An Inertial Method for Split Common Fixed Point Problems in Hilbert Spaces

Abstract

In this paper, we consider the split common fixed point problem in Hilbert spaces. By using the inertial technique, we propose a new algorithm for solving the problem. Under some mild conditions, we establish two weak convergence theorems of the proposed algorithm. Moreover, the stepsize in our algorithm is independent of the norm of the given linear mapping, which can further improve the performance of the algorithm.

1. Introduction

In recent years, there has been growing interest in the study of the split common fixed point problem because of its various applications in signal processing and image reconstruction [1–3]. More specifically, the problem consists in finding satisfyingwhere and stand for the fixed point sets of mappings and , respectively, and is a bounded linear mapping. Here, and are two Hilbert spaces. In particular, if we let the mappings in (1) be the projections, then it is reduced to the well-known split feasibility problem (SFP): find such thatwhere and are two nonempty closed convex subsets and is a bounded linear mapping; see, e.g., [1, 4–7].

There are several algorithms for solving the split common fixed point problem. Among them, Censor and Segal [8] introduced an algorithm as where stands for the identity mapping, is the adjoint mapping of , and the stepsize is a constant in . In particular, when and , then the above algorithm is reduced to the well-known CQ algorithm for solving the split feasibility problem [4]. Note that this choice of the stepsize requires the exact value or estimation of the norm . To avoid the calculation of , Cui and Wang [9] proposed a variable stepsize as

It is readily seen that the above choice of the stepsize does not need any prior knowledge of the linear operator. Recently, Wang [10] introduced a new method for solving (1) aswhere the stepsize is set as

Recently, the above algorithms were further extended to the general case; see, e.g., [2, 10–17].

The inertial method was first introduced in [18], and now, it has been successfully applied to solving various optimization problems arising from some applied sciences [19, 20]. In particular, this method was also applied for solving the split feasibility problem [21, 22]. By applying the inertial technique, Dang et al. [21] recently proposed the inertial relaxed CQ algorithm, which is defined aswhere and . It is clear that the constant stepsize requires the estimation of the norm . To avoid the estimation of the norm, Gibali et al. [23] modified the above stepsize aswith . It is shown that the inertial relaxed CQ algorithm converges weakly toward a solution of the SFP provided that . The main advantage of the inertial method is that it can indeed speed up the convergence of the original algorithm. It is thus natural to extend it to the split common fixed point problem. Recently, Cui et al. [24] proposed a modified algorithm of (3) aswhere and is defined as in (6). It was shown that algorithm (9) converges weakly to a solution of the problem provided that .

In this paper, we aim to continue the study of the split common fixed point problem in Hilbert spaces. Motivated by the inertial method, we propose a new algorithm for solving the split common fixed point problem that greatly improves the performance of the original algorithm. Moreover, the stepsize in our algorithm is independent of the norm . Under some mild conditions, we establish two weak convergence theorems of the proposed algorithm.

2. Preliminary

In the following, we shall assume that problem (1) is consistent, that is, its solution set denoted by is nonempty. The notation “” stands for strong convergence, “” weak convergence, and the set of weak cluster points of a sequence . Let be a nonempty closed convex subset. For a mapping defined on , we let be its fixed point set and be its complement.

Definition 1. A mapping is said to be nonexpansive if is called quasi-nonexpansive if , and

Definition 2. Let be a mapping with . Then, is said to be demiclosed at 0 if, for any in , there holds the following implication:It is well known that if is a nonexpansive mapping, then is demiclosed at 0; see [25].

Lemma 1 (see [25]). If is quasi-nonexpansive, then

Lemma 2 (see [25]). Assume that is a sequence in such that(i)For each , the limit of exists(ii)Any weak cluster point of belongs to Then, is weakly convergent to an element in .

Lemma 3 (see [18]). Let and be two nonnegative real sequences such that andwhere . Then, the sequence is convergent.

Lemma 4 (see [25]). Let and . It then follows that

3. The Proposed Algorithm

Algorithm 1. Let be arbitrary. Given , choose , and setIf , then stop; otherwise, update the next iteration viawhere

Remark 1. In comparison, our stepsize (18) is independent of the norm so that the calculation or estimation of is avoided.

Remark 2. If for some , then is a solution of the problem. To see this, let . It then follows from Lemma 1 that , andCombining these inequalities yieldsThis yields , which implies .
If we let in (16), then we get a new algorithm for problem (1).

Algorithm 2. Let be arbitrary. Given , if , then stop; otherwise, update the next iteration viawhere

4. Convergence Analysis

In this section, we shall establish the convergence of the proposed algorithm. By Remark 2, we may assume that Algorithm 1 generates an infinite iterative sequence. To proceed, we first prove the following lemma.

Lemma 5. Let and be the sequences generated by Algorithm 1. Let . Then, for any , it follows that

Proof. Since is quasi-nonexpansive, we haveIn view of (18), we haveTo finish the proof, it suffices to note thatThis completes the proof.

Theorem 1. Assume that is quasi-nonexpansive such that is demiclosed at 0, and is quasi-nonexpansive such that is demiclosed at 0. If, for each , such that(c1), then the sequence generated by Algorithm 1 converges weakly to an element in .

Proof. We first show that the sequence is convergent for any . From Lemma 4, we deduceBy Lemma 5, this yieldsLet . Then, the above inequality can be rewritten asBy condition (c1), we then apply Lemma 3 to deduce that is convergent, and so is the sequence .
We next show that each weak cluster point of belongs to . Since is convergent, this implies that converges to 0 as . It then follows from (29) thatNote that by condition (c1). By passing to the limit in the above inequality, we have converging to 0 so thatMoreover, it is clear that is bounded; thus, the set is nonempty. Now, take any , and take a subsequence such that it weakly converges to . On the contrary, we deduce from (c1) thatso that also weakly converges to and weakly converges to . Since and are both demiclosed at 0, this together with (31) indicates and ; that is, is an element in .
Finally, by Lemma 2, the sequence converges weakly to a solution of problem (1).

Remark 3. We now construct a sequence satisfying condition (c1). For each , letWe next study the convergence of Algorithm 1 under another condition. To proceed, we need the following lemma.

Lemma 6. Let and be the sequences generated by Algorithm 1. For any , let . If is nondecreasing, thenwhere is defined as in Lemma 5.

Proof. In view of (17) and (18), we getIt then follows from inequality (25) thatMoreover, it follows from (27) thatOn the contrary, we haveSubstituting this into (21), we haveSince is nondecreasing, this impliesFrom the definition of , we get the desired inequality.

Theorem 2. Assume that is quasi-nonexpansive such that is demiclosed at 0, and is quasi-nonexpansive such that is demiclosed at 0. If(c2) is nondecreasing and converges to , then the sequence generated by Algorithm 1 converges weakly to an element in .

Proof. We first show that is convergent for each . It then follows from Lemma 6 and the range of thatso that is nonincreasing. From the definition of , we getBy induction, we haveThus, is bounded. Moreover, from the definition of ,which implies that is bounded from below, and thus, it is convergent. Passing to the limit in (41) yieldsOn the contrary,from which it follows thatHere, we used the fact (by the definition of ) thatThus, is convergent.
We next show that the sequence converges weakly to a solution of problem (1). By Lemma 2, it suffices to show that each weak cluster point of belongs to . Moreover, it is clear that is bounded; thus, the set is nonempty. Now, take any . On the contrary, we deduce from (16) and (45) thatIn a similar way, we deduce that and ; that is, is an element in . Hence, the proof is complete.
If we let , then it satisfies (c1) and (c2). As a result, we get the following conclusion.

Corollary 1. Assume that is quasi-nonexpansive such that is demiclosed at 0, and is quasi-nonexpansive such that is demiclosed at 0. Then, the sequence generated by Algorithm 2 converges weakly to an element in .

5. Concluding Remarks

The main contribution of this paper is to propose a new algorithm for solving the split common fixed point problem in Hilbert spaces. There are two advantages of the proposed algorithm. Compared with the original algorithm for solving the problem, our proposed algorithm is faster in convergence rate. Furthermore, the stepsize in the proposed algorithm is independent of the norm of the given linear mapping, which can further improve its performance.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the Natural Science Foundation of China (no. 11701154) and Key Scientific Research Projects of Universities in Henan Province (nos. 19B110010 and 20A110029).