- Open Access
Damped projection method for split common fixed point problems
© Cui et al.; licensee Springer 2013
- Received: 20 October 2012
- Accepted: 4 March 2013
- Published: 22 March 2013
The paper deals with the split common fixed-point problem (SCFP) introduced by Censor and Segal. Motivated by Eicke’s damped projection method, we propose a cyclic iterative scheme and prove its strong convergence to a solution of SCFP under some mild assumptions. An application of the proposed method to multiple-set split feasibility problems is also included.
- Initial Guess
- Strong Convergence
- Nonlinear Operator
- Common Fixed Point
- Real Sequence
where satisfies (i) , ; (ii) ; (iii) . It is clear that such an algorithm is an extension of (3). However, algorithm (4) fails to include the original one (3) because of condition (iii).
where is known as the step-size. They proved that if U and T in (6) are directed operators, then λ should be chosen in . Some further generations of this algorithm were studied by Moudafi  for demicontractive operators and by Wang-Xu  for finitely many directed operators.
We note that the existing algorithms for SCFP have only weak convergence in the framework of infinite-dimensional spaces (see [18, 19]). However, as pointed by Bauschke and Combettes , norm convergence of the algorithm is much more desirable than weak convergence in some applied sciences. It is therefore of interest to seek modifications of these algorithms so that strong convergence is guaranteed. Following the damped projection method, we propose in this paper a new iterative scheme and prove its strong convergence to a solution of SCFP. An application of our method to multiple-set split feasibility problems is also included. This enables us to cover some recent results on split feasibility problems.
Throughout this paper, I denotes the identity operator on ℋ, the set of fixed points of an operator T, ‘→’ strong convergence, and ‘⇀’ weak convergence. Given a positive integer p, denote by the mod function taking values in .
Definition 1 An operator is called nonexpansive if , ; firmly nonexpansive if , .
Clearly, firm nonexpansiveness implies nonexpansiveness. It is well known that nonexpansive operators are demiclosed at zero (cf. ).
Definition 3 Let be an operator with . Then T is called directed if , , ; ν-demicontractive with if , , .
Lemma 1 (Bauschke-Combettes )
Lemma 2 (Wang-Xu )
whenever is nonempty.
for all and .
Hence the proof is complete. □
We end this section by a useful lemma.
Lemma 4 (Xu )
- (i), ,
- (ii)or .
In this section, we consider the following problem.
where p is a positive integer and , are two classes of directed operators such that and are demiclosed at zero for every .
where , and , , are properly chosen real sequences.
- (ii), ,
If the solution set of problem (11) denoted by Ω is nonempty, then the sequence generated by (12) converges strongly to .
By induction, the sequence is bounded, and so is .
Adding up (14)-(16), we thus get inequality (13).
Finally, we prove . To see this, let be a subsequence such that it includes all elements in with the property: each of them is less than or equal to the term after it. Following an idea developed by Maingé , we consider two possible cases on such a sequence.
We therefore apply Lemma 4 to conclude .
which immediately implies . Consequently, follows from (17) and the proof is complete. □
where and are directed operators so that and are demiclosed at zero.
- (i), ,
converges strongly to , whenever such point exists.
In this section, we extend our result to SCFP for demicontractive operators recently considered by Moudafi .
where p is a positive integer and , are respectively -demicontractive and -demicontractive operator so that and are demiclosed at zero for every .
The following lemma states a relation between directed and demicontractive operators.
Lemma 5 Let and . If T is ν-demicontractive, then is directed.
Then the result follows from Lemma 1. □
where , and . By using the previous lemma, we can easily extend our result to demicontractive operators.
- (i), ,
If the solution set of problem (22) denoted by Ω is nonempty, then the sequence generated by (23) converges strongly to .
Remark 1 Theorem 3 also holds true if we relax hypothesis (ii) above as .
where , , and , , are properly chosen real sequences.
- (ii), ,
If the solution set of MSFP denoted by Ω is nonempty, then the sequence generated by (25) converges strongly to .
Proof We note that the metric projection is firmly nonexpansive, which implies is directed and is demiclosed at zero. Hence, by using Theorem 1, one can immediately get the desired result. □
Remark 2 Theorem 4 covers [, Theorem 3.1], and we relax the condition on as . Moreover, the choice of variable is more flexible than the fixed one. Also, we cover the result of  and remove one condition posed on : either or .
We would like to express our sincere thanks to the referees for their valuable suggestions. This work is supported by the National Natural Science Foundation of China, Tianyuan Foundation (11226227), the Basic Science and Technological Frontier Project of Henan (122300410268) and the Foundation of Henan Educational Committee (12A110016).
- Censor Y, Elfving T: A multiprojection algorithm using Bregman projections in a product space. Numer. Algorithms 1994, 8(2):221–239. 10.1007/BF02142692MathSciNetView ArticleGoogle Scholar
- Censor Y, Elfving T, Kopf N, Bortfeld T: The multiple-sets split feasibility problem and its applications for inverse problems. Inverse Probl. 2005, 21(6):2071–2084. 10.1088/0266-5611/21/6/017MathSciNetView ArticleGoogle Scholar
- Byrne C: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Probl. 2003, 20: 103.MathSciNetView ArticleGoogle 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(10):2353–2365. 10.1088/0031-9155/51/10/001View ArticleGoogle Scholar
- Byrne C: Iterative oblique projection onto convex sets and the split feasibility problem. Inverse Probl. 2002, 18(2):441–453. 10.1088/0266-5611/18/2/310MathSciNetView ArticleGoogle Scholar
- Lopez G, Martin V, Xu HK: Perturbation techniques for nonexpansive mappings with applications. Nonlinear Anal., Real World Appl. 2009, 10(4):2369–2383. 10.1016/j.nonrwa.2008.04.020MathSciNetView ArticleGoogle Scholar
- Qu B, Xiu N: A note on the CQ algorithm for the split feasibility problem. Inverse Probl. 2005, 21(5):1655–1665. 10.1088/0266-5611/21/5/009MathSciNetView ArticleGoogle Scholar
- Schöpfer F, Schuster T, Louis A: An iterative regularization method for the solution of the split feasibility problem in Banach spaces. Inverse Probl. 2008., 24(5): Article ID 055008Google Scholar
- Wang F, Xu H: Choices of variable steps of the CQ algorithm for the split feasibility problem. Fixed Point Theory 2011, 12: 489–496.MathSciNetGoogle Scholar
- Xu HK: A variable Krasnosel’skii-Mann algorithm and the multiple-set split feasibility problem. Inverse Probl. 2006, 22(6):2021–2034. 10.1088/0266-5611/22/6/007View ArticleGoogle Scholar
- Yang Q: The relaxed CQ algorithm solving the split feasibility problem. Inverse Probl. 2004, 20(4):1261–1266. 10.1088/0266-5611/20/4/014View ArticleGoogle Scholar
- Xu HK: Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces. Inverse Probl. 2010., 26(10): Article ID 105018Google Scholar
- Fenghui W, Hong-Kun X: Approximating curve and strong convergence of the CQ algorithm for the split feasibility problem. J. Inequal. Appl. 2010., 2010: Article ID 102085Google Scholar
- Eicke B: Iteration methods for convexly constrained ill-posed problems in Hilbert space. Numer. Funct. Anal. Optim. 1992, 13(5–6):413–429. 10.1080/01630569208816489MathSciNetView ArticleGoogle Scholar
- Engl HW, Hanke M, Neubauer A 375. In Regularization of Inverse Problems. Springer, Berlin; 1996.View ArticleGoogle Scholar
- Dang Y, Gao Y: The strong convergence of a KM-CQ-like algorithm for a split feasibility problem. Inverse Probl. 2010., 27: Article ID 015007Google Scholar
- Censor Y, Segal A: The split common fixed point problem for directed operators. J. Convex Anal. 2009, 16(2):587–600.MathSciNetGoogle Scholar
- Moudafi A: The split common fixed-point problem for demicontractive mappings. Inverse Probl. 2010., 26(5): Article ID 055007Google Scholar
- Wang F, Xu HK: Cyclic algorithms for split feasibility problems in Hilbert spaces. Nonlinear Anal. 2011, 74(12):4105–4111. 10.1016/j.na.2011.03.044MathSciNetView ArticleGoogle Scholar
- Bauschke HH, Combettes PL: A weak-to-strong convergence principle for Fejé-monotone methods in Hilbert spaces. Math. Oper. Res. 2001, 26(2):248–264. 10.1287/moor.126.96.36.19958MathSciNetView ArticleGoogle Scholar
- Goebel K, Kirk WA 28. In Topics in Metric Fixed Point Theory. Cambridge University Press, Cambridge; 1990.View ArticleGoogle Scholar
- Xu HK: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 2002, 66(1):240–256. 10.1112/S0024610702003332View ArticleGoogle Scholar
- Maingé PE: Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization. Set-Valued Anal. 2008, 16(7):899–912. 10.1007/s11228-008-0102-zMathSciNetView ArticleGoogle Scholar
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