MPEMathematical Problems in Engineering1563-51471024-123XHindawi10.1155/2021/99617349961734Research ArticleHypergraphical Metric Spaces and Fixed Point Theoremshttps://orcid.org/0000-0001-9912-9960LiXiaodong1KhanFarhan2https://orcid.org/0000-0002-6466-3992AliGohar3GulLubna3https://orcid.org/0000-0003-3904-8442SarwarMuhammad2AhmadAli1Huanghe Jiaotong UniversityJiaozuo 454950HenanChina2Department of MathematicsUniversity of MalakandChakdara Dir (L)Khyber PakhtunkhwaPakistanuom.edu.pk3Department of MathematicsIslamia College PeshawarKhyber PakhtunkhwaPakistanicp.edu.pk202127620212021233202166202127620212021Copyright © 2021 Xiaodong Li et al.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 . About basic notions of graph theory, we refer to the readers  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  deliberated fixed point theory by using graph. Jachymsky  replaced the order structure with a graph structure on a metric space and studied the well-known Banach contraction principle. Aleomraninejad et al.  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.  investigated some fixed point theorems in b-metric space endowed with graph. Argoubi et al.  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.  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.  presented some fixed point results for set contractions on metric spaces with a directed graph. In 2017, Debanath and Neog  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.  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 MetricsDefinition 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.

Hypergraph GH 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 hypergraph GH 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+eɛ|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.

Let GH=v,ξ be a directed hypergraph and vi,vjv is a directed path from s to t in GH, which represent the sequence πs,t of the form πst=v1,e1,v2,e2,,en1,vn:n>0 such that viv,i1,2,3,,n and ejξ,j1,2,3,,n1. v1sv1Te1 and vn=tvnHei1Tei,i2,,n1.

Definition 5.

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

Definition 6.

A hypergraph in which we assign numerical value, i.e., nonnegative real numbers 0 to their edges is called labeled graph.

Definition 7.

(see ). Let ζ set endowed with graph Gm and dGm:ζζR be a function satisfying the following condition:

GM1.dGma,b=0, if a=b

GM2.dGma,b>0, if ab

GM3.dGma,b=dGmb,a,a,bζ

GM4.apbGm, capbGm implies dGma,bdGma,c+dGmc,b,a,b,cζ

Then, the mapping dGm 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.

Suppose ζ 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:ξξR satisfying the following condition:

HGM1.dGHei,ej=0, if ei=ej

HGM2.dGHei,ej>0, if eiej

HGM3.dGHei,ej=dGHej,ei,ei,ejξ

HGM4.eipejGHekeipejGH implies dGHei,ejdGHei,ek+dGHek,ej,ei,ej,ekξ

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.

Let ζ=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:ξξR+ by(1)dGHei,ej=0,if ei=ej,5A,if ei,ejv1,v2eiej,3A,if ei,ejv1,v3eiej,A,if ei,ejv2,v3eiej,4A,if ei,ejv1,v4,v5,v6eiej,6A,if ei,ejv2,v4,v5,v6eiej,2A,if ei,ejv4,v5,v6,v2eiej,where A>1 is the positive real number. Evidently dGH is not a graphical metric because(2)dGHv1,v2dGHv1,v3+dGHv3,v2,since 5A>3A+A.

On the other hand,(3)dGHv1,v2dGHv1,v4,v5,v6+dGHv4,v5,v6,v2.

In this case, we have 5A4A+6A. Therefore, dGH is the hypergraphical metric space.

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

Example 2.

Let X=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,ej1,1eieji,jN. Define a mapping dGH:ξξR+ by(4)dGHei,ej=0,if ei=ej;eiej,if ei,ej0,1eiej;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.

Let ξ,dGH be hypergraphical metric space. An open ball BGHe,ɛ with center e and radius ϵ is defined as(5)BGHe,ɛ=e:epeGH,dGHe,e<ɛ.

Since ξGHΔ, therefore, we have eBGHe,ɛ. Hence, BGHe,ɛ is nonempty eξ and ɛ>0. The collection(6)B=BGHe,ɛ:eξ,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 eS there exist an ɛ>0 such that BGHe,ɛS; of course, a subset T of ξ is called closed if its complement Tc is open.

Lemma 1.

Every open ball in ξ is an open set.

Proof.

Let eBGHe,ɛ for some eξ and ɛ>0. Let α=ɛdGHe,e>0 and eBGHe,α; by definition, we have epeGH and epeGH and so that epeGH. Now, from Property (4) of hypergraphical metric space, dGHe,edGHe,e+dGHe,e<α+dGHe,e=ɛdGHe,e+dGHe,e=ɛ. Hence, BGHe,αBGHe,ɛ. Hence, every open ball in ξ is an open set.

Definition 10.

Suppose ξ,dGH is hypergraphical metric space and en be a sequence in ξ, then en is called convergent and converges to eξ if for given ɛ>0 there nN such that dGHen,eɛ,n>n. Obviously the sequence en is convergent and converges to e if and only if limndGHen,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.

let X 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 2A0, where 2A=1/2n:n. We define ξGH=ei,ej:eiej,i,j. Define a mapping dGH:ξ×ξ+ by(7)dGHei,ej=0,if ei=ej;ei×ej,if ei,ej2Aeiej;12,otherwise.

Clearly, dGH is a hypergraphical metric on ξ. Now, let us consider the sequence en in ξ where en=1/2n,n; then, for any fixed k, we have(8)dGH12n,12k=12n+k0,n.

Therefore, the sequence 1/2n converges to 1/2k for every fixed k.

Lemma 2.

Let ξ,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 every eξ, the singleton set e is a closed subset of ξ or the set ξe is an open subset of ξ. For this, let us suppose eξe, then clearly ee 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 eBGHe,ɛ, then dGHe,e<ɛ which is contradiction to . Hence, BGHe,ɛξe is open, and hence hypergraphical metric space is not Hausdorff.

Remark 3.

Let ξ,dGH be hypergraphical metric space in previous remark, then 1/2 is limit point of the sequence en=1/2nξ, but for any k, if k>1, we have limndGH1/2n,1/2k=0dGH1/2,1/2k. Therefore, a hypergraphical metric does not need to be continuous.

Definition 11.

Let ξ,dGH be hypergraphical metric space, and enξ is a sequence. Then, en is called Cauchy if for given ɛ>0 there exist n belong to such that dGHen,em<ɛ,n,m>n; obviously the sequence en is Cauchy sequence limndGHen,em=0.

Definition 12.

A hypergraphical metric space ξ,dGH is called complete if each Cauchy sequence in ξ converges in ξ. Suppose GH is another hypergraph such that each eξGH is subset of VGH, that is, eVGH, 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 hypergraph GH is considered to be directed. We include directed path (p) between edges and denote by eGHl=eξ: directed path from to of length l.

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 ξ,dGH is hypergraphical metric space and F:ξξ is a mapping and GH is subhypergraph of GH such that ξGHΔ. 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 eξGHFeξGH

GHC2: there exists α0,1, such that for eiejξGH and Fei,FejξGH, dGHFei,FejαdGHei,ej for all eiejξGH

Here, we assign the hypergraphical distance between the edges of GH, and hypergraphical contraction decreases the distance by factor α0,1. The sequence en having earliest value e0ξ is called F-picard sequence if en=Fen1,n. Further, we suppose that GH is a subhypergraph of GH such that ξGHΔ. 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.

Suppose ξ,dGH is GH-complete hypergraphical metric space and F:ξξ be a GH,GH-hypergraphical contraction and also satisfies the following conditions. 1 There exist e0ξ such that Fe0e0GHl, for some l. 2 If GH-termwise connected F-picard sequence en converges in ξ, then a limit eξ of en exists and n0, such that en,eξGH,n>n0.

Then, there exist eξ such that the F-picard sequence en of initial value e0 is GH-termwise connected and converges to e and Fe.

Proof.

Suppose e0ξ such that Fe0e0GHl for some l and en is F-picard sequence having initial value e0, then eii=0l is a path such that e0=e0,Fe0=el, and ei1,eiξGH for i=1,2,3,,l. As F is a GH,GH-hypergraphical contraction, we have(9)Fei1,FeiξGH,for i=1,2,3,,l.

Therefore, Teii=0l represent a path from Fe0=Feo=e1 to Tel=F2e0=e2 of length l and so e2e1GHl; proceeding similarly, we get the path Tneii=0l from Fne0=Fne0=en to Fnel=FnFe0=en+1 of length l. Hence, en+1enGHl,n; thus, en is a GH-termwise connected sequence. Since Fnei1,FneiξGH for i=1,2,3,,l and n. Using condition GHC2, we have(10)dGHFnei1,FneiαndGHei1,ei.

Since GH is a subgraph of GH, en and is a termwise connected sequence in GH, by using (10), the following relation holds n, m>n:(11)dGHen,en+1=dGHFne0,Fn+1e0i=1lαndGHei1,ei=αnFl.where Fl=i=1ldGHei1,ei. Again as the sequence en is GH-termwise connected, therefore, n,m with m>n, we have(12)dGHen,emi=nm1dGHei,ei+1i=nm1αiFl=i=nm1αin+nFl=αni=nm1αinFl=αn1α.

Since α0,1, we obtain limn,mdGHen,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 eξ and n0N, such that en,eξGH,n>n0 and limndGHen,e=0. Thus, the sequence en converges to eξ. Now, if en,eξGH for all n>n0 by using GHC2, we obtain(13)dGHen+1,Fe=dGHFen,FeαdGHen,e,for all n>n0,since limndGHen,e=0.

Therefore,(14)limndGHen+1,Fe=0.

A similar result holds if e,enξGH, and hence the sequence en converges to both e and Fe.

If we replace ξ by the set of vertices instead of edges, we get the following corollary.

Corollary 1.

Suppose ξ,dGH is GHcomplete graphical metric space and F:ξξ be a GH,GH-graphical contraction and also satisfies the following conditions.

There exist x0ξ such that Fx0x0GHl. For some lN.

If GH-termwise connected, F-picard sequence xn converges in ξ. Then, a limit xξ of xn exists and n0N, such that xn,xξGH,n>n0.

Then, there exist xξ such that the F-picard sequence en of earliest value x0 is GH-termwise connected and converges to x and Fx.

Remark 4.

Corollary 1 is the result of Shukla .

Remark 5.

Theorem 1 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.

Suppose ζ 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 2A0. Here, 2A is the set, that is, 2A=1/2n:nN and ξGH=ei,ej:eiej and i,jN. Define a mapping dGH:ξξR+ by(15)dGHei,ej=0,if ei=ej;eiej,if ei,ej2Aeiej;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: ξξ by(16)Fe=e2,if e2A;12,if e=0;

It should be noted that F is hypergraphical contraction having α=1/4,e2A, and we have e,FeξGH, which implies that FeeGHl. 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.

Suppose ξ,dGH is hypergraphical metric space and F:ξξ 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 eiξ and ejFξ, then ei=ej.

We represent all fixed point of a set by FixF, and notation for this is ξF=eξ:e,FeξGH .

Remark 6.

If we chose ξGH=ξξ, then it is clear to check that quadruple ξ,dGH,GH,F satisfies property P for arbitrary subhypergraph GH.

Example 5.

Let X, GH, and dGH be those which is used in Example 1. And ξGH=Δei,ej:ei,ej0,1,eieji,jN.(17)Fe=e,if Q0,1;1,otherwise;

Then, the quadruple ξ,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.

Suppose ξ,dGH is GH complete hypergraphical metric space and F:ξξ is GH,GHhypergraphical contraction, it holds the following: (1) there exist e0ξ such that Te0e0GHl for some lN; (2) if a GHtermwise connected Tpicard sequence en converges in ξ, then there eξ of en which is limit point and n0N such that en,eξGH for all n>n0; then, there exist eξ 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 sequence en having earliest value e0 converges to e and Te. As eξ and FeFξ, 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 property P; it is very important to note that in Example 5, every condition of above result holds except property P. However, Fix F=Φ. Therefore, in the above theorem property, P remains unused.

5. Applications

Let I>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=eX:0<infp0,Iep and ep1,t0,I. Here, we have GH=GH and ξGH=Δei,ejei,ejBHeipejpp0,I. Consider that hypergraphical metric space which is given below, that is, dGH:ξξR is given by(18)dGHeiej=0,if ei=ej,supp0,Iln1eipejp,if ei,ejBHeiej,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,I0,I0,+ and h:0,IRR are continuous functions. Mapping βC0,I,R is called lower solution of (19) if βp0Igp,qhq,βqdq,p0,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:ξξ 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:

hq,:RR is increasing function on (0, 1] for every q0,I. Moreover, infp0,Igp,q>0 and gp,qhq,1I1.

There exist α0,1 and ρ1,+ such that for ei, ejξ and ei,ejξ,q,r0,I.

(21)hq,eqhr,ejreiqejrα,0I0Igp,qgp,rdqdrρ,p0,I.

Then, existence of a lower solution of (19) in BH confirms the existence of solution of (19) in ξ.

Proof.

By condition (b), for ei, ejξ also ei,ejξ, and P0,I, we derived(22)ln1FeipFejp=ln10I0Igp,qgp,rhq,eiqhr,ejrdqdr  ln1infp0,Ieipejpα0I0Igp,qgp,rdqdr=ln1FeipFejp=ln10I0Igp,qgp,rhq,eiqhr,ejrdqdr  ln10I0Igp,qgp,reiqejrαdqdr  ln1infp0,Ieipejpα0I0Igp,qgp,rdqdr  ln1ρinfp0,IeipejpααdGHeip,ejpzz.

Then, we have(23)dGHFei,Fej=supp0,Iln1FeipFejpαdGHeiP,ejP.

Further, for ei, ejBH, and eipejp,p0,I and from condition (a) we have infp0,IFeip>0 and(24)Feip=0Igp,qhq,eiqdq0Igp,qh1,1dq1,Fe1p=0Igp,qfq,eiqdq0Igp,qhq,ejqdq=Fejp.

Consequently, the existence of lower solution of equation (19), i.e., βBH 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.

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.

AltunI.SimsekH.Some fixed point theorems on ordered metric spaces and applicationFixed Point Theory and Applications2010201011162146910.1155/2010/6214692-s2.0-77950458379SintunavaratW.Fixed point results in b-metric spaces approach to the existence of a solution for nonlinear integral equationsSerie A. Matemáticas2016110258560010.1007/s13398-015-0251-52-s2.0-84981499481KarapinarE.NoorwaliM.Dragomir and Gosa type inequalities on b-metric spacesJournal of Inequalities and applications201929NageshH. M.GirishV. R.On the entire Zagreb indices of the line graph and line cut-vertex graph of the subdivision graphOpen Journal of Mathematical Sciences20204147047510.30538/oms2020.0137NumanM.ButtS. I.TaimurA.Super cyclic antimagic covering for some families of graphsOpen Journal of Mathematical Sciences202151AsifF.ZahidZ.ZafarS.Leap Zagreb and leap hyper-Zagreb indices of Jahangir and Jahangir derived graphsEngineering and Applied Science Letter20203218echeniqueF.A short and constructive proof of Tarski’s fixed-point theoremInternational Journal of Game Theory200533221521810.1007/s0018204001922-s2.0-33644530065JacekJ.The contraction principle for mappings on a metric with a graphProceeding of the American Mathematical Society2008136413591373AleomraninejadS.RezapourS.ShahzadN.Some fixed point results on a metric space with a graphTopology and its Applications20121593SamreenM.KamranT.ShahzadN.Some fixed point theorems in b-Metric space endowed with graphAbstract and Applied Analysis201320139967132ArgoubiH.SametB.TuriniciM.SametB.TuriniciM.Fixed Point results on a metric space endowed with a finite number of graphs and applicationsCzechoslovak Mathematical Journal201464124125010.1007/s10587-014-0097-62-s2.0-84906056418ShuklaS.Graphical metric space: a generalized setting in fixed point theorySerie A, Matematicas2016111310.1007/s13398-016-0316-02-s2.0-85020398781AbbasM.AlfuraidanM. R.KhanA. R.NazirT.Fixed point results for set-contractions on metric spaces with a directed graphFixed Point Theory and Applications20152015110.1186/s13663-015-0263-z2-s2.0-84922329141NeogM.DebnathP.Fixed points of set valued mappings in terms of start point on a metric space endowed with a directed graphMathematics20175ChuensupantharatN.KumamP.ChauhanV.SinghD.Graphic contraction mapping via graphical b-metric spaces with applicationsArticle in the Bulletin of the Malaysian Society Series20182RegmiS.ArgyrosI. K.GeorgeS.Convergence analysis for a fast class of multi-step Chebyshev-Halley-type methods under weak conditionsOpen Journal of Mathematical Sciences202141344310.30538/oms2021.0143RegmiS.ArgyrosC.ArgyrosC.ArgyrosI. K.GeorgeS.On some iterative methods with frozen derivatives for solving equationsOpen Journal of Mathematical Sciences20215120921710.30538/oms2021.0158