Journal of Function Spaces
Volume 2016 (2016), Article ID 4093524, 7 pages
http://dx.doi.org/10.1155/2016/4093524

Research Article

Strong Convergence Theorems for an Implicit Iterative Algorithm for the Split Common Fixed Point Problem

Huimin He,¹ Sanyang Liu,¹ and Rudong Chen²

¹School of Mathematics and Statistics, Xidian University, Xi’an 710071, China
²Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China

Received 13 September 2016; Revised 11 November 2016; Accepted 21 November 2016

Academic Editor: Ismat Beg

Copyright © 2016 Huimin He 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.

Abstract.

The aim of this paper is to construct a novel implicit iterative algorithm for the split common fixed point problem for the demicontractive operators , , and  , , where , and we obtain the sequence which strongly converges to a solution of this problem, and the solution satisfies the variational inequality. , , where denotes the set of all solutions of the split common fixed point problem.

1. Introduction

The split feasibility problem (SFP) is to find a point

where is a nonempty closed convex subset of a Hilbert space , is a nonempty closed convex subset of a Hilbert space , and is a bounded linear operator.

This problem was proposed by Censor and Elfving [1] in 1994.

Since the SFP can extensively be applied in fields such as intensity-modulated radiation therapy, signal processing, and image reconstruction, then the SFP has received so much attention by so many scholars; see [2–23].

In 1994, Censor and Elfving [1] proposed the original algorithm in ,

where and are nonempty closed convex subsets of , in the finite-dimensional is a matrix, and is the projection operator from onto .

As we know, the computation of the inverse is not easy if the inverse of existed. So, the algorithm (2) does not become popular.

In 2002 and 2004, Byrne [2, 3] gave the so-called algorithm as follows:

where with taken as the largest eigenvalue of the operator and and denote the projection operators from and onto the sets , , respectively.

For the stepsize of algorithm (3) is fixed and closely related to spectral radius of , then the projection operators and are not easily calculated usually.

The split common fixed point problem (SCFP) is to find a point

where and , and and denote the fixed point sets of and .

This problem was proposed by Censor and Segal [12] in 2009. Note that the SCFP is closely related to SFP and it is a particular case of SFP.

In 2009, Censor and Segal [12] introduced the original algorithm for directed operators as follows:

where the step size satisfies , and they obtained that weakly converges to a solution of the SCFP (4) if the solution of SCFP exists. But it is obvious that the choice of the step size depends on the norm of operator, , which is the disadvantage of this algorithm.

The next two years, some extension results on the operators are obtained, such as Moudafi (2010) [24], Moudafi (2011) [25], and Wang and Xu (2011) [14].

In order to overcome this disadvantage, Cui and Wang [26] proposed the following algorithm in 2014:

where and the step size is chosen by the following way:

and they proved that the sequence converges weakly to a solution of the SCFP (4). Note that the advantage of this algorithm is that the step size searches automatically and does not depend on the norm of operator .

Recently, Byrne et al. [27] introduced the split common null point problem (SCNPP) for set-valued maximal monotone mappings in Hilbert spaces. Given set-valued mappings , , and , , respectively, and the bounded linear operators , , the SCNPP is formulated as follows:

As we know, the SCNPP generalizes the split common fixed point problem and the split variational inequality problem [28, 29].

Motivated by the viscosity idea of [30], in this paper, we construct a novel algorithm for demicontractive operators to approximate the solution of the SCFP (4), that is, the following implicit iterative algorithm:

where and the step size is also chosen as (7).

The research highlight of this paper is that the strong convergence of the SCFP (4) is constructed; that is to say the sequence generated by (9) converges strongly to a solution of the SCFP.

2. Preliminaries

Throughout this paper, we denote the set of all solutions of the SCFP (4) by . We use to indicate that converges weakly to . Similarly, symbolizes the sequence which converges strongly to .

Let , , and be Hilbert spaces endowed with the inner product and norm , and and are nonempty closed convex subsets of and , respectively.

Some concepts and lemmas are given in the following and they are useful in proving our main results.

Definition 1. A operator is said to be
(i) nonexpansive if

(ii) quasi-nonexpansive if

(iii) directed if

(iv) -demicontractive with if

Note that (12) is equivalent to

Definition 2. Let be an operator, then is said to be demiclosed at zero, if for any in , the following implication holds

As we know, the nonexpansive mappings are demiclosed at zero [31].

Definition 3. Let be a nonempty closed convex subset of a Hilbert space , the metric (nearest point) projection from to is defined as follows: Given , is the only point in with the property

Lemma 4 (see [32]). Let be a nonempty closed convex subset of a Hilbert space , is a nonexpansive mapping from onto and is characterized as: Given , there holds the inequality

Lemma 5 (see [32]). Let be a Hilbert space, then the following inequality holds,

Lemma 6 (Cui and Wang [26]). Let be a bounded linear operator and a -demicontractive operator with . If , then

(a), .(b)In addition, for

 where , and

Lemma 7 (Cui and Wang [26]). Let be a -demicontractive operator with . Denote for . Then for any and ,

3. Main Results

Proposition 8. Based on the definitions in preliminaries, the classes of -demicontractive operators, directed operators, quasi-nonexpansive operators, and nonexpansive operators have close relations. We can visually use the following Venn diagram (Figure 1) to denote their relations.

Figure 1: The relations of -demicontractive operator, directed operator, quasi-nonexpansive operator, and nonexpansive operator.

Proof. From Definition 1, the following conclusion is obtained easily.

(i)The nonexpansive operator is quasi-nonexpansive operator.(ii)The quasi-nonexpansive operator is -demicontractive operator.(iii)The directed operator is -demicontractive operator.

Next, we give the novel implicit algorithm to solve the SCFP (4) for demicontractive operators. In the sequel, the assumptions are given as follows:

(i) is a -demicontractive operator with .(ii) is a -demicontractive operator with .(iii)Both and are demiclosed at zero.

Algorithm 9. Choose an initial guess arbitrarily. Let be a fixed contraction on with coefficient (), . Assume that the th iteration has been constructed. Then the th iteration is via the following formula:

where is the adjoint of bounded linear operator and the step size is chosen in the following way:

Theorem 10. Assume the solution set of the SCFP (4) . If satisfies and , then the sequence generated by implicit algorithm (22) converges strongly to a point , and ; that is, satisfies the following variational inequality:

Proof. The proof is divided into three steps.
Step 1. We show that is bounded.
Denote and take ; it follows from (22) that

(i) If . Then ; from (21), we get

Thus

Hence

So, is bounded, so is .
(ii) If . It follows from (19) and (21) that we get

Thus,

Combining with (30) and (25), we get (28). So, is bounded, so is .
Step 2. We show that there exists a subsequence such that as , and solves the variational inequality (24).
By the reflexivity of Hilbert space and the boundedness of , there exists a weakly convergence subsequence such that , as .
First, we show that , as .
Next, we denote by .
(i) If . From (18) and (21), we have

Hence

So

For as , the above inequality implies that

(ii) If . From (18), (19) and (21), we have

Then, (33) is obtained. By the similar proofs of the case of , we conclude that

Second, we show that
(i) If . From (31), we get

Hence

For the case , then it is clear we obtain

From (37) and the demiclosedness of at zero, then

Since is bounded linear operator, then is weak continuity; then

From (39) and the demiclosedness of at zero, then

Hence, by (40) and (42).
(ii) If . From (35), we get

So, we have

Take , we have

Moreover,

Hence, from (46)

By , we have

So,

For , then by (50).
From (45) and the demiclosedness of at zero, then

From (48) and the demiclosedness of at zero, then

So, by (51) and (52).
Third, we show that solves the variational inequality (24).
Indeed, from (22), we get

The above equality and (30) imply that

Since

take the limit through and we obtain

Step 3. We show that as .
To show that as , we only need to show that any subsequence of converges strongly to .
Assuming the above conclusion does not hold, that is to say, there exists another subsequence , which converges strongly to as . Similarly, we know solves the variational inequality

Replacing with in (56) and replacing with in (57), we obtain

Adding up the above variational inequality yields

Thus . This is contradicting with the assumption , so converges strongly to .
The proof is completed.

4. Applications

In this section, we consider some special cases as the applications of Theorem 10.

Based on the relations of -demicontractive operators, directed operators, and quasi-nonexpansive operators (Proposition 8), the following corollaries are obtained easily.

Corollary 11. Let and be quasi-nonexpansive operators and and be demiclosed at zero. Assume the SCFP (4) is consistent . If satisfies and , then the sequence generated by implicit algorithm (22) converges strongly to a point , and ; that is, satisfies the following variational inequality (24).

Corollary 12. Let and be directed operators and and be demiclosed at zero. Assume the SCFP (4) is consistent . If satisfies and , then the sequence generated by implicit algorithm (22) converges strongly to a point , and ; that is, satisfies the following variational inequality (24).

Corollary 13. Let be a directed operator, be a quasi-nonexpansive operator, and and be demiclosed at zero. Assume the SCFP (4) is consistent . If satisfies and , then the sequence generated by implicit algorithm (22) converges strongly to a point , and ; that is, satisfies the following variational inequality (24).

Corollary 14. Let be a directed operator, be a -demicontractive operator, and and be demiclosed at zero. Assume the SCFP (4) is consistent . If satisfies and , then the sequence generated by implicit algorithm (22) converges strongly to a point , and ; that is, satisfies the following variational inequality (24).

Corollary 15. Let be a quasi-nonexpansive operator, be a -demicontractive operator, and and be demiclosed at zero. Assume the SCFP (4) is consistent . If satisfies and , then the sequence generated by implicit algorithm (22) converges strongly to a point , and ; that is, satisfies the following variational inequality (24).

5. Conclusions

In this paper, the research highlights that the strong convergence of the SCFP (4) is constructed. We construct a novel implicit algorithm for demicontractive operator to solve the split common fixed points problem SCFP, and we prove that the sequence strongly converges to a solution of the SCFP. These results further complete the theory of the SCFP, and some relevant work can be extended in the future.

Competing Interests

The authors declare that they have no competing interests.

Authors’ Contributions

This work was carried out by the three authors, in collaboration. Moreover, the three authors have read and approved the final manuscript.

Acknowledgments

This paper was funded by Fundamental Research Funds for the Central Universities (no. JB150703), National Science Foundation for Young Scientists of China (no. 11501431), and National Science Foundation for Tian Yuan of China (no. 11426167).

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