Multiple-set split feasibility problems for κ-asymptotically strictly pseudo-nonspreading mappings in Hilbert spaces
© Quan and Chang; licensee Springer. 2014
Received: 19 October 2013
Accepted: 30 January 2014
Published: 13 February 2014
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.
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  for modeling inverse problems. The () can be used in various disciplines such as medical image reconstruction , 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.
In 1967, Browder and Petryshyn  introduced the concept of κ-strictly pseudo-nonspreading mapping.
Definition 1.1 
Clearly, every nonspreading mapping is κ-strictly pseudo-nonspreading.
The class of asymptotically strict pseudo-contractions was introduced by Qihou  in 1996. Kim and Xu , Inchan and Nammanee , Zhou  Cho , and Ge  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  introduced a class of nonspreading type mappings which is more general than the class studied in  in Hilbert spaces and proved some weak and strong convergence theorems in real Hilbert spaces. Recently, Chang et al.  studied the multiple-set split feasibility problem for an asymptotically strict pseudo-contraction in the framework of infinite-dimensional Hilbert spaces.
Definition 1.2 
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.
Example 1.4 Now, we give an example of κ-asymptotically strict pseudo-contractive mapping.
where is a sequence in such that .
, and .
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.
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.
Therefore inequality (1.3) holds.
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 , Osilike and Isiogugu , Chang et al. , and many others.
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 .
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 .
- (i)For all and for all ,
- (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.
Since , we have for each . Hence . This shows that is closed.
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 .
Since , it follows from (2.2) and (2.3) that . So and . This completes the proof. □
3 Main results
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 .
This shows that the limit exists.
(II) Now we prove that the limit exists.
(III) Now, we prove that , .
(V) Finally, we prove that , , and it is a solution of problem () (1.1).
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 .
This completes the proof of Theorem 3.1. □
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 .
that is, and both converge strongly to the point . This completes the proof of Theorem 3.2. □
In this section we shall utilize the results presented in Section 3 to study the hierarchical variational inequality problem.
Hence from Theorem 3.1 we have the following theorem.
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. □
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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