Browder and Göhde fixed point theorem for monotone nonexpansive mappings
- Buthinah Abdullatif Bin Dehaish^{1}Email author and
- Mohamed Amine Khamsi^{2, 3}
https://doi.org/10.1186/s13663-016-0505-8
© Bin Dehaish and Khamsi 2016
Received: 26 August 2015
Accepted: 27 January 2016
Published: 20 February 2016
Abstract
Let X be a Banach space or a complete hyperbolic metric space. Let C be a nonempty, bounded, closed, and convex subset of X and \(T: C \rightarrow C\) be a monotone nonexpansive mapping. In this paper, we show that if X is a Banach space which is uniformly convex in every direction or a uniformly convex hyperbolic metric space, then T has a fixed point. This is the analog to Browder and Göhde’s fixed point theorem for monotone nonexpansive mappings.
Keywords
MSC
1 Introduction
‘The theory of fixed points is one of the most powerful tools of modern mathematics’ said Felix Browder, who gave a new impetus to the modern fixed point theory via the development of nonlinear functional analysis as an active and vital branch of mathematics. The flourishing field of fixed point theory started in the early days of topology (the work of Poincaré, Lefschetz-Hopf, and Leray-Schauder). For example, the existence problems are usually translated into a fixed point problem like the existence of solutions to elliptic partial differential equations, or the existence of closed periodic orbits in dynamical systems, and more recently the existence of answer sets in logic programming.
Recently a new direction has been discovered dealing with the extension of the Banach contraction principle [1] to metric spaces endowed with a partial order. Ran and Reurings [2] successfully carried out the first attempt see also [3]. In particular, they showed how this extension is useful when dealing with some special matrix equations. A similar approach was carried out by Nieto and Rodríguez-López [4] and used such arguments in solving some differential equations. In [5] Jachymski gave a more general unified version of these extensions by considering graphs instead of a partial order. In this paper, we investigate the existence of fixed points of monotone nonexpansive mappings. In particular, we prove that if X is a uniformly convex hyperbolic metric space, then any monotone nonexpansive mapping defined on a nonempty bounded convex subset has a fixed points.
In terms of content, this paper overlaps in places with the popular books on fixed point theory by Aksoy and Khamsi [6], by Goebel and Kirk [7], by Dugundji and Granas [8], by Khamsi and Kirk [9], and by Zeidler [10]. Material on the general theory of Banach space geometry and hyperbolic geometry is drawn from many sources but the books by Beauzamy [11], by Diestel [12], by Goebel and Reich [13], and by Bridson and Haefliger [14] are worth of special mention.
2 Preliminaries
The extension of the Banach contraction principle in metric spaces endowed with a partial order was initiated by Ran and Reurings [2] (see also [15–18]). In order to discuss such extension, we will need to assume that the metric space \((M,d)\) is endowed with a partial order ⪯. We will say that \(x, y \in M\) are comparable whenever \(x \preceq y\) or \(y \preceq x\). Next we give the definition of monotone mappings.
Definition 2.1
Next we give the definition of monotone Lipschitzian mappings.
Definition 2.2
Note that monotone Lipschitzian mappings are not necessarily continuous. They usually have a good topological behavior of comparable elements but not the entire set on which they are defined.
3 Monotone nonexpansive mappings in hyperbolic metric spaces
The fixed point theory for nonexpansive mappings finds its root in the works of Browder [19], Göhde [20], and Kirk [21] published in the same year 1965. Basically it took four decades to extend the contractive condition to the case of mappings with Lipschitz constant \(k =1\). It was clear from the start that such mappings have a different behavior from contraction mappings. The first results obtained in 1965 were discovered in Banach spaces. It took a few decades to extend the fixed point theory of nonexpansive mappings to nonlinear domains. Similarly and following the extension of the Banach contraction principle to the case of metric spaces endowed with a partial order, it was natural to try to investigate the case of nonexpansive mappings into such metric spaces.
In this section, we will establish Browder and Göhde’ s fixed point theorem for monotone nonexpansive mappings. The setting will be uniformly convex hyperbolic metric spaces.
Definition 3.1
Remark 3.1
[28]
- (i)
Let us observe that \(\delta(r,0)=0\), and \(\delta(r,\varepsilon)\) is an increasing function of ε for every fixed r.
- (ii)For \(r_{1}\leq r_{2}\) we have$$ 1- \frac{r_{2}}{r_{1}} \biggl(1-\delta \biggl(r_{2}, \varepsilon \frac {r_{1}}{r_{2}} \biggr) \biggr)\leq\delta(r_{1}, \varepsilon). $$
- (iii)If \((M,d)\) is uniformly convex, then \((M,d)\) is strictly convex, i.e., wheneverfor any \(x,y, a \in M\), then we must have \(x=y\).$$d \biggl(\frac{1}{2}x \oplus\frac{1}{2} y, a \biggr) = d(x,a) = d(y,a) $$
Among the nice properties satisfied by uniformly convex hyperbolic metric space \((X,d)\) is the property (R) [29] which says that if \(\{C_{n}\}\) is a decreasing sequence of nonempty, bounded, convex, and closed subsets of X, then \(\bigcap_{n \geq1} C_{n} \neq\emptyset\). The following technical lemma will be useful.
Lemma 3.1
Proof
In order to proceed, we will need the following fundamental result. Its origin may be found in [7, 32].
Proposition 3.1
Proof
Now we are ready to state the main result of this section.
Theorem 3.1
Let \((X, d, \preceq)\) be a partially ordered hyperbolic metric space as described above. Assume \((X,d)\) is uniformly convex. Let C be a nonempty convex closed bounded subset of X not reduced to one point. Let \(T: C \rightarrow C\) be a monotone nonexpansive mapping. Assume there exists \(x_{0} \in C\) such that \(x_{0}\) and \(T(x_{0})\) are comparable. Then T has a fixed point.
Proof
In the next section, we will show how to weaken the uniform convexity property when we assume that X is a vector space.
4 Monotone nonexpansive mappings in Banach spaces
Lemma 4.1
Proof
Since Banach spaces are hyperbolic metric spaces, we have a similar conclusion to Proposition 3.1.
Proposition 4.1
Using the same ideas in the proof of Theorem 3.1, we get the following fixed point result.
Theorem 4.1
Let \((X, \|\cdot\|, \preceq)\) be a partially ordered Banach such that order intervals are closed and convex. Assume X is uniformly convex in every direction. Let C be a nonempty weakly compact convex subset of X not reduced to one point. Let \(T: C \rightarrow C\) be a monotone nonexpansive mapping. Assume there exists \(x_{0} \in C\) such that \(x_{0}\) and \(T(x_{0})\) are comparable. Then T has a fixed point.
Declarations
Acknowledgements
This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks technical and financial support of DSR.
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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