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
MSC
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
References
- Moudafi, A.: Alternating CQ-algorithms for convex feasibility and split fixed-point problems. J. Nonlinear Convex Anal. 15, 809–818 (2014) MathSciNetMATHGoogle Scholar
- Attouch, H., Bolte, J., Redont, P., Soubeyran, A.: Alternating proximal algorithms for weakly coupled minimization problems. Applications to dynamical games and PDE’s. J. Convex Anal. 15, 485–506 (2008) MathSciNetMATHGoogle Scholar
- Censor, Y., Bortfeld, T., Martin, B., Trofimov, A.: A unified approach for inversion problems in intensity-modulated radiation therapy. Phys. Med. Biol. 51, 2353–2365 (2006) View ArticleGoogle Scholar
- Byrne, C.: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Probl. 20, 103–120 (2004) MathSciNetView ArticleMATHGoogle Scholar
- Censor, Y., Elfving, T.: A multiprojection algorithm using Bregman projections in a product space. Numer. Algorithms 8, 221–239 (1994) MathSciNetView ArticleMATHGoogle Scholar
- Censor, Y., Elfving, T., Kopf, N., Bortfeld, T.: The multiple-sets split feasibility problem and its applications for inverse problems. Inverse Probl. 21, 2071–2084 (2005) MathSciNetView ArticleMATHGoogle Scholar
- Censor, Y., Gibali, A., Reich, S.: Algorithms for the split variational inequality problem. Numer. Algorithms 59, 301–323 (2012) MathSciNetView ArticleMATHGoogle Scholar
- Qu, B., Xiu, N.: A note on the CQ algorithm for the split feasibility problem. Inverse Probl. 21, 1655–1665 (2005) MathSciNetView ArticleMATHGoogle Scholar
- Xu, H.K.: A variable Krasnosel’skiĭ–Mann algorithm and the multiple-set split feasibility problem. Inverse Probl. 22, 2021–2034 (2006) View ArticleMATHGoogle Scholar
- Xu, H.K.: Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces. Inverse Probl. 26, 105018 (2010) MathSciNetView ArticleMATHGoogle Scholar
- Yang, Q.: The relaxed CQ algorithm solving the split feasibility problem. Inverse Probl. 20, 1261–1266 (2004) MathSciNetView ArticleMATHGoogle Scholar
- Masad, E., Reich, S.: A note on the multiple-set split convex feasibility problem in Hilbert space. J. Nonlinear Convex Anal. 8, 367–371 (2007) MathSciNetMATHGoogle Scholar
- Yao, Y., Yao, Z., Abdou, A.A.N., Cho, Y.J.: Self-adaptive algorithms for proximal split feasibility problems and strong convergence analysis. Fixed Point Theory Appl. 2015, 205 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Wen, M., Peng, J., Tang, Y.: A cyclic and simultaneous iterative method for solving the multiple-sets split feasibility problem. J. Optim. Theory Appl. 166, 844–860 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Qin, X., Yao, J.C.: Projection splitting algorithms for nonself operators. J. Nonlinear Convex Anal. 18(5), 925–935 (2017) MathSciNetMATHGoogle Scholar
- Cho, S.Y.: Generalized mixed equilibrium and fixed point problems in a Banach space. J. Nonlinear Sci. Appl. 9, 1083–1092 (2016) MathSciNetView ArticleMATHGoogle Scholar
- Cho, S.Y., Qin, X., Yao, J.C., Yao, Y.H.: Viscosity approximation splitting methods for monotone and nonexpansive operators in Hilbert spaces. J. Nonlinear Convex Anal. 19, 251–264 (2018) Google Scholar
- Yao, Y.H., Liou, Y.C., Yao, J.C.: Split common fixed point problem for two quasi-pseudocontractive operators and its algorithm construction. Fixed Point Theory Appl. 2015, 127 (2015) View ArticleMATHGoogle Scholar
- Yao, Y.H., Liou, Y.C., Yao, J.C.: Iterative algorithms for the split variational inequality and fixed point problems under nonlinear transformations. J. Nonlinear Sci. Appl. 10, 843–854 (2017) MathSciNetView ArticleGoogle Scholar
- Yao, Y.H., Yao, J.C., Liou, Y.C., Postolache, M.: Iterative algorithms for split common fixed points of demicontractive operators without priori knowledge of operator norms. Carpathian J. Math. in press Google Scholar
- Yao, Y.H., Agarwal, R.P., Postolache, M., Liou, Y.C.: Algorithms with strong convergence for the split common solution of the feasibility problem and fixed point problem. Fixed Point Theory Appl. 2014, 183 (2014) MathSciNetView ArticleMATHGoogle Scholar
- Byrne, C.: Iterative oblique projection onto convex subsets and the split feasibility problem. Inverse Probl. 18, 441–453 (2002) View ArticleMATHGoogle Scholar
- Censor, Y., Segal, A.: The split common fixed point problem for directed operators. J. Convex Anal. 16, 587–600 (2009) MathSciNetMATHGoogle Scholar
- Wang, F., Xu, H.: Cyclic algorithms for split feasibility problems in Hilbert spaces. Nonlinear Anal. 74, 4105–4111 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Tang, Y.C., Liu, L.W.: Several iterative algorithms for solving the split common fixed point problem of directed operators with applications. Optimization 65(1), 53–65 (2016) MathSciNetView ArticleMATHGoogle Scholar
- Moudafi, A., Al-Shemas, E.: Simultaneous iterative methods for split equality problems and application. Trans. Math. Program. Appl. 1, 1–11 (2013) Google Scholar
- Chang, S.S., Wang, L., Zhao, Y.: On a class of split equality fixed point problems in Hilbert spaces. J. Nonlinear Var. Anal. 1, 201–212 (2017) Google Scholar
- Tang, J., Chang, S.S., Dong, J.: Split equality fixed point problem for two quasi-asymptotically pseudocontractive mappings. J. Nonlinear Funct. Anal. 2017, Article ID 26 (2017) Google Scholar
- Dong, Q.L., He, S., Zhao, J.: Solving the split equality problem without prior knowledge of operator norms. Optimization 64(9), 1887–1906 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Wu, Y., Chen, R., Shi, L.Y.: Split equality problem and multiple-sets split equality problem for quasi-nonexpansive multi-valued mappings. J. Inequal. Appl. 2014, 428 (2014) MathSciNetView ArticleMATHGoogle Scholar
- Byrne, C., Moudafi, A.: Extensions of the CQ algorithm for the split feasibility and split equality problems. Documents De Travail 18(8), 1485–1496 (2013) MathSciNetGoogle Scholar
- Zhao, J.: Solving split equality fixed-point problem of quasi-nonexpansive mappings without prior knowledge of operators norms. Optimization 64(12), 2619–2630 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Zhao, J., He, S.: Viscosity approximation methods for split common fixed-point problem of directed operators. Numer. Funct. Anal. Optim. 36(4), 528–547 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Zhao, J., Wang, S.: Viscosity approximation methods for the split equality common fixed point problem of quasi-nonexpansive operators. Acta Math. Sci. 36(5), 1474–1486 (2016) MathSciNetView ArticleMATHGoogle Scholar
- Wang, F.: A new iterative method for the split common fixed point problem in Hilbert spaces. Optimization 66(3), 407–415 (2017) MathSciNetView ArticleMATHGoogle Scholar
- Matinez-Yanes, C., Xu, H.K.: Strong convergence of the CQ method for fixed point processes. Nonlinear Anal. 64, 2400–2411 (2006) MathSciNetView ArticleMATHGoogle Scholar
- Hao, Y., Cho, S.Y., Qin, X.: Some weak convergence theorems for a family of asymptotically nonexpansive nonself mappings. Fixed Point Theory Appl. 2010, 218573 (2010) MathSciNetView ArticleMATHGoogle Scholar
- Bauschke, H.H.: The approximation of fixed points of composition of nonexpansive mappings in Hilbert space. J. Math. Anal. Appl. 202, 150–159 (1996) MathSciNetView ArticleMATHGoogle Scholar