Fixed point results for setcontractions on metric spaces with a directed graph
 Mujahid Abbas^{1},
 Monther Rashed Alfuraidan^{2}Email author,
 Abdul Rahim Khan^{2} and
 Talat Nazir^{3}
https://doi.org/10.1186/s136630150263z
© Abbas et al.; licensee Springer. 2015
Received: 25 August 2014
Accepted: 26 December 2014
Published: 1 February 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.
Keywords
MSC
1 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]
 (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 \(i\in \{1,2,\ldots,k\}\).
 (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]
 (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].
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
 (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.
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.
Definition 1.4
 (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 \(E(G_{0})=X\times X\).
 (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]
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 \(d(u,v)<\varepsilon\).
2 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
 (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 (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 \(X_{T}\neq\emptyset\).
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
Proof
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
 (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 haveObviously, (1.1) is satisfied in the cases (a), (b), and (c).$$ H \bigl(T ( A ) ,T ( B ) \bigr)=H \bigl(\{0,1\},\{0,1\} \bigr)=0. $$
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)
Declarations
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.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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