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Research Article | Open Access
Meixia Li, Xueling Zhou, Wenchao Wang, "Internal Perturbation Projection Algorithm for the Extended Split Equality Problem and the Extended Split Equality Fixed Point Problem", Journal of Function Spaces, vol. 2020, Article ID 6034754, 15 pages, 2020. 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
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.
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 , 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  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  proposed the following iterative algorithm: where with . They studied the weak convergence of scheme (4).
Recently, Che et al.  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  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  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.  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.
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 . In the real Hilbert space , for , the following relations hold: (1)(2)(3)(4) for
Definition 4 . A mapping is said to be demiclosed at if, for any sequence which converges weakly to and with
Lemma 5 . 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 . 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 . 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 . 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
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.
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
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
Since , then
From the definition of , we have
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 .