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Multiple-set split feasibility problems for κ-asymptotically strictly pseudo-nonspreading mappings in Hilbert spaces
Journal of Inequalities and Applications volume 2014, Article number: 69 (2014)
Abstract
Some weak and strong convergence theorems for solving multiple-set split feasibility problems for κ-asymptotically strictly pseudo-nonspreading mappings in infinite-dimensional Hilbert spaces are proved. The results presented in the paper extend and improve the corresponding results of Xu (Inverse Probl. 22(6):2021-2034, 2006), Osilike and Isiogugu (Nonlinear Anal. 74:1814-1822, 2011), Chang et al. (Abstr. Appl. Anal. 2012:491760, 2012), and others.
MSC:47H05, 47H09, 49M05.
1 Introduction
Throughout this article, we always assume that , are real Hilbert spaces; ‘→’ and ‘⇀’ denote strong and weak convergence, respectively.
The split feasibility problem () in finite dimensional spaces was first introduced by Censor and Elfving [1] for modeling inverse problems. The () can be used in various disciplines such as medical image reconstruction [2], image restoration, computer tomography, and radiation therapy treatment planning [3–5]. The multiple-set split feasibility problem () was studied in [4–7].
Let be a bounded linear operator, and , , be two finite families of mappings, and , where and are the sets of fixed points of and , respectively.
The so-called multiple set split feasibility problem is
In the sequel, we use Γ to denote the set of solutions of the problem () (1.1), that is,
Let H be a real Hilbert space and K be a nonempty closed convex subset of H. Following Kohsaka and Takahashi [8–11], a mapping is said to be nonspreading if
It is to see that the above inequality is equivalent to
In 1967, Browder and Petryshyn [12] introduced the concept of κ-strictly pseudo-nonspreading mapping.
Definition 1.1 [12]
Let H be a real Hilbert space. A mapping is said to be κ-strictly pseudo-nonspreading if there exists such that
Clearly, every nonspreading mapping is κ-strictly pseudo-nonspreading.
The class of asymptotically strict pseudo-contractions was introduced by Qihou [13] in 1996. Kim and Xu [14], Inchan and Nammanee [15], Zhou [16] Cho [17], and Ge [18] proved that the class of asymptotically strict pseudo-contractions is demiclosed at the origin and also obtained some weak convergence theorems for the class of mappings. In 2012, Osilike and Isiogugu [19] introduced a class of nonspreading type mappings which is more general than the class studied in [11] in Hilbert spaces and proved some weak and strong convergence theorems in real Hilbert spaces. Recently, Chang et al. [7] studied the multiple-set split feasibility problem for an asymptotically strict pseudo-contraction in the framework of infinite-dimensional Hilbert spaces.
Definition 1.2 [7]
Let H be a real Hilbert space, we say that the mapping is a κ-asymptotically strict pseudo-contraction if there exists a constant and a sequence with () such that
holds for all .
In this article we introduce the following class of κ-asymptotically strictly pseudo-nonspreading mappings which is more general than that of κ-strictly pseudo-nonspreading mappings and κ-asymptotically strict pseudo-contractions.
Definition 1.3 Let H be a real Hilbert space. A mapping is said to be κ-asymptotically strictly pseudo-nonspreading if there exists a constant and a sequence with () such that
Example 1.4 Now, we give an example of κ-asymptotically strict pseudo-contractive mapping.
Let C be a unit ball in a real Hilbert , and let be a mapping defined by
where is a sequence in such that .
It is proved in Goebel and Kirk [20] that
-
(i)
, ;
-
(ii)
, and .
Define , , , then
Letting , then , , we have
This implies that T is a κ-asymptotically strict pseudo-contractive mapping.
Example 1.5 Now, we give an example of κ-asymptotically strictly pseudo-nonspreading mapping.
Let with the norm be defined by
and let be an orthogonal subspace of X (i.e., , we have ). It is obvious that C is a nonempty closed convex subset of X. For each , we define a mapping by
Next we prove that T is a κ-asymptotically strictly pseudo-nonspreading mapping.
In fact, for any , we have the following cases.
Case 1. If and , then we have , , and so then inequality (1.3) holds.
Case 2. If and , then we have that , . This implies that
Therefore inequality (1.3) holds.
Case 3. If and , then we have , . Hence we have
Thus inequality (1.3) still holds. Therefore the mapping defined by (1.5) is a κ-asymptotically strictly pseudo-nonspreading mapping.
The purpose of this article is under suitable conditions to prove some weak and strong convergence theorems for solving multiple-set split feasibility problem (1.1) for a κ-asymptotically strictly pseudo-nonspreading mapping in infinite-dimensional Hilbert spaces. The results presented in the paper extend and improve the corresponding results of Xu [6], Osilike and Isiogugu [19], Chang et al. [7], and many others.
2 Preliminaries
In the sequel, we first recall some definitions, notations, and conclusions which will be needed in proving our main results.
Let E be a real Banach space. A mapping T with domain and range in E is said to be demiclosed at origin if whenever is a sequence in converging weakly to a point and converging strongly to 0, then .
A Banach space E is said to have the property if, for any sequence with , we have
for all with .
It is well known that each Hilbert space possesses the Opial property.
A mapping is said to be semicompact if for any bounded sequence with , there exists a subsequence such that converges strongly to some point .
A mapping is said to be uniformly L-Lipschitzian if there exists a constant such that
Let K be a nonempty closed convex subset of a real Hilbert space H. The metric projection is a mapping such that for each , is the unique point in K such that , . It is known that for each ,
Lemma 2.1 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 H which converges weakly to , then
Lemma 2.2 Let K be a nonempty closed convex subset of a real Hilbert space H, and let be a continuous κ-asymptotically strictly pseudo-nonspreading mapping. If , then it is a closed and convex subset.
Proof Let be a sequence such that . Now we prove that . In fact, since T is κ-asymptotically strictly pseudo-nonspreading, for each , we have
Taking the limit as in the above inequality, we have
Since , we have for each . Hence . This shows that is closed.
Now we prove that is convex. In fact, let , and , we prove that . Since and , by using Lemma 2.1(i), we have
Taking on both sides of the above inequality, we have
Since , we have
and so , i.e., . This completes the proof. □
Lemma 2.3 Let K be a nonempty closed convex subset of a real Hilbert space H, and let be a continuous κ-asymptotically strictly pseudo-nonspreading mapping. Then is demiclosed at 0, that is, if and , then .
Proof Since is weak convergence, is bounded. For each , define by
From Lemma 2.1(iii), we have
Thus we have
In particular, for each ,
On the other hand, we have
Since and T is a κ-asymptotically strictly pseudo-nonspreading mapping, taking on both sides of the above equality, we get
By virtue of and (), we have
On the other hand, it follows from (2.1) that
Since , it follows from (2.2) and (2.3) that . So and . This completes the proof. □
3 Main results
Theorem 3.1 Let , , A, , , C, Q be the same as in multiple set split feasibility problem (1.1). For each , let be a uniformly -Lipschitzian and -asymptotically strictly pseudo-nonspreading mapping, be a uniformly -Lipschitzian and -asymptotically strictly pseudo-nonspreading mapping. Let be the sequence generated by
where γ is a constant and , λ is the spectral of the operator , and is a sequence in with . If , then the sequence converges weakly to a point .
Proof The proof is divided into five steps.
(I) We first prove the limit exists for any .
Since , we have and . It follows from (3.1) that
Because is a -asymptotically strictly pseudo-nonspreading mapping, for any , we have
Taking , we have
Therefore we have
Simplifying the above inequality, we have that
It follows from (3.2) and (3.3) that
On the other hand,
Since is a -asymptotically strictly pseudo-nonspreading mapping and noting , we have
Again since
hence from (3.6) and (3.7) we have that
By virtue of (3.8) we have
It follows from (3.9) that
Substituting (3.10) into (3.5) and then substituting the resulting inequality into (3.4), we have
This shows that the limit exists.
(II) Now we prove that the limit exists.
By (3.11) we have
This implies that
and
It follows from (3.5), (3.12), and (3.13) that the limit exists and
(III) Now, we prove that , .
In fact, it follows from (3.1) that
This together with (3.12) and (3.13) shows that
Similarly, it follows from (3.1), (3.12), and (3.15) that
(IV) We prove that, for each ,
In fact, it follows from (3.13) that
Since is uniformly -Lipschitzian continuous, it follows from (3.16) and (3.18) that
Similarly, we can prove that for each ,
Since is uniformly -Lipschitzian continuous, in the same way as above, we can also prove that
(V) Finally, we prove that , , and it is a solution of problem () (1.1).
In fact, since is bounded, there exists a subsequence such that . Hence, for any positive integer , there exists a subsequence with such that . Again from (3.17) we have that
Since is demiclosed at zero, it follows that . By the arbitrariness of , we have
Moreover, from (3.1) and (3.13) we have . Since A is a linear bounded 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 k, it follows that . This together with shows that , that is, is a solution to the problem () (1.1).
Next we prove that and .
In fact, assume that there exists another subsequence such that with . Consequently, by virtue of the existence of and the Opial property of a Hilbert space, we have
This is a contradiction. Therefore, . By (3.1) and (3.13), we have
This completes the proof of Theorem 3.1. □
Theorem 3.2 Let , , A, , , C, Q be the same as in Theorem 3.1. For each , let be a uniformly -Lipschitzian and -asymptotically strictly pseudo-nonspreading mapping, be a uniformly -Lipschitzian and -asymptotically strictly pseudo-nonspreading mapping. Let be the sequence generated by
where γ is a constant and , λ is the spectral of the operator , and is a sequence in with . If and if there exists a positive integer j 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 (3.17) that
Therefore, there exists a subsequence of , which (for the sake of convenience) we still denote by , such that . Since , , and so . By virtue of exists, we know that
that is, and both converge strongly to the point . This completes the proof of Theorem 3.2. □
4 Applications
In this section we shall utilize the results presented in Section 3 to study the hierarchical variational inequality problem.
Let H be a real Hilbert space, , , be uniformly -Lipschitzian and -asymptotically strictly pseudo-nonspreading mappings with . Let be a nonspreading mapping. The so-called hierarchical variational inequality problem for a finite family of mappings with respect to the mapping T is to find an such that
It is easy to see that (4.1) is equivalent to the following fixed point problem:
where is the metric projection from H onto ℱ. Letting and (the fixed point set of ) and (the identity mapping on H), problem (4.2) is equivalent to the following multi-set split feasibility problem:
Hence from Theorem 3.1 we have the following theorem.
Theorem 4.1 Let H, , T, C, Q be the same as above. Let , be the sequences defined by
where γ is a constant and , and is a sequence in with . If , then converges weakly to a solution of hierarchical variational inequality problem (4.1).
Proof In fact, by the assumption that T is a nonspreading mapping, T is κ-strictly pseudo-nonspreading with . Taking and in Theorem 3.1, by the same method as that given in Theorem 3.1, we can prove that converges weakly to a point , which is a solution of hierarchical variational inequality problem (4.1) immediately. □
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Acknowledgements
The authors would like to express their thanks to the referees and the editors for their helpful comments and advices. This work was supported by the National Research Foundation of Yibin University (No. 2011B07) and by the Scientific Research Fund Project of Sichuan Provincial Education Department (No. 12ZB345) and the National Natural Sciences Foundation of China (Grant No. 11361170).
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Quan, J., Chang, Ss. Multiple-set split feasibility problems for κ-asymptotically strictly pseudo-nonspreading mappings in Hilbert spaces. J Inequal Appl 2014, 69 (2014). https://doi.org/10.1186/1029-242X-2014-69
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DOI: https://doi.org/10.1186/1029-242X-2014-69