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On monotone pointwise contractions in Banach and metric spaces
- Afrah AN Abdou^{1}Email author and
- Mohamed A Khamsi^{2, 3}
https://doi.org/10.1186/s13663-015-0381-7
© Abdou and Khamsi 2015
- Received: 26 January 2015
- Accepted: 10 July 2015
- Published: 4 August 2015
Abstract
In this work we define the new concept of monotone pointwise contraction mappings in Banach and metric spaces. Then we prove the existence of fixed points of such mappings.
Keywords
- fixed point
- \(\operatorname{CAT}(0)\) spaces
- hyperbolic metric spaces
- monotone mapping
- pointwise contraction
MSC
- 47H09
- 46B20
- 47H10
- 47E10
1 Introduction
The notion of asymptotic pointwise mappings was introduced in [1–3]. The ultrapower technique was useful in proving some related fixed point results. In [3], the authors gave simple and elementary proofs for the existence of fixed point theorems for asymptotic pointwise mappings without the use of ultrapowers. In [4], most of these results were extended to metric spaces. In this work, we introduce the new concept of monotone mappings in Banach and metric spaces. Indeed recently a new direction has been discovered dealing with the extension of the Banach contraction principle to metric spaces endowed with a partial order. The first attempt was successfully carried out by Ran and Reurings [5]. In particular, they showed how this extension is useful when dealing with some special matrix equations. Another similar approach was carried out by Nieto and Rodríguez-López [6] and they used such arguments in solving some differential equations. In [7], Jachymski gave a more general unified version of these extensions by considering graphs instead of a partial order.
In this work, we investigate the fixed point theory of pointwise contraction mappings to the case of monotone mappings. In particular, we will extend the main result of [3] to the case of monotone mappings. Our approach is new and different from the ideas found in [5, 6].
For more on metric fixed point theory, the reader may consult the book [8].
2 Preliminaries
- (i)
\(x\leq y\Rightarrow x+z\leq y+z\) for all \(x,y,z\in X\);
- (ii)
\(x\leq y\Rightarrow\alpha x\leq\alpha y\) for all \(x,y\in X\) and \(\alpha\in\mathbb{R}^{+}\).
- (i)
\([a,\rightarrow) = \{x \in X; a \preceq x\}\),
- (ii)
(\(\leftarrow,a] = \{x \in X; x \preceq a\}\),
Definition 1
- (a)
monotone if \(T(x) \leq T(y)\) whenever \(x \leq y\),
- (b)monotone pointwise Lipschitzian, if T is monotone and for any \(x \in X\), there exists \(k(x) \in[0,+\infty)\) such thatfor any \(y \in X\) such that x and y are comparable, i.e., \(x \leq y\) or \(y \leq x\).$$\bigl\Vert T(x)-T(y)\bigr\Vert \leq k(x) \|x-y\|, $$
It is clear that pointwise contractive behavior was introduced to extend the contractive behavior in Banach contraction principle [9].
Example 1
- (i)
\((0,x)\) and \((1,y)\) are not comparable for any \(x, y \in K\);
- (ii)
\((\varepsilon, x) \preceq(\varepsilon, y)\) if and only if \(x \leq y\) (using the natural pointwise order in \(\ell^{2}\)), for any \(\varepsilon\in\{0,1\}\) and \(x, y \in K\).
The central fixed point result for pointwise contraction mappings is the following theorem [1, 2].
Theorem 2.1
Let K be a weakly compact convex subset of a Banach space and suppose \(T: K\rightarrow K\) is a pointwise contraction. Then T has a unique fixed point, \(x_{0}\). Moreover, the orbit \(\{T^{n}(x)\}\) converges to \(x_{0}\), for each \(x \in M\).
Note that if T is a monotone pointwise contraction, then it is not necessarily continuous by contrast to the pointwise contraction case. Since the main focus of this paper is the fixed point problem, we have the following result.
Lemma 2.1
Let \((X, \|\cdot\|, \leq)\) be as above. Let C be a nonempty subset of X. Let \(T: C \rightarrow X\) be a monotone pointwise contraction. If \(a \in \operatorname{Fix}(T)\), then for any \(x \in X\) comparable to a, i.e., \(a \leq x\) or \(x \leq a\), we have \(\{T^{n}(x)\}\) converges to a. In particular, if a and b are two comparable fixed points of T, then we must have \(a = b\).
Proof
The crucial part in dealing with pointwise contractions is the existence of the fixed point. Usually it takes more assumptions than the classical Banach contraction principle.
3 Existence of fixed point of monotone pointwise contractions in Banach spaces
Theorem 3.1
Let \((X, \|\cdot\|, \leq)\) be as above. Let C be a weakly compact nonempty convex subset of X. Let \(T: C \rightarrow C\) be a monotone pointwise contraction. Assume there exists \(x_{1} \in C\) such that \(x_{1} \leq T(x_{1})\). Consider the sequence \(\{x_{n}\}\) defined by (1). Then \(\{x_{n}\}\) is weakly convergent and \(\lim_{n \rightarrow\infty} \|x_{n} - T(x_{n})\|=0\).
Proof
Remark 1
Let ω be the weak limit of \(\{x_{n}\}\). Using the properties of the order intervals, we conclude that \(x_{n} \leq\omega\), for any \(n \geq1\). The monotonicity of T implies \(T(x_{n}) \leq T(\omega)\), for any \(n \geq1\). Since \(\{x_{n}\}\) is a quasi-fixed point sequence of T, i.e., \(\lim_{n \rightarrow+\infty} \|x_{n} - T(x_{n})\| = 0\), we conclude that \(\{T(x_{n})\} \) also weakly converges to ω, which implies \(\omega\leq T(\omega)\).
Next we prove the main result of this work, which can be seen as an analog to Theorem 2.1.
Theorem 3.2
Let \((X, \|\cdot\|, \leq)\) be as above. Let C be a weakly compact nonempty convex subset of X. Let \(T: C \rightarrow C\) be a monotone pointwise contraction. Assume there exists \(x_{1} \in C\) such that \(x_{1} \leq T(x_{1})\). Then T has a fixed point \(z \in C\) and \(\{T^{n}(x_{1})\}\) converges to z.
Proof
Note that the conclusion of Theorem 3.2 is still valid if we assume \(T(x_{1}) \leq x_{1}\). In this case the Krasnoselskii sequence \(\{x_{n}\}\) will be monotone decreasing and its limit will be less than \(x_{1}\).
4 Existence of fixed point of monotone pointwise contractions in metric spaces
In this section, we discuss the analog of Theorem 3.2 in metric spaces as the authors of [4] did. The approach of both the linear and the nonlinear cases uses an intersection property of convex sets [3] of admissible sets [4]. Since the proof of the main result of the previous section is based on the Krasnoselskii iteration, we will need some kind of convex combination. This is the reason why we do need some kind of metric convexity.
Obviously, normed linear spaces are hyperbolic spaces. As nonlinear examples, one can consider the Hadamard manifolds [14], the Hilbert open unit ball equipped with the hyperbolic metric [15], and the \(\operatorname{CAT}(0)\) spaces [16–18]. We will say that a subset C of a hyperbolic metric space X is convex if \([x,y]\subset C\) whenever x, y are in C.
Definition 2
Let \((X,d)\) be a hyperbolic metric space. Let C be a nonempty convex closed subset of X. We say that C is weakly compact if and only if for any decreasing sequence \(\{C_{n}\}\) of closed nonempty convex subsets of C, we see that \(\bigcap_{n \geq1} C_{n}\) is not empty.
- (i)
\([a,\rightarrow) = \{x \in X; a \preceq x\}\),
- (ii)
\((\leftarrow,a] = \{x \in X; x \preceq a\}\),
Definition 3
- (a)
monotone if \(T(x) \leq T(y)\) whenever \(x \leq y\),
- (b)monotone pointwise Lipschitzian, if T is monotone and for any \(x \in X\), there exists \(k(x) \in[0,+\infty)\) such thatfor any \(y \in X\) such that x and y are comparable.$$d\bigl(T(x),T(y)\bigr) \leq k(x) d(x,y), $$
The following technical lemma is the metric version of the linear case above.
Lemma 4.1
- (i)
\(x_{n} \leq x_{n+1} \leq T(x_{n})\), for any \(n \geq1\);
- (ii)
\(\lim_{n \rightarrow\infty} \|x_{n} - T(x_{n})\|=0\).
The metric version of Theorem 3.2 is the following.
Theorem 4.1
Let \((X, d, \leq)\) be as above. Let C be a weakly compact nonempty convex subset of X. Let \(T: C \rightarrow C\) be a monotone pointwise contraction. Assume there exists \(x_{1} \in C\) such that \(x_{1} \leq T(x_{1})\). Then T has a fixed point \(z \in C\) and \(\{T^{n}(x_{1})\}\) converges to z.
Proof
Note that the conclusion of Theorem 4.1 is still valid if we assume \(T(x_{1}) \leq x_{1}\).
Declarations
Acknowledgements
This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks DSR technical and financial support.
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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