New algorithms designed for the split common fixed point problem of quasi pseudocontractions

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R E S E A R C H

Open Access

New algorithms designed for the split

common ﬁxed point problem of

quasi-pseudocontractions

Li-Jun Zhu

1

_{, Yeong-Cheng Liou}

2,3*

_{, Jen-Chih Yao}

3,4

_{and Yonghong Yao}

5

*_{Correspondence:}

[email protected]

2_{Department of Information}

Management, Cheng Shiu University, Kaohsiung, 833, Taiwan

3_{Center for Fundamental Science,}

Kaohsiung Medical University, Kaohsiung, 807, Taiwan Full list of author information is available at the end of the article

Abstract

In this paper, we study the split common fixed point problem, which is to find a fixed point of a quasi-pseudocontractive mapping in one space whose image under a linear transformation is a fixed point of anther quasi-pseudocontractive mapping in the image space. We design and analyze a new iterative algorithm for solving this split common fixed point problem. A weak convergence theorem is given.

MSC: 49J53; 49M37; 65K10; 90C25

Keywords: split common ﬁxed point; quasi-pseudocontractive mappings; weak convergence

1 Background and motivation

LetCandQbe nonempty closed convex subsets of real Hilbert spacesHandH, respec-tively. The split feasibility problem is formulated as ﬁnding a pointx∗with the property

x∗∈C and Ax∗∈Q, (.)

whereA:H→H is a bounded linear operator. The split feasibility problem in ﬁnite-dimensional Hilbert spaces was ﬁrst introduced by Censor and Elfving [] for modeling in-verse problems which arise from phase retrievals and in medical image reconstruction []. A special case of the split feasibility problem (.) is whenQ={b}is singleton and then (.) is reduced to the convexly constrained linear inverse problem []

x∗∈C and Ax∗=b, (.)

which has received considerable attention.

The well-known projected Landweber algorithm [] is widely used to solve (.). This algorithm generates a sequence{xn}in such a way that we have

• initialization:xselected inHarbitrarily, and • iteration:

xn+=PC

xn+γAT(b–Axn)

, (.)

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wherePCdenotes the nearest point projection fromHontoC,γ > is a parameter

such that <γ < /A_{, and}_AT_{is the transpose of}_A_.

When the system (.) is reduced to the unconstrained linear system

Ax∗=b, (.)

then the projected Landweber algorithm [] is turned to the Landweber algorithm:

xn+=xn+γAT(b–Axn). (.)

The simultaneous algebraic reconstruction technique is a typical example of the Landwe-ber algorithm (.) when the system (.) is ﬁnite-dimensional.

The first iterative algorithm for solving the split feasibility problem (.) in the finite-dimensional case is proposed by Censor and Elfving [] who define a sequencexnby the

recursion:

xn+=A–PQ

PA(C)(Axn)

, n≥, (.)

whereCandQare closed convex sets ofRn_{, and}_A_{is an}_n_×_n_{matrix of full rank. Here}

A(C) ={y∈Rn_:_y₌_Ax,_x_∈_C_}_{is the image of}_C_{under the matrix}_A.

Because of the presence of the inverseA–_{, the algorithm (.) has not become popular.} A more popular algorithm that solves the split feasibility problem (.) is the so-called CQ algorithm introduced by Byrne []. This algorithm, which does not involveA–_{, generates} a sequence{xn}as follows:

xn+=PC

xn–γAT(I–PQ)Axn

, n≥, (.)

where  <γ < /A_and_P

Qdenotes the nearest point projection fromHontoQ.

Conse-quently, Xu [] extend the above results from the ﬁnite-dimensional spaces to the inﬁnite-dimensional spaces.

In the case whereCandQin (.) are the intersections of finitely many fixed point sets of nonlinear operators, problem (.) is called by Censor and Segal [] the split common fixed point problem. More precisely, the split common fixed point problem requires one to seek an elementx∗∈Hsatisfying

x∗∈

m

i=

Fix(Si) and Ax∗∈ n

j=

Fix(Tj), (.)

whereFix(Si) andFix(Tj) denote the ﬁxed point sets of two classes of nonlinear operators

Si:H→H andTj:H→H. In this situation, Byrne’s CQ algorithm does not work because the metric projection onto fixed point sets is generally not easy to calculate. To solve the two-set split common fixed point problem, motivated by the algorithms (.) and (.), Censor and Segal [] proposed the following iterative method: For any initial guess x∈H, define{xn}recursively by

xn+=U

xn–λA∗(I–T)Axn

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whereUandT are directed operators andλ>  is known as the step-size. They proved that if λ∈(,_A), then (.) converges to a split common ﬁxed point x∗∈ ={x∈

Fix(U);Ax∈Fix(T)}. Consequently, Moudaﬁ [] extended (.) to the following relaxed

algorithm:

⎧ ⎨ ⎩

un=xn–γA∗(I–T)Axn,

xn+= ( –αn)un+αnU(un), n∈N,

where UandT are demicontractive operators,β∈(, ),γ ∈(,–μ_λ ) withλbeing the spectral radius of the operatorA∗Aandαn∈(, ) is relaxation parameter. We note that the

classes of directed and demicontractive operators are important classes since they include the orthogonal projections and the subgradient projectors. For some other related work, please refer to [–] and [].

In the present paper, our main motivation is to extend the classes of directed and demicontractive operators to the class of quasi-pseudocontractions because the class of quasi-pseudocontractions includes the classes of directed and demicontractive operators as special cases. Interest in pseudocontractive mappings stems mainly from their ﬁrm connection with the class of monotone operators. We present a uniﬁed framework for the study of this problem and this class of operators. We propose an iterative algorithm and study its convergence.

2 Notations and lemmas

LetHbe a real Hilbert space with inner product·,·and norm · , respectively. LetC be a nonempty closed convex subset ofH.

Recall that a mappingT:C→Cis called

• nonexpansive ifTx–Ty ≤ x–yfor allx,y∈C;

• quasi-nonexpansive ifTx–x∗ ≤ x–x∗for allx∈Candx∗∈Fix(T);

• ﬁrmly nonexpansive ifTx–Ty_{≤ x}_–_y_–_(I_–_T)x_{– (I}_–_T_)y_{for all}_x,_y_∈_C_;

• ﬁrmly quasi-nonexpansive ifTx–x∗_≤_x_–_x∗_–_Tx_–_x_{for all}_x_∈_C_and x∗∈Fix(T);

• strictly pseudocontractive ifTx–Ty_≤_x_–_y₊_k_(I_–_T)x_{– (I}_–_T_)y_{for all} x,y∈C, wherek∈[, );

• directed ifTx–x∗,Tx–x ≤for allx∈Candx∗∈Fix(T);

• demicontractive ifTx–x∗_≤_x_–_x∗₊_k_Tx_–_x_{for all}_x_∈_C_and_x∗_∈_Fix_(T₎_,

wherek∈[, ).

The concept of directed operators was introduced by Bauschke and Combettes [] who proved thatT:C→Cis directed if and only if

Tx–x∗≤x–x∗–Tx–x

for allx∈C andx∗∈Fix(T). It can be seen easily that the class of directed operators coincides with that of ﬁrmly quasi-nonexpansive mappings.

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fundamental because they include many types of nonlinear operators arising in applied mathematics and optimization; see for example [] and references therein.

Recall also that a mappingT:C→Cis called pseudocontractive if

Tx–Ty,x–y ≤ x–y

for allx,y∈C. It is well known thatTis pseudocontractive if and only if

Tx–Ty≤ x–y+(I–T)x– (I–T)y

for allx,y∈CandT:C→Cis said to be quasi-pseudocontractive if

_Tx_–_x∗

≤x–x∗+Tx–x (.)

for allx∈Candx∗∈Fix(T).

It is obvious that the class of quasi-pseudocontractive mappings includes the class of demicontractive mappings.

A mappingT:C→Cis calledL-Lipschitzianif there existsL>  such that

Tx–Ty ≤Lx–y

for allx,y∈C.

Usually, the convergence of ﬁxed point algorithms requires some additional smoothness properties of the mappingT such as demiclosedness.

Recall that a mappingTis said to be demiclosed if, for any sequence{xn}which weakly

converges tox, and if the sequence˜ {T(xn)}strongly converges toz, we haveT(˜x) =z.

Observe also that the nonexpansive operators are both quasi-nonexpansive and strictly pseudocontractive maps and are well known for being demiclosed. For the pseudocon-tractions, the following demiclosedness principle is well known.

Lemma .([]) Let H be a real Hilbert space,C a closed convex subset of H.Let U:

C→C be a continuous pseudocontractive mapping.Then

(i) Fix(U)is a closed convex subset ofC, (ii) (I–U)is demiclosed at zero.

In the next section, we will need to impose the demiclosedness to the quasi-pseudocon-tractions.

It is well known that in a real Hilbert spaceH, the following equality holds:

tx+ ( –t)y=tx+ ( –t)y–t( –t)x–y (.)

for allx,y∈Handt∈[, ].

Lemma .([]) Let H be a Hilbert space and let{un}be a sequence in H such that there

exists a nonempty set⊂H satisfying the following:

(i) for everyu∈,limnun–uexists,

(ii) any weak-cluster point of the sequence{un}belongs in.

Then there exists x†_∈_{such that}_{u

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In the sequel we shall use the following notations:

. ωw(un) ={x:∃unj→xweakly}denote the weakω-limit set of{un};

. un xstands for the weak convergence of{un}tox;

. un→xstands for the strong convergence of{un}tox. 3 Main results

In this section, we will focus our attention on the following general two-operator split common ﬁxed point problem:

ﬁndx∗∈Csuch thatAx∗∈Q, (.)

whereA:H→His a bounded linear operator,U:H→His a quasi-pseudocontractive mapping andT :H→H is a quasi-pseudocontractive mapping with nonempty ﬁxed point setsFix(U) =CandFix(T) =Q, and we denote the solution set of the two-operator split common ﬁxed point problem by

={x∈C;Ax∈Q}.

Algorithm . Foru∈H, deﬁne a sequence{un}as follows:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

xn=un+γ νA∗[ηI+ ( –η)T(( –β)I+βT) –I]Aun,

yn= ( –ξn)xn+ξnUxn,

un+= [ – ( –δn)αn]xn+ ( –δn)αnUyn

(.)

for alln∈N, whereγ,ν,η, andβare four constants,{αn},{δn}, and{ξn}are three sequences

in [, ].

Now, we demonstrate the convergence analysis of the algorithm (.).

Theorem . Let H and Hbe two real Hilbert spaces.Let A:H→H be a bounded

linear operator.Let U:H→Hand T:H→Hbe two L-Lipschitzian quasi-pseudocon-tractions with nonemptyFix(U) =C andFix(T) =Q.Assume T–I and U–I are demiclosed atand=∅.If the parametersγ,ν,η,β,{αn},{δn},and{ξn}satisfy the following control

conditions:

(C):  <ν< and <γ <λν,whereλis the spectral radius of the operatorA∗A;

(C):  <lim infn→∞αn≤lim supn→∞αn< ;

(C):  <  –η≤β<√_+_L_+ and <a≤ –δn≤ξn<



√

+L_+ for alln∈N.

Then the sequence{un}generated by algorithm(.)weakly converges to a split common

ﬁxed pointμ∈.

Remark . Without loss of generality, we may assume that the Lipschitz constantL> .

It is obvious thatβ<√ 

+L_+ <



Lfor alln≥.

Sinceξn<√_+_L_+, we have

 – ξn–ξnL> 

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Proposition . Let the mapping T:H→Hbe L-Lipschitzian with L> .Then

Fix(T) =FixT( –β)I+βT

for allβ∈(,

L).

Proof As a matter of fact,Fix(T)⊂Fix(T(( –β)I+βT)) is obvious. Next, we show thatFix(T(( –β)I+βT))⊂Fix(T).

Take anyx∗∈Fix(T((–β)I+βT)). We haveT((–β)I+βT)x∗=x∗. SetS= (–β)I+βT. We haveTSx∗=x∗. WriteSx∗=y∗. ThenTy∗=x∗. Now we showx∗=y∗. In fact,

_x∗_–_y∗₌_Ty∗_–_Sx∗

=Ty∗– ( –β)x∗–βTx∗ =βTy∗–Tx∗

≤βLy∗–x∗.

Sinceβ< _L, we deduce y∗=x∗∈Fix(S) =Fix(T). Thus,x∗∈Fix(T). Hence, Fix(T(( – β)I+βT))⊂Fix(T). Therefore,Fix(T(( –β)I+βT)) =Fix(T).

Proposition .

ηx+ ( –η)T( –β)I+βTx–x∗≤x–x∗

for all x∈Hand all x∈Fix(T).

Proof Sincex∗∈Fix(T), we have from (.)

T( –β)I+βTx–x∗≤( –β)x–x∗+βTx–x∗

+( –β)I+βTx–T( –β)I+βTx (.)

and

Tx–x∗≤x–x∗+Tx–x (.)

for allx∈H.

By (.), (.), and (.), we obtain

T( –β)I+βTx–x∗

≤( –β)x–x∗+βTx–x∗

+( –β)I+βTx–T( –β)I+βTx

=( –β)x–T( –β)I+βTx+βTx–T( –β)I+βTx

+( –β)x–x∗+βTx–x∗

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–β( –β)x–Tx+ ( –β)x–x∗+βTx–x∗–β( –β)x–Tx ≤( –β)x–x∗+βx–x∗+x–Tx

– β( –β)x–Tx+ ( –β)x–T( –β)I+βTx

+βTx–T( –β)I+βTx.

Noting thatT isL-Lipschitzian andx– (( –β)I+βT)x=β(x–Tx), we have

T( –β)I+βTx–x∗

≤( –β)x–x∗+βx–x∗+x–Tx

– β( –β)x–Tx+ ( –β)x–T( –β)I+βTx+βLx–Tx

=x–x∗+ ( –β)x–T( –β)I+βTx–β – β–βLx–Tx. (.)

Sinceβ<√ 

+L_+, we have

 – β–βL> .

From (.), we can deduce

T( –β)I+βTx–x∗≤x–x∗+ ( –β)x–T( –β)I+βTx (.)

for allx∈Handx∗∈Fix(T). Hence,

ηx+ ( –η)T( –β)I+βTx–x∗

≤ηx–x∗+ ( –η)T( –β)I+βTx–x∗

=ηx–x∗+ ( –η)T( –β)I+βTx–x∗

–η( –η)T( –β)I+βTx–x

≤ηx–x∗+ ( –η)x–x∗+ ( –β)x–T( –β)I+βTx

–η( –η)T( –β)I+βTx–x

=x–x∗+ ( –η)( –β–η)T( –β)I+βTx–x. (.)

By (C) and (.), we deduce

ηx+ ( –η)T( –β)I+βTx–x∗≤x–x∗.

Proposition . Let the mapping T:H→H be L-Lipschitzian with L> .If T–I is

demiclosed at,then T(( –β)I+βT) –I is also demiclosed atwhenβ∈(,_L).

Proof Let the sequence{xn} ⊂Hsatisfyingxn x˜andxn–T(( –β)I+βT)xn→. Next,

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From Proposition ., we only need to prove thatx˜∈Fix(T). As a matter of fact, since T isL-Lipschitzian, we have

xn–Txn ≤xn–T

( –β)I+βTxn+T

( –β)I+βTxn–Txn

≤xn–T

( –β)I+βTxn+βLxn–Txn.

It follows that

xn–Txn ≤



 –βLxn–T

( –β)I+βTxn.

Hence,

lim

n→∞xn–Txn= .

Applying the demiclosedness ofT, we immediately deducex˜∈Fix(T).

Next, we prove Theorem ..

Proof Letx∗∈. Then we getx∗∈Fix(U) andAx∗∈Fix(T). From (.) and (.), we have

un+–x∗ 

= – ( –δn)αn

xn+ ( –δn)αnUyn–x∗



=( –αn)

xn–x∗

+αn

δnxn+ ( –δn)Uyn–x∗



= ( –αn)xn–x∗



+αnδnxn+ ( –δn)Uyn–x∗



–αn( –αn)δnxn+ ( –δn)Uyn–xn



=αn

δnxn–x∗



+ ( –δn)Uyn–x∗



–δn( –δn)Uyn–xn

+ ( –αn)xn–x∗



. (.)

Sincex∗∈Fix(U), we have from (.)

Ux–x∗≤x–x∗+x–Ux (.)

for allx∈C.

By a similar argument to that of (.), we obtain

Uyn–x∗≤xn–x∗+ ( –ξn)xn–Uyn. (.)

Substituting (.) to (.) and noting that  –ξn≤δn, we have un+–x∗



≤( –αn)xn–x∗

 +αn

δnxn–x∗



+ ( –δn)xn–x∗



+ ( –ξn)xn–Uyn

–δn( –δn)Uyn–xn



= ( –αn)xn–x∗



+αnxn–x∗



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

≤xn–x∗–αn( –αn)δnxn+ ( –δn)Uyn–xn. (.)

Sinceλis the spectral radius of the operatorAA∗, we deduce

ηI+ ( –η)T( –β)I+βT–IAun,AA∗

ηI+ ( –η)T( –β)I+βT–IAun

≤ληI+ ( –η)T( –β)I+βT–IAun

 .

This together with (.) implies that

xn–x∗



=un–x∗+γ νA∗

ηI+ ( –η)T( –β)I+βT–IAun



=un–x∗



+ γ νA∗ηI+ ( –η)T( –β)I+βT–IAun,un–x∗

+γνA∗ηI+ ( –η)T( –β)I+βT–IAun

=un–x∗+ γ ν

A∗ηI+ ( –η)T( –β)I+βT–IAun,un–x∗

+γνηI+ ( –η)T( –β)I+βT–IAun,

AA∗ηI+ ( –η)T( –β)I+βT–IAun

≤un–x∗



+ γ νA∗ηI+ ( –η)T( –β)I+βT–IAun,un–x∗

+γνληI+ ( –η)T( –β)I+βT–IAun



. (.)

By Proposition . and noting thatAx∗∈Fix(T), we have

ηI+ ( –η)T( –β)I+βTAun–Ax∗≤Aun–Ax∗.

At the same time, we have the following equality in Hilbert spaces:

x–y=x+y– x,y. (.)

In (.), picking upx= [ηI+ ( –η)T(( –β)I+βT) –I]Aunandy= [ηI+ ( –η)T(( –

β)I+βT)]Aun–Ax∗we deduce Aun–Ax∗



=ηI+ ( –η)T( –β)I+βT–IAun

–ηI+ ( –η)T( –β)I+βTAun–Ax∗

=ηI+ ( –η)T( –β)I+βT–IAun

+ηI+ ( –η)T( –β)I+βTAun–Ax∗

– ηI+ ( –η)T( –β)I+βT–IAun,

ηI+ ( –η)T( –β)I+βTAun–Ax∗

≤ηI+ ( –η)T( –β)I+βT–IAun



+Aun–Ax∗



– ηI+ ( –η)T( –β)I+βT–IAun,

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It follows that

ηI+ ( –η)T( –β)I+βT–IAun,

≤

ηI+ ( –η)T

( –β)I+βT–IAun

 .

Thus,

A∗ηI+ ( –η)T( –β)I+βT–IAun,un–x∗

=ηI+ ( –η)T( –β)I+βT–IAun,Aun–Ax∗

=ηI+ ( –η)T( –β)I+βT–IAun,

+ηI+ ( –η)T( –β)I+βT–IAun,

Aun–

ηI+ ( –η)T( –β)I+βTAun

≤

ηI+ ( –η)T



–ηI+ ( –η)T( –β)I+βT–IAun



= –

ηI+ ( –η)T



. (.)

From (.), (.), and (.), we get

un+–x∗≤un–x∗–γ ν( –λγ ν)ηI+ ( –η)T

( –β)I+βT–IAun

–αn( –αn)δnxn+ ( –δn)Uyn–xn. (.)

We deduce immediately that

un+–x∗≤un–x∗.

Hence,limn→∞un–x∗exists. This implies that{un}is bounded. Consequently, we have

≤γ ν( –λγ ν)ηI+ ( –η)T( –β)I+βT–IAun



≤un–x∗



–un+–x∗ 

→.

Therefore,

lim

n→∞ηI+ ( –η)T

( –β)I+βT–IAun= . (.)

Since{un}is bounded,ωw(un)=∅. We can takeμ∈ωw(un), that is, there exists{unj}such

thatω–lim_j→∞unj=μ. SinceT –Iis demiclosed at , by Proposition ., we see that

T(( –β)I+βT) –Iis also demiclosed at . Then, from (.), we obtain

ηI+ ( –η)T( –β)I+βT–IAμ= .

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From (.), we deduce

αn( –αn)δnxn+ ( –δn)Uyn–xn≤un–x∗–un+–x∗→.

This together with (C) implies that

lim

n→∞δnxn+ ( –δn)Uyn–xn=n→∞lim( –δn)Uyn–xn= .

Noticing that  –δn≥a, we get immediately

lim

n→∞Uyn–xn= .

SinceUisL-Lipschitzian, we have

Uxn–xn ≤ Uxn–Uyn+Uyn–xn

≤Lxn–yn+Uyn–xn

=LξnUxn–xn+Uyn–xn.

It follows that

Uxn–xn ≤

  –Lξn

Uyn–xn.

Sinceξn<√_+_L_+<



L, we deduce

lim

n→∞Uxn–xn= . (.)

From (.) and (.), we have lim_n→∞xn–un= . Thus,ω–limj→∞xnj =μ. By the

demiclosedness ofU–Iat  and (.), we getμ∈Fix(U). Hence,μ∈Fix(U). Therefore, μ∈.

Note that there is no more than one weak-cluster point of{un}. In fact, if we assume there

exists another{unk}such thatω–limk→∞unk=μ˜=μ, then we can deduceμ˜ ∈Fix(U).

Now we showμ˜=μ. By the Opial property of Hilbert space, we have

lim inf

k→∞ unk–μ˜<lim inf_k→∞ unk–μ=_n→∞limun–μ

=lim inf

j→∞ unj–μ<lim inf_j→∞ unj–μ˜

= lim

n→∞un–μ˜

=lim inf

k→∞ unk–μ˜.

This is a contradiction. Hence, the weak convergence of the whole sequence{un}follows

by applying Lemma . with=. This completes the proof.

Remark . Since the class of quasi-pseudocontractions contains the demicontractive

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Corollary . Let H and Hbe two real Hilbert spaces.Let A:H→Hbe a bounded linear operator.Let U:H→Hand T:H→Hbe two L-Lipschitzian demicontractive mappings with nonemptyFix(U) =C andFix(T) =Q.Assume T–I and U–I are demi-closed atand=∅.If the parametersγ,ν,η,β,{αn},{δn}and{ξn}satisfy the following

control conditions:

(C):  <ν< and <γ <λν,whereλis the spectral radius of the operatorA∗A;

(C):  <lim infn→∞αn≤lim supn→∞αn< ;

(C):  <  –η≤β<√_+_L_+ and <a≤ –δn≤ξn<



√

+L_+ for alln∈N.

Then the sequence{un}generated by algorithm(.)weakly converges to a split common

ﬁxed pointμ∈.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the ﬁnal manuscript.

Author details

1_{School of Mathematics and Information Science, Beifang University of Nationalities, Yinchuan, 750021, China.} 2_{Department of Information Management, Cheng Shiu University, Kaohsiung, 833, Taiwan.}3_{Center for Fundamental}

Science, Kaohsiung Medical University, Kaohsiung, 807, Taiwan.4_{Department of Mathematics, King Abdulaziz University,}

P.O. Box 80203, Jeddah, 21589, Saudi Arabia.5_{Department of Mathematics, Tianjin Polytechnic University, Tianjin, 300387,}

China.

Acknowledgements

Li-Jun Zhu was supported in part by NNSF of China (61362033). Yeong-Cheng Liou was supported in part by the Grants NSC 101-2628-E-230-001-MY3 and NSC 103-2923-E-037-001-MY3. Jen-Chih Yao was partially supported by the Grant NSC 103-2923-E-037-001-MY3.

Received: 22 April 2014 Accepted: 15 July 2014 Published:19 Aug 2014

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