On monotone pointwise contractions in Banach spaces with a graph
 Monther Rashed Alfuraidan^{1}Email author
https://doi.org/10.1186/s1366301503906
© Alfuraidan 2015
Received: 21 February 2015
Accepted: 24 July 2015
Published: 12 August 2015
Abstract
In this work, we give a new definition of Gmonotone pointwise contraction mappings in metric spaces endowed with a graph. Then we obtain sufficient conditions for the existence of a fixed point for such mappings. The proofs are based on the crucial inequality (GK).
Keywords
MSC
1 Introduction
The notion of asymptotic pointwise mappings was introduced in [1–4]. The use of ultrapower technique was useful in proving some related fixed point results. In the paper [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 [5], most of these results were extended to metric spaces. In this paper, we introduce the new concept of Gmonotone mappings in Banach 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 by Ran and Reurings [6]. In particular, they show how this extension is useful when dealing with some special matrix equations. Another similar approach was carried by Nieto and RodríguezLópez [7] who used such arguments in solving some differential equations. In [8], Jachymski gave a more general unified version of these extensions by considering graphs instead of a partial order. Recently, the author [9] showed the existence of fixed points for monotone multivalued mappings on a metric space with a graph.
In this work, we investigate the fixed point theory of pointwise Gmonotone contraction mappings. In particular, we will extend the main result of [3] to the case of Gmonotone mappings. Our approach is new and different from the ideas found in [6, 7]. This work was inspired by [10].
For more on metric fixed point theory, the reader may consult the book [11].
2 Graph basic definitions
The terminology of graph theory instead of partial ordering gives a wider picture and yields interesting generalization of the Banach contraction principle. In this section, we give the basic graph theory definitions and notations which will be used throughout.
Given a digraph \(G = (V ,E)\), a (di)path of G is a sequence \(a_{0}, a_{1}, \ldots, a_{n}, \ldots\) with \((a_{i}, a_{i+1} )\in E(G)\) for each \(i = 0, 1, 2, \ldots\) . A finite path \((a_{0}, a_{1},\ldots, a_{n})\) is said to have length \(n+1\) for \(n \in\mathbb{N}\). A closed directed path of length \(n>1\) from x to y, i.e., \(x=y\), is called a directed cycle. An acyclic digraph is a digraph that has no directed cycle. A digraph is connected if there is a finite (di)path joining any two of its vertices and it is weakly connected if \(\widetilde{G}\) is connected.
Definition 2.1
Definition 2.2
Let \((X, \\cdot\)\) be a Banach space. ω is called a weakcluster point of a sequence \((x_{n})_{n\in\mathbb{N}}\) in X if there exists a subsequence \((x_{\phi(n)})_{n\in\mathbb{N}}\) such that \((x_{\phi(n)})_{n\in\mathbb{N}}\) converges weakly to ω.
As Jachymski [8] did, we introduce the following property.
 (∗):

for any sequence \((x_{n})_{n \in\mathbb{N}}\) in X such that \((x_{n}, x_{n+1})\in E(G)\) for \(n \in\mathbb{N}\) and ω is a weakcluster point of \((x_{n})_{n\in\mathbb{N}}\), then there exists a subsequence \((x_{\phi(n)})_{n\in\mathbb{N}}\) which converges weakly to ω and \((x_{\phi(n)}, \omega)\in E(G)\) for every \(n \geq1\).

for any sequence \((x_{n})_{n \geq1}\) in X such that \((x_{n}, x_{n+1})\in E(G)\) for \(n \geq1\) and ω is a weakcluster point of \((x_{n})_{n\geq1}\), we have \((x_{n}, \omega)\in E(G)\) for every \(n \geq1\).
Let us finish this section with the following example of a transitive cyclic digraph which can not be generated by a partial order. Therefore our approach is different from the one used in [10] which is based on the use of a partial order in Banach and metric spaces.
Example 2.1

\((x,y)\in E(G)\) if and only if \(x_{i} \leq y_{i}\), for \(i =2, \ldots \) , where \(x=(x_{n})\) and \(y=(y_{n})\) are in \(l_{2}\).
3 Monotone pointwise contraction mappings
Let us start this section by defining Gmonotone pointwise Lipschitzian mappings.
Definition 3.1
 (1)
Gmonotone if T is edge preserving, i.e., \((T(x),T(y))\in E(G)\) whenever \((x,y)\in E(G)\) for any \(x, y \in C\).
 (2)Gmonotone pointwise Lipschitzian if T is Gmonotone and for any \(x \in X\), there exists \(k(x) \in[0,+\infty)\) such that$$d\bigl(T(x),T(y)\bigr) \leq k(x) d(x,y) \quad \text{for any }y\in C \text{ such that } (x,y)\in E(\widetilde{G}). $$
If \(k(x) \in[0,1)\) for any \(x \in X\), then T is said to be a Gmonotone pointwise contraction mapping. If \(k(x) \leq1\) for any \(x \in X\), then T is said to be a Gmonotone nonexpansive mapping. A fixed point of T is any element \(x \in C\) such that \(T(x) =x\). The set of all fixed points of T is denoted by \(\operatorname{Fix}(T)\).
It is clear that the pointwise contractive concept was introduced to extend the contractive behavior in the Banach contraction principle.
Example 3.1
 (1)
\((0, x)\) and \((1, y)\) are not connected for any \(x, y \in K\);
 (2)
\((\varepsilon, x)\) and \((\varepsilon, y)\) are connected if and only if \(x \leq y\) (using the natural pointwise order in \(l^{2}\)) for any \(\varepsilon\in\{0, 1\}\) and \(x, y \in K\).
For more examples on fixed points of multivalued mappings on metric spaces endowed with a graph, see [13].
The fundamental fixed point result for pointwise contraction mappings is the following theorem.
Theorem 3.1
Let C be a weakly compact convex subset of a Banach space and suppose that \(T: C\rightarrow C\) is a pointwise contraction. Then T has a unique fixed point z. Moreover, the orbit \((T^{n}(x))_{n \geq1}\) converges to z for each \(x \in C\).
Note that if T is a Gmonotone pointwise Lipschitzian mapping, then it is not necessarily continuous by contrast to the case of pointwise Lipschitzian mappings. Since the main focus of this paper is about the existence of the fixed points, we have the following result.
Theorem 3.2
Let \((X, d)\) be a metric space and G be a reflexive digraph defined on X. Let C be a nonempty subset of X. Let \(T: C \rightarrow C\) be a Gmonotone pointwise contraction. If \(a \in \operatorname{Fix}(T)\), then for any \(x \in X\) such that \((a,x)\in E(G)\), we have \((T^{n}(x))_{n \geq1}\) converges to a. In particular, if a and b are two fixed points of T and \((a, b)\in E(G)\), then we must have \(a = b\).
Proof
Remark 3.1
In both Banach and metric spaces [1, 5], the pointwise contraction mappings have at most one fixed point. But in the case of Gmonotone pointwise contraction mappings, we may have more than one fixed point. Indeed Jachymski [8] proved that Gcontractions have a fixed point in each component of elements that are compatible. Since we do not assume the weak connectivity of the digraph G, we may have more than one component which implies the possibility to have more than one fixed point.
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.
4 Existence of fixed point of monotone pointwise contractions
 (CG)If \((x,y)\in E(G)\) and \((w,z) \in E(G)\), thenfor all \(x,y,w,z\in X\) and \(\alpha\in\mathbb{R}^{+}\).$$\bigl(\alpha x + (1\alpha) w,\alpha y + (1\alpha) z\bigr)\in E(G) $$
Lemma 4.1
 (i)
If \((x_{1} , T(x_{1}))\in E(G)\), then we have \((x_{n}, x_{n+1}) \in E(G)\) for any \(n \geq1\).
 (ii)
If \((T(x_{1}), x_{1})\in E(G)\), then we have \((x_{n+1}, x_{n}) \in E(G)\) for any \(n \geq1\).
Proof
In order to show that the main property satisfied by the sequence is defined by (KIS), we need the following result which may be found in [14, 15]. We will give its proof here.
Lemma 4.2
Proof
As a direct consequence of Lemma 4.2, we get the following result.
Theorem 4.1
Let \((X, \\cdot\)\) be a Banach space and G be a reflexive digraph defined on X. Assume that \(E(G)\) has properties (∗) and (CG). Let C be a bounded nonempty convex subset of X. Let \(T: C \rightarrow C\) be a Gmonotone nonexpansive mapping. Assume that there exists \(x_{1} \in C\) such that \((x_{1} , T(x_{1}))\in E(\widetilde{G})\). Consider the sequence \((x_{n})_{n \geq1}\) defined by (KIS). Then we have \(\lim_{n \rightarrow \infty} \x_{n}  T(x_{n})\=0\).
Proof
Theorem 4.2
Let \((X, \\cdot\)\) be a Banach space and G be a reflexive digraph defined on X. Assume that \(E(G)\) has properties (∗) and (CG). Assume X satisfies the large Opial property. Let C be a weakly compact nonempty convex subset of X. Let \(T: C \rightarrow C\) be a Gmonotone pointwise contraction. Assume that there exists \(x_{1} \in C\) such that \((x_{1}, T(x_{1}))\in E(G)\). Then T has a fixed point.
 (i)
\([a, \rightarrow) = \{x \in X; (a,x) \in E(G)\}\),
 (ii)
\((\leftarrow, a] = \{x \in X; (x,a) \in E(G)\}\)
Theorem 4.3
Let \((X, \\cdot\)\) be a Banach space and G be a reflexive transitive digraph defined on X. Assume that \(E(G)\) has property (CG) and Gintervals are closed. Let C be a weakly compact nonempty convex subset of X. Let \(T: C \rightarrow C\) be a Gmonotone pointwise contraction. Assume that there exists \(x_{1} \in C\) such that \((x_{1}, T(x_{1}))\in E(\widetilde{G})\). Then T has a fixed point.
Proof
Declarations
Acknowledgements
The author acknowledges King Fahd University of Petroleum and Minerals for supporting this research.
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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