A simultaneous iterative method for split equality problems of two finitefamilies of strictly pseudononspreading mappings without prior knowledge ofoperator norms
© Che and Li; licensee Springer. 2015
Received: 1 July 2014
Accepted: 8 October 2014
Published: 5 January 2015
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 theT-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.
which allows asymmetric and partial relations between the variables x andy. 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 . 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].
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. and Zhao et al. 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.
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 . 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.
To establish our results, we need the following technical lemmas.
Lemma 2.1 ()
The following definition will be useful for our results.
In 2009, Kangtunyakarn and Suantai  introduced T-mapping generated by and as follows.
Using the above definition, we have the following important lemma.
Lemma 2.2LetCbe a nonempty convex subset of real Banach space. Let be a finite family of -strictly pseudononspreading mappings ofCinto itself with , and let be real numbers such that for every . IfTis theT-mapping generated by and , then andTis a quasi-nonexpansive mapping.
3 Main results
Now, we are in a position to prove our convergence results in this section.
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 .
The following conclusions can be obtained from Theorem 3.1 immediately.
We now pay attention to applying our simultaneous iterative algorithms to some convexand nonlinear analysis notions; see, for example, .
4.1 Split feasibility problem
4.2 Variational problems via resolvent mappings
and the algorithm is applied to the following form.
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).
- 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
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