An Iterative Algorithm for the Split Equality and Multiple-Sets Split Equality Problem

Abstract

The multiple-sets split equality problem (MSSEP) requires finding a point , such that , where and are positive integers, and are closed convex subsets of Hilbert spaces , , respectively, and , are two bounded linear operators. When , the MSSEP is called the split equality problem (SEP). If  , then the MSSEP and SEP reduce to the well-known multiple-sets split feasibility problem (MSSFP) and split feasibility problem (SFP), respectively. One of the purposes of this paper is to introduce an iterative algorithm to solve the SEP and MSSEP in the framework of infinite-dimensional Hilbert spaces under some more mild conditions for the iterative coefficient.

1. Introduction and Preliminaries

1.1. Introduction

Let and be nonempty closed convex subsets of real Hilbert spaces and , respectively, and let be a bounded linear operator. The multiple-sets split feasibility problem (MSSFP) is to find a point satisfying the property: if such point exists. If , then the MSSFP reduce to the well-known split feasibility problem (SFP).

The SFP and MSSFP were first introduced by Censor and Elfving [1] and Censor et al. [2], respectively, which attract many authors’ attention due to its applications in signal processing [1] and intensity-modulated radiation therapy [2]. Various algorithms have been invented to solve it; see [1–8], e.t.

Recently, Moudafi [9] propose a new split equality problem (SEP): let , , and be real Hilbert spaces; let , be two nonempty closed convex sets; and let , be two bounded linear operators. Find , satisfying When , SEP reduces to the well-known SFP.

Naturally, we propose the following multiple-sets split equality problem (MSSEP) requiring to find a point , such that where and are positive integers; and are closed convex subsets of Hilbert spaces , , respectively, and , are two bounded linear operators.

In the paper [9], Moudafi gave an alternating CQ-algorithm and relaxed alternating CQ-algorithm iterative algorithm for solving the split equality problem.

We use to denote the solution set of SEP, that is, and assume consistency of SEP so that is closed, convex, and nonempty.

Let in and define by ; then has the matrix form The SEP problem can be reformulated as finding with or solving the following minimization problem: In paper [10], we used the well-known Tychonov regularization that got some algorithms to converge strongly to the minimum-norm solution of the SEP.

Note that the convergence of the above algorithms depends on the exact requirements of the iterative coefficient. Therefore, the aim of this paper is to introduce an iterative algorithm to solve the SEP and MSSEP in the framework of infinite-dimensional Hilbert spaces under some more mild conditions for the iterative coefficient.

Throughout the rest of this paper, denotes the identity operator on Hilbert space and is the set of the fixed points of an operator . An operator on a Hilbert space is nonexpansive if, for each and in , . is said to be averaged, if there exists and a nonexpansive operator such that .

Let denote the projection from onto a nonempty closed convex subset of ; that is, It is well known that is characterized by the following inequality: and is nonexpansive and averaged.

We now collect some elementary facts which will be used in the proofs of our main results.

Lemma 1 (see [11, 12]). Let be a Banach space, a closed convex subset of , and a nonexpansive mapping with . If is a sequence in weakly converging to and if converges strongly to , then .

Lemma 2 (see [13]). Let be a Hilbert space and a sequence in such that there exists a nonempty set satisfying the following.(i)For every , exists.(ii)Any weak-cluster point of the sequence belongs to .
Then, there exists such that weakly converges to .

Lemma 3 (see [4]). Let and be averaged operators and suppose that is nonempty. Then .

The following lemma is vital in our main results.

Lemma 4. Let , where with being the spectral radius of the self-adjoint operator on . Then we have the following:(1) (i.e., is nonexpansive) and averaged;(2), ;(3) if and only if is a solution of the variational inequality ,  for all .

Proof. (1) It is easily proved that ; we only prove that is averaged. Indeed, choose , such that ; then , where is a nonexpansive mapping. That is to say, is averaged.
(2) If , it is obvious that . Conversely, assuming that , we have . Hence ; then ; we get that . It follows .
Now we prove . By , is obvious. On the other hand, since , and both and are averaged, from Lemma 3, we have .
(3) Consider

2. Iterative Algorithm for SEP

In this section, we establish an iterative algorithm that converges weakly to a solution of SEP.

Algorithm 5. Choose an arbitrary initial point , and sequence is generated by the following iteration: where and with being the spectral radius of the self-adjoint operator on .

To prove its convergence we need the following lemma.

Lemma 6. The sequence generated by algorithm (10) is Féjer-monotone with respect to ; that is to say, for every , if and .

Proof. Let and choose ; by Lemma 4, , and we have Moreover, we have Hence, we can get that It follows that , for all , .

Theorem 7. If  , then the sequence generated by algorithm (10) converges weakly to a solution of SEP (2).

Proof. Let be a solution of SEP; according to Lemma 6, we can get that the sequence is monotonically decreasing and converges to some positive real. Since and , by (14), we have
Since is Féjer-monotonicity, it follows that is bounded. Let be a weak-cluster point of and let be the sequence of indices, such that converges weakly to . By Lemma 1, we can get that . It follows that .
Since , it follows that converges weakly to . On the other hand, . Using Lemma 1 again, we obtain that . That is to say, .
Hence . By Lemma 4, we get that is a solution of SEP (2).
The weak convergence of the whole sequence holds true since all conditions of the well-known Opial’s lemma (Lemma 2) are fulfilled with .

3. Iterative Algorithm for MSSEP

In this section, we establish an iterative algorithm that converges weakly to a solution of MSSEP.

We use to denote the solution set of MSSEP, that is, and assume consistency of MSSEP so that is closed, convex, and nonempty.

Without loss of generality, we assume that . In fact, if , let , for .

Let in and define by ; then has the following matrix form: The original problem now can be reformulated as finding with , or, more generally, minimizing the function over .

Algorithm 8. For an arbitrary initial point , sequence is generated by the following iteration: where , is a sequence in , and with being the spectral radius of the self-adjoint operator on .

The proof of the following lemma is similar to Lemma 4, and we omit its proof.

Lemma 9. Let , where with being the spectral radius of the self-adjoint operator on . Then we have , , and .

To prove its convergence we also need the following lemma.

Lemma 10. Any sequence generated by algorithm (18) is the Féjer-monotone with respect to ; namely, for every , provided that is a sequence in and .

Proof. Let and take ; by Lemma 9, , for all , and we have Moreover, all the same to the proof of Lemma 6, we have Hence, we have It follows that , for all , .

Theorem 11. If , then the sequence generated by algorithm (18) converges weakly to a solution of MSSEP (3).

Proof. From (22) and the fact that and , we obtain that Therefore,
Since is Féjer-monotone, it follows that is bounded. Let be a weak-cluster point of . Taking a subsequence of such that converges weakly to , then, by Lemma 1, we can get that ; it follows that .
Let ; it follows that converges weakly to .
Since it follows that On the other hand Hence We can get that and for all .
Moreover, for any , Thus, for all . Using Lemma 1 again, we obtain that . That is to say, for all .
Hence . By Lemma 9, we obtain that is a solution of MSSEP (3).
The weak convergence of the whole sequence holds true since all conditions of the well-known Opial’s lemma (Lemma 2) are fulfilled with .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This research was supported by NSFC Grants no. 11071279; no. 11226125; and no. 11301379.