Internal Perturbation Projection Algorithm for the Extended Split Equality Problem and the Extended Split Equality Fixed Point Problem

Li, Meixia; Zhou, Xueling; Wang, Wenchao

doi:https://doi.org/10.1155/2020/6034754

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Recent Advances in Function Spaces and its Applications in Fractional Differential Equations 2020

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Research Article | Open Access

Volume 2020 | Article ID 6034754 | https://doi.org/10.1155/2020/6034754

Internal Perturbation Projection Algorithm for the Extended Split Equality Problem and the Extended Split Equality Fixed Point Problem

Meixia Li,¹Xueling Zhou,²and Wenchao Wang²

Academic Editor: Liguang Wang

Received03 Mar 2020

Accepted11 Apr 2020

Published03 Jun 2020

Abstract

In this article, we study the extended split equality problem and extended split equality fixed point problem, which are extensions of the convex feasibility problem. For solving the extended split equality problem, we present two self-adaptive stepsize algorithms with internal perturbation projection and obtain the weak and the strong convergence of the algorithms, respectively. Furthermore, based on the operators being quasinonexpansive, we offer an iterative algorithm to solve the extended split equality fixed point problem. We introduce a way of selecting the stepsize which does not need any prior information about operator norms in the three algorithms. We apply our iterative algorithms to some convex and nonlinear problems. Finally, several numerical results are shown to confirm the feasibility and efficiency of the proposed algorithms.

1. Introduction

Let , and be three real Hilbert spaces and and be two nonempty, closed, and convex sets. The split feasibility problem (SFP) is formulated where is a bounded linear operator. The SFP was first introduced by Censor and Elfving [1], which was used in modeling various inverse problems arising from phase retrievals and medical image reconstruction and further studied by many researchers. See, for instance, [2–10].

Moudafi [11, 12] introduced the following split equality feasibility problem (SEFP), which is where and are two bounded linear operators. Obviously, if and , then (2) reduces to (1). The split equality feasibility problem (2) allows asymmetric and partial relations between the variables and . In order to solve SEFP, many researchers proposed their suggestions, such as [13–17] and references therein. Moudafi [11] introduced the following iterative method:

Under some suitable conditions, he proved that the sequence weakly converges to the solution of (2) in Hilbert spaces. In addition, Yu and Wang [18] proposed the following iterative algorithm: where with . They studied the weak convergence of scheme (4).

Recently, Che et al. [19] proposed the following extended split equality problem (ESEP) which is an extension of the convex feasibility problem. Let be a real Hilbert space. For , assume are nonempty closed convex subsets of real Hilbert spaces , respectively. The extended split equality problem is where are linear operators. They presented the following simultaneous iterative algorithm:

Under some suitable conditions, they obtained the weak convergence of (6).

In order to avoid using the projection, Moudafi [11] introduced and studied the following problem. Let and be nonlinear operators such that and , where and denote the sets of fixed points of and , respectively. If and , then SEFP (2) reduces which is called the split equality fixed point problem (SEFPP). Many scholars have studied this issue, such as [20–22].

To solve problem (7), Che and Li [23] proposed the following iterative algorithm:

They established the weak convergence of scheme (8) under the conditions that the operators and are quasinonexpansive mappings.

Similarly, Che et al. [19] proposed the following extended split equality fixed point problem (ESEFPP), which is and presented the following simultaneous iterative algorithm: where are the -mapping generated by which is a finite family of -strictly pseudononspreading. They obtained the weak convergence of (10).

Motivated by the works mentioned above, we continue to study the ESEP (5) and ESEFPP (9) with internal perturbation projection and do not need any prior information about the operator norms. The paper is organized as follows. In Section 2, we introduce some preliminaries to be employed in the subsequent analysis. In Section 3, we present two simultaneous iterative algorithms to solve ESEP (5) and establish the weak and the strong convergence of the proposed algorithms, respectively. We propose a simultaneous iterative algorithm to solve ESEFPP (9) and obtain the weak convergence of the proposed algorithm in Section 4. In Section 5, we apply our iterative algorithms to some convex and nonlinear problems. In the concluding section, several numerical results are shown to confirm the effectiveness of our algorithms.

2. Preliminaries

In this paper, we use and to denote the strong convergence and the weak convergence, respectively. We use to stand for the weak -limit set of . For any , there exists a unique nearest point in , denoted by , such that

It is well known that is nonexpansive and firmly nonexpansive. And has the following well-known properties.

Lemma 1. Let be a nonempty, closed, and convex set. The following conclusions hold: (1) (2) (3),

Definition 2. A mapping is said to be (1)nonexpansive if (2)quasinonexpansive if (3)firmly nonexpansive if (4)firmly quasinonexpansive if (5)-strictly pseudononspreading if

Lemma 3 [24]. In the real Hilbert space , for , the following relations hold: (1)(2)(3)(4) for

Definition 4 [25]. A mapping is said to be demiclosed at if, for any sequence which converges weakly to and with

Lemma 5 [26]. Let be a sequence of real numbers that does not decrease at infinity in the sense that there exists a subsequence of which satisfies for all . Define the integer sequence , for (such that ) is as follows: Then, there hold the following properties: (1) and (2) and

Lemma 6 [27]. Assume that is a sequence of nonnegative real numbers such that where is a sequence in and is a sequence in such that (1) and (2) or Then,

Definition 7 [19]. Let be a nonempty closed and convex subset of real Banach space. Let be a finite family of mappings of into itself. For let where and We define a mapping as follows: Such a mapping is called the -mapping generated by and

Lemma 8 [19]. Let be a nonempty closed and convex subset of real Banach space. Let be a finite family of -strictly pseudononspreading mappings of into itself with and For let where and Assume that for and for and If is the -mapping generated by and then and is a quasinonexpansive mapping.

3. Iterative Algorithms for ESEP

In this section, we introduce two simultaneous iterative algorithms with internal perturbation projection to solve ESEP (5) and define the solution set of ESEP (5) as

Algorithm 9. Initialization: take arbitrary.
Iteration step: for a given current iterate , we calculate the next iterate by where the stepsize is chosen in such a way that if , then where the index set small enough and , set and go to (15). Otherwise, for the iteration stops.

Remark 10. Note that in (16), the choice of the stepsize is independent of the norm , for . Furthermore, we will show from Lemma 11 that is well defined.

Lemma 11. Assume the solution set of ESEP (5) is nonempty, then defined by (16) is well defined.

Proof. Let , then Noting that for , we have As a result, Summing the above equalities and applying Lemma 1 (3) as well as the condition one has Consequently, for , we have , then which leads that is well defined.

Theorem 12. Let be a real Hilbert space. For , assume that are nonempty closed convex subsets of real Hilbert spaces , and are bounded linear operators with their adjoint operators . Then, the sequence generated by Algorithm 9 converges weakly to a solution of ESEP (5). Furthermore, for and as .

Proof. Let and For from (15) and Lemma 3, we have According to Lemma 1 and (21) can be written as Note that Summing (23) for from to , we can obtain Let we have Therefore, the sequence is nonincreasing and lower bounded by 0. Hence, converges to some finite limit, suppose as , and sequence is bounded. Letting and taking the limit in the two sides of (25), for we can obtain that which implies Since , then From the definition of , we have then, which implies From (32), for we can get Consequently, the sequence is asymptotically regular.
Let then for there exists a subsequence of which converges weakly to . From (29) and the lower semicontinuity of the squared norm, we have which implies
Furthermore, it follows from (29) and the lower semicontinuity of the squared norm where which means that That is, Obviously, Hence,
In the sequel, we will show the uniqueness of the weak cluster points of . Assume that is another weak cluster point of . It follows from the definition of that Without loss of generality, we can suppose , and then Reversing the role of and , we can obtain Adding (39) and (40), we deduce which yields that . Hence, the sequence weakly converges to a solution of ESEP (5), which completes the proof.

In the following, we introduce another simultaneous iterative algorithm with internal perturbation projection to solve ESEP (5) and prove the strong convergence of the algorithm.

Algorithm 13. Initialization: let arbitrary.
Iteration step: for a given current iterate , we calculate the next iterate by where is the same as Algorithm 9,

Theorem 14. Let be a real Hilbert space. For , let be nonempty closed convex subsets of real Hilbert spaces , respectively. are bounded linear operators with their adjoint operators , respectively. If , and then the sequence generated by Algorithm 13 converges strongly to a solution of ESEP (5) denoted by . Furthermore, for and as .

Proof. Let , which means Let Similar to the proof of the Theorem 12, we have the following inequality From Lemma 3, we have By induction, we can get which implies that sequence is bounded by setting . In the sequel, we divide the proof into two cases.

Case 1. Assume that there exists integer such that is decreasing sequence for all , which implies exists. Consequently, (45) can be rewritten as Similar to the proof of Theorem 12, taking limit in the two sides of (47) for , , we can obtain For another, since is bounded, has a convergent subsequence. Without loss of generality, we have . From (42), for we have which implies is bounded. Since , for we can infer Combining (49) and (50), we have Similar to the proof of Theorem 12, we can get . Since is asymptotically regular, we can get . Depending on the property of the projection, we have which combining with Lemma 6 and (52), we conclude the sequence converges strongly to

Case 2. Assume is not a decreasing sequence. Then suppose that there exists a subsequence of , for such that . For all (such that ), define a integer sequence as follows: It is clear that is a nondecreasing sequence satisfying , and
Similar to the proof of Case 1, we can obtain Taking the limit in the two sides of (55) for , according to Case 1, we have That implies . Depending on the property of the projection, we have From (53), we can get Since , we have which implies , that is, By (58), we obtain which implies By Lemma 5, we have , which means the sequence converges strongly to Thus, we complete the proof.

4. Iterative Algorithm for ESEFPP

In this section, we introduce a simultaneous iterative algorithm with internal perturbation projection to solve ESEFPP (9) and define the solution set of ESEFPP (9) as

Algorithm 15. Initialization: let , arbitrary.
Iteration step: for a given current iterate , we calculate the next iterate by where the stepsize is chosen in such a way that if , then where the index set small enough and , set and go to (63). Otherwise, for the iteration stops.

Theorem 16. Let be a real Hilbert space. For , let be bounded linear operators with their adjoint operators , respectively. And are quasinonexpansive mappings and demiclosed at 0. Then, the sequence generated by Algorithm 15 weakly converges to a solution of ESEFPP (9). Furthermore, for , and as .

Proof. Let and For from (63) and Lemma 3 (2), we have It follows from then (66) can be rewritten as Since are quasinonexpansive mappings, by (68), we have Note that Summing (69) for from to , we can obtain Let . According to the definition of , we have Therefore, the sequence is a nonincreasing sequence and lower bounded by 0. As a result converges to some finite limit, suppose as , and sequence is bounded. Letting and taking limit in the two sides of (71) for , we obtain that Since , we have From the definition of we have which implies Now, let us prove that is asymptotically regular. Indeed, from we have Consequently, which yields is asymptotically regular.
Let then there exists a subsequence of which weakly converges to . For we can obtain Due to the demiclosednesses of at and , it yields , which implies . Furthermore, it follows from (74) and the lower semicontinuity of the squared norm, we have where which means that That is, Obviously, Finally, following the proof of Theorem 12, we can prove the uniqueness of the weak cluster points of . Thus, we complete the proof.

Similar to the proof of Theorem 16, from Definition 7 and Lemma 8, we have the following corollary.

Corollary 17. Let be a real Hilbert space. For , let be bounded linear operators with their adjoint operators and be nonempty closed convex subsets of real Hilbert spaces . Let be a finite family of -strictly pseudononspreading mappings of into itself with and For let where and Assume that for and for and If are the -mapping generated by and and demiclosed at , then the sequence generated by Algorithm 15 weakly converges to a solution of ESEFPP (9). Furthermore, for , , and as .

5. Application

In this section, we apply our iterative algorithms to some convex and nonlinear problems, for example, [11, 28].

5.1. Split Equality Feasibility Problem

Note that ESEP (5) reduces to SEFP (2) as . Then, we can obtain the following algorithm to study SEFP (2) from Algorithm 9.

Algorithm 18. Initialization: let arbitrary.
Iteration step: for a given current iterate , we calculate the next iterate by where the stepsize is chosen as Algorithm 9. Set and go to (83). Otherwise, for , iteration stops.

5.2. Split Equality Fixed Point Problem

Note that ESEFPP (9) reduces to SEFPP (7) as . Then, we can obtain the following algorithm to study SEFPP (7) from Algorithm 15.

Algorithm 19. Initialization: let , arbitrary.
Iteration step: for a given current iterate , we calculate the next iterate by where the stepsize is chosen as Algorithm 15. Set and go to (84). Otherwise, for iteration stops.

5.3. Extend Split Equation Problem

Similar to Algorithm 15, we have another algorithm to solve ESEP (5).

Algorithm 20. Initialization: let , arbitrary.
Iteration step: for a given current iterate , we calculate the next iterate by where the stepsize is chosen as Algorithm 9. Set and go to (85). Otherwise, for iteration stops.

6. Numerical Examples

We are in a position to show some numerical examples to demonstrate the performance and convergence of Algorithms 9 and 13. The whole programs are written in MATLAB 2017b. All the numerical results are carried out on a personal Lenovo computer with Intel(R) Core(TM) i7-7500U CPU 2.70 GHz RAM 4.00 GB. We denote the vector with all elements 1 by in what follows.

Example 21. Let to find such that

Example 22. Let where are all generated randomly.

In the Example 22, we consider , and four initial values.

Case 1.

Case 2.

Case 3.

Case 4.

We take , , in Algorithms 9 and 13. In the following tables and figures, we denote Algorithm 5.4 in reference [19] by ECCL. And we set “” and “” to express the number of iteration and CPU time in seconds, respectively. We use as the stop criterion, where . The numerical results can be seen from Table 1 and Figures 1–5.

We take the initial point , , and in the Example 21. Figures 1–3 express the behaviors of , and , for , of Algorithms 9 and 13 and ECCl for Example 21, respectively.

Furthermore, for testing the stationary property of iterative number, we carry 500 experiments for different initial points which are presented randomly, such as in Example 21 for Algorithm 13 and ECCL, the results can be found in Figures 4 and 5, respectively.

Table 1 shows the numerical results of Example 22 when Algorithms 9 and 13 and ECCL select different initial points and matrix dimensions.

From Figures 1–3 and Table 1, we can see that the iterative numbers of Algorithms 9 and 13 are less than ECCL. From Figures 4 and 5 and Table 1, we can see that the stationary property of the iterative number and CPU time of Algorithm 13 are exceeded by ECCL. And Algorithm 2 is better than Algorithm 13 in these aspects.

Data Availability

All data generated or analysed during this study are included in this manuscript.

Conflicts of Interest

The authors declare that they have no competing interests regarding the present manuscript.

Authors’ Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Acknowledgments

This project is supported by the Natural Science Foundation of China (Grant Nos. 11401438 and 11571120) and Shandong Provincial Natural Science Foundation (Grant Nos. ZR2017LA002 and ZR2019MA022).

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Copyright © 2020 Meixia Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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