFP-) is nonempty, there existsw¯ ∈H such thatAw¯ ∈

F(U). SinceBis aν-inverse-strongly monotone mapping ofQinto H, it follows from

Lemma .(iii) and (iv) that

By Lemma .(iii), for someκ>_, we know that

I–U=I–JσG(I–σB) isκ-ism. ()

In Theorem . [], Moudafi showed that

A∗_(I–U)Ais

κ

R-ism. ()

By Lemma .(iii) and  <λ<_R, we know that

I–λA∗(I–U)Ais averaged ()

for someκ>_. Since

¯

x=I–λA∗(I–U)A

¯

x. ()

This implies

A∗(I–U)Ax¯= . ()

We know thatU(Ax) =¯ Ax+¯ w, withA∗w= , which combined with the fact thatU(Aw) =¯

Aw¯ yields

U(Ax) –¯ U(Aw)¯ =Ax¯+w–Aw¯=Ax¯–Aw¯+w. ()

SinceU=JσG(I–σB) is a nonexpansive mapping and we have equation (), we have

w= .

This implies that

Ax¯=Fix(U) =FixJσG(I–σB)

. ()

This shows thatx¯is a solution of (SFP-).

In the following theorem, we consider the multiple set split monotonic variational in-clusion problem (MSSMVIP-):

Findx¯∈Hsuch thatx¯∈G–()∩G– (),Ax¯∈(B+G)–(),

andAx¯∈

B+G–().

That is,

Findx¯∈Hsuch thatx¯∈Fix

JG

λ

∩FixJG

r

,Ax¯∈Fix(U) andAx¯∈Fix(U)

whereU=JσG(I–σB), U=JG

σ

I–σB

.

Theorem . Let T:C→Hbe a-generalized hybrid mapping.Suppose thatis the

solutions set of(MSSMVIP-)withFix(T)∩=∅.Takeμ∈Ras follows:

 <μ<γ¯ L.

Let{xn} ⊂Hbe defined by

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

x∈C chosen arbitrarily,

yn=JG

λ (I–λA∗(I–U)A)JrG(I–rA∗(I–U)A)xn,

sn=_nn_k–_=Tk_yn,

xn+=αnxn+ ( –αn)(βnθn+ ( –βnV)sn),

(.)

where U=JσG(I–σB),U=JG

σ(I–σB),{αn} ⊂(, ),{βn} ⊂(, ),r∈(,∞)andλ∈ (,∞).We have

(i)  <lim infn→∞αn≤lim supn→∞αn< ; (ii) lim_n→∞βn= ,and

∞

n=βn=∞; (iii)  <λ<_R

, <r< 

R, <σ< νand <σ _{< ν;}

(iv) lim_n→∞θn=θfor someθ∈H.

Thenlimn→∞xn=x,¯ wherex¯=PFix(T)∩(x¯–Vx¯+θ).This pointx is also a unique solution¯

of the hierarchical variational inequality:

Vx¯–θ,q–x¯ ≥, ∀q∈Fix(T)∩.

Proof LetF=A∗(I–U)A,F=A∗(I–U)A,λn=λandrn=rfor alln∈Nin Theo-rem .. It follow from TheoTheo-rem .(ii) thatFiis μi

Ri-ivm for someμi>



 and eachi= , .

Then algorithm (.) in Theorem . follows immediately from algorithm (.) in Theo-rem ..

SinceFix(T)∩is nonempty, there existsw¯∈Csuch that ¯

w∈ Fix(T)∩FixJG

λ

∩FixI–λA∗_(I–U)A

∩FixJG

r

∩FixI–rA∗_(I–U)A

. ()

This implies that

¯

w∈ Fix(T)∩FixJG

λ

I–λA∗_(I–U)A

∩FixJG

r

I–rA∗_(I–U)A

. ()

That is,

¯

w∈Fix(T)∩FixJG

λ (I–λF)

∩FixJG

r (I–rF)

. ()

Hence,

¯

It follows from Theorem . thatlimn→∞xn=x, where¯ ¯

x=PFix(T)∩(F+G)–_{∩(F+G)}–_(x¯–Vx¯+θ).

This pointx¯is also a unique solution of the hierarchical variational inequality:

Vx¯–θ,q–x¯ ≥, ∀q∈Fix(T)∩(F+G)–∩(F+G)–.

If

w∈Fix(T)∩(G+F)–∩(G+F)–. ()

By equations (), (), and (), we know that

w=Tw, w=JG

λ

I–λA∗_(I–U)A

w, w=JG

r

I–rA∗_(I–U)A

w. ()

By=∅, equation (), and Lemma .(v), we have

w=Tw, w=JG

λ w, w=

I–λA∗_(I–U)A

w, w=JG

r w,

w=I–rA∗_(I–U)A

w.

()

It follows from Theorem .(ii) thatwis a solution of (MSSMVIP-). Therefore,w∈

Fix(T)∩and

Fix(T)∩(G+F)–∩(G+F)–⊆Fix(T)∩.

Conversely, ifw∈Fix(T)∩, by equations (), (), (), and (), we know thatw∈

Fix(T)∩(G+F)–∩(G+F)– and

Fix(T)∩⊆Fix(T)∩(G+F)–∩(G+F)–.

Therefore,Fix(T)∩=Fix(T)∩(G+F)–∩(G+F)– and the proof is completed.

Remark . Moudafi [] studied a weak convergence theorem for the split monotone variational inclusion problem, while Theorem . is a strong convergence theorem for the multiply sets split monotone variational inclusion problem.

By Theorem ., we study the mathematical programming problem with (MSSMVIP-) and fixed point set constraints.

Theorem . In Theorem.,let h:C→Rbe a convex Gâteaux differential function with Gâteaux derivative V,and the assumption(iv)is replaced bylimn→∞θn= .Then

lim_n→∞xn=x,¯ where x¯=PFix(T)∩(x¯ –Vx).¯ This point x is also a unique solution of¯

the mathematical programming problem with(MSSMVIP-)and fixed point constraints:

Proof Letθ=  in Theorem ., by Theorem ., we see that

Vx,¯ q–x¯ ≥, ∀q∈F(T)∩. ()

Sinceh:C→Ris a convex Gâteaux differential function with Gâteaux dirivitiveV, we obtain

Vx,¯ y–x¯=lim

t→

h(x¯+t(y–x)) –¯ h(x)¯ t

=lim

t→

h(( –t)x¯+ty) –h(x)¯ t

≤lim

t→

( –t)h(x) +¯ th(y) –h(x)¯ t

=h(y) –h(x)¯ ()

for ally∈C. By equations () and (), it is easy to see thath(x)¯ ≤h(q) for allq∈Fix(T)∩

.

If we puth(x) =_xin Theorem ., thenV=I, and we have the following minimum norm of common solutions for (MSSMVIP-) andFix(T).

Theorem . In Theorem.,let the iteration process{xn+}be replaced by

xn+=αnxn+ ( –αn)

βnθn+ ( –βn)sn

, n∈N.

Thenlimn→∞xn=x,¯ wherex¯=PFix(T)∩().This pointx is also a unique minimum solu-¯

tion ofFix(T)∩:minx∈Fix(T)∩x.

The multiple sets split variational inequality problem (MSSMVIP-) is defined as fol-lows:

Findx¯∈Hsuch thatx¯∈G–()∩G– (), ()

and

B(Ax),¯ y–Ax¯

≥ for ally∈Q, ()

B(Ax),¯ y–Ax¯

≥ for ally∈Q. ()

That is,

Findx¯∈Hsuch thatx¯∈Fix

JG

λ

∩FixJG

r

()

and

Ax¯∈Fix

PQ(I–σB) and Ax¯∈Fix

PQI–σB. ()

Theorem . In Theorem ., let U =JσG(I –σB), U =JG

σ(I–σB)be replaced by U =PQ(I–σB), U=PQ(I –σB), respectively. Suppose that the set of solutions for

(MSSMVIP-)is andFix(T)∩=∅.Thenlimn→∞xn=x,¯ wherex¯=PFix(T)∩(x¯–

Vx¯+θ).This pointx is also a unique solution of the hierarchical variational inequality:¯

Vx¯–θ,q–x¯ ≥, ∀q∈Fix(T)∩.

Proof LetG=∂iQandG=∂iQ in Theorem ., then, by equation (), we haveJσG(I– σB) =PQ(I–σB),J_σG(I–σB) =PQ(I–σB). SinceFix(T)∩=∅, there existsw¯ ∈C

such that we can find

¯

w∈Hsuch thatw¯∈Fix

JG

λ

∩FixJG

r

()

and

Aw¯∈Fix

PQ(I–σB)

and Aw¯ ∈Fix

PQ

I–σB. ()

This implies that

Aw¯∈Fix

JσG(I–σB)

and Aw¯∈Fix

J_σGI–σB. ()

Therefore,w¯∈Fix(T)∩=∅. It follows from Theorem . thatlimn→∞xn=x, where¯ x¯=

PFix(T)∩(x¯–Vx¯+θ). This pointx¯is also a unique solution of the hierarchical variational

inequality:

Vx¯–θ,q–x¯ ≥, ∀q∈Fix(T)∩.

By equations (), (), and (),limn→∞xn=x, where¯ x¯=PFix(T)∩(x¯–Vx¯+θ). This

pointx¯is also a unique solution of the hierarchical variational inequality:

Vx¯–θ,q–x¯ ≥, ∀q∈Fix(T)∩. Remark . Censoret al.[] studied a weak convergence theorem for the split varia-tional inequalities problem with the addivaria-tional assumption, while Theorem . studies a strong convergence theorem for multiply sets split variational inequalities problem with-out this additional assumption.

By Theorem ., we study a mathematical programming problem with (MSSMVIP-) and fixed point set constraints.

Theorem . In Theorem.,let h:C→Rbe a convex Gâteaux differential function with Gâteaux derivative V,and the assumption(iv)is replaced bylim_n→∞θn= .Then

limn→∞xn=x,¯ wherex¯=PFix(T)∩(x¯–Vx).¯ This pointx is also a unique solution of the¯

following mathematical programming problem with(MSSMVIP-)and fixed point con-straints:

min

x∈Fix(T)∩

Proof By Theorem . and following the same argument as in the proof of Theorem .,

we see that the proof is complete.

In the following theorem, we consider the following split monotonic variational inclu-sion problem (MSSMVIP-):

Findx¯∈Hsuch thatx¯∈G–(),x¯∈(G+F)–, andAx¯∈

B+G–().

That is,

Findx¯∈Hsuch thatx¯∈Fix

JG

λ (I–λF)

∩FixJG

r

, andAx¯∈Fix(U)

whereU=JG

σ

I–σB

.

Letbe the solutions set of (MSSMVIP-).

Theorem . Let T:C→Hbe a-generalized hybrid mapping.Suppose thatis the

solutions set of(MSSMVIP-)withFix(T)∩=∅.Takeμ∈Ras follows:

 <μ<γ¯ L.

Let{xn} ⊂Hbe defined by

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

x∈C chosen arbitrarily,

yn=JG

λ (I–λF)JrG(I–rA∗(I–U)A)xn,

sn=_nn_k–_=Tk_yn,

xn+=αnxn+ ( –αn)(βnθn+ ( –βnV)sn),

(.)

where U=JG

σ(I–σB),{αn} ⊂(, ),{βn} ⊂(, ),r∈(,∞)andλ∈(,∞). (i)  <lim infn→∞αn≤lim sup_n→∞αn< ;

(ii) limn→∞βn= ,and∞_n=βn=∞; (iii)  <λ< κ, <r<_R_ and <σ< ν;

(iv) lim_n→∞θn=θfor someθ∈H.

Thenlim_n→∞xn=x,¯ wherex¯=PFix(T)∩(x¯–Vx¯+θ).This pointx is also a unique solution¯

of the hierarchical variational inequality:

Vx¯–θ,q–x¯ ≥, ∀q∈Fix(T)∩.

Proof LetF=A∗(I–U)A,λn=λandrn=rfor alln∈Nin Theorem .. It follows from Theorem .(ii) thatF is μ_Rii-ivm for someμi>_. Then algorithm (.) in Theorem . follows immediately from algorithm (.) in Theorem ..

SinceFix(T)∩is nonempty, there existsw¯∈Csuch that ¯

w∈Fix(T)∩FixJG

λ (I–λF)

∩FixJG

r

∩FixI–rA∗_(I–U)A

. ()

This implies that

¯

w∈Fix(T)∩FixJG

λ (I–λF)

∩FixJG

r

I–rA∗_(I–U)A

That is,

¯

w∈Fix(T)∩FixJG

λ (I–λF)

∩FixJG

r (I–rF)

. ()

Hence,

¯

w∈Fix(T)∩(G+F)–∩(G+F)–=∅. ()

It follows from Theorem . thatlimn→∞xn=x, where¯

¯

x=PFix(T)∩(F+G)–_{∩(F+G)}–_(x¯–Vx¯+θ).

This pointx¯is also a unique solution of the hierarchical variational inequality:

Vx¯–θ,q–x¯ ≥, ∀q∈Fix(T)∩(F+G)–∩(F+G)–.

If

w∈Fix(T)∩(G+F)–∩(G+F)–. ()

That is,

w=Tw, w=JG

λ (I–F)w and w=JrG

I–rA∗_(I–U)A

w. ()

By=∅, equation () and Lemma .(v), we have

w=Tw, w=JG

λ (I–λF)w, w=J G

r w, w=

I–rA∗_(I–U)A

w. ()

By=∅, equation (), and Theorem .(ii), we see thatwis a solution of (MSSMVIP-).

Therefore, w∈Fix(T)∩ and Fix(T)∩(G+F)–∩(G+F)–⊆Fix(T)∩.

Conversely, if w∈Fix(T)∩, by equations (), (), (), and (), we know that

w∈Fix(T)∩(G+F)–∩(G+F)– and

Fix(T)∩⊆Fix(T)∩(G+F)–∩(G+F)–.

Therefore,Fix(T)∩=Fix(T)∩(G+F)–∩(G+F)– and the proof is completed.

Remark . Theorem . also improve Theorem . [].

For eachi= , , letfi:C×C→Rbe a bifunction satisfying the conditions (A)-(A). The system of mixed type equilibria problem (MSSMVIP-) is defined as follows.

Findx¯∈Csuch thatf(x,¯ x) + x–x,¯ Fx¯ ≥ andf(x,¯ x) + x–x,¯ Fx¯ ≥

for allx∈C.

Theorem . Let T :C→H be a -generalized hybrid mapping.For each i= , ,let

fi:C×C→Rbe a bifunction satisfying the conditions(A)-(A),and J Af

λ ,J Af

r ,defined as Lemma..Suppose thatis the solutions set of(MSSMVIP-)with(Af+F)

–_∩

(Af+F)–∩Fix(T)=∅.Let{xn} ⊂H be defined by ⎧

⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

x∈C chosen arbitrarily,

yn=J Af

λ (I–λF)J

Af

r (I–rF)xn, sn=_nn_k–_=Tkyn,

xn+=αnxn+ ( –αn)(βnθn+ ( –βnV)sn),

(.)

where{αn} ⊂(, ),{βn} ⊂(, ),r∈(,∞)andλ∈(,∞).Assume that: (i)  <lim infn→∞αn≤lim supn→∞αn< ;

(ii) limn→∞βn= ,and

∞

n=βn=∞; (iii)  <a≤λ≤b< κ, <a≤r≤b<κ;

(iv) lim_n→∞θn=θfor someθ∈H.

Thenlimn→∞xn=x,¯ wherex¯=PFix(T)∩(x¯–Vx¯+θ).This pointx is also a unique solution¯

of the hierarchical variational inequality:

Vx¯–θ,q–x¯ ≥, ∀q∈Fix(T)∩.

Proof For eachi= , , letAfi be as in Lemma .. By Lemma ., we see thatAfi is a

maximal monotone operator with the domain ofAfi⊂C. Furthermore, for anyx∈Hand

r> , the resolventTfi

λ officoincides with the resolvent ofAfi,i.e.,

Tfi

λx= (I+λAfi)

–_{x. ()}

Fori= , , letGi=Afiin Theorem .. By equation (), we have

Tfi

λx= (I+λAf) –_x=JG

λ x, Trfx= (I+rAf) –_x=JG

r x. ()

Then algorithm (.) in Theorem . follows immediately from algorithm (.) in Theo-rem ..

By equation (), we haveFix(Tf

λ) =A–f =Fix(J

Af

λ ) andFix(T f

r ) =A–_f =Fix(J Af

r ). It follows from Theorem . thatlimn→∞xn=x, where¯

¯

x=PFix(T)∩(x¯–Vx¯+θ).

This pointx¯is also a unique solution of the hierarchical variational inequality:

Vx¯–θ,q–x¯ ≥, ∀q∈Fix(T)∩.

Herew∈(G+F)–∩(G+F)–. That is,w∈(Af+F)

–_∩(Af +F)

–_{. That is,}

w∈Fix(JAf

λ (I–λF))∩Fix(J

Af_

r (I–rF)). That is,