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Fixed point theorems and iterative approximations for monotone nonexpansive mappings in ordered Banach spaces
Fixed Point Theory and Applications volume 2016, Article number: 73 (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.
1 Introduction
Let T be a mapping with domain \(D(T)\) and range \(R(T)\) in a Banach space E. Then T is called nonexpansive if
for all \(x,y\in D(T)\). The fixed point set of T is denoted by \(F(T):= \{x\in K; T x = x\}\).
In 2010, Aoyama et al. [1] introduced a class of λhybrid mappings, that is, a mapping T is called a λhybrid mapping in Hilbert space H if
for all \(x,y\in D(T)\). They showed a fixed point theorem and an ergodic theorem for such a mapping. Clearly, a nonexpansive mapping is a 1hybrid mapping. In 2011, Aoyama and Kohsaka [2] also introduced the concept of αnonexpansive mapping, that is, a mapping T is called αnonexpansive if \(\alpha<1\) and
for all \(x,y\in D(T)\). Obviously, a nonexpansive mapping is 0nonexpansive and a λhybrid mapping is \(\frac{1\lambda }{2\lambda}\)nonexpansive if \(\lambda<2\) in a Hilbert space H (for more details, see [2]).
The following classical result for nonexpansive mappings was showed to still hold for αnonexpansive 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 αnonexpansive mapping. Then \(F(T)\ne\emptyset\) if and only if \(\{T^{n}x\}\) is bounded for some \(x \in C\).
Very recently, Bachar and Khamsi [3] introduced the concept of a monotone nonexpansive mapping in a Banach space E endowed with the partial order ‘≤’ and investigated common approximate fixed points of monotone nonexpansive semigroups. A mapping \(T:D(T)\to R(T)\) is called monotone nonexpansive if T is monotone (\(Tx\leq Ty\) whenever \(x\leq y\)) and
for all \(x,y\in D(T)\) with \(x\leq y\). Clearly, a monotone nonexpansive mapping may be discontinuous.
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 \(y^{*}\in F(T)\).
In order to finding a fixed point of a nonexpansive mapping T, Mann [4] introduced the following iteration scheme which is referred to as the Mann iteration: for any \(x_{1}\in D(T)\),
for each \(n\geq1\), where \(\beta_{n}\in[0,1]\) is a sequence with some conditions. Subsequently, many mathematical workers have been investigated the convergence of the Mann iteration and its modified version for nonexpansive mappings and pseudocontractions. For example, see [5–14]. However, there are not many convergence theorems of such an iteration in an ordered Banach space \((E,\leq)\). Recently, Dehaish and Khamsi [15] obtained the weak convergence of the Mann iteration for a monotone nonexpansive mapping provided \(\alpha _{n}\in[a,b]\subset(0,1)\). But their results do not entail \(\beta _{n}=\frac{1}{n+1}\).
Motivated by the above results, we consider the weak convergence of the Mann iteration scheme for a monotone nonexpansive mapping T under the condition
which contain \(\beta_{n}=\frac{1}{n+1}\) as a special case.
2 Preliminaries and basic results
Let P be a closed convex cone of a real Banach space E. A partial order ‘≤’ with respect to P in E is defined as follows:
for all \(x,y\in E\).
Throughout this paper, let E be a Banach space with the norm ‘\(\ \cdot\\)’ and the partial order ‘≤’. Let \(F(T)=\{x\in H: Tx=x\}\) denote the set of all fixed points of a mapping T. An order interval \([x,y]\) for all \(x,y\in E\) is given by
Obviously, the order interval \([x,y]\) is closed and convex. In fact, let \(z_{1},z_{2}\in[x,y]\). Then \(z_{1}x\in P\), \(z_{2}x\in P\), \(yz_{1}\in P\), and \(yz_{2}\in P\); and so, for any \(t\in(0,1)\),
Thus \(tz_{1}+(1t)z_{2}\in[x,y]\), that is, \([x,y]\) is convex. Let \(\{z_{n}\} \subset[x,y]\) with \(\lim_{n\to\infty}z_{n}=z\). Then, for each \(n\geq 1\), \(z_{n}x\in P\) and \(yz_{n}\in P\), and hence we have
that is, \(x\leq z\leq y\) and so \(z\in[x,y]\), that is, \([x,y]\) is closed. Then the convexity of the order interval \([x,y]\) implies that
for all \(x,y\in E\) with \(x\leq y\).
Definition 2.1
Let K be a nonempty closed and convex subset of a Banach space E. A mapping \(T : K \to E\) is said to be:

(1)
monotone [3] if \(Tx\leq Ty\) for all \(x, y \in K\) with \(x\leq y\);

(2)
monotone nonexpansive [3] if T is monotone and
$$\TxTy\\leq\xy\ $$for all \(x, y \in K\) with \(x\leq y\).
A Banach space E is said to be:

(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 \(\xy\\geq\varepsilon\).
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)
For any real numbers \(q>1\) and \(r>0\), a Banach space E is uniformly convex if and only if there exists a continuous strictly increasing convex function \(g:[0,+\infty)\to[0,+\infty)\) with \(g(0)=0\) such that
for all \(x,y\in B_{r}(0)=\{x\in E; \x\\leq r\}\) and \(t\in[0,1]\), where \(\omega(q,t)=t^{q}(1t)+t(1t)^{q}\). In particular, take \(q=2\) and \(t=\frac{1}{2}\),
The following conclusion is well known.
Lemma 2.3
(Takahashi [17], Theorem 1.3.11)
Let K be a nonempty closed convex subset of a reflexive Banach space E. Assume that \(\varphi:K\to R\) is a proper convex lower semicontinuous and coercive function. Then the function φ attains its minimum on K, that is, there exists \(x\in K\) such that
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
Let \(x_{1}=x\) and \(x_{n+1}=Tx_{n}=T^{n}x\). Then \(x_{1}=x\leq Tx=x_{2}\), and so,
By analogy, we must have
Let \(K_{n}=\{z\in K:x_{n}\leq z\}\) for all \(n\geq1\). Clearly, for each \(n\geq1\), \(K_{n}\) is closed convex and \(y\in K_{n}\) and so \(K_{n}\) is nonempty too. Let \(K^{*}=\bigcap_{n=1}^{\infty}K_{n}\). Then \(K^{*}\) is a nonempty closed convex subset of K. Since \(\{x_{n}\}\) is bounded, we can define a function \(\varphi:K^{*}\to[0,+\infty)\) as follows:
for all \(z\in K^{*}\). From Lemma 2.3, it follows that there exists \(y^{*}\in K_{1}\) such that
Now, we show \(y^{*}=Ty^{*}\). In fact, by the definition of \(K^{*}\), we obtain
Then we have \(x_{n+1}=Tx_{n}\leq Ty^{*}\) by the monotonicity of T and hence, for each \(n\geq1\), \(x_{n}\leq Ty^{*}\). So we have \(Ty^{*}\in K^{*}\). From the convexity of \(K^{*}\), it follows that \(\frac{y^{*}+Ty^{*}}{2}\in K^{*}\) and so, by (3.1), we have
On the other hand, we have
Combining (3.2) and (3.3), we have
It follows from Lemma 2.2 (\(q=2\) and \(t=\frac{1}{2}\)) and (3.4) that
Noticing (3.2), we have
and so \(g(\y^{*}Ty^{*}\)=0\). Thus we have \(y^{*}=Ty^{*}\) by the property of g. This yields the desired conclusion. This completes the 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
Let \(x_{1}=x\), \(x_{n+1}=Tx_{n}=T^{n}x\), and let \(K_{n}=\{z\in K:z\leq x_{n}\}\) for all \(n\geq1\). Using the same proof technique of Theorem 3.1, it is easy to obtain
for all \(n\geq1\) and \(K^{*}=\bigcap_{n=1}^{\infty}K_{n}\) is a nonempty closed convex subset of K. The remainder of the proof is the same as ones of Theorem 3.1 and so we omit it. □
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
Let \(x_{1}=x\) and \(x_{n+1}=Tx_{n}=T^{n}x\). Then it follows from the conditions \(p=Tp\) and \(p\leq x\) (or \(x\leq p\)) that \(p=Tp\leq Tx_{n}=x_{n+1}\) (or \(x_{n+1}=Tx_{n}\leq Tp=p\)) for all \(n\geq1\) and so
and so \(\x_{n}p\\leq\xp\\) for all \(n\geq1\) and hence the sequence \(\{T^{n}x\}\) is bounded. This completes the 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
It follows from the definition of T that \(0\leq p\) for all \(p\in F(T)\). Then the conclusion that \(\{T^{n}0\}\) is bounded directly follows from Theorem 3.5. For each \(x\in P\), it is obvious that \(0\leq x\) and hence, by the monotonicity of T, we have
It follows from the definition of a monotone nonexpansive mapping that
and so the sequence \(\{T^{n}x\}\) is bounded. The desired conclusion follows. This completes the 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.
Proof
The conclusions directly follow from Theorems 3.4 and 3.6. □
3.2 The convergence of the Mann iteration
In this section, for a monotone nonexpansive mapping T, we consider the Mann iteration sequence defined by
for each \(n\geq1\), where \(\{\beta_{n}\}\) in \((0,1)\) satisfies the following condition:
Clearly, the above condition contains \(\beta_{n}=\frac{1}{n+1}\) as a special case.
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)
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 the sequence \(\{x_{n}\}\) is defined by (3.5) and \(x_{1}\leq Tx_{1}\) (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

(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
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 the sequence \(\{x_{n}\}\) is defined by (3.5) and \(x_{1}\leq Tx_{1}\) (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
Proof
It follows from Lemma 3.8 that
for all \(n\geq1\). Then it follows from the nonexpansiveness of T, \(p=Tp\), and an application of Lemma 2.2 (\(q=2\) and \(t=\beta _{n}\)) that
and so
Therefore, we have
Now, we claim that there exists a subsequence \(\{x_{n_{k}}\}\) such that
Suppose that the conclusion is not true. Then, for all subsequences \(\{ x_{n_{k}}\}\) such that \(\lim_{k\to\infty}g(\x_{n_{k}}Tx_{n_{k}}\)>0\), we have
Thus there exists a positive number a and a positive integer N such that \(g(\x_{n}Tx_{n}\)>a>0\) for all \(n>N\). Consequently, we have
and hence, by the condition \(\sum_{n=1}^{\infty}\beta_{n}(1\beta _{n})=+\infty\), we obtain
This contradicts (3.6). So (3.7) holds and hence, by the property of g, we have
On the other hand, we have
Therefore, the sequence \(\{\x_{n}Tx_{n}\\}\) is monotonically nonincreasing and hence it follows that \(\lim_{n\to\infty}\ x_{n}Tx_{n}\\) exists. This yields the desired conclusion. This completes the proof. □
Recall that a Banach space E is said to satisfy Opial’s condition [12] if a sequence \(\{x_{n}\}\) with \(\{x_{n}\}\) weakly converges to a point \(x\in E\) implies
for all \(y\in E\) with \(y\neq x\).
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
It follows from Lemma 3.8 and Theorem 3.9 that \(\{x_{n}\}\) is bounded and
Then there exists a subsequence \(\{x_{n_{k}}\}\subset\{x_{n}\}\) such that \(\{x_{n_{k}}\}\) weakly converges to a point \(x^{*}\in K\). Following Lemma 3.8, we have \(x_{1}\leq x_{n_{k}}\leq x^{*}\) (or \(x^{*}\leq x_{n_{k}}\leq x_{1}\)) for all \(k\geq1\). In particular, we have
Now, we claim \(x^{*}=Tx^{*}\). In fact, suppose that this is not true. Then, from the nonexpansiveness of T and Opial’s condition, it follows that
which is a contradiction. Thus, by Lemma 3.8(2), it follows that the limit \(\lim_{n\to\infty}\x_{n}x^{*}\\) exists.
Now, we show that \(\{x_{n}\}\) weakly converges to the point \(x^{*}\). Suppose that this is not true. Then there exists a subsequence \(\{x_{n_{j}}\}\) that converges weakly to a point \(z\in K\) and \(z\ne x^{*}\). Similarly, it follows that \(z=Tz\) and \(\lim_{n\to\infty}\x_{n}z\\) exists. It follows from Opial’s condition that
This is a contradiction and hence \(x^{*}=z\). This completes the proof. □
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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 January3 March 2016. He thanks Professor Poom Kumam and the university for their hospitality and support.
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Song, Y., Kumam, P. & Cho, Y.J. Fixed point theorems and iterative approximations for monotone nonexpansive mappings in ordered Banach spaces. Fixed Point Theory Appl 2016, 73 (2016). https://doi.org/10.1186/s136630160563y
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DOI: https://doi.org/10.1186/s136630160563y