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Strong convergence theorems for a class of split feasibility problems and fixed point problem in Hilbert spaces
- Jinhua Zhu^{1},
- Jinfang Tang^{1} and
- Shih-sen Chang^{2}Email author
https://doi.org/10.1186/s13660-018-1881-x
© The Author(s) 2018
- Received: 6 July 2018
- Accepted: 14 October 2018
- Published: 23 October 2018
Abstract
In this paper we consider a class of split feasibility problem by focusing on the solution sets of two important problems in the setting of Hilbert spaces. One of them is the set of zero points of the sum of two monotone operators and the other is the set of fixed points of mappings. By using the modified forward–backward splitting method, we propose a viscosity iterative algorithm. Under suitable conditions, some strong convergence theorems of the sequence generated by the algorithm to a common solution of the problem are proved. At the end of the paper, some applications and the constructed algorithm are also discussed.
Keywords
- Split feasibility
- Maximal monotone operators
- Inverse strongly monotone operator
- Fixed point problems
- Strong convergence theorems
MSC
- 26A18
- 47H04
- 47H05
- 47H10
1 Introduction
2 Preliminaries
Throughout this paper, we denote by \(\Bbb {N}\) the set of positive integers, and by \(\Bbb {R}\) the set of real numbers. Let H be a real Hilbert space with the inner product \(\langle \cdot,\cdot \rangle \) and norm \(\|\cdot \|\), respectively. When \(\{x_{n}\}\) is a sequence in H, we denote the weak convergence of \(\{x_{n}\}\) to x in H by \(x_{n}\rightharpoonup x\).
We now collect some important conclusions and properties, which will be needed in proving our main results.
Lemma 2.1
- (i)
The composition of finitely many averaged mappings is averaged. In particular, if \(T_{i}\) is \(\alpha_{i}\)-averaged, where \(\alpha_{i} \in (0,1)\) for \(i =1,2\), then the composition \(T_{1}T_{2}\) is α-averaged, where \(\alpha =\alpha_{1}+\alpha_{2}-\alpha_{1} \alpha_{2}\).
- (ii)
If A is β-ism and \(\gamma \in (0,\beta ]\), then \(T:= I-\gamma A\) is firmly nonexpansive.
- (iii)
A mapping \(T:H\to H\) is nonexpansive if and only if \(I-T\) is \(\frac{1}{2}\)-ism.
- (iv)
If A is β-ism, then, for \(\gamma >0\), γA is \(\frac{\beta }{\gamma }\)-ism.
- (v)
T is averaged if and only if the complement \(I-T\) is β-ism for some \(\beta >\frac{1}{2}\). Indeed, for \(\alpha \in (0,1)\), T is α-averaged if and only if \(I-T\) is \(\frac{1}{2 \alpha }\)-ism.
Lemma 2.2
([17])
Let \(T=(1-\alpha)A+\alpha N\) for some \(\alpha \in (0,1)\). If A is β-averaged and N is nonexpansive then T is \(\alpha +(1-\alpha)\beta \)-averaged.
Lemma 2.3
([20])
Lemma 2.4
([12])
- (i)
\(L^{*}(I-T)L\) is \(\frac{1}{2\|L\|^{2}}\)-ism,
- (ii)
For \(0< r<\frac{1}{\|L\|}\),
- (iia)
\(I-rL^{*}(I-T)L\) is \(r\|L\|^{2}\)-averaged,
- (iib)
\(J^{B}_{\lambda }(I-rL^{*}(I-T)L)\) is \(\frac{1+r\|L\|^{2}}{2}\)-averaged, for \(\lambda >0\),
- (iii)
If \(r=\|L\|^{-2}\), then \(I-rL^{*}(I-T)L\) is nonexpansive.
Lemma 2.5
([21])
Lemma 2.6
([22])
Let C be a closed convex subset of a Hilbert space H and let T be a nonexpansive mapping of C into itself. Then \(U:=I-T\) is demiclosed, i.e., \(x_{n}\rightharpoonup x_{0}\) and \(Ux_{n}\to y_{0}\) imply \(Ux_{0}=y_{0}\).
Lemma 2.7
([10])
- (i)
\(z\in \Omega^{A+B}_{L,T}\),
- (ii)
\(z=J^{B}_{\lambda }((I_{\lambda }-A)-\gamma L^{*}(I-T)L)z\),
- (iii)
\(0\in L^{*}(I-T)Lz+(A+B)z\),
Lemma 2.8
([23])
- (i)
\(\sum^{\infty }_{n=1}\beta_{n}=\infty\);
- (ii)
\(\limsup_{n\to \infty }\frac{\delta_{n}}{\beta_{n}}\leq 0\) or \(\sum^{\infty }_{n=1}|\delta_{n}|<\infty \).
3 Main results
We are now in a position to give the main result of this paper.
Lemma 3.1
- (i)
\(0< a\leq \lambda_{n}\leq b_{1}<\frac{\beta }{2}\),
- (ii)
\(0< a\leq \gamma_{n}\leq b_{2}<\frac{1}{2\|L\|^{2}}\), for some \(a,b_{1},b_{2}\in \Bbb {R}\).
Proof
Theorem 3.2
- (i)
\(0< a\leq \lambda_{n}\leq b_{1}<\frac{\beta }{2}\), and \(\Sigma^{\infty }_{n=1}|\lambda_{n}-\lambda_{n-1}|<\infty\),
- (ii)
\(0< a\leq \gamma_{n}\leq b_{2}<\frac{1}{2\|L\|^{2}}\), and \(\Sigma^{\infty }_{n=1}|\gamma_{n}-\gamma_{n-1}|<\infty\), for some \(a, b_{1}, b_{2} \in \Bbb {R}\),
Proof
Furthermore, it follows from (3.13) and (3.14) that \(\{u_{n}\}\), \(\{x_{n}\}\) and \(\{S(u_{n})\}\) have the same asymptotical behavior, so \(\{u_{n}\}\) also converges weakly to x̂. Since S is nonexpansive, by (3.13) and Lemma 2.6, we obtain that \(\hat{x}\in F(S)\). Thus \(\hat{x}\in \Omega^{A+B}_{L,T}\cap F(S)\).
If \(A:=0\), the zero operator, then the following result can be obtained from Theorem 3.2 immediately.
Corollary 3.3
If \(H_{1}=H_{2}\), \(L=I\), then by applying Theorem 3.2, we can obtain the following result.
Corollary 3.4
4 Applications
In this section, we will utilize the results presented in the paper to study variational inequality problems, convex minimization problem and split common fixed point problem in Hilbert spaces.
4.1 Application to variational inequality problem
Theorem 4.1
4.2 Application to convex minimization problem
Theorem 4.2
4.3 Application to split common fixed point problem
This problem is called the split common fixed point problem (SCFP), and was studied by many authors (see [25–28], for example). By using Theorem 3.2, we can obtain the following result.
Theorem 4.3
Proof
We consider \(B:=0\), the zero operator. The required result follows from the fact that the zero operator is monotone and continuous, hence it is maximal monotone. Moreover, in this case, we see that \(J^{B}_{\lambda }\) is the identity operator on \(H_{1}\), for each \(\lambda >0\). Thus algorithm (3.4) reduces to (4.8), by setting \(A:=I-V\) and \(B:=0\). □
Declarations
Acknowledgements
The authors would like to express their thanks to the Editor and the Referees for their helpful comments.
Availability of data and materials
Not applicable.
Funding
The first author was supported by Scientific Research Fund of Sichuan Provincial Department of Science and Technology (2015JY0165), the second author was supported by Scientific Research Fund of Sichuan Provincial Education Department (16ZA0331) and the third author was supported by The Natural Science Foundation of China Medical University, Taichung, Taiwan.
Authors’ contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Competing interests
None of the authors have any competing interests in the 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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