General split equality problems in Hilbert spaces

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R E S E A R C H

Open Access

General split equality problems in Hilbert

spaces

Rudong Chen

*

_{, Jie Wang and Huiwen Zhang}

*_{Correspondence:}

[email protected]

Department of Mathematics, Tianjin Polytechnic University, Tianjin, 300387, China

Abstract

A new convex feasibility problem, the split equality problem (SEP), has been proposed by Moudaﬁ and Byrne. The SEP was solved through the ACQA and ARCQA algorithms. In this paper the SEPs are extended to inﬁnite-dimensional SEPs in Hilbert spaces and we established the strong convergence of a proposed algorithm to a solution of general split equality problems (GSEPs).

Keywords: general split equality problem; strong convergence; the minimum norm solution

1 Introduction

In the present paper, we are concerned with the general split equality problem (GSEP) which is formulated as ﬁnding pointsxandywith the property:

x∈ ∞

i=

Ciandy∈ ∞

j=

Qj, such thatAx=By, (.)

whereCiandQjare two nonempty closed convex subsets of real Hilbert spacesHand

H, respectively,Halso is a Hilbert space,A:H→H,B:H→Hare two bounded

linear operators.

It generalizes the split equality problem (SEP), which is to ﬁndx∈C,y∈Qsuch that Ax=By[], as well as the split feasibility problem (SFP). WhenB=I, the SEP becomes a SFP. As we know, the SEP has received much attention due to its applications in im-age reconstruction, signal processing, and intensity-modulated radiation therapy, see for instance [–].

To solve the SEP, Byrne and Moudafi put forward the alternating CQ-algorithm (ACQA) and the relaxed alternating CQ-algorithm (RACQA). For an exhaustive study of ACQA and RACQA, see for instance [, ]. The approximate SEP (ASEP), which is only to find approximate solutions to SEP, is also proposed and solved through the simultaneous iter-ative algorithm (SSEA), the relaxed SSEA (RSSEA) and the perturbed SSEA (PSSEA) by Byrne and Moudafi, see for example [, ].

This paper aims at a study of an iterative algorithm improved by Eslamian [] for the GSEP in the Hilbert space. We show the strong convergence of the presented algorithms to a solution of the GSEP, and we obtain an algorithm which strongly converges to the minimum norm solution of the GSEP.

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2 Preliminaries

For the sake of simplicity, we will denote byH a real Hilbert space with inner product ·,· and norm · . Let Cbe a nonempty closed convex subset of H. LetT :H →H be an operator on H. Recall thatT is said to be nonexpansive ifTx–Ty ≤ x–y, ∀x,y∈H. A typical example of nonexpansivity is the orthogonal projectionPC fromH onto a nonempty closed convex subsetC⊆Hdeﬁned byx–PCx=minx–y,y∈C. It is well known thatPCxis characterized by the relation

PCx∈C, x–PCx,y–PCx ≤, ∀y∈C.

Lemma . Let S=C×Q in RN_×_RM₌_RI_,_{where I}₌_N₊_M._Deﬁne

G=A –B, w=

x y

, and so G∗G=

A∗A –A∗B –B∗A B∗B

,

then w∗ =_yx∗∗solves the SEP if and only if w∗ solves the ﬁxed point equation PS(I–

γG∗G)w∗=w∗.

Lemma . Let H be a Hilbert space.Then for any given sequence{xn}in H,any given sequence{λn}∞_n_=of positive numbers with∞_n_=λn= and for any positive integer i,j with i<j,

∞

n= λnxn



≤ ∞

n=

λnxn–λiλjxi–xj.

Lemma . Let H be a Hilbert space.For every x and y in H,the following inequality holds:

x+y≤ x+ y,x+y.

Lemma . Let C be a nonempty closed convex subset of H,and let T:C→C be a non-expansive mapping withFix(T)=∅.Then T is demiclosed on C,that is,if xnx∈C and xn–Txn→,then x=Tx.

Lemma . Assume{an}is a sequence of nonnegative real numbers such that an+≤( –

γn)an+δnwhere{γn}is a sequence in(, )and{δn}is a sequence such that

(a) ∞_n_=γn=∞;

(b) lim sup_n→∞δn/γn≤or

∞

n=|δn|<∞.

Thenlimn→∞an= .

Lemma . Let{tn}be a sequence of real numbers that does not decrease at inﬁnity,in the sense that there exists a subsequence{tnj}j≥of{tn}such that

{tnj}<{tnj+} for all j≥.

Also consider the sequence of the integers{τ(n)}n≥ndeﬁned by

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Then{τ(n)}n≥nis a nondecreasing sequence verifyinglim_n→∞τ(n) =∞,and for all n≥n,

the following two estimates hold:

tτ(n)≤tτ(n)+, tn≤tτ(n)+.

3 Main results

LetCi⊆RN _{and let}_Qi_⊆_RM_{be closed, nonempty convex sets, and let}_A,_B_be_J_×_N_and J×Mreal matrices, respectively. LetSi=Ci×Qi. Deﬁne

G=A –B, w=

x y

,

then

G∗G=

A∗A –A∗B –B∗A B∗B

.

The problem (.) can also be formulated as ﬁndingw∈S=+_i_=∞SiwithGw=  or with minimizing the functionGwoverw∈S[].

Proposition . w∗=x_y∗∗solves the GSEP(.)if and only if

w∗∈

+∞

i=

PS_iI–λn,iG∗G

w∗.

Proof Assume that there existsw∗satisfyingw∗∈+_i_=∞PSi(I–λn,iG∗G)w∗, then for anyi∈

[, +∞), we havew∗=PSi(I–λn,iG∗G)w∗. We usexandyto expressw∗=PSi(I–λn,iG∗G)w∗:

x∗=PCix∗–λn,iA∗

Ax∗–By∗, (.)

y∗=PCiy∗+λn,iB∗

Ax∗–By∗. (.)

By Lemma ., for anyi∈[, +∞), there existx∗∈Ci andy∗∈Qi, such thatAx∗=By∗. Therefore, there existx∗∈+_i_=∞Ciandy∗∈

+∞

i= Qi, such thatAx∗=By∗, that is to say,w∗ solves GSEP (.).

Assume thatw∗solves GSEP (.), such thatGw∗= , that is, for anyi∈[, +∞), we have x∗∈Ciandy∗∈Qi, such thatAx∗=By∗. SubstitutingAx∗=By∗ into (.) and (.), we obtain for anyi∈[, +∞),w∗=PSi(I–λn,iG∗G)w∗. Therefore,w∗solvesw∗∈

+∞

i= PSi(I–

λn,iG∗G)w∗.

Theorem . Assume that the GSEP has a nonempty solution set.Suppose that f is a self k-contraction mapping of H,k∈(, ),and let{wn}be a sequence generated by

wn+=αnwn+βnf(wn) +

∞

i= γn,iPSi

I–λn,iG∗G

wn, n≥, (.)

whereαn+βn+

_∞

i=γn,i= .If the sequences{αn},{βn},{γn,i},and{λn,i}satisfy the following

conditions:

(i) lim_n→∞βn= and

∞

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(ii) lim inf_n→∞αnγn,i> ,for eachi∈N,

(iii) {λn,i} ⊂(,_L),for eachi∈N,whereL=ρ(G∗G),

then the sequence{wn}strongly converges to w∗,where w∗=Pf(w∗),w∗= x∗

y∗

.

Proof We ﬁrst prove that{wn}is bounded. Letz∈; actually, by Lemma .,z∈equals the ﬁxed point equationz=PS_i(I–λn,iG∗G)z. Note that for eachi∈N, {λn,i} ⊂(,_L),

whereL=ρ(G∗G), then the operatorPSi(I–λn,iG∗G) is nonexpansive. We also know that f is ak-contraction mapping, then

wn+–z=

αnwn+βnf(wn) +

∞

i= γn,iPSi

I–λn,iG∗G

wn–z

≤αnwn–z+βnf(wn) –z+ ∞

i= γn,iPSi

I–λn,iG∗G

wn–z

≤αnwn–z+βnf(wn) –z

+ ∞

i= γn,iPSi

I–λn,iG∗G

wn–PSiI–λn,iG∗G

z

≤αnwn–z+βnf(wn) –z+ ∞

i=

γn,iwn–z

= ( –βn)wn–z+βnf(wn) –z

≤( –βn)wn–z+βnf(wn) –f(z)+βnf(z) –z ≤( –βn)wn–z+βnkwn–z+βnf(z) –z

= – ( –k)βn

wn–z+ ( –k)βn 

 –kf(z) –z

≤max

wn–z, 

 –kf(z) –z

.

Then, from the upper deduction we havewn–z ≤max{wn––z,_–_kf(z) –z}and

wn+–z ≤( –βn)wn–z+βnf(wn) –z

≤max

wn–z, 

 –kf(z) –z

≤ · · ·

≤max

w–z,



 –kf(z) –z

.

We can conclude that{wn},{f(wn)}are bounded. Furthermore, from (.) and Lemma . we get

wn+–z =

∞

i= γn,iPSi

I–λn,iG∗G

wn–z



=

αn(wn–z) +βn

f(wn) –z+ ∞

i= γn,i

PSiI–λn,iG∗G

wn–z

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≤αnwn–z+βnf(wn) –z



+ ∞

i= γn,iPSi

I–λn,iG∗G

wn–z

–αnγn,iPSi

I–λn,iG∗G

wn–wn

≤αnwn–z+βnf(wn) –z



+ ∞

i=

γn,iwn–z

–αnγn,iPSi

I–λn,iG∗G

wn–wn

= ( –βn)wn–z+βnf(wn–z)



–αnγn,iPSi

I–λn,iG∗G

wn–wn.

It follows that

αnγn,iPSi

I–λn,iG∗G

wn–wn

≤ wn–z–wn+–z+βnf(wn) –z. (.)

In order to show that{wn} →w∗, we consider two cases.

Case : Suppose that{wn–w∗}is a monotone sequence. Sincewn–w∗is bounded, wn–w∗is convergent. Take the limit on both sides for (.), becauselim_n→∞βn=  and lim inf_n→∞αnγn,i> , and we getlimn→∞PSi(I–λn,iG

∗_G)wn_–_wn_{= ,}_∀_i_∈_N_.

We ﬁrst prove there exists a uniquew∗∈, such thatw∗=Pf(w∗). SincePis

nonex-pansive andf is a selfk-contraction mapping, we get

P(f)(w) –P(f)(w)≤f(w) –f(w)≤kw–w;

therefore, there exists a uniquew∗∈, such thatw∗=Pf(w∗).

Next, we show that{wn} →w∗. Using Lemma ., we get

_wn_+_–_w∗

=

∞

i= γn,iPS

I–λn,iG∗G

wn–w∗

 = αn

wn–w∗+βn

f(wn) –w∗+ ∞

i= γn,i

PSI–λn,iG∗G

wn–w∗



≤αn

wn–w∗

+ ∞

i= γn,i

PS

I–λn,iG∗G

wn–w∗



+ βn

f(wn) –w∗,wn+–w∗

≤( –βn)wn–w∗+ βn

f(wn) –fw∗,wn+–w∗

+ βn

fw∗–w∗,wn+–w∗

≤( –βn)wn–w∗+ βnkwn–w∗wn+–w∗

+ βn

≤( –βn)wn–w∗+βnkwn–w∗+wn+–w∗ 

+ βn

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By induction, we obtain

wn+–w∗ 

≤( –βn)+βnk

 –βnk wn–w ∗

+ βn  –βnk

=  – βn+βnk

 –βnk wn–w ∗

+ β



n

 –βnkwn–w ∗

+ βn  –βnk

≤

 –( –k)βn  –βnk

wn–w∗

+( –k)βn  –βnk

βnM ( –k)+

  –k

≤( –ηn)wn–w∗+ηnδn,

where ηn= (–_–βk_nk)βn, δn={_(–βnM_k₎ +_–_kf(w∗) –w∗,wn+–w∗}andM=sup{wn–w∗, n≥}.

Since limn→∞βn = ,

∞

n=βn = , we have

∞

n=ηn = ∞. Next, we will prove lim sup_n→∞δn≤. Actually, _(–βnM_k₎ → (sincelimn→∞βn= ), so we just need to prove lim sup_n→∞f(w∗) –w∗,wn–w∗ ≤. Take a subsequence{wnk}in{wn}, such that

lim n→∞

fw∗–w∗,wn_k–w∗=lim sup n→∞

fw∗–w∗,wn–w∗.

Since{wn_k}is bounded, there exists a subsequence{wn_kj}converging weakly tov. Suppose thatwn_k vandλn,i→λi∈(,_G∗_G), according to Lemma .,v∈. Sincev∈and w∗=Pf(w∗),

lim sup n→∞

fw∗–w∗,wn–w∗= lim n→∞

fw∗–w∗,wnk–w∗

=fw∗–w∗,v–w∗≤,

as desired.

Therefore,∞_n_=ηn=∞andlim supn→∞δn≤ hold. All conditions of Lemma . are satisﬁed. Thereforewn+–w∗ →,wn→w∗.

Case : If{wn–w∗}is not a monotone sequence, we could deﬁne an integer sequence {τ(n)}by

τ(n) =maxk≤n:wk–w∗≤wk+–w∗.

It is easy to see that{τ(n)}is nondecreasing and whenn→ ∞we getτ(n)→ ∞. For all n≥nwe obtainwτ(n)–w∗<wτ(n)+–w∗. Then{wτ(n)–w∗}is a monotone sequence

and according to Case , we havelim_n→∞wτ(n)–w∗=  andlimn→∞wτ(n)+–w∗= .

Finally, from Lemma ., we get

≤wn–w∗≤maxwn–w∗,wτ(n)–w∗≤wτ(n)+–w∗→, n→ ∞.

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For every n≥, w∗ ∈ solves the GSEP if and only if w∗ solves the ﬁxed point equationw∗=PSi(I–λn,iG∗G)w∗,i∈N. Actually, we have provedlimn→∞wn–PSi(I– λn,iG∗G)wn=  andwn→w∗. Thenw∗=PSi(I–λn,iG∗G)w∗,i∈N, that is,w∗∈solves the GSEP.

Therefore, the sequence{wn}strongly converges tow∗=Pf(w∗). This completes the

proof.

Corollary . We deﬁne a sequence{wn}iteratively

wn+=αnwn+

∞

i= γn,iPSi

I–λn,iG∗G

wn, n≥, (.)

whereαn+

∞

i=γn,i⊂(, ).If{αn},{γn,i},{λn,i}satisfy the following conditions:

(i) lim_n→∞(αn+

∞

i=γn,i) = and

∞

n=( –αn–

∞

i=γn,i) =∞,

(ii) lim inf_n→∞αnγn,i> ,for eachi∈N,

then {wn} converges strongly to a point w∗ which is the minimum norm solution of GSEP(.).

Proof Letf =  in (.), then we get (.). We have provedwn+→w∗=Pf(w∗) in

Theo-rem .. Then,

fw∗–w∗,z–w∗=fw∗–Pf

w∗,z–Pf

w∗≤.

Hence,f(w∗) –w∗,z–w∗ ≤. Sincef = , then–w∗,z–w∗ ≤, for allz∈, that is,

w∗≤w∗,z≤w∗· z ⇒ w∗≤ z.

Thus,w∗is the minimum norm solution of GSEP (.). This completes the proof.

Let{Ti}∞_i_=:H→Hbe a countable family of nonexpansive mappings with∞_i_=F(Ti)=∅ and letT:H→Hbe a nonexpansive mapping. Consider the variational inequality prob-lem of ﬁnding a common ﬁxed point of{Ti}with respect to a nonexpansive mappingTis to

ﬁndx∗∈ ∞

i=

F(Ti), such thatx∗–Tx∗,x∗–x≤, ∀x∈ ∞

i=

F(Ti). (.)

It is easy to see that (.) equals the following ﬁxed point problem:

ﬁndx∗∈ ∞

i=

F(Ti), such thatx∗=P∞ i=F(Ti)Tx

∗_. _(.)

LettingCi=F(Ti), Qj=F(PF(Tj)T),A=I,B=I, then the upper problem (.) is

trans-formed into GSEP (.):

ﬁndx∈ ∞

i=

Ciandy∈ ∞

j=

Qj, such thatAx=By.

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Theorem . Ifαn+βn+

∞

i=γn,i= and the sequences{αn},{βn},{γn,i},and{λn,i}satisfy

the following conditions:

(i) lim_n→∞βn= and

∞

n=βn=∞,

(ii) lim infn→∞αnγn,i> ,for eachi∈N,

the sequence{wn}deﬁned by(.)converges strongly to a point w∗which solves the following variational inequality with w∗∈:

fw∗–w∗,z–w∗≤ for all z∈.

Proof We know from the proof of Theorem . that the sequence{wn}deﬁned by (.) converges strongly tow∗=Pf(w∗), which solves the GSEP. Also since GSEP (.) equals

(.),w∗solves the variational inequality problem (.). Sincew∗=Pf(w∗), by (.) and

(.), we havef(w∗) –w∗,z–w∗ ≤. Actually, sincef is a selfk-contraction mapping, k∈(, ), thenf is also a nonexpansive mapping. That is to say, the condition in (.), thatTis a nonexpansive mapping, is satisﬁed. Therefore,{wn}deﬁned by (.) converges strongly to a solution off(w∗) –w∗,z–w∗ ≤. This completes the proof.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and signiﬁcantly in writing this paper. All authors read and approved the ﬁnal manuscript.

Acknowledgements

We wish to thank the referees for their helpful comments and suggestions. This research was supported by NSFC Grant No. 11071279.

Received: 17 October 2013 Accepted: 24 January 2014 Published:13 Feb 2014

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