Hierarchical problems with applications to mathematical programming with multiple sets split feasibility constraints

R E S E A R C H

Open Access

Hierarchical problems with applications to

mathematical programming with multiple

sets split feasibility constraints

Zenn-Tsun Yu

_{and Lai-Jiu Lin}

*_{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 establish a strong convergence theorem for hierarchical problems, an equivalent relation between a multiple sets split feasibility problem and a fixed point problem. As applications of our results, we study the solution of mathematical programming with fixed point and multiple sets split feasibility constraints,

mathematical programming with fixed point and multiple sets split equilibrium constraints, mathematical programming with fixed point and split feasibility constraints, mathematical programming with fixed point and split equilibrium constraints, minimum solution of fixed point and multiple sets split feasibility problems, minimum norm solution of fixed point and multiple sets split equilibrium problems, quadratic function programming with fixed point and multiple set split feasibility constraints, mathematical programming with fixed point and multiple set split feasibility inclusions constraints, mathematical programming with fixed point and split minimax constraints.

Keywords: hierarchical problems; multiple sets split feasibility problem; fixed point problem; mathematical programming; quadratic function programming; minimum norm solution

1 Introduction

The split feasibility problem (SFP) in finite dimensional Hilbert spaces was first introduced by Censor and Elfving [] for modeling inverse problems which arise from phase retrievals and in medical image reconstruction. Since then, the split feasibility problem (SFP) has received much attention due to its applications in signal processing, image reconstruction, with particular progress in intensity-modulated radiation therapy, approximation theory, control theory, biomedical engineering, communications, and geophysics. For examples, one can refer to [–] and related literature. Since then, many researchers have studied (SFP) in finite dimensional or infinite dimensional Hilbert spaces. For example, one can see [, –].

A special case of problem (SFP) is the convexly constrained linear inverse problem in the finite dimensional Hilbert space []:

(CLIP) Findx¯∈Csuch thatAx¯=b, whereb∈H,

which has extensively been investigated by using the Landweber iterative method []

xn+:=xn+γAT(b–Axn), n∈N.

In , Byrne [] first introduced the so-called CQ algorithm which generates a se-quence{xn}by the following recursive procedure:

xn+=PC

xn–ρnA∗(I–PQ)Axn

, ()

where the stepsizeρnis chosen in the interval (, /A), andPCandPQare the metric

projections ontoC⊆RnandQ⊆Rm, respectively. Compared with Censor and Elfving’s algorithm [] where the matrix inverseAis involved, the CQ algorithm () seems more easily executed since it only deals with metric projections with no need to compute matrix inverses.

In , Xu [] modified Byrne’s CQ algorithm and proved the weak convergence the-orem in infinite Hilbert spaces for their modified algorithm.

Let Cbe a nonempty closed convex subset of a real Hilbert space H with the inner product·,·and the norm · . A mappingT:C→His said to be nonexpansive ifTx–

Ty ≤ x–yfor allx,y∈C;T is said to be a quasi-nonexpansive mapping ifFix(T)=∅ andTx–y ≤ x–yfor allx∈Candy∈Fix(T), we denote byFix(T) ={x∈C:Tx=x}

the set of fixed points ofT.A:C→His called strongly positive if

x,Ax ≥αx, ∀x∈C.

Letf be a contraction onH and{αn}be a sequence in [, ]. In , Xu [] proved

that under some condition on{αn}, the sequence{xn}generated by

xn+=αnfxn+ ( –αn)Txn

strongly converges tox∗inFix(T), which is the unique solution of the variational inequality

(I–f)x∗,x–x∗≥ for allx∈Fix(T).

Xu [] also studied the following minimization problem over the set of fixed points of a nonexpansive operatorTon a real Hilbert spaceH:

min x∈Fix(T)



Bx,x–a,x,

whereais a given point inHandBis a strongly positive bounded linear operator onH. In [], Xu proved that the sequence{xn}defined by the following iterative method

xn+= (I–αnB)Txn+αna

converges strongly to the unique solution of the minimization problem of a quadratic func-tion. In [], Marinoet al.considered the following iterative method:

They proved that the sequence generated by () converges strongly to the fixed pointx∗

ofTwhich solves the following:

(A–γf)x∗,x–x∗≥

for allx∈Fix(T). For some more related works, see [–] and the references therein. In this paper, we establish a strong convergence theorem for hierarchical problems, an equivalent relation between a multiple sets split feasibility problem and a fixed point prob-lem. As applications of our results, we study the solution of mathematical programming with fixed point and multiple sets split feasibility constraints, mathematical programming with fixed point and multiple sets split equilibrium constraints, mathematical program-ming with fixed point and split feasibility constraints, mathematical programprogram-ming with fixed point and split equilibrium constraints, minimum solution of fixed point and mul-tiple sets split feasibility problems, minimum norm solution of fixed point and mulmul-tiple sets split equilibrium problems, quadratic function programming with fixed point and multiple set split feasibility constraints, mathematical programming with fixed point and multiple set split feasibility inclusions constraints, mathematical programming with fixed point and split minimax constraints.

2 Preliminaries

Throughout this paper, letNbe the set of positive integers and letRbe the set of real num-bers,Hbe a (real) Hilbert space with the inner product·,·and the norm·, respectively, and letCbe a nonempty closed convex subset ofH. We denote the strong convergence and the weak convergence of{xn}tox∈Hbyxn→xandxnx, respectively. For each x,y∈Handλ∈[, ], we have

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

Hence, we also have

x–y,u–v=x–v+y–u–x–u–y–v ()

for allx,y,u,v∈H.

Forα> , a mappingA:H→His calledα-inverse-strongly monotone (α-ism) if

x–y,Ax–Ay ≥αAx–Ay, ∀x,y∈H.

If  <λ≤α,A:H→His anα-inverse-strongly monotone mapping, thenI–λA:H→ His nonexpansive. A mappingT:C→His said to be a firmly nonexpansive mapping if

Tx–Ty≤ x–y–(I–T)x– (I–T)y

for everyx,y∈C. LetT:C→Hbe a mapping. Thenp∈Cis called an asymptotic fixed point ofT[] if there exists{xn} ⊆Csuch thatxnp, andlimn→∞xn–Txn= . We

monotone if there existsγ¯>  such thatx–y,Vx–Vy ≥ ¯γx–y_{for allx,y∈H. SuchV}

is also calledγ¯-strongly monotone. A nonlinear operatorV:H→His called Lipschitzian continuous if there existsL>  such thatVx–Vy ≤Lx–yfor allx,y∈H. SuchVis also calledL-Lipschitzian continuous.

LetBbe a mapping ofHinto H_{. The effective domain ofBis denoted byD(B), that is,} D(B) ={x∈H:Bx=∅}. A multi-valued mappingBis said to be a monotone operator on

H ifx–y,u–v ≥ for allx,y∈D(B),u∈Bx, andv∈By. A monotone operatorBon

H is said to be maximal if its graph is not properly contained in the graph of any other monotone operator onH. For a maximal monotone operatorBonHandr> , we may define a single-valued operatorJr= (I+rB)–:H→D(B), which is called the resolvent of Bforr, and letB– ={x∈H: ∈Bx}.

The following lemmas are needed in this paper.

Lemma .[] Let H and H be two real Hilbert spaces, A:H→H be a bounded linear operator,and A∗be the adjoint of A.Let C be a nonempty closed convex subset of H, and let G:H→H be a firmly nonexpansive mapping.Then A∗(I–G)A is a _A-ism, that is,

 AA

∗_{(I–G)Ax–A}∗_(I–G)Ay

≤x–y,A∗(I–G)Ax–A∗(I–G)Ay

for all x,y∈H.

Lemma .[] Let C be a nonempty closed convex subset of a real Hilbert space H.Let G:H→H be a firmly nonexpansive mapping.Suppose thatFix(G)=∅.Thenx–Gx,Gx–

w ≥for each x∈H and each w∈Fix(G).

A mappingT:H→His said to be averaged ifT= ( –α)I+αS, whereα∈(, ) andS:

H→His a nonexpansive mapping. 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}ni=are averaged and have a common fixed point,then

n

i=Fix(Ti) =Fix(T· · ·Tn).

Lin and Takahashi [] gave the following results in a Hilbert spaces.

Lemma .[] Let PCbe the metric projection of H onto C,and let V be aγ¯-strongly monotone and L-Lipschitzian continuous operator withγ¯ > and L> .Let t≥satisfy

γ¯>tL_{and > tγ¯.Then we know that}

z=PC(I–tV)z ⇔ Vz,y–z ≥ ⇔ z=PC(I–V)z.

By Lemma ., we have the following lemma.

Lemma . Let V:H→H be aγ¯-strongly monotone and L-Lipschitzian continuous op-erator 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 exists 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 ≥, ∀q∈C.

Lemma .[] Let B be a maximal monotone mapping on H.Let Jrbe the resolvent of B defined by Jr= (I+rB)–for each r> .Then the following hold:

(i) For eachr> ,Jris single-valued and firmly nonexpansive; (ii) For eachr> ,D(Jr) =HandFix(Jr) ={x∈D(B) : ∈Bx};

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

s–t

s Jsx–Jtx,Jsx–x ≥ Jsx–Jtx 

for all s,t> and x∈H.In particular,

Jsx–Jtx ≤ | s–t|

s Jsx–x

for all s,t> and x∈H.

Letα,β∈R,Tbe a generalized hybrid mapping [] ifαTx–Ty+ ( –α)Ty–x≤

βTx–y_{+ ( –β)x–y}_{for allx,y∈C.}

Lemma .[] Let C be a nonempty closed convex subset of a real Hilbert space H.Let T:C→H be a generalized hybrid mapping,then F(T) =F(Tˆ).

Remark . IfTis a generalized hybrid mapping withFix(T)=∅. By the definition ofT

and Lemma ., we have thatTis a quasi-nonexpansive mapping withF(T) =F(Tˆ).

Lemma .[] Let{an} be a sequence of real numbers such that there exists a subse-quence {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 for all

(sufficiently large)numbers k∈N:

amk ≤amk+ and ak≤amk+.

In fact,mk=max{j≤k:aj<aj+}.

numbers with∞_n=un<∞,{tn}be a sequence of real numbers withlim suptn≤.Suppose that an+≤( –αn)an+αntn+unfor each n∈N.Thenlimn→∞an= .

We know that the equilibrium problem is to findz∈Csuch that

(EP) g(z,y)≥ for eachy∈C,

whereg:C×C→Ris a bifunction. This problem includes fixed point problems, opti-mization problems, variational inequality problems, Nash equilibrium problems, minimax inequalities, and saddle point problems as special cases. (For examples, one can see [] and related literatures.)

The solution set of equilibrium problem (EP) is denoted byEP(g). For solving the equi-librium problem, let us assume that the bifunctiong:C×C→Rsatisfies the following 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.

We have the following result from Blum and Oettli [].

Theorem .[] Let C be a nonempty closed convex subset of a real Hilbert space H.Let g:C×C→Rbe a bifunction which satisfies conditions(A)-(A).Then,for each r> 

and each x∈H,there exists z∈C such that

g(z,y) +

ry–z,z–x ≥

for all y∈C.

In , Combettes and Hirstoaga [] established the following important properties of a resolvent operator.

Theorem .[] Let C be a nonempty closed convex subset of a real Hilbert space H,and let g:C×C→Rbe a function satisfying conditions(A)-(A).For r> ,define Trg:H→C by

T_rgx= z∈C:g(z,y) +

ry–z,z–x ≥,∀y∈C

for all x∈H.Then the following hold:

(i) Trg is single-valued;

(ii) Trgis firmly nonexpansive,that is,Trgx–Trgy≤ x–y,Trgx–Trgyfor allx,y∈H; (iii) {x∈H:Trgx=x}={x∈C:g(x,y)≥,∀y∈C};

(iv) {x∈C:g(x,y)≥,∀y∈C}is a closed and convex subset ofC.

3 Convergence theorems of hierarchical problems

LetHbe a real Hilbert space, and letIbe an identity mapping onH,Cbe a nonempty closed convex subset ofH. For eachi= , , letκi>  and letBibe aκi-inverse-strongly

monotone mapping ofCintoH. LetGibe a maximal monotone mapping onHsuch that

the domain ofGiis included inCfor eachi= , . LetJλ= (I+λG)–andTr= (I+rG)–

for eachλ>  andr> . Let{θn} ⊂Hbe a sequence. LetVbe aγ¯-strongly monotone and L-Lipschitzian continuous operator withγ¯>  andL> . Throughout this paper, we use these notations and assumptions unless specified otherwise.

The following strong convergence theorem for hierarchical problems is one of our main results of this paper.

Theorem . Let T:C→H be a quasi-nonexpansive mapping withFix(T) =Fix(Tˆ)such that F(T)∩(B+G)–∩(B+G)–=∅.Takeμ∈Ras follows:

 <μ<γ¯

L.

Let{xn} ⊂H be defined by

(.)

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

x∈C chosen arbitrarily,

yn=Jλn(I–λnB)Trn(I–rnB)xn,

sn=Tyn,

xn+=αnxn+ ( –αn)(βnθn+ ( –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)∩(B+G)–∩(B+G)–(¯x–Vx¯+θ).This pointx is also¯ a unique solution of the following hierarchical variational inequality:

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

Proof Take anyx¯∈Fix(T)∩(B+G)–∩(B+G)– and letx¯be fixed. Thenx¯=Jλn(I–

λnB)¯xandx¯=Trn(I–rnB)¯x. Letun=Trn(I–rnB)xn. For eachn∈N, we have

un–x¯

=Trn(I–rnB)xn–Trn(I–rnB)x¯ 

≤(xn–x¯) –rn(Bxn–Bx¯) 

≤ xn–x¯– rnxn–x¯,Bxn–Bx¯+rnBxn–Bx¯

≤ xn–x¯– rnκBxn–Bx¯+rnBxn–Bx¯

≤ xn–x¯–rn(κ–rn)Bxn–Bx¯

and

yn–x¯

=Jλn(I–λnB)un–Jλn(I–λnB)¯x



≤(un–x¯) –λn(Bun–Bx¯) 

≤ un–x¯– λnun–x¯,Bun–Bx¯+λnBun–Bx¯

≤ un–x¯– λnκBun–Bx¯+λnBun–Bx¯

≤ un–x¯–λn(κ–λn)Bun–Bx¯

≤ un–x¯

≤ xn–x¯. ()

SinceTis a quasi-nonexpansive mapping, we obtain that

sn–x¯=Tyn–x¯ ≤ yn–x¯ ≤ un–x¯ ≤ xn–x¯. ()

Letzn=βnθn+ (I–βnV)sn, we have that

zn–x¯=βnθn+ (I–βnV)sn–x¯

≤βnθn–Vx¯+(I–βnV)(sn–x¯)

≤βnθn–Vx¯+(I–βnV)sn– (I–βnV)¯x. ()

Putτ=γ¯–L_μ, we have that

(I–βnV)sn– (I–βnV)¯x 

=sn–x¯– βnsn–x¯,Vsn–Vx¯+βnVsn–Vx¯

≤ sn–x¯– βnγ¯sn–x¯+βnLsn–x¯

≤ – βnγ¯+βnL _s

n–x¯

≤ – βnτ–βn

Lμ–βnL

sn–x¯

≤ – βnτ+βnτ

sn–x¯

≤( –βnτ)xn–x¯.

Since  –βnτ> , we obtain that

(I–βnV)sn– (I–βnV)¯x≤( –βnτ)xn–x¯. ()

We have from () and () that

Thus, we obtain from the definition ofxnand () that

xn+–x¯=αnxn+ ( –αn)

βnθn+ (I–βnV)sn

–x¯

≤αnxn–x¯+ ( –αn)βnθn+ (I–βnV)sn

–x¯

≤αnxn–x¯+ ( –αn)

βnθn–Vx¯+ ( –βnτ)xn–x¯

≤ – ( –αn)βnτ

xn–x¯+βn( –αn)τ

θn–Vx¯

τ

≤max xn–x¯,

θn–Vx¯

τ

≤maxxn–x¯,M

,

whereM=max{θn–Vx¯

τ ,n∈N}. By induction, we deduce

xn–x¯ ≤max

x–x¯,M

.

This implies that the sequence{xn}is bounded. Furthermore,{un},{zn},{yn}and{sn}are

bounded.

By the definition of{xn}, we have that

xn+–xn=αnxn+ ( –αn)

βnθn+ (I–βnV)sn

–xn

= ( –αn)

βnθn+ (I–βnV)sn

–xn

= ( –αn)[βnθn–βnVsn+sn–xn]. ()

By (), we have that

xn+–xn,xn–x¯

=( –αn)[βnθn–βnVsn+sn–xn],xn–x¯

= ( –αn)βnθn,xn–x¯– ( –αn)βnVsn,xn–x¯+ ( –αn)sn–xn,xn–x¯. ()

By () and (), we have that

xn+–x¯–xn–x¯–xn+–xn

= ( –αn)βnθn,xn–x¯– ( –αn)βnVsn,xn–x¯

+ ( –αn)

sn–x¯–xn–x¯–sn–xn

. ()

By () and (), we have that

xn+–x¯–xn–x¯–xn+–xn

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

By (), we obtain that

xn+–xn

≤( –αn)

βnθn–Vsn+sn–xn 

= ( –αn)

β_nθn–Vsn+sn–xn+ βnθn–Vsnsn–xn

. ()

By () and (), we have that

xn+–x¯–xn–x¯

≤( –αn)βnθn,xn–x¯– ( –αn)βnVsn,xn–x¯– ( –αn)sn–xn

+ ( –αn)

β_nθn–Vsn+sn–xn+ βnθn–Vsnsn–xn

≤( –αn)βnθn,xn–x¯– ( –αn)βnVsn,xn–x¯– ( –αn)αnsn–xn

+ ( –αn)

β_nθn–Vsn+ βnθn–Vsnsn–xn

.

Hence, we obtain that

xn+–x¯–xn–x¯+ ( –αn)αnsn–xn

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

+ ( –αn)

β_nθn–Vsn+ βnθn–Vsnsn–xn

. ()

We will divide the proof into two cases as follows.

Case : There exists a natural numberNsuch thatxn+–x¯ ≤ xn–x¯for eachn≥N.

So,limn→∞xn–x¯exists. Hence, it follows from (), (i), and (ii) that

lim

n→∞sn–xn= . ()

By (), (), (i), and (ii), we have that

lim

n→∞xn+–xn= . ()

We also have that

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

By (), (iv), and (ii), we have that

lim

n→∞zn–sn= . ()

By () and (), we have that

lim

By () and (), we have that

sn–x¯≤ un–x¯≤ xn–x¯–rn(κ–rn)Bun–Bx¯.

Therefore,

rn(κ–rn)Bun–Bx¯≤ xn–x¯–sn–x¯

≤ xn–sn

sn–x¯+xn–x¯

. ()

Thus, by (), (), and (iii), we have that

lim

n→∞Bun–Bx¯= . ()

SinceTrnis firmly nonexpansive, we have from () that

un–x¯

= Trn(I–rnB)xn–Trn(I–rnB)¯x



≤un–x¯, (I–rnB)xn– (I–rnB)¯x

≤un–x¯,xn–x¯– rnun–x¯,Bxn–Bx¯

≤ un–x¯+xn–x¯–un–xn

– rnxn–x¯,Bxn–Bx¯– rnun–xn,Bxn–Bx¯

≤ un–x¯+xn–x¯–un–xn

– λnκBxn–Bx¯+ rnxn–un,Bxn–Bx¯

≤ un–x¯+xn–x¯–un–xn+ rnxn–unBxn–Bx¯. ()

By () and (), we have that

sn–x¯≤ un–x¯

≤ xn–x¯–un–xn+ rnxn–unBxn–Bx¯.

Therefore,

un–xn

≤ xn–x¯–sn–x¯+ rnxn–unBxn–Bx¯

≤ xn–sn

xn–x¯+sn–x¯

+ rnxn–unBxn–Bx¯. ()

Thus, by (), (), and (), we have that

lim

n→∞un–xn= . ()

By () and (), we have that

Therefore,

λn(κ–λn)Bun–Bx¯≤ xn–x¯–sn–x¯

≤ xn–sn

sn–x¯+xn–x¯

. ()

Thus, by (), (), and (iii), we have that

lim

n→∞Bun–Bx¯= . ()

SinceJλnis firmly nonexpansive, we have from () that

yn–x¯

= Jλn(I–λnB)un–Jλn(I–λnB)¯x



≤yn–x¯, (I–λnB)un– (I–λnB)¯x

≤yn–x¯,un–x¯– λn

yn–x¯,Bun–Bx¯

≤ yn–x¯+un–x¯–yn–un

– λnun–x¯,Bun–Bx¯– λnyn–un,Bun–Bx¯

≤ yn–x¯+un–x¯–yn–un

– λnκBun–Bx¯+ λnun–yn,Bun–Bx¯

≤ yn–x¯+un–x¯–yn–un+ λnun–ynBun–Bx¯. ()

By () and (), we have that

sn–x¯≤ yn–x¯

≤ un–x¯–yn–un+ λnun–ynBun–Bx¯

≤ xn–x¯–yn–un+ λnun–ynBun–Bx¯.

Therefore,

yn–un

≤ xn–x¯–sn–x¯+ λnun–ynBun–Bx¯

≤ xn–sn

xn–x¯+sn–x¯

+ λnun–ynBun–Bx¯. ()

Thus, by (), (), and (), we have that

lim

n→∞yn–un= . ()

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

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

¯

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

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

We want to show that

lim sup

n→∞

Vx¯–θ,zn–x¯ ≥.

Without loss of generality, there exists a subsequence{znk}of{zn}such thatznk wfor

somew∈Hand

lim sup

n→∞

Vx¯–θ,zn–x¯= lim k→∞

(Vx¯–θ,znk–x¯

. ()

By () and (), we have that

lim

n→∞un–zn= 

andunk w. 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 () and Lemma .,

we have that

un_kj–Jλ¯(I–λ¯B)un_kj

≤un_kj–Jλ

nkj(I–λnkjB)unkj+Jλ_nkj(I–λ¯B)unkj –Jλ¯(I–λ¯B)un_kj

+Jλ_nkj(I–λn_kjB)un_kj –Jλ_nkj(I–λ¯B)un_kj

≤ un_kj –yn_kj+|λn_kj–λ¯|Bun_kj

+| λn_kj –λ¯|

¯

λ Jλ(¯ I–λ¯B)unkj – (I–λ¯B)unkj→. () By (),un_kj w, Lemma . and .,w∈Fix(Jλ¯(I–λ¯B)) = (B+G)–. Without loss

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

{rn_kj}converges to a number¯r∈[a,b]. By () and Lemma ., we have that

xn_kj–Tr¯(I–rB¯ )xn_kj

≤xn_kj –Tr_nkj(I–rn_kjB)xn_kj+Tr_nkj(I–rn_kjB)xn_kj –Tr_nkj(I–rB¯ )xn_kj

+Tr_nkj(I–rB¯ )un_kj–Tr¯(I–rB¯ )un_kj

≤ xn_kj –un_kj+|rn_kj–r¯|Bun_kj

+|

rn_kj –r¯|

¯

By (),xn_kjw, Lemma ., we have thatw∈Fix(Tr¯(I–rB¯ )) = (B+G)–. From (),

(), and (), we have that

Tyn–yn=sn–yn ≤ sn–xn+xn–un+un–yn →.

SinceFix(T) =Fix(Tˆ), we have fromTynj–ynj → andyn_kjwthatw∈F(T). Hence,

w∈Fix(T)∩(B+G)–∩(B+G)–. So, we have from () and () that

lim sup n→∞

Vx¯–θ,zn–x¯

= lim k→∞

Vx¯–θ,znk–x¯

=Vx¯–θ,w–x¯ ≥. ()

Letzn=βnθn+ ( –βnV)sn. Then it follows from () that

zn–x¯ =βnθn+ ( –βnV)sn–x¯ 

=βn(θn–Vx¯) + ( –βnV)(sn–x¯)

≤( –βnV)(sn–x¯) 

+ βnθn–Vx¯,zn–x¯

≤( –βnτ)xn–x¯+ βnθn–Vx¯,zn–x¯. ()

Thus, we obtain from the definition ofxnand () that

xn+–x¯

=αnxn+ ( –αn)

βnθn+ ( –βnV)sn

–x¯

≤αnxn–x¯+ ( –αn)βnθn+ ( –βnV)sn

–x¯ 

≤αnxn–x¯+ ( –αn)

( –βnτ)xn–x¯+ βnθn–Vx¯,zn–x¯

≤αn+ ( –αn)( –βnτ)

xn–x¯+ βn( –αn)θn–Vx¯,zn–x¯

≤ – ( –αn)

βnτ– (βnτ)

xn–x¯+ βn( –αn)θn–Vx¯,zn–x¯

≤ – ( –αn)βnτ

xn–x¯+ ( –αn)(βnτ)xn–x¯

+ βn( –αn)θn–θ,zn–x¯+ βn( –αn)θ–Vx¯,zn–x¯

≤ – ( –αn)βnτ

xn–x¯+ ( –αn)βnτ

βnτxn–x¯



+θn–θ,zn–x¯

τ +

θ–Vx¯,zn–x¯

τ

. ()

By (), (), assumptions, and Lemma ., we know thatlimn→∞xn=x¯, where

¯

x=PFix(T)∩(B+G)–∩(B+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

Hence, it follows from () and () that

( –αmj)αmjsmj–xmj



≤( –αmj)βmjθmj,xmj–x¯– ( –αmj)βmjVsmj,xmj–x¯

+ ( –αmj)

_β

mjθmj–Vsmj

_{+ β}

mjθmj–Vsmjsmj–xmj

()

for eachj∈N. Hence, it follows from (), (i), and (ii) that

lim

j→∞smj–xmj= . ()

We want to 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¯=klim→∞Vx¯–θ,zmjk –x¯. ()

With the similar argument as in the proof of Case , we havew∈Fix(T)∩(B+G)–∩

(B+G)–. So, we have from () and () that

lim sup j→∞

Vx¯–θ,zmj–x¯= lim

k→∞Vx¯–θ,zmjk –x¯=Vx¯–θ,w–x¯ ≥. ()

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

xmj+–x¯



≤ – ( –αmj)βmjτ

xmj–x¯

_{+ ( –α}

mj)(βmjτ)

_x mj–x¯



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

Fromxmj–x¯ ≤ xmj+–x¯, we have that

( –αmj)βmjτxmj–x¯



≤( –αmj)(βmjτ)

_x mj–x¯

_{+ β}

mj( –αmj)θn–θ,zmj–x¯

+ βmj( –αmj)θ–Vx¯,zmj–x¯. ()

Since ( –αmj)βmj> , we have that

τxmj–x¯

_≤β

mjτxmj–x¯

_{+ θ}

n–θ,zmj–x¯+ θ–Vx¯,zmj–x¯. ()

By (), (), and assumptions, we know that

lim

By (), (), and assumptions, we know that

lim

j→∞xmj+–xmj= .

Thus, we have that

lim

j→∞xmj+–x¯= . ()

By () and (), we have that

lim

j→∞xj–x¯ ≤j→∞limxmj+–x¯= .

Therefore, the proof is completed.

LetCandQbe nonempty closed convex subsets of real Hilbert spacesHandH,

re-spectively. LetIdenote the identy mapping onHand onH. LetGibe a maximal

mono-tone mapping onH such that the domain ofGi is included inC for eachi= , . Let Jλ= (I+λG)–andTr= (I+rG)–for eachλ>  andr> . LetAi:H→Hbe a bounded

linear operator, and letA∗_i be the adjoint ofAifor eachi= , . Now, we recall the following

multiple sets split feasibility problem:

(MSFPFF) Findx¯∈Hsuch thatx¯∈Fix(Jλn)∩Fix(Trn),Ax¯∈Fix(F), andAx¯∈Fix(F) for eachn∈N.

In order to study the convergence theorems for the solution set of multiple sets split feasibility problem (MSFPFF), we must give an essential result in this paper.

Theorem . Given anyx¯∈H. (i) Ifx¯is a solution of(MSFPFF),then

Jλn(I–ρnA∗(I–F)A)Trn(I–σnA∗(I–F)A)¯x=x¯for eachn∈N. (ii) Suppose thatJλn(I–ρnA∗(I–F)A)Trn(I–σnA∗(I–F)A)¯x=x¯with

 <ρn<_A

+, <σn<



A+for eachn∈Nand the solution set of(MSFPFF)is nonempty.Thenx¯is a solution of(MSFPFF).

Proof (i) Suppose thatx¯∈His a solution of (MSFPFF). Thenx¯∈Fix(Jλn)∩Fix(Trn),Ax¯∈

Fix(F), andAx¯∈Fix(F) for eachn∈N. It is easy to see that

Jλn

I–ρnA∗(I–F)A

Trn

I–σnA∗(I–F)A

¯

x=x¯

for eachn∈N.

(ii) Since the solution set of (MSFPFF) is nonempty, there existsw¯ ∈H such thatw¯ ∈ Fix(Jλn)∩Fix(Trn),Aw¯∈Fix(F), andAw¯ ∈Fix(F). So,

¯

w∈Fix(Jλn)∩Fix

I–ρnA∗(I–F)A

∩Fix(Trn)∩Fix

I–σnA∗(I–F)A

=∅. ()

By Lemma ., we have that

A∗_(I–F)Ais

 A

For eachn∈N, by (),  <ρn<_A

+, and Lemma .(ii), (iii), we know that

I–ρnA∗(I–F)Ais averaged. ()

By Lemma . again, we have that

A∗_(I–F)Ais

 A

-ism. ()

For eachn∈N, by (),  <σn<_A

+, and Lemma .(ii), (iii), we know that

I–σnA∗(I–F)Ais averaged. ()

On the other hand, for eachn∈N, sinceJλn, andTrnare firmly nonexpansive mappings,

it is easy to see that

JλnandTrnare



 averaged. ()

Hence, by (), (), (), and Lemma .(v), we have that for eachn∈N,

¯

x∈FixJλn

I–ρA∗_(I–F)A

Trn

I–σA∗_(I–F)A

=Fix(Jλn)∩Fix

I–ρA∗_(I–F)A

∩Fix(Trn)∩Fix

I–σA∗_(I–F)A

.

This implies that for eachn∈N,

¯

x=Jλn

I–ρA∗_(I–F)A

¯

x and x¯=Trn

I–σA∗_(I–F)A

¯

x.

By Lemma ., for eachn∈N,

¯

x–ρA∗_(I–F)Ax¯

–x¯,x¯–w≥ for eachw∈Fix(Jλn),

¯

x–ρA∗_(I–F)Ax¯

–x¯,x¯–w≥ for eachw∈Fix(Trn).

That is, for eachn∈N,

A∗_(I–F)Ax¯,x¯–w

≤ for eachw∈Fix(Jλn),

A∗_(I–F)Ax¯,x¯–w

≤ for eachw∈Fix(Trn).

()

For eachn∈N, by () and the fact thatA∗_i is the adjoint ofAifor eachi= , ,

Ax¯–FAx¯,Ax¯–Aw ≤ for eachw∈Fix(Jλn),

Ax¯–FAx¯,Ax¯–Aw ≤ for eachw∈Fix(Trn).

()

On the other hand, by Lemma . again,

Ax¯–FAx¯,v–FAx¯ ≤ for eachv∈Fix(F),

Ax¯–FAx¯,v–FAx¯ ≤ for eachv∈Fix(F).

For eachn∈N, by () and (),

Ax¯–FAx¯,v–FAx¯+Ax¯–Aw ≤,

Ax¯–FAx¯,v–FAx¯+Ax¯–Aw ≤

()

for eachw∈Fix(Jλn)∩Fix(Trn),v∈Fix(F), andv∈Fix(F).

That is, for eachn∈N,

Ax¯–FAx¯≤ Ax¯–FAx¯,Aw–v,

Ax¯–FAx¯≤ Ax¯–FAx¯,Aw–v

()

for eachw∈Fix(Jλn)∩Fix(Trn),v∈Fix(F), andv∈Fix(F).

Sincew¯ is a solution of multiple sets split feasibility problem (MSFPFF), we know that

¯

w∈Fix(Jλn)∩Fix(Trn),Aw¯∈Fix(F), andAw¯∈Fix(F) for eachn∈N. So, it follows from

() thatAx¯=Fix(F) andAx¯=Fix(F). Furthermore,x¯∈Fix(Jλn) andx¯∈Fix(Trn) for

eachn∈N. Therefore,x¯is a solution of (MSFPFF).

Applying Theorem . and Theorem ., we can find the solution of the following hier-archical problem.

Theorem . Let T:C→H be a quasi-nonexpansive mapping with Fix(T) =Fix(Tˆ).

Let C and Q be two nonempty closed convex subsets of real Hilbert spaces H and H, respectively.For each i= , ,let Fi be a firmly nonexpansive mapping of Hinto H,let Ai:H→Hbe a bounded linear operator,and let A∗i be the adjoint of Ai.Suppose that the solution set of(MSFPFF)isandFix(T)∩=∅.Let{xn} ⊂H be defined by

(.)

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

x∈C chosen arbitrarily,

yn=Jλn(I–λnA∗(I–F)A)Trn(I–rnA∗(I–F)A)xn, sn=Tyn,

xn+=αnxn+ ( –αn)(βnθn+ ( –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<_A

+,and <a≤rn≤b<



A+; (iv) limn→∞θn=θfor someθ∈H.

Thenlim_n→∞xn=x¯,wherex¯=PFix(T)∩(x¯–Vx¯+θ).This pointx is also a unique solution¯ of the following hierarchical problem:Findx¯∈Fix(T)∩such that

Vx¯–θ,q–x¯ ≥, ∀q∈Fix(T)∩.

Proof Since Fi is firmly nonexpansive, it follows from Lemma . that we have that A∗_i(I–Fi)Ai:C→H is_A

i-ism for eachi= , . For eachi= , , putBi=A

∗

i(I–Fi)Ai

Since the solution set of (MSFPFF) is nonempty, by (), we have for eachn∈N

¯

w∈FixJλn

I–λnA∗(I–F)A

∩FixTrn

I–rnA∗(I–F)A

=∅. ()

This implies that for eachn∈N,

¯

w∈FixJλn(I–λnB)

∩FixTrn(I–rnB)

=∅. ()

So,

¯

w∈(B+G)–∩(B+G)–=∅. ()

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

¯

x=PFix(T)∩(B+G)–∩(B+G)–(x¯–Vx¯+θ).

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

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

that is, for eachn∈N,

¯

x=Jλn(I–λnB)¯x=Jλn

I–λnA∗(I–F)A

¯

x ()

and

¯

x=Trn(I–rnB)x¯=Trn

I–rnA∗(I–F)A

¯

x. ()

This implies that for eachn∈N,

¯

x=Jλn

I–λnA∗(I–F)A

Trn

I–rnA∗(I–F)A

¯

x. ()

By assumptions, (), and Theorem .(ii), we know thatx¯is a solution of (MSFPFF).

Fur-thermore,x¯∈Fix(T). Therefore,x¯∈Fix(T)∩. By the same argument as (), (), and (), we also have

Vx¯–θ,q–x¯ ≥, ∀q∈Fix(T)∩.

Therefore, the proof is completed.

4 Applications to mathematical programming with multiple sets split feasibility constraints

By Theorem ., we obtain mathematical programming with fixed point and multiple sets split feasibility constraints.

Theorem . Let T:C→H be a quasi-nonexpansive mapping withFix(T) =Fix(Tˆ).In Theorem.,let h:C→Rbe a convex Gâteaux differential function with Gâteaux deriva-tive V.Let

=q∈H:q∈Fix(Jλn)∩Fix(Trn),Aq∈Fix(F),Aq∈Fix(F),n∈N

.

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

mathematical programming with fixed point and multiple sets split feasibility constraints:

minq∈Fix(T)∩h(q).

Proof Putθ=  in Theorem .. Then, by Theorem ., there existsx¯∈Fix(T)∩such that

Vx¯,q–x¯ ≥, ∀q∈F(T)∩. ()

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

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 () and (), it is easy to see thath(x¯)≤h(q) for allq∈Fix(T)∩.

We can apply Theorem . to study the mathematical programming of a quadratic func-tion with fixed point and multiple sets split feasibility constraints.

Theorem . Let T:C→H be a quasi-nonexpansive mapping withFix(T) =Fix(Tˆ).In Theorem.,let B:C→C be a strongly positive self-adjoint bounded linear operator and a∈H.Let

=q∈H:q∈Fix(Jλn)∩Fix(Trn),Aq∈Fix(F),Aq∈Fix(F),n∈N

.

Let{xn} ⊂H be defined by

(.)

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

x∈C chosen arbitrarily,

yn=Jλn(I–λnA∗(I–F)A)Trn(I–rnA∗(I–F)A)xn, sn=Tyn,

xn+=αnxn+ ( –αn)(βnθn+ (sn–βn(B(sn) –a)))

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

∞

n=βn=∞; (iii)  <a≤λn≤b<_A

+,and <a≤rn≤b<



A+; (iv) limn→∞θn= .

Thenlimn→∞xn=x¯.This pointx is also a unique solution of the mathematical program-¯ ming of a quadratic function with fixed point and multiple sets split feasibility constraints:

minq∈Fix(T)∩_Bq,q–a,q.

Proof Leth:C→Rbe defined by

h(x) = 

Bx,x–a,x.

It is easy to see thathis a convex function. SinceBis a strongly positive self-adjoint oper-ator, there existsη>  such thatBx,x ≥ηx_{. This implies that}

Bx–By,x–y=B(x–y),x–y≥ηx–y. ()

Therefore,

Bx,x+By,y ≥ Bx,y+By,x. ()

From this we can show thathis a convex function. Indeed, for anyx,y∈Cand anyλ∈ [, ]. It follows from () that

hλx+ ( –λ)y

= 

Bλx+ ( –λ)y,λx+ ( –λ)y–a,λx+ ( –λ)y

= 

λBx+ ( –λ)By,λx+ ( –λ)y–a,λx+ ( –λ)y

= λ

_Bx,x+

λ( –λ)Bx,y+ 

λ( –λ)By,x+  ( –λ)

_By,y

–λa,x– ( –λ)a,y

≤ λ

_Bx,x+

λ( –λ)

Bx,x+By,y+ ( –λ)

_By,y

–λa,x– ( –λ)a,y

≤

λBx,x+ 

( –λ)By,y–λa,x– ( –λ)a,y ≤λh(x) + ( –λ)h(y).

LetV(x) =B(x) –afor allx∈C. It is easy to see thatV is the Gâteaux derivative of h. Indeed, for anyu∈H,x∈Cand anyt∈[, ]. SinceBis a self-adjoint bounded linear operator, we see that for eachu∈H,

h(x)(u) =lim t→

h(x+tu) –h(x)

=lim t→ 

B(x+tu),x+tu–a,x+tu– 

Bx,x+a,x t

=lim t→ 

tBu+Bx,tu+x–a,tu+x– 

Bx,x+a,x t

=lim t→  [t

_{Bu,u+tBu,x+tBx,u+Bx,x] –ta,u–a,x–}

Bx,x+a,x t

=lim t→

 

tBu,u+Bu,x+Bx,u–a,u

=Bx,u–a,u=Bx–a,u=Vx,u.

Therefore,V is the Gâteaux derivative ofh. SinceBis a strongly positive bounded linear operator inH, we have that

Vx–Vy=Bx–a–By+a=Bx–By ≤ Bx–y.

This implies thatV is Lipschitz, and we have that

Vx–Vy,x–y=Bx–a–By+a,x–y=B(x–y),x–y≥ηx–y.

This implies thatV is strongly monotone. Therefore, Theorem . follows from

Theo-rem ..

Theorem . Let T:C→H be a quasi-nonexpansive mapping with Fix(T) =Fix(Tˆ).

In Theorem.,let h:C→Rbe a convex Gâteaux differential function with Gâteaux derivative V.Letibe a maximal monotone mapping on Hsuch that the domain ofiis included in Q for each i= , ,where Q is a closed convex subset of H.Let

=q∈H:q∈G– ∩G–,Aq∈– ,Aq∈– 

.

Then lim_n→∞xn = x¯. This point x is also a unique solution of the mathematical¯ programming with fixed point and multiple sets split feasibility problem constraints:

minq∈Fix(T)∩h(q).

Proof LetJλ= (I+λG)–,Tr= (I+rG)–,F= (I+λ)–, andF= (I+r)–for each

λ>  andr>  in Theorem .. ThenFix(Jλ) =G–

 ,Fix(Tr) =G– ,Fix(F) =– , and Fix(F) =– . Therefore, Theorem . follows from Theorem ..

Takahashiet al.[] showed the following result.

Lemma .[] Let C be a nonempty closed convex subset of a Hilbert space H,and let g:C×C→Rbe a bifunction satisfying conditions(A)-(A).Define Agas follows:

(L.) Agx= ⎧ ⎨ ⎩

{z∈H:g(x,y)≥ y–x,z,∀y∈C}, ∀x∈C,

∅, ∀x∈/C.

ThenEP(g) =A–

Now, we consider the following multiple sets split equilibrium problem:

(MSEP) Findx¯∈Hsuch thatx¯∈EP(g)∩EP(g),Ax¯∈EP(f)andAx¯∈EP(f).

Applying Theorems . and ., Lemma ., we can find the minimum norm solution of (MSEP) and mathematical programming with fixed point and multiple sets split equi-librium constraints.

Theorem . Let T:C→H be a quasi-nonexpansive mapping with Fix(T) =Fix(Tˆ).

Let C and Q be nonempty closed convex subsets of Hilbert spaces Hand H,respectively. Let g:C×C→R,g:C×C→R,f:Q×Q→R,and f:Q×Q→Rbe bifunctions satisfying conditions(A)-(A),and let Tg

λn,T

g

rn,T

f

λn,T

f

rn be the resolvent of g,g,f,f, respectively,forλn> ,rn> .For i= , ,let Ai:H→H be a bounded linear operator, and let A∗_i be the adjoint of Ai.Let h:C→Rbe a convex Gâteaux differential function with Gâteaux derivative V.Suppose thatis the solution set of(MSEP)andFix(T)∩=∅.

Let{xn} ⊂H be defined by

(.) ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

x∈C chosen arbitrarily,

yn=Tλgn(I–λnA

∗

(I–T f

λn)A)T

g

rn(I–rnA∗(I–T f

rn)A)xn,

sn=Tyn,

xn+=αnxn+ ( –αn)(βnθn+ ( –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<_A_+,and <a≤rn≤b<_A_+; (iv) limn→∞θn= .

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

the mathematical programming with fixed point and multiple sets split equilibrium con-straints:min_q∈Fix(T)∩h(q).

Proof DefineAgas (L.). By Lemma ., we know thatEP(g) =A–g  andAgis a maximal

monotone operator with the domain ofAg⊂C. Furthermore, for anyx∈Handr> , the

resolventTrgofgcoincides with the resolvent ofAg,i.e.,Trgx= (I+rAg)–x. By Theorem ., Tf

λn,T

f

rnare firmly nonexpansive mappings.

PutG=Ag,G=Ag,F=T

f

λnandF=T

f

rnin Theorem .. ThenJλnx= (I+λnAg)

–_x=

Tg

λnx, Trnx= (I+rnAg)

–_x=Tg

rnx. By Theorem ., we have that Fix(Jλn) =Fix(T

g

λn) =

EP(g), Fix(Trn) =Fix(T

g

rn) =EP(g), Fix(F) =Fix(T

f

λn) =EP(f) andFix(F) =Fix(T

f

rn) =

EP(f). Therefore, the solution set of (MSEP) coincides with the solution set of (MSFPFF).

Therefore, by Theorem ., we get the result.

The following unique minimum norm common solution of a fixed point problem and multiple sets split equilibrium problem is a special case of Theorem ..

Corollary . Let C and Q be nonempty closed convex subsets of Hilbert spaces Hand H,respectively.Let g:C×C→R,g:C×C→R,f:Q×Q→R,and f:Q×Q→R be bifunctions satisfying conditions(A)-(A),and let Tg

λn,T

g

rn,T

f

λn,T

f

rnbe the resolvent of