A damped algorithm for the split feasibility and fixed point problems

R E S E A R C H

Open Access

A damped algorithm for the split feasibility

and fixed point problems

Cun-lin Li

_{, Yeong-Cheng Liou}

_{and Yonghong Yao}

*_{Correspondence:}

[email protected]

3_{Department of Mathematics,}

Tianjin Polytechnic University, Tianjin, 300387, China Full list of author information is available at the end of the article

Abstract

The purpose of this paper is to study the split feasibility problem and the fixed point problem. We suggest a damped algorithm. Convergence theorem is proven.

MSC: 47J25; 47H09; 65J15; 90C25

Keywords: split feasibility problem; fixed point problem; nonexpansive mapping; damped algorithm

1 Introduction

LetCandQbe two closed convex subsets of two Hilbert spacesHandH, respectively,

and letA:H→Hbe a bounded linear operator. Finding a pointx∗satisfies

x∗∈C and Ax∗∈Q. (.)

This problem, referred to as the split problem, has been studied by some authors. See,e.g., [–] and []. Some algorithms for solving (.) have been presented. One is Byrne’s CQ algorithm []

xn+=PC

xn–τA∗(I–PQ)Axn

, n∈N,

whereτ∈(,

L) withLbeing the largest eigenvalue of the matrixA∗A,Iis the unit matrix or operator, andPCandPQdenote the orthogonal projections ontoCandQ, respectively. Motivated by Byrne’s CQ algorithm, Xu [] suggested a single step regularized method

xn+=PC

( –αnγn)xn–γnA∗(I–PQ)Axn

, n∈N. (.)

Very recently, Dang and Gao [] introduced the following damped projection algorithm

xn+= ( –βn)xn+βnPC

( –αn)

xn–τA∗(I–PQ)Axn

, n∈N.

If every closed convex subset of a Hilbert space is the fixed point set of its associating projection, then the split feasibility problem becomes a special case of the split common fixed point problem of finding a pointx∗with the property

x∗∈Fix(U) and Ax∗∈Fix(T).

This problem was first introduced by Censor and Segal [], who invented an algorithm, which generates a sequence{xn}according to the iterative procedure

xn+=U

xn–γA∗(I–T)Axn

, n∈N.

Recently, Cui, Su and Wang [] extended the damped projection algorithm to the split common fixed point problems. For some related work, please refer to [] and [, ].

Motivated by these results, the purpose of this paper is to study the following split fea-sibility problem and fixed point problem

Findx∗∈C∩Fix(T) such thatAx∗∈Q∩Fix(S), (.) whereS:Q→QandT:C→Care two nonexpansive mappings. We suggest a damped algorithm for solving (.). Convergence theorem is proven.

2 Preliminaries

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

Definition . A mappingT:C→Cis callednonexpansiveif

Tx–Ty ≤ x–y

for allx,y∈C.

We will useFix(T) to denote the set of fixed points ofT, that is,Fix(T) ={x∈C:x=Tx}.

Definition . We callPC:H→Cthe metric projection if for eachx∈H

x–PC(x)=inf

x–y:y∈C.

It is well known that the metric projectionPC:H→Cis characterized by

x–PC(x),y–PC(x)

≤

for allx∈H,y∈C. From this, we can deduce thatPCis firmly-nonexpansive, that is,

PC(x) –PC(y)



≤x–y,PC(x) –PC(y)

(.)

for allx,y∈H. HencePCis also nonexpansive.

It is well known that in a real Hilbert spaceH, the following two equalities hold

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

for allx,y∈Handt∈[, ], and

for allx,y∈H. It follows that

x+y≤ x+ y,x+y (.)

for allx,y∈H.

Lemma .[] Let C be a closed convex subset of a real Hilbert space H,and let S:C→C be a nonexpansive mapping.Then,the mapping I–S is demiclosed.That is,if{xn}is a

sequence in C such that xn→x∗weakly and(I–S)xn→y strongly,then(I–S)x∗=y. Lemma .[] Assume that{an}is a sequence of nonnegative real numbers such that

an+≤( –γn)an+δn, n∈N,

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

() ∞_n=γn=∞;

() lim sup_n→∞δn

γn≤or

∞

n=|δn|<∞.

Thenlimn→∞an= . 3 Main results

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

respectively. LetA:H→H be a bounded linear operator with its adjointA∗. LetS: Q→QandT :C→Cbe two nonexpansive mappings. We use to denote the set of solutions of (.), that is,={x∗|x∗∈C∩Fix(T),Ax∗∈Q∩Fix(S)}. Now, we present our

algorithm.

Algorithm . Forx∈Harbitrarily, let{xn}be a sequence defined by

xn+=TPC

( –αn)

xn–δA∗(I–SPQ)Axn

for alln∈N, (.)

where{αn}n∈Nand{βn}n∈Nare two real number sequences in (, ) andδ∈(,_A). Theorem . Suppose=∅.Assume the sequence{αn}n∈Nsatisfies three conditions

(C) limn→∞αn= ;

(C) ∞_n=αn=∞;

(C) lim_n→∞αn+ αn = .

Then the sequence{xn},generated by algorithm(.),converges strongly to x∗=P().

Proof For the convenience, we writezn=PQAxn,yn= ( –αn)(xn–δA∗(I–SPQ)Axn) and

un=PC(( –αn)(xn–δA∗(I–SPQ)Axn)) for alln∈N. Thusun=PCynfor alln∈N. Let x∗ = P(). Hence, x∗ ∈ C ∩ Fix(T) and Ax∗ ∈ Q ∩ Fix(S). By the firmly-nonexpansivity ofPCandPQ, we can deduce the following conclusions

zn–Ax∗=PQAxn–PQAx∗≤Axn–Ax∗, (.) un–x∗=PCyn–PCx∗≤yn–x∗, (.) Szn–Ax∗



≤zn–Ax∗



≤Axn–Ax∗



–zn–Axn, (.)

and

zn+–zn=PQAxn+–PQAxn ≤ Axn+–Axn. (.)

From (.) and (.), we have

xn+–x∗=Tun–x∗≤un–x∗≤yn–x∗. (.)

Using (.), we get

yn–x∗



=( –αn)

xn–x∗+δA∗(Szn–Axn)

–αnx∗



≤( –αn)(xn–x∗+δA∗(Szn–Axn)



+αnx∗



= ( –αn)xn–x∗+δA∗(Szn–Axn)



+ δxn–x∗,A∗(Szn–Axn)

+αnx∗



. (.)

SinceAis a linear operator with its adjointA∗, we have

xn–x∗,A∗(Szn–Axn)

=Axn–x∗

,Szn–Axn

=Axn–Ax∗+Szn–Axn– (Szn–Axn),Szn–Axn

=Szn–Ax∗,Szn–Axn

–Szn–Axn. (.)

Again using (.), we obtain

Szn–Ax∗,Szn–Axn

=

Szn–Ax

∗

+Szn–Axn–Axn–Ax∗



. (.)

By (.), (.) and (.), we get

xn–x∗,A∗(Szn–Axn)

= 

Szn–Ax

∗

+Szn–Axn–Axn–Ax∗

–Szn–Axn

≤ 

Axn–Ax

∗

–zn–Axn+Szn–Axn

–Axn–Ax∗



–Szn–Axn

= –

zn–Axn

_–

Szn–Axn

_{. (.)}

Substituting (.) into (.), we deduce

yn–x∗



≤( –αn)

xn–x∗



+δASzn–Axn

+ δ

–

zn–Axn

_–

Szn–Axn



+αnx∗



= ( –αn)xn–x∗



–δzn–Axn

+αnx∗



≤( –αn)xn–x∗+αnx∗. (.)

It follows from (.) that

xn+–x∗ 

≤yn–x∗



≤( –αn)xn–x∗+αnx∗ ≤maxxn–x∗,x∗

.

The boundedness of the sequence{xn}yields.

Next, we estimatexn+–xn. Setvn=xn–δA∗(I–SPQ)Axn. According to (.) and (.), we have

vn+–vn=xn+–xn+δ

A∗(SPQ–I)Axn+–A∗(SPQ–I)Axn



=xn+–xn+δA∗

(SPQ–I)Axn+– (SPQ–I)Axn



+ δxn+–xn,A∗

(SPQ–I)Axn+– (SPQ–I)Axn

≤ xn+–xn+δASzn+–Szn– (Axn+–Axn)



+ δAxn+–Axn,Szn+–Szn– (Axn+–Axn)

=xn+–xn+δASzn+–Szn– (Axn+–Axn)



+ δSzn+–Szn,Szn+–Szn– (Axn+–Axn)

– δSzn+–Szn– (Axn+–Axn)



=xn+–xn+δASzn+–Szn– (Axn+–Axn)



+δSzn+–Szn+Szn+–Szn– (Axn+–Axn)

–Axn+–Axn

– δSzn+–Szn– (Axn+–Axn)

=xn+–xn+

δA–δSzn+–Szn– (Axn+–Axn)



+δSzn+–Szn–Axn+–Axn

≤ xn+–xn+

δA–δSzn+–Szn– (Axn+–Axn)



+δzn+–zn–Axn+–Axn

. (.)

Sinceδ∈(,_A), we derive by virtue of (.) and (.) that

vn+–vn ≤ xn+–xn. (.)

From (.) and (.), we have

xn+–xn ≤ yn+–yn

=( –αn+)vn+– ( –αn)vn

≤( –αn+)vn+–vn+|αn+–αn|vn ≤( –αn+)xn+–xn+|αn+–αn|vn.

It follows that

xn+–xn ≤ |

αn+–αn|

αn+

vn.

This, together with condition (C), implies that

lim

n→∞xn+–xn= . (.)

That is,

lim

n→∞xn–Tun= . (.)

Using the firmly-nonexpansiveness ofPC, we have

un–x∗



=PCyn–x∗



≤yn–x∗–PCyn–yn =yn–x∗



–un–yn. (.)

Thus,

xn+–x∗≤un–x∗ ≤yn–x∗



–un–yn

≤( –αn)xn–x∗



+αnx∗



–un–yn. (.)

It follows that

un–yn≤xn–x∗



–xn+–x∗ 

+αnx∗



≤xn–x∗+xn+–x∗xn+–xn+αnx∗



.

This, together with (.) and (C), implies that

lim

n→∞un–yn= . (.)

Returning to (.) and using (.), we have

xn+–x∗ 

≤yn–x∗



≤( –αn)xn–x∗



+ ( –αn)

δA–δSzn–Axn

– ( –αn)δzn–Axn+αnx∗



Hence,

( –αn)

δ–δA_Sz

n–Axn+ ( –αn)δzn–Axn

≤xn–x∗



–xn+–x∗ 

+αnx∗



≤xn–x∗+xn+–x∗xn+–xn+αnx∗



,

which implies that

lim

n→∞Szn–Axn=nlim→∞zn–Axn= . (.)

So,

lim

n→∞Szn–zn= . (.)

Note that

yn–xn=δA∗(SPQ–I)Axn+αnvn ≤δASzn–Axn+αnvn.

It follows from (.) that

lim

n→∞xn–yn= . (.)

From (.), (.) and (.), we get

lim

n→∞xn–Txn= . (.)

Now, we show that

lim sup n→∞

x∗,yn–x∗

≥.

Choose a subsequence{yni}of{yn}such that

lim sup n→∞

x∗,yn–x∗

=lim i→∞

x∗,yni–x

∗_{. (.)}

Since the sequence{yni}is bounded, we can choose a subsequence{yn_ij}of{yni}such that

yn_ij z. For the sake of convenience, we assume (without loss of generality) thatyniz.

Consequently, we derive from the above conclusions that

xniz, uniz, AxniAz and zniAz. (.)

To this end, we deduce thatz∈C∩Fix(T) andAz∈Q∩Fix(S). That is to say, z∈. Therefore,

lim sup n→∞

x∗,yn–x∗

= lim i→∞

x∗,yni–x

∗

= lim i→∞

x∗,z–x∗

≥. (.)

Finally, we prove thatxn→x∗. From (.), we have xn+–x∗≤yn–x∗

=( –αn)

vn–x∗

–αnx∗



≤( –αn)vn–x∗



– αn

x∗,yn–x∗

≤( –αn)xn–x∗



– αn

x∗,yn–x∗

. (.)

Applying Lemma . and (.) to (.), we deduce thatxn→x∗. The proof is

com-pleted.

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 Management, Beifang University of Nationalities, Yinchuan, 750021, China.}2_{Department of Information}

Management, Cheng Shiu University, Kaohsiung, 833, Taiwan.3_{Department of Mathematics, Tianjin Polytechnic}

University, Tianjin, 300387, China.

Acknowledgements

Cun-lin Li was supported in part by NSFC 71161001-G0105. Yeong-Cheng Liou was supported in part by NSC 101-2628-E-230-001-MY3 and NSC 101-2622-E-230-005-CC3. Yonghong Yao was supported in part by NSFC 11071279, NSFC 71161001-G0105 and LQ13A010007.

Received: 27 May 2013 Accepted: 29 July 2013 Published: 14 August 2013 References

1. Byrne, C: Iterative oblique projection onto convex subsets and the split feasibility problem. Inverse Probl.18, 441-453 (2002)

2. Censor, Y, Elfving, T: A multiprojection algorithm using Bregman projections in a product space. Numer. Algorithms8, 221-239 (1994)

3. Ceng, LC, Ansari, QH, Yao, JC: An extragradient method for split feasibility and fixed point problems. Comput. Math. Appl.64, 633-642 (2012)

4. Wang, F, Xu, HK: Approximating curve and strong convergence of the CQ algorithm for the split feasibility problem. J. Inequal. Appl.2010, Article ID 102085 (2010)

5. Dang, Y, Gao, Y: The strong convergence of a KM-CQ-like algorithm for a split feasibility problem. Inverse Probl.27, 015007 (2011)

6. Xu, HK: Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces. Inverse Probl.26, 105018 (2010)

7. Yao, Y, Wu, J, Liou, YC: Regularized methods for the split feasibility problem. Abstr. Appl. Anal.2012, Article ID 140679 (2012)

8. Yao, Y, Kim, TH, Chebbi, S, Xu, HK: A modified extragradient method for the split feasibility and fixed point problems. J. Nonlinear Convex Anal.13, 383-396 (2012)

9. Yao, Y, Postolache, M, Liou, YC: Strong convergence of a self-adaptive method for the split feasibility problem. Fixed Point Theory Appl.2013, 201 (2013)

10. Censor, Y, Segal, A: The split common fixed point problem for directed operators. J. Convex Anal.16, 587-600 (2009) 11. Cui, H, Su, M, Wang, F: Damped projection method for split common fixed point problems. J. Inequal. Appl.2013, 123

12. Moudafi, A: A note on the split common fixed-point problem for quasi-nonexpansive operators. Nonlinear Anal.74, 4083-4087 (2011)

13. He, ZH: The split equilibrium problems and its convergence algorithms. J. Inequal. Appl.2012, 162 (2012) 14. He, ZH, Du, WS: On hybrid split problem and its nonlinear algorithms. Fixed Point Theory Appl.2013, 47 (2013) 15. Geobel, K, Kirk, WA: Topics in Metric Fixed Point Theory. Cambridge Studies in Advanced Mathematics, vol. 28.

Cambridge University Press, Cambridge (1990)

16. Xu, HK: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc.66, 240-256 (2002) doi:10.1186/1029-242X-2013-379