Weak- and strong-convergence theorems of solutions to split feasibility problem for nonspreading type mapping in Hilbert spaces
© Chang et al.; licensee Springer. 2014
Received: 1 October 2013
Accepted: 17 December 2013
Published: 9 January 2014
The purpose of this article is to study the weak- and strong-convergence theorems of solutions to split a feasibility problem for a family of nonspreading-type mapping in Hilbert spaces. The main result presented in this paper improves and extends some recent results of Censor et al., Byrne, Yang, Moudafi, Xu, Censor and Segal, Masad and Reich, and others. As an application, we solve the hierarchical variational inequality problem by using the main theorem.
MSC:47J05, 47H09, 49J25.
Throughout this paper, we assume that H is a real Hilbert space, D is a nonempty and closed convex subset of H. In the sequel, we denote by ‘’ and ‘’ the strong and weak convergence of , respectively. Denote by the set of all positive integers and by the set of fixed points of a mapping .
- (1)is said to be nonexpansive if
- (2)T is said to be quasi-nonexpansive if is nonempty and(1.1)
- (3)T is said to be nonspreading if(1.2)
- (4)T is said to be k-strictly pseudo-nonspreading , if there exists a constant such that(1.4)
if T is nonspreading and , then T is quasi-nonexpansive;
if T is nonspreading, then it is k-strictly pseudo-nonspreading with . But the converse is not true from the following example. Thus, we know that the class of k-strictly pseudo-nonspreading mappings is more general than the class of nonspreading mappings.
Example 1.3 
Then T is a k-strictly pseudo-nonspreading mapping, but it is not nonspreading.
In 2010, Kurokawa and Takahashi  obtained a weak mean ergodic theorem of Baillon’s type  for nonspreading mappings in Hilbert spaces. They further proved a strong-convergence theorem somewhat related to Halpern’s type  for this class of mappings using the idea of mean convergence in Hilbert spaces.
In 2011, Osilike and Isiogugu  first introduced the concept of k-strictly pseudo-nonspreading mappings and proved a weak mean convergence theorem of Baillon’s type similar to the ones obtained in . Furthermore, using the idea of mean convergence, a strong-convergence theorem similar to the one obtained in  is proved which extends and improves the main theorems of  and an affirmative answer given to an open problem posed by Kurokawa and Takahashi  for the case where the mapping T is averaged.
On the other hand, the split feasibility problem (SFP) in finitely dimensional spaces was first introduced by Censor and Elfving  for modeling inverse problems which arise from phase retrievals and in medical image reconstruction . Recently, it has been found that the (SFP) can also be used in various disciplines such as image restoration, computer tomography and radiation therapy treatment planning [7–9].
where , is a constant.
For this purpose, we first recall some definitions, notations and conclusions which will be needed in proving our main results.
is said to be demiclosed at 0, if, for any sequence with , , then .
T is said to be semicompact, if, for any bounded sequence , , then there exists a subsequence such that converges strongly to some point .
Lemma 2.2 
If , then is closed and convex;
is demiclosed at zero.
- (1)For all and for all ,(2.1)
- (2)For all ,
Lemma 2.4 
Lemma 2.5 
If and , then the limit exists.
is demiclosed at zero;
is a quasi-nonexpansive mapping.
Proof Since , the conclusions (1), (2) are obvious.
This completes the proof of Lemma 2.6. □
Lemma 2.7 
for every , exists;
each weak-cluster point of the sequence is in W.
Then there exists such that weakly converges to .
3 Weak- and strong-convergence theorems
and are two real Hilbert spaces, is a bounded linear operator and is the adjoint of A;
is an infinite family of -strictly pseudo-nonspreading mappings with ;
is a finite family of -strictly pseudo-nonspreading mappings with ;
Now we are in a position to give the following main theorem.
, for each ;
for each , ;
both and converge weakly to some point ;
in addition, if there exists some positive integer m such that is semicompact, then both and converge strongly to .
Proof First we prove the conclusion (I).
Hence is also bounded.
Hence conclusion (3.10) is proved.
Step 4. Now we show that every weak-cluster point of the sequence is in Γ.
These show that .
Step 5. Summing up the above arguments, we have proved that: (i) for each , the limits and exist (see equation (3.2)); (ii) every weak-cluster point of the sequence (or ) is in Γ. Taking and (or ) in Lemma 2.7, therefore all conditions in Lemma 2.7 are satisfied. By using Lemma 2.7, , and . This completes the proof of the conclusion (I).
Next we prove the conclusion (II).
i.e., and both converge strongly to the point . This completes the proof of Theorem 3.1. □
for the mappings, we extend the mappings from nonexpansive mappings, or demi-contractive mappings, to the more general family of k-strictly pseudo-nonspreading mappings;
for the algorithms, we propose some new hybrid iterative algorithms which are different from the ones given in [5–7, 9, 17, 18, 21, 22]. Under suitable conditions, some weak- and strong-convergence results for the algorithms are proved.
If we put in Theorem 3.1, we immediately get the following.
, for each ;
- (b)for each , . Let
the sequence converges weakly to some point ;
in addition, if there exists some positive integer m such that is semicompact, then the sequence converges strongly to .
In this section we utilize the results presented in Section 3 to study the hierarchical variational inequality problem.
Hence from Theorem 3.1 we have the following theorem.
, for each ;
for each , ;
If , then converges weakly to a solution of the hierarchical variational inequality problem (4.1). In addition, if one of the mappings is semicompact, then both and converge strongly to a solution of the hierarchical variational inequality problem (4.1).
Proof In fact, by the assumption that T is a nonspreading mapping, hence by Remark 1.2, T is a ρ-strictly pseudo-nonspreading with . Taking and in Theorem 3.1, all conditions in Theorem 3.1 are satisfied. The conclusions of Theorem 4.1 can immediately be obtained from Theorem 3.1. □
Remark 4.2 If (the identity mapping), then we can get the results of Corollary 3.3.
This work was supported by the Basic Science Research Program through the National Research Foundation (NRF) Grant funded by Ministry of Education of the republic of Korea (2013R1A1A2054617).
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