General Split Feasibility Problems in Hilbert Spaces

Abstract

Introducing a general split feasibility problem in the setting of infinite-dimensional Hilbert spaces, we prove that the sequence generated by the purposed new algorithm converges strongly to a solution of the general split feasibility problem. Our results extend and improve some recent known results.

1. Introduction

Let and be infinite-dimensional real Hilbert spaces, and let be a bounded linear operator. Let and be the families of nonempty closed convex subsets of and , respectively.

(a) The convex feasibility problem (CFP) is formulated as the problem of finding a point with the property:

(b) The split feasibility problem (SEP) is formulated as the problem of finding a point with the property: where and are nonempty closed convex subsets of and , respectively.

(c) The multiple-set split feasibility problem (MSSFP) is formulated as the problem of finding a point with the property: Note that (MSSFP) reduces to (SEP) if we take .

There is a considerable investigation on CFP in view of its applications in various disciplines such as image restoration, computer tomograph, and radiation therapy treatment planning [1]. The split feasibility problem SFP in the setting of finite-dimensional Hilbert spaces was first introduced by Censor and Elfving [2] for modelling inverse problems which arise from phase retrievals and in medical image reconstruction [3]. Since then, a lot of work has been done on finding a solution of SFP and MSSFP; see, for example, [2–25]. Recently, it is found that the SFP can also be applied to study the intensity-modulated radiation therapy; see, for example, [6, 16] and the references therein. Very recently, Xu [8] considered the SFP in the setting of infinite-dimensional Hilbert spaces.

The original algorithm given in [2] involves the computation of the inverse provided it exists. In [8], Xu studied some algorithm and its convergence. In particular, he applied Mann’s algorithm to the SFP and purposed an algorithm which is proved to be weakly convergent to a solution of the SFP. He also established the strong convergence result, which shows that the minimum-norm solution can be obtained. In [7], Wang and Xu purposed the following cyclic algorithm to solve MSSFP: where mod, (mod function take values in ), and . They show that the sequence convergence weakly to a solution of MSSFP provided the solution exists. To study strong convergence to a solution of MSSFP, first we introduce a general form of the split feasibility problem for infinite families as follows.

(d) General split feasibility problem (GSFP) is to find a point such that We denote by the solution set of GSFP.

In this paper, using viscosity iterative method defined by Moudafi [21], we propose an algorithm for finding the solutions of the general split feasibility problem in a Hilbert space. We establish the strong convergence of the proposed algorithm to a solution of GSFP.

2. Preliminaries

Throughout the paper, we denote by a real Hilbert space with inner product and norm . Let be a sequence in and . Weak convergence of to is denoted by , and strong convergence by . Let be a closed and a convex subset of . For every point , there exists a unique nearest point in , denoted by . This point satisfies The operator is called the metric projection or the nearest point mapping of onto . The metric projection is characterized by the fact that and Recall that a mapping is called nonexpansive if It is well known that is a nonexpansive mapping. It is also known that satisfies Opial’s condition, that is, for any sequence with , the inequality holds for every with .

Lemma 1. Let be a Hilbert space. Then, for all

Lemma 2 (see [22]). Let be a Hilbert space, and let be a sequence in . Then, for any given sequence with and for any positive integer with ,

Lemma 3 (see [23]). Assume that is a sequence of nonnegative real numbers such that where , , and satisfy the following conditions: (i), ,(ii) or ,(iii) for all with .Then, .

Lemma 4 (see [24]). Let be a sequence of real numbers such that there exists a subsequence of such that for all . Then, there exists a nondecreasing sequence such that , and the following properties are satisfied by all (sufficiently large) numbers : In fact

Lemma 5 (demiclosedness principle [25]). Let be a nonempty closed and convex subset of a real Hilbert space . Let be a nonexpansive mapping such that . Then, is demiclosed on , that is, if , and , then .

3. Main Result

In the following result, we propose an algorithm and prove that the sequence generated by the proposed method converges strongly to a solution of the GSFP.

Theorem 6. Let and be real Hilbert spaces, and let be a bounded linear operator. Let and be the families of nonempty closed convex subsets of and , respectively. Assume that GSFP (5) has a nonempty solution set . Suppose that is a self -contraction mapping of , and let be a sequence generated by as where . If the sequences , , , and satisfy the following conditions: (i) and ,(ii)for each , , (iii)for each , and ,then, the sequence converges strongly to , where .

Proof. First, we show that is bounded. In fact, let . Since , the operators are nonexpansive, and hence we have which implies that is bounded, and we also obtain that is bounded. Next, we show that for each , By using Lemma 2, for every and , we have that Hence, for each , we have
Next, we show that there exists a unique such that . We observe that for each , solves the GSFP (5) if and only if solves the fixed point equation that is, the solution sets of fixed point equation (20) and GSFP (5) are the same (see for details [8]). Note that if , then the operators are nonexpansive. Since the fixed point set of nonexpansive operators is closed and convex, the projection onto the solution set is well defined whenever . We observe that is a contraction of into itself. Indeed, since is nonexpansive, Hence, there exists a unique element such that .
In order to prove that as , we consider two possible cases.
Case 1. Assume that is a monotone sequence. In other words, for large enough, is either nondecreasing or nonincreasing. Since is bounded we have is convergent. Since and is bounded, from (19) we get that By assuming that , we obtain Now, we show that To show this inequality, we choose a subsequence of such that Since is bounded, there exists a subsequence of which converges weakly to . Without loss of generality, we can assume that and for each . From (23), we have Notice that for each , is nonexpansive. Thus, from Lemma 5, we have . Therefore, it follows that Finally, we show that . Applying Lemma 1, we have that This implies that where and . It is easy to see that , and . Hence, by Lemma 3, the sequence converges strongly to .
Case 2. Assume that is not a monotone sequence. Then, we can define an integer sequence for all (for some large enough) by Clearly, is a nondecreasing sequence such that as and for all , From (19), we obtain that Following an argument similar to that in Case 1, we have And by similar argument, we have where , and . Hence, by Lemma 3, we obtain and . Now, from Lemma 4, we have Therefore, converges strongly to .

For finite collections we have the following consequence of Theorem 6.

Theorem 7. Let and be real Hilbert spaces, and let be a bounded linear operator. Let be a family of nonempty closed convex subsets in , and let be a family of nonempty closed convex subsets in . Assume that MSSFP has a nonempty solution set . Let be an arbitrary element in , and let be a sequence generated by and where . If the sequences , , , and satisfy the following conditions: (i) and ,(ii)for all , , (iii)for all , and then the sequence converges strongly to , where .