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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 817943, 7 pages
http://dx.doi.org/10.1155/2013/817943
Research Article

The Split Feasibility Problems for Countable Families of Asymptotically Strict Pseudocontractions

College of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan 650221, China

Received 15 May 2013; Accepted 22 August 2013

Academic Editor: Somyot Plubtieng

Copyright © 2013 Wei-Qi Deng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

An up-to-date algorithm for solving the split feasibility problem for countable families of asymptotically strict pseudocontractions is introduced in the framework of Hilbert spaces. Our results greatly improve and extend those of other authors whose related research studies are restricted to the situation of at most finitely many such mappings.

1. Introduction

The split feasibility problem (SFP) in finite dimensional spaces was first introduced by Censor and Elfving [1] for modeling inverse problems which arise from phase retrievals and in medical image reconstruction [2]. Recently, it has been found that the SFP can also be used in various disciplines such as image restoration, computer tomograph, and radiation therapy treatment planning [35].

The split feasibility problem in an infinite dimensional Hilbert space can be found in [2, 4, 68].

Let and be two real Hilbert spaces with inner product and the corresponding norm . Let and be nonempty closed convex subsets of and , respectively. The purpose of this paper is to introduce and study the following multiple-set split feasibility problem for an infinite family of asymptotically strict pseudocontractions (MSSFP) in the framework of infinite-dimensional Hilbert spaces. Find such that where is a bounded linear operator.

In the sequel, we use to denote the set of solutions of the problem (MSSFP), that is,

2. Preliminaries

We first recall some definitions, notations, and conclusions which will be needed in proving our main results.

Let be a Banach space. A mapping is said to be demiclosed at origin, if for any sequence with and , then , where denotes that converges weakly to .

A Banach space is said to satisfy Opial’s condition, if for any sequence in , implies that It is well known that every Hilbert space satisfies Opial's condition.

Definition 1. Let be a real Hilbert space, be a mapping from into itself and the fixed point set of be nonempty.(1)is called a -asymptotically strict pseudocontraction if there exists a constant and a sequence with such that Especially, if for each in (4) and there exists a such that then is called a -strict pseudocontraction.(2) is said to be uniformly -Lipschitzian if there exists a constant such that (3) is said to be semicompact if for any bounded sequence with , there exists a subsequence of such that converges strongly to a point .

Remark 2. If we put in (4), then the mapping is asymptotically nonexpansive.
If we put in (5), then the mapping is nonexpansive.
Each -asymptotically strict pseudocontraction and each -strict pseudocontraction both are demiclosed at origin [9].

In 2011, Moudafi [10] proposed the following iterative algorithm for solving split common fixed problem of quasinonexpansive mappings: for arbitrarily chosen , and proved that converges weakly to a split common fixed point , where and are two quasinonexpansive mappings, is a bounded linear operator, and denotes the adjoint of .

Motivated and inspired by the studies of Moudafi [10, 11] and Chang et al. [12], in this paper, we introduce an algorithm for solving the split feasibility problems for countable families of asymptotically strict pseudocontractions and prove some strong and weak convergence theorems for such mappings in Hilbert spaces. The results extend those of the authors [12] whose related research studies are restricted to the situation of at most finite families of such mappings.

By using the well-known inequality in Hilbert spaces, we can easily show the following proposition, whose proof is omitted.

Proposition 3 (see [12]). Let be a -asymptotically strict pseudocontraction. If , then for each and , the following inequalities hold and they are equivalent:

Lemma 4 (see [13]). Let , , and be the sequences of nonnegative real numbers satisfying If and , then the exists.

Lemma 5 (see [14]). Let be a nonempty closed convex subset of a real Hilbert space and a nonexpansive mapping from into itself. If has a fixed point, then is demiclosed at zero, where is the identity mapping of .

Lemma 6 (see [15]). The unique solutions to the positive integer equation are where denotes the maximal integer that is not larger than .

3. Main Results

In the sequel, we assume that the following conditions are satisfied:(a) and are two real Hilbert spaces, is a bounded linear operator, and denotes the adjoint of ;(b) is a sequence of uniformly -Lipschitzian and -asymptotically strict pseudocontractions and is a sequence of uniformly -Lipschitzian and -asymptotically strict pseudocontractions satisfying the following conditions: (1)   , ; (2) and ; (3)for each , , and

The multiple-set split feasibility problem for infinite families of nonlinear mappings and is to find a point whose set of solutions is denoted by .

Lemma 7. Let and be the same as those mentioned above. Let be the following sequence generated by an arbitrarily chosen where , with and being the solutions to the positive integer equation: ; that is, for each , there exist unique and such that is a sequence in , and is a constant satisfying the following condition: (4) for all and , where is a positive constant. If , for any , then(I) and exist and have the same values;(II) for each , there exists a corresponding subsequence of such that where .

Proof. (I) Taking , that is, and , and using (15) and (9), we have where Further, letting , , , in (10) and noting , we have Substituting (22) into (21) and simplifying it, we have Substituting (20) and (23) into (19) and simplifying it, we have Again, substituting (24) into (18) and simplifying it, we have
By condition (4), we have where Note that . Hence, from Lemma 4, we know that the following limit exists: We now prove that for each , the limit exists. In fact, from (25) and (28), it follows that This, combined with condition (4), implies that Therefore, it follows from (19), (28), and (32) that exists.
(II) We firstly prove that and . As a matter of fact, it follows from (15) that In view of (31) and (32), we have that Similarly, it follows from (15), (32), and (34) that Next, for each , we consider the corresponding subsequence of . For example, by Lemma 6 and the definition of , we have and . Note that , that is, , and whenever . Set . Since are uniformly -Lipschitzian and for , we have, for each and , Thus, it follows from (31) and (35) that, for each , Similarly, we have, for each , This completes the proof.

Theorem 8. Let and be the same as those in Lemma 7. Suppose that is a sequence defined by (15). If and there exist mappings and and nondecreasing functions with and for all such that and for all , then converges strongly to some member of .

Proof. By Lemma 7, there exists a subsequence of such that Since for all , by taking as on both sides in the inequality above, we have which implies by the definition of the function .
Now we will show that is a Cauchy sequence. By Lemma 7, there exists a constant such that for any and all . And for any , there exists a positive integer such that for all and . Then, for any and and , we have Taking the infimum in the above inequalities for all yields that which implies that is a Cauchy sequence. Therefore, there exists a such that as since is complete. Firstly, we show that . shows that , which implies that since . Secondly, we show that . Since converges to and for all , then . This implies that because of the closedness of , and so . It finally follows from the existence of that as . This completes the proof.

Example 9. Let with the standard norm and . Let be two sequences of mappings defined by It is easily shown that is uniformly -Lipschitzian and a sequence of -asymptotically strict pseudocontractions. We now prove that the sequence defined by (15) converges strongly to some member of . Let for all with and . If , we then have where with . Define a nondecreasing function by . Since , we then have Similarly, we also can define a nondecreasing function with such that for some , which implies that, by Lemma 7 and Theorem 8, as .

Theorem 10. Let and be the same as those in Lemma 7. Let and be two sequences of nonexpansive mappings. Let be the following sequence generated by an arbitrarily chosen where is a sequence in for some ; ; with satisfying the positive integer equation: . Then converges weakly to an .

Proof. It is clear that both and are asymptotically strict pseudocontractions. Then, by the proof of Lemma 7, we have In addition, we also have which implies that, by induction, for any nonnegative integer , For each , since it follows from (49) and (52) that Now, for each , we claim that As a matter of fact, setting where , , we obtain that Then, since as , it follows from (52) and (54) that (55) holds obviously. Similarly, we have, for each , Next, since is bounded, there exists a subsequence such that (some point in ). From (55) we have for each . By Lemma 5, each is demiclosed at zero, so we know that . Moreover, it follows from (48) and (50) that Since is a linear bounded operator, it yields that . In view of (58) we have Again since each is demiclosed at zero, we know that . This implies that .
Note that each Hilbert space possesses Opial property, which guarantees that the weakly subsequential limit of is unique. Consequently, converges weakly to the point . Since , we know that converges weakly to . The proof is completed.

Remark 11. Note that, from Remark 2(3), the class of -asymptotically strict pseudocontractions is demiclosed at zero. Then, together with nonexpansiveness replaced by Lipschitz continuity, the two sequences of nonexpansive mappings and in Theorem 10 can be extended to -asymptotically strict pseudocontractions as in Lemma 7.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author is very grateful to the referees for their useful suggestions, by which the contents of this paper has been improved. This work is supported by the General Project of Scientific Research Foundation of Yunnan University of Finance and Economics (YC2013A02).

References

  1. Y. Censor and T. Elfving, “A multiprojection algorithm using Bregman projections in a product space,” Numerical Algorithms, vol. 8, no. 2–4, pp. 221–239, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. C. Byrne, “Iterative oblique projection onto convex sets and the split feasibility problem,” Inverse Problems, vol. 18, no. 2, pp. 441–453, 2002. View at Publisher · View at Google Scholar · View at MathSciNet
  3. Y. Censor, T. Bortfeld, B. Martin, and A. Trofimov, “A unified approach for inversion problems in intensity-modulated radiation therapy,” Physics in Medicine and Biology, vol. 51, no. 10, pp. 2353–2365, 2006. View at Publisher · View at Google Scholar
  4. Y. Censor, T. Elfving, N. Kopf, and T. Bortfeld, “The multiple-sets split feasibility problem and its applications for inverse problems,” Inverse Problems, vol. 21, no. 6, pp. 2071–2084, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. Y. Censor, A. Motova, and A. Segal, “Perturbed projections and subgradient projections for the multiple-sets split feasibility problem,” Journal of Mathematical Analysis and Applications, vol. 327, no. 2, pp. 1244–1256, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. H. K. Xu, “A variable Krasnosel'skii–Mann algorithm and the multiple-set split feasibility problem,” Inverse Problems, vol. 22, no. 6, pp. 2021–2034, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  7. Q. Yang, “The relaxed CQ algorithm solving the split feasibility problem,” Inverse Problems, vol. 20, no. 4, pp. 1261–1266, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. J. Zhao and Q. Yang, “Several solution methods for the split feasibility problem,” Inverse Problems, vol. 21, no. 5, pp. 1791–1799, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. T. H. Kim and H. K. Xu, “Convergence of the modified Mann's iteration method for asymptotically strict pseudo-contractions,” Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 9, pp. 2828–2836, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. A. Moudafi, “A note on the split common fixed-point problem for quasi-nonexpansive operators,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 12, pp. 4083–4087, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. A. Moudafi, “The split common fixed-point problem for demicontractive mappings,” Inverse Problems, vol. 26, no. 5, Article ID 055007, 6 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. S. S. Chang, Y. J. Cho, J. K. Kim, W. B. Zhang, and L. Yang, “Multiple-set split feasibility problems for asymptotically strict pseudocontractions,” Abstract and Applied Analysis, vol. 2012, Article ID 491760, 12 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. K. Aoyama, Y. Kimura, W. Takahashi, and M. Toyoda, “Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space,” Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 8, pp. 2350–2360, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, vol. 28 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 1990. View at Publisher · View at Google Scholar · View at MathSciNet
  15. W. Q. Deng and P. Bai, “An implicit iteration process for common fixed points of two infinite families of asymptotically nonexpansive mappings in Banach spaces,” Journal of Applied Mathematics, vol. 2013, Article ID 602582, 6 pages, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet