Fixed Point Results for 
 C
 -Contractive Mappings in Generalized Metric Spaces with a Graph

In this paper, we establish fixed point theorems for Chatterjea contraction mappings on a generalized metric space endowed with a graph. Our results extend, generalize, and improve many of existing theorems in the literature. Moreover, some examples and an application to matrix equations are given to support our main result.


Introduction
Fixed point theorems for contraction mappings and their generalizations play a crucial role in the determination of the existence and uniqueness of solutions of certain problems in mathematics and applied sciences, such as variational and linear inequalities, mathematical models, optimization, and mathematical economics. In 1922, Banach [1] proved the contraction principle, today named after him, which states any contraction on a complete metric space has a unique fixed point. In 1972, Chatterjea [2] proved that a selfmapping on a complete metric space X has a unique fixed point whenever there exists 0 ≤ k < 1/2 such that On the other hand, different generalizations of the usual notion of metric space were proposed by a number of mathematicians (see [3,4]). Recently, Jleli and Samet [5] introduced a new concept of generalized metric space that, in fact, recovers various topological spaces. The class of such metric spaces is larger than the class of standard metric spaces, than the class of b-metric spaces, than that of dislocated metric spaces, than that of dislocated b-metric spaces, and than the class of modular spaces with the Fatou property. The interested reader is referred to [5] for further details.
This work is the continuation of [6]. Motivated by the ideas recently introduced in [7][8][9][10][11][12]), we extend the Chatterjea fixed point theorem to the setting of generalized metric spaces with a graph. As corollaries, we obtain Chatterjea fixed point theorems in the setting of partially ordered metric spaces. Furthermore, we generalize the common fixed point result given in [13]. We provide an example to illustrate our main result.

Preliminaries
We recall the definition of generalized metric space and some related topological concepts, as introduced firstly by Jleli and Samet in [5].
Definition 1 [5]. Let X be a nonempty set and D : X × X⟶½0, +∞ be a given mapping.
For every x ∈ X, define the set We say that D is a generalized metric on X if it satisfies the following conditions: (D 1 ) For every ðx, yÞ ∈ X × X, Dðx, yÞ = 0 implies x = y (D 2 ) For every ðx, yÞ ∈ X × X, Dðx, yÞ = Dðy, xÞ (D 3 ) There exists C > 0 such that for all ðx, yÞ ∈ X × X, if there exists fx n g ∈ CðD, X, xÞ, then D x, y ð Þ≤ Clim sup n⟶∞ D x n , y ð Þ: ð3Þ The pair ðX, DÞ is called a generalized metric space.
(i) A sequence fx n g in a generalized metric space ðX, DÞ is said to be D-convergent to x ∈ X if fx n g ∈ CðD, X, xÞ The space ðX, DÞ is said to be D-compact if every sequence in X has a D-convergent subsequence to some element in X The basic concepts, notation, and terminology related to graph theory can be found, for example, in [14,15]. A directed graph or digraph G consists of a nonempty set VðGÞ, whose elements are called the vertices of G, and a set EðGÞ ⊂ VðGÞ × VðGÞ, called the set of directed edges of G. The diagonal of the cartesian product VðGÞ × VðGÞ will be denoted by Δ. A digraph is said to be reflexive if EðGÞ contains all loops, i.e., if Δ ⊂ EðGÞ. G is said to be transitive if, for any x, y, z ∈ VðGÞ Given a digraph G = ðV, EÞ, a directed path in G is a sequence of vertices. a 0 ,a 1 ,...,a n ⋯ , with ða i , a i+1 Þ ∈ EðGÞ for each i ∈ ℕ. A finite path ða 0 , a 1 , ⋯, a n Þ is said to have length n. The transitive closure of G is the digraph G ′ such that VðG ′ Þ = VðGÞ and that ði, jÞ is an edge in G ′ if there is a directed path from i to j in G.
We say that a vertex x in VðGÞ is isolated if for any vertex y in VðGÞ such that x ≠ y, neither ðx, yÞ ∈ EðGÞ nor ðy, xÞ ∈ EðGÞ.
In the sequel, given a graph G, G −1 will stand for it, that is, for the graph obtained from G by reversing the direction of its edges. Thus, In addition,G will stand for the undirected graph obtained from G by ignoring the direction of its edges. In other words, Throughout this paper, the triplet ðX, D, GÞ will stand for the generalized metric space ðX, DÞ endowed with a reflexive digraph G such that VðGÞ = X. In [16], Alfuraidan et al. introduced the idea of G-monotonicity of sequences and the G-completeness of the metric space. Specifically, Definition 3 [16]. Let G be a digraph. A sequence fx n g ∈ VðGÞ is said to be The preceding notion of G-completeness can naturally be extended to the setting of generalized metric spaces as follows: Definition 4. A generalized metric space ðX, DÞ is said to be G -complete if any D-Cauchy, G-monotone sequence fx n g ⊂ VðGÞ is D-convergent to an element in VðGÞ.

Remark 5.
It is shown in [16] (Example 3.3) that G-completeness is finer than usual completeness.
The following definitions of some useful types of continuity are borrowed from [11]. Definition 6. A self-mapping T on the generalized metric space X is called (i) Subsequentially continuous, if for every sequence fx n g ⊂ X, D-convergent to x ∈ X, there exists a subsequence fx n q g of fx n g such that fTx n q gD-converges to Tx (as q ⟶ ∞) (ii) Orbitally G-continuous, if for all x, y ∈ VðGÞ and any sequence fk n g of positive integers The following property, initially introduced in [17] for partially ordered sets and in [11] for metric spaces with a graph, is often assumed to relax continuity assumptions.
Property (JNRL). The digraph G is said to satisfy the property (JNRL), if for any G-monotone increasing (decreasing) sequence fx n g, which D-converges to some x ∈ VðGÞ, it holds that ðx n , xÞ ∈ EðGÞ (ðx, x n Þ ∈ EðGÞ), for any n ∈ ℕ.
Let ðX, D, GÞ be a generalized metric space endowed with a reflexive graph. Motivated by [11,18], we define G-Chatterjea mappings on a generalized metric space ðX, DÞ with a graph, as follows: Journal of Function Spaces Definition 7. A mapping T : X ⟶ X is said to be a G -Chatterjea mapping if the following conditions are satisfied: (i) T is G-monotone (edge-preserving), that is, if: (ii) There exists k ∈ ½0, 1/2Þ such that for every ðx, yÞ ∈ E ðGÞ, Remark 8. It follows immediately from the above definition that: (i) If T is a G-Chatterjea mapping, then T is both a G −1 -Chatterjea and aG-Chatterjea mapping (ii) Any Chatterjea mapping is a G 0 -Chatterjea mapping, where the complete graph G 0 is defined by VðG 0 Þ = X and EðG 0 Þ =X × X The following example shows that a G-Chatterjea mapping is not necessarily a Chatterjea mapping. Example 1. Let X = f0, 1, 2, 3g. Consider the function D defined on X by Dðx, yÞ = ðx − yÞ 2 . It can be shown that D is a generalized metric with constant C ≥ 2.
On the other hand, consider the digraph G with VðGÞ = X and edges It can be easily seen that f is a G-Chatterjea mapping with constant k ∈ ½1/9, 1/2Þ.

Main Results
In this section, we extend the fixed point theorems for G -Chatterjea mappings to the setting of a generalized metric space with a digraph.
Let T : X ⟶ X be a mapping. Let : n ∈ ℕg. The following technical lemmas are necessary for the proof of the main result in this work.
Proof. Without loss of generality, assume that ðx 0 , The following notation will be used in the sequel: Lemma 10. Under the assumptions of Lemma 9, if T is a G -Chatterjea mapping with constant k ∈ ½0, 1/2Þ, then (ii) For every ðm, nÞ ∈ ℕ * 2 such that m ≤ n, we have where δ 0 = δðD, T, x 0 Þ. Proof.
(i) The proof of this statement follows from the application of two-dimensional induction on p = n + m, for every p ≥ 2 Since it is clear that the inequality (13) holds for p = 2 with ðm, nÞ = ð1, 1Þ.
and that It follows from inequality (13) that Theorem 11. Let ðX, D, GÞ be a generalized, G-complete metric space and T : X ⟶ X be a G-Chatterjea mapping with constant k ∈ ½0, 1/2Þ. Suppose that there exists x 0 ∈ X such that δðD, T, x 0 Þ < ∞, that ðx 0 , Tx 0 Þ ∈ EðGÞ, and that the subgraph G½O T ðx 0 Þ is transitive. Under these assumptions, the sequence fT n x 0 g converges to some ω ∈ X. Moreover, if one of the following conditions ðiÞ − ðiiiÞ holds, namely Proof. Without loss of generality, it may be assumed that ðx 0 , Tx 0 Þ ∈ EðGÞ. Select ðm, nÞ ∈ ℕ * × ℕ * such that m ≤ n. From Lemma 9, it is clear that ðT m x 0 , T n x 0 Þ ∈ EðGÞ. If T is a G-Chatterjea mapping, Lemma 10 yields Thus, fT n x 0 g is a D-Cauchy sequence. Since ðX, D, GÞ is G-complete, the sequence fT n x 0 gD-converges to some ω ∈ X.
(i) It follows from the subsequential continuity assumption on T that there exists a subsequence fT n q x 0 g such that fT n q +1 x 0 gD-converges to Tω as n q ⟶ ∞. The uniqueness of the limit yields Tω = ω (ii) Assume that T is orbitally G-continuous. Since fT n x 0 gD-converges to ω and ðT n x 0 , T n+1 x 0 Þ ∈ EðGÞ, it follows that TðT n x 0 Þ ⟶ Tω. Likewise, TðT n x 0 Þ = T n+1 x 0 ⟶ ω. Hence, ω = Tω (iii) Assume that G satisfies Property (JNRL) and that Dðx 0 , TωÞ < ∞. Since fT n x 0 g is G-increasing and it D-converges to ω ∈ X, it follows that ðT n x 0 , ωÞ ∈ EðGÞ, for any n ∈ ℕ Select n ∈ ℕ, n ≥ 1. If T is a G-Chatterjea mapping, then necessarily It follows by induction on n, that for any n ≥ 1, Journal of Function Spaces Let j ∈ 1, n. Since fT p x 0 g p≥n D-converges to ω using ðD 3 Þ, it follows that Applying Lemma 10, we obtain Then, Finally, inequality (23) becomes Since Dðx 0 , TωÞ < ∞, it follows that lim n⟶∞ DðT n x 0 , TωÞ = 0. Therefore, fT n x 0 gD-converges to Tω. Uniqueness of the limit yields Tω = ω.
The following example illustrates Theorem 11.
It can be easily verified that ðD 1 Þ and ðD 2 Þ hold. For the validity of ðD 3 Þ, observe first that for all x ≠ 0, we have Cð D, X, xÞ = ∅. If x = 0, then there exists a sequence fx n g such that lim n⟶∞ Dðx n , xÞ = 0. Consider the sets P ≔ fn ∈ ℕ : x n ≠ 0g and Q ≔ fn ∈ ℕ : x n = 0g. We distinguish three cases: If P is finite, then there exists C ≥ 1 such that for any y ∈ X it holds that If Q is finite, then there exists C ≥ 6 such that for any y ∈ X If P and Q are infinite, there exist two increasing functions φ, ψ : ℕ ⟶ ℕ such that, for all n ∈ ℕ, x φðnÞ ≠ 0, x ψðnÞ = 0, and fx n g = fx φðnÞ g ∪ fx ψðnÞ g. Then, for any y ∈ X lim sup n⟶∞ D x φ n ð Þ , y = y j j 3 and lim sup n⟶∞ D x ψ n ð Þ , y = 2 y j j: Thus, D is a generalized metric with C ≥ 6. Note that X is not a D-compact space. Indeed, let fx n g n∈ℕ * be a sequence of X such that x n = 1 − 1/n and suppose that there exists a subsequence fx φðnÞ g of fx n g wich D-converges to an element x in X. Since lim n⟶∞ Dðx φðnÞ , xÞ = 0, we have Thus, |x | = −1. Contradiction. Consider the graph G on X consisting of the transitive closure of the graph represented in Figure 1.
Note that Let us prove that the space ðX, DÞ is G-complete. Let fx n g be a G-monotone, D-Cauchy sequence in X. We have two cases: 5 Journal of Function Spaces Case 1. If there exists n 0 ∈ ℕ such that x n = 0, for any n ≥ n 0 . We have lim n⟶∞ Dðx n , 0Þ = 0. Therefore, the sequence fx n gD -converges to 0.
Next, we present a version of Theorem 11 in the setting of a partially ordered generalized metric space. Let ðX, D,≤Þ be a generalized metric space endowed with a partial order. We define the directed graph G ≤ on X as follows: VðG ≤ Þ = X and EðG ≤ Þ = fðx, yÞ ∈ X × X : x ≤ yg. In this setting, we say that T : X ⟶ X is a monotone Chatterjea mapping if it is a G ≤ -Chatterjea mapping. We also say that T is orbitally monotone continuous if T is orbitally G ≤ -continuous. The generalized metric space ðX, D,≤Þ satisfies the Property (JNRL) if whenever fx n g is a decreasing (respectively increasing) sequence such that x n ⟶ x in X, then for all n ∈ ℕ, x ≤ x n (respectively x n ≤ x). Theorem 13. Let ðX, D,≤Þ be a generalized D-complete metric space endowed with a partial order and T : X ⟶ X be a monotone Chatterjea mapping with constant k ∈ ½0, 1/2Þ. Suppose that there exists x 0 ∈ X such that δðD, T, x 0 Þ < ∞ and that either x 0 ≤ Tx 0 or Tx 0 ≤ x 0 . Then, the sequence fT n x 0 g converges to some ω ∈ X. Moreover, if any one of the conditions ðiÞ − ðiiiÞ in Theorem 11 holds, then ω is a fixed point of T.

Journal of Function Spaces
Proof. Since the subgraph G ≤ ½O T ðx 0 Þ is transitive, Theorem 13 is a direct consequence of Theorem 11.
We remark that Theorem 3.9 in [19] is a corollary of the preceding theorem, from which it can be derived simply by removing the ordering.
We next set to show that the fixed point result given in Theorem 11 is, in fact, a generalization of the analogue common fixed point theorem established in [13]. To this effect, we state and prove the following lemma, introduced by Haghi et al. in [20].

Lemma 15.
Let X be a nonempty set and f : X ⟶ X a function. Then, there exists a subset E ⊂ X such that f ðEÞ = f ðXÞ. Moreover, f : E ⟶ X is one-to-one.
Let T, S : X ⟶ X be two self mappings. We recall the definition of G-Chatterjea S-contraction and the property (P) given in [13].
Definition 16. We say that T is G-Chatterjea S-contraction if there exists k ∈ ½0, 1/2Þ such that for every x, y ∈ VðGÞ, it holds that We recall that x * is said to be a point of coincidence of T and S, if there exists a in X such that x * = Ta = Sa.
Property (P). The digraph G is said to satisfy the property (P) for T and S, if whenever x * , y * are points of coincidence of T and S in VðGÞ, then ðx * , y * Þ ∈ EðGÞ and Dðx * , y * Þ<∞.
Suppose that TðXÞ ⊆ SðXÞ. If x 0 ∈ X is arbitrary, we can choose a point x 1 in X such that Tx 0 = Sx 1 . Proceeding in this manner, assuming that x n in X is given, we can define x n+1 ∈ X by the recurrence relation Tx n = Sx n+1 , n = 0, 1, 2,: ⋯ By CðT, SÞ, we denote the set of all elements x 0 of X such that ðSx n , Sx m Þ ∈ EðGÞ, for n, m = 1, 2, ⋯. The following notation will be used in the sequel: Corollary 17. Let ðX, DÞ be a generalized metric space endowed with a reflexive digraph G. Assume that VðGÞ = X, that G has no parallel edges, and that it satisfies the (JNRL) property. Let T and S be two self mappings on X such that T is a G-Chatterjea S-contraction, SðXÞ is a D-complete subspace of X and that TðXÞ ⊆ SðXÞ.
(1) Suppose that there exists x 0 ∈ CðT, SÞ such that δðD, S, T, x 0 Þ < ∞. Then, the sequence fSx n g defined by (46) D-converges to x * = Sa, with a ∈ X. Moreover, if DðTx 0 , TaÞ < ∞, then x * is a point of coincidence of T and S in X (2) In addition, T and S have a unique point of coincidence in X if the digraph G has the property (P) for T and S. Finally, if T and S are weakly compatible, then T and S have a unique common fixed point in X Since S is one-to-one on X 0 , F is well defined. Let u, v ∈ Y. There exist x, y ∈ X such that u = Sx and v = Sy. If ðu, vÞ ∈ EðGÞ, then ðSx, SyÞ ∈ EðGÞ. Since T is a G -Chatterjea S-contraction, there exists k ∈ ½0, 1/2Þ such that i.e., DðFðSxÞ, FðSyÞÞ ≤ kðDðFðSxÞ, SyÞ + DðSx, FðSyÞÞÞ: Then Consequently, F is a G-Chatterjea mapping on Y.
By virtue of Theorem 11, the sequence fSx n g = fF n−1 y 0 g D-converges to x * = Sa with a ∈ Y ⊂ X.
Moreover, we have 7 Journal of Function Spaces and since G satisfies property (JNRL), on account of Theorem 11, x * is a fixed point of F. Hence Ta = FðSaÞ = Fx * = x * = Sa, and x * is a point of coincidence of T and S in X, as claimed.
Assume next that there exists another point of coincidence y * ∈ SðXÞ, that b ∈ X, and that y * = Sb = Tb = FðSbÞ = Fy * . Since the digraph G has the property (P) for T and S, then ðx * , y * Þ ∈ EðGÞ and Dðx * , y * Þ < ∞. By Proposition 12, necessarily x * = y * , which implies that T and S have a unique point of coincidence in X. It follows from [21] (Proposition 1.4) that if T and S are weakly compatible, then T and S have a unique common fixed point in X.

Application
In this section, we study the existence and uniqueness of solution for the following general nonlinear matrix equation in the set of all n × n Hermitian-positive definite matrices P ðnÞ: where A is n × n nonsingular matrix, A * is the Hermitian transpose of the matrix A, the matrix B is n × n positive define matrix, and F : EðnÞ ⟶ EðnÞ is a self-adjoint operator such that EðnÞ is a nonempty subset of P ðnÞ. This type of matrix equation arises in control theory, ladder networks, dynamic programming, stochastic filtering and statistics, etc. For M, N ∈ P ðnÞ, we denote We denote by k:k the spectral norm kAk = ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρðA * AÞ p = kA * k, where ρðA * AÞ is the largest eigenvalue of A * A. We recall that the Thompson metric is defined on P ðnÞ by: such that where W ðA/BÞ = inf fλ > 0 : A ≤ λBg = λ max ðA −ð1/2Þ B A −ð1/2Þ Þ. It is easy to verify that ðP ðnÞ, dÞ is a complete metric space (see [22]). In the sequel, we consider the space P ðnÞ endowed by the Thompson generalized metric D defined by for any A, B ∈ P ðnÞ. In the following lemmas, we extend some properties of the Thompson metric given in [23] to the Thompson generalized metric space.