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Fixed point theorems and iterative approximations for monotone nonexpansive mappings in ordered Banach spaces
- Yisheng Song^{1},
- Poom Kumam^{2}Email author and
- Yeol Je Cho^{2, 3}Email author
https://doi.org/10.1186/s13663-016-0563-y
© Song et al. 2016
- Received: 13 January 2016
- Accepted: 17 June 2016
- Published: 29 June 2016
Abstract
In this paper, we prove some existence theorems of fixed points of a monotone nonexpansive mapping T in a Banach space E with the partial order ‘≤’, where a such mapping may be discontinuous. In particular, in finite dimensional spaces, such a mapping T has a fixed point in E if and only if the sequence \(\{T^{n}0\}\) is bounded in E. In order to find a fixed point of such a mapping T, we prove the weak convergence of the Mann iteration scheme under the condition \(\sum_{n=1}^{\infty}\beta_{n}(1-\beta_{n})=\infty\), which entails \(\beta _{n}=\frac{1}{n+1}\) as a special case.
Keywords
- ordered Banach space
- fixed point
- monotone nonexpansive mapping
- λ-hybrid mapping
- Mann iterative scheme
MSC
- 47H06
- 47J05
- 47J25
- 47H10
- 47H17
- 49J40
- 47H04
- 65J15
1 Introduction
The following classical result for nonexpansive mappings was showed to still hold for α-nonexpansive mappings in a uniformly convex Banach space E.
Theorem 1.1
([2])
Let C be a nonempty and closed convex subset of uniformly convex Banach space E and \(T:C\to C\) be an α-nonexpansive mapping. Then \(F(T)\ne\emptyset\) if and only if \(\{T^{n}x\}\) is bounded for some \(x \in C\).
In this paper, we show the following existence theorem of fixed points for a monotone nonexpansive mapping T.
Theorem 1.2
Let K be a nonempty and closed convex subset of a uniformly convex Banach space \((E,\leq)\) and \(T : K\to K\) be a monotone nonexpansive mapping. Assume that there exists \(x\in K\) such that \(x\leq Tx\) (or \(Tx\leq x\)) and the sequence \(\{T^{n}x\}\) is bounded. Then \(F(T)\ne\emptyset\) and \(x\leq y^{*}\) (or \(y^{*}\leq x\)) for some \(y^{*}\in F(T)\).
2 Preliminaries and basic results
Definition 2.1
- (1)
strictly convex if \(\|\frac{x+y}{2}\|<1\) for all \(x,y\in E\) with \(\|x\|=\|y\|=1\) and \(x\neq y\);
- (2)
uniformly convex if, for all \(\varepsilon\in (0,2]\), there exists \(\delta>0\) such that \(\frac{\|x+y\|}{2}<1-\delta\) for all \(x,y\in E\) with \(\|x\|=\|y\|=1\) and \(\|x-y\|\geq\varepsilon\).
The following inequality was showed by Xu [16] in a uniformly convex Banach space E, which is known as Xu’s inequality.
Lemma 2.2
(Xu [16], Theorem 2)
The following conclusion is well known.
Lemma 2.3
(Takahashi [17], Theorem 1.3.11)
3 Main results
3.1 Existence of fixed points
In this section, we prove some existence theorems of fixed points of a monotone nonexpansive mapping in a uniformly convex Banach space \((E,\leq)\).
Theorem 3.1
Let K be a nonempty and closed convex subset of a uniformly convex Banach space \((E,\leq)\) and \(T : K\to K\) be a monotone nonexpansive mapping. Assume that there exists \(x\in K\) such that \(x\leq Tx\), the sequence \(\{T^{n}x\}_{n=1}^{\infty}\) is bounded. Then \(F(T)\ne\emptyset\) and \(y'\geq x\) for some \(y'\in F(T)\).
Proof
Theorem 3.2
Let K be a nonempty and closed convex subset of a uniformly convex Banach space \((E,\leq)\) and \(T : K\to K\) be a monotone nonexpansive mapping. Assume that there exists \(x\in K\) such that \(Tx\leq x\), the sequence \(\{T^{n}x\}_{n=1}^{\infty}\) is bounded and all \(n\geq1\). Then \(F(T)\ne\emptyset\) and \(y'\leq x\) for some \(y'\in F(T)\).
Proof
Theorem 3.3
Let E be a uniformly convex Banach space with the partial order ‘≤’ with respect to closed convex cone P and \(T : P\to P\) be a monotone nonexpansive mapping. Assume that the sequence \(\{T^{n}0\}_{n=1}^{\infty}\) is bounded. Then \(F(T)\ne\emptyset\).
Proof
It follows from the definition of the partial order ‘≤’ that \(0\leq T0\). Then the conclusions directly follow from Theorem 3.1. □
Denote \(\mathbb{R}^{m}=\{(r_{1}, r_{2},\ldots, r_{m}): r_{i}\in \mathbb{R}, i=1,2,\ldots,m\}\) and \(\mathbb{R}^{m}_{+}=\{(r_{1}, r_{2},\ldots, r_{m}): r_{i}\geq0, i=1,2,\ldots,m\}\), where \(\mathbb{R}\) is the set of all real numbers.
Theorem 3.4
Let \(T : \mathbb{R}^{m}_{+}\to\mathbb {R}^{m}_{+}\) be a monotone nonexpansive mapping. Assume that the sequence \(\{T^{n}0\}_{n=1}^{\infty}\) is bounded. Then \(F(T)\ne\emptyset\).
Proof
Let \(T^{n}0=(r^{(n)}_{1},r^{(n)}_{2},\ldots,r^{(n)}_{m})\in\mathbb {R}^{m}_{+}\). It follows from the boundedness of the sequence \(\{T^{n}0\}\) that there exist a positive real number r such that \(r^{(n)}_{i}\leq r\) for all n and \(i=1,2,\ldots,m\). Take \(y=(r,r,\ldots,r)\). So the conclusions directly follow from Theorem 3.3. □
Theorem 3.5
Let K be a nonempty and closed convex subset of a Banach space \((E,\leq)\) and \(T : K\to K\) be a monotone nonexpansive mapping. Assume that \(F(T)\ne\emptyset\) and there exist \(x\in K\) and \(p\in F(T)\) such that \(p\leq x\) (or \(x\leq p\)). Then the sequence \(\{T^{n}x\}\) is bounded.
Proof
Theorem 3.6
Let E be a Banach space with the partial order ‘≤’ with respect to closed convex cone P and \(T : P\to P\) be a monotone nonexpansive mapping. Assume that \(F(T)\ne \emptyset\). Then the sequence \(\{T^{n}0\}\) is bounded. Furthermore, the sequence \(\{T^{n}x\}\) is bounded for all \(x\in P\).
Proof
Theorem 3.7
Let \(T : \mathbb{R}^{m}_{+}\to\mathbb {R}^{m}_{+}\) be a monotone nonexpansive mapping. Then \(F(T)\ne\emptyset\) if and only if the sequence \(\{T^{n}0\}\) is bounded.
3.2 The convergence of the Mann iteration
The following lemma is showed by Dehaish and Khamsi [15], where the conclusion (3) is obtained from the proof of Lemma 3.1 in [15].
Lemma 3.8
(Dehaish and Khamsi [15], Lemmas 3.1 and 3.2)
- (1)
\(\{x_{n}\}\) is bounded and \(x_{n}\leq x_{n+1}\leq Tx_{n}\) (or \(Tx_{n}\leq x_{n+1}\leq x_{n}\));
- (2)
\(\lim_{n\to\infty}\|x_{n}-p\|\) exists;
- (3)
\(x_{n}\leq x\) (or \(x\leq x_{n}\)) for all \(n\geq1\) provided \(\{x_{n}\}\) weakly converges to a point \(x\in K\).
Theorem 3.9
Proof
Next, we show the weak convergence of the sequence \(\{x_{n}\}\) defined by (3.5). The proof is similar to the ones of Dehaish and Khamsi [15], but, for more details, we give the proof.
Theorem 3.10
Let K be a nonempty and closed convex subset of a uniformly convex Banach space \((E,\leq)\) and \(T : K\to K\) be a monotone nonexpansive mapping. Assume that E satisfies Opial’s condition and the sequence \(\{x_{n}\}\) is defined by (3.5) with \(x_{1}\leq Tx_{1}\) (or \(Tx_{1}\leq x_{1}\)). If \(F(T)\ne\emptyset\) and \(p\leq x_{1}\) (or \(x_{1}\leq p\)) for some \(p\in F(T)\), then \(\{x_{n}\}\) weakly converges to a fixed point \(x^{*}\) of T.
Proof
Declarations
Acknowledgements
The work was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission (NRU59 Grant No. 59000399). Also, the work was supported by the National Natural Science Foundation of P.R. China (Grant No. 11571095), Program for Innovative Research Team (in Science and Technology) in University of Henan Province (14IRTSTHN023). Yeol Je Cho was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and future Planning (2014R1A2A2A01002100). Moreover, this work was carried out while Yeol Je Cho was visiting Theoretical and Computational Science Center (TaCS), Science Laboratory Building, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangkok, Thailand, during 15 January-3 March 2016. He thanks Professor Poom Kumam and the university for their hospitality and support.
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Authors’ Affiliations
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