- Research
- Open Access

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

- Xin-Fang Zhang
^{1}Email author, - Lin Wang
^{1}Email author, - Zhao Li Ma
^{2}Email author and - Li Juan Qin
^{3}Email author

**2015**:1

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

© Zhang et al.; licensee Springer. 2015

**Received:**12 April 2014**Accepted:**11 December 2014**Published:**7 January 2015

## 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* and*Q* 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
.

where is a bounded linear operator.

*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:

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 [2–7].

*SCFP*) formappings

*S*and

*T*is to find a point with the property

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].

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*?

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]

*Let**E**be a real uniformly convex Banach space*, *K**bea nonempty closed subset of**E*, *and let*
*be an asymptoticallynonexpansive mapping*. *Then*
*isdemiclosed at zero*, *that is*, *if*
*convergesweakly to a point*
*and*
,*then*
.

*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 ,

## 2 Main results

**Theorem 2.1**

*Let*

*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*:

*where*
*denotesthe adjoint of**A*,
*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 .

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

Step 2. We prove for any .

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

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

*m*,

*n*with , from and (1.6), we have

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

Step 4. We will show that .

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 2.2**

*Let*

*be aHilbert space*,

*bean asymptotically nonexpansive mapping with a sequence*

*satisfying*.

*The sequence*

*is defined as follows*: ,

*where*
*and*
*satisfies*
.*If*
, *then*
*converges strongly to a fixedpoint*
*of**T*.

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

**Corollary 2.3**

*Let*

*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*:

*where*
*denotesthe adjoint of**A*,
*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.

- (i)
, ;

- (ii)
, , .

Therefore
is anasymptotically nonexpansive mapping from *C* into itself with
.

*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
. Then*A* 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]

*Let*

*C*

*be a nonempty closed convex subset of a Hilbertspace*

*H*,

*and let*

*be a bifunctionsatisfying*(A1)-(A4).

*Let*

*and*.

*Then there exists*

*suchthat*

**Lemma 3.2**[21]

*Then*

- (1)
*is single*-*valued*; - (2)
- (3)
;

- (4)
*is nonempty*,*closed and convex*.

**Theorem 3.3**

*Let*

*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*:

*where*
*denotesthe adjoint of**A*,
,
*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. □

*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 into

*R*. The split equilibrium problem (

*SEP*) is to

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
,
, then*F* is a bi-function from
into*R* 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
into*R* with
.

Clearly, when , we have . So .

**Corollary 3.6**

*Let*

*and*

*be twoHilbert spaces*,

*bea bounded linear operator*,

*be a bifunctionsatisfying*

*and*

*be a bifunctionsatisfying*.

*Taking*,

*for arbitrarily chosen*,

*the sequence*

*is defined asfollows*:

*where*
*denotesthe adjoint of**A*,
,
*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
.

*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:

Hence from Theorem 2.1 we have the following theorem.

**Theorem 3.8**

*Let*

*H*, , ,

*C*

*and*

*Q*

*be the same as above*.

*Let*

*and*,

*and let the sequence*

*be defined asfollows*:

*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

## References

- 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 - 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 - 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 - Censor Y, Seqal A:
**The split common fixed point problem for directed operators.***J. Convex Anal.*2009,**16:**587–600.MathSciNetMATHGoogle Scholar - 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 - 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 - 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 - 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 - 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 - Moudafi A:
**The split common fixed point problem for demi-contractive mappings.***Inverse Probl.*2010.,**26:**Article ID 055007Google Scholar - 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 - 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 - 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 - 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 - 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 - 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 - Quan J, Chang SS, Zhang X:
**Multiple-set split feasibility problems for**k**-strictly pseudononspreadingmapping in Hilbert spaces.***Abstr. Appl. Anal.*2013.,**2013:**Article ID 342545 10.1155/2013/342545Google Scholar - 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 - 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 - 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 - Combettes PL, Hirstoaga SA:
**Equilibrium programming in Hilbert spaces.***J. Nonlinear Convex Anal.*2005,**6:**117–136.MathSciNetMATHGoogle Scholar - He Z:
**The split equilibrium problem and its convergence algorithms.***J. Inequal. Appl.*2012.,**2012:**Article ID 162Google Scholar

## Copyright

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.