Abstract
In this paper, we are concerned with the multiple-sets split common fixed-point problems whenever the involved mappings are demicontractive. We first study several properties of demicontractive mappings and particularly their connection with directed mappings. By making use of these properties, we propose some new iterative methods for solving multiple-sets split common fixed-point problems, as well as multiple-sets spit feasibility problems. Under mild conditions, we establish their weak convergence of the proposed methods.
1. Introduction
The split common fixed-point problem (SCFP) requires finding an element in a fixed-point set such that its image under a linear transformation belongs to another fixed-point set. Formally, it consists in finding such thatwhere is a bounded linear mapping from a Hilbert space into another Hilbert space , and and are respectively the fixed-point sets of nonlinear mappings and . Specially, if and are both metric projections, then problem (1) is reduced to the well-known split feasibility problem (SFP) [1]. Actually, the SFP can be formulated as finding such thatwhere and are nonempty closed convex sets, and mapping is as above. These two problems recently have been extensively investigated since they play an important role in various areas including signal processing and image reconstruction [2–6].
We assume throughout the paper that problem (1) is consistent, which means that its solution set is nonempty. Censor and Segal [7] studied problem (1) when and are directed mappings. In this situation, they proposed the following method:where is the conjugate of , stands for the identity mapping, and is a properly chosen stepsize. It is shown that if is chosen in , then (7) converges weakly to a solution of (1). Subsequently, this result was extended to more general cases (see, e.g., [8–17]). Since the choice of the stepsize is related to , thus to implement (7), one has to compute (or at least estimate) the norm , which is generally not easy in practice. A way avoiding this is to adopt variable stepsize which ultimately has no relation with [9, 10, 18]. In this connection, Wang and Cui [10] proposed the following stepsize:On the other hand, Wang [19] proposed a new method:where is chosen such that
It is clear that the selection of stepsizes (8) and (6) does not rely on the norm , which in turn improves the performance of the original algorithm. Assume that and are both directed such that and are demiclosed at 0. It is shown that the sequence generated by (7) and (8) or (5) and (6) converges weakly to a solution of problem (1).
Now, let us consider the multiple-sets split common fixed-point problem (MSCFP) that is more general than the SCFP. Formally, it consists in finding such thatwhere and are two positive integers, is a bounded linear mapping from a Hilbert space into another Hilbert space , and and are respectively the fixed-point sets of nonlinear mappings and . Specially, if these nonlinear mappings are all metric projections, problem (7) is reduced to the well-known MSFP [20]. Actually, it can be formulated as the problem of finding such thatwhere and are two positive integers, is as above, and and are two classes of nonempty convex closed subsets.
Inspired by the works mentioned above, we are aimed to introduce and analyze iterative methods for solving the MSCFP in Hilbert spaces. We first study several properties of demicontractive mappings and especially find its connection with the directed mapping. By making use of these properties, we propose a new iterative algorithm for solving the MSCFP, as well as MSFP. Under mild conditions, we obtain the weak convergence of the proposed algorithm. Our results extend the related works from the case of two-sets to the case of multiple-sets.
2. Preliminary
Throughout the paper, assume that are real Hilbert spaces, and denotes its fixed-point set of a mapping . The following formula plays an important role in the subsequent analysis.
Lemma 1 (see [21]). Let and . It then follows that
We next recall the definition of several important classes of nonlinear mappings.
Definition 1 (see [21]). Let be a mapping from into .(i) is nonexpansive if(ii) is firmly nonexpansive if(iii) is -strictly pseudocontractive if
Definition 2 (see [21]). Let be a mapping with .(i) is quasinonexpansive if(ii) is directed if(iii) is -demicontractive if
It is clear that a directed mapping is -demicontractive, while a quasinonexpansive mapping is 0-demicontractive. It is also clear that a firmly nonexpansive mapping is -strictly pseudocontractive, while a nonexpansive mapping is 0-strictly pseudocontractive.
It is well known that a mapping is firmly nonexpansive if and only if is nonexpansive (cf. [21]). Analogously, we can easily get the following lemma, which presents a characteristic of directed mappings by using quasinonexpansive mappings.
Lemma 2. A mapping is directed if and only if is quasinonexpansive.
We now study properties of demicontractive mappings.
Lemma 3 (see [22]). Let be -demicontractive with . Then, the following hold.(i);(ii).
Lemma 4. For each , assume that is -demicontractive with . Let , where . If is nonempty, then
Proof. We first show . Pick . It then follows thatSince is chosen arbitrarily, we have .
It suffices to show that . Fix and choose any . Since and is -demicontractive, we haveThus, . Since , we have for all . Moreover, since is chosen arbitrarily, we get . Hence, the proof is complete.
Lemma 5. For each , assume that is -demicontractive with . Let , where . If is nonempty, then is directed. Moreover, if for each , is demiclosed at 0, then is also demiclosed at 0.
Proof. By Lemma 4, we have . By Lemma 2, it suffices to show that is quasinonexpansive. To this end, fix any . By Lemma 1 and the property of demicontractions thathence for all . It then follows thatThus, is quasinonexpansive, which implies is directed.
Let us now prove the second assertion. By Lemma 4, we have . Let be such that and as . Fix . Since is -demicontractive, we haveSince , we have , which, by our hypothesis, implies for all , that is, . By Lemma 4, the proof is complete.
Finally, we end this section by recalling two weak convergence theorems of iterative methods for approximating a solution of the two-sets SCFP (1).
Theorem 1 (see [10], Theorem 3.1). (Assume that and are both directed such that and are both demiclosed at 0. Then, the sequence , generated by (7) and (8), converges weakly to a solution of problem (1).
Theorem 2 (see [19], Theorem 3.4). Assume that and are both directed such that and are both demiclosed at 0. Then, the sequence , generated by (5) and (6), converges weakly to a solution of problem (1).
3. The Case for Demicontractive Mappings
In this section, we are concerned with the multiple-sets split common feasibility problem and we assume that (7) is consistent, which means that its solution set is nonempty. First, motivated by (7) and (8), we propose the first algorithm for solving problem (7).
Algorithm 1. Let be arbitrary and choose with , with . Given , update the next iteration viawhere if ; otherwise,
Theorem 3. Assume that and are respectively and -demicontractive such that and are demiclosed at 0 for and . Then, the sequence , generated by Algorithm 1, converges weakly to a solution of (7).
Proof. Let and . Thus, we can rewrite Algorithm 1 aswhere if ; otherwise,By Lemma 5, and are both directed such as and are demiclosed at 0. It then follows from Theorem 1 that weakly converges to a point that satisfies and . Moreover, by Lemma 4, we conclude that and , that is, is a solution of problem (7).
Motivated by (5) and (6), we propose the second algorithm for solving problem (7).
Algorithm 2. Let be arbitrary and choose with , with . Given , ifthen stop; otherwise, update the next iteration viawhere
Theorem 4. Assume that and are respectively and -demicontractive such that and are demiclosed at 0 for and . Then, the sequence , generated by Algorithm 2, converges weakly to a solution of (7).
Proof. Let and . Thus, we can rewrite Algorithm 2 as , whereBy Lemma 5, and are both directed such as and are demiclosed at 0. It then follows from Theorem 2 that weakly converges to a point that satisfies and . Moreover, by Lemma 4, we conclude that and , that is, is a solution of problem (7).
4. Multiple-Sets Split Feasibility Problem
In this section, we apply the previous result to approximate a solution of the multiple-sets split feasibility problem (MSFP). Also, we assume that problem (8) is consistent, which means that its solution set is nonempty. By applying Algorithm 1, we obtain the first algorithm for solving (8).
Algorithm 3. Let be arbitrary and choose with , with . Given , update the next iteration viawhere if ; otherwise,
Theorem 5. The sequence , generated by Algorithm 3, converges weakly to a solution of (2).
Proof. It suffices to notice that both and are -demicontractive, which implies for all . Applying Theorem 3 yields the desired assertion.
Next, we propose the second algorithm for solving (8) by applying Algorithm 2.
Algorithm 4. Let be arbitrary and choose with , with . Given , ifthen stop; otherwise, update the next iteration viawhere
Theorem 6. The sequence , generated by Algorithm 4, converges weakly to a solution of (8).
Proof. It suffices to notice that both and are -demicontractive, which implies for all . Applying Theorem 4 yields the desired assertion.
5. Conclusion
In this paper, we consider the MSCFP whenever the involved mappings are demicontractive. We obtained several properties of demicontractive mappings and particularly their connection with directed mappings. These properties enable us to propose some new iterative methods for solving MSCFP, as well as MSFP. Under mild conditions, we establish their weak convergence of the proposed methods. Our results extend the existing works from the case of two-sets to the case of multiple-sets.
Data Availability
No data were used to support this study.
Conflicts of Interest
The author declares no conflicts of interest.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Grant No. 12101286).