- Research
- Open Access
Solving the multiple-set split equality common fixed-point problem of firmly quasi-nonexpansive operators
- Jing Zhao^{1, 2}Email author and
- Haili Zong^{1, 2}
https://doi.org/10.1186/s13660-018-1668-0
© The Author(s) 2018
- Received: 12 January 2018
- Accepted: 28 March 2018
- Published: 13 April 2018
Abstract
In this paper, we propose parallel and cyclic iterative algorithms for solving the multiple-set split equality common fixed-point problem of firmly quasi-nonexpansive operators. We also combine the process of cyclic and parallel iterative methods and propose two mixed iterative algorithms. Our several algorithms do not need any prior information about the operator norms. Under mild assumptions, we prove weak convergence of the proposed iterative sequences in Hilbert spaces. As applications, we obtain several iterative algorithms to solve the multiple-set split equality problem.
Keywords
- The multiple-set split equality common fixed-point problem
- The multiple-set split equality problem
- Firmly quasi-nonexpansive mapping
- Weak convergence
- Iterative algorithms
- Hilbert space
MSC
- 47H09
- 47H10
- 47J05
- 54H25
1 Introduction
2 Preliminaries
2.1 Concepts
Throughout this paper, we always assume that H is a real Hilbert space with the inner product \(\langle\cdot,\cdot\rangle\) and the norm \(\Vert \cdot \Vert \). Let I denote the identity operator on H. Denote the fixed-point set of an operator T by \(F(T)\). We denote by → the strong convergence and by ⇀ the weak convergence. We use \(\omega_{w}(x_{k})=\{x:\exists x_{k_{j}} \rightharpoonup x\}\) to stand for the weak ω-limit set of \(\{x_{k}\}\) and use Γ to stand for the solution set of MSECFP (1.1).
Definition 2.1
- (i)
nonexpansive if \(\Vert Tx-Ty \Vert \leq \Vert x-y \Vert \) for all \(x,y\in H\);
- (ii)
firmly nonexpansive if \(\Vert Tx-Ty \Vert ^{2}\leq \Vert x-y \Vert ^{2}- \Vert (x-y)-(Tx-Ty) \Vert ^{2}\) for all \(x,y\in H\);
- (iii)firmly quasi-nonexpansive (i.e., directed operator) if \(F(T)\neq\emptyset\) andor equivalently$$\Vert Tx-q \Vert ^{2}\leq \Vert x-q \Vert ^{2}- \Vert x-Tx \Vert ^{2} $$for all \(x\in H\) and \(q\in F(T)\).$$\langle x-q,x-Tx\rangle\geq \Vert x-Tx \Vert ^{2} $$
Definition 2.2
An operator \(T:H\rightarrow H\) is called demiclosed at the origin if, for any sequence \(\{x_{n}\}\) which weakly converges to x, and if the sequence \(\{Tx_{n}\}\) strongly converges to 0, then \(Tx=0\).
Remark 2.1
It is easily seen that a firmly nonexpansive operator is nonexpansive. Firmly quasi-nonexpansive operators contain firmly nonexpansive operators with a nonempty fixed-point set. A projection operator is firmly nonexpansive.
2.2 Mathematical model
2.3 The well-known lemmas
The following lemmas will be helpful for our main results in the next section.
Lemma 2.1
Lemma 2.2
([36])
Lemma 2.3
([37])
Lemma 2.4
([38])
Let E be a uniformly convex Banach space, K be a nonempty closed convex subset of E, and \(T:K\rightarrow K\) be a nonexpansive mapping. Then \(I-T\) is demi-closed at origin.
3 Parallel and cyclic iterative algorithms
In this section, we introduce parallel and cyclic iterative algorithms and prove the weak convergence for solving MSECFP (1.1) of firmly quasi-nonexpansive operators. In our algorithms, the selection of the step size does not need any prior information of the operator norms \(\Vert A \Vert \) and \(\Vert B \Vert \).
- (A1)
The problem is consistent, namely its solution set Γ is nonempty;
- (A2)
Both \(U_{i}\) and \(T_{j}\) are firmly quasi-nonexpansive operators, and both \(I-U_{i}\) and \(I-T_{j}\) are demiclosed at origin (\(1\leq i\leq p\), \(1\leq j\leq r\)).
- (A3)
The sequences \(\{\alpha_{n}^{i}\}_{i=1}^{p}\), \(\{\beta_{n}^{j}\}_{j=1}^{r}\subset[0,1]\) such that \(\sum_{i=1}^{p}\alpha_{n}^{i}=1\) and \(\sum_{j=1}^{r}\beta_{n}^{j}=1\) for every \(n\geq0\), \(j(n)=n(\operatorname {mod}r)+1\), \(i(n)=n(\operatorname {mod}p)+1\).
Algorithm 3.1
Remark 3.1
Note that in (3.2) the choice of the step size \(\tau_{n}\) is independent of the norms \(\Vert A \Vert \) and \(\Vert B \Vert \). The value of τ does not influence the considered algorithm, it was introduced just for the sake of clarity.
Lemma 3.1
\(\tau_{n}\) defined by (3.2) is well defined.
Proof
Theorem 3.1
Assume that \(\liminf_{n\rightarrow\infty}\alpha_{n}^{i}>0 (1\leq i \leq p)\) and \(\liminf_{n\rightarrow\infty}\beta_{n}^{j}>0 (1\leq j \leq r)\). Then the sequence \(\{(x_{n}, y_{n})\}\) generated by Algorithm 3.1 weakly converges to a solution \((x^{\ast},y^{\ast})\) of MSECFP (1.1). Moreover, \(\Vert Ax_{n}-By_{n} \Vert \rightarrow 0\), \(\Vert x_{n+1}-x_{n} \Vert \rightarrow0\), and \(\Vert y_{n+1}-y_{n} \Vert \rightarrow0\) as \(n\rightarrow\infty\).
Proof
Next, we propose the cyclic iterative algorithm for solving MSECFP (1.1) of firmly quasi-nonexpansive operators.
Algorithm 3.2
Theorem 3.2
The sequence \(\{(x_{n}, y_{n})\}\) generated by Algorithm 3.2 weakly converges to a solution \((x^{\ast},y^{\ast})\) of MSECFP (1.1). Moreover, \(\Vert Ax_{n}-By_{n} \Vert \rightarrow 0\), \(\Vert x_{n}-x_{n+1} \Vert \rightarrow0\), and \(\Vert y_{n}-y_{n+1} \Vert \rightarrow0\) as \(n\rightarrow\infty\).
Proof
Now, we give applications of Theorem 3.1 and Theorem 3.2 to solve MSEP (1.2). Assume that the solution set S of MSEP (1.2) is nonempty. Since the orthogonal projection operator is firmly nonexpansive, by Lemma 2.4 we have the following results for solving MSEP (1.2).
Corollary 3.1
Corollary 3.2
4 Mixed cyclic and parallel iterative algorithms
Now, for solving MSECFP (1.1) of firmly quasi-nonexpansive operators, we introduce two mixed iterative algorithms which combine the process of cyclic and simultaneous iterative methods. In our algorithms, the selection of the step size does not need any prior information of the operator norms \(\Vert A \Vert \) and \(\Vert B \Vert \), and the weak convergence is proved. We go on making use of assumptions (A1)–(A3).
Algorithm 4.1
Theorem 4.1
Assume that \(\liminf_{n\rightarrow\infty}\alpha_{n}^{i}>0\) (\(1\leq i \leq p\)). Then the sequence \(\{(x_{n}, y_{n})\}\) generated by Algorithm 4.1 weakly converges to a solution \((x^{\ast},y^{\ast})\) of MSECFP (1.1). Moreover, \(\Vert Ax_{n}-By_{n} \Vert \rightarrow 0\), \(\Vert x_{n+1}-x_{n} \Vert \rightarrow0\), and \(\Vert y_{n+1}-y_{n} \Vert \rightarrow0\) as \(n\rightarrow\infty\).
Proof
Next, we propose another mixed cyclic and parallel iterative algorithm for solving MSECFP (1.1) of firmly quasi-nonexpansive operators.
Algorithm 4.2
Similar to the proof of Theorem 4.1, we can get the following result.
Theorem 4.2
Assume that \(\liminf_{n\rightarrow\infty}\beta_{n}^{j}>0\) (\(1\leq j \leq r\)). Then the sequence \(\{(x_{n}, y_{n})\}\) generated by Algorithm 4.2 weakly converges to a solution \((x^{\ast},y^{\ast})\) of MSECFP (1.1) of firmly quasi-nonexpansive operators. Moreover, \(\Vert Ax_{n}-By_{n} \Vert \rightarrow 0\), \(\Vert x_{n}-x_{n+1} \Vert \rightarrow0\) and \(\Vert y_{n}-y_{n+1} \Vert \rightarrow0\) as \(n\rightarrow\infty\).
Finally, we obtain two mixed iterative algorithms to solve MSEP (1.2). Assume that the solution set S of MSEP (1.2) is nonempty.
Corollary 4.1
Corollary 4.2
5 Results and discussion
To avoid computing the norms of the bounded linear operators, we introduce parallel and cyclic iterative algorithms with self-adaptive step size to solve MSECFP (1.1) governed by firmly quasi-nonexpansive operators. We also propose two mixed iterative algorithms and do not need the norms of bounded linear operators. As applications, we obtain several iterative algorithms to solve MSEP (1.2).
6 Conclusion
In this paper, we have considered MSECFP (1.1) of firmly quasi-nonexpansive operators. Inspired by the methods for solving SCFP (2.1) and MSCFP (1.3), we introduce parallel and cyclic iterative algorithms for solving MSECFP (1.1). We also present two mixed iterative algorithms which combine the process of parallel and cyclic iterative methods. In our several iterative algorithms, the step size is chosen in a self-adaptive way and the weak convergence is proved.
Declarations
Funding
This work was supported by the National Natural Science Foundation of China (No. 61503385) and Open Fund of Tianjin Key Lab for Advanced Signal Processing (No. 2017ASP-TJ03).
Authors’ contributions
All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests regarding the present manuscript.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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