# Common Fixed Points in Generalized Metric Spaces with a Graph

Authors: Karim Chaira; Mustapha Kabil; Abdessamad Kamouss
Journal: Journal of Mathematics
 (2019)

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## Abstract

The aim of this paper is to prove the existence and uniqueness of points of coincidence and common fixed points for a pair of self-mappings defined on generalized metric spaces with a graph. Our results improve and extend several recent results of metric fixed point theory.


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## Body

## 1. Introduction

The study of common fixed points of mappings satisfying certain contractive conditions has been at the center of vigorous research activity. In 1976, Jungck [1] proved a common fixed point theorem for commuting maps such that one of them is continuous. In 1982, Sessa [2] generalized the concept of commuting maps to weakly commutating pair of self-mappings. In 1986, Jungck generalized this idea, first to compatible mappings [3] and then in 1996 to weakly compatible mappings [4]. Using the weakly compatibility, several authors established coincidence points results for various classes of mappings on metric spaces with Fatou property (see [5, 6]). In 2011, Haghi et al. showed that some coincidence point and common fixed point generalizations in fixed point theory are not real generalizations as they could easily be obtained from the corresponding fixed point theorems.Recently, Jleli and Samet introduced a new concept of generalized metric spaces (also known as JS-metric spaces) recovering various topological spaces including standard metric spaces, b-metric spaces, dislocated metric spaces, and modular spaces (see [7]).Motivated by the ideas given in some recent works on metric space with a graph [8–21], we extend some common fixed point theorems of Banach, Chatterjea, and Kannan contractions in standard metric spaces, b-metric spaces, dislocated metric spaces, and modular spaces (see, for example, [20, 22–24]) to common fixed point theorems in generalized metric spaces with a graph. As corollaries, we obtain some results in generalized metric spaces for these contractions. Finally, some examples are given to illustrate our results.


## 2. Some Basic Concepts

In this section, we give some basic notations, definitions, and useful results in generalized metric spaces endowed with a graph.LetX be a nonempty set and D:X×X→0,+∞ be a given mapping.For everyx∈X, let us define the set (1)CD,X,x=xn⊂X:limn→∞⁡Dxn,x=0.Definition 1 (see [7]).
We say thatD is a generalized metric on X if it satisfies the following conditions:
D 1 for every x,y∈X×X, we have Dx,y=0⟹x=y;
( D 2 ) for every x,y∈X×X, we have D(x,y)=D(y,x);
( D 3 ) there exists C>0 such that if (x,y)∈X×X, xn∈C(D,X,x), then (2)Dx,y≤Climsupn→∞⁡Dxn,y.In this case, we say that the pair (X,D) is a generalized metric space.
Obviously, if the setC(D,X,x) is empty for every x∈X, then X,D is a generalized metric space if and only if D1 and D2 are satisfied. A sequence xn in a generalized metric space (X,D) is said to be D-convergent to x∈X if xn∈C(D,X,x). Note that if the set C(D,X,x) is not empty for some x∈X, then Dx,x=0.
A sequencexn in a generalized metric space (X,D) is said to be a D-Cauchy sequence if limm,n→∞⁡Dxn,xm=0. Note that, in generalized metric spaces, a sequence has at most one limit and a D-convergent sequence may not be D-Cauchy sequence. Moreover, (X,D) is said to be D-complete if every D-Cauchy sequence in X is D-convergent to some element in X.
The next example shows that aD-convergent sequence may not be a D-Cauchy sequence.Example 2.
LetX=R+∪0,∞, and let D:X×X→[0,∞] be defined as follows: (3)Dx,y=x+yifatleastoneofxoryis01+x+y3otherwise.Now we check the axioms of a generalized metric space:
D 1   D ( x , y ) = 0 ⇒ either x+y=0 or 1+x+y/3=0. Thus, x+y=0 and x=y=0.
D 2 It is clear that, for all x,y∈X, D(x,y)=D(y,x).
D 3 For all x≠0, we have C(D,X,x)=⌀.
Ifx=0, then we can always find a sequence xn such that limn→∞⁡Dxn,x=0. Taking C≥3, we have D(0,y)=y≤Climsupn→∞⁡D(xn,y).
Now, in this structure, let us consider the sequencexn where xn=1/n for all n in N∗. Then,(4)limn→∞⁡Dxn,0=limn→∞⁡1n+0=0⇒xnD-convergesto0.But, (5)limm,n→∞⁡Dxn,xm=limm,n→∞⁡1+1/n+1/m3≠0.This shows that xn is a D-convergent sequence but not a D-Cauchy sequence. Note that for any x∈X such that x≠0, D(x,x)≠0.Now, we recall some preliminaries from graph theory which are needed for the sequel. The basic concepts related to a graph may be found in any textbook on graph theory, see, for example, [25, 26]. A directed graph or digraph G is determined by a nonempty set V(G) of its vertices and the set E(G)⊂V(G)×V(G) of its directed edges. Let Δ denote the diagonal of the Cartesian product V(G)×V(G). A digraph is said to be reflexive if the set E(G) of its edges contains all loops; i.e., Δ⊂E(G). By G-1 we denote the graph obtained from G by reversing the direction of edges and by G~ we denote the undirected graph obtained from G by ignoring direction of the edges.Throughout this paper, let(X,D) be a generalized metric space. Consider a directed graph G such that the set V(G) of its vertices coincides with X, and the set E(G) of its edges contains all loops. We assume that G has no parallel edges. Moreover, we will use the concept of increasing or decreasing sequences in the sense of a digraph. Therefore, the following definitions are needed.Definition 3 (see [27]).
LetG be a digraph. A sequence xn∈V(G) is said to be (1)
G-increasing, if(xn,xn+1)∈E(G) for all n∈N;(2)
G-decreasing, if(xn+1,xn)∈E(G) for all n∈N;(3)
G-monotone, if it is either G-increasing or G-decreasing.
We will need to assume a property introduced in [28, 29] in partially ordered sets and in [19] in metric spaces with a graph. The digraph G is said to satisfy the (JNRL) property, if for any G-increasing sequence (resp., G-decreasing sequence) xn which D-converges to some x∈V(G), we have (xn,x)∈E(G) (resp., (x,xn)∈E(G)) for any n∈N. Let T and S be two self-mappings on X. The following definitions and proposition will be needed in the sequel.Definition 4 (see [4, 30]).
If there existsx∈X such that Sx=Tx=y, then x is called a coincidence point of S and T, while y is called a point of coincidence (or coincidence value) of T and S. If Sx=Tx=x, then x is called a common fixed point of S and T.
The pair of mappingsT and S is said to be weakly compatible if they commute at their coincidence points.
The digraphG is said to satisfy the property (P) for T and S, if x∗,y∗ are points of coincidence of T and S in V(G), then (x∗,y∗)∈E(G~) and D(x∗,y∗)<∞.Proposition 5 (see [30]).
LetS and T be weakly compatible self-mappings on a nonempty set X. If S and T have a unique coincidence point x, then x is the unique common fixed point of S and T.Now, we introduce G-Banach, G-Chatterjea, and G-Kannan S-contractions mappings in generalized metric spaces.Definition 6.
LetT, S:X→X be two self-mappings. We say that T is (1)
G-Banach S-contraction if there existsk∈[0,1) such that for every x,y∈V(G)(6)Sx,Sy∈EG~⟹DTx,Ty≤kDSx,Sy;(2)
G-Chatterjea S-contraction if there existsk∈[0,1/2) such that for every x,y∈V(G)(7)Sx,Sy∈EG~⟹DTx,Ty≤kDTx,Sy+DSx,Ty;(3)
G-Kannan S-contraction if there existsk∈[0,1/2) such that for every x,y∈V(G)(8)Sx,Sy∈EG~⟹DTx,Ty≤kDTx,Sx+DSy,Ty.
The numberk is called the constant of T.


## 3. Main Results

In this section we establish common fixed point theorems for a pair of weakly compatible self-mappingsS and T such that T is a G-Banach, G-Chatterjea, or G-Kannan S-contraction in the framework of generalized metric spaces with a reflexive digraph G.Throughout this section we assume that(X,D) is a generalized metric space with C≥1, and G is a reflexive directed graph such that V(G)=X, E(G)⊇Δ and the graph G has no parallel edges.LetT and S be two self-mappings on X such that T(X)⊆S(X).Ifx0∈X is arbitrary, we can choose a point x1 in X such that Tx0=Sx1. Continuing in this way, for a value xn in X, we can find xn+1∈X such that(9)Txn=Sxn+1,n=0,1,2,…By C(T,S), we denote the set of all elements x0 of X such that (Sxn,Sxm)∈E(G~) for n,m=1,2,…. The following notation is useful in the sequel:(10)δD,S,T,x0=sup⁡DSxp,Sx1:p≥2.Theorem 7.
Let(X,D) be a generalized metric space endowed with a reflexive digraph G such that V(G)=X and G has no parallel edges and satisfies the (JNRL) property. Let T and S be two self-mappings on X such that T is a G-Banach S-contraction, S(X) is a D-complete subspace of X, and T(X)⊆S(X).
Suppose that there existsx0∈C(T,S) such that δ(D,S,T,x0)<∞; then T and S have a point of coincidence x∗∈X. Moreover, T and S have a unique point of coincidence in X if the digraph G has the property(P) for T and S.
Furthermore, ifT and S are weakly compatible, then T and S have a unique common fixed point in X.Proof.
Suppose that there existsx0 in C(T,S) such that δ(D,S,T,x0)<∞.
Letm and n be in N such that n≥m≥2. Since (Sxn-1,Sxm-1)∈E(G~) and T is a G-Banach S-contraction, then(11)DTxn-1,Txm-1=DSxn,Sxm≤kDSxn-1,Sxm-1,and then, by induction on m, we can get (12)DSxn,Sxm≤km-1DSxn-m+1,Sx1≤km-1δD,S,T,x0.Then D(Sxn,Sxm)→0 when n,m→∞, which implies that Sxn is a D-Cauchy sequence in S(X).
SinceS(X) is a D-complete subspace of X, then the sequence SxnD-converges to some x∗∈S(X). Thus there exists a∈X such that x∗=Sa.
Letn≥2. Since G satisfies the (JNRL) property, then (Sxn-1,Sa)∈E(G~) and since T is a G-Banach S-contraction, then (13)DSxn,Ta=DTxn-1,Ta≤kDSxn-1,Sa=kDSxn-1,x∗.Since limn→∞⁡D(Sxn-1,x∗)=0, then limn→∞⁡D(Sxn,Ta)=0.
With the uniqueness of the limit we getTa=x∗=Sa. Thus x∗ is a point of coincidence of S and T (a is a coincidence point of S and T).
Suppose that there exists another point of coincidencey∗∈S(X) and b∈X such that y∗=Sb=Tb. With the property (P), we have (Sa,Sb)∈E(G~) and D(x∗,y∗)<∞.
SinceT is a G-Banach S-contraction, then we have D(Sa,Sb)=D(Ta,Tb)≤kD(Sa,Sb); thus (1-k)D(Sa,Sb)≤0; then D(Sa,Sb)=0, proving that x∗=Sa=Sb=y∗. Since S and T are weakly compatible, then by Proposition 5, x∗ is the unique common fixed point of S and T.In the next theorem, we establish a common fixed point result for G-Chatterjea S-contraction.Theorem 8.
Let(X,D) be a generalized metric space endowed with a reflexive digraph G such that V(G)=X and G has no parallel edges and satisfies the (JNRL) property. Let T and S be two self-mappings on X such that T is a G-Chatterjea S-contraction and S(X) is a D-complete subspace of X and T(X)⊆S(X). (1)
Suppose that there existsx0∈C(T,S) such that δ(D,S,T,x0)<∞; then the sequence Sxn defined by (9) D-converges to x∗=Sa with a∈X. Moreover, if D(Tx0,Ta)<∞, then x∗ is a point of coincidence of T and S in X.(2)
Moreover,T and S have a unique point of coincidence in X if the digraph G has the property(P) for T and S. Furthermore, if T and S are weakly compatible, then T and S have a unique common fixed point in X.Proof.
(i) Suppose that there exists x0 in C(T,S) such that δ(D,S,T,x0)<∞. Let m and n be in N such that n≥m≥2. Let us prove that(14)DSxn,Sxm≤∑j=m-1n+m-3kjj-1m-2en+m-1-j+∑j=n-1n+m-3kjj-1n-2en+m-1-j,with en+m-1-j=D(Sxn+m-1-j,Sx1).
We prove statement (14) by two-dimensional induction on p=n+m for every p≥4.
Observe that(15)DSx2,Sx2≤kDSx1,Sx2+DSx2,Sx1≤∑j=11kjj-10e3-j+∑j=11kjj-10e3-j.Thus, inequality (14) holds for p=4 with (m,n)=(2,2).
Now, assume that inequality (14) holds for any (m′,n′)∈N2 such that n′≥m′≥2 and n′+m′=p, and let (m,n)∈N2 such that n≥m≥2 and n+m=p+1.
SinceT is a G-Chatterjea S-contraction and (Sxn-1,Sxm-1)∈E(G~), then (16)DSxn,Sxm=DTxn-1,Txm-1≤kDSxn-1,Txm-1+DTxn-1,Sxm-1=kDSxn-1,Sxm+DSxn,Sxm-1.Thus,(17)DSxn,Sxm≤kDSxn-1,Sxm+DSxn,Sxm-1.Since n+(m-1)=p and (n-1)+m=p, the inductive hypothesis gives (18)kDSxn-1,Sxm≤∑j=m-1n+m-4j-1m-2kj+1en+m-2-j+∑j=n-2n+m-4j-1n-3kj+1en+m-2-jkDSxn,Sxm-1≤∑j=m-2n+m-4j-1m-3kj+1en+m-2-j+∑j=n-1n+m-4j-1n-2kj+1en+m-2-j.From inequality (17), we get (19)DSxn,Sxm≤∑j=m-1n+m-4j-1m-3+j-1m-2kj+1en+m-2-j+km-1en+∑j=n-1n+m-4j-1n-3+j-1n-2kj+1en+m-2-j+kn-1em≤∑j=m-1n+m-4jm-2kj+1en+m-2-j+∑j=n-1n+m-4jn-2kj+1en+m-2-j+km-1en+kn-1em≤∑j=mn+m-3j-1m-2kjen+m-1-j+∑j=nn+m-3j-1n-2kjen+m-1-j+km-1en+kn-1em≤∑j=m-1n+m-3j-1m-2kjen+m-1-j+∑j=n-1n+m-3j-1n-2kjen+m-1-j,proving that inequality (14) holds for (n,m)∈N∗ such that n+m=p+1.
Sinceen+m-1-j≤δ0=δ(D,S,T,x0), j-1m-2≤2j-1, and j-1n-2≤2j-1, then, for any n,m∈N such that n≥m≥2, we have (20)DSxn,Sxm≤12δ0∑j=m-1n+m-32kj+∑j=n-1n+m-32kj≤12δ02km-11-2k+2kn-11-2k≤δ02km-11-2k.Then D(Sxn,Sxm)→0 when n and m→∞, which implies that Sxn is a D-Cauchy sequence. Since S(X) is a D-complete subspace of X, then the sequence SxnD-converges to some x∗∈S(X); that is, there exists a∈X such that x∗=Sa. Since G satisfies the (JNRL) property, then (Sxn-1,Sa)∈E(G~). Since T is a G-Chatterjea S-contraction, then for any n≥2 we have (21)DSxn,x∗≤Climsupp→∞⁡DSxn,Sxp≤Climsupp→∞⁡δ02kn-11-2k=Cδ02kn-11-2kNow we prove that for any integer n≥2 we have(22)DSxn,Ta≤kn-1DTx0,Ta+Cδ01-2k2kn-1For n=2, (23)DSx2,Ta=DTx1,Ta≤kDSx1,Ta+DTx1,Sa≤kDTx0,Ta+DSx2,x∗≤kDTx0,Ta+Cδ01-2k2k1.Let n≥2. Suppose that inequality (22) holds for n and prove that it holds for n+1. (24)DSxn+1,Ta=DTxn,Ta≤kDSxn,Ta+DTxn,Sa≤kDSxn,Ta+DSxn+1,Sa≤knDTx0,Ta+kCδ01-2k2kn-1+kCδ01-2k2kn≤knDTx0,Ta+Cδ01-2k2kn.Since D(Tx0,Ta)<∞, then limn→∞⁡D(Sxn,Ta)=0. With the uniqueness of the limit we get Ta=x∗=Sa. Thus x∗ is a point of coincidence of S and T (a is a coincidence point of S and T).
(ii) Assume that there exists another point of coincidence y∗∈S(X) and b∈X such that y∗=Sb=Tb, (Sa,Sb)∈E(G~), and D(x∗,y∗)<∞.
SinceT is a G-Chatterjea S-contraction, then we have (25)DSa,Sb=DTa,Tb≤kDSa,Tb+DTa,Sb≤kDSa,Sb+DSa,Sb, and thus (1-2k)D(Sa,Sb)≤0, and then D(Sa,Sb)=0 and x∗=Sa=Sb=y∗. Since S and T are weakly compatible, then, by Proposition 5, x∗ is the unique common fixed point of S and T.In the next theorem, we establish a common fixed point result for G-Kannan S-contraction.Theorem 9.
Let(X,D) be a generalized metric space endowed with a reflexive digraph G such that V(G)=X and G has no parallel edges and satisfies the (JNRL) property. Let T and S be two self-mappings on X such that T is a G-Kannan S-contraction with constant k∈0,inf1/2,1/C and S(X) is a D-complete subspace of X and T(X)⊆S(X). (1)
Suppose that there existsx0∈C(T,S) such that δ(D,S,T,x0)<∞; then the sequence Sxn defined by (9) D-converges to x∗=Sa with a∈X. Moreover, if D(Sa,Ta)<∞ then x∗ is a point of coincidence of T and S in X.(2)
Moreover,T and S have a unique point of coincidence in X if the digraph G has the property(P) for T and S. Furthermore, if T and S are weakly compatible, then T and S have a unique common fixed point in X.Proof.
(i) Suppose that there exists x0 in C(T,S) such that δ0=δ(D,S,T,x0)<∞. Let m and n be in N such that n≥m≥2.
SinceT is a G-Kannan S-contraction and (Sxn-1,Sxm-1)∈E(G~), then we have (26)DSxn,Sxn-1=DTxn-1,Txn-2≤kDSxn-1,Txn-1+DTxn-2,Sxn-2≤kDSxn-1,Sxn+DSxn-1,Sxn-2,which implies that(27)DSxn,Sxn-1≤k1-kDSxn-1,Sxn-2.Then, by induction, we can get(28)DSxn,Sxn-1≤k1-kn-2DSx2,Sx1≤βn-2DSx2,Sx1,where β=k/1-k. Now by using (8) we get (29)DSxn,Sxm≤kDSxn-1,Sxn+DSxm,Sxm-1≤kβn-2+βm-2DSx2,Sx1.Since D(Sx2,Sx1)<δ0, then D(Sxn,Sxm)→0 when n and m→∞, which implies that Sxn is a D-Cauchy sequence. Since S(X) is a D-complete subspace of X, then the sequence SxnD-converges to some x∗∈S(X). Thus there exists a∈X such that x∗=Sa.
SinceG satisfies the (JNRL) property, then (Sxn-1,Sa)∈E(G~). Since T is a G-Kannan S-contraction, then for any n≥2 we have (30)DSxn,Ta=DTxn-1,Ta≤kDTxn-1,Sxn-1+DTa,Sa≤kDSxn,Sxn-1+DTa,Sa≤kδ0βn-2+DTa,Sa.
Taking superior limit asn→∞, we get (31)DSa,Ta≤Climsupn→∞⁡DSxn,Ta≤kCDSa,Ta.Thus (1-kC)D(Sa,Ta)≤0. Since D(Sa,Ta)<∞, then Ta=x∗=Sa. Thus x∗ is a point of coincidence of S and T (a is a coincidence point of S and T).
(ii) Suppose that there exists another point of coincidence y∗∈S(X) and b∈X such that y∗=Sb=Tb. With the property (P), we have (Sa,Sb)∈E(G~) and D(x∗,y∗)<∞.
SinceT is a G-Kannan S-contraction, then we have (32)DSa,Sb=DTa,Tb≤kDTa,Sa+DSb,Tb≤0,and then Sa=Sb=x∗=y∗.
SinceS and T are weakly compatible, then, by Proposition 5, x∗ is the unique common fixed point of S and T.Example 10.
LetX=R and define D:X×X→R+ by D(x,y)=x-y2 for all x,y∈X. Then (X,D) is a D-complete generalized metric space with the coefficient C≥2. Consider the graph G on X defined by V(G)=X and(33)EG=Δ∪0,110n:n=0,1,…∪110m,110n:m>nandn,m=0,1,….The digraph G satisfies the (JNRL) property; in fact let yn be a G-monotone sequence in V(G) which D-converges to some y∈V(G). (1)
Letyn be a G-increasing sequence; i.e., (yn,yn+1)∈E(G) for any n∈N; then we have the following:
(a)
Ify0=a such that a≠0 and a≠1/10p,∀p∈N, then yn=a for all n∈N.(b)
Ify0=0, then yn is a null sequence or yn is a stationary sequence.(c)
Ify0=1/10p with p∈N, then yn is a stationary sequence.In the three cases, we have(yn,y)∈E(G).(2)
Letyn be a G-decreasing sequence; i.e., (yn+1,yn)∈E(G) for any n∈N; then we have the following:
(a)
Ify0=a such that a≠1/10p,∀p∈N, then yn is a constant sequence.(b)
Ify0=1/10p with p∈N, then yn is a stationary sequence or yn=1/10pn for any n∈N such that pn is a increasing integer sequence and ynD-converges to 0.In both cases, we have(y,yn)∈E(G).LetT and S be two self-mappings on X defined by(34)Tx=x2ifx≠271ifx=27.and Sx=5x for all x∈X. Obviously, T(X)⊆S(X)=X.Letx,y∈X such that (Sx,Sy)∈E(G).(1)
Ifx=y, then Sx=Sy and so (Sx,Sy)∈E(G). Thus D(Tx,Ty)=0.(2)
Ifx=0,y=1/5.10n, then Sx=0, Sy=1/10n and so (Sx,Sy)∈E(G).(35)DTx,Ty=1102n+2DSx,Sy=1102nDSx,Ty+Dfx,gy=101102n+2DTx,Sx+Dfy,gy=81102n+2(3)
Ifx=1/5.10m, y=1/5.10n, then Sx=1/10m, Sy=1/10n and so (Sx,Sy)∈E(G~).(36)DTx,Ty=102n+102m-2.10n+m102n+m+1DSx,Sy=102n+102m-2.10n+m102n+mDSx,Ty+DTx,Sy=101102n+102m-40.10n+m102n+m+1DTx,Sx+DTy,Sy=81102n+102m102n+m+1In all cases we have(37)DTx,Ty≤k1DSx,Sywithk1∈1100,12DTx,Ty≤k2DSx,Ty+DTx,Sywithk2∈1101,12DTx,Ty≤k3DTx,Sx+DTy,Sywithk3∈181,12proving that T is a G-Banach, G-Chatterjea, and G-Kannan S-contraction, respectively.Forx0=1/5 we have δ(D,S,T,x0)=1/100<∞ and we can verify that x0∈C(T,S). In fact, (9) gives that Sx1=T1/5=1/10 and so Sx2=Tx1=1/102. Proceeding in this way, we get Sxn=1/10n for n=1,2,… and hence (Sxm,Sxn)=(1/10m,1/10n)∈E(G) for m,n=1,2,…. Then SxnD-converges to 0. Since D(Tx0,T0)=1/100<∞ and D(S0,T0)=0<∞, then 0=T0=S0 is a point of coincidence of T and S. Obviously G satisfies the property (P) for T and S.Furthermore,T and S are weakly compatible. Thus, we have all the conditions of Theorems 7, 8, and 9, proving that T and S have a unique common fixed point in X which is 0.The next remark shows that we cannot relax the weak compatibility condition in Theorem7.Remark 11.
In Example10, if we take Sx=5x-9 for all x∈X, and x0=2, then we can easily verify that T is a G-Banach S-contraction, x0∈C(T,S) and T(2)=S(2)=1; i.e., 2 is a coincidence point of S and T. Since TS2≠ST2, then T and S are not weakly compatible. However, we can check that all other conditions of Theorem 7 are satisfied. Despite the fact that T and S have a unique point of coincidence 1, they do not have any common fixed point.Remark 12.
(1)
By takingG=G0, where G0 is the complete graph, i.e., G0=(X,X×X), we get corollaries for Theorems 7, 8, and 9 in the framework of generalized metric space without using graph for Banach, Chatterjea, and Kannan S-contractions. (2)
Since standard metric spaces and b-metric spaces are particular generalized metric spaces, then our results can be viewed as generalizations and extensions of corresponding results in [20, 22–24] and several other comparable results. (3)
By takingS=IX in Theorems 7, 8, and 9, we obtain versions of Banach, Chatterjea, and Kannan fixed point theorems in generalized metric spaces with a graph.



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