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Remarks on monotone multivalued mappings on a metric space with a graph
Journal of Inequalities and Applications volume 2015, Article number: 202 (2015)
Abstract
Let \((X,d)\) be a metric space and \(J: X\rightarrow2^{X}\) be a multivalued mapping. In this work, we discuss the definition of Gcontraction mappings introduced by Beg et al. (Comp. Math. Appl. 60:12141219, 2010) and show that it is restrictive and fails to give the main result of (Beg et al. in Comp. Math. Appl. 60:12141219, 2010). In this work, we give a new definition of the Gcontraction and obtain sufficient conditions for the existence of fixed points for such mappings.
Introduction
Fixed point theorems for monotone singlevalued mappings in a metric space endowed with a partial ordering have been widely investigated. These theorems are hybrids of the two most fundamental and useful theorems in fixed point theory: the Banach contraction principle [1], Theorem 2.1, and Tarski’s fixed point theorem [2, 3]. Generalizing the Banach contraction principle for multivalued mapping to metric spaces, Nadler [4] obtained the following result.
Theorem 1.1
([4])
Let \((X,d)\) be a complete metric space. Denote by \(\mathcal {CB}(X)\) the set of all nonempty closed bounded subsets of X. Let \(F: X \rightarrow \mathcal {CB}(X)\) be a multivalued mapping. If there exists \(k\in[0,1)\) such that
for all \(x,y\in X\), where H is the PompeiuHausdorff metric on \(CB(X)\), then F has a fixed point in X.
A number of extensions and generalizations of Nadler’s theorem were obtained by different authors; see for instance [5, 6] and references cited therein. The Tarski theorem was extended to multivalued mappings by different authors; see [7–9]. The existence of fixed points for singlevalued mappings in partially ordered metric spaces was initially considered by Ran and Reurings in [10], who proved the following result.
Theorem 1.2
([10])
Let \((X,\preceq)\) be a partially ordered set such that every pair \(x,y\in X\) has an upper and lower bound. Let d be a metric on X such that \((X,d)\) is a complete metric space. Let \(f: X\rightarrow X\) be a continuous monotone (either order preserving or order reversing) mapping. Suppose that the following conditions hold:

1.
There exists \(k\in[0,1)\) with
$$d\bigl(f(x),f(y)\bigr)\leq k d(x,y),\quad \textit{for all }x \succeq y. $$ 
2.
There exists an \(x_{0} \in X\) with \(x_{0} \preceq f(x_{0})\) or \(x_{0} \succeq f(x_{0})\).
Then f is a Picard Operator (PO), that is, f has a unique fixed point \(x^{*}\in X\) and for each \(x\in X\), \(\lim_{n\rightarrow \infty} f^{n}(x)=x^{*}\).
After this, various authors considered the problem of existence of a fixed point for contraction mappings in partially ordered metric spaces; see [11–14] and references cited therein. Nieto et al. in [14] extended the ideas of [10] to prove the existence of solutions to some differential equations. Recently, two results have appeared, giving sufficient conditions for f to be a PO, if \((X,d)\) is endowed with a graph. The first result in this direction was given by Jachymski and Lukawska [15, 16], who generalized the results of [12, 14, 17, 18] to singlevalued mapping in metric spaces with a graph instead of partial ordering.
The aim of this paper is twofold: first to give a correct definition of monotone multivalued mappings, second to extend the conclusion of Theorem 1.2 to the case of monotone multivalued mappings in metric spaces endowed with a graph.
Preliminaries
It seems that the terminology of graph theory instead of partial ordering gives a clearer picture and yields an interesting generalization of the Banach contraction principle. Let us begin this section with such a terminology for metric spaces as will be used throughout.
Let G be a directed graph (digraph) with the set of vertices \(V(G)\) and the set of edges \(E(G)\) contains all the loops, i.e. \((x,x) \in E(G)\) for any \(x \in V(G)\). We also assume that G has no parallel edges (arcs) and so we can identify G with the pair \((V(G),E(G))\). Our graph theory notations and terminology are standard and can be found in all graph theory books, like [19] and [20]. Moreover, we may treat G as a weighted graph (see [20], p.309]) by assigning to each edge the distance between its vertices. By \(G^{1}\) we denote the conversion of a graph G, i.e., the graph obtained from G by reversing the direction of edges. Thus we have
A digraph G is called an oriented graph if whenever \((u,v)\in E(G)\), then \((v,u)\notin E(G)\). The letter \(\widetilde{G}\) denotes the undirected graph obtained from G by ignoring the direction of edges. Actually, it will be more convenient for us to treat \(\widetilde{G}\) as a directed graph for which the set of its edges is symmetric. Under this convention,
We call \((V',E')\) a subgraph of G if \(V'\subseteq V(G)\), \(E'\subseteq E(G)\), and for any edge \((x,y)\in E'\), \(x, y\in V'\).
If x and y are vertices in a graph G, then a (directed) path in G from x to y of length N is a sequence \((x_{i})_{i=1}^{i=N}\) of \(N + 1\) vertices such that \(x_{0} = x\), \(x_{N} = y\), and \((x_{n1},x_{n})\in E(G)\) for \(i = 1,\ldots,N\). A graph G is connected if there is a directed path between any two vertices. G is weakly connected if \(\widetilde{G}\) is connected. If G is such that \(E(G)\) is symmetric and x is a vertex in G, then the subgraph \(G_{x}\) consisting of all edges and vertices which are contained in some path beginning at x is called the component of G containing x. In this case \(V(G_{x}) =[x]_{G}\), where \([x]_{G}\) is the equivalence class of the relation \(\mathcal{R}\) defined on \(V(G)\) by the rule
Clearly \(G_{x}\) is connected.
Definition 2.1
([21])
Let \((X,d)\) be a metric space and \(\mathcal {CB} (X)\) be the class of all nonempty closed and bounded subsets of X. The PompeiuHausdorff distance [21] on \(\mathcal {CB} (X)\) is defined by
for \(U,W\in \mathcal {CB} (X)\), where \(d(u,W):= \inf_{w\in W}d(u,w)\). The mapping H is said to be a PompeiuHausdorff metric induced by d.
Definition 2.2
([4])
Let \((X,d)\) be a metric space and \(\mathcal {CB} (X)\) be the class of all nonempty closed and bounded subsets of X. A multivalued map \(J : X \rightarrow \mathcal {CB} (X)\) is called contractive if there exists \(k \in[0,1)\) such that
for all \(x, y \in X\).
Example 2.1
Let \(I=[0,1]\) denote the unit interval of real numbers (with the usual metric) and let \(f: I\rightarrow I\) be given by
Define \(F: I\rightarrow2^{I}\) by \(F(x)=\{0\}\cup\{f(x)\}\) for each \(x\in I\). It is easy to verify that F is a multivalued contraction mapping with set of fixed points \(\{0,\frac{2}{3}\}\).
Example 2.2
Let \(I^{2}=\{(x,y) : 0\leq x \leq1 \mbox{ and } 0\leq y \leq 1\}\), and let \(F: I^{2} \rightarrow \mathcal {CB}(I^{2})\) be defined by \(F(x,y)\) is the line segment in \(I^{2}\) from the point \((\frac{1}{2}x,0)\) to the point \((\frac{1}{2}x,1)\) for each \((x,y)\in I^{2}\). It is easy to see that F is a multivalued contraction mapping with the set of fixed points \(\{(0,y) : 0\leq y \leq1 \}\).
Next we introduce the concept of monotone multivalued mappings. In [22], the authors offered the following definition.
Definition 2.3
([22], Def. 2.6)
Let \(F: X \rightsquigarrow X\) be a set valued mapping with nonempty closed and bounded values. The mapping F is said to be a Gcontraction if there exists \(k\in[0, 1)\) such that
and such that if \(u\in F(x)\) and \(v\in F(y)\) are such that
then \((u, v) \in E (G)\).
In particular, this definition implies that if \(u\in F(x)\) and \(v\in F(y)\) are such that
then \((u, v) \in E (G)\), which is very restrictive. In fact, in the proof of Theorem 3.1 in [22], there is absolutely no reason for \((x_{1},x_{2}) \in E(G)\). Definition 2.4 of Gcontraction multivalued mappings, inspired by the definition of contraction multivalued mappings in [23, 24], is more appropriate. In the sequel, we assume that \((X,d)\) is a metric space, and G is a directed graph (digraph) with the set of vertices \(V(G)=X\) and the set of edges \(E(G)\) contains all the loops, i.e. \((x,x) \in E(G)\), for any \(x \in X\).
Definition 2.4
A multivalued mapping \(T: X \rightarrow2^{X}\) is said to be monotone increasing Gcontraction if there exists \(\alpha\in[0,1)\) such that for any \(u, w \in X\) with \((u,w)\in E(G)\) and any \(U \in T(u)\) there exists \(W \in T(w)\) such that
Property 1
For any sequence \((x_{n})_{n\in\mathbb{N}}\) in X, if \(x_{n} \rightarrow x\) and \((x_{n}, x_{n+1})\in E(G)\) for \(n\in\mathbb{N}\), then \((x_{n}, x)\in E(G)\).
Main results
We begin with the following theorem, which gives the existence of a fixed point for monotone multivalued mappings in metric spaces endowed with a graph.
Theorem 3.1
Let \((X,d)\) be a complete metric space and suppose that the triple \((X,d,G)\) has property 1. Let \(T:X \rightarrow{ \mathcal{C}B}(X)\) be a monotone increasing Gcontraction mapping and \(X_{T}:=\{x\in X; (x,u)\in E(G)\textit{ for some }u\in T(x)\}\). If \(X_{T}\neq\emptyset\), then the following statements hold:

(1)
For any \(x\in X_{T}\), \(T_{[x]_{\widetilde{G}}}\) has a fixed point.

(2)
If G is weakly connected, then T has a fixed point in G.

(3)
If \(X':=\bigcup\{[x]_{\widetilde{G}} : x\in X_{T}\}\), then \(T_{X'}\) has a fixed point in X.

(4)
If \(T(X)\subseteq E(G)\) then T has a fixed point.

(5)
\(\operatorname {Fix}T\neq\emptyset\) if and only if \(X_{T}\neq\emptyset\).
Proof
1. Let \(x_{0} \in X_{T}\), then there exists \(x_{1} \in T(x_{0})\) such that \((x_{0}, x_{1})\in E(G)\). Since T is monotone increasing Gcontraction, there exists \(x_{2} \in T(x_{1})\), \((x_{1},x_{2})\in E(G)\), such that
where \(\alpha< 1\) is associated to the definition of T being monotone increasing Gcontraction. Without loss of generality, we may assume \(\alpha> 0\). By induction, we construct a sequence \(\{x_{n}\}\) such that \(x_{n+1} \in T(x_{n})\), \((x_{n},x_{n+1})\in E(G)\), and
for any \(n \geq1\). Since \(\sum_{n=0}^{\infty} d(x_{n},x_{n+1}) \leq d(x_{0},x_{1})\sum_{n=0}^{\infty} \alpha^{n} <\infty\), we conclude that \(\{x_{n}\}\) is a Cauchy sequence, and hence converges to some \(x \in X\) since X is a complete metric space. We claim that \(x \in T(x)\), i.e. x is a fixed point of T. Indeed using the definition of Gcontraction of T, there exists \(y_{n} \in T(x)\) such that \((x_{n+1}, y_{n}) \in E(G)\) and
for any \(n \geq1\). Hence
for any \(n \geq1\). This implies that \(\{y_{n}\}\) converges to x. Since \(T(x)\) is closed, we get \(x \in T(x)\) as claimed. As \((x_{n},x)\in E(G)\), for every \(n\geq0\), we conclude that \((x_{0},x_{1},\ldots,x_{n},x)\) is a path in G and so \(x\in[x_{0}]_{\widetilde{G}}\).
2. Since \(X_{T}\neq\emptyset\), there exists an \(x_{0}\in X_{T}\), and since G is weakly connected, then \([x_{0}]_{\widetilde {G}}=X\) and by 1, mapping T has a fixed point.
3. It follows easily from 1 and 2.
4. \(T(X) \subseteq E(G)\) implies that all \(x\in X\) are such that there exists some \(y\in T(x)\) with \((x,y)\in E(G)\); so \(X_{T}=X\) and by 2 and 3, T has a fixed point.
5. Assume \(\operatorname {Fix}T\neq\emptyset\). This implies that there exists an \(x \in \operatorname {Fix}T\) such that \(x\in T(x)\). \(\triangle\subseteq E(G)\) therefore \((x,x)\in E(G)\), which implies that \(x\in X_{T}\). So \(X_{T}\neq \emptyset\). Conversely if \(X_{T}\neq\emptyset\), then \(\operatorname {Fix}T\neq\emptyset\), follows from 2 and 3. □
Remark 3.1
The missing information in Theorem 3.1 is the uniqueness of the fixed point. In fact, we do have a partial positive answer to this question. Indeed if \(\bar{u}\) and \(\bar{w}\) are two fixed points of T such that \((\bar {u},\bar{w})\in E(G)\), then we must have \(\bar{u} = \bar{w}\). In general T may have more than one fixed point.
Remark 3.2
If we assume G is such that \(E(G):=X\times X\) then clearly G is connected and our Theorem 3.1 gives Nadler’s theorem [4].
The following is a direct consequence of Theorem 3.1.
Corollary 3.1
Let \((X, d)\) be a complete metric space and the triple \((X,d,G)\) have the Property 1. If G is weakly connected then every Gcontraction \(T: X\rightarrow \mathcal {CB}(X)\) such that \((x_{0}, x_{1})\in E(G)\), for some \(x_{1}\in T(x_{0})\), has a fixed point.
Example 3.1
Let \(X=\{0,1,2,3,4\}=V (G ) \) and
Let \(V ( G ) \) be endowed with metric \(d:X\times X\rightarrow \mathbb{R} ^{+}\) defined by
The graph of G is shown in Figure 1.
The PompeiuHausdorff weights assigned to \(U,W\in CB ( X ) \) are
Define \(T: X \rightarrow \mathcal {CB}(X)\) as follows:
Note that, for all \(x,y\in X\) with edge between x and y, there is an edge between \(T(x)\) and \(T(y)\). Also there is a path between x and y implies that there is a path between \(T(x)\) and \(T(y)\). Moreover, T is a Gcontraction with all other assumptions of Theorem 3.1 satisfied and T has 0 as a fixed point.
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Acknowledgements
The author is grateful to King Fahd University of Petroleum and Minerals for supporting this research. He would also like to thank Professor M. A. Khamsi who read carefully the earlier versions of this paper and suggested some improvements.
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Alfuraidan, M.R. Remarks on monotone multivalued mappings on a metric space with a graph. J Inequal Appl 2015, 202 (2015). https://doi.org/10.1186/s1366001507126
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DOI: https://doi.org/10.1186/s1366001507126
MSC
 47H09
 46B20
 47H10
 47E10
Keywords
 directed graph
 connected/weakly connected graph
 fixed point
 metric space
 monotone multivalued contraction mapping
 PompeiuHausdorff distance