1 Introduction

Fixed point theorems for monotone single-valued 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

$$H\bigl(F(x),F(y)\bigr)\leq k d(x,y) $$

for all \(x,y\in X\), where H is the Pompeiu-Hausdorff 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 single-valued 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. 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. 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 single-valued 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.

2 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

$$E\bigl(G^{-1}\bigr)=\bigl\{ (y,x)| (x,y)\in E(G)\bigr\} . $$

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,

$$E(\widetilde{G})=E(G)\cup E\bigl(G^{-1}\bigr). $$

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 \((x_{n-1},x_{n})\in E(G)\) 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

$$y\ {\mathcal{R}}\ z \mbox{ if there is a (directed) path in }G\mbox{ from }y\mbox{ to }z. $$

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 Pompeiu-Hausdorff distance [21] on \(\mathcal {CB} (X)\) is defined by

$$H(U,W):= \max\Bigl\{ \sup_{w\in W} d(w,A), \sup _{u\in U} d(u,W)\Bigr\} , $$

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 Pompeiu-Hausdorff 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

$$H\bigl(J(x),J(y)\bigr)\leq k d(x,y), $$

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

$$ f(x)= \textstyle\begin{cases} \frac{1}{2}x+\frac{1}{2}, & 0\leq x\leq\frac{1}{2}, \\ -\frac{1}{2}x+\frac{1}{2}, & \frac{1}{2}\leq x \leq1. \end{cases} $$

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 G-contraction if there exists \(k\in[0, 1)\) such that

$$H\bigl(F(x), F(y)\bigr) \leq k d(x, y),\quad \mbox{for all }(x,y)\in E(G) $$

and such that if \(u\in F(x)\) and \(v\in F(y)\) are such that

$$d(u,v)\leq k d(x,y)+\alpha,\quad \mbox{for each }\alpha> 0, $$

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

$$d(u,v)\leq k d(x,y), $$

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 G-contraction 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

([23, 24])

A multivalued mapping \(T: X \rightarrow2^{X}\) is said to be monotone increasing G-contraction 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

$$(U,W) \in E(G) \quad \mbox{and}\quad d(U,W) \leq\alpha d(u,v). $$

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 \((x_{n}, x)\in E(G)\).

3 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 G-contraction 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. (1)

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

  2. (2)

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

  3. (3)

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

  4. (4)

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

  5. (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 G-contraction, there exists \(x_{2} \in T(x_{1})\), \((x_{1},x_{2})\in E(G)\), such that

$$d(x_{1},x_{2}) \leq\alpha d(x_{0},x_{1}), $$

where \(\alpha< 1\) is associated to the definition of T being monotone increasing G-contraction. 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

$$d(x_{n},x_{n+1}) \leq\alpha d(x_{n},x_{n-1}) \leq\alpha^{n} d(x_{0},x_{1}), $$

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 G-contraction of T, there exists \(y_{n} \in T(x)\) such that \((x_{n+1}, y_{n}) \in E(G)\) and

$$d(x_{n+1},y_{n}) \leq\alpha d(x_{n},x), $$

for any \(n \geq1\). Hence

$$d(y_{n},x) \leq d(y_{n},x_{n+1}) + d(x_{n+1},x) \leq\alpha d(x_{n},x) + d(x_{n+1},x), $$

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 G-contraction \(T: X\rightarrow \mathcal {CB}(X)\) 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

$$\begin{aligned} E ( G ) =&\bigl\{ (0,0), ( 1,1 ),(2,2),(3,3),(0,1), ( 0,2 ),(0,3),(1,2), ( 1,3 ),(2,3)\bigr\} . \end{aligned}$$

Let \(V ( G ) \) be endowed with metric \(d:X\times X\rightarrow \mathbb{R} ^{+}\) defined by

$$\begin{aligned} d( 0,0) =&d( 1,1) =d(2,2)=d ( 3,3 ) =0, \\ d ( 0,1 ) =&d ( 1,0 ) =\frac{1}{4}, \\ d ( 0,2 ) =&d ( 2,0 ) =d ( 1,2 ) =d ( 2,1 )=d ( 1,3 ) =\cdots=d(3,2) = \frac{4}{5}. \end{aligned}$$

The graph of G is shown in Figure 1.

Figure 1
figure 1

G : Pompeiu-Hausdorff weighted graph.

Full size image

The Pompeiu-Hausdorff weights assigned to \(U,W\in CB ( X ) \) are

$$ H(U,W)= \textstyle\begin{cases} \frac{1}{4} & \text{if }U,W\subseteq\{0,1\}\text{ with }U\neq W, \\ \frac{4}{5} & \text{if }U\text{ or }W\text{ (or both)}\nsubseteq\{0,1\}\text{ with }U\neq W, \\ 0 & \text{if }U=W. \end{cases} $$

Define \(T: X \rightarrow \mathcal {CB}(X)\) as follows:

$$ T(x)= \textstyle\begin{cases} \{0\} & \text{if }x\in\{0,1\}, \\ \{1\} & \text{if }x\in\{2,3\}.\end{cases} $$

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 G-contraction with all other assumptions of Theorem 3.1 satisfied and T has 0 as a fixed point.