Solutions for a variational inclusion problem with applications to multiple sets split feasibility problems

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R E S E A R C H

Open Access

Solutions for a variational inclusion problem

with applications to multiple sets split

feasibility problems

Lai-Jiu Lin

1*

_{, Yew-Dann Chen}

1,2

_{and Chih-Sheng Chuang}

3

*_{Correspondence:}

maljlin@cc.ncue.edu.tw

1_{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 the set of common solutions for two variational inclusion problems in a real Hilbert space and establish a strong convergence theorem of this problem. As applications, we study unique minimum norm solutions of the following problems: multiple sets split feasibility problems, system of convex constrained linear inverse problems, convex constrained linear inverse problems, split feasibility problems, convex feasibility problems. We establish iteration processes of these problems and show the strong convergence theorems of these iteration processes. MSC: 47J20; 47J25; 47H05; 47H09

Keywords: ﬁrmly nonexpansive mapping; proximal-contraction; multiple sets split feasibility problem; maximum monotone; convex feasibility problem

1 Introduction

LetC,C, . . . ,Cmbe nonempty closed convex subsets of a real Hilbert spaceH. The well-known convex feasibility problem (CFP) is to ﬁndx∗∈Hsuch that

x∗∈C∩C∩ · · · ∩Cm.

The convex feasibility problem has received a lot of attention due to its diverse applica-tions in mathematics, approximation theory, communicaapplica-tions, geophysics, control theory, biomedical engineering. One can refer to [, ].

IfCandCare closed vector spaces of a real Hilbert spaceH, in , von Neumann showed that any sequence{xn}is generated from the method of alternating projections:

x∈H,x=PCx,x=PCx,x=PCx, . . . . Then{xn}converges strongly to somex¯∈ C∩C. IfCandCare nonempty closed convex subsets ofH, Bregman [] showed that the sequence{xn}generated from the method of alternating projection converges weakly to a point inC∩C. Hundal [] showed that the strong convergence fails ifCandCare nonempty closed convex subsets ofH. Recently, Boikanyoet al.[] proposed the following process:

xn+=JβGn

αnu+ ( –αn)xn+en

, n= , , . . . . (.)

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and

xn=JρGn

λnu+ ( –λn)xn–+en

, n= , , . . . , (.)

whereu,x∈Hare given arbitrarily,GandGare two set-valued maximal monotone op-erators withJG

β = (I+βG)–, and{αn},{λn},{βn},{ρn},{en},{en}are sequences. Boikanyo

et al.[] proved that the sequence{xn}converges strongly to a pointx¯∈G–()∩G–() under suitable conditions.

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

x∗∈Csuch thatAx∗∈Q,

whereC,Qare nonempty closed convex subsets of real Hilbert spacesH,H, respec-tively.A:H→H is a bounded linear operator. The split feasibility problem (SFP) in finite dimensional real Hilbert spaces was first introduced by Censor and Elfving [] for modeling inverse problems which arise from medical image reconstruction. Since then, the split feasibility problem (SFP) has received much attention due to its applications in signal processing, image reconstruction, approximation theory, control theory, biomed-ical engineering, communications, and geophysics. For examples, one can refer to [, , –] and related literature. A special case of problem (SFP) is the convexly constrained linear inverse problem in a finite dimensional real Hilbert space []:

(CLIP) Findx¯∈Csuch thatAx¯=b,

whereCis a nonempty closed convex subset of a real Hilbert spaceHandbis a given element of a real Hilbert spaceH, which has extensively been investigated by using the Landweber iterative method []:

xn+:=xn+γAT(b–Axn), n∈N.

Let C,C, . . . ,Cm be nonempty closed convex subsets of H, let Q,Q, . . . ,Qm be nonempty closed convex subsets ofH, and letA,A, . . . ,Am:H→Hbe bounded lin-ear operators. The well-known multiple sets split feasibility problem (MSSFP) is to ﬁnd

x∗∈Hsuch that

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

The multiple sets split feasibility problem (MSSFP) contains convex feasibility problem (CFP) and split feasibility problem (SFP) as special cases [, , ]. Indeed, Censoret al.

[] ﬁrst studied this type of problem. Xu [] and Lopezet al.[] also studied this type of problem. In , Boikanyo and Moroşanu [] gave the following algorithm:

vn+:=anu+bnvn+cnJαGnvn, n∈N∪ {},

vn:=fnu+gnvn–+hnJβGnvn–, n∈N,

(.)

whereuis given inH, andG,G are two set-valued maximal monotone mappings on

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proved that{vn}in (.) converges strongly to somex¯∈G–()∩G–() under suitable conditions.

Motivated by the above works, we consider the following algorithm:

⎧ ⎪ ⎨ ⎪ ⎩

v∈His chosen arbitrarily,

vn+:=anu+bnvn+cnJδGn(I–δnB)vn, n∈N∪ {},

vn:=fnu+gnvn–+hnJγGn(I–γnB)vn–, n∈N,

(.)

where G,G are two set-valued maximal monotone mappings on a real Hilbert space

H,B,B:C→H are two mappings,{an},{bn},{cn},{fn},{gn}, and{hn}are sequences in [, ]. We show that the sequence{vn}generated by (.) converges strongly to some

¯

x∈(B+G)–()∩(B+G)–() under suitable conditions.

Algorithm (.) contains as special cases the inexact proximal point algorithm (e.g., [, ]) and the generalized contraction proximal point algorithm (e.g., []). Our conclusion extends and uniﬁes Boikanyo and Moroşanu’s result in [], Wang and Cui’s result in [] becomes a special case. For details, one can see Section .

In this paper, we ﬁrst study the set of common solutions for two variational inclusion problems in a Hilbert space and establish a strong convergence theorem of this problem. As applications, we study unique minimum norm solutions of the following problems: multiple sets split feasibility problems, system of convex constrained linear inverse prob-lems, convex constrained linear inverse probprob-lems, split feasibility probprob-lems, convex feasi-bility problems. We establish iteration processes of these problems and show strong con-vergence theorems of these iteration processes.

2 Preliminaries

Throughout this paper, letH,H,H,Hdenote real Hilbert spaces with the inner product

·,· and the norm · ; letNbe the set of all natural numbers andR+_{be the set of all} pos-itive real numbers. A set-valued mappingAwith domainD(A) onHis called monotone ifu–v,x–y ≥ for anyu∈Ax,v∈Ayand for allx,y∈D(A). A monotone operatorAis called maximal monotone if its graph{(x,y) :x∈D(A),y∈Ax}is not properly contained in the graph of any other monotone mapping. The set of all zero points ofAis denoted by

A–_(),_i.e._,_A–_{() =}_{_x_∈_H_{: }_∈_Ax_}_{. In what follows, we denote the strong convergence} and the weak convergence of{xn}tox∈Hbyxn→xandxnx, respectively. In order to facilitate our discussion, in the next section, we recall some facts. The following equality is easy to check:

αx+βy+γz

=αx+βy+γz–αβx–y–αγx–z–βγy–z (.) for eachx,y,z∈Handα,β,γ∈[, ] withα+β+γ = . Besides, we also have

x+y≤ x+ y,x+y (.)

for each x,y∈H. Let C be a nonempty closed convex subset of H, and a mapping

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is said to be quasi-nonexpansive if Fix(T)=∅andTx–y ≤ x–yfor allx∈Cand

y∈Fix(T). A mappingT:C→His said to be ﬁrmly nonexpansive if

Tx–Ty≤ x–y– (I–T)x– (I–T)y 

for everyx,y∈C. Besides, it is easy to see thatFix(T) is a closed convex subset ofCifT:

C→His a quasi-nonexpansive mapping. A mappingT:C→His said to beα -inverse-strongly monotone (α-ism) if

x–y,Tx–Ty ≥αTx–Ty

for allx,y∈Handα> .

The following lemmas are needed in this paper.

Lemma .[] Let A:H→Hbe a bounded linear operator,and let A∗be the adjoint

of A.Suppose that C is a nonempty closed convex subset ofH,and that G:C→His a

ﬁrmly nonexpansive mapping.Then A∗(I–G)A is_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 ofH,and let G:C→Hbe a ﬁrmly nonexpansive mapping.Suppose thatFix(G)is nonempty.Thenx–Gx,Gx–w ≥for each x∈Hand each w∈Fix(G).

LetCbe a nonempty closed convex subset ofH. Then, for eachx∈H, there is a unique elementx¯∈Csuch thatx–x¯=min_y∈Cx–y. Here, we setPCx=x¯andPCis said to be the metric projection fromHontoC.

Lemma .[] Let C be a nonempty closed convex subset ofH,and let PCbe the metric

projection fromHonto C.Thenx–PCx,PCx–y ≥for each x∈Hand each y∈C.

For a set-valued maximal monotone operatorGonHandr> , we may deﬁne an oper-atorJG

r :H→HwithJrG= (I+rG)–which is called the resolvent mapping ofGforr.

Lemma .[, ] Let G:H→Hbe a set-valued maximal monotone mapping.Then we have that

(i) for eachα> ,JG

α is a single-valued and ﬁrmly nonexpansive mapping.

(ii) D(JG

α) =HandFix(JαG) =G–().

(iii) x–JαGx ≤x–JβGxfor eachx∈Hand allα,β∈(,∞)with <α≤β. Lemma .[] Let G be a set-valued maximal monotone operator onH.For a> ,we deﬁne the resolvent J_aG= (I+aG)–.Then the following holds:

JαGx–JβGx



≤α–β

α

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for allα,β> and x∈H.In fact,

JαGx–JβGx ≤ |α–β|

α J

G

αx–x

for allα,β> and x∈H.

Lemma .[] Let{Sn}be a sequence of real numbers that does not decrease at

inﬁn-ity,in the sense that there exists a subsequence{Sni}i≥of{Sn}such that Sni<Sni+for all

i∈N.Consider the sequence{τ(n)}n≥n deﬁned byτ(n)=max{n≤k≤n:τk<τk+}for some suﬃciently large number n∈N.Then{τ(n)}n≥nis a nondecreasing sequence with

τ(n)→ ∞as n→ ∞,and for all n≥n,

Sτ(n)≤Sτ(n)+ and Sn≤Sτ(n)+.

In fact,max{Sτ(n),Sn} ≤Sτ(n)+.

Lemma .[] Let{an}and{bn}be two sequences in[, ].Let{en}be a sequence of

non-negative real numbers.Let{tn}and{kn}be two sequences of real numbers.Let{Sn}n∈Nbe

a sequence of nonnegative real numbers with

Sn+≤( –an)( –bn)Sn+antn+bnkn+en

for each n∈N.Assume that:

∞

n=

an=∞, lim sup

n→∞ tn≤, lim supn→∞ kn≤, and ∞

n=

en=∞.

Thenlimn→∞Sn= .

A mappingT:H→His said to be averaged ifT= ( –α)I+αS, whereS:H→His a nonexpansive mapping andα∈(, ).

Lemma .[] Let C be a nonempty closed convex subset of Hand T:C→Hbe a mapping.Then the following hold:

(i) Tis a nonexpansive mapping if and only ifI–T is _-inverse-strongly monotone

(_-ism).

(ii) IfSisν-ism,thenγS is_γν-ism.

(iii) Sis averaged if and only ifI–Sisν-ismfor someν>  .

Indeed,Sisα-averaged if and only ifI–Sis _(_α₎-ismforα∈(, ). (iv) If S and T are averaged,then the compositionST is also averaged.

(v) If the mappings {Ti}ni=are averaged and have a common ﬁxed point,then n

i=Fix(Ti) =Fix(T· · ·Tn)for eachn∈N.

Lemma .[] Let T be a nonexpansive self-mapping on a nonempty closed convex sub-set C ofH,and let{xn}be a sequence in C.If xnw andlimn→∞xn–Txn= ,then

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LetCbe a nonempty closed convex subset ofH. The indicator functionιCdeﬁned by

ιCx=

⎧ ⎨ ⎩

, x∈C,

∅, x∈/C

is a proper lower semicontinuous convex function and its subdiﬀerential∂ιCdeﬁned by

∂ιCx=

z∈H:y–x,z ≤ιC(y) –ιC(x),∀y∈H

is a maximal monotone operator []. Furthermore, we also deﬁne the normal coneNCu ofCatuas follows:

NCu=

z∈H:z,v–u ≤,∀v∈C.

We can deﬁne the resolventJ_λ∂iCof∂iCforλ> ,i.e.,

J_λ∂iCx= (I+λ∂iC)–x

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 that

u=Jλ∂iCx ⇔ x∈u+λ∂iCu

⇔ x–u∈λNCu

⇔ x–u,y–u ≤, ∀y∈C ⇔ u=PCx.

3 Main results

LetC,QandQbe nonempty closed convex subsets ofH,HandH, respectively. For eachi= ,  andκi> , letBibe aκi-inverse-strongly monotone mapping ofCintoH, and letGibe a set-valued maximal monotone mapping onHsuch that the domain ofGi is included inCfor eachi= , . LetFbe a ﬁrmly nonexpansive mapping ofHintoH andFbe a ﬁrmly nonexpansive mapping ofHintoH. NoteJλG= (I+λG)–andJrG= (I+rG)–for eachλ>  andr> . LetA:H→Hbe a bounded linear operator,A:

H→Hbe a bounded linear operator, andA∗i be the adjoint ofAifori= , . Throughout this paper, we use these notations unless speciﬁed otherwise.

Theorem . Suppose that(B+G)–()∩(B+G)–()is nonempty,and{an},{bn},{cn},

{fn},{gn},{hn}are sequences in[, ]such that an+bn+cn= ,fn+gn+hn= ,  <an< ,

and <fn< for each n∈N.For an arbitrarily ﬁxed u∈H,deﬁne a sequence{vn}by

⎧ ⎪ ⎨ ⎪ ⎩

v∈His chosen arbitrarily,

vn+:=anu+bnvn+cnJδGn(I–δnB)vn, n∈N∪ {},

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Thenlim_n→∞vn=P(B+G)–()∩(B+G)–()u provided the following conditions are satisﬁed:

(i) lim_n→∞an=limn→∞fn= ;

(ii) either∞_n=an=∞or

∞

n=fn=∞;

(iii)  <a≤δn≤b< κand <f≤γn≤g< κfor eachn∈Nand for some

a,b,f,g∈R+_;

(iv) lim inf_n→∞cn> ,lim infn→∞hn> .

Proof Given any ﬁxed¯v:=P(B+G)–()∩(B+G)–()u. Thenv¯=J

G

δn(I–δnB)v¯andv¯=J G γn(I–

γnB)v¯. SinceBisκ-ism and κ_δ_n>_, it follows from Lemma . thatI–δnBis averaged. HenceI–δnBis nonexpansive. By Lemma .,JδGn(I–δnG) andJ

G

γn(I–γnG) are averaged. HenceJG

δn(I–δnG) andJ G

γn(I–γnG) are nonexpansive. By condition (iv), we may assume that there exist positive real numberscandhsuch thatcn≥>  andhn≥h>  for each

n∈N. For eachn∈N, we have that

vn–v¯

= fnu+gnvn–+hnJγGn(I–γnB)vn––v¯

≤fnu–v¯+gnvn––¯v+hn JγGn(I–γnB)vn––v¯

≤fnu–v¯+gnvn––¯v+hnvn––¯v

=fnu–v¯+ (gn+hn)vn––v¯

=fnu–v¯+ ( –fn)vn––v¯ (.)

and

vn+–v¯

= anu+bnvn+cnJδGn(I–δnB)vn–v¯

≤anu–v¯+bnvn–v¯+cn JδGn(I–δnB)vn–v¯

≤anu–v¯+bnvn–v¯+cnvn–v¯

=anu–v¯+ (bn+cn)vn–v¯

≤( –an)fnu–v¯+ ( –fn)vn––v¯

+anu–v¯ by (.)

=fn( –an)u–v¯+( –an)( –fn)vn––v¯+anu–v¯

= – ( –an)( –fn)u–v¯+ ( –an)( –fn)vn––v¯

≤maxu–v¯,vn––v¯

≤maxu–v¯,v–v¯

.

By the mathematical induction method, we know that{vn–},{vn}and{vn}are bounded sequences. By Lemma ., we have that

JG

δn(I–δnB)vn–v¯ 

= JG

δn(I–δnB)vn–J G

δn(I–δnB)v¯ 

≤ (I–δnB)vn– (I–δnB)¯v 

– I–JG

δn(I–δnB)

vn–

I–JG

δn(I–δnB)

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≤ vn–v¯– I–JδGn(I–δnB)

vn–

I–JG

δn(I–δnB)

¯ v 

=vn–¯v– vn–JδGn(I–δnB)vn 

for eachn∈N. Hence,

vn+–v¯

= an(u–v¯) +bn(vn–v¯) +cn

JG

≤ bn(vn–v¯) +cn

JG

+ anu–v¯,vn+–v¯

= ( –an) b_n(vn–v¯) +cn

JG

+ anu–v¯,vn+–v¯

= ( –an)b_nvn–v¯+cn J G

–b_nc_n vn–JδGn(I–δnB)vn 

+ anu–v¯,vn+–v¯

≤bnvn–v¯+cn JδGn(I–δnB)vn–v¯ 

+ anu–v¯,vn+–v¯

≤bnvn–v¯+cn

vn–v¯– vn–JδGn(I–δnB)vn 

+ anu–v¯,vn+–v¯

= (bn+cn)vn–v¯+ anu–v¯,vn+–v¯ –cn vn–JδGn(I–δnB)vn 

, (.)

where

b_n:= bn

bn+cn

, c_n:= cn

bn+cn .

Similarly, we have

vn–v¯≤(gn+hn)vn––v¯+ fnu–v¯,vn–v¯

–hn JγGn(I–γnB)vn––vn– 

. (.)

Consequently, it follows from (.) and (.) that

vn+–v¯

≤(bn+cn)vn–v¯+ anu–v¯,vn+–v¯ –cn JδGn(I–δnB)vn–vn 

≤( –an)( –fn)vn––¯v+ fnu–v¯,vn–v¯

–hn JγGn(I–γnB)vn––vn– 

+ anu–¯v,vn+–¯v

–cn JδGn(I–δnB)vn–vn 

= ( –an)( –fn)vn––v¯+ fn( –an)u–v¯,vn–v¯

–hn( –an) JG

γn(I–γnB)vn––vn– 

+ anu–v¯,vn+–v¯

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For eachn∈N, setSn:=vn––v¯. ThenSn+=vn+–v¯and (.) become

Sn+≤( –an)( –fn)Sn+ anu–v¯,vn+–v¯ + fn( –an)u–v¯,vn–¯v

–hn( –an) JG

≤( –an)( –fn)Sn+ anu–v¯,vn+–v¯ + fn( –an)u–v¯,vn–¯v. (.)

Case : {Sn} is eventually decreasing,i.e., there exists a natural numberN such that

vn+–v¯ ≤ vn––v¯for eachn≥N. So,{Sn}is convergent andlimn→∞vn––v¯ exists. For alln∈N,cn≥c,hn≥hand (.), we have that

≤c JG

δn(I–δnB)vn–vn 

+h( –an) JG

≤( –an)( –fn)Sn–Sn++ anu–v¯,vn+–v¯ + fn( –an)u–¯v,vn–v¯ . (.)

Noting via condition (i) and the fact that{vn}is bounded that

lim n→∞

( –an)( –fn)Sn–Sn+

= ,

lim

n→∞anu–v¯,vn+–v¯ =n→∞lim fn( –an)u–v¯,vn–¯v = ,

we conclude from (.) that

lim n→∞

c JG

δn(I–δnB)vn–vn 

+h( –an) JG

= .

Therefore,

lim n→∞ J

G

δn(I–δnB)vn–vn =_n→∞lim J G

γn(I–γnB)vn––vn– = . (.) Since{vn}is a bounded sequence inH, there is a subsequence{v(n)k}of{vn}such that

v(n)k x¯∈Hand lim sup

n→∞ u–v¯,vn–v¯ =k→∞limu–v¯,v(n)k–v¯ =u–v¯,x¯–v¯ .

On the other hand,  <a≤δn≤b< κ, there exists a subsequence{δn_kj}of{δn}such that

{δn_kj}converges to a numberδ¯∈[a,b]. By Lemma ., we have

v(n)_kj –JδG¯(I–δ¯B)v(n)kj

≤ v(n)_kj–JδG_nkj (I–δn_kjB)v(n)_kj

+ JG

δ_nkj(I–δ¯B)v(n)_kj–JδG¯(I–δ¯B)v(n)kj + JG

δ_nkj(I–δn_kjB)v(n)_kj –J

G

(10)

≤ v(n)_kj–JδG

nkj(I–δnkjB)v(n)kj +|δnkj–δ¯|Bunkj +|

δn_kj–δ¯|

¯

δ J

G

¯

δ (I–δ¯B)v(n)kj– (I–δ¯B)v(n)kj →. (.) By (.) and Lemma .,x¯∈Fix(JG

¯

δ (I–δ¯B)) = (B+G)

–_{(). Since  <}_c_≤_γ

n≤d< κ, there exists a subsequence{γn_kj}of{γn}such that{γn_kj}converges to a numberγ¯∈[c,d]. We have that

vn+–vn= anu+bnvn+cnJδGn(I–δnB)vn–vn

≤anu–vn+cn JδGn(I–δnB)vn–vn (.) and

vn–vn–= fnu+gnvn–+hnJγGn(I–γnB)vn––vn–

≤fnu–vn–+hn JγGn(I–γnB)vn––vn– . (.) Since{vn}is bounded, we conclude from (.), (.), (.), and conditions (i), (ii) that

lim

n→∞vn+–vn= . (.)

By Lemma ., we have

v(n)_kj+–Jγ¯(I–γ¯B)v(n)_kj+

≤ v(n)_kj+–Jγ_nkj(I–γn_kjB)v(n)_kj+ + Jγ

nkj(I–γnkjB)v(n)kj+–Jγnkj(I–γ¯B)v(n)kj+ + Jγ_nkj(I–γ¯B)v(n)_kj+–Jr¯(I–rB¯ )v(n)_kj+

≤ v(n)_kj+–Jγ_nkj(I–γn_kjB)v(n)_kj+ +|γn_kj –γ¯|Bv(n)_kj+

+|

γn_kj–γ¯|

¯

γ Jγ¯(I–γ¯B)v(n)kj+– (I–γ¯B)v(n)kj+ →. (.)

Sincelimn→∞vn+–vn= , we know thatv(n)_kj+x¯. By (.) and Lemma ., we know thatJG

¯

γ (I–γ¯B)x¯=x¯. So,x¯∈(B+G)–(). This shows thatx¯∈(B+G)–()∩ (B+G)–(). It follows fromv¯:=P(B+G)–()∩(B+G)–()uand Lemma . that

lim sup

n→∞ u–v¯,vn–v¯ =u–v¯,x¯–v¯ ≤. (.)

By (.) and (.),

lim sup

n→∞ u–v¯,vn+–v¯ =lim sup

n→∞

u–¯v,vn+–vn +u–v¯,vn–v¯

≤lim sup

(11)

Applying Lemma . to inequality (.) withtn= u–v¯,vn+–¯v andkn= ( –an)u–

¯

v,vn–¯v, we obtain from (.) and (.) and conditions (i), (ii) thatlimn→∞Sn= . That is, lim_n→∞vn–=¯v. And then it follows from (.) that limn→∞vn=v¯. Thus, lim_n→∞vn=v¯.

Case : Suppose that{Sn}is not an eventually decreasing sequence. Let{Sni}be a sub-sequence of{Sn}such thatSni≤Sni+for alli≥, also consider the sequence of integers

{τ(n)}n≥n, deﬁned byτ(n) =max{k≤n,Sk<Sk+}, for somen(nis a suﬃciently large

number). Then{τ(n)}n≥n is a nondecreasing sequence aslimn→∞τ(n) =∞, and for all n≥n, one has that

Sτ(n)≤Sτ(n)+ and Sn≤Sτ(n)+. (.)

That is,max{Sτ(n),Sn} ≤Sτ(n)+. For suchn≥n, it follows from (.) that

vτ(n)––v¯ ≤ vτ(n)+–v¯ (.)

and

vn––v¯ ≤ vτ(n)+–¯v. (.)

From (.) and (.), we obtain

Sτ(n)≤Sτ(n)+

≤( –aτ(n))( –fτ(n))Sτ(n)+ aτ(n)u–¯v,vτ(n)+–v¯

+ fτ(n)( –aτ(n))u–v¯,vτ(n)–v¯

–cτ(n) JδGτ(n)(I–δτ(n)B)vτ(n)–vτ(n)



–hτ(n) JγG_τ(n)(I–γτ(n)B)vτ(n)––vτ(n)–



≤( –aτ(n))( –fτ(n))Sτ(n)+ aτ(n)u–¯v,vτ(n)+–v¯

+ fτ(n)( –aτ(n))u–v¯,vτ(n)–v¯ . (.)

Just as the argument of Case , we have

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

lim_n→∞vτ(n)–JδGτ(n)(I–δτ(n)B)vτ(n)= ,

lim_n→∞vτ(n)––JγGτ(n)(I–γτ(n)B)vτ(n)–= ,

lim sup_n→∞u–v¯,vτ(n)+–v¯ ≤, lim sup_n→∞u–v¯,vτ(n)–v¯ ≤, lim_n→∞vτ(n)+–vτ(n)= .

(.)

By (.) and (.), we have

Sτ(n)+≤( –aτ(n))( –fτ(n))Sτ(n)++ aτ(n)u–v¯,vτ(n)+–v¯

(12)

for alln≥n. This implies that

Sτ(n)+≤( –aτ(n))( –fτ(n))Sτ(n)++aτ(n)Kvτ(n)+–vτ(n)

+ fτ(n)( –aτ(n)) +aτ(n)

u–¯v,vτ(n)–¯v

for alln≥n, whereK= u–v¯. Furthermore, we have

Sτ(n)+≤

aτ(n)Kvτ(n)+–vτ(n)

aτ(n)+fτ(n)( –aτ(n))

+ u–v¯,vτ(n)–v¯

≤Kvτ(n)+–vτ(n)+ u–v¯,vτ(n)–v¯ . (.)

Hence, it follows from (.) and (.) that

lim

n→∞Sτ(n)+= . (.)

By (.) and (.), we know thatlim_n→∞Sn= . Then, just as the argument in the proof of Case , we obtainlimn→∞vn=v¯. Therefore, the proof is completed.

Remark .

(i) If we putB=B= in Theorem ., then Theorem . is reduced to Theorem  in

[].

(ii) Boikanyoet al.[] showed that strong convergence theorem of a proximal point algorithm with error can be obtained from strong convergence of a proximal point algorithm without errors. Therefore, in Theorem ., we study strong convergence of variational inclusion problems without error.

As a simple consequence of Theorem ., we have the following theorem.

Theorem . Suppose that(B+G)–()is nonempty,and{an},{bn},{cn}are sequences

in[, ]such that an+bn+cn= ,  <an< for each n∈N.For an arbitrary ﬁxed u∈H,

deﬁne a sequence{xn}by

x∈His chosen arbitrarily,

xn+:=anu+bnxn+cnJδGn(I–δnB)xn, n∈N∪ {}.

Thenlim_n→∞vn=P(B+G)–()u if the following conditions are satisﬁed:

(i) lim_n→∞an= ,∞n=an=∞;

(ii)  <a≤δn≤b< κfor eachn∈Nand for somea,b∈R+;

(iii) lim inf_n→∞cn> .

Proof Setfn= ,B= ,gn+hn= ,{gn}and{hn}are sequences in [, ], andG=∂ιH.

Deﬁne a sequencevnby

(13)

and

vn=vn–=xn–.

Then

vn+=anu+bnvn+cnJδGn(I–δnB)vn, n∈N∪ {},

and

vn=fnu+gnvn–+hnvn–=fnu+gnvn–+hnJ

∂ι_H 

γn vn–, n∈N∪ {}.

Since

G–_ () = (∂ι_H)– =FixJ∂ιH γn

=Fix(P_H) =H,

it is easy to see that

(B+G)–() =

(B+G)–()∩H

=(B+G)–()∩(B+G)–()

=∅.

Then Theorem . follows from Theorem ..

Remark . Following the same argument as in Remark  in [], we see that Theo-rems . and . contain [, Theorem .], [, Theorem ], [, TheoTheo-rems -] and many other recent results as special cases.

4 Applications

Now, we recall the following multiple sets split feasibility problem (MSSFP-A):

Findx¯∈Hsuch thatx¯∈G– ()∩G– (),Ax¯∈Fix(F) andAx¯∈Fix(F).

Theorem .[] Given anyx¯∈H,{λn}and{γn}are sequences in(,∞).

(i) Ifx¯is a solution of(MSSFP-A),then

JG

λn(I–ρnA ∗

(I–F)A)J G

rn (I–σnA∗(I–F)A)x¯=x¯for eachn∈N.

(ii) Suppose thatJG λn(I–ρnA

∗

(I–F)A)J G

rn (I–σnA∗(I–F)A)x¯=x¯with

 <ρn<_A__+, <σn<_A__+ for eachn∈Nand the solution set of(MSSFP-A)is

nonempty.Thenx¯is a solution of(MSSFP-A).

In order to study the convergence theorems for the solution set of multiple split feasi-bility problem (MSSFP-A), we need the following problems and the following essential tool which is a special case of Theorem . in []:

(SFP-) Findx¯∈Hsuch thatx¯∈Fix

JG

ρn

andAx¯∈Fix(F).

Lemma . Given anyx¯∈H.

(i) Ifx¯is a solution of(SFP-),thenJG

ρn(I–ρnA ∗

(14)

(ii) Suppose thatJG ρn(I–ρnA

∗

(I–F)A)x¯=x¯with <ρn<_A

+for eachn∈N,and

the solution set of(SFP-)is nonempty.Then(I–ρnA∗(I–F)A)and

JG

ρn(I–ρnA ∗

(I–F)A)are averaged andx¯is a solution of(SFP-).

Proof (i) Suppose thatx¯∈His a solution of (SFP-). Thenx¯∈Fix(JρGn),Ax¯∈Fix(F) for eachn∈N. It is easy to see that

JG ρn

I–ρnA∗(I–F)A

¯

x=x¯, n∈N.

(ii) Since the solution set of (SFP-) is nonempty, there exists w¯ ∈H such thatw¯ ∈ Fix(JG

ρn),Aw¯ ∈Fix(F). Thenw¯ ∈G –

 (). If we putG=GandF=F, we get that the solution set of (MSSFP-A) is nonempty. By Lemma . we have that

A∗_(I–F)Ais 

A

-ism. (.)

By (.),  <ρn<_A

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

I–ρnA∗(I–F)Ais averaged for eachn∈N. (.) On the other hand, for eachn∈N,JG

ρn is a ﬁrmly nonexpansive mappings, it is easy to see that

JG ρn is



-averaged. (.)

Hence, by (.), (.) and Lemma .(iv) and (v), we see that

JG ρn

is averaged.

Since

JG ρn

¯ x=x¯,

so

JG ρn

JG

ρn

¯ x=x¯.

Then Lemma . follows from Theorem . by takingG=G,F=Fandρn=rn.

Remark . From the following result, we know that Lemma . is more useful than The-orem ..

Theorem . Suppose that the solution setAof(MSSFP-A)is nonempty and{an},{bn},

{cn},{fn},{gn},{hn}are sequences in[, ]such that an+bn+cn= ,fn+gn+hn= ,  <an< ,

and <fn< for each n∈N.For an arbitrarily ﬁxed u∈H,a sequence{vn}is deﬁned by

⎧ ⎪ ⎨ ⎪ ⎩

vn+:=anu+bnvn+cnJρGn(I–ρnA ∗

(I–F)A)vn, n∈N∪ {},

vn:=fnu+gnvn–+hnJσGn(I–σnA ∗

(15)

Thenlim_n→∞vnis a unique solution of the following optimization problem:

min q∈A

q–u

provided the following conditions are satisﬁed:

∞

n=fn=∞;

(iii)  <a≤ρn≤b<_A

+, <c≤σn≤d<



A+ for eachn∈N,and for some a,b,c,d∈R+_;

(iv) lim infn→∞cn> ,lim infn→∞hn> .

Proof SinceFiis ﬁrmly nonexpansive, it follows from Lemma . thatA∗i(I–Fi)Aiis_Ai 

-ism for eachi= , . For eachi= , , putBi=A∗i(I–Fi)Aiin Lemma .. Then algorithm in Theorem . follows immediately from algorithm in Theorem .. SinceAis nonempty, by Lemma ., we have that

¯

w∈FixJG ρn

∩FixJG σn

I–σnA∗(I–F)A

=∅ (.)

for eachn∈N. This implies that

¯

w∈FixJG

ρn(I–ρnB)

∩FixJG

σn(I–σnB)

=∅ (.)

for eachn∈N. Hence,

¯

w∈(B+G)–()∩(B+G)–()

=∅.

By Theorem .,lim_n→∞vn=x¯, wherex¯=P(B+G)–()∩(B+G)–()u. That is,

¯

x∈(B+G)–()∩(B+G)–()

and

¯x–u ≤ q–u

for allq∈(B+G)–()∩(B+G)–(). Since

¯

x∈(B+G)–()∩(B+G)–(),

we know that

¯

x∈FixJG

ρn(I–ρnB)

∩FixJG

σn(I–σnB)

.

That is,

¯

x∈FixJG ρn

(16)

and

¯

x∈FixJG σn

I–σnA∗(I–F)A

.

By Lemma ., we get thatx¯∈A. Similarly, ifq∈(B+G)–()∩(B+G)–(), then

q∈A. Thereforex¯=PAu. This shows thatlimn→∞vnis a unique solution of the

opti-mization problem

min q∈A

q–u.

Therefore, the proof is completed.

In the following theorem, we study the following multiple sets split feasibility problem (MSSMVIP-A):

Findx¯∈Hsuch thatx¯∈C,Ax¯∈QandAx¯∈Q.

LetAdenote the solution set of (MSSMVIP-A). The following theorem is a special case of Theorem .. Hence, it is also a special case of Theorem ..

Theorem . Suppose thatAis nonempty,and that{an},{bn},{cn},{fn},{gn},{hn}are

sequences in[, ]with an+bn+cn= ,fn+gn+hn= ,  <an< ,and <fn< for each

n∈N.For an arbitrary ﬁxed u∈H,a sequence{vn}is deﬁned by

⎧ ⎪ ⎨ ⎪ ⎩

vn+:=anu+bnvn+cnPC(I–ρnA∗(I–PQ)A)vn, n∈N∪ {},

vn:=fnu+gnvn–+hnPC(I–σnA∗(I–PQ)A)vn–, n∈N.

min q∈A

q–u

(i) limn→∞an=limn→∞fn= ;

∞

n=fn=∞;

(iii)  <a≤ρn≤b<_A__+, <c≤σn≤d<_A__+ for eachn∈Nand for some a,b,c,d∈R+_;

Proof PutG=G=∂ιC,F=PQ, andF=PQ. ThenG,Gare two set-valued maximum monotone mappings,F andFare ﬁrmly nonexpansive mappings. SinceJρ∂ιnC=PC and

J∂ιC

σn =PC, we haveFix(F) =Fix(PQ) =Q,Fix(F) =Fix(PQ) =Q

_,_Fix₍_J∂ιC

ρn ) =Fix(PC) =C andFix(J∂ιC

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In the following theorem, we study the following split feasibility problem (MSSMVIP-A):

Findx¯∈Hsuch thatx¯∈C∩Q,Ax¯∈QwhereQis a nonempty closed

subset ofH.

LetAdenote the solution set of problem (MSSMVIP-A). The following is also a spe-cial case of Theorem ..

Theorem . Suppose that Qis a nonempty closed convex subset ofH,Ais nonempty,

and{an},{bn},{cn},{fn},{gn},{hn}are sequences in[, ]with an+bn+cn= ,fn+gn+hn= ,  <an< ,and <fn< for each n∈N.For an arbitrary ﬁxed u∈H,a sequence{vn}is

deﬁned by

⎧ ⎪ ⎨ ⎪ ⎩

vn+:=anu+bnvn+cnPC(I–ρnA∗(I–PQ)A)vn, n∈N∪ {},

vn:=fnu+gnvn–+hnPC(I–σn(I–PQ))vn–, n∈N.

Thenlimn→∞vnis a unique solution of the following optimization problem:

min q∈A

q–u

∞

n=fn=∞;

(iii)  <a≤ρn≤b<_A__+, <c<σn<d<_ for eachn∈Nand for some

a,b,c,d∈R+_;

Proof PutA=IandH=H in Theorem .. Then Theorem . follows from

Theo-rem ..

In the following theorem, we study the following convex feasibility problem (MSSMVIP-A):

Findx¯∈Hsuch thatx¯∈C∩Q∩Q, whereQ,Qare nonempty closed

subsets ofH.

LetAdenote the solution set of (MSSMVIP-A). The following is a special case of Theorem ..

Theorem . Suppose that Q and Qare nonempty closed convex subsets ofH,Ais

nonempty,and{an},{bn},{cn},{fn},{gn},{hn}are sequences in[, ]with an+bn+cn= ,

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a sequence{vn}is deﬁned by

⎧ ⎪ ⎨ ⎪ ⎩

vn+:=anu+bnvn+cnPC(I–ρn(I–PQ))vn, n∈N∪ {},

vn:=fnu+gnvn–+hnPC(I–σn(I–PQ))vn–, n∈N.

min q∈A

q–u

(i) limn→∞an=limn→∞fn= ;

∞

n=fn=∞;

(iii)  <a<ρn<b<_, <c<σn<d<_ for eachn∈Nand for somea,b,c,d∈R+;

Proof PutA=A=IandH=H=Hin Theorem .. Then Theorem . follows from

Theorem ..

In the following theorem, we study the following convex feasibility problem (MSSMVIP-A):

Findx¯∈Hsuch thatx¯∈Q∩Q, whereQandQare nonempty closed

convex subsets ofH.

LetAdenote the solution set of (MSSMVIP-A).

The following existent theorem of a convex feasibility problem follows immediately from Theorem ..

Theorem . Suppose that Q and Qare nonempty closed convex subsets ofH,Ais

nonempty,and{an},{bn},{cn},{fn},{gn},{hn}are sequences in[, ]with an+bn+cn= ,

fn+gn+hn= ,  <an< ,and <fn< for each n∈N.For an arbitrary ﬁxed u∈H.Deﬁne

a sequence{vn}by

⎧ ⎪ ⎨ ⎪ ⎩

vn+:=anu+bnvn+cn(I–ρn(I–PQ))vn, n∈N∪ {},

vn:=fnu+gnvn–+hn(I–σn(I–PQ))vn–, n∈N.

Thenlimn→∞vnis a unique solution of the following optimization problem:

min q∈A

q–u

∞

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(iii)  <a<ρn<b<_, <c<σn<d<_ for eachn∈Nand for somea,b,c,d∈R+;

Proof PutC=H, thenPC=PH. Then Theorem . follows from Theorem ..

In the following theorem, we study the following system of convexly constrained linear inverse problem (SCCLIP):

Findx¯∈Hsuch thatx¯∈C,Ax¯=bandAx¯=b, whereb∈Handb∈H.

LetAdenote the solution set of (SCCLIP).

Theorem . Suppose thatAis nonempty,and b∈H,b∈H.Let{an},{bn},{cn},{fn},

{gn},and{hn}be sequences in[, ]with an+bn+cn= ,fn+gn+hn= ,  <an< ,and  <fn< for each n∈N.For an arbitrary ﬁxed u∈H,a sequence{vn}is deﬁned by

⎧ ⎪ ⎨ ⎪ ⎩

vn+:=anu+bnvn+cnPC(vn–ρnA∗(Avn–b)), n∈N∪ {},

vn:=fnu+gnvn–+hnPC(vn––σnA∗(Avn––b)), n∈N.

min q∈A

q–u

∞

n=fn=∞;

+, <c≤σn≤d<



A+ for eachn∈Nand for some a,b,c,d∈R+_;

Proof PutQ={b}andQ={b}. Then Theorem . follows from Theorem ..

In the following theorem, we study the following convexly constrained linear inverse problem (CCLIP):

Findx¯∈Hsuch thatx¯∈C∩QandAx¯=b, whereb∈HandQis a nonempty

closed convex subset ofH.

LetAdenote the solution set of (CCLIP).

Theorem . Suppose that Qis a nonempty closed convex subset ofH.Ais nonempty,

b∈H,and{an},{bn},{cn}, {fn},{gn},{hn} are sequences in[, ]with an+bn+cn= ,

fn+gn+hn= ,  <an< ,and <fn< for each n∈N.For an arbitrary ﬁxed u∈H,

a sequence{vn}is deﬁned by

⎧ ⎪ ⎨ ⎪ ⎩

vn+:=anu+bnvn+cnPC(vn–ρn(Avn–b)), n∈N∪ {},

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min q∈A

q–u

∞

n=fn=∞;

+, <c<σn<d<



 for eachn∈N,and for some

a,b,c,d∈R+_;

Remark . The iteration in Theorem . is diﬀerent from the Landweber iterative method []:

xn+:=xn+γAT(b–Axn), n∈N.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

L-JL designed and coordinated this research project and revised the paper. Y-DC carried out the project, drafted and revised the manuscript. C-SC coordinated the project and revised the paper.

Author details

1_{Department of Mathematics, National Changhua University of Education, Changhua, 50058, Taiwan.}2_{Department of}

Chemical and Materials Engineering, Southern Taiwan University of Science and Technology, Tainan, 71005, Taiwan.

3_{Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, Taiwan.}

Acknowledgements

Prof. C-S Chuang was supported by the National Science Council of Republic of China while he work on the publish, and Y-D Chen was supported by Southern Taiwan University of Science and Technology.

Received: 3 April 2013 Accepted: 7 November 2013 Published:06 Dec 2013

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