Journal of Function Spaces
Volume 2017 (2017), Article ID 5306802, 11 pages
https://doi.org/10.1155/2017/5306802

Research Article

Weak and Strong Convergence Theorems for the Multiple-Set Split Equality Common Fixed-Point Problems of Demicontractive Mappings

Yaqin Wang,¹ Tae-Hwa Kim,² and Xiaoli Fang¹

¹Department of Mathematics, Shaoxing University, Shaoxing 312000, China
²Department of Applied Mathematics, College of Natural Sciences, Pukyong National University, Busan 48513, Republic of Korea

Correspondence should be addressed to Tae-Hwa Kim; taehwa@pknu.ac.kr

Received 29 March 2017; Accepted 19 October 2017; Published 24 December 2017

Academic Editor: Xinguang Zhang

Copyright © 2017 Yaqin Wang 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.

We consider mixed parallel and cyclic iterative algorithms in this paper to solve the multiple-set split equality common fixed-point problem which is a generalization of the split equality problem and the split feasibility problem for the demicontractive mappings without prior knowledge of operator norms in real Hilbert spaces. Some weak and strong convergence results are established. The results obtained in this paper generalize and improve the recent ones announced by many others.

1. Introduction

For modeling inverse problems which arise from phase retrieval and in medical image reconstruction, in 1994, Censor and Elfving [1] firstly introduced the following split feasibility problem (SFP) in finite-dimensional Hilbert spaces.

Let and be nonempty closed convex sets of the Hilbert spaces and , respectively, and let be a bounded linear operator. The split feasibility problem (SFP) is formulated to

It has been found that the SFP can be used in many areas such as image restoration, computer tomograph, and radiation therapy treatment planning. Some methods have been proposed to solve split feasibility problems; see, for instance, [2–5].

Assuming that the SFP (1) has a solution, it is not hard to see that

where and are the metric projections from onto and from onto , respectively, is a positive constant, and denotes the adjoint of . This implies that SFP can be solved by using fixed-point algorithms.

In 2013, Moudafi and Al-Shemas [6] introduced the following new split feasibility problem, which is called the split equality fixed-point problem (SEFP). Let be real Hilbert spaces, let and be two bounded linear operators, and let and be two firmly quasi-nonexpansive mappings. The SEFP in [6] is to

where and denote the sets of the fixed points of mappings and , respectively. The goal is to cover many situations, for instance, in decomposition methods for PDFs and applications in game theory and in intensity-modulated radiation therapy (IMRT).

For solving the SEFP (3), Moudafi and Al-Shemas [6] introduced the following simultaneous iterative method:

for firmly quasi-nonexpansive mappings and , where stand for the spectral radii of and , respectively.

In 2014, Zhao [7] introduced the following simultaneous Mann iterative algorithm:

where the step size does not depend on the operator norms and . And she proved the weak convergence of this algorithm (5) to solve SEFP (3) governed by quasi-nonexpansive operators and .

Recently, the multiple-set split equality common fixed-point problem (MSECFP) studied by Zhao and Wang [8] is to

where are integers. They introduced the following two mixed iterative algorithms for solving the MSECFP (6) of quasi-nonexpansive mappings:

and

where the step size does not depend on the operator norms and . And they proved the weak convergence of such algorithms.

Very recently, Wang and Kim [9] introduced the following iterative scheme for solving SEFP (3) of demicontractive mappings,

and obtain a strong convergence result with no compactness assumptions on the spaces or the mappings and with no extra conditions on the fixed-point sets.

Inspired and motivated by the works mentioned above, we consider the mixed parallel and cyclic iterative algorithms for MSECFP (6) of demicontractive mappings which are a generalization of quasi-nonexpansive mappings without prior knowledge of operator norms in Hilbert spaces. Under some mild assumptions, we prove weak and strong convergence results of such algorithms for solving MSECFP (6).

2. Preliminaries

Throughout this paper, we always assume that are real Hilbert spaces and let and be the set of positive integers and real numbers, respectively. In what follows, we denote strong and weak convergence in the space by “" and “", respectively, and the set of the fixed points of a mapping by . Also, we use to represent weak -limit set of .

Let be a nonempty closed convex subset of a Hilbert space . The metric (or nearest point) projection from onto is defined as follows: given , the unique point satisfies the property

It is well known [10] that is a nonexpansive mapping and is characterized by the inequality

Definition 1. Let be a real Hilbert space. A mapping is said to be

(i)nonexpansive if ;(ii)quasi-nonexpansive if and if ;(iii)firmly nonexpansive if

or equivalently

(iv) firmly quasi-nonexpansive if and

(v) -demicontractive if and there exists a constant such that

Remark 2. Notice that every 0-demicontractive mapping is exactly quasi-nonexpansive. In particular, we say that it is quasi-strictly pseudocontractive [11] if . Moreover, if , every -demicontractive mapping becomes quasi-nonexpansive. Therefore, it is sufficient to only take in (v) of Definition 1 in Hilbert spaces. However, as seen in (iv) of Definition 1, every firmly quasi-nonexpansive mapping (often called to be a directed operator [12])) is obviously -demicontractive.
It is worth noting that the class of demicontractive mappings is more general than the class of quasi-nonexpansive mappings and the class of firmly quasi-nonexpansive mappings.

Definition 3. Let be a real Hilbert space. An operator is called demiclosed at origin for any sequence which converges weakly to , and if the sequence converges strongly to 0, then

As a special case of the demicloseness principle on uniformly convex Banach spaces given by [13], we know that if is a nonempty closed convex subset of a Hilbert space and is a nonexpansive mapping, then the mapping is demiclosed on . Now, the following question naturally arises: If is quasi-nonexpansive, is still demiclosed on ? The answer is negative even at 0 as follows.

Example 4 (see [9]; Example  2.11). The mapping   : is defined by

Then, is a quasi-nonexpansive mapping, but is not demiclosed at 0.

Definition 5. Let be a nonempty closed convex subset of a real Hilbert space . A sequence is Fejér-monotone relative to the target set (simply -Fejérian) if

Lemma 6 (see [14]). Let be a real Hilbert space. Then, for all and ,

Since the square of the norm is convex, we have the following lemma.

Lemma 7. Let be a real Hilbert space. Then,

for , and

Lemma 8 (see [15]). Let be -demicontractive with and set for . Then, is quasi-nonexpansive for and

Lemma 9 (see [11]; Proposition ). Assume that is a closed convex subset of a Hilbert space . Let be a mapping. If is a -demicontractive mapping (which is also called -quasi-strict pseudocontraction in [11]), then the fixed-point set is closed and convex.

Lemma 10 (see [16]). Let and be Banach spaces and be a continuous linear operator from to . Then, is weakly continuous.

Lemma 11 (see [17, 18]). Let be a nonempty closed convex subset of a real Hilbert space . Suppose that is -Fejérian. Then, if and only if .

Lemma 12 (see [19]; Lemma ). 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 sequence of integers as follows:

where such that . Then, there hold the following properties:

(i) and ;(ii) and

Lemma 13 (see [20]). Assume that is a sequence of nonnegative real numbers such that

where is a sequence in and is a sequence in such that

(a) and ;(b) or

Then, .

3. Main Results

In this section, we study two mixed parallel and cyclic iterative algorithms for MSECFP (6) of demicontractive mappings where the step sizes do not depend on the operator norms and , and we prove the weak and strong convergence of such algorithms. Denote the solution set of MSECFP (6) by ; that is,

Given two positive integers and , the -mod function takes values in the set and the -mod function takes values in the set as

for for some integers and , and

for for some integers and

Put . Define the inner product on as follows:

It is easy to see that is a real Hilbert space and

Note also that if is a sequence in , there holds the following relation:

Algorithm 14. Let be arbitrary and be integers. Let such that and . Assume that the th iterate has been constructed and , and then we calculate the th iterate via the formulas

Assume the step size is chosen in such a way that

for all and some small enough , where the index set ,

otherwise, being any nonnegative value). If , then we take and

Algorithm 15. Let be arbitrary and be integers. Let such that and . Assume that the th iterate has been constructed and , and then we calculate the th iterate via the formulas

Assume that are the same as in Algorithm 14 and for the step size is chosen as (30); otherwise, being any nonnegative value). If , then we take and

Now, we will see from Lemma 16 that is well defined and bounded. The proof of the following lemma is also added for completing the proof of Lemma in [7] for the sake of convenience.

Lemma 16. Assume the solution set of (6) is nonempty. Then, defined by (30) is well defined and bounded.

Proof. Take , that is, , , and We have

By adding the above two equalities and by taking into account, we obtain

Consequently, for , that is, , we have or . And since

we have

Thus, we can choose small enough . This causes to be well defined. From (30), we obtain that and is a fixed positive number, so is bounded.

Assumption 17. Assume that

(i)the solution set of (6) is nonempty;(ii) and are -demicontractive and -demicontractive, respectively;(iii) and are demiclosed at origin.

Let and . Clearly, is -demicontractive for all and is -demicontractive for all .

Lemma 18. Given two bounded linear operators and , let and be -demicontractive and -demicontractive, respectively. Assume that the solution set of (6) is nonempty. Then, is a nonempty closed convex set.

Proof. By Lemma 9, we have that and are both closed convex subsets, and since and are both linear, it is easy to see that is a closed convex subset in .

Lemma 19 (see [21]; Lemma ). Let be a bounded sequence of a Hilbert space . Let be a positive integer and . If and , then, for any , there exists a subsequence of depending on such that and .

Theorem 20. Let Assumption 17 be satisfied. Given two bounded linear operators and , if the following conditions are satisfied:

(a) and ,(b),

then the sequence generated by Algorithm 14 converges weakly to a solution of (6).

Proof. First, we claim that the sequence is -Fejérian. Indeed, taking , that is, , , and , by Algorithm 14, we obtain

Similarly, we have

It follows from the above inequalities and that

Taking , we have for every and . It follows from (29) that

Using Lemma 8 for any , we have

Thus, by (43) and (44) and Lemma 7, we obtain

By (ii) of Assumption 17 and from (29) and Lemma 6, we also have

Then, it follows from (42), (45), and (46) that

Hence, is -Fejérian.
Next, we show that, for all , , , , and . In fact, from (50), we see that the sequence is decreasing and bounded below by 0. Consequently, it converges to some finite limit. So, the sequences , , , and are also bounded. Then, from (49) and the assumption on in (30), it follows that

which together with (48) implies that

Also, from (47) and condition (a), it follows that

This, combined with condition (a), implies that, for any ,

It follows from (47) and condition (b) that

Since

it follows from Lemma 16 that is bounded, so by (52) we have

Similarly, we have

It follows from (55) and (58) that

Finally, we claim that . To this end, let . From (28), it follows that , . Since by (57), we use (54) and Assumption 17(iii) to derive that . Given fixed , since and , now we apply Lemma 19 to get a subsequence of , depending on , such that and for all . Moreover, it turns out that by (58) and

by virtue of (55). Using Assumption 17(iii) yields for any ; that is, . On the other hand, by Lemma 10, we have , which together with weakly lower semicontinuity of the norm implies that

that is, ; hence, . Since is obviously a closed convex set by Lemma 18 and we have shown that in is -Fejérian and , by Lemma 11, we conclude that the sequence generated by Algorithm 14 converges weakly to a point of .

Let be a metric projection from onto We denote the origins of and by and , respectively.

Theorem 21. Let Assumption 17 be satisfied. Given two bounded linear operators and , if the following conditions are satisfied:

(1) and ;(2);(3) and ,

then the sequence generated by Algorithm 15 converges strongly to a solution of (6).

Proof. Set . By (11), we readily see

Since , we have and such that Taking , we have for every and . Similar to the proof of (45) and (46), we obtain

It follows from (33) and (42) that

By (63), (64), and (66), we have

Now, by setting , from the above inequality, we obtain

By induction, we have

which implies that is bounded. Hence, ,, and are all bounded. By (63)–(65), we have

Then, it follows from (70) that

Now, we will divide the proof into two cases to establish strong convergence.
Case 1. Suppose that there exists such that for all . In this case, exists and then Hence, using (71) and the assumptions on , and , we obtain

The last equality together with condition (1) implies that

From (33) and (73), we have

Then, from (33), (75), and (76) and condition (3), we obtain

which together with (75) implies that

It follows from (33), (72), and (77) that

This, combined with (72), yields

Since is bounded in , and moreover

Indeed, for any , hence , by virtue of (28). Since by (79), it immediately follows from (75) and Assumption 17(iii) that . On the other hand, since , now we use Lemma 19 to derive that, for any , there exists a subsequence such that and for all . From (81), it results that . From (72), we have

Hence, it follows from (iii) of Assumption 17 that for any ; that is, . Mimicking the idea in the proof of Theorem 20, we conclude that ; hence, .
By (33), we observe

Similarly, we have

Then, from (42), (63), (64), (84), and (85), it follows that

By the boundedness of in , there exists a point and a subsequence of in such that and

Then, it immediately follows from (76), (77), and (82) that . Thus, and (87) and (62) yield

Letting , . From Lemma 13, (86) and (88), and the assumptions on , we conclude that the sequence converges strongly to .
Case 2. Suppose that there exists a subsequence such that for all In this case, let be a mapping for all (for some large enough) by

Then, we have from Lemma 12 that and, clearly, is a nondecreasing sequence such that as . From (71), we obtain

Conditions (1)–(3) imply

Now, using the sequence in place of the bounded sequence in and mimicking the proof of Case , we could obtain that

With the help of (86) and , we obtain

which implies that

Therefore, from , (92), and the above inequality, we have

In view of (93), it also results that

As a direct consequence of Lemma 12, we have

Since , the sequence generated by Algorithm 15 converges strongly to , completing the proof.

First, we shall give an example which satisfies all the conditions of the solution set of MSECFP (6) and the mappings and in Theorems 20 and 21.

Example 22. Let and let and be arbitrarily fixed. Let be defined by and for all . Then, it is easy to see that and . Thus, . Also, is -demicontractive and is -demicontractive by Example in [9], where and ; then, and are demiclosed at 0 by Remark in [9].

Next, we give an example which satisfies condition (a) in Theorem 20 and condition (1) in Theorem 21.

Example 23. We can take , = .

Remark 24. The following two mistakes in the proof of Theorem in [9] should be modified as in the proof of Theorem 21:

(i)In Case , the part “for any and ” and in Case the part “for any and ” should be modified to “for any ” and “for any ,” respectively.(ii)For proving of Theorem in [9], we should adopt the proving method of (87) and (88) in Theorem 21.

Remark 25. We can also consider the following two algorithms:

and

Under the suitable conditions, using methods similar to Theorems 20 and 21, we can prove the weak and strong convergence of the sequences generated by these two algorithms, respectively.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

Tae-Hwa Kim is supported by a research grant from Pukyong National University (year 2017). Yaqin Wang is supported by the National Natural Science Foundation of China (nos. 11401388 and 11671365). Xiaoli Fang is supported by Zhejiang Provincial Natural Science Foundation of China (nos. LQ13A010007 and LY14A010006).

References

Y. Censor and T. Elfving, “A multiprojection algorithm using Bregman projections in a product space,” Numerical Algorithms, vol. 8, no. 2-4, pp. 221–239, 1994.

C. Byrne, “Iterative oblique projection onto convex sets and the split feasibility problem,” Inverse Problems, vol. 18, no. 2, pp. 441–453, 2002.

Y. Yao, R. P. Agarwal, M. Postolache, and Y.-C. Liou, “Algorithms with strong convergence for the split common solution of the feasibility problem and fixed point problem,” Fixed Point Theory and Applications, vol. 2014, no. 1, article no. 183, 14 pages, 2014.

Y. Yao, M. Postolache, and Y.-C. Liou, “Strong convergence of a self-adaptive method for the split feasibility problem,” Fixed Point Theory and Applications, vol. 2013, article no. 201, 12 pages, 2013.

Y. Yao, W. Jigang, and Y.-C. Liou, “Regularized methods for the split feasibility problem,” Abstract and Applied Analysis, vol. 2012, Article ID 140679, 13 pages, 2012.

A. Moudafi and E. Al-Shemas, “Simultaneous iterative methods for split equality problem,” Transactions on Mathematical Programming and Applications, vol. 1, no. 2, pp. 1–11, 2013.

J. Zhao, “Solving split equality fixed-point problem of quasi-nonexpansive mappings without prior knowledge of operators norms,” Optimization. A Journal of Mathematical Programming and Operations Research, vol. 64, no. 12, pp. 2619–2630, 2015.

J. Zhao and S. Wang, “Mixed iterative algorithms for the multiple-set split equality common fixed-point problems without prior knowledge of operator norms,” Optimization. A Journal of Mathematical Programming and Operations Research, vol. 65, no. 5, pp. 1069–1083, 2016.

Y. Wang and T. H. Kim, “Simultaneous iterative algorithm for the split equality fixed-point problem of demicontractive mappings,” Journal of Nonlinear Sciences and Applications, vol. 10, no. 1, pp. 154–165, 2017.

W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, Japan, 2000.

G. Marino and H.-K. Xu, “Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 329, no. 1, pp. 336–346, 2007.

Y. Censor and A. Segal, “The split common fixed point problem for directed operators,” Journal of Convex Analysis, vol. 16, no. 2, pp. 587–600, 2009.

K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, vol. 28 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 1990.

C. Martinez-Yanes and H.-K. Xu, “Strong convergence of the CQ method for fixed point iteration processes,” Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal, vol. 64, no. 11, pp. 2400–2411, 2006.

A. Moudafi, “The split common fixed-point problem for demicontractive mappings,” Inverse Problems, vol. 26, no. 5, Article ID 055007, pp. 587–600, 2010.

M. C. Joshi and R. K. Bose, Some topics in nonlinear functional analysis, John Wiley and Sons, New York, NY, USA, 1985.

P. L. Combettes, Fejér-Monotonicity in Convex Optimization, in Encyclopedia of Optimization, C. A. Floudas and P. M. Pardalos, Eds., vol. 2, Springer-Verlag, New York, NY, USA, 2001.

S.-W. Han and T.-H. Kim, “ϕ-Fejér-monotone sequences and their convergence theorems,” Journal of Nonlinear and Convex Analysis, vol. 17, no. 2, pp. 211–223, 2016.

P.-E. Maingé, “Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization,” Set-Valued and Variational Analysis, vol. 16, no. 7-8, pp. 899–912, 2008.

H.-K. Xu, “Iterative algorithms for nonlinear operators,” Journal Of The London Mathematical Society-Second Series, vol. 66, no. 1, pp. 240–256, 2002.

Y. Wang, T.-H. Kim, X. Fang, and H. He, “The split common fixed-point problem for demicontractive mappings and quasi-nonexpansive mappings,” Journal of Nonlinear Sciences and Applications, vol. 10, no. 6, pp. 2976–2985, 2017.