# Fixed Point Results forC-Contractive Mappings in Generalized Metric Spaces with a Graph

Authors: Karim Chaira; Mustapha Kabil; Abdessamad Kamouss
Journal: Journal of Function Spaces
 (2021)

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## Abstract

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.


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## Body

## 1. 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 self-mapping on a complete metric space X has a unique fixed point whenever there exists 0≤k<1/2 such that
(1)dTx,Ty≤kdx,Ty+dy,Tx,forallx,y∈X.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 [7–12]), 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.


## 2. 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].
LetX be a nonempty set and D:X×X⟶0,+∞ be a given mapping.
For everyx∈X, define the set
(2)CD,X,x=xn⊂X:limn⟶∞Dxn,x=0.We say thatD is a generalized metric on X if it satisfies the following conditions:(D1) For every x,y∈X×X, Dx,y=0 implies x=y(D2) For every x,y∈X×X, Dx,y=Dy,x(D3) There exists C>0 such that for all x,y∈X×X, if there exists xn∈CD,X,x, then
(3)Dx,y≤Climsupn⟶∞Dxn,y.The pairX,D is called a generalized metric space.Definition 2 [5].
(i)
A sequencexn in a generalized metric space X,D is said to be D-convergent to x∈X if xn∈CD,X,x(ii)
A sequencexn in a generalized metric space X,D is said to be a D-Cauchy sequence if limm,n⟶∞Dxn,xm=0(iii)
The spaceX,D is said to be D-complete if every D-Cauchy sequence in X is D-convergent to some element in X(iv)
The spaceX,D is said to be D-compact if every sequence in X has a D-convergent subsequence to some element in XThe 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 VG, whose elements are called the vertices of G, and a set EG⊂VG×VG, called the set of directed edges of G. The diagonal of the cartesian product VG×VG will be denoted by Δ. A digraph is said to be reflexive if EG contains all loops, i.e., if Δ⊂EG. G is said to be transitive if, for any x,y,z∈VG(4)x,y∈EGandy,z∈EGx,z∈EG.Given a digraphG=V,E, a directed path in G is a sequence of vertices.a0,a1,...,an⋯, with ai,ai+1∈EG for each i∈ℕ. A finite path a0,a1,⋯,an is said to have length n. The transitive closure of G is the digraph G′ such that VG′=VG and that i,j is an edge in G′ if there is a directed path from i to j in G.We say that a vertexx in VG is isolated if for any vertex y in VG such that x≠y, neither x,y∈EG nor y,x∈EG.In the sequel, given a graphG, G−1 will stand for it, that is, for the graph obtained from G by reversing the direction of its edges. Thus,
(5)EG−1=x,y∈X×X:y,x∈EG.In addition,G~ will stand for the undirected graph obtained from G by ignoring the direction of its edges. In other words,
(6)EG~=EG∪EG−1.Throughout this paper, the tripletX,D,G will stand for the generalized metric space X,D endowed with a reflexive digraph G such that VG=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].
LetG be a digraph. A sequence xn∈VG is said to be
(i)
G-increasing, if xn,xn+1∈EG for all n∈ℕ(ii)
G-decreasing, if xn+1,xn∈EG for all n∈ℕ(iii)
G-monotone, if it is either G-increasing or G-decreasingThe preceding notion ofG-completeness can naturally be extended to the setting of generalized metric spaces as follows:Definition 4.
A generalized metric spaceX,D is said to be G-complete if any D-Cauchy, G-monotone sequence xn⊂VG is D-convergent to an element in VG.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-mappingT on the generalized metric space X is called
(i)
Subsequentially continuous, if for every sequencexn⊂X, D-convergent to x∈X, there exists a subsequence xnq of xn such that TxnqD-converges to Tx (as q⟶∞)(ii)
OrbitallyG-continuous, if for all x,y∈VG and any sequence kn of positive integers(7)Tknx⟶yandTknx,Tkn+1x∈EG~implyTTknx⟶Ty.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 xn, which D-converges to some x∈VG, it holds that xn,x∈EG (x,xn∈EG), for any n∈ℕ.LetX,D,G be a generalized metric space endowed with a reflexive graph. Motivated by [11, 18], we define G-Chatterjea mappings on a generalized metric space X,D with a graph, as follows:Definition 7.
A mappingT: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:(8)Tx,Ty∈EG,foreveryx,y∈EG,(ii)
There existsk∈0,1/2 such that for every x,y∈EG,(9)DTx,Ty≤kDTx,y+Dx,Ty.Remark 8.
It follows immediately from the above definition that:(i)
IfT is a G-Chatterjea mapping, then T is both a G−1-Chatterjea and a G~-Chatterjea mapping(ii)
Any Chatterjea mapping is aG0-Chatterjea mapping, where the complete graph G0 is defined by VG0=X and EG0=X×XThe following example shows that aG-Chatterjea mapping is not necessarily a Chatterjea mapping.Example 1.
LetX=0,1,2,3. Consider the function D defined on X by Dx,y=x−y2. It can be shown that D is a generalized metric with constant C≥2.
Consider the mappingf:X⟶X defined by
(10)f0=1f1=f2=0,f3=1.SinceDf0,f2=1 and Df0,2)+D(0,f2=1, f is not a Chatterjea mapping.On the other hand, consider the digraphG with VG=X and edges
(11)EG=0,0,1,1,2,2,3,3,0,3,1,2,1,3,2,3.It can be easily seen thatf is a G-Chatterjea mapping with constant k∈1/9,1/2.


## 3. Main Results

In this section, we extend the fixed point theorems forG-Chatterjea mappings to the setting of a generalized metric space with a digraph.LetT:X⟶X be a mapping. Let x0∈X. Let GOTx0 be the subgraph of G induced on the orbit OTx0≔Tnx0:n∈ℕ. The following technical lemmas are necessary for the proof of the main result in this work.Lemma 9.
LetT:X⟶X be a G-monotone mapping and suppose that there exists x0∈X such that x0,Tx0∈EG (respectively, Tx0,x0∈EG) and that the subgraph GOTx0 is transitive. Then, Tnx0 is a G-increasing (respectively, G-decreasing) sequence and Tmx0,Tnx0∈EG (respectively, Tnx0,Tmx0∈EG) for any m,n∈ℕ such that m≤n.Proof.
Without loss of generality, assume thatx0,Tx0∈EG. Since T is G-monotone, it follows that Tx0,T2x0∈EG. Induction on n yieldsTnx0,Tn+1x0∈EG for all n∈ℕ. Therefore, Tnx0 is a G-monotone increasing sequence. Since Tmx0,Tm+1x0,Tm+1x0,Tm+2x0,⋯,Tn−1x0,Tnx0∈EG and GOTx0istransitive, it follows that Tmx0,Tnx0∈EG.
The following notation will be used in the sequel:(12)δD,T,x0≔supDTix0,x0:i∈ℕ∗.Lemma 10.
Under the assumptions of Lemma9, if T is a G-Chatterjea mapping with constant k∈0,1/2, then
(i)
For everym,n∈ℕ∗×ℕ∗, we have(13)DTnx0,Tmx0≤δ0∑mn+m−1kjj−1m−1+∑nn+m−1kjj−1n−1,(ii)
For everym,n∈ℕ∗2 such that m≤n, we have(14)DTnx0,Tmx0≤δ01−2k2km,where δ0=δD,T,x0.Proof.
(i)
The proof of this statement follows from the application of two-dimensional induction onp=n+m, for every p≥2
Since(15)DT1x0,T1x0≤kDTx0,x0+kDx0,Tx0≤δ02k,
it is clear that the inequality (13) holds for p=2 with m,n=1,1.
Assume next that inequality (13) holds for any m′,n′∈ℕ∗2 be chosen in such a way that n′+m′=p; let m,n∈ℕ∗2 with n+m=p+1.
SinceT is a G-Chatterjea mapping and Tn−1x0,Tm−1x0∈EG~, it holds that
(16)DTnx0,Tmx0≤kDTnx0,Tm−1x0+DTn−1x0,Tmx0.
Sincen+m−1=p and n−1+m=p, the inductive hypothesis yields
(17)DTnx0,Tmx0≤kδ0∑j=mn+m−2kjj−1m−1+∑j=n−1n+m−2kjj−1n−2+∑j=m−1n+m−2kjj−1m−2+∑j=nn+m−2kjj−1n−1≤kδ0∑j=mn+m−2kjj−1m−1+j−1m−2+km−1+∑j=nn+m−2kjj−1n−1+j−1n−2+kn−1≤kδ0∑j=mn+m−2kjjm−1+∑j=nn+m−2kjjn−1+km−1+kn−1≤kδ0∑j=m−1n+m−2kjjm−1+∑j=n−1n+m−2kjjn−1≤δ0∑j=mn+m−1kjj−1m−1+∑j=nn+m−1kjj−1n−1,that inequality (13) holds for n,m∈ℕ∗ such that n+m=p+1.
(ii)
Letn,m∈ℕ∗; assume that m≤n. Since j−1m−1≤2j−1 for any j∈m,n+m−1 and j−1n−1≤2j−1 for any j∈n,n+m−1, it follows that(18)∑j=mn+m−1kjj−1m−1≤12∑j=mn+m−12kj≤121−2k2km,and that
(19)∑j=nn+m−1kjj−1n−1≤121−2k2kn≤121−2k2km.
It follows from inequality (13) that
(20)DTnx0,Tmx0≤δ01−2k2km.Theorem 11.
LetX,D,G be a generalized, G-complete metric space and T:X⟶X be a G-Chatterjea mapping with constant k∈0,1/2. Suppose that there exists x0∈X such that δD,T,x0<∞, that x0,Tx0∈EG~, and that the subgraph GOTx0 is transitive. Under these assumptions, the sequence Tnx0 converges to some ω∈X. Moreover, if one of the following conditions i−iii holds, namely
(i)
T is subsequentially continuous(ii)
T is orbitally G-continuous(iii)
G satisfies property (JNRL) and Dx0,Tω<∞then ω is a fixed point of T.Proof.
Without loss of generality, it may be assumed thatx0,Tx0∈EG. Select m,n∈ℕ∗×ℕ∗ such that m≤n. From Lemma 9, it is clear that Tmx0,Tnx0∈EG. If T is a G-Chatterjea mapping, Lemma 10 yields
(21)DTnx0,Tmx0≤δ0/1−2k2km.
Thus,Tnx0 is a D-Cauchy sequence. Since X,D,G is G-complete, the sequence Tnx0D-converges to some ω∈X.
(i)
It follows from the subsequential continuity assumption onT that there exists a subsequence Tnqx0 such that Tnq+1x0D-converges to Tω as nq⟶∞. The uniqueness of the limit yields Tω=ω(ii)
Assume thatT is orbitally G-continuous. Since Tnx0D-converges to ω and Tnx0,Tn+1x0∈EG, it follows that TTnx0⟶Tω. Likewise, TTnx0=Tn+1x0⟶ω. Hence, ω=Tω(iii)
Assume thatG satisfies Property (JNRL) and that Dx0,Tω<∞. Since Tnx0 is G-increasing and it D-converges to ω∈X, it follows that Tnx0,ω∈EG, for any n∈ℕ
Selectn∈ℕ, n≥1. If T is a G-Chatterjea mapping, then necessarily
(22)DTnx0,Tω≤kDTn−1x0,Tω+kDTnx0,ω≤k2DTn−2x0,Tω+k2DTn−1x0,ω+kDTnx0,ω.
It follows by induction onn, that for any n≥1,
(23)DTnx0,Tω≤knDx0,Tω+∑j=1nkjDTn+1−jx0,ω.
Letj∈1,n. Since Tpx0p≥nD-converges to ω using D3, it follows that
(24)DTn+1−jx0,ω≤Climsupp⟶∞DTn+1−jx0,Tpx0.
Applying Lemma10, we obtain
(25)DTn+1−jx0,ω≤Cδ01−2klimsupp⟶∞2kn+1−j≤Cδ01−2k2kn+1−j.
Then,(26)kjDTn+1−jx0,ω≤Cδ01−2kkj2kn+1−j≤Cδ01−2k2kn+112j.
Hence(27)∑j=1nkjDTn+1−jx0,ω≤Cδ01−2k2kn+1∑j=1n12j≤Cδ01−2k2kn+1.
Finally, inequality (23) becomes
(28)DTnx0,Tω≤knDx0,Tω+Cδ01−2k2kn+1.
SinceDx0,Tω<∞, it follows that limn⟶∞DTnx0,Tω=0. Therefore, Tnx0D-converges to Tω. Uniqueness of the limit yields Tω=ω.Proposition 12.
Suppose thatT is G-Chatterjea. If T has two fixed points ω and ω′ in X, such that Dω,ω′<∞ and ω,ω′∈EG, then ω=ω′.Proof.
Suppose thatω,ω′∈X are two fixed points of T such that Dω,ω′<∞. Since T is a G-Chatterjea mapping, we have
(29)Dω,ω′=DTω,Tω′≤kDTω,ω′+Dω,Tω′,which implies that
(30)Dω,ω′≤2kDω,ω′.
Hence(31)1−2kDω,ω′≤0.
Therefore,Dω,ω′=0, i.e., ω=ω′.The following example illustrates Theorem11.Example 2.
LetX be the open interval −1,1. Consider the function D defined on X as follows:
(32)Dx,y=2∣x∣+∣y∣ifeitherofx=0ory=0,∣x∣+∣y∣3otherwise.It can be easily verified thatD1 and D2 hold. For the validity of D3, observe first that for all x≠0, we have CD,X,x=∅. If x=0, then there exists a sequence xn such that limn⟶∞Dxn,x=0. Consider the sets P≔n∈ℕ:xn≠0 and Q≔n∈ℕ:xn=0. We distinguish three cases:IfP is finite, then there exists C≥1 such that for any y∈X it holds that
(33)D0,y=2∣y∣≤2C∣y∣=Climsupn⟶∞Dxn,y.IfQ is finite, then there exists C≥6 such that for any y∈X(34)D0,y=2y≤C∣y∣3=Climsupn⟶∞Dxn,y.IfP and Q are infinite, there exist two increasing functions φ,ψ:ℕ⟶ℕ such that, for all n∈ℕ, xφn≠0, xψn=0, and xn=xφn∪xψn. Then, for any y∈X(35)limsupn⟶∞Dxφn,y=y3andlimsupn⟶∞Dxψn,y=2y.Thus,D is a generalized metric with C≥6. Note that X is not a D-compact space. Indeed, let xnn∈ℕ∗ be a sequence of X such that xn=1−1/n and suppose that there exists a subsequence xφn of xn wich D-converges to an element x in X. Since limn⟶∞Dxφn,x=0, we have
(36)limn⟶∞xφn+x=0.Thus,∣x∣=−1. Contradiction.Consider the graphG on X consisting of the transitive closure of the graph represented in Figure 1.Figure 1
All loops and isolated vertices are not represented.Note that(37)EG=Δ⋃−1n2n,0,−1m2m,−1n2n:n,m∈ℕ∗andn>m.Let us prove that the spaceX,D is G-complete. Let xn be a G-monotone, D-Cauchy sequence in X. We have two cases:Case 1.
If there existsn0∈ℕ such that xn=0, for any n≥n0. We have limn⟶∞Dxn,0=0. Therefore, the sequence xnD-converges to 0.Case 2.
If there existsn0∈ℕ and a nondecreasing sequence pn⊂ℕ∗ such that xn=−1/2pn, for any n≥n0. We have limn⟶∞Dxn,0=limn⟶∞1/31/2pn=0. Therefore, the sequence xn is D-convergent to 0.(Note that in the case where there exists a nonzero elementα in X such that xn=α for any n∈ℕ, the sequence xn is not D-Cauchy).Now, consider the self-mappingT on X defined by
(38)Tx=−x2ifx∈X∩ℚ,x5otherwise.One can easily see that:(39)T−1n2n,T0=−1n+12n+1,0∈EG,foranyn∈ℕ∗,T−1m2m,T−1n2n=−1m+12m+1,−1n+12n+1∈EG,foranyn,m∈ℕ∗withn>m.It is therefore apparent thatT is G-monotone. For x0=−1/2, we have that x0,Tx0∈EG, GOTx0 is transitive and
(40)δT,D,x0=supD−1i2i,−12:i∈ℕandi≥2=14.Pickx,y∈X such that x,y∈EG. If x=y∈X∩ℚ, then
(41)DTx,Tx=∣x∣3≤kx=kDTx,x+Dx,Tx.Observe that ifx=y∈X\ℚ, then
(42)DTx,Tx=2∣x∣15≤k12x15=kDTx,x+Dx,Tx.On the other hand, ifx,y=−1n/2n,0, then
(43)DT−1n2n,T0=12n≤k32n=kDT−1n2n,0+D−1n2n,T0.Finally, in the casex,y=−1m/2m,−1n/2n), then
(44)DT−1m2m,T−1n2n=2n+2m3×2n+m+1≤k2n+2m2n+m+1=kDT−1m2m,−1n2n+D−1m2m,T−1n2n.In all cases,DTx,Ty≤kDTx,y+Dy,Tx. Thus, T is a G-Chatterjea mapping with constant k∈1/3,1/2.The sequenceTnx0=−1n+1/2n+1 is G-increasing, D-convergent to 0, and furthermore, −1n+1/2n+1,0∈EG. Hence, G satisfies Property (JNRL). On account of Theorem 11, T has a fixed point, namely 0.Next, we present a version of Theorem11 in the setting of a partially ordered generalized metric space. Let X,D,≤ be a generalized metric space endowed with a partial order. We define the directed graph G≤ on X as follows: VG≤=X and EG≤=x,y∈X×X:x≤y. 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 X,D,≤ satisfies the Property (JNRL) if whenever xn is a decreasing (respectively increasing) sequence such that xn⟶x in X, then for all n∈ℕ, x≤xn (respectively xn≤x).Theorem 13.
LetX,D,≤ be a generalized D-complete metric space endowed with a partial order and T:X⟶X be a monotone Chatterjea mapping with constant k∈0,1/2. Suppose that there exists x0∈X such that δD,T,x0<∞ and that either x0≤Tx0orTx0≤x0. Then, the sequence Tnx0 converges to some ω∈X. Moreover, if any one of the conditions i−iii in Theorem 11 holds, then ω is a fixed point of T.Proof.
Since the subgraphG≤OTx0 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.Corollary 14.
LetX,D be a D-complete generalized metric space and T:X⟶X be a Chatterjea contraction with constant k∈0,1/2. Suppose that there exists x0∈X such that δD,T,x0<∞. Then, Tnx0 converges to some ω∈X. If Dx0,Tω<∞, then ω is a fixed point of T with Dω,ω=0. Moreover, if ω′∈X is another fixed point of T such that Dω,ω′<∞, then ω=ω′.Proof.
TakingG=G0, where G0 is the complete graph, i.e., VG0=X and EG0=X×X, the proof follows from Theorem 11 and Proposition 12.We next set to show that the fixed point result given in Theorem11 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.
LetX be a nonempty set and f:X⟶X a function. Then, there exists a subset E⊂X such that fE=fX. Moreover, f:E⟶X is one-to-one.LetT, 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 thatT is G-Chatterjea S-contraction if there exists k∈0,1/2 such that for every x,y∈VG, it holds that
(45)Sx,Sy∈EG~DTx,Ty≤kDTx,Sy+DSx,Ty.We recall thatx∗ 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 VG, then x∗,y∗∈EG~ and Dx∗,y∗<∞.Suppose thatTX⊆SX. If x0∈X is arbitrary, we can choose a point x1 in X such that Tx0=Sx1. Proceeding in this manner, assuming that xn in X is given, we can define xn+1∈X by the recurrence relation
(46)Txn=Sxn+1,n=0,1,2,.⋯ByCT,S, we denote the set of all elements x0 of X such that Sxn,Sxm∈EG~, for n,m=1,2,⋯. The following notation will be used in the sequel:
(47)δD,S,T,x0=supDSxp,Sx1:p≥2.Corollary 17.
LetX,D be a generalized metric space endowed with a reflexive digraph G. Assume that VG=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, SX is a D-complete subspace of X and that TX⊆SX.
(1)
Suppose that there existsx0∈CT,S such that δD,S,T,x0<∞. Then, the sequence Sxn defined by (46) D-converges to x∗=Sa, with a∈X. Moreover, if DTx0,Ta<∞, 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 XProof.
By Lemma15, there exists X0⊂X such that SX0=SX=Y; moreover, S:X0⟶X is one-to-one. Define the mapping F:Y⟶Y as
(48)FSx=Tx.
SinceS is one-to-one on X0, F is well defined.
Letu, v∈Y. There exist x, y∈X such that u=Sx and v=Sy. If u,v∈EG~, then Sx,Sy∈EG~. Since T is a G-Chatterjea S-contraction, there exists k∈0,1/2 such that
(49)DTx,Ty≤kDTx,Sy+DSx,Ty,
i.e.,DFSx,FSy≤kDFSx,Sy+DSx,FSy. Then
(50)DFu,Fv≤kDFu,v+Du,Fv.
Consequently,F is a G-Chatterjea mapping on Y.
Suppose that there existsx0∈CT,S such that δD,S,T,x0<∞. Setting y0=Sx1, it is clear that Fy0=FSx1=Tx1=Sx2. It follows easily by induction that Fpy0=Sxp+1, for any p∈ℕ. Moreover,
(51)δD,F,y0=supDFiy0,y0:i≥1=supDFp−1y0,y0:p≥2=supDSxp,Sx1:p≥2=δD,S,T,x0<∞.
FromSxn,Sxm∈EG~ for n,m=1,2,⋯, it follows that Fiy0,Fjy0∈EG~ for any i, j in ℕ. Hence, GOFy0 is transitive. Furthermore, Sx1,Sx2=y0,Fy0∈EG~.
By virtue of Theorem11, the sequence Sxn=Fn−1y0D-converges to x∗=Sa with a∈Y⊂X.
Moreover, we have(52)Dy0,Fx∗=DSx1,FSa=DTx0,Ta<∞,and since G satisfies property (JNRL), on account of Theorem 11, x∗ is a fixed point of F. Hence Ta=FSa=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 coincidencey∗∈SX, that b∈X, and that y∗=Sb=Tb=FSb=Fy∗. Since the digraph G has the property (P) for T and S, then x∗,y∗∈EG~ and Dx∗,y∗<∞. 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.


## 4. Application

In this section, we study the existence and uniqueness of solution for the following general nonlinear matrix equation in the set of alln×n Hermitian-positive definite matrices Pn:
(53)Xq−A∗FXsA=B,q>2,s∈0,1,X∈Pn,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:En⟶En is a self-adjoint operator such that En is a nonempty subset of Pn. This type of matrix equation arises in control theory, ladder networks, dynamic programming, stochastic filtering and statistics, etc.ForM,N∈Pn, we denote
(54)M≺N⇔N−Mispositivedefinite.We denote by. the spectral norm A=ρA∗A=A∗, where ρA∗A is the largest eigenvalue of A∗A. We recall that the Thompson metric is defined on Pn by:
(55)d:Pn×Pn⟶ℝ+,such that
(56)dA,B=maxlnWAB,lnWBA=lnA−1/2BA−1/2,where WA/B=infλ>0:A≤λB=λmaxA−1/2BA−1/2. It is easy to verify that Pn,d is a complete metric space (see [22]). In the sequel, we consider the space Pn endowed by the Thompson generalized metric D defined by
(57)DA,B=lnA−1/2BA−1/22,for any A,B∈Pn. In the following lemmas, we extend some properties of the Thompson metric given in [23] to the Thompson generalized metric space.Lemma 18.
LetD:Pn×Pn⟶ℝ+ be a Thompson generalized metric on the open convex cone Pn; then, for any A,B∈Pn and nonsingular matrix M, we have the following conditions:
(i)
DA,B=DA−1,B−1=DM∗AM,M∗BM, where A−1,B−1 are the inversion of matrices A and B, respectively(ii)
DAr,Br≤r2DA,B,r∈−1,1(iii)
DM∗ArM,M∗BrM≤r2DA,B,r∈−1,1Proof.
Letd be the Thompson metric defined by (56). Since DA,B=dA,B2, we get i, ii, and iii by the invariant under the matrix inversion, congruence transformations for nonsingular matrix M, and the nonpositive curvature property of the Thompson metric d.Lemma 19.
For anyA,B,C,D∈Pn,
(58)DA+C,B+D≤maxDA,B,DC,D.
Especially,DA+C,B+C≤DA,B.Proof.
Letd be the Thompson metric defined by (56). By using [23] (Lemma 2.1), we have dA+C,B+D≤maxdA,B,dC,D. Since DA,B=dA,B2, we deduce our result.
We endowPn by the graph G defined by:
(59)VG=PnandEG=Δ∪M,N∈Pn×Pn,M≺N.
We give a graphical version of [24] (Lemma 4.3) in Pn endowed with the graph G.Lemma 20.
For anyA,B∈Pn, if A,B∈EG, then Ar,Br∈EG for all r∈]0,1], and Br,Ar∈EG for all r∈[−1,0[.Proof.
LetA,B∈Pn such that A,B∈EG. If A=B then Ar,Br∈EG for all r∈]0,1]. If A≺B, then by using the Löwner-Heinz inequality [25, 26], we get Ar≺Br. Thus, Ar,Br∈EG for all r∈]0,1].Theorem 21.
LetX0∈Pn and En=X∈Pn:X0,X∈EG. If the operator F is nondecreasing and for all X,Y∈En such that X,Y∈EG, we have:
(60)lnFX−1/2FYFX−1/2≤lnX−1/2B+A∗FYsA1/qX−1/2and X0q,B∈EG; then, the matrix equation (53) has a unique solution.Proof.
LetT:En⟶En be a mapping defined by
(61)TX=B+A∗FXsA1/q,X∈En.
LetX∈En. Since X0,X∈EG and F is nondecreasing,
(62)B+A∗FX0sA,B+A∗FXsA∈EG.
AsX0q,B∈EG and A∗FX0sA∈Pn, then X0q,B+A∗FX0sA∈EG. By Lemma 20, we have
(63)X0,B+A∗FXsA1/q=X0,TX∈EG.
Thus,TX∈En and so T is well defined.
LetX,Y∈En such that X,Y∈EG, we have TYq−TXq=A∗FYs−FXsA, then TXq,TYq∈EG. Since 0≤1/q<1, by Lemma 20 we have TX,TY∈EG.
LetX,Y be two elements in En such that X,Y∈EG. By using Lemmas 18 and 19 we have
(64)DFX,FY≥1s2DFXs,FYs≥1s2DA∗FXsA,A∗FYsA≥1s2DB+A∗FXsA,B+A∗FYsA≥1s2DTXq,TYq≥sq2DTX,TY.
Thus,(65)DTX,TY≤sq2DFX,FY.
IflnFX−1/2FYFX−1/2≤lnX−1/2B+A∗FYsA1/qX−1/2, then
(66)DFX,FY≤DX,TY.
From (65) and (66), we get
(67)DTX,TY≤sq2DX,TY+DY,TX.
Thus, there existsk=s/q2∈0,1/2 such that T is a G-Chatterjea mapping on En.
SinceX0q,B∈EG, X0q,B+A∗FX0sA∈EG. Thus, X0,TX0∈EG.
Next, we show thatEn satisfies the Property (JNRL) for the generalized metric D. In fact, let Xkk be a nondecreasing sequence of En which converges to X∈En. If the set k∈ℕ:Xk=X0 is infinite, there exists a nondecreasing function ϕ:ℕ⟶ℕ such that Xϕk=X0, ∀k∈ℕ, then X=X0∈En. If not, Xk≠X0, for large integer k. Fix m∈ℕ arbitrary. For all k>m,
(68)Xm,Xk∈EG⟹Xk−Xm∈Pn⟹Xk∈Pn+Xm.
SincePn+Xm is closed, X∈Pn+Xm. Thus, Xm,X∈EG, for all m∈ℕ. Thus, according to Theorem 11, we can show that there exists X∗∈En such that TX∗=X∗ which is a solution of the matrix equation (53).
If equation (53) has another solution Y∗ such that X∗,Y∗∈EG~, then using Proposition 12, we have Y∗=X∗.


## 5. Conclusion

Summarizing the present work enhances the area in many directions(1)
Using Theorem13, we can improve the following results
(i)
Theorem 2 in [27](ii)
Theorem 2.12 in [18](iii)
Theorem 3.9 in [19](iv)
Theorem 8 in [13](2)
Establish or improve Chatterjea fixed point theorems in the setting of standard metric spaces, dislocated metric spaces,b-metric spaces, and modular spaces with the Fatou property, also in these spaces endowed with a partial order and more generally with a graph



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*Source: 8840347-2021-02-11.xml*