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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.

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.

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 [

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 [

Shukhla et al. [

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 [

Hypergraph is real generalization of graph. The edges of hypergraph connect any number of nodes. Formally it is a pair, i.e.,

Hypergraph

The size of directed hypergraph

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

The edges which connect other edges are called hyperdelta edges; that is, vertices of these edges are also edges and denoted by

A hypergraph in which we assign numerical value, i.e., nonnegative real numbers

(see [

Then, the mapping

By combining the concept of hypergraph and graphical metric space, we introduced the following notion of hypergraphical metric spaces.

Suppose

Then,

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.

Let

On the other hand,

In this case, we have

Not every hypergraphical metric space is metric. Let us provide an example as follows.

Let

Then,

Let

Since

Every open ball in

Let

Suppose

. The limit of a sequence in hypergraphical metric space may not be unique as clear from the following example.

let

Clearly,

Therefore, the sequence

Let

We want to show that for every

Let

Let

A hypergraphical metric space

In this paper, we suppose that hypergraph

In this section, we provide fixed point results in hypergraphical metric space; for this, we need various definitions to support our main results.

Suppose

Here, we assign the hypergraphical distance between the edges of

Suppose

Then, there exist

Suppose

Therefore,

Since

Since

Therefore,

A similar result holds if

If we replace

Suppose

There exist

If

Then, there exist

Corollary

Theorem

Suppose

Then,

F:

It should be noted that

Suppose

We represent all fixed point of a set by Fix

If we chose

Let

Then, the quadruple

Suppose

From Theorem 3.2, F-picard sequence

In the above result, fixed point of F exists due to property

Let

So,

Consider that coming conditions hold the following:

There exist

Then, existence of a lower solution of (

By condition (b), for

Then, we have

Further, for

Consequently, the existence of lower solution of equation (

The data used to support this research are included within the paper.

The authors declare that they have no conflicts of interest.

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.