- Research
- Open Access

# A simultaneous iterative method for split equality problems of two finitefamilies of strictly pseudononspreading mappings without prior knowledge ofoperator norms

- Haitao Che
^{7}Email author and - Meixia Li
^{7}Email author

**2015**:1

https://doi.org/10.1186/1687-1812-2015-1

© Che and Li; licensee Springer. 2015

**Received:**1 July 2014**Accepted:**8 October 2014**Published:**5 January 2015

## Abstract

In this article, we first introduce the concept of *T*-mapping of a finitefamily of strictly pseudononspreading mapping
, and we show that the fixed point set of the*T*-mapping is the set of common fixed points of
and *T* is a quasi-nonexpansive mapping.Based on the concept of a *T*-mapping, we propose a simultaneousiterative algorithm to solve the split equality problem with a way of selectingthe stepsizes which does not need any prior information about the operatornorms. The sequences generated by the algorithm weakly converge to a solution ofthe split equality problem of two finite families of strictly pseudononspreadingmappings. Furthermore, we apply our iterative algorithms to some convex andnonlinear problems.

**MSC:** 47H09, 47H05, 47H06, 47J25.

## Keywords

- fixed point
- feasibility problem
- simultaneous iterative method
- weak convergence
- strictly pseudononspreading mapping

## 1 Introduction

which allows asymmetric and partial relations between the variables *x* and*y*. The interest is to cover many situations, for instance indecomposition methods for PDEs, applications in game theory and inintensity-modulated radiation therapy (IMRT). In decision sciences, this allows oneto consider agents who interplay only via some components of their decisionvariables, for further details, the interested reader is referred to [13]. In IMRT, this amounts to envisage a weak coupling between the vector ofdoses absorbed in all voxels and that of the radiation intensity, for furtherdetails, the interested reader is referred to [13, 14].

*i.e.*, (1.1) has a solution), if solves (1.1), then it solves the following fixedpoint equation system:

where , and are the spectral radii of and , respectively. The weak convergence of the sequence to a solution of (1.1) under some conditions wasproved.

for firmly quasi-nonexpansive operators *U* and *T*, where
,
and
are the spectral radiuses of
and
, respectively.

Observe that in the algorithms (1.3) and (1.5) mentioned above, the determination ofthe stepsize
depends on the operator (matrix) norms
and
(or the largest eigenvalues of
and
). To implement the alternating algorithm (1.3) andthe simultaneous algorithm (1.5), one has first to compute (or, at least, estimate)operator norms of *A* and *B*, which is in general not easy inpractice.

To overcome this difficulty, Lopez *et al.*[16] and Zhao *et al.*[17] presented useful method for choosing the stepsizes which do not needprior knowledge of the operator norms for solving the split feasibility problems andmultiple-set split feasibility problems, respectively.

The advantage of our choice (1.6) of the stepsizes lies in the fact that no priorinformation about the operator norms of *A* and *B* is required, andstill convergence is guaranteed.

*i.e.*,

**Algorithm 1.1**Let , be arbitrary and be real number sequences in . Assume that the

*k*th iterate , has been constructed and , then we calculate th iterate via the formula

where the stepsize is chosen by (1.6). If , then is a solution of the problem (1.4) and the iterativeprocess stops. Otherwise, we set and go on to (1.7) to evaluate the next iterate .

**Remark 1.1** Notice that in (1.6) the choice of the stepsize
is independent of the norms
and
.

## 2 Preliminaries

*H*be a real Hilbert space with innerproduct and induced norm , and denote by

*C*be a nonempty closed convexsubset of

*H*. Let be a mapping. A point is said to be a fixed point of

*T*provided . we use to denote the fixed point set. We write to indicate that the sequence converges weakly to

*x*, implies that converges strongly to

*x*. We use to stand for the weak

*ω*-limit set of . For any , there exists a unique nearest point in

*C*,denoted by , such that

Before proceeding, we need to introduce a few concepts.

**Remark 2.1** It is easy to see that
and
. Furthermore,
is well known to contain resolvents and projectionoperators, and
includes subgradient projection operators [18]. *T* is a nonspreading mapping if and only if *T* is a0-strictly pseudononspreading mapping.

The so-called demiclosedness principle plays an important role in our argument.

A mapping
is called demiclosed at the origin if for anysequence
which weakly converges to *x*, and if thesequence
strongly converges to 0, then
.

To establish our results, we need the following technical lemmas.

**Lemma 2.1** ([19])

The following definition will be useful for our results.

In 2009, Kangtunyakarn and Suantai [20] introduced *T*-mapping generated by
and
as follows.

**Definition 2.1**Let

*C*be a nonempty convex subset of real Banachspace. Let be a finite family of mappings of

*C*intoitself, and let be real numbers such that for every . We define a mapping as follows:

Such a mapping *T* is called the *T*-mapping generated by
and
.

Using the above definition, we have the following important lemma.

**Lemma 2.2***Let**C**be a nonempty convex subset of real Banach space*. *Let*
*be a finite family of*
-*strictly pseudononspreading mappings of**C**into itself with*
, *and let*
*be real numbers such that*
*for every*
. *If**T**is the**T*-*mapping generated by*
*and*
, *then*
*and**T**is a quasi*-*nonexpansive mapping*.

*Proof*It is easy to deduce that . Next, we claim that . Let and . Assume that , for , it follows from being a finite family of -strictly pseudononspreading mappings of

*C*into itself that

it yields . Applying the same argument, we can conclude that and , for .

It follows that
. Therefore,
, that is,
. Hence,
. From the definition of *T* and (2.7), we findthat *T* is a quasi-nonexpansive mapping. □

**Proposition 2.1***Let**C**be a closed convex subset of a real Hilbert space**H*. *If**T**is a quasi*-*nonexpansive mapping from**C**into itself*, *then*
*is closed and convex*.

## 3 Main results

Now, we are in a position to prove our convergence results in this section.

**Theorem 3.1**

*Let*, ,

*be real Hilbert spaces*.

*Given two bounded linear operators*, .

*Let*

*be a finite family of*-

*strictly pseudononspreading mappings of*

*C*

*into itself with*,

*and Let*

*be a finite family of*-

*strictly pseudononspreading mappings of*

*Q*

*into itself with*.

*Suppose that*

*U*

*is defined by*(2.5)

*which is generated by*

*and*,

*with*

*for every*,

*and suppose that*

*T*

*is defined by*(2.5)

*which is generated by*

*and*,

*with*

*for every*,

*respectively*.

*Assume that*

*and*

*are demiclosed at the origin*.

*If the solution set*Γ

*of*(1.4)

*is nonempty and for small enough*

*and*,

*then the sequence*
*generated by Algorithm* 1.1 *weakly converges to a solution*
*of* (1.4). *Moreover*,
,
, *and*
*as*
.

Inequalities (3.1) and (3.2) lead to and is bounded.

Now, we show that . Indeed, as is shown below, we break up the proof bydistinguishing two cases.

Conversely, suppose that there exists such that , for all , following the above process, we obtain theresults.

for all . We can divide the sequence into two sequences: one satisfies , which is denoted by and the other sequence satisfies , which is denoted by . Following the process of Case 1, we show that theresults hold for the subsequences with and . Thus, we obtain .

which yields is asymptotically regular. Similarly, and is asymptotically regular, too.

(3.15) and (3.17) mean that . In the same way as above, we can also show that as .

which means and . Hence, the sequence weakly converges to a solution of the problem (1.4),which completes the proof. □

The following conclusions can be obtained from Theorem 3.1 immediately.

**Theorem 3.2**

*Let*, ,

*be real Hilbert spaces*.

*Given two bounded linear operators*, .

*Let*

*U*

*be a*

*ρ*-

*strictly pseudononspreading mapping of*

*C*

*into itself and Let*

*T*

*be a*

*τ*-

*strictly pseudononspreading mapping of*

*Q*

*into itself*.

*Assume that*

*and*

*are demiclosed at the origin*.

*If the solution set*Γ

*of*(1.4)

*is nonempty and for small enough*

*and*,

*then the sequence*
*generated by Algorithm* 1.1 *weakly converges to a solution*
*of* (1.4). *Moreover*,
,
, *and*
*as*
.

**Theorem 3.3**

*Let*, ,

*be real Hilbert spaces*.

*Given two bounded linear operators*, .

*Let*

*U*

*be a nonspreading mapping of*

*C*

*into itself and let*

*T*

*be a nonspreading mapping of*

*Q*

*into itself*.

*Assume that*

*and*

*are demiclosed at the origin*.

*If the solution set*Γ

*of*(1.4)

*is nonempty and for small enough*

*and*,

*then the sequence*
*generated by Algorithm* 1.1 *weakly converges to a solution*
*of* (1.4). *Moreover*,
,
, *and*
*as*
.

## 4 Applications

We now pay attention to applying our simultaneous iterative algorithms to some convexand nonlinear analysis notions; see, for example, [21].

### 4.1 Split feasibility problem

*C*and

*Q*be nonempty closed convex subset of real Hilbertspaces and , respectively. The split feasibility problem(SFP) is to find a point

where is a bounded linear operator. The SFP was firstintroduced by Censor and Elfving [22] for modeling inverse problems which arise from phase retrievals andin medical image reconstruction [23].

If , , then Algorithm 1.1 becomes:

where the stepsize is chosen by (1.6). If , then is a solution of the problem (4.1) and theiterative process stops. Otherwise, we set and go on to (4.1) to evaluate the next iterate .

Furthermore, if and , then we obtain the following simultaneousiterative algorithm for solving SFP (4.1).

where the stepsize is chosen by (1.6). If , then is a solution of the problem (4.1) and theiterative process stops. Otherwise, we set and go on to (4.3) to evaluate the next iterate .

### 4.2 Variational problems via resolvent mappings

*M*areexactly fixed points of its resolvent mapping. If and , where is another maximal monotone operator, the problemunder consideration is nothing but

and the algorithm is applied to the following form.

where the stepsize is chosen by (1.6). If , then is a solution of the problem (4.4) and theiterative process stops. Otherwise, we set and go on to (4.5) to evaluate the next iterate .

## Declarations

### Acknowledgements

We thank Prof. Yiju Wang for his careful reading of the manuscript and thank theanonymous referees and the editor for their constructive comments andsuggestions, which greatly improved this article. This project is supported bythe Natural Science Foundation of China (Grant Nos. 11401438, 11171180,11171193, 11126233), and Project of Shandong Province Higher Educational Scienceand Technology Program (Grant No. J14LI52).

## Authors’ Affiliations

## References

- Aleyner A, Reich S:
**Block-iterative algorithms for solving convex feasibility problems in Hilbertand in Banach spaces.***J. Math. Anal. Appl.*2008,**343**(1)**:**427–435. 10.1016/j.jmaa.2008.01.087MathSciNetView ArticleMATHGoogle Scholar - Bauschke HH, Borwein JM:
**On projection algorithms for solving convex feasibility problems.***SIAM Rev.*1996,**38**(3)**:**367–426. 10.1137/S0036144593251710MathSciNetView ArticleMATHGoogle Scholar - Byrne C:
*A Unified Treatment of Some Iterative Algorithms in Signal Processing andImage Reconstruction*. Dekker, New York; 1984.Google Scholar - Masad E, Reich S:
**A note on the multiple-set split convex feasibility problem in Hilbertspace.***J. Nonlinear Convex Anal.*2007,**8:**367–371.MathSciNetMATHGoogle Scholar - Yao Y, Chen R, Marino G, Liou YC:
**Applications of fixed point and optimization methods to the multiple-setssplit feasibility problem.***J. Appl. Math.*2012.,**2012:**Article ID 927530Google Scholar - Qu B, Xiu N:
**A note on the CQ algorithm for the split feasibility problem.***Inverse Probl.*2005,**21:**1655–1665. 10.1088/0266-5611/21/5/009MathSciNetView ArticleMATHGoogle Scholar - Qu B, Xiu N:
**A new halfspace-relaxation projection method for the split feasibilityproblem.***Linear Algebra Appl.*2008,**428**(5)**:**1218–1229.MathSciNetView ArticleMATHGoogle Scholar - Xu HK:
**Iterative methods for the split feasibility problem in infinite-dimensionalHilbert spaces.***Inverse Probl.*2010,**26**(10)**:**5018–5034.View ArticleGoogle Scholar - Censor Y, Elfving T, Kopf N, Bortfled T:
**The multi-sets split feasibility problem and it applications to inverseproblems.***Inverse Probl.*2005,**21:**2071–2084. 10.1088/0266-5611/21/6/017View 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 - Zhao J, Yang Q:
**Several solution methods for the split feasibility problem.***Inverse Probl.*2005,**21:**1791–1799. 10.1088/0266-5611/21/5/017MathSciNetView ArticleMATHGoogle Scholar - Moudafi A:
**Alternating CQ-algorithm for convex feasibility and split fixed-pointproblems.***J. Nonlinear Convex Anal.*2014,**15:**809–818.MathSciNetMATHGoogle Scholar - Attouch H, Bolte J, Redont P, Soubeyran A:
**Alternating proximal algorithms for weakly coupled minimization problems.Applications to dynamical games and PDEs.***J. Convex Anal.*2008,**15:**485–506.MathSciNetMATHGoogle Scholar - Censor Y, Bortfeld T, Martin B, Trofimov A:
**A unified approach for inversion problems in intensity-modulated radiationtherapy.***Phys. Med. Biol.*2006,**51:**2353–2365. 10.1088/0031-9155/51/10/001View ArticleGoogle Scholar - Moudafi A, Al-Shemas E:
**Simultaneous iterative methods for split equality problems andapplications.***Trans. Math. Program. Appl.*2013,**1:**1–11.Google Scholar - Lopez G, Martin-Marquez V, Wang F, Xu H:
**Solving the split feasibility problem without prior knowledge of matrixnorms.***Inverse Probl.*2012.,**27:**Article ID 085004Google Scholar - Zhao J, Zhang J, Yang Q:
**A simple projection method for solving the multiple-sets split feasibilityproblem.***Inverse Probl. Sci. Eng.*2013,**21:**537–546. 10.1080/17415977.2012.712521MathSciNetView ArticleGoogle Scholar - Maruster S, Popirlan C:
**On the Mann-type iteration and convex feasibility problem.***J. Comput. Appl. Math.*2008,**24:**390–396.MathSciNetView ArticleMATHGoogle Scholar - Chang SS:
**Some problems and results in the study of nonlinear analysis.***Nonlinear Anal.*1997,**30**(7)**:**4197–4208. 10.1016/S0362-546X(97)00388-XMathSciNetView ArticleMATHGoogle Scholar - Kangtunyakarn A, Suantai S:
**A new mapping for finding common solutions of equilibrium problems and fixedpoint problems of finite family of nonexpansive mappings.***Nonlinear Anal.*2009,**71:**4448–4460. 10.1016/j.na.2009.03.003MathSciNetView ArticleMATHGoogle Scholar - Rockafellar RT, Wets R
**Grundlehren der Mathematischen Wissenschafte 317.**In*Variational Analysis*. Springer, Berlin; 1998.View ArticleGoogle Scholar - Censor Y, Elfving T:
**A multiprojection algorithm using Bregman projections in a product space.***Numer. Algorithms*1994,**8:**221–239. 10.1007/BF02142692MathSciNetView ArticleMATHGoogle Scholar - Byrne C:
**Iterative oblique projection onto convex subsets and the split feasibilityproblem.***Inverse Probl.*2002,**18:**441–453. 10.1088/0266-5611/18/2/310View ArticleMATHGoogle 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 theoriginal work is properly credited.