## Advances on Multivalued Operators and Related Fixed Point Problems

View this Special IssueResearch Article | Open Access

# Multiple-Set Split Feasibility Problems for -Strictly Pseudononspreading Mapping in Hilbert Spaces

**Academic Editor:**E. Karapinar

#### Abstract

The purpose of this paper is to prove some weak and strong convergence theorems for solving the multiple-set split feasibility problems for -strictly pseudononspreading mapping in infinite-dimensional Hilbert spaces by using the proposed iterative method. The main results presented in this paper extend and improve the corresponding results of Xu et al. (2006), of Osilike et al. (2011), and of many other authors.

#### 1. Introduction and Preliminaries

Censor and Elfving first introduced the split feasibility problem (SFP) [1] in finite dimensional spaces for modeling inverse problems. The SFP can be used in various disciplines such as medical image reconstruction [2], image restoration, computer tomograph, and radiation therapy treatment planning [3–5]. The multiple-set split feasibility problem (MSSFP) was studied in [4–6].

In the sequel, we always assume that , are two real Hilbert spaces and denote by “” and “” the strong and weak convergence, respectively.

The so-called *multiple-set split feasibility problem* (MSSFP) is to find such that , where is a bounded linear operator, and , are the families of mappings, and , , and , where and denote the sets of fixed points of and , respectively. In the sequel, we use to denote the set of solutions of the MSSFP; that is,

Recently, Kohsaka and Takahashi [7, 8] introduced an important class of mappings which is called the class of * nonspreading mappings*.

*Definition 1 (see [7, 8]). *Let to be a nonempty closed convex subset of a Hilbert space . A mapping is said to be nonspreading, if

In [9], Iemoto and Takahashi proved that this definition is equivalent to the following.

*Definition 2 (see [9]). *Let be a nonempty closed convex subset of a Hilbert space . A mapping is said to be nonspreading, if

Browder and Petryshyn [10] proposed the following -*strictly pseudononspreading mapping*.

*Definition 3 (see [10]). *Let be a real Hilbert space. We say that a mapping is -strictly pseudononspreading if there exists , such that
Clearly every nonspreading mapping is -strictly pseudononspreading.

Osilike and Isiogugu [11] introduced a class of nonspreading type mappings which is more general than that of the mappings studied in [12] in Hilbert spaces and proved some weak and strong convergence theorems in real Hilbert spaces. Recently, the split feasibility problem also was considered in the work by Deepho and Kumam [13, 14] and Sunthrayuth et al. [15], and some weak and strong convergence theorems are proved in real Hilbert spaces.

The purpose of this paper is to study the multiple-set split feasibility problem (MSSFP) for -strictly pseudononspreading mappings in the framework of infinite dimensional Hilbert spaces.

In the sequel, we recall some definitions, notations, and conclusions which will be needed in proving our main results.

*Definition 4 (see [3]). *Let be a real Banach space. A mapping with domain and range in is said to be demiclosed at origin if for any sequence in which converges weakly to a point and converges strongly to 0, then .

*Definition 5. *A Banach space is said to have * Opial property* if, for any sequence with , we have
for all with .

*Remark 6. *It is well known that each Hilbert space possesses Opial property.

*Definition 7. *A mapping is said to be semicompact, if, for any bounded sequence with , there exists a subsequence such that converges strongly to some point .

Lemma 8 (see [11]). *Let H be a real Hilbert space; then the following results hold.*(i)*For all , and for all ,
*(ii)*.*(iii)*If is a sequence in which converges weakly to , then
*

*Definition 9. *Let be a nonempty closed convex subset of a real Hilbert space . The * metric projection* is a mapping such that, for each , is the unique point in such that , . It is known that, for each ,

Lemma 10 (see [11]). *Let be a nonempty closed convex subset of a real Hilbert space , and let be a -strictly pseudononspreading mapping. If , then it is closed and convex.*

Lemma 11 (see [11]). *Let be a nonempty closed convex subset of a real Hilbert space , and let be a -strictly pseudononspreading mapping. Then is demiclosed at 0.*

#### 2. Main Results

Theorem 12. *Let , , , , , , be the same as aforementioned. For each , let be a -strictly pseudononspreading mapping and let be a -strictly pseudononspreading mapping. Let be the sequence generated by
**
where with being the spectral of the operator and , and is a sequence in with . If (where is the set of solutions of the MSSFP defined by (1)), then the sequence converges weakly to a point .*

*Proof. *The proof is divided into four steps.

(I) We first prove that exists for any .

Since , and . It follows from (9) that

Because is -strictly pseudononspreading, for each , we have

Taking , we have

This implies that

Thus it yields that

Substituting (14) into (10) and simplifying, we have

On the other hand,

Since is -strictly pseudononspreading and noting that , we have

This leads to

By (18), we have

It follows from (19) that

By using (15), (16), (19), and (20), we have

This shows that exists.

(II) We now prove that exists.

In fact, by (21), we have

This implies that

By virtue of (16), (23), and (24), it follows that exists and .

(III) Now, we prove that and .

In fact, it follows from (9) that

This together with (23) and (24) leads to .

Similarly, it follows from (9), (23), and (25) that

(IV) Finally, we prove that and , which is a solution of the MSSFP.

In fact, since is bounded, there exists a subsequence such that . Hence, for any positive integer , there exists a subsequence with such that . Again, by (24) we know that , as ; therefore, we have that

Since is demiclosed at zero, it follows that . By the arbitrariness of , we have

Moreover, from (9) and (24), we have . Since is a bounded linear operator, it follows that . For any positive integer , there exists a subsequence with such that and . Since is demiclosed at zero, we have . By the arbitrariness of , it follows that . This together with shows that ; that is, is a solution to the MSSFP.

Now, we prove that and .

Suppose on the contrary that there exists another subsequence such that with . Consequently, by virtue of the existence of and the Opial property of Hilbert space, we have

This is a contradiction. Therefore . By (9) and (24), we have

Therefore, the conclusion follows.

This completes the proof of Theorem 12.

Theorem 13. *Let , , , , , , be the same as in Theorem 12. For each , let be a -strictly pseudononspreading mapping, and be a -strictly pseudononspreading mapping. Let be the sequence generated by
**
where with being the spectral of the operator and , and is a sequence in with . If and if there exists a positive integer such that is semicompact, then the sequence converges strongly to a point .*

*Proof. *Without loss of generality, we can assume that is semicompact. It follows from (27) that

Therefore, there exists a subsequence of (for the sake of convenience, we still denote it by ), such that . Since , , and so . By virtue of the existence of , we know that

That is, and both converge strongly to the point . This completes the proof of Theorem 13.

#### Conflict of Interests

The authors declare that they have no conflict of interests.

#### Authors’ Contribution

All the authors contributed equally and significantly in writing this paper. All the authors read and approved the final paper.

#### Acknowledgments

This work was supported by the Natural Science Foundation of Yibin University (no. 2011B07) and the Scientific Research Fund Project of Sichuan Provincial Education Department (no. 12ZB345) and the Fund of Science and Technology Department of Guizhou Province ([2013] 2223).

#### References

- 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 - 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 Site | Google Scholar - 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 - 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 | Zentralblatt MATH | MathSciNet - H.-K. Xu, “A variable Krasnoselskii-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 - F. Kohsaka and W. Takahashi, “Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces,”
*Archiv der Mathematik*, vol. 91, no. 2, pp. 166–177, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - F. Kohsaka and W. Takahashi, “Existence and approximation of fixed points of firmly nonexpansive-type mappings in Banach spaces,”
*SIAM Journal on Optimization*, vol. 19, no. 2, pp. 824–835, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. Iemoto and W. Takahashi, “Approximating common fixed points of nonexpansive mappings and nonspreading mappings in a Hilbert space,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 71, no. 12, pp. e2082–e2089, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - F. E. Browder and W. V. Petryshyn, “Construction of fixed points of nonlinear mappings in Hilbert space,”
*Journal of Mathematical Analysis and Applications*, vol. 20, pp. 197–228, 1967. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - M. O. Osilike and F. O. Isiogugu, “Weak and strong convergence theorems for nonspreading-type mappings in Hilbert spaces,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 74, no. 5, pp. 1814–1822, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Y. Kurokawa and W. Takahashi, “Weak and strong convergence theorems for nonspreading mappings in Hilbert spaces,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 73, no. 6, pp. 1562–1568, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - J. Deepho and P. Kumam, “A modified Halpern's iterative scheme for solving split feasibility problems,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 876069, 8 pages, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - J. Deepho and P. Kumam, “Split feasibility and fixed-point problems for asymptotically quasi-nonexpansive mappings,”
*Journal of Inequalities and Applications*, vol. 2013, p. 322, 2013. View at: Publisher Site | Google Scholar - P. Sunthrayuth, Y. J. Cho, and P. Kumam, “General iterative algorithms approach to variational inequalities and minimum-norm fixed point for minimization and split,”
*OPSEARCH*, 2013. View at: Publisher Site | Google Scholar

#### Copyright

Copyright © 2013 Jing Quan et al. 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.