On the split common fixed point problem for strict pseudocontractive and asymptotically nonexpansive mappings in Banach spaces
- Jinfang Tang^{1},
- Shih-sen Chang^{2, 3}Email author,
- Lin Wang^{3} and
- Xiongrui Wang^{1}
https://doi.org/10.1186/s13660-015-0832-z
© Tang et al. 2015
Received: 14 June 2015
Accepted: 18 September 2015
Published: 29 September 2015
Abstract
In this paper, we prove a weak convergence theorem and a strong convergence theorem for split common fixed point problem involving a quasi-strict pseudo contractive mapping and an asymptotical nonexpansive mapping in the setting of two Banach spaces. Our results are new and seem to be the first outside Hilbert spaces.
Keywords
MSC
1 Introduction
The SFP in finite-dimensional Hilbert spaces was first introduced by Censor and Elfving [1] for modeling inverse problems which arise from phase retrievals and in medical image reconstruction [2]. The split common fixed point problems in Hilbert spaces were introduced by Moudafi [3] in 2010. Since then, various algorithms have been invented to solve SFP and SCFP [4–16]. In 2014, Cui and Wang [17] investigated the split common fixed point problems of τ-quasi-strict pseudocontractive mappings in the setting of two Hilbert spaces.
In 2015, Takahashi [18], Takahashi and Yao [19] first attempted to introduce and consider the split feasibility problem and split common null point problem in the setting of a Banach space. By using hybrid methods and Halpern-type methods and under suitable conditions, some strong and weak convergence theorems for such kinds of problems are obtained. The results presented in [18] and [19] seem to be the first outside Hilbert spaces.
Motivated and inspired by the research going on in the direction of split feasibility problems and split common fixed point problems, we have the purpose in this article to consider and study the split common fixed point problem for a τ-quasi-strict pseudocontractive mapping and asymptotical nonexpansive mappings in the setting of two Banach spaces. We construct an iterative scheme to approximate a solution for such kind of split common fixed point problem in the setting of two Banach spaces. Our results are new and seem to be the first outside Hilbert spaces on this problem.
2 Preliminaries
Throughout this paper, we assume that E is a real Banach space with the dual \(E^{*}\) and C is a nonempty closed convex subset of E. Let T be a mapping. We denote by \(F(T)\) the set of fixed points of T. We denote by ‘→’ and ‘⇀’ strong convergence and weak convergence, respectively.
Lemma 2.1
Lemma 2.2
Definition 2.3
- (i)T is said to be \(\{k_{n}\}\)-asymptotically nonexpansive if there exists a sequence \(\{k_{n}\} \subset[1, \infty)\) with \(k_{n} \to1\) such that$$\bigl\Vert T^{n} x-T^{n} y\bigr\Vert \leq k_{n} \|x-y\|,\quad \forall n \ge1, x,y\in C; $$
- (ii)T is said to be τ-strict pseudocontractive if there exists a constant \(\tau\in[0,1)\) such that$$ \|Tx-Ty\|^{2}\leq\|x-y\|^{2}+\tau\bigl\Vert (I-T)x-(I-T)y \bigr\Vert ^{2}, \quad \forall x,y\in C; $$(2.1)
- (iii)T is said to be τ-quasi-strict pseudocontractive if \(F(T)\neq\emptyset\) and there exists a constant \(\tau\in[0,1)\) such that$$ \|Tx-p\|^{2}\leq\|x-p\|^{2}+\tau\bigl\Vert (I-T)x\bigr\Vert ^{2},\quad \forall p\in F(T),x\in C. $$(2.2)
Example of \(\{k_{n}\}\)-asymptotically nonexpansive mapping
- (i)
\(\|Tx-Ty\|\leq2\|x-y\|\), \(\forall x,y \in C\);
- (ii)
\(\|T^{n}x-T^{n} y\|\leq2\prod_{i=2}^{n}a_{j}\|x-y\|\), \(\forall n\geq2\) and \(x,y \in C\).
Example of τ-strict pseudocontractive mapping
Now, we give an example of a τ-strict pseudocontractive mapping.
In fact, for any \(x, y \in C\), we have the following.
Case 1. If \(\prod_{i=1}^{\infty}x_{i}< 0\) and \(\prod_{i=1}^{\infty}y_{i}< 0\), then we have \(T x=x\), \(T y=y\), and so inequality (2.1) holds.
Definition 2.4
- (1)
Let \(T: C\to C\) be a mapping with \(F(T)\neq\emptyset\). Then T is said to be demiclosed at zero if for any \(\{x_{n}\}\subset C\) with \(x_{n}\rightharpoonup x\) and \(\|x_{n} - Tx_{n}\| \to0\), \(x=Tx\).
- (2)Let E be a Banach space. E is said to have the Opial property if for any sequence \(\{x_{n}\}\) in E with \(x_{n} \rightharpoonup x^{*}\), for any \(y\in E\) with \(y \neq x^{*}\), we have$$\liminf_{n \to\infty}\bigl\Vert x_{n} - x^{*}\bigr\Vert < \liminf_{n \to\infty}\|x_{n} - y\|. $$
Definition 2.5
A mapping \(T:C\to C\) is said to be semi-compact if for any bounded sequence \(\{x_{n}\}\subset C\) such that \(\|x_{n}-Tx_{n}\|\to0\) (\(n\to\infty\)), there exists a subsequence \(\{x_{n_{j}}\}\) of \(\{x_{n}\}\) such that \(\{x_{n_{j}}\}\) converges strongly to \(x^{*}\in C\).
Lemma 2.6
3 Main results
- (1)
\(E_{1}\) is a real uniformly convex and 2-uniformly smooth Banach space having the Opial property and the best smoothness constant k satisfying \(0< k<\frac{1}{\sqrt{2}}\).
- (2)
\(E_{2}\) is a real Banach space.
- (3)
\(A:E_{1} \to E_{2}\) is a bounded linear operator and \(A^{*}\) is the adjoint of A.
- (4)
\(S: E_{1}\to E_{1}\) is an \(\{l_{n}\}\)-asymptotical nonexpansive mapping with \(\{l_{n}\} \subset(1, \infty)\) and \(l_{n} \to1\). \(T: E_{2}\to E_{2}\) is a τ-quasi-strict pseudocontractive mapping with \(F(S)\neq\emptyset \) and \(F(T)\neq\emptyset\), and T is demiclosed at zero.
Remark
It follows from condition (1) that \(E_{1}\) is a real smooth, strictly convex and reflexive Banach space. Therefore, as is well known, the normalized duality mapping \(J_{1}: E_{1} \to2^{E_{1}^{*}}\) is single-valued, one-to-one and onto. And \(J_{1}^{-1}: E_{1}^{*} \to2^{E_{1}}\) is also single-valued, one-to-one and onto.
Theorem 3.1
- (I)
If \(\Gamma=\{p\in F(S):Ap\in F(T)\}\neq\emptyset\) (the set of solutions of (SCFP) (3.1)), then the sequence \(\{x_{n}\}\) converges weakly to a point \(x^{*}\in\Gamma\).
- (II)
In addition, if \(\Gamma=\{p\in F(S):Ap\in F(T)\}\neq\emptyset\) and S is semi-compact, then \(\{x_{n}\}\) converges strongly to a point \(x^{*}\in\Gamma\).
Proof
Now we prove conclusion (I).
We divide the proof into four steps.
Step 1. We first show that the limit \(\lim_{n\to\infty}\|x_{n}-p\|\) exists for each \(p\in\Gamma\).
Step 2. We prove that \(\lim_{n\to\infty}\|x_{n+1}-x_{n}\|=0\) and \(\lim_{n\to\infty}\|z_{n+1}-z_{n}\|=0\).
Step 3. We prove that \(\lim_{n\to\infty}\|z_{n}-Sz_{n}\|=0\).
Step 4. We prove that \(\{x_{n}\}\) converges weakly to \(x^{*}\in\Gamma\).
Now we prove that \(\{x_{n}\}\) converges weakly to \(x^{*}\in\Gamma\).
Next, we prove conclusion (II).
Since \(\lim_{n\to\infty}\|z_{n}-Sz_{n}\|=0\) and S is semi-compact, there exists a subsequence \(\{z_{n_{k}}\}\) of \(\{z_{n}\}\) such that \(\{z_{n_{k}}\}\) converges strongly to \(\mu^{*}\in E_{1}\). By using (3.14), we know that the subsequence \(\{x_{n_{k}}\}\) of \(\{x_{n}\}\) converges strongly to \(\mu^{*}\), too. Due to \(\{x_{n}\}\) converging weakly to \(x^{*}\), we have \(\mu^{*}=x^{*}\). Since \(\lim_{n\to\infty}\|x_{n}-x^{*}\|\) exists and \(\lim_{n_{k}\to\infty}\|x_{n_{k}}-x^{*}\|=0\), we know that \(\{x_{n}\}\) converges strongly to \(x^{*}\in\Gamma\). This completes the proof of conclusion (II). □
4 Application to hierarchical variational inequality problem in Banach spaces
In this section we shall utilize the results presented in Section 3 to study the hierarchical variational inequality problem in Banach spaces.
Let E be a strictly convex and real reflexive Banach space and K be a nonempty closed and convex subset of E. Then, for any \(x \in E\), there exists a unique element \(z \in K\) such that \(\|x - z\| \le\|x - y\|\), \(\forall y \in K\). Putting \(z = P_{K} x\), we call \(P_{K}\) the metric projection of E onto K.
Lemma 4.1
[21]
- (i)
\(z = P_{K} x\);
- (ii)
\(\langle z - y, J(x - z)\rangle\ge0\), \(\forall y \in K\),
Definition 4.2
Hence from Theorem 3.1 we have the following.
Theorem 4.3
- (I)
If \(\Gamma_{1}\) (the set of solutions of hierarchical variational inequality problem (4.1)) is nonempty, then the sequence \(\{ x_{n}\}\) converges weakly to a point \(x^{*}\in\Gamma_{1}\).
- (II)
In addition, if \(\Gamma_{1}\) is nonempty and S is semi-compact, then \(\{x_{n}\}\) converges strongly to a point \(x^{*}\in\Gamma_{1}\).
5 A numerical example
Theorem 5.1
Declarations
Acknowledgements
This work was supported by the Scientific Research Fund of Sichuan Provincial Department of Science and Technology (No. 2015JY0165) and the Scientific Research Project of Yibin University (No. 2013YY06) and the National Natural Science Foundation of China (No. 11361070).
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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