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General split equality problems in Hilbert spaces
Fixed Point Theory and Applications volume 2014, Article number: 35 (2014)
A new convex feasibility problem, the split equality problem (SEP), has been proposed by Moudafi and Byrne. The SEP was solved through the ACQA and ARCQA algorithms. In this paper the SEPs are extended to infinite-dimensional SEPs in Hilbert spaces and we established the strong convergence of a proposed algorithm to a solution of general split equality problems (GSEPs).
In the present paper, we are concerned with the general split equality problem (GSEP) which is formulated as finding points x and y with the property:
where and are two nonempty closed convex subsets of real Hilbert spaces and , respectively, also is a Hilbert space, , are two bounded linear operators.
It generalizes the split equality problem (SEP), which is to find , such that , as well as the split feasibility problem (SFP). When , the SEP becomes a SFP. As we know, the SEP has received much attention due to its applications in image reconstruction, signal processing, and intensity-modulated radiation therapy, see for instance [2–5].
To solve the SEP, Byrne and Moudafi put forward the alternating CQ-algorithm (ACQA) and the relaxed alternating CQ-algorithm (RACQA). For an exhaustive study of ACQA and RACQA, see for instance [6, 7]. The approximate SEP (ASEP), which is only to find approximate solutions to SEP, is also proposed and solved through the simultaneous iterative algorithm (SSEA), the relaxed SSEA (RSSEA) and the perturbed SSEA (PSSEA) by Byrne and Moudafi, see for example [1, 8].
This paper aims at a study of an iterative algorithm improved by Eslamian  for the GSEP in the Hilbert space. We show the strong convergence of the presented algorithms to a solution of the GSEP, and we obtain an algorithm which strongly converges to the minimum norm solution of the GSEP.
For the sake of simplicity, we will denote by H a real Hilbert space with inner product and norm . Let C be a nonempty closed convex subset of H. Let be an operator on H. Recall that T is said to be nonexpansive if , . A typical example of nonexpansivity is the orthogonal projection from H onto a nonempty closed convex subset defined by , . It is well known that is characterized by the relation
Lemma 2.1 Let in , where . Define
then solves the SEP if and only if solves the fixed point equation .
Lemma 2.2 Let H be a Hilbert space. Then for any given sequence in H, any given sequence of positive numbers with and for any positive integer i, j with ,
Lemma 2.3 Let H be a Hilbert space. For every x and y in H, the following inequality holds:
Lemma 2.4 Let C be a nonempty closed convex subset of H, and let be a nonexpansive mapping with . Then T is demiclosed on C, that is, if and , then .
Lemma 2.5 Assume is a sequence of nonnegative real numbers such that where is a sequence in and is a sequence such that
Lemma 2.6 Let be a sequence of real numbers that does not decrease at infinity, in the sense that there exists a subsequence of such that
Also consider the sequence of the integers defined by
Then is a nondecreasing sequence verifying , and for all , the following two estimates hold:
3 Main results
Let and let be closed, nonempty convex sets, and let A, B be and real matrices, respectively. Let . Define
The problem (1.1) can also be formulated as finding with or with minimizing the function over .
Proposition 3.1 solves the GSEP (1.1) if and only if
Proof Assume that there exists satisfying , then for any , we have . We use x and y to express :
By Lemma 2.1, for any , there exist and , such that . Therefore, there exist and , such that , that is to say, solves GSEP (1.1).
Assume that solves GSEP (1.1), such that , that is, for any , we have and , such that . Substituting into (3.1) and (3.2), we obtain for any , . Therefore, solves . □
Theorem 3.2 Assume that the GSEP has a nonempty solution set Ω. Suppose that f is a self k-contraction mapping of H, , and let be a sequence generated by
where . If the sequences , , , and satisfy the following conditions:
, for each ,
, for each , where ,
then the sequence strongly converges to , where , .
Proof We first prove that is bounded. Let ; actually, by Lemma 2.1, equals the fixed point equation . Note that for each , , where , then the operator is nonexpansive. We also know that f is a k-contraction mapping, then
Then, from the upper deduction we have and
We can conclude that , are bounded.
Furthermore, from (3.3) and Lemma 2.2 we get
It follows that
In order to show that , we consider two cases.
Case 1: Suppose that is a monotone sequence. Since is bounded, is convergent. Take the limit on both sides for (3.4), because and , and we get , .
We first prove there exists a unique , such that . Since is nonexpansive and f is a self k-contraction mapping, we get
therefore, there exists a unique , such that .
Next, we show that . Using Lemma 2.3, we get
By induction, we obtain
where , and .
Since , , we have . Next, we will prove . Actually, (since ), so we just need to prove . Take a subsequence in , such that
Since is bounded, there exists a subsequence converging weakly to v. Suppose that and , according to Lemma 2.4, . Since and ,
Therefore, and hold. All conditions of Lemma 2.5 are satisfied. Therefore , .
Case 2: If is not a monotone sequence, we could define an integer sequence by
It is easy to see that is nondecreasing and when we get . For all we obtain . Then is a monotone sequence and according to Case 1, we have and . Finally, from Lemma 2.6, we get
Therefore, the sequence converges strongly to .
For every , solves the GSEP if and only if solves the fixed point equation , . Actually, we have proved and . Then , , that is, solves the GSEP.
Therefore, the sequence strongly converges to . This completes the proof. □
Corollary 3.3 We define a sequence iteratively
where . If , , satisfy the following conditions:
, for each ,
, for each , where ,
then converges strongly to a point which is the minimum norm solution of GSEP (1.1).
Proof Let in (3.3), then we get (3.5). We have proved in Theorem 3.2. Then,
Hence, . Since , then , for all , that is,
Thus, is the minimum norm solution of GSEP (1.1). This completes the proof. □
Let be a countable family of nonexpansive mappings with and let be a nonexpansive mapping. Consider the variational inequality problem of finding a common fixed point of with respect to a nonexpansive mapping T is to
It is easy to see that (3.6) equals the following fixed point problem:
Letting , , , , then the upper problem (3.7) is transformed into GSEP (1.1):
Therefore, GSEP (1.1) equals (3.6). Hence, we have the following result.
Theorem 3.4 If and the sequences , , , and satisfy the following conditions:
, for each ,
, for each , where ,
the sequence defined by (3.3) converges strongly to a point which solves the following variational inequality with :
Proof We know from the proof of Theorem 3.2 that the sequence defined by (3.3) converges strongly to , which solves the GSEP. Also since GSEP (1.1) equals (3.6), solves the variational inequality problem (3.6). Since , by (3.7) and (3.6), we have . Actually, since f is a self k-contraction mapping, , then f is also a nonexpansive mapping. That is to say, the condition in (3.6), that T is a nonexpansive mapping, is satisfied. Therefore, defined by (3.3) converges strongly to a solution of . This completes the proof. □
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We wish to thank the referees for their helpful comments and suggestions. This research was supported by NSFC Grant No. 11071279.
The authors declare that they have no competing interests.
All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
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Chen, R., Wang, J. & Zhang, H. General split equality problems in Hilbert spaces. Fixed Point Theory Appl 2014, 35 (2014). https://doi.org/10.1186/1687-1812-2014-35
- general split equality problem
- strong convergence
- the minimum norm solution