Research Article | Open Access

Mohammad Eslamian, Abdul Latif, "General Split Feasibility Problems in Hilbert Spaces", *Abstract and Applied Analysis*, vol. 2013, Article ID 805104, 6 pages, 2013. https://doi.org/10.1155/2013/805104

# General Split Feasibility Problems in Hilbert Spaces

**Academic Editor:**Qamrul Hasan Ansari

#### Abstract

Introducing a general split feasibility problem in the setting of infinite-dimensional Hilbert spaces, we prove that the sequence generated by the purposed new algorithm converges strongly to a solution of the general split feasibility problem. Our results extend and improve some recent known results.

#### 1. Introduction

Let and be infinite-dimensional real Hilbert spaces, and let be a bounded linear operator. Let and be the families of nonempty closed convex subsets of and , respectively.

(a) The *convex feasibility problem* (CFP) is formulated as the problem of finding a point with the property:

(b) The *split feasibility problem* (SEP) is formulated as the problem of finding a point with the property:
where and are nonempty closed convex subsets of and , respectively.

(c) The *multiple-set split feasibility problem* (MSSFP) is formulated as the problem of finding a point with the property:
Note that (MSSFP) reduces to (SEP) if we take .

There is a considerable investigation on CFP in view of its applications in various disciplines such as image restoration, computer tomograph, and radiation therapy treatment planning [1]. The split feasibility problem SFP in the setting of finite-dimensional Hilbert spaces was first introduced by Censor and Elfving [2] for modelling inverse problems which arise from phase retrievals and in medical image reconstruction [3]. Since then, a lot of work has been done on finding a solution of SFP and MSSFP; see, for example, [2–25]. Recently, it is found that the SFP can also be applied to study the intensity-modulated radiation therapy; see, for example, [6, 16] and the references therein. Very recently, Xu [8] considered the SFP in the setting of infinite-dimensional Hilbert spaces.

The original algorithm given in [2] involves the computation of the inverse provided it exists. In [8], Xu studied some algorithm and its convergence. In particular, he applied Mann’s algorithm to the SFP and purposed an algorithm which is proved to be weakly convergent to a solution of the SFP. He also established the strong convergence result, which shows that the minimum-norm solution can be obtained. In [7], Wang and Xu purposed the following cyclic algorithm to solve MSSFP: where mod, (mod function take values in ), and . They show that the sequence convergence weakly to a solution of MSSFP provided the solution exists. To study strong convergence to a solution of MSSFP, first we introduce a general form of the split feasibility problem for infinite families as follows.

(d) *General split feasibility problem* (GSFP) is to find a point such that
We denote by the solution set of GSFP.

In this paper, using viscosity iterative method defined by Moudafi [21], we propose an algorithm for finding the solutions of the general split feasibility problem in a Hilbert space. We establish the strong convergence of the proposed algorithm to a solution of GSFP.

#### 2. Preliminaries

Throughout the paper, we denote by a real Hilbert space with inner product and norm . Let be a sequence in and . Weak convergence of to is denoted by , and strong convergence by . Let be a closed and a convex subset of . For every point , there exists a unique nearest point in , denoted by . This point satisfies The operator is called the metric projection or the nearest point mapping of onto . The metric projection is characterized by the fact that and Recall that a mapping is called nonexpansive if It is well known that is a nonexpansive mapping. It is also known that satisfies Opial’s condition, that is, for any sequence with , the inequality holds for every with .

Lemma 1. *Let be a Hilbert space. Then, for all *

Lemma 2 (see [22]). *Let be a Hilbert space, and let be a sequence in . Then, for any given sequence with and for any positive integer with ,
*

Lemma 3 (see [23]). *Assume that is a sequence of nonnegative real numbers such that
**
where , , and satisfy the following conditions: *(i)*, ,*(ii)* or ,*(iii)* for all with .**Then, .*

Lemma 4 (see [24]). *Let be a sequence of real numbers such that there exists a subsequence of such that for all . Then, there exists a nondecreasing sequence such that , and the following properties are satisfied by all (sufficiently large) numbers :
**
In fact
*

Lemma 5 (demiclosedness principle [25]). *Let be a nonempty closed and convex subset of a real Hilbert space . Let be a nonexpansive mapping such that . Then, is demiclosed on , that is, if , and , then . *

#### 3. Main Result

In the following result, we propose an algorithm and prove that the sequence generated by the proposed method converges strongly to a solution of the GSFP.

Theorem 6. *Let and be real Hilbert spaces, and let be a bounded linear operator. Let and be the families of nonempty closed convex subsets of and , respectively. Assume that GSFP (5) has a nonempty solution set . Suppose that is a self -contraction mapping of , and let be a sequence generated by as
**
where . If the sequences , , , and satisfy the following conditions: *(i)* and ,*(ii)*for each , , *(iii)*for each , and ,**then, the sequence converges strongly to , where . *

*Proof. *First, we show that is bounded. In fact, let . Since , the operators are nonexpansive, and hence we have
which implies that is bounded, and we also obtain that is bounded. Next, we show that for each ,
By using Lemma 2, for every and , we have that
Hence, for each , we have

Next, we show that there exists a unique such that . We observe that for each , solves the GSFP (5) if and only if solves the fixed point equation
that is, the solution sets of fixed point equation (20) and GSFP (5) are the same (see for details [8]). Note that if , then the operators are nonexpansive. Since the fixed point set of nonexpansive operators is closed and convex, the projection onto the solution set is well defined whenever . We observe that is a contraction of into itself. Indeed, since is nonexpansive,
Hence, there exists a unique element such that .

In order to prove that as , we consider two possible cases.*Case *1*.* Assume that is a monotone sequence. In other words, for large enough, is either nondecreasing or nonincreasing. Since is bounded we have is convergent. Since and is bounded, from (19) we get that
By assuming that , we obtain
Now, we show that
To show this inequality, we choose a subsequence of such that
Since is bounded, there exists a subsequence of which converges weakly to . Without loss of generality, we can assume that and for each . From (23), we have
Notice that for each , is nonexpansive. Thus, from Lemma 5, we have . Therefore, it follows that
Finally, we show that . Applying Lemma 1, we have that
This implies that
where
and . It is easy to see that , and . Hence, by Lemma 3, the sequence converges strongly to .*Case *2*.* Assume that is not a monotone sequence. Then, we can define an integer sequence for all (for some large enough) by
Clearly, is a nondecreasing sequence such that as and for all ,
From (19), we obtain that
Following an argument similar to that in Case 1, we have
And by similar argument, we have
where , and . Hence, by Lemma 3, we obtain and . Now, from Lemma 4, we have
Therefore, converges strongly to .

For finite collections we have the following consequence of Theorem 6.

Theorem 7. *Let and be real Hilbert spaces, and let be a bounded linear operator. Let be a family of nonempty closed convex subsets in , and let be a family of nonempty closed convex subsets in . Assume that MSSFP has a nonempty solution set . Let be an arbitrary element in , and let be a sequence generated by and
**
where . If the sequences , , , and satisfy the following conditions: *(i)* and ,*(ii)*for all , , *(iii)*for all , and
**then the sequence converges strongly to , where . *

#### Acknowledgment

This paper is funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The authors therefore, acknowledge with thanks the DSR for technical and financial support.

#### References

- P. L. Combettes, “The convex feasibility problem in image recovery,” in
*Advances in Imaging and Electron Physics*, P. Hawkes, Ed., vol. 95, pp. 155–270, Academic Press, New York, NY, USA, 1996. View at: Google Scholar - 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 Site | Google Scholar | Zentralblatt MATH | MathSciNet - 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 Site | Google Scholar | MathSciNet - L.-C. Ceng, Q. H. Ansari, and J.-C. Yao, “Relaxed extragradient methods for finding minimum-norm solutions of the split feasibility problem,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 75, no. 4, pp. 2116–2125, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - L.-C. Ceng, Q. H. Ansari, and J.-C. Yao, “An extragradient method for solving split feasibility and fixed point problems,”
*Computers & Mathematics with Applications*, vol. 64, no. 4, pp. 633–642, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - 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 Site | Google Scholar | Zentralblatt MATH | MathSciNet - F. Wang and H.-K. Xu, “Cyclic algorithms for split feasibility problems in Hilbert spaces,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 74, no. 12, pp. 4105–4111, 2011. View at: Publisher Site | Google Scholar | MathSciNet - H.-K. Xu, “Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces,”
*Inverse Problems*, vol. 26, no. 10, Article ID 105018, p. 17, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - 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 Site | Google Scholar | MathSciNet - 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 Site | Google Scholar | MathSciNet - Q. Yang, “The relaxed CQ algorithm solving the split feasibility problem,”
*Inverse Problems*, vol. 20, no. 4, pp. 1261–1266, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - H. H. Bauschke and J. M. Borwein, “On projection algorithms for solving convex feasibility problems,”
*SIAM Review*, vol. 38, no. 3, pp. 367–426, 1996. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Y. Alber and D. Butnariu, “Convergence of Bregman projection methods for solving consistent convex feasibility problems in reflexive Banach spaces,”
*Journal of Optimization Theory and Applications*, vol. 92, no. 1, pp. 33–61, 1997. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - B. Qu and N. Xiu, “A note on the $CQ$ algorithm for the split feasibility problem,”
*Inverse Problems*, vol. 21, no. 5, pp. 1655–1665, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - 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 Site | Google Scholar | Zentralblatt MATH | MathSciNet - Y. Censor and A. Segal, “The split common fixed point problem for directed operators,”
*Journal of Convex Analysis*, vol. 16, no. 2, pp. 587–600, 2009. View at: Google Scholar | Zentralblatt MATH | MathSciNet - Y. Yao, W. Jigang, and Y.-C. Liou, “Regularized methods for the split feasibility problem,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 140679, 13 pages, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Y. Dang and Y. Gao, “The strong convergence of a KM-CQ-like algorithm for a split feasibility problem,”
*Inverse Problems*, vol. 27, article 015007, p. 9, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - F. Wang and H.-K. Xu, “Approximating curve and strong convergence of the $CQ$ algorithm for the split feasibility problem,”
*Journal of Inequalities and Applications*, vol. 2010, Article ID 102085, 13 pages, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - L.-C. Ceng, Q. H. Ansari, and J.-C. Yao, “Mann type iterative methods for finding a common solution of split feasibility and fixed point problems,”
*Positivity*, vol. 16, no. 3, pp. 471–495, 2012. View at: Publisher Site | Google Scholar | MathSciNet - A. Moudafi, “Viscosity approximation methods for fixed-points problems,”
*Journal of Mathematical Analysis and Applications*, vol. 241, no. 1, pp. 46–55, 2000. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S.-S. Chang, J. K. Kim, and X. R. Wang, “Modified block iterative algorithm for solving convex feasibility problems in Banach spaces,”
*Journal of Inequalities and Applications*, vol. 2010, Article ID 869684, 14 pages, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - H.-K. Xu, “Iterative algorithms for nonlinear operators,”
*Journal of the London Mathematical Society*, vol. 66, no. 1, pp. 240–256, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - P.-E. Maingé, “Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization,”
*Set-Valued Analysis*, vol. 16, no. 7-8, pp. 899–912, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - K. Goebel and W. A. Kirk,
*Topics in Metric Fixed Point Theory*, vol. 28, Cambridge University Press, Cambridge, UK, 1990. View at: Publisher Site | MathSciNet

#### Copyright

Copyright © 2013 Mohammad Eslamian and Abdul Latif. 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.