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Fixed point results for setcontractions on metric spaces with a directed graph
Fixed Point Theory and Applications volume 2015, Article number: 14 (2015)
Abstract
In this paper, we establish the existence of fixed points for setvalued mappings satisfying certain graph contractions with setvalued domain endowed with a graph. These results unify, generalize, and complement various known comparable results in the literature.
Introduction and preliminaries
Existence of fixed points in ordered metric spaces has been studied by Ran and Reurings [1]. Recently, many researchers have obtained fixed point results for single and setvalued mappings defined on partially ordered metrics spaces (see, e.g., [2–6]). Jachymski and Jozwik [7] introduced a new approach in metric fixed point theory by replacing the order structure with a graph structure on a metric space. In this way, the results proved in ordered metric spaces are generalized (see also [8] and the references therein); in fact, in 2010, GwozdzLukawska and Jachymski [9], developed the HutchinsonBarnsley theory for finite families of mappings on a metric space endowed with a directed graph. Abbas and Nazir [10] obtained some fixed point results for power graph contraction pair endowed with a graph. Bojor [11] proved fixed point theorem of φcontraction mapping on a metric space endowed with a graph. Recently, Bojor [12] proved fixed point theorems for Reich type contractions on metric spaces with a graph. For more results in this direction, we refer to [13–17] and the references mentioned therein. The reader interested in fixed point results of partial metric spaces is referred to [2, 10, 18]. In this paper, we prove fixed point results for setvalued maps, defined on the family of closed and bounded subsets of a metric space endowed with a graph and satisfying graph ϕcontractive conditions. These results extend and strengthen various known results in [7, 8, 11, 19–21].
Consistent with Jachymski [8], let \((X,d)\) be a metric space and Δ denotes the diagonal of \(X\times X\). Let G be a directed graph, such that the set \(V(G)\) of its vertices coincides with X and \(E(G)\) be the set of edges of the graph which contains all loops, that is, \(\Delta \subseteq E(G)\). Also assume that the graph G has no parallel edges and, thus, one can identify G with the pair \((V(G),E(G))\).
Definition 1.1
[8]
An operator \(f:X\rightarrow X\) is called a Banach Gcontraction or simply a Gcontraction if

(a)
f preserves edges of G; for each \(x,y\in X\) with \((x,y)\in E(G)\), we have \((f(x),f(y))\in E(G)\),

(b)
f decreases weights of edges of G; there exists \(\alpha \in (0,1)\) such that for all \(x,y\in X\) with \((x,y)\in E(G)\), we have \(d(f(x),f(y))\leq\alpha d(x,y)\).
If x and y are vertices of G, then a path in G from x to y of length \(k\in \mathbb{N} \) is a finite sequence \(\{x_{n}\}\) (\(n\in\{0,1,2,\ldots,k\}\)) of vertices such that \(x_{0}=x\), \(x_{k}=y\), and \((x_{i1},x_{i})\in E(G)\) for \(i\in \{1,2,\ldots,k\}\).
Notice that a graph G is connected if there is a directed path between any two vertices and it is weakly connected if \(\widetilde{G}\) is connected, where \(\widetilde{G}\) denotes the undirected graph obtained from G by ignoring the direction of the edges. Denote by \(G^{1}\) the graph obtained from G by reversing the direction of the edges. Thus,
It is more convenient to treat \(\widetilde{G}\) as a directed graph for which the set of its edges is symmetric; under this convention, we have
If G is such that \(E(G)\) is symmetric, then for \(x\in V(G)\), the symbol \([x]_{G}\) denotes the equivalence class of the relation R defined on \(V(G)\) by the rule:
Recall that if \(f:X\rightarrow X\) is an operator, then by \(F_{f}\) we denote the set of all fixed points of f. We set also
Jachymski and Jozwik [7] used the following property:

(P)
for any sequence \(\{x_{n}\}\) in X, if \(x_{n}\rightarrow x\) as \(n\rightarrow\infty\) and \((x_{n},x_{n+1}) \in E(G)\), then \((x_{n},x)\in E(G)\).
Theorem 1.2
[7]
Let \((X,d)\) be a complete metric space and let G be a directed graph such that \(V(G)=X\). Let \(E(G)\) and the triplet \((X,d,G)\) have property (P). Let \(f:X\rightarrow X\) be a Gcontraction. Then the following statements hold:

(1)
\(F_{f}\neq\emptyset\) if and only if \(X_{f}\neq \emptyset\);

(2)
if \(X_{f}\neq\emptyset\) and G is weakly connected, then f is a Picard operator, i.e., \(F_{f}=\{x^{\ast}\}\) and sequence \(\{f^{n}(x)\}\rightarrow x^{\ast}\) as \(n\rightarrow\infty\), for all \(x\in X \);

(3)
for any \(x\in X_{f}\), \(f_{[ x]_{\widetilde{G}}}\) is a Picard operator;

(4)
if \(X_{f}\subseteq E(G)\), then f is a weakly Picard operator, i.e., \(F_{f}\neq\emptyset\) and, for each \(x\in X\), we have sequence \(\{f^{n}(x)\}\rightarrow x^{\ast}(x)\in F_{f}\) as \(n\rightarrow\infty\).
For a detailed discussion concerning Picard and weakly Picard operators, we refer to Rus [22, 23] and to Berinde [24, 25].
Let \((X,d)\) be a metric space and let \(CB(X)\) be the class of all nonempty closed and bounded subsets of X. For \(A,B\in CB(X)\), let
where \(d(x,B)=\inf\{d(x,b):b\in B\}\) is the distance of a point x to the set B. The mapping H is said to be the PompeiuHausdorff metric induced by d.
Throughout this paper, we assume that a directed graph G has no parallel edge and G is a weighted graph in the sense that each vertex x is assigned the weight \(d(x,x)=0\) and each edge \((x,y)\) is assigned the weight \(d(x,y)\). Since d is a metric on X, the weight assigned to each vertex x to vertex y need not be zero and, whenever a zero weight is assigned to some edge \((x,y)\), it reduces to a loop \((x,x)\) having weight 0. Further, in PompeiuHausdorff metric induced by metric d, the PompeiuHausdorff weight assigned to each \(U,V\in CB ( X ) \) need not be zero (that is, \(H ( U,V ) \neq0\)) and, whenever a zero PompeiuHausdorff weight is assigned to some \(U,V\in CB ( X ) \), it reduces to \(U=V\).
Definition 1.3
Let A and B be two nonempty subsets of X. Now we treat some terminology:

(a)
by ‘there is an edge between A and B’, we mean there is an edge between some \(a\in A\) and \(b\in B\) which we denote by \((A,B)\subset E ( G ) \).

(b)
by ‘there is a path between A and B’, we mean that there is a path between some \(a\in A\) and \(b\in B\).
In \(CB(X)\), we define a relation R in the following way:
For \(A,B\in CB(X)\), we have \(ARB\) if and only if there is a path between A and B.
We say that the relation R on \(CB ( X ) \) is transitive if there is a path between A and B, and there is a path between B and C, then there is a path between A and C.
For \(A\in CB(X)\), the equivalence class of A induced by R is denoted by
Now we consider the mapping \(T:CB(X)\rightarrow CB(X)\) instead of \(T:X\rightarrow X\) or \(T:X\rightarrow CB(X)\) to study fixed points of graph contraction mappings.
For a mapping \(T:CB ( X ) \rightarrow CB ( X ) \), we define the following set:
Definition 1.4
Let \(T:CB(X)\rightarrow CB(X)\) be a setvalued mapping. The mapping T is said to be a graph ϕcontraction if the following conditions hold:

(i)
There is an edge between A and B implies there is an edge between \(T(A)\) and \(T(B)\) for all \(A,B\in CB(X)\).

(ii)
There is a path between A and B implies there is a path between \(T(A)\) and \(T(B)\) for all \(A,B\in CB(X)\).

(iii)
There exists an upper semicontinuous and nondecreasing function \(\phi:\mathbb{\mathbb{R} }^{+}\rightarrow\mathbb{\mathbb{R} }^{+}\) with \(\phi(t)< t\) for each \(t>0\) such that there is an edge between A and B implies
$$ H \bigl( T ( A ) ,T ( B ) \bigr) \leq\phi \bigl(H(A,B) \bigr)\quad \mbox{for all }A,B\in CB ( X ) . $$(1.1)
Example 1.5

(1)
Any constant mapping \(T:CB(X)\rightarrow CB(X)\) is a graph ϕcontraction for \(\Delta\subset E(G)\).

(2)
Any graph ϕcontraction map for a graph G is also a graph ϕcontraction for graph \(G_{0}\), where the graph \(G_{0}\) is defined by \(E(G_{0})=X\times X\).
It is obvious if \(T:CB(X)\rightarrow CB(X)\) is a graph ϕcontraction for graph G, then T is also graph ϕcontraction for the graphs \(G^{1}\) and \(\widetilde{G}\).
A graph G is said to have property:
 (P^{∗}):

if for any sequence \(\{X_{n}\}\) in \(CB(X)\) with \(X_{n}\rightarrow X\) as \(n\rightarrow\infty\), there exists an edge between \(X_{n}\) and \(X_{n+1}\) for \(n\in \mathbb{N} \), implies that there is a subsequence \(\{X_{n_{k}}\}\) of \(\{X_{n}\}\) with an edge between \(X_{n_{k}}\) and X for \(n\in \mathbb{N} \).
Definition 1.6
Let \(T:CB(X)\rightarrow CB(X)\). The set \(A\in CB(X)\) is said to be a fixed point of T if \(T(A)=A\). The set of all fixed points of T is denoted by \(F ( T ) \).
A subset Γ of \(CB ( X ) \) is said to be complete if for any set \(X,Y\in\Gamma\), there is an edge between X and Y.
Definition 1.7
[19]
A metric space \((X,d)\) is called an εchainable metric space for some \(\varepsilon >0\) if for given \(x,y\in X\), there is \(n\in \mathbb{N} \) and a sequence \(\{x_{n}\}\) such that
We need of the following lemma of Nadler [21] (see also [26]).
Lemma 1.8
If \(U,V\in CB(X)\) with \(H(U,V)<\varepsilon \), then for each \(u\in U\) there exists an element \(v\in V\) such that \(d(u,v)<\varepsilon\).
Fixed point results
In this section, we obtain several fixed point results for setvalued selfmaps on \(CB(X)\) satisfying certain graph contraction conditions.
Theorem 2.1
Let \((X,d)\) be a complete metric space endowed with a directed graph G such that \(V(G)=X\) and \(E(G)\supseteq \Delta\). If \(T:CB ( X ) \rightarrow CB ( X ) \) is a graph ϕcontraction mapping such that the relation R on \(CB ( X ) \) is transitive, then following statements hold:

(i)
If \(F ( T ) \) is complete, then the PompeiuHausdorff weight assigned to the \(U,V\in F(T)\) is 0.

(ii)
\(X_{T}\neq\emptyset\) provided that \(F ( T ) \neq \emptyset\).

(iii)
If \(X_{T}\neq\emptyset\) and the weakly connected graph G satisfies the property (P^{∗}), then T has a fixed point.

(iv)
\(F ( T ) \) is complete if and only if \(F ( T ) \) is a singleton.
Proof
To prove (i), let \(U,V\in F ( T ) \). Suppose that the PompeiuHausdorff weight assign to the U and V is not zero. Since T is a graph ϕcontraction, we have
a contradiction. Hence (i) is proved.
To prove (ii), let \(F ( T ) \neq\emptyset\). Then there exists \(U\in CB(X)\) such that \(T(U)=U\). Since \(\Delta\subseteq E(G)\) and U is nonempty, we conclude that \(X_{T}\neq\emptyset\).
To prove (iii), let \(U\in X_{T}\). As T is a graph ϕcontraction and \(A,B\in CB(X)\), it follows by the hypothesis \(CB(X)\subseteq [ A]_{\widetilde{G}}=P ( X ) \), where \(P(X)\) denotes the power set of X and so, \(T(A)\in [ A]_{\widetilde{G}}\). Now for \(A\in CB(X)\) and \(B\in [ A]_{\widetilde{G}}\), there exists a path \(\{x_{i}\}_{i=0}^{n}\) from some \(x\in A\) and to \(y\in T ( A ) \), that is, \(x_{0}=x\) and \(x_{n}=y\) and \((x_{i1},x_{i})\in E(\widetilde{G})\), for \(i=1,2,\ldots,n\), such that \(x_{0}\in A_{0}=A\), \(x_{1}\in A_{1},\ldots, x_{n}\in A_{n}=T ( A ) \), where each \(A_{i}\in CB(X)\). Since T is also a graph ϕcontraction for graph \(\widetilde{G}\), for \(i=1,2,\ldots,n\), we have
and so we obtain
for all \(n\in \mathbb{N} \). Now for \(m,n\in \mathbb{N} \) with \(m>n\),
On taking the upper limit as \(n,m\rightarrow\infty\), we get \(H(T^{n} ( A ) ,T^{m} ( A ) )\) converges to 0. Since \((X,d)\) is complete, we have \(T^{n} ( A ) \rightarrow U^{\ast}\) as \(n\rightarrow\infty\) for some \(U^{\ast}\in CB ( X ) \). There exists an edge between U and \(T(U)\), the fact that T is a graph ϕcontraction yields the result that there is an edge between \(T^{n}(U)\) and \(T^{n+1}(U)\) for all \(n\in \mathbb{N} \). By property (P^{∗}), there exists a subsequence \(\{ T^{n_{k}}(U)\}\) such that there is an edge between \(T^{n_{k}}(U)\) and \(U^{\ast}\) for every \(n\in \mathbb{N} \). By the transitivity of the relation R, there is a path in G (and hence also in \(\widetilde{G}\)) between U and \(U^{\ast}\). Thus \(U\in [ U]_{\widetilde{G}}\). Now
Now \(T^{n_{k}} ( U ) \rightarrow U^{\ast}\) as \(n\rightarrow \infty\) implies, on taking the upper limit as \(n\rightarrow\infty\), \(T^{n_{k}+1} ( U ) \rightarrow T ( U^{\ast} ) \) as \(n\rightarrow\infty\). Thus we obtain \(U^{\ast}=T ( U^{\ast } ) \).
Finally to prove (iv), suppose the set \(F ( T ) \) is complete. We are to show that \(F(T) \) is singleton. Assume to the contrary that there exist \(U,V\in CB ( X ) \) such that \(U,V\in F ( T ) \) and \(U\neq V\). By completeness of \(F ( T ) \), there exists an edge between U and V. As T is a graph ϕcontraction, so we have
a contradiction. Hence \(U=V\).
Conversely, if \(F(T)\) is singleton, then obviously \(F(T)\) is complete. □
The following corollary is a direct consequence of Theorem 2.1(iii).
Corollary 2.2
Let \((X,d)\) be a complete metric space endowed with a directed graph G such that \(V(G)=X\) and \(E(G)\supseteq \Delta\). If G is weakly connected, then graph ϕcontraction mapping \(T:CB ( X ) \rightarrow CB(X)\) with \((A_{0},A_{1})\subset E(G)\) for some \(A_{1}\in T ( A_{0} ) \), has a fixed point.
Corollary 2.3
Let \((X,d)\) be a εchainable complete metric space for some \(\varepsilon>0\), \(T:CB ( X ) \rightarrow CB(X)\) and \(\phi:\mathbb{R}^{+}\rightarrow\mathbb {R}^{+}\) be an upper semicontinuous and nondecreasing function with \(\phi (t)< t \) for each \(t>0\) with
If
then T has a fixed point.
Proof
By Lemma 1.8, from \(H ( A,B ) <\varepsilon\), we have for each \(a\in A\), an element \(b\in B\) such that \(d(a,b)<\varepsilon\). Consider the graph G as \(V(G)=X\) and
Then the εchainability of \((X,d)\) implies that G is connected. For \((A,B)\subset E(G)\), we have from the hypothesis
This implies that T is a graph ϕcontraction mapping.
Also, G has property (P^{∗}). Indeed, if \(\{X_{n}\}\) in \(CB(X)\) with \(X_{n}\rightarrow X\) as \(n\rightarrow\infty\) and \(( X_{n},X_{n+1} ) \subset E ( G ) \) for \(n\in \mathbb{N} \), implies that there is a subsequence \(\{X_{n_{k}}\}\) of \(\{X_{n}\}\) such that \(( X_{n_{k}},X ) \subset E ( G ) \) for \(n\in \mathbb{N} \). So by Theorem 2.1(iii), T has a fixed point. □
Example 2.4
Let \(X=\{0,1,2,\ldots,n1\}=V ( G ) \) and
Let \(V ( G ) \) be endowed with metric \(d:X\times X\rightarrow \mathbb{R} ^{+}\) defined by
The PompeiuHausdorff weights (for \(n=4\)) assigned to \(A,B\in CB ( X ) \) are shown in Figure 1.
Furthermore,
Define \(T:CB ( X ) \rightarrow CB(X)\) as follows:
Note that, for all \(A,B\in CB(X)\) with edge between A and B, there is an edge between \(T(A)\) and \(T(B)\). Also there is a path between A and B implies that there is a path between \(T(A)\) and \(T(B)\).
Define \(\phi:[0,\infty)\rightarrow [0,\infty)\) by
An easy computation shows that ϕ is continuous on \([0,\infty)\) and \(\phi(t)< t\) for all \(t>0\).
Now for all \(A,B\in CB ( X ) \), we consider the following cases:

(a)
For \(A,B\subseteq\{0,1\}\), we have \(H(T ( A ) ,T ( B ) )=0\).

(b)
If \(A\subseteq \{ \{0\},\{1\},\{0,1\} \} \) and \(B\varsubsetneq \{ \{0\},\{1\},\{0,1\} \} \), then we have
$$\begin{aligned}[b] H \bigl(T ( A ) ,T ( B ) \bigr) &=H \bigl(\{0\},\{0,1\} \bigr)= \frac{1}{n}\\ &<\frac{4n}{5n+5}=\phi \biggl( \frac{n}{n+1} \biggr) =\phi \bigl(H(A,B) \bigr). \end{aligned} $$ 
(c)
In the case \(A,B\varsubsetneq \{ \{0\},\{1\},\{0,1\} \} \), we have
$$ H \bigl(T ( A ) ,T ( B ) \bigr)=H \bigl(\{0,1\},\{0,1\} \bigr)=0. $$Obviously, (1.1) is satisfied in the cases (a), (b), and (c).
Hence for all \(A,B\in CB ( X ) \) having an edge between A and B, (1.1) is satisfied and so T is a graph ϕcontraction. Thus all the conditions of Theorem 2.1 are satisfied. Moreover, \(\{0\}\) is the fixed point of T and \(F ( T ) \) is complete.
Remark 2.5

(1)
If \(E(G):=X\times X\), then clearly G is connected and Theorem 2.1 improves and generalizes Theorem 2.5 in [19], Theorems 2.12.3 in [11] and Theorem 3.1 in [7].

(2)
Theorem 2.1 with the graph G improves and generalizes Theorem 2.1 in [20] from single valued to setvalued mappings.

(3)
If \(E(G):=X\times X\), then clearly G is connected and our Corollary 2.2 extends and generalizes Theorem 2.5 in [19], Theorem 3.2 in [21], and Theorem 3.1 in [7].

(4)
If \(E(G):=X\times X\), then clearly G is connected and our Corollary 2.3 improves and generalizes Theorem 3.2 in [21] and Theorem 3.1 in [7].

(5)
Corollary 2.3 extends and improves the Banach contraction theorem and Theorem 5.1 in [27].
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Acknowledgements
The authors M Abbas and AR Khan are grateful to King Fahd University of Petroleum and Minerals for supporting research project IN 121023. The author MR Alfuraidan is grateful to King Fahd University of Petroleum and Minerals for supporting this research.
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Abbas, M., Alfuraidan, M.R., Khan, A.R. et al. Fixed point results for setcontractions on metric spaces with a directed graph. Fixed Point Theory Appl 2015, 14 (2015). https://doi.org/10.1186/s136630150263z
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DOI: https://doi.org/10.1186/s136630150263z
MSC
 47H10
 54H25
 54E50
Keywords
 fixed point
 setvalued mapping
 setvalued domain
 directed graph
 graph ϕcontraction