Weak Convergence Theorems on the Split Common Fixed Point Problem for Demicontractive Continuous Mappings

Abstract

We are concerned with the split common fixed point problem in Hilbert spaces. We propose a new method for solving this problem and establish a weak convergence theorem whenever the involved mappings are demicontractive and Lipschitz continuous. As an application, we also obtain a new method for solving the split equality problem in Hilbert spaces.

1. Introduction

The split common fixed point problem (SCFP) is an inverse problem that aims to find an element in a fixed point set such that its image under a linear transformation belongs to another fixed point set. More specifically, given two Hilbert spaces and , the SCFP consists in finding such thatwhere is a bounded linear mapping and and are, respectively, the fixed point sets of nonlinear mappings and . Particularly, if and are both metric projections, the SCFP is reduced to the well-known split feasibility problem (SFP). Actually, the SFP can be formulated as the problem of finding a point such thatwhere and are nonempty closed convex sets, and mapping is as above. These two problems have been extensively investigated since they play an important role in various areas including signal processing and image reconstruction [1–5].

We assume throughout the paper that problem is consistent, which means that its solution set, denoted by , is nonempty. Censor and Segal [6] studied the SCFP when and are firmly quasi-nonexpansive mappings, and proposed the following method:where is a properly chosen stepsize. It is shown that if is chosen in , then the sequence generated by method (2) converges weakly to a solution of problem . Subsequently, this result was extended to quasi-nonexpansive operators [7], demicontractive operators [8, 9], two groups of finitely many firmly quasi-nonexpansive mappings [10, 11], and the more general common null point problem [12]. Also, some variants of method (2) have been considered in [13–16].

Since the choice of the stepsize is related to , thus, to implement method (3), one has to compute (or at least estimate) the norm , which is generally not easy in practice. A way to avoid this is to adopt variable stepsize which ultimately has no relation with [8, 10, 17–19]. Wang [18] recently proposed a new method for solving the SCFP:where is chosen such thatWang proved that if mappings and are firmly quasi-nonexpansive, then the sequence generated by (3)-(4) converges weakly to a solution of problem . Wang and Xu [19] recently proposed another choice of the stepsize:where is chosen such thatThey proved that if mappings and are nonexpansive, then the sequence generated by (3) and (5)-(6) converges weakly to a solution of problem . It is clear that these choices of the stepsize do not rely on the norm .

In this paper, we first extend the above result for method (3) from nonexpansive mappings to demicontractive continuous mappings. By using properties of product spaces, we change the split equality problem into a special split common fixed point problem. As a result, based on our extension, we obtain a new method for solving the split equality problem in Hilbert spaces.

2. Preliminaries

Throughout the paper, , are Hilbert spaces, is the identity operator, “" stands for strong convergence, and “" stands for weak convergence. For a mapping , is the set of the fixed points of , , and .

Definition 1. Let be a nonlinear mapping. (i) is called firmly nonexpansive, if (ii) is called nonexpansive, if (iii) is called strictly pseudo-contractive, if there exists such that (iv) is called -Lipschitz continuous, if there exists such that

Definition 2. Let be a nonlinear mapping with . (i) is called firmly quasi-nonexpansive, if (ii) is called quasi-nonexpansive, if (iii) is called k-demicontractive, if there exists such that

Note that the class of strictly pseudo-contractive mappings properly includes the class of nonexpansive mappings, while the class of nonexpansive mappings properly includes the class of firmly nonexpansive mappings. And the class of demicontractive mappings properly includes the class of quasi-nonexpansive mappings, while the class of quasi-nonexpansive mappings properly includes the class of firmly quasi-nonexpansive mappings.

Definition 3 (demiclosedness property). Let be a sequence in and be a mapping. Then is said to have demiclosedness property if the following implication holds: It is known that strictly pseudo-contractive mappings possess the demiclosedness property [20]. In particular, both nonexpansive and firmly nonexpansive mappings possess such a property.

Lemma 4 (see [20]). Let be a -strictly pseudo-contractive mapping. Then is demicontractive and Lipschitz continuous and moreover has the demiclosedness property.

The metric projection from onto a nonempty closed convex subset is defined by which is characterized byIt is well known that the metric projection is firmly nonexpansive.

Definition 5. Let be a nonempty closed convex subset in . (i)A sequence in is Fejér-monotone with respect to if (ii)A sequence in is quasi Fejér-monotone with respect to if  where satisfies

Lemma 6 (see [21]). A quasi Fejér-monotone sequence (with respect to ) is weakly convergent to if and only if every weak cluster point of belongs to .

Lemma 7 (see [22]). Let and be positive real sequences such that If , or , then the limit of the sequence exists.

3. The Case for Demicontractive Continuous Mappings

In this section, we consider the SCFP for demicontractive continuous mappings. Under this situation, we shall prove that the sequence generated by (3) and (5)-(6) still converges weakly to a solution of problem .

Lemma 8. Let , , and , where , , and are mappings defined in . Assume that is -demicontractive and -Lipschitz continuous, is -demicontractive and -Lipschitz continuous, and and are demiclosed at the origin. For any , we have the following: (i).(ii).(iii) is -Lipschitz continuous with .(iv)If and as , then

Proof. (i) It is readily seen that . To see the converse, let and fix any . Since and are demicontractive, we have Adding up these two inequalities, we havewhich yields , that is, . This implies .
(ii) Let . It follows from (20) that which yields the desired inequality.
(iii) Let We have (iv) We note that is bounded by its weak convergence. By inequality (20), we havewhich implies that Since , this by the demiclosedness property implies On the other hand, for any , we have Hence , which yields . Altogether, .

Theorem 9. Let . Assume that is -demicontractive and -Lipschitz continuous, is -demicontractive and -Lipschitz continuous, and and are demiclosed at the origin. If condition (6) holds, then the sequence , generated by (3) and (5), converges weakly to a solution of problem .

Proof. Let , , and . It then follows from Lemma 8 that By (5), we havein particular, By our hypothesis (6), this implies that is quasi Fejér-monotone with respect to .
Next, we deduce from (27) and the boundedness of (guaranteed by the quasi-Fejér-monotonicity) that Thus, we haveIn view of (6), this impliesOn the other hand, since then we have In light of (30) and (6), we have By Lemma 7, exists, and further we have by (31). Hence, by Lemma 8, we conclude that every weak cluster point of belongs to .
Finally, we deduce from Lemma 6 that converges weakly to a solution of problem .

Corollary 10. Assume that and are both strictly pseudo-contractive mappings. If condition (6) holds, then the sequence , generated by (3) and (5), converges weakly to a solution of problem .

Proof. It follows from Lemma 4 and Theorem 9.

Remark 11. It is readily seen that the above result holds true for nonexpansive and firmly nonexpansive mappings. As a result, it extends the results in [19] from nonexpansive mappings to demicontractive continuous mappings.

4. New Methods for the Split Equality Problem

The split equality problem (SEP) is an inverse problem that requests findingwhere and are two bounded linear mappings, while and are two nonlinear mappings. The SEP was first introduced by Moudafi and Al-Shemas [23], and they proposed the following iterative method: Under some certain conditions, they proved the weak convergence of the iterative sequence generated by method (34).

Our method is actually motivated by (3), since problem can be regarded as a special SCFP: find such thatwhere , , and is the projection onto the set Motivated by (3), we now propose a new method for solving problem . For an arbitrary initial guess , define recursively bywhere is chosen as

In what follows, we will show the SEP amounts to problem (35). Now consider the product space , in which the inner product and the norm are, respectively, defined by where , with ,

Lemma 12. Let , , and and be as in problem . Define a mapping by Then we have the following: (i) is a bounded linear mapping.(ii).

Proof. (i) Let . Since and are both linear, we have on the other hand, we have which implies . Thus is linear and bounded.
(ii) For , we have This gives , which implies that Hence the lemma is proved.

Lemma 13. Assume that is -demicontractive and -Lipschitz continuous, is -demicontractive and -Lipschitz continuous, and and are demiclosed at the origin. Define a mapping by where is in with Then (i);(ii) is demicontractive and Lipschitz continuous;(iii) is demiclosed at the origin.

Proof. It is easy to check (i). For (ii), fix It follows that which implies that is demicontractive. On the other hand, we have where is in with , This implies that is Lipschitz continuous.
To show (iii), let be a sequence such that it converges weakly to and converges strongly to . This implies that and , which, by the demiclosedness of , gives . Similarly, we have , so that So the lemma is proved.

Theorem 14. Assume that and are two demicontractive and Lipschitz continuous mappings such that and are demiclosed at the origin. If condition (6) is fulfilled, then the sequence generated by (36), (37), and (48) converges weakly to a solution of problem .

Proof. Let , , be defined as in the previous lemmas, and be the projection onto the set Then method (36) can be rewritten as whereNote that is firmly nonexpansive. By Lemma 13, all assumptions in Theorem 9 are fulfilled. Hence, by applying Theorem 9, we conclude that converges weakly to some such that and , which clearly yields , , and . Hence the theorem is proved.

5. Conclusions

We studied the split common fixed point problem in Hilbert spaces. We proposed a new method for solving such a problem and established a weak convergence theorem whenever the involved mappings are demicontractive and continuous. We also obtained a new method for solving the split equality problem in Hilbert spaces.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by Program for Science and Technology Innovation Talents in the Universities of Henan Province (Grant no. 15HASTIT013).