Strong Convergence on the Split Feasibility Problem by Mixing -Mapping

Abstract

In this paper, we concern with the split feasibility problem (SFP) in real Hilbert space whenever the sets involved are nonempty, closed, and convex. By mixing -mapping with the viscosity, we introduce a new iterative algorithm for solving the split feasibility problem, and we prove that our proposed algorithm is convergent strongly to a solution of the split feasibility problem.

1. Introduction

Throughout this article, we assume that and are two real Hilbert spaces. The split feasibility problem (SFP) was introduced by Censor and Elfving [1], and it is formulated as finding a point in such that is in , namely,where and are nonempty, closed, and convex subsets of real Hilbert spaces and , respectively, and is a bounded linear operator from to .

Many inverse problems arising from various fields of science and technology, such as intensity-modulated radiation therapy [2], signal processing, and image reconstruction, can be summarized as SFP. Due to its applications, many algorithms have been invented to solve SFP (see, for instance, [3–11]).

To solve problem (1), in 2002, Byren [3] introduced a popular algorithm which is called the CQ-algorithm as follows:where is the identity operator on , and denote the metric projection onto the closed convex subsets and , respectively, and is the adjoint operator of and . In 2018, Wang [10] proposed his algorithm as follows:where , , and were given byin which and are two given convex functions, andin which .

To obtain strong convergence theorem, Wang [10] modified his algorithm as follows:where , , and were given as the same to the weak convergence theorem; is a sequence in which is chosen so that

On the other hand, another problem which is similar to the split feasibility problem is the proximal split feasibility problem (PSFP), and the proximal split feasibility problem is to find a point satisfying the property:where and are two proper and lower semicontinuous convex functions, is a linear bounded operator, and , .

To solve problem (8), in 2014, Moudafi and Thakur [12] introduced the split proximal algorithm by the following iterative scheme:where the stepsize was the suitable positive real number sequence, and they also proved the weak convergence of the sequence generated by the above equation to a solution of problem (8). In 2015, Shehu et al. [13] introduced a viscosity-type algorithm for solving proximal split feasibility problems as follows:where is a contraction mapping. They also proved a strong convergence of the sequence generated by iterative scheme (10) in Hilbert spaces. Recently, Shehu and Iyiola [14] introduced the following algorithm for solving split proximal problems and fixed point problems in Hilbert spaces:where is a -strictly pseudocontractive mapping. They also showed that, under certain assumptions imposed on the parameters, the sequence was generated by the algorithm that they introduced converges strongly to .

If we defined and as indicated functions of sets and , where and are nonempty, closed, and convex sets of and , respectively, then the proximal split feasibility problem (8) becomes the split feasibility problem (1). In this paper, inspired and motivated by these works that have been done, we focus on the split feasibility problem in Hilbert spaces.

The rest of this paper is organized as follows. In Section 2, we review some definitions and lemmas that we need. In Section 3, we introduce a new iterative algorithm based on the viscosity method and -mapping which is defined in Section 2 for finding a solution of the split feasibility problem and prove a strong convergence theorem under some mild conditions.

2. Preliminaries

Throughout this paper, let be a real Hilbert space with the inner product and norm . We denote by the identity operator on and by the set of all weak cluster points of . The notation    stands for strong convergence and    stands for weak convergence.

Definition 1. (see [15]). Let be a nonlinear mapping. Then, is(1)Nonexpansive if(2)Firmly nonexpansive if

Definition 2. Let be a nonempty closed convex subset of . Then, an orthogonal projection is defined by

Lemma 1. Let be a nonempty closed convex subset of , then(1).(2) and both are (firmly) nonexpansive.(3).(4).

Definition 3. Let be an operator with . If for any in , and , we can obtain , then we claim that is demiclosed at zero.

Lemma 2. If is a nonexpansive operator, then is demiclosed at zero.

Lemma 3. For all , and , we have(1).(2).

Definition 4. If for any and , let be a nonexpansive mapping on and be real numbers with . We define a mapping on for each bythen we say that is the -mapping generated by and .

Lemma 4. (see [16]). Let be a W-mapping, then is a nonexpansive mapping.

Lemma 5. (see [17]). Assume that is a sequence of nonnegative real number such thatwhere , , and satisfy the following conditions:(1);(2)either or ;(3) for all and .Then, .

3. Main Results

To introduce our iterative algorithm for solving the split feasibility problem in real Hilbert spaces, firstly, we shall assume that problem (1) is consistent, namely, its solution set, denoted by , is nonempty. Secondly, we need to define a special -mapping as follows:where . From Lemma 4, we know that is nonexpansive.

Now, we will introduce our iterative algorithms for the split feasibility problem.

Algorithm 1. Given an initial point , let be a contraction mapping with . Assume that has been constructed and compute by the following iterative scheme:where and satisfy(1);(2).

Theorem 1. Let be the sequence generated by Algorithm 1, then converges strongly to a solution of the SFP (1), where .

Proof. Since is a contraction mapping with and the fact that is nonexpansive, it is clear that is also a contraction mapping. By Banach fixed point theorem, there exists , such that .
Since , that is, and . By the definition of , we have . In what follows, we will divide the proof into four steps.
Firstly, we prove that the sequence is bounded.
From (18) and Lemmas 1 and 3, we haveBy condition (1), we have , so . Therefore, from (18), we obtainBy introduction, we obtainfor all . The above inequality implies that the sequence is bounded. Combining with (18), we know that , , and are also bounded.
Secondly, we show that the following inequality holds:where andwith .
From equations (18) and (19) and Lemma 3, we haveSo, inequality (22) holds.
Thirdly, we show that is finite.
Since is bounded, we haveThis implies that . Next, we will show that by contraction. If we assume that , then there exists , such that for all . From (22), we obtainBy introduction, we haveSince , , then there exists , such that . Combining with the last inequality, we havewhich is contradicted with the fact that is nonnegative. Thus, . So, is finite.
Lastly, we show that .
Since is finite, there exists a subsequence such thatSince is bounded, without loss of generality, we may assume the limit of exists. From (29), we may also assume the following limit exists:These conditions and imply . So, we obtainthat is,Next, we prove that any weak cluster point of the sequence is a solution of the SFP (1).
Since is bounded, let be a weak cluster point of the sequence ; without loss of generality, we assume that ; then, we obtain . From the fact that and are nonexpansive, Lemma 2 implies and are demiclosed at zero; from (32), we obtain and , i.e., , hence .
Finally, we show that .
From (18) and the definition of , we knowand, from (32), we haveSo,This implies that any weak cluster point of also belongs to . Without loss of generality, we assume that converges weakly to . Now, combing (29), Lemma 1, and the fact that , we can obtainFrom Lemma 5, we get , which ends the proof.
From Theorem 1, we obtain the following subresult on the split feasibility problem (1).

Algorithm 2. Given an initial point , let be fixed. Assume that has been constructed and compute by the following iterative scheme:where and satisfy(1);(2).

Corollary 1. Let be the sequence generated by Algorithm 2, then converges strongly to a solution of the SFP (1), where .

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed to each part of this work equally, and they all read and approved the final manuscript.

Acknowledgments

This research was supported by the National Natural Science Foundation of China (nos. 11601339 and 11701154), the Natural Science Foundation of the Department of Education, Henan Province (nos. 19A110020 and 20A110020), the Graduate Education Reform and Quality Improvement Project of Henan Province, Higher Education Teaching Reform and Practice Project (postgraduate education) of Henan Normal University (no. YJS2019JG01), and Program for Graduate Innovative Research of Henan Normal University (no. YL201919).