# Hypergraphical Metric Spaces and Fixed Point Theorems

Authors: Xiaodong Li; Farhan Khan; Gohar Ali; Lubna Gul; Muhammad Sarwar
Journal: Mathematical Problems in Engineering
 (2021)

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## 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.


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## Body

## 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.,GH=ζ,ξ in which ζ represents set of vertices and ξ is a set of nonempty subsets of ζ called hyperedges or simply edges.Definition 2.
HypergraphGH is said to be directed hypergraph if GH=V,ξ where V≠∅ is a finite set and is known as the set of nodes of GH and ξ is the set of directed hyperedges, where a hyperedge or hyperarc e=Te,He is a directed hyperedge with Te>0 and He>0 and both are disjoint. He and Te represent head and tail, respectively, where hyperedge ends and starts and contains set of nodes.Definition 3.
The size of directed hypergraphGH 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., GH=V+∑e∈ɛ|Te|+He.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.
LetGH=v,ξ be a directed hypergraph and vi,vj∈v is a directed path from s to t in GH, which represent the sequence πs,t of the form πs⟶t=v1,e1,v2,e2,…,en−1,vn:n>0 such that vi∈v,∀i∈1,2,3,…,n and ej∈ξ,∀j∈1,2,3,…,n−1. v1∈s⇒v1∈Te1 and vn=t⟹vn∈Hei−1∩Tei,∀i∈2,…,n−1.Definition 5.
The edges which connect other edges are called hyperdelta edges; that is, vertices of these edges are also edges and denoted byΔ.Definition 6.
A hypergraph in which we assign numerical value, i.e., nonnegative real numbers0∞ to their edges is called labeled graph.Definition 7.
(see [12]). Let ζ≠∅ set endowed with graph Gm and dGm:ζ∗ζ⟶R be a function satisfying the following condition:GM1.dGma,b=0,ifa=bGM2.dGma,b>0,ifa≠bGM3.dGma,b=dGmb,a,∀a,b∈ζGM4.apbGm, c∈apbGmimpliesdGma,b≤dGma,c+dGmc,b,∀a,b,c∈ζ
Then, the mappingdGm is called a graphical metric on ζ, and the pair ζ,dGm 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.
Supposeζ be a nonempty set endowed with hypergraph GH such that VGH=ζ and let ξ represent hyperedges of GH such that each hyperedge e represents nonempty subset of ζ. Suppose the mapping dGH:ξ∗ξ⟶R satisfying the following condition:HGM1.dGHei,ej=0,ifei=ejHGM2.dGHei,ej>0,ifei≠ejHGM3.dGHei,ej=dGHej,ei,∀ei,ej∈ξHGM4.eipejGHek∈eipejGHimpliesdGHei,ej≤dGHei,ek+dGHek,ej,∀ei,ej,ek∈ξ
Then,dGH is called a hypergraphical metric on ξ, and ξ,dGH 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.
Letζ=v1,v2,v3,v4,v5,v6 be the set of vertices, and let ξ=v1,v2,v3,v4,v5,v6 which is composed by edges of hypergraph GH. Now, let us define a function dGH:ξ∗ξ⟶R+ by(1)dGHei,ej=0,ifei=ej,5A,ifei,ej∈v1,v2ei≠ej,3A,ifei,ej∈v1,v3ei≠ej,A,ifei,ej∈v2,v3ei≠ej,4A,ifei,ej∈v1,v4,v5,v6ei≠ej,6A,ifei,ej∈v2,v4,v5,v6ei≠ej,2A,ifei,ej∈v4,v5,v6,v2ei≠ej,where A>1 is the positive real number. Evidently dGH is not a graphical metric because(2)dGHv1,v2≰dGHv1,v3+dGHv3,v2,since 5A>3A+A.
On the other hand,(3)dGHv1,v2≤dGHv1,v4,v5,v6+dGHv4,v5,v6,v2.
In this case, we have5A≤4A+6A. Therefore, dGH is the hypergraphical metric space.Not every hypergraphical metric space is metric. Let us provide an example as follows.Example 2.
LetX=0,1; here, X interval means the weight of edges of GH, where GH be the hypergraph such that its edges can be defined as ξGH=ΔUei,ej:ei,ej∈1,1ei≤ej∧i,j∈N. Define a mapping dGH:ξ∗ξ⟶R+ by(4)dGHei,ej=0,ifei=ej;ei∗ej,ifei,ej∈0,1ei≠ej;ei+ej,otherwise.Then,dGH is a hypergraphical metric on ξ and ξ,dGH is a hypergraphical metric space obviously where dGH not a metric on ξ.Definition 9.
Letξ,dGH be hypergraphical metric space. An open ball BGHe,ɛ with center e and radius ϵ is defined as(5)BGHe,ɛ=e′:epe′GH,dGHe,e′<ɛ.
SinceξGH⊒Δ, therefore, we have e∈BGHe,ɛ. Hence, BGHe,ɛ is nonempty ∀e∈ξ and ɛ>0. The collection(6)B=BGHe,ɛ:e∈ξ,ɛ>0,which is the neighborhood system for the topology TGH on ξ induced by the hypergraphical metric dGH. A subset S of ξ is called open if for every e∈S there exist an ɛ>0 such that BGHe,ɛ⊂S; of course, a subset T of ξ is called closed if its complement Tc is open.Lemma 1.
Every open ball inξ is an open set.Proof.
Lete′∈BGHe,ɛ for some e∈ξ and ɛ>0. Let α=ɛ−dGHe,e′>0 and e″∈BGHe′,α; by definition, we have epe′GH and e′pe″GH and so that epe″GH. Now, from Property (4) of hypergraphical metric space, dGHe″,e≤dGHe″,e′+dGHe′,e<α+dGHe′,e=ɛ−dGHe′,e+dGHe′,e=ɛ. Hence, BGHe′,α⊂BGHe,ɛ. Hence, every open ball in ξ is an open set.Definition 10.
Supposeξ,dGH is hypergraphical metric space and en be a sequence in ξ, then en is called convergent and converges to e∈ξ if for given ɛ>0 there ∃n∘∈N such that dGHen,e≤ɛ,∀n>n∘. Obviously the sequence en is convergent and converges to e if and only if limn⟶∞dGHen,e=0.Remark 2.
. The limit of a sequence in hypergraphical metric space may not be unique as clear from the following example.Example 3.
letX be the set of vertices of hypergraph, and we take ξGH to be the set of subsets of X such that each subset represents an edge of the hypergraph GH. Now, we labeled some edges from the set 2A∪0, where 2A=1/2n:n∈ℕ. We define ξGH=ei,ej:ei≤ej,i,j∈ℕ. Define a mapping dGH:ξ×ξ⟶ℝ+ by(7)dGHei,ej=0,ifei=ej;ei×ej,ifei,ej∈2Aei≠ej;12,otherwise.
Clearly,dGH is a hypergraphical metric on ξ. Now, let us consider the sequence en in ξ where en=1/2n,∀n∈ℕ; then, for any fixed k∈ℕ, we have(8)dGH12n,12k=12n+k⟶0,n⟶∞.
Therefore, the sequence1/2n converges to 1/2k for every fixed k∈ℕ.Lemma 2.
Letξ,dGH be a hypergraphical metric space with induced hypergraphical topology TGH. Then, TGH is T1 but not generally Hausdorff, i.e., T2.Proof.
We want to show that for everye∈ξ, the singleton set e is a closed subset of ξ or the set ξ−e is an open subset of ξ. For this, let us suppose e′∈ξ−e, then clearly e′≠e and dGHe,e′>0. Now, let us take dGHe,e′=2ɛ>0⟶∗. Then, clearly e does not belong to BGHe′,ɛ. Suppose on contrary that e∈BGHe′,ɛ, then dGHe,e′<ɛ which is contradiction to ∗. Hence, BGHe′,ɛ⊂ξ−e is open, and hence hypergraphical metric space is not Hausdorff.Remark 3.
Letξ,dGH be hypergraphical metric space in previous remark, then 1/2 is limit point of the sequence en=1/2n∈ξ, but for any k∈ℕ, if k>1, we have limn⟶∞dGH1/2n,1/2k=0≠dGH1/2,1/2k. Therefore, a hypergraphical metric does not need to be continuous.Definition 11.
Letξ,dGH be hypergraphical metric space, and en∈ξ is a sequence. Then, en is called Cauchy if for given ɛ>0 there exist n∘ belong to ℕ such that dGHen,em<ɛ,∀n,m>n∘; obviously the sequence en is Cauchy sequence ⇔limn⟶∞dGHen,em=0.Definition 12.
A hypergraphical metric spaceξ,dGH is called complete if each Cauchy sequence in ξ converges in ξ. Suppose GH′ is another hypergraph such that each e∈ξGH′ is subset of VGH′, that is, e⊆VGH′, then ξ,dGH is called GH′-complete if every GH′ termwise connected Cauchy sequence in ξ converges in ξ.In this paper, we suppose that hypergraphGH is considered to be directed. We include directed path (p) between edges and denote by eGHl=e′∈ξ:directed path from to of lengthl.


## 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.
Supposeξ,dGH is hypergraphical metric space and F:ξ⟶ξ is a mapping and GH∗ is subhypergraph of GH such that ξGH∗⊇Δ. Then, F is said GH,GH∗-hypergraphical contraction on ξ if the conditions given below are satisfied.GHC1:F preserves edges in GH∗ such that e∈ξGH∗⇒Fe∈ξGH∗GHC2: there exists α∈0,1, such that for eiej∈ξGH∗ and Fei,Fej∈ξGH∗, dGHFei,Fej≤αdGHei,ej for all eiej∈ξGH∗Here, we assign the hypergraphical distance between the edges ofGH∗, and hypergraphical contraction decreases the distance by factor α∈0,1. The sequence en having earliest value e0∈ξ is called F-picard sequence if en=Fen−1,∀n∈ℕ. Further, we suppose that GH∗ is a subhypergraph of GH such that ξGH∗⊇Δ. The next theorem is the dominant outcome which gives sufficient conditions for the convergence of picard sequence yielded by GH,GH′-hypergraphical contraction on GH′-complete hypergraphical metric space.Theorem 1.
Supposeξ,dGH is GH′-complete hypergraphical metric space and F:ξ⟶ξ be a GH,GH′-hypergraphical contraction and also satisfies the following conditions. 1 There exist e0∈ξ such that Fe0∈e0GH′l, for some l∈ℕ. 2 If GH′-termwise connected F-picard sequence en converges in ξ, then a limit e′∈ξ of en exists and n0∈ℕ, such that en,e′∈ξGH′,∀n>n0.Then, there existe∗∈ξ such that the F-picard sequence en of initial value e0 is GH′-termwise connected and converges to e∗ and Fe∗.Proof.
Supposee0∈ξ such that Fe0∈e0GH′l for some l∈ℕ and en is F-picard sequence having initial value e0, then ei′i=0l is a path such that e0=e0′,Fe0=el′, and ei−1′,ei′∈ξGH′ for i=1,2,3,…,l. As F is a GH,GH′-hypergraphical contraction, we have(9)Fei−1′,Fei′∈ξGH′,fori=1,2,3,…,l.
Therefore,Tei′i=0l represent a path from Fe0′=Feo=e1 to Tel′=F2e0=e2 of length l and so e2∈e1GH′l; proceeding similarly, we get the path Tnei′i=0l from Fne0′=Fne0=en to Fnel′=FnFe0=en+1 of length l. Hence, en+1∈enGH′l,∀n∈ℕ; thus, en is a GH′-termwise connected sequence. Since Fnei−1′,Fnei′∈ξGH′ for i=1,2,3,…,l and n∈ℕ. Using condition GHC2, we have(10)dGHFnei−1′,Fnei′≤αndGHei−1′,ei′.
SinceGH′ is a subgraph of GH, en and is a termwise connected sequence in GH′, by using (10), the following relation holds ∀n∈ℕ, m>n:(11)dGHen,en+1=dGHFne0,Fn+1e0≤∑i=1lαndGHei−1′,ei′=αnFl.where Fl=∑i=1ldGHei−1′,ei′. Again as the sequence en is GH′-termwise connected, therefore, n,m∈ℕ with m>n, we have(12)dGHen,em≤∑i=nm−1dGHei,ei+1≤∑i=nm−1αiFl=∑i=nm−1αi−n+nFl=αn∑i=nm−1αi−nFl=αn1−α.
Sinceα∈0,1, we obtain limn,m⟶∞dGHen,em=0. Therefore, en is a Cauchy sequence in ξ. From GH′-completeness of ξ, the sequence en converges in ξ. And from condition (2), there exist e∗∈ξ and n0∈N, such that en,e∗∈ξGH′,∀n>n0 and limn⟶∞dGHen,e∗=0. Thus, the sequence en converges to e∗∈ξ. Now, if en,e∗∈ξGH′ for all n>n0 by using GHC2, we obtain(13)dGHen+1,Fe∗=dGHFen,Fe∗≤αdGHen,e∗,for alln>n0,since limn⟶∞dGHen,e∗=0.
Therefore,(14)limn⟶∞dGHen+1,Fe∗=0.
A similar result holds ife∗,en∈ξGH′, and hence the sequence en converges to both e∗ and Fe∗.If we replaceξ by the set of vertices instead of edges, we get the following corollary.Corollary 1.
Supposeξ,dGH is GH′−complete graphical metric space and F:ξ⟶ξ be a GH,GH′-graphical contraction and also satisfies the following conditions.(1)
There existx0∈ξ such that Fx0∈x0GH′l. For some l∈N.(2)
IfGH′-termwise connected, F-picard sequence xn converges in ξ. Then, a limit x′∈ξ of xn exists and n0∈N, such that xn,x′∈ξGH′,∀n>n0.
Then, there existx∗∈ξ such that the F-picard sequence en of earliest value x0 is GH′-termwise connected and converges to x∗ and Fx∗.Remark 4.
Corollary1 is the result of Shukla [12].Remark 5.
Theorem1 confirms only convergent of a picard sequence yielded from a GH,GH′-hypergraphical contraction on a GH′-complete hypergraphical metric space. Next example displays that no one should appreciate this theorem as an existence theorem in GH′-complete hypergraphical metric space.Example 4.
Supposeζ be the nonempty set of vertices of hypergraph GH and ξGH be the set of subset of ζ such that each subset represents an edge of the hypergraph GH. Note (here, GH means weighted hypergraph) that we labeled some edges of GH from set 2A∪0. Here, 2A is the set, that is, 2A=1/2n:n∈N and ξGH=ei,ej:ei≤ejandi,j∈N. Define a mapping dGH:ξ∗ξ⟶R+ by(15)dGHei,ej=0,ifei=ej;ei∗ej,ifei,ej∈2Aei≠ej;12,otherwise.Then,dGH is hypergraphical metric on ξ and ξ,dGH is GH-complete hypergraphical metric space; now, here we define a mapping.F:ξ⟶ξ by(16)Fe=e2,ife∈2A;12,ife=0;It should be noted thatF is hypergraphical contraction having α=1/4,∀e∈2A, and we have e,Fe∈ξGH, which implies that Fe∈eGH′l. Also any GH-termwise connected and convergent sequence in ξ is constant or monotonic decreasing subsequence with respect to usual order of the sequence 1/2n and having at least one limit e′ such that property (2) of contraction theorem hold surely. However, there is no fixed point in ξ of F. As mentioned, that convergent sequence’s limit may not be unique in hypergraphical metric space GH. Therefore, we provide one more definition that is as follows.Definition 14.
Supposeξ,dGH is hypergraphical metric space and F:ξ⟶ξ is a mapping, then property P holds for the quadruple x,dGH,GH′,T, that is:
P: whenever a GH′-termwise connected, F-picard sequence xn having limits ei and ej where ei∈ξ and ej∈Fξ, then ei=ej.We represent all fixed point of a set by Fix∗F, and notation for this is ξF=e∈ξ:e,Fe∈ξGH′ .Remark 6.
If we choseξGH=ξ∗ξ, then it is clear to check that quadruple ξ,dGH,GH′,F satisfies property P for arbitrary subhypergraph GH′.Example 5.
LetX, GH, and dGH be those which is used in Example 1. And ξGH′=Δ∪ei,ej:ei,ej∈0,1,ei≤ej∧i,j∈N.(17)Fe=e,if∈Q∩0,1;1,otherwise;Then, the quadrupleξ,dGH,GH′,F has the property P. In the next theorem, we want to give enough condition for the existence of fixed point of a GH,GH′-graphical contraction.Theorem 2.
Supposeξ,dGH is GH′−complete hypergraphical metric space and F:ξ⟶ξ is GH,GH′−hypergraphical contraction, it holds the following: (1) there exist e0∈ξ such that Te0∈e0GH′l for some l∈N; (2) if a GH′−termwise connected T−picard sequence en converges in ξ, then there ∃e′∈ξ of en which is limit point and n0∈N such that en,e′∈ξGH′ for all n>n0; then, there exist e∗∈ξ such that the T-picard sequence en having earliest value e0 is GH′-termwise connected and converges to e∗ and Te∗. Also, if the quadruple ξ,dGH,GH′,F satisfies property P, then there must be fixed point of F in ξ.Proof.
From Theorem 3.2, F-picard sequenceen having earliest value e0 converges to e∗ and Te∗. As e∗∈ξ and Fe∗∈Fξ, therefore, by property P, it is essential that Te∗=e∗. Hence, F has fixed point e∗ which is a fixed point of T.Remark 7.
In the above result, fixed point of F exists due to propertyP; it is very important to note that in Example 5, every condition of above result holds except property P. However, FixF=Φ. Therefore, in the above theorem property, P remains unused.


## 5. Applications

LetI>0 and ζ=C0,1,R represent set of weights of edges which is in the form of real continuous function on weighted interval 0,I. We give a special application of fixed point theory for examining integral equations of ζ; 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 BH=e∈X:0<infp∈0,Iepandep≤1,t∈0,I. Here, we have GH=GH′ and ξGH=Δei,ejei,ej∈BHeip≤ejp∀p∈0,I. Consider that hypergraphical metric space which is given below, that is, dGH:ξ∗ξ⟶R is given by(18)dGHeiej=0,ifei=ej,supp∈0,Iln1eip⋅ejp,ifei,ej∈BHei≠ej,1,otherwise.So,ξ,dGH is GH′-complete hypergraphical metric space. Here, we suppose following integral equation:(19)eip=∫0Igp,qhq,eiqdq,where g:0,I∗0,I⟶0,+∞ and h:0,I∗R⟶R are continuous functions. Mapping β∈C0,I,R is called lower solution of (19) if βp≤∫0Igp,qhq,βqdq,p∈0,I. Here, we want to prove that the existence of lower solution of Z1 conforms the existence of solution of (19). Let us suppose that the operator F:ξ⟶ξ is defined by(20)Feip=∫0Igp,qhq,eiqdq,and sufficient conditions are provided for existence of fixed point of (20) in ξ, and obviously that fixed point is solution of (19).Theorem 3.
Consider that coming conditions hold the following:(a)
…hq,∗:R⟶R is increasing function on (0, 1] for every q∈0,I. Moreover, infp∈0,Igp,q>0 and gp,qhq,1≤I−1.(b)
There existα∈0,1 and ρ∈1,+∞ such that for ei, ej∈ξ and ei,ej∈ξ,∀q,r∈0,I.(21)hq,eqhr,ejr≥eiqejrα,∫0I∫0Igp,qgp,rdqdr≥ρ,∀p∈0,I.
Then, existence of a lower solution of (19) in BH confirms the existence of solution of (19) in ξ.Proof.
By condition (b), forei, ej∈ξ also ei,ej∈ξ, and P∈0,I, we derived(22)ln1Feip⋅Fejp=ln1∫0I∫0Igp,qgp,rhq,eiqhr,ejrdqdr≤ln1infp∈0,Ieipejpα∫0I∫0Igp,qgp,rdqdr=ln1Feip⋅Fejp=ln1∫0I∫0Igp,qgp,rhq,eiqhr,ejrdqdr≤ln1∫0I∫0Igp,qgp,reiqejrαdqdr≤ln1infp∈0,Ieipejpα∫0I∫0Igp,qgp,rdqdr≤ln1ρinfp∈0,Ieipejpα≤αdGHeip,ejpzz.
Then, we have(23)dGHFei,Fej=supp∈0,Iln1FeipFejp≤αdGHeiP,ejP.
Further, forei, ej∈BH, and eip≤ejp,∀p∈0,I and from condition (a) we have infp∈0,IFeip>0 and(24)Feip=∫0Igp,qhq,eiqdq≤∫0Igp,qh1,1dq≤1,Fe1p=∫0Igp,qfq,eiqdq≤∫0Igp,qhq,ejqdq=Fejp.
Consequently, the existence of lower solution of equation (19), i.e., β∈BH implies that property of Theorem 3 holds. Also, the quadruple ξ,dGH,GH′,F 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 ξ.



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