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Multiple-Set Split Feasibility Problems for -Strictly Pseudononspreading Mapping in Hilbert Spaces
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)  in finite dimensional spaces for modeling inverse problems. The SFP can be used in various disciplines such as medical image reconstruction , 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,
In , Iemoto and Takahashi proved that this definition is equivalent to the following.
Definition 2 (see ). Let be a nonempty closed convex subset of a Hilbert space . A mapping is said to be nonspreading, if
Browder and Petryshyn  proposed the following -strictly pseudononspreading mapping.
Definition 3 (see ). 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  introduced a class of nonspreading type mappings which is more general than that of the mappings studied in  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. , 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 ). 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 ). 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 ). 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 ). 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.
All the authors contributed equally and significantly in writing this paper. All the authors read and approved the final paper.
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 ( 2223).
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