Abstract
Hypergraph is a generalization of graph in which an edge can join any number of vertices. Hypergraph is used for combinatorial structures which generalize graphs. In this research work, the notion of hypergraphical metric spaces is introduced, which generalizes many existing spaces. Some fixed point theorems are studied in the corresponding spaces. To show the authenticity of the established work, nontrivial examples and applications are also provided.
1. Introduction
Graph theory has been used to study the various concepts of navigation in an arbitrary space. A work place can be denoted as a vertex in the language of graph theory, and edges denote the connections between these places (vertices). Hypergraph is a generalization of graph in which an edge can join any number of vertices. Hypergraph is used for combinatorial structures which generalize graphs. The applications of hypergraph can be found in Engineering sciences, many areas of Computer Science, and almost all areas of Mathematics.
Moreover, directed hypergraphs are used in computer science, particularly in the development of data mining, software testing, image segmentation and processing, information security, and communication networks.
2. Preliminaries
Frechet et al. initiated the concept of metric spaces in 1906, which open the door for entering into a more waste and new field in the world of mathematics. Upon this foundation, different researchers introduced different generalized metric spaces and studied various fixed point results with applications. In this way, we refer some recent developments in [1–3]. About basic notions of graph theory, we refer to the readers [4–6] and references therein.
In 1736, Leonhard Euler put the framework of graph theory by studying the historical problem of seven bridges of Konigsberg and prefigured the concept of topology. Echinique [7] deliberated fixed point theory by using graph. Jachymsky [8] replaced the order structure with a graph structure on a metric space and studied the well-known Banach contraction principle. Aleomraninejad et al. [9] gave the concept of some fixed point results on metric space with a graph, in which they presented some iterative results for G-contractive and G-nonexpansive mappings on graphs. Samreen et al. [10] investigated some fixed point theorems in b-metric space endowed with graph. Argoubi et al. [11] presented some fixed point results and its applications by considering self-mappings defined on a metric space endowed with a finite number of graphs.
Shukhla et al. [12] gave the concept of graphical metric space which is a generalized setting in fixed point theory and established some fixed point results with applications. Abbas et al. [13] presented some fixed point results for set contractions on metric spaces with a directed graph. In 2017, Debanath and Neog [14] initiated the concept of start point on a metric space endowed with a directed graph. They offered the alternate concept of start point in a directed graph and provided the characterizations which are necessary for a directed graph having start point. Kumam et al. [15] presented graphic contraction mapping in b-metric space and established some fixed point results with applications.
Motivated by the above results, combining the notion of hypergraph and metric, we introduced hypergraphical metric space which generalized the concept of graphical metric space. In hypergraphical metric space, vertices of graph are replaced by edges. Some conclusions, examples, and an application to integral equation are also presented to authenticate the acceptation and unifying power of obtained generalizations. For iterative numerical schemes, the interesting readers can refer the recent papers [16, 17].
3. Hypergraph and Hypergraphical Metrics
Definition 1. Hypergraph is real generalization of graph. The edges of hypergraph connect any number of nodes. Formally it is a pair, i.e., in which represents set of vertices and is a set of nonempty subsets of called hyperedges or simply edges.
Definition 2. Hypergraph is said to be directed hypergraph if where is a finite set and is known as the set of nodes of and is the set of directed hyperedges, where a hyperedge or hyperarc is a directed hyperedge with and and both are disjoint. and represent head and tail, respectively, where hyperedge ends and starts and contains set of nodes.
Definition 3. The size of directed hypergraph is defined as the sum of the tail and head nodes of each hyperedge together with the number of nodes of the hypergraph, i.e., .
Definition 4. A directed path in a directed hypergraph is a sequence of nodes and hyperedges such that each edge points from a node in the sequence to its successor in the sequence.
Let be a directed hypergraph and is a directed path from s to t in , which represent the sequence of the form such that and . and .
Definition 5. The edges which connect other edges are called hyperdelta edges; that is, vertices of these edges are also edges and denoted by .
Definition 6. A hypergraph in which we assign numerical value, i.e., nonnegative real numbers to their edges is called labeled graph.
Definition 7. (see [12]). Let set endowed with graph and be a function satisfying the following condition: , Then, the mapping is called a graphical metric on , and the pair is called graphical metric space.
By combining the concept of hypergraph and graphical metric space, we introduced the following notion of hypergraphical metric spaces.
Definition 8. Suppose be a nonempty set endowed with hypergraph such that and let represent hyperedges of such that each hyperedge represents nonempty subset of . Suppose the mapping satisfying the following condition: Then, is called a hypergraphical metric on , and is said to be hypergraphical metric space.
Remark 1. We noted that hypergraphical metric space is the real generalization of graphical metric space; that is, every graphical metric space is hypergraphical metric but converse is not true.
Example 1. Let be the set of vertices, and let which is composed by edges of hypergraph . Now, let us define a function bywhere is the positive real number. Evidently is not a graphical metric becausesince .
On the other hand,In this case, we have . Therefore, is the hypergraphical metric space.
Not every hypergraphical metric space is metric. Let us provide an example as follows.
Example 2. Let ; here, interval means the weight of edges of , where be the hypergraph such that its edges can be defined as . Define a mapping by
Then, is a hypergraphical metric on and is a hypergraphical metric space obviously where not a metric on .
Definition 9. Let be hypergraphical metric space. An open ball with center and radius is defined asSince , therefore, we have . Hence, is nonempty and . The collectionwhich is the neighborhood system for the topology on induced by the hypergraphical metric . A subset of is called open if for every there exist an such that ; of course, a subset of is called closed if its complement is open.
Lemma 1. Every open ball in is an open set.
Proof. Let for some and . Let and ; by definition, we have and and so that . Now, from Property (4) of hypergraphical metric space, . Hence, . Hence, every open ball in is an open set.
Definition 10. Suppose is hypergraphical metric space and be a sequence in , then is called convergent and converges to if for given there such that . Obviously the sequence is convergent and converges to if and only if .
Remark 2. . The limit of a sequence in hypergraphical metric space may not be unique as clear from the following example.
Example 3. let be the set of vertices of hypergraph, and we take to be the set of subsets of such that each subset represents an edge of the hypergraph . Now, we labeled some edges from the set , where . We define . Define a mapping byClearly, is a hypergraphical metric on . Now, let us consider the sequence in where ; then, for any fixed , we haveTherefore, the sequence converges to for every fixed .
Lemma 2. Let be a hypergraphical metric space with induced hypergraphical topology . Then, is but not generally Hausdorff, i.e., .
Proof. We want to show that for every , the singleton set is a closed subset of or the set is an open subset of . For this, let us suppose , then clearly and . Now, let us take . Then, clearly does not belong to . Suppose on contrary that , then which is contradiction to . Hence, is open, and hence hypergraphical metric space is not Hausdorff.
Remark 3. Let be hypergraphical metric space in previous remark, then is limit point of the sequence , but for any , if , we have . Therefore, a hypergraphical metric does not need to be continuous.
Definition 11. Let be hypergraphical metric space, and is a sequence. Then, is called Cauchy if for given there exist belong to such that ; obviously the sequence is Cauchy sequence .
Definition 12. A hypergraphical metric space is called complete if each Cauchy sequence in converges in . Suppose is another hypergraph such that each is subset of , that is, , then is called -complete if every termwise connected Cauchy sequence in converges in .
In this paper, we suppose that hypergraph is considered to be directed. We include directed path (p) between edges and denote by .
4. Main Results
In this section, we provide fixed point results in hypergraphical metric space; for this, we need various definitions to support our main results.
Definition 13. Suppose is hypergraphical metric space and is a mapping and is subhypergraph of such that . Then, is said -hypergraphical contraction on if the conditions given below are satisfied. preserves edges in such that : there exists , such that for and , for all
Here, we assign the hypergraphical distance between the edges of , and hypergraphical contraction decreases the distance by factor . The sequence having earliest value is called -picard sequence if . Further, we suppose that is a subhypergraph of such that . The next theorem is the dominant outcome which gives sufficient conditions for the convergence of picard sequence yielded by -hypergraphical contraction on -complete hypergraphical metric space.
Theorem 1. Suppose is -complete hypergraphical metric space and be a -hypergraphical contraction and also satisfies the following conditions. There exist such that , for some . If -termwise connected -picard sequence converges in , then a limit of exists and , such that .
Then, there exist such that the -picard sequence of initial value is -termwise connected and converges to and .
Proof. Suppose such that for some and is -picard sequence having initial value , then is a path such that , and for . As is a -hypergraphical contraction, we haveTherefore, represent a path from to of length and so ; proceeding similarly, we get the path from to of length . Hence, ; thus, is a -termwise connected sequence. Since for and . Using condition , we haveSince is a subgraph of , and is a termwise connected sequence in , by using (10), the following relation holds , :where . Again as the sequence is -termwise connected, therefore, with , we haveSince , we obtain . Therefore, is a Cauchy sequence in . From -completeness of , the sequence converges in . And from condition (2), there exist and , such that and . Thus, the sequence converges to . Now, if for all by using , we obtainsince .
Therefore,A similar result holds if , and hence the sequence converges to both and .
If we replace by the set of vertices instead of edges, we get the following corollary.
Corollary 1. Suppose is graphical metric space and be a -graphical contraction and also satisfies the following conditions.(1)There exist such that . For some .(2)If -termwise connected, F-picard sequence converges in . Then, a limit of exists and , such that .Then, there exist such that the -picard sequence of earliest value is -termwise connected and converges to and .
Remark 4.
Corollary 1 is the result of Shukla [12].
Remark 5. Theorem 1 confirms only convergent of a picard sequence yielded from a -hypergraphical contraction on a -complete hypergraphical metric space. Next example displays that no one should appreciate this theorem as an existence theorem in -complete hypergraphical metric space.
Example 4. Suppose be the nonempty set of vertices of hypergraph and be the set of subset of such that each subset represents an edge of the hypergraph . Note (here, means weighted hypergraph) that we labeled some edges of from set . Here, is the set, that is, and . Define a mapping by
Then, is hypergraphical metric on and is -complete hypergraphical metric space; now, here we define a mapping.
F: by
It should be noted that is hypergraphical contraction having , and we have , which implies that . Also any -termwise connected and convergent sequence in is constant or monotonic decreasing subsequence with respect to usual order of the sequence and having at least one limit such that property (2) of contraction theorem hold surely. However, there is no fixed point in of . As mentioned, that convergent sequence’s limit may not be unique in hypergraphical metric space . Therefore, we provide one more definition that is as follows.
Definition 14. Suppose is hypergraphical metric space and is a mapping, then property holds for the quadruple , that is:
: whenever a -termwise connected, F-picard sequence having limits and where and , then .
We represent all fixed point of a set by Fix, and notation for this is .
Remark 6. If we chose , then it is clear to check that quadruple satisfies property for arbitrary subhypergraph .
Example 5. Let X, , and be those which is used in Example 1. And .
Then, the quadruple has the property . In the next theorem, we want to give enough condition for the existence of fixed point of a -graphical contraction.
Theorem 2. Suppose is hypergraphical metric space and is hypergraphical contraction, it holds the following: (1) there exist such that for some ; (2) if a termwise connected Tpicard sequence converges in , then there of which is limit point and such that for all ; then, there exist such that the T-picard sequence having earliest value is -termwise connected and converges to and . Also, if the quadruple satisfies property , then there must be fixed point of F in .
Proof. From Theorem 3.2, F-picard sequence having earliest value converges to and . As and , therefore, by property , it is essential that . Hence, F has fixed point which is a fixed point of T.
Remark 7. In the above result, fixed point of F exists due to property ; it is very important to note that in Example 5, every condition of above result holds except property . However, . Therefore, in the above theorem property, remains unused.
5. Applications
Let and represent set of weights of edges which is in the form of real continuous function on weighted interval . We give a special application of fixed point theory for examining integral equations of ; we show that according to certain condition, the actuality of a lower or upper solution of an integral equation ensures the solution of integral equation. Let . Here, we have and . Consider that hypergraphical metric space which is given below, that is, is given by
So, is -complete hypergraphical metric space. Here, we suppose following integral equation:where and are continuous functions. Mapping is called lower solution of (19) if . Here, we want to prove that the existence of lower solution of conforms the existence of solution of (19). Let us suppose that the operator is defined byand sufficient conditions are provided for existence of fixed point of (20) in , and obviously that fixed point is solution of (19).
Theorem 3. Consider that coming conditions hold the following:(a) is increasing function on (0, 1] for every . Moreover, and .(b)There exist and such that for , and .Then, existence of a lower solution of (19) in confirms the existence of solution of (19) in .
Proof. By condition (b), for , also , and , we derivedThen, we haveFurther, for , , and and from condition (a) we have andConsequently, the existence of lower solution of equation (19), i.e., implies that property of Theorem 3 holds. Also, the quadruple has property (p). Hence, all conditions of Theorem 2.8 are satisfied. Thus, the operator F has a fixed point which is solution of integral equation (19) in .
Data Availability
The data used to support this research are included within the paper.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
X. Li analyzed the results and finalized the paper. G. Ali supervised the work. L. Gul proved the main results. F. Khan wrote the first draft of the paper. M. Sarwar verified the results.