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Weak- and strong-convergence theorems of solutions to split feasibility problem for nonspreading type mapping in Hilbert spaces
Fixed Point Theory and Applications volume 2014, Article number: 11 (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 .
Definition 1.1 Let be a mapping.
is said to be nonexpansive if
T is said to be quasi-nonexpansive if is nonempty and(1.1)
T is said to be nonspreading if(1.2)
It is easy to prove that equation (1.2) is equivalent to
T is said to be k-strictly pseudo-nonspreading , if there exists a constant such that(1.4)
Remark 1.2 It follows from Definition 1.1 that
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 
Let ℛ denote the set of real numbers with the usual norm. Let be a mapping defined by
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].
The purpose of this paper is to introduce the following multiple-set split feasibility problem (MSSFP) for an infinite family of k-strictly pseudo-nonspreading mappings and a finite family of ρ-strictly pseudo-nonspreading mappings in infinitely dimensional Hilbert spaces, i.e., to find such that
where , are two real Hilbert spaces, is a bounded linear operator, is an infinite family of -strictly pseudo-nonspreading mappings and is a finite family of -strictly pseudo-nonspreading mappings, and . Also we wish to study the weak and strong convergence of the following iterative sequence to a solution of problem (1.6):
where , is a constant.
In the sequel we denote Γ the set of solutions of (MSSFP) equation (1.6), i.e.,
For this purpose, we first recall some definitions, notations and conclusions which will be needed in proving our main results.
Definition 2.1 Let E be a real Banach space, and be a mapping.
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 
Let H be a real Hilbert space, D be a nonempty and closed convex subset of H, and be a k-strictly pseudo-nonspreading mapping.
If , then is closed and convex;
is demiclosed at zero.
Lemma 2.3 Let H be a real Hilbert space. Then the following statements hold:
For all and for all ,(2.1)
For all ,
Lemma 2.4 
Let E be a uniformly convex Banach space, be a closed ball with center 0 and radius . Then for any given sequence and any given number sequence with , , there exists a strictly increasing continuous and convex function with such that for any , ,
Lemma 2.5 
Let , and be sequences of nonnegative real numbers satisfying
If and , then the limit exists.
Lemma 2.6 Let D be a nonempty and closed convex subset of H and be a k-strictly pseudo-nonspreading mapping with . Let , . Then the following conclusions hold:
is demiclosed at zero;
is a quasi-nonexpansive mapping.
Proof Since , the conclusions (1), (2) are obvious.
Now we prove the conclusion (3). In fact, since T is a k-strictly pseudo-nonspreading mapping, it follows from Lemma 2.3 that
If , then . Hence from equation (2.4),
This completes the proof of Lemma 2.6. □
Lemma 2.7 
Let H be a Hilbert space and be a sequence in H such that there exists a nonempty set satisfying:
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
For solving the multiple-set split feasibility problem (MSSFP) equation (1.6), we assume that the following conditions are satisfied:
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.
Theorem 3.1 Let , , A, , , , C, Q, k, ρ be the same as above. Let be a sequence generated by
where , , is a constant, and satisfy the following conditions:
, for each ;
for each , ;
Let (the set of solutions of (MSSFP) equation (1.6) defined by equation (1.7)). Then we have the following:
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).
Step 1. We prove that the sequences , and are bounded and, for each , the following limits exist and
In fact, for given , by the definition of Γ,
Therefore, we have
Since is a family of k-strictly pseudo-nonspreading mappings, by Lemma 2.2, is closed and convex. It follows from Lemma 2.6 that, for each and ,
Further, since is a finite family of ρ-strictly pseudo-nonspreading mappings, we have
By condition (c), , therefore we have
This implies that the limit exists. It follows from equations (3.8) and (3.3) that the limit exists also, and
Therefore, and are bounded. Since for each , is quasi-nonexpansive, we have
Hence is also bounded.
Step 2. Now we prove that for any given positive integer , the following conclusions hold:
Therefore, we have
By conditions (b) and (c) we have
Since g is continuous and strictly increasing with , from equation (3.12) we have
Hence conclusion (3.10) is proved.
Step 3. Now, we prove that
In fact, it follows from equation (3.1) that
By virtue of equations (3.1) and (3.10), one has
This together with equations (3.10) and (3.15) shows that
Similarly, we have
Step 4. Now we show that every weak-cluster point of the sequence is in Γ.
Indeed, since is a bounded sequence in , there exists a subsequence such that . It follows from equation (3.10) that
By Lemma 2.2, is demiclosed at zero. Since , this implies that is also demiclosed at zero. Hence . By the arbitrariness of , we have
On the other hand, it follows from equations (3.1) and (3.10) that
Since A is a bounded linear operator, this implies that . Also, by equation (3.10)
Hence for any given positive integer , there exists a subsequence with such that
Since , and by Lemma 2.2, is demiclosed at 0. This implies that . By the arbitrariness of ,
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).
Without loss of generality, we may assume that is semicompact. Since , this implies that is also semicompact. In view of equation (3.10), we have
Therefore, there exists a subsequence of such that . Since , we have and so . By virtue of equation (3.9), we have
i.e., and both converge strongly to the point . This completes the proof of Theorem 3.1. □
Remark 3.2 Theorem 3.1 improves and extends the corresponding results of Censor et al. [5, 8, 9], Byrne , Yang , Moudafi , Xu , Censor and Segal , Masad and Reich , Deepho and Kumam [19, 20] and others in the following aspects:
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.
Corollary 3.3 Let H, , k be the same as above. Let be a sequence generated by
where , , is a constant, satisfy the following conditions:
, for each ;
for each , . Let
Then we have the following:
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.
Let H be a real Hilbert space, , be a countable family of -strictly pseudo-nonspreading mappings with , and
Let be a nonspreading mapping. The so-called hierarchical variational inequality problem for a countable family of mappings with respect to mapping T is to find an such that
It is easy to see that equation (4.1) is equivalent to the following fixed point problem: to find such that
where is the metric projection from H onto ℱ. Letting and (the fixed point set of ) and (the identity mapping on H), then the problem (4.2) is equivalent to the following multi-set split feasibility problem: to find such that
Hence from Theorem 3.1 we have the following theorem.
Theorem 4.1 Let H, , T, C, Q, k be the same as above. Let , be the sequences defined by
where , , , and satisfy the following conditions:
, 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.
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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).
The authors declare that they have no competing interests.
The main idea of this paper is proposed by JKK and SSC. JKK and SSC prepared the manuscript initially and performed all the steps of the proofs in this research. All authors read and approved the final manuscript.
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Chang, S., Kim, J.K., Cho, Y.J. et al. Weak- and strong-convergence theorems of solutions to split feasibility problem for nonspreading type mapping in Hilbert spaces. Fixed Point Theory Appl 2014, 11 (2014) doi:10.1186/1687-1812-2014-11
- split feasibility problem
- convex feasibility problem
- k-strictly pseudo-nonspreading mapping
- Opial’s condition