Algorithmic and analytical approach to the split common fixed points problem
© Zhu et al.; licensee Springer. 2014
Received: 14 March 2014
Accepted: 25 July 2014
Published: 18 August 2014
The split problem, especially the split common fixed point problem, has been studied by many authors. In this paper, we study the split common fixed point problem for the pseudo-contractive mappings and the quasi-nonexpansive mappings. We suggest and analyze an iterative algorithm for solving this split common fixed point problem. A weak convergence theorem is given.
MSC:49J53, 49M37, 65K10, 90C25.
where with L being the largest eigenvalue of the matrix , I is the unit matrix or operator and and denote the orthogonal projections onto C and Q, respectively. In the case of nonlinear constraint sets, orthogonal projections may demand a great amount of work of solving a nonlinear optimization problem to minimize the distance between the point and the constraint set. However, it can easily be estimated by linear approximation using the current constraint violation and the sub-gradient at the current point. This was done by Yang, in his recent paper , where he proposed a relaxed version of the CQ-algorithm in which orthogonal projections are replaced by sub-gradient projections, which are easily executed when the sets C and Q are given as lower level sets of convex functions; see also . There are a large number of references on the CQ method for the split feasibility problem in the literature; see, for instance, [10–25].
It is our main purpose in this paper to develop algorithms for the split common fixed point for the pseudo-contractive and quasi-nonexpansive mappings. Weak convergence theorem is given. Our results improve and develop previously discussed feasibility problems and related algorithms.
Let H be a real Hilbert space with inner product and norm , respectively. Let C be a nonempty closed convex subset of H.
for all .
We will use to denote the set of fixed points of S, that is, .
for all .
Interest in pseudo-contractive mappings stems mainly from their firm connection with the class of nonlinear accretive operators. It is a classical result, see Deimling , that if S is an accretive operator, then the solutions of the equations correspond to the equilibrium points of some evolution systems.
for all .
Remark 2.4 We call T nonexpansive if and T is contractive if .
Remark 2.6 It is obvious that if T is nonexpansive with , then T is quasi-nonexpansive.
Usually, the convergence of fixed point algorithms requires some additional smoothness properties of the mapping T such as demi-closedness.
Definition 2.7 A mapping T is said to be demi-closed if, for any sequence which weakly converges to , and if the sequence strongly converges to z, then .
for all and .
Lemma 2.8 ()
is a closed convex subset of C,
is demi-closed at zero.
Lemma 2.9 ()
for every , exists,
any weak-cluster point of the sequence belongs in Ω.
Then there exists such that weakly converges to .
denote the weak ω-limit set of ;
stands for the weak convergence of to x;
stands for the strong convergence of to x.
3 Main results
where , with λ being the largest eigenvalue of the matrix .
where , are relaxation parameters.
Inspired by their works, we introduce the following algorithm.
for all , where γ and ν are two constants, , , and are three sequences in .
In the sequel, we assume the parameters satisfy the following restrictions.
(R1): and , where λ is the largest eigenvalue of the matrix ;
(R3): for all , where L is the Lipschitz constant of U.
Remark 3.2 Without loss of generality, we may assume that the Lipschitz constant . It is obvious that for all . Since , we have for all .
Theorem 3.3 Let and be two real Hilbert spaces. Let be a bounded linear operator. Let be a pseudo-contractive mapping with Lipschitzian constant L and be a quasi-nonexpansive mapping with nonempty and . Assume that is demi-closed at 0 and . Then the sequence generated by algorithm (3.3) weakly converges to a split common fixed point .
for all .
Hence, . Therefore, . Since there is no more than one weak-cluster point, the weak convergence of the whole sequence follows by applying Lemma 2.9 with . This completes the proof. □
Hence, T is a continuous quasi-nonexpansive mapping but not nonexpansive.
for all .
Hence, U is a Lipschitzian pseudo-contractive mapping but it is not nonexpansive.
The authors are grateful to the five reviewers for their valuable comments and suggestions. Li-Jun Zhu was supported in part by NNSF of China (61362033) and NZ13087. Yeong-Cheng Liou was supported in part by NSC 101-2628-E-230-001-MY3 and NSC 101-2622-E-230-005-CC3.
- Bauschke HH, Borwein JM: On projection algorithms for solving convex feasibility problems. SIAM Rev. 1996, 38: 367–426. 10.1137/S0036144593251710View ArticleMathSciNetGoogle Scholar
- Combettes PL: The convex feasibility problem in image recovery. 95. In Advances in Imaging and Electron Physics. Edited by: Hawkes P. Academic Press, New York; 1996:155–270.Google Scholar
- Stark H (Ed): Image Recovery Theory and Applications. Academic Press, Orlando; 1987.Google Scholar
- Censor Y, Bortfeld T, Martin B, Trofimov A: A unified approach for inversion problems in intensity modulated radiation therapy. Phys. Med. Biol. 2006, 51: 2353–2365. 10.1088/0031-9155/51/10/001View ArticleGoogle Scholar
- Censor Y, Elfving T: A multiprojection algorithm using Bregman projections in a product space. Numer. Algorithms 1994, 8: 221–239. 10.1007/BF02142692View ArticleMathSciNetGoogle Scholar
- Byrne C: Bregman-Legendre multidistance projection algorithms for convex feasibility and optimization. In Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications. Edited by: Butnariu D, Censor Y, Reich S. Elsevier, Amsterdam; 2001:87–100.View ArticleGoogle Scholar
- Byrne C: Iterative oblique projection onto convex subsets and the split feasibility problem. Inverse Probl. 2002, 18: 441–453. 10.1088/0266-5611/18/2/310View ArticleMathSciNetGoogle Scholar
- Yang Q: The relaxed CQ algorithm solving the split feasibility problem. Inverse Probl. 2004, 20: 1261–1266. 10.1088/0266-5611/20/4/014View ArticleGoogle Scholar
- Zhao J, Yang Q: Several solution methods for the split feasibility problem. Inverse Probl. 2005, 21: 1791–1799. 10.1088/0266-5611/21/5/017View ArticleGoogle Scholar
- Ceng LC, Ansari QH, Yao JC: An extragradient method for split feasibility and fixed point problems. Comput. Math. Appl. 2012, 64: 633–642. 10.1016/j.camwa.2011.12.074View ArticleMathSciNetGoogle Scholar
- Censor Y, Motova A, Segal A: Perturbed projections and subgradient projections for the multiple-sets split feasibility problem. J. Math. Anal. Appl. 2007, 327: 1244–1256. 10.1016/j.jmaa.2006.05.010View ArticleMathSciNetGoogle Scholar
- Censor Y, Segal A: The split common fixed point problem for directed operators. J. Convex Anal. 2009, 16: 587–600.MathSciNetGoogle Scholar
- Dang Y, Gao Y: The strong convergence of a KM-CQ-like algorithm for a split feasibility problem. Inverse Probl. 2011., 27: Article ID 015007 10.1088/0266-5611/27/1/015007Google Scholar
- Deepho J, Kumam P: A modified Halpern’s iterative scheme for solving split feasibility problems. Abstr. Appl. Anal. 2012., 2012: Article ID 876069 10.1155/2012/876069Google Scholar
- López G, Martín-Márquez V, Xu HK: Iterative algorithms for the multiple-sets split feasibility problem. In Biomedical Mathematics: Promising Directions in Imaging, Therapy Planning and Inverse Problems. Edited by: Censor Y, Jiang M, Wang G. Medical Physics Publishing, Madison; 2009:243–279.Google Scholar
- Moudafi A: A note on the split common fixed-point problem for quasi-nonexpansive operators. Nonlinear Anal. 2011, 74: 4083–4087. 10.1016/j.na.2011.03.041View ArticleMathSciNetGoogle Scholar
- Wang F, Xu HK: Approximating curve and strong convergence of the CQ algorithm for the split feasibility problem. J. Inequal. Appl. 2010., 2010: Article ID 102085 10.1155/2010/102085Google Scholar
- Wang F, Xu HK: Cyclic algorithms for split feasibility problems in Hilbert spaces. Nonlinear Anal. 2011, 74: 4105–4111. 10.1016/j.na.2011.03.044View ArticleMathSciNetGoogle Scholar
- Xu HK: A variable Krasnosel’skii-Mann algorithm and the multiple-set split feasibility problem. Inverse Probl. 2006, 22: 2021–2034. 10.1088/0266-5611/22/6/007View ArticleGoogle Scholar
- Xu HK: Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces. Inverse Probl. 2010., 26: Article ID 105018 10.1088/0266-5611/26/10/105018Google Scholar
- Yao Y, Kim TH, Chebbi S, Xu HK: A modified extragradient method for the split feasibility and fixed point problems. J. Nonlinear Convex Anal. 2012, 13: 383–396.MathSciNetGoogle Scholar
- Yao Y, Wu J, Liou YC: Regularized methods for the split feasibility problem. Abstr. Appl. Anal. 2012., 2012: Article ID 140679 10.1155/2012/140679Google Scholar
- Yu X, Shahzad N, Yao Y: Implicit and explicit algorithms for solving the split feasibility problem. Optim. Lett. 2012, 6: 1447–1462. 10.1007/s11590-011-0340-0View ArticleMathSciNetGoogle Scholar
- Zhao J, Yang Q: Self-adaptive projection methods for the multiple-sets split feasibility problem. Inverse Probl. 2011., 27: Article ID 035009 10.1088/0266-5611/27/3/035009Google Scholar
- Zhang W, Han D, Li Z: A self-adaptive projection method for solving the multiple-sets split feasibility problem. Inverse Probl. 2009., 25: Article ID 115001 10.1088/0266-5611/25/11/115001Google Scholar
- Deimling K: Zeros of accretive operators. Manuscr. Math. 1974, 13: 365–374. 10.1007/BF01171148View ArticleMathSciNetGoogle Scholar
- Zhou H: Strong convergence of an explicit iterative algorithm for continuous pseudo-contractions in Banach spaces. Nonlinear Anal. 2009, 70: 4039–4046. 10.1016/j.na.2008.08.012View ArticleMathSciNetGoogle Scholar
- Bauschke HH, Combettes PL: A weak-to-strong convergence principle for Fejér-monotone methods in Hilbert spaces. Math. Oper. Res. 2001, 26: 248–264. 10.1287/moor.18.104.22.16858View ArticleMathSciNetGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.