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The strong convergence theorems for split common fixed point problem ofasymptotically nonexpansive mappings in Hilbert spaces

Journal of Inequalities and Applications20152015:1

https://doi.org/10.1186/1029-242X-2015-1

  • Received: 12 April 2014
  • Accepted: 11 December 2014
  • Published:

Abstract

In this paper, an iterative algorithm is introduced to solve the split common fixedpoint problem for asymptotically nonexpansive mappings in Hilbert spaces. Theiterative algorithm presented in this paper is shown to possess strong convergencefor the split common fixed point problem of asymptotically nonexpansive mappingsalthough the mappings do not have semi-compactness. Our results improve and developprevious methods for solving the split common fixed point problem.

MSC: 47H09, 47J25.

Keywords

  • split common fixed point problem
  • asymptotically nonexpansive mapping
  • strong convergence
  • Hilbert space
  • algorithm

1 Introduction and preliminaries

Throughout this paper, let and bereal Hilbert spaces whose inner product and norm are denoted by and , respectively; let C andQ be nonempty closed convex subsets of and ,respectively. A mapping is said to benonexpansive if for any .A mapping is said to bequasi-nonexpansive if for any and , where is the set of fixed pointsof T. A mapping is calledasymptotically nonexpansive if there exists a sequence satisfying such that for any .A mapping is semi-compact if, forany bounded sequence with , there exists asubsequence such that converges strongly to somepoint .

The split feasibility problem (SFP) is to find a point with the property
(1.1)

where is a bounded linear operator.

Assuming that SFP (1.1) is consistent (i.e., (1.1) has a solution), itis not hard to see that solves(1.1) if and only if it solves the following fixed point equation:
(1.2)

where and arethe (orthogonal) projections onto C and Q, respectively, isany positive constant, and denotes the adjoint of A.

The SFP in finite-dimensional Hilbert spaces was first introduced by Censor andElfving [1] for modeling inverse problems whicharise from phase retrievals and in medical image reconstruction [2]. Recently, it has been found that the SFP can also beused in various disciplines such as image restoration, computer tomograph, and radiationtherapy treatment planning [27].

Let and be two mappings satisfying and ,respectively; let be a bounded linear operator. The split common fixed point problem (SCFP) formappings S and T is to find a point with the property
(1.3)

We use Γ to denote the set of solutions of SCFP (1.3), that is, .

Since each closed and convex subset may be considered as a fixed point set of aprojection on the subset, hence the split common fixed point problem (SCFP) isa generalization of the split feasibility problem (SFP) and the convexfeasibility problem (CFP) [5].

Split feasibility problems and split common fixed point problems have been studied bysome authors [815]. In 2010,Moudafi [10] proposed the following iterationmethod to approximate a split common fixed point of demi-contractive mappings: forarbitrarily chosen ,

and he proved that converges weakly to a split common fixedpoint , where and are two demi-contractive mappings, is a bounded linear operator.

Using the iterative algorithm above, in 2011, Moudafi [9] also obtained a weak convergence theorem for the split commonfixed point problem of quasi-nonexpansive mappings in Hilbert spaces. After that, someauthors also proposed some iterative algorithms to approximate a split common fixedpoint of other nonlinear mappings, such as nonspreading type mappings [16], asymptotically quasi-nonexpansive mappings[12], κ-asymptoticallystrictly pseudononspreading mappings [17],asymptotically strictly pseudocontraction mappings [18]etc., but they just obtained weak convergence theoremswhen those mappings do not have semi-compactness. This naturally brings us to thefollowing question.

Can we construct an iterative scheme which can guarantee the strong convergence forsplit common fixed point problems without assumption ofsemi-compactness?

In this paper, we introduce the following iterative scheme. Let , ,the sequence is defined as follows:
(1.4)

where and are two asymptotically nonexpansive mappings, is a bounded linear operator, denotes the adjoint of A. Under some suitable conditions on parameters, theiterative scheme is shown to converge strongly to a splitcommon fixed point of asymptotically nonexpansive mappings and without the assumption of semi-compactness on and .

The following lemma and results are useful for our proofs.

Lemma 1.1[19]

LetEbe a real uniformly convex Banach space, Kbea nonempty closed subset ofE, and let be an asymptoticallynonexpansive mapping. Then isdemiclosed at zero, that is, if convergesweakly to a point and ,then .

Let C be a closed convex subset of a real Hilbert space H. denotes the metric projection of H onto C. It is well known that ischaracterized by the properties: for and ,
(1.5)
and
(1.6)
In a real Hilbert space H, it is also well known that
(1.7)
and
(1.8)

2 Main results

Theorem 2.1Let and be twoHilbert spaces, bea bounded linear operator, bean asymptotically nonexpansive mapping with the sequence satisfying ,and bean asymptotically nonexpansive mapping with the sequence satisfying , and , respectively.Let , ,and let the sequence be defined as follows:
(2.1)

where denotesthe adjoint ofA, and satisfies , , .If , then converges stronglyto .

Proof We will divide the proof into five steps.

Step 1. We first show that is closedand convex for any .

Since ,so isclosed and convex. Assume that is closedand convex. For any ,since

we know that is closed and convex. Therefore is closedand convex for any .

Step 2. We prove for any .

Let , then from (2.1) we have
(2.2)
where
(2.3)
Substituting (2.3) into (2.2), we can obtain that
(2.4)
In addition, it follows from (2.1) that
(2.5)

Therefore, from (2.4) and (2.5), we know that and for any .

Step 3. We will show that is a Cauchy sequence.

Since and ,then
(2.6)
It means that is bounded. For any , by using(1.6), we have
which implies that . Thus is nondecreasing. Therefore, by the boundednessof , exists. For somepositive integers m, n with , from and (1.6), we have
(2.7)

Since exists, it followsfrom (2.7) that . Therefore is a Cauchy sequence.

Step 4. We will show that .

Since ,we have
(2.8)
(2.9)
(2.10)
Notice that , it follows from (2.4)that
thus, since is bounded and ,from (2.8) we have
(2.11)
On the other hand, since
we have
Since and ,we know that
(2.12)
In addition, since , we know that . So from
we can obtain that
(2.13)
Similarly, we have
(2.14)

Step 5. We will show that converges strongly to an element ofΓ.

Since is a Cauchy sequence, we may assume that ,from (2.8) we have ,which implies that .So it follows from (2.13) and Lemma 1.1 that .

In addition, since A is a bounded linear operator, we have that . Hence, itfollows from (2.14) and Lemma 1.1 that . This means that and converges strongly to . The proof iscompleted. □

In Theorem 2.1, as and ,we have the following result.

Corollary 2.2Let be aHilbert space, bean asymptotically nonexpansive mapping with a sequence satisfying .The sequence is defined as follows: ,
(2.15)

where and satisfies .If , then converges strongly to a fixedpoint ofT.

In Theorem 2.1, when and aretwo nonexpansive mappings, the following result holds.

Corollary 2.3Let and be twoHilbert spaces, bea bounded linear operator, and betwo nonexpansive mappings such that and , respectively.Let , ,and let the sequence be defined as follows:
(2.16)

where denotesthe adjoint ofA, and satisfies .If , then converges stronglyto .

Remark 2.4 When and aretwo quasi-nonexpansive mappings and and are demiclosed at zero, Corollary 2.3 also holds.

Example 2.5 Let C be a unit ball in a real Hilbert space ,and let be amapping defined by
It is proved in Goebel and Kirk [20] that
  1. (i)

    , ;

     
  2. (ii)

    , , .

     
Taking , , itis easy to see that .So we can take ,and , ,then

Therefore is anasymptotically nonexpansive mapping from C into itself with .

Let D be an orthogonal subspace of with thenorm and the inner product for and . For each , we define amapping by

It is easy to show that or for any .Therefore is anasymptotically nonexpansive mapping from D into itself with since for anysequence with .

Obviously, C and D are closed convex subsets of and ,respectively. Let be defined by for . ThenA is a bounded linear operator with adjoint operator for . Clearly, , .

Taking , , , , and , . Itfollows from Theorem 2.1 that converges strongly to .

3 Applications and examples

Application to the equilibrium problem

Let H be a real Hilbert space, C be a nonempty closed and convexsubset of H, and let the bifunction satisfy the following conditions:

(A1) , ;

(A2) , ;

(A3) For all , ;

(A4) For each , thefunction is convex and lower semi-continuous.

The so-called equilibrium problem for F is to find a point such that for all .The set of its solutions is denoted by .

Lemma 3.1[21]

LetCbe a nonempty closed convex subset of a HilbertspaceH, and let be a bifunctionsatisfying (A1)-(A4). Let and .Then there exists suchthat

Lemma 3.2[21]

Assume that satisfies(A1)-(A4). For and ,define a mapping as follows:
Then
  1. (1)

    is single-valued;

     
  2. (2)
    is firmly nonexpansive, that is, for all ,
     
  3. (3)

    ;

     
  4. (4)

    is nonempty, closed and convex.

     
Theorem 3.3Let and be twoHilbert spaces, bea bounded linear operator, bea nonexpansive mapping, be a bifunctionsatisfying (A1)-(A4). Assume that and .Taking ,for arbitrarily chosen ,the sequence is defined asfollows:
(3.1)

where denotesthe adjoint ofA, , and satisfies .If , then thesequence converges strongly toa point .

Proof It follows from Lemma 3.2 that , is nonempty, closed and convex and is a firmly nonexpansive mapping. Hence all conditions in Corollary 2.3 aresatisfied. The conclusion of Theorem 3.3 can be directly obtained fromCorollary 2.3. □

Let and be two real Hilbert spaces. Let C be a closed convex subset of ,K be a closed convex subset of , be a bounded linear operator. Assume that F is a bi-function from into R and G is a bi-function from intoR. The split equilibrium problem (SEP) is to
(3.2)
and
(3.3)

Let denote the solution setof the split equilibrium problem SEP.

Example 3.4[22]

Let , and . Let for all R, then A is a bounded linear operator. Let and bedefined by and , respectively. Clearly, and . So .

Example 3.5[22]

Let with the standard norm and with the norm for some . and . Define a bi-function ,where , , thenF is a bi-function from intoR with . For each ,let ,then A is a bounded linear operator from into .In fact, it is also easy to verify that and for some and .Now define another bi-function G as follows: for all .Then G is a bi-function from intoR with .

Clearly, when , we have . So .

Corollary 3.6Let and be twoHilbert spaces, bea bounded linear operator, be a bifunctionsatisfying and be a bifunctionsatisfying .Taking ,for arbitrarily chosen ,the sequence is defined asfollows:
(3.4)

where denotesthe adjoint ofA, , and satisfies .If , then thesequence converges strongly toa point .

Remark 3.7 Since Example 3.4 and Example 3.5 satisfy the conditionsof Corollary 2.3, the split equilibrium problems in Example 3.4 andExample 3.5 can be solved by algorithm (3.4).

Application to the hierarchial variational inequality problem

Let H be a real Hilbert space, and be two nonexpansive mappings from H to H such that and .

The so-called hierarchical variational inequality problem for nonexpansive mapping with respect to a nonexpansive mapping isto find a point such that
(3.5)
It is easy to see that (3.5) is equivalent to the following fixed point problem:
(3.6)
where is the metric projection from H onto . Letting and (the fixed point set ofthe mapping )and (theidentity mapping on H), then problem (3.6) is equivalent to the followingsplit feasibility problem:
(3.7)

Hence from Theorem 2.1 we have the following theorem.

Theorem 3.8LetH, , ,CandQbe the same as above.Let and ,and let the sequence be defined asfollows:
(3.8)

where and satisfies .If , then thesequence converges strongly toa solution of the hierarchical variational inequality problem (3.5).

Declarations

Acknowledgements

The authors would like to express their thanks to the reviewers and editors for theirhelpful suggestions and advice. This work was supported by the National NaturalScience Foundation of China (Grant No. 11361070) and the Scientific ResearchFoundation of Postgraduate of Yunnan University of Finance and Economics.

Authors’ Affiliations

(1)
College of Statistics and Mathematics, Yunnan University of Finance and Economics, Long Quan Road, Kunming, China
(2)
School of Information Engineering, The College of Arts and Sciences, Yunnan NormalUniversity, Long Quan Road, Kunming, 650222, China
(3)
Department of Mathematics, Kunming University, Pu Xin Road No. 2, Kunming Economicand Technological Development Zone, Kunming, 650214, China

References

  1. Byrne C: Iterative oblique projection onto convex subsets and the split feasibilityproblems.Inverse Probl. 2002, 18:441–453. 10.1088/0266-5611/18/2/310View ArticleMATHGoogle Scholar
  2. Censor Y, Elfving T: A multiprojection algorithm using Bregman projection in a product space.Numer. Algorithms 1994, 8:221–239. 10.1007/BF02142692MathSciNetView ArticleMATHGoogle Scholar
  3. Censor Y, Elfving T, Kopf N, Bortfeld T: The multiple-sets split feasibility problem and its applications.Inverse Probl. 2005, 21:2071–2084. 10.1088/0266-5611/21/6/017MathSciNetView ArticleMATHGoogle Scholar
  4. Censor Y, Seqal A: The split common fixed point problem for directed operators.J. Convex Anal. 2009, 16:587–600.MathSciNetMATHGoogle Scholar
  5. Censor Y, Bortfeld T, Martin B, Trofimov T: A unified approach for inversion problem in intensity-modulated radiationtherapy.Phys. Med. Biol. 2006, 51:2353–2365. 10.1088/0031-9155/51/10/001View ArticleGoogle Scholar
  6. Censor Y, Motova A, Segal A: Perturbed projections and subgradient projections for the multiple-sets splitfeasibility problems.J. Math. Anal. Appl. 2007, 327:1244–1256. 10.1016/j.jmaa.2006.05.010MathSciNetView ArticleMATHGoogle Scholar
  7. Lopez G, Martin V, Xu HK: Iterative algorithms for the multiple-sets split feasibility problem. In Biomedical Mathematics: Promising Directions in Imaging, Therapy Planning andInverse Problems. Edited by: Censor Y, Jiang M, Wang G. Medical Physics Publishing, Madison; 2009:243–279.Google Scholar
  8. Dang Y, Gao Y: The strong convergence of a KM-CQ-like algorithm for a split feasibilityproblem.Inverse Probl. 2011., 27: Article ID 015007Google Scholar
  9. Moudafi A: A note on the split common fixed point problem for quasi-nonexpansiveoperators.Nonlinear Anal. 2011, 74:4083–4087. 10.1016/j.na.2011.03.041MathSciNetView ArticleMATHGoogle Scholar
  10. Moudafi A: The split common fixed point problem for demi-contractive mappings.Inverse Probl. 2010., 26: Article ID 055007Google Scholar
  11. Maruster S, Popirlan C: On the Mann-type iteration and convex feasibility problem.J. Comput. Appl. Math. 2008, 212:390–396. 10.1016/j.cam.2006.12.012MathSciNetView ArticleMATHGoogle Scholar
  12. Qin LJ, Wang L, Chang SS: Multiple-set split feasibility problem for a finite family of asymptoticallyquasi-nonexpansive mappings.Panam. Math. J. 2012,22(1):37–45.MATHGoogle Scholar
  13. Wang F, Xu HK: Approximation curve and strong convergence of the CQ algorithm for the splitfeasibility problem.J. Inequal. Appl. 2010., 2010: Article ID 102085 10.1155/2010/102085Google Scholar
  14. Xu HK: A variable Krasnosel’skii-Mann algorithm and the multiple-set splitfeasibility problem.Inverse Probl. 2006, 22:2021–2034. 10.1088/0266-5611/22/6/007View ArticleMATHGoogle Scholar
  15. Yang Q: The relaxed CQ algorithm for solving the split feasibility problem.Inverse Probl. 2004, 20:1261–1266. 10.1088/0266-5611/20/4/014MathSciNetView ArticleMATHGoogle Scholar
  16. Chang SS, Kim JK, Cho YJ, Sim JY: Weak-and strong-convergence theorems of solution to split feasibility problem fornonspreading type mapping in Hilbert spaces.Fixed Point Theory Appl. 2014., 2014: Article ID 11Google Scholar
  17. Quan J, Chang SS, Zhang X: Multiple-set split feasibility problems fork-strictly pseudononspreadingmapping in Hilbert spaces.Abstr. Appl. Anal. 2013., 2013: Article ID 342545 10.1155/2013/342545Google Scholar
  18. Chang SS, Cho YJ, Kim JK, Zhang WB, Yang L: Multiple-set split feasibility problems for asymptotically strictpseudocontractions.Abstract and Applied Analysis 2012., 2012: Article ID 491760 10.1155/2012/491760Google Scholar
  19. Chang SS, Cho YJ, Zhou H: Demi-closed principle and weak convergence problems for asymptoticallynonexpansive mappings.J. Korean Math. Soc. 2001, 38:1245–1260.MathSciNetMATHGoogle Scholar
  20. Goebel K, Kirk WA: A fixed point theorem for asymptotically nonexpansive mappings.Proc. Am. Math. Soc. 1972, 35:171–174. 10.1090/S0002-9939-1972-0298500-3MathSciNetView ArticleMATHGoogle Scholar
  21. Combettes PL, Hirstoaga SA: Equilibrium programming in Hilbert spaces.J. Nonlinear Convex Anal. 2005, 6:117–136.MathSciNetMATHGoogle Scholar
  22. He Z: The split equilibrium problem and its convergence algorithms.J. Inequal. Appl. 2012., 2012: Article ID 162Google Scholar

Copyright

© Zhang et al.; licensee Springer. 2015

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative CommonsAttribution License (http://creativecommons.org/licenses/by/4.0), which permitsunrestricted use, distribution, and reproduction in any medium, provided the originalwork is properly credited.

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