# On Generalized Rationalα−Geraghty Contraction Mappings in G−Metric Spaces

Authors: N. Priyobarta; Bulbul Khomdram; Yumnam Rohen; Naeem Saleem
Journal: Journal of Mathematics
 (2021)

---

## Abstract

In this paper, we discuss about various generalizations ofα−admissible mappings. Furthermore, we extend the concept of α−admissible to generalize rational α−Geraghty contraction in G−metric space. With this new contraction mapping, we establish some fixed-point theorems in G−metric space. The obtained result is verified with an example.


---

## Body

## 1. Introduction and Preliminaries

Samet et al. [1] make a remarkable contribution by introducing α−admissible and α−ψ−contractive mappings. They also showed that the most celebrated result, the Banach contraction principle and various other results, are consequences of their results. Geraghty [2] also made an improvement of the Banach contraction principle. Mustafa and Sims [3] introduced the concept of G−metric space and established the Banach contraction principle. In this paper, considering both the concepts of Samet et al. [1] and Geraghty [2], we introduce generalized rational α−Geraghty contraction in the framework of G−metric space and establish some theorems on fixed points.Mustafa and Sims [3] give the following definition.Definition 1.
(see [3]). Let U be a nonempty set, and G:U×U×U⟶ℝ+ satisfies the following:(i)
Gξ,θ,ϕ=0 if and only if ξ=θ=ϕ.(ii)
0<Gξ,ξ,θ for all ξ,θ∈U with ξ≠θ.(iii)
Gξ,ξ,θ≤Gξ,θ,ϕ for all ξ,θ,ϕ∈U with θ≠ϕ.(iv)
Gξ,θ,ϕ=Gξ,ϕ,θ=Gθ,ϕ,ξ=⋯ (symmetry in all three variables).(v)
Gξ,θ,ϕ≤Gξ,t,t+Gt,θ,ϕ, for all ξ,θ,ϕ,t∈U.
Here,G is known as generalized metric or G−metric. The pair U,G is known as G−metric space.
The following example shows the relation between metric space andG−metric space given by Mustafa and Sims [3].Example 1.
(see [3]).(i)
LetU,m be an ordinary metric space, then(1)Gξ,θ,ϕ=13mξ,θ+mθ,ϕ+mξ,ϕ,is aG−metric on U.(ii)
LetU,m be an ordinary metric space, then(2)Gξ,θ,ϕ=maxmξ,θ,mθ,ϕ,mξ,ϕ,is aG−metric on U.(iii)
LetG be a G−metric on U, then(3)mξ,θ=Gξ,θ,θ+Gξ,ξ,θ,    is a metric on U.Definition 2.
(see [3]). Consider a G−metric space U,G and a sequence ξn of points of U. ξn is said to be G−convergent to ξ∈U provided limn,m⟶+∞Gξn,ξm,ξ=0; that is, there exists K∈ℕ satisfying Gξn,ξm,ξ<ε and m,n≥K where ε>0. Here, ξ is known as the limit of the sequence ξn and is denoted as ξn⟶ξ or limn⟶+∞ξn=ξ.Proposition 1.
(see [3]). In a G−metric space U,G, we have the following equivalent statements:(i)
ξn is G convergent to ξ.(ii)
Gξn,ξn,ξ⟶0 when n⟶+∞.(iii)
Gξn,ξ,ξ⟶0 when n⟶+∞.(iv)
Gξn,ξm,ξ⟶0 when n,m⟶+∞.Definition 3.
(see [3]). In a G−metric space U,G, the sequence ξn is said to be G−Cauchy if for any ε>0, there exists K∈ℕ satisfying Gξn,ξm,ξl<ε and n,m,l≥K; that is, Gξn,ξm,ξl⟶0 when n,m,l⟶+∞.Proposition 2.
(see [3]). In a G−metric space U,G, we have the following equivalent statements:(i)
The sequenceξn is G−Cauchy.(ii)
For anyε>0, there exists K∈ℕ such that Gξn,ξn,ξm<ε for all n,m≥K.Definition 4.
(see [3]). A G−metric space U,G is said to be G−complete if every G−Cauchy sequence is G−convergent in U,G.Lemma 1.
(see [3]). In a G−metric space U,G, for ξ,θ,ϕ,t∈U, we have the following:(i)
IfGξ,θ,ϕ=0, then ξ=θ=ϕ.(ii)
Gξ,θ,ϕ≤Gξ,ξ,θ+Gξ,ξ,ϕ.(iii)
Gξ,θ,θ≤2Gθ,ξ,ξ.(iv)
Gξ,θ,ϕ≤Gξ,t,ϕ+Gt,θ,ϕ.(v)
Gξ,θ,ϕ≤2/3Gξ,θ,t+Gξ,t,ϕ+Gt,θ,ϕ.(vi)
Gξ,θ,ϕ≤Gξ,t,t+Gθ,t,t+Gϕ,t,t.Definition 5.
(see [3]). In a G−metric space U,G, a mapping R:U⟶U is known as G−continuous if Rξn is G−convergent to Rξ, where ξn is any G−convergent sequence converging to ξ.
Here firstly, we recall the definition ofα-admissible mappings and its generalizations in metric space and G-metric space.Definition 6.
(see [1]). Let R be a self-mapping on a metric space U,d, and let α:U×U⟶0,+∞ be a function. R is said to be an α-admissible if ξ,θ∈U, and αξ,θ≥1 makes αRξ,Rθ≥1.Example 2.
(see [1]). Consider U=0,+∞ and define R:U⟶U and α:U×U⟶0,+∞ by Rξ=5ξ for all ξ,θ∈U and(4)αξ,θ=eθ/ξ,ifξ≥θ,ξ≠0,0,ifξ<θ.
Then,R is α-admissible.Definition 7.
(see [4]). Let R,S:U⟶U and α:U×U⟶0,+∞. It is said that the pair R,S is α-admissible if ξ,θ∈U such that αξ,θ≥1, then we have αRξ,Sθ≥1 and αSξ,Rθ≥1.Definition 8.
(see [5]). Let R:U⟶U and α:U×U⟶−∞,+∞. It is said that R is a triangular α-admissible mapping if the following holds:(T1)αξ,θ≥1 makes αRξ,Rθ≥1,ξ,θ∈U.(T2)αξ,ϕ≥1, αϕ,θ≥1, makes αξ,θ≥1,ξ,θ,ϕ∈U.Definition 9.
(see [4]). Let R,S:U⟶U and α:U×U⟶0,+∞. It is said that a pair R,S is a triangular α-admissible mapping if the following holds:(T1)αξ,θ≥1 makes αRξ,Sθ≥1 and αSξ,Rθ≥1,ξ,θ∈U.(T2)αξ,ϕ≥1, αϕ,θ≥1, makes αξ,θ≥1,ξ,θ,ϕ∈U.Definition 10.
(see [6]). Let R be a self-mapping on a metric space U,m and let α,η:U×U⟶0,+∞ be two functions. It is said that R is α-admissible mapping with respect to η if ξ,θ∈U, and αξ,θ≥ηξ,θ implies αRξ,Rθ≥ηRξ,Rθ.
It can be noted that ifηξ,θ=1, then the above definition becomes Definition 6. If we take αξ,θ=1, then R is said to be an η-subadmissible mapping.Lemma 2.
(see [5]). Let R:U⟶U be a triangular α-admissible mapping. Let us take ξ0∈U such that αξ0,Rξ0≥1. Form a sequence ξn as ξn+1=Rξn. Then, αξn,ξm≥1, where m,n∈N∪0, n<m.Lemma 3.
(see [7]). Let R,S:U⟶U be triangular α-admissible mapping. Let us take ξ0∈U such that αξ0,Rξ0≥1. Form sequences ξ2i+1=Rξ2i and ξ2i+2=Sξ2i+1, where i=0,1,2,…. Then, αξn,ξm≥1, where m,n∈N∪0, n<m.Alghamdi and Karapinar [8] generalized the concept of α-admissible mappings in the context of G-metric space and called it β-admissible. The definition of β-admissible given by Alghamdi and Karapinar is defined as follows.Definition 11.
(see [8]). Let R:U⟶U and β:U×U×U⟶0,+∞, then R is said to be β-admissible if for all ξ,θ,ϕ∈U then(5)βξ,θ,ϕ≥1impliesβRξ,Rθ,Rϕ≥1.
Alghamdi and Karapinar [8] introduced G−β−ψ contractive mappings of type-I and type-II. They also introduced G−β−ψ contractive mappings of type-A. They also gave the relation between these different types of G−β−ψ contractions and equivalent Banach contractions.
Alghamdi and Karapinar [9] further generalized the results of Alghamdi and Karapinar [8] by introducing generalized G−β−ψ contractive mappings of type-I and type-II.
Kutbi et al. [10] defined rectangular G−α−admissible mapping. They also defined weak α−ψ−ϕ contractive mappings to establish some coincidence point theorems for coupled and tripled in Gb-metric space.Definition 12.
(see [10]). Let U,G be a G-metric space and let R,S:U⟶U and α:U3⟶0,+∞. R is said to be a rectangular G−α−admissible mapping with respect to S if the following holds:(i)
αSξ,Sθ,Sϕ≥1 implies αRξ,Rθ,Rϕ≥1, ξ,θ,ϕ∈U.(ii)
αSξ,Sθ,Sθ≥1 and αSθ,Sϕ,Sϕ≥1 imply αSξ,Sθ,Sϕ≥1, ξ,θ,ϕ∈U.
Hussain et al. [11] generalized the concept of rectangular G−α−admissible mappings used to obtain coupled and tripled fixed-point theorems.
Hussain et al. [12] established a generalized form of α−admissible mappings in order to prove coincidence points and common fixed points in the framework of G-metric spaces. Furthermore, several authors obtained different kinds of generalization of Banach contraction principle in different spaces (see for details [13–20]).Definition 13.
(see [12]). Let U be an arbitrary set, α:U×U×U⟶0,+∞, and R:U⟶U. The mapping R is called an α-dominating map on U if αξ,Rξ,Rξ≥1 or αξ,ξ,Rξ≥1 for each ξ in U.Definition 14.
(see [12]). In an arbitrary set U, let R,S:U⟶U be given mappings and α:U×U×U⟶0,+∞ be a function. The pair R,S is said to be partially weakly G−α−admissible if and only if αRξ,SRξ,SRξ≥1 for all ξ∈U.Definition 15.
(see [12]). In an arbitrary set U, let R,S:U⟶U be given mappings and α:U×U×U⟶0,+∞ be a function. The pair R,S is said to be partially weakly G−α−admissible with respect to T if and only if for all ξ∈U, αRξ,Sθ,Sθ≥1, where θ∈T−1Rξ.
In the above definition, ifR=S, R is said to be partially weakly G−α−admissible (or α−admissible of rank 3) with respect to T.
IfT=IU (the identity mapping on U), then the above definition becomes the definition of partially weakly G−α−admissible pair.
Ansari et al. [21] also studied α-admissible mappings in G-metric space by introducing G-η-subadmissible mapping and α-dominating map. They also introduced η-subdominating map, α-regular in the framework of G-metric space, and partially weakly G-α-admissible and partially weakly G-η-subadmissible mappings.Definition 16.
(see [21]). Let U,G be a G-metric space, and let R be a self-mapping on U and η:U×U×U⟶0,+∞ be a function. R is said to be a G−η−subadmissible (or η-subadmissible of rank 3) mapping if(6)ξ,θ,ϕ∈U,ηξ,θ,ϕ≤1,impliesηRξ,Rθ,Rϕ≤1.Definition 17.
(see [21]). Let U be an arbitrary set, η:U×U×U⟶0,+∞, and R:U⟶U. A mapping R is called an η-subdominating map on U if ηξ,Rξ,Rξ≤1 or αξ,ξ,Rξ≤1 for each ξ in U.Definition 18.
(see [21]). In a G-metric space U,G, let R,S:U⟶U be given mappings and η:U×U×U⟶0,+∞ be a function. The pair R,S is said to be partially weakly G−η−subadmissible (or η−subadmissible of rank 3) if and only if ηRξ,SRξ,SRξ≤1 for all ξ∈U.Definition 19.
(see [21]). In a G-metric space U,G, let R,S:U⟶U be given mappings and η:U×U×U⟶0,+∞ be a function. The pair R,S is said to be partially weakly G−η−subadmissible (or η−subadmissible of rank 3) with respect to T if and only if for all ξ∈U, αRξ,Sθ,Sθ≥1, where θ∈T−1Rξ.
Hussain et al. [22] defined G−α,ψ−Meir–Keeler contractive mapping and used it in proving fixed-point theorems in the framework of G-metric spaces.Definition 20.
(see [22]). Let U,G be a G-metric space and ψ∈Ψ. Let R:U⟶U be an α-admissible mapping satisfying the following: for each ε>0, there exists δ>0 such that ε≤ψGξ,θ,ϕ<ε+δ implies αξ,ξαθ,θαϕ,ϕψGRξ,Rθ,Rϕ<ε for all ξ,θ,ϕ∈U. Then, R is known as a G−α,ψ−Meir–Keeler contractive mapping.In the above definition,Ψ is the collection of nondecreasing functions ψ:0,+∞⟶0,+∞ continuous in t such that ψt=0 if and only if t=0 and ψt+s≤ψt+ψs.The concept ofα-admissible mappings is extended to S-metric space by Zhou et al. [23] and called it γ-admissible. They are defined as follows.Definition 21.
(see [23]). Let R:U⟶U and γ:U3⟶0,+∞, then R is said to be γ-admissible if for all ξ,θ,ϕ∈U:(7)γξ,θ,ϕ≥1impliesγRξ,Rθ,Rϕ≥1.
They also extendedγ-admissibility for two mappings. Furthermore, they also introduced concepts of various contractive mappings viz. type A, type B, type C, type D, and type E.
Bulbul et al. [24] also derived the concept of generalized S−β−ψ contractive-type mappings on the line of generalized G-β-γ contractive-type mappings. Nabil et al. [25] also defined the concept of α-admissible mappings in Sb-metric space.
From these, what we observe is thatβ-admissible was for the first time used by Samet et al. [1] to represent α-admissible while dealing with coupled fixed point-related problems. Phiangsungnoen et al. [26] also used the name β-admissible mapping in order to represent α-admissible for fuzzy mappings. On the contrary, β-admissible of Alghamdi and Karapinar [9] and γ-admissible of Zhou et al. [23] are all extended versions of α-admissible mappings in G-metric space and S-metric space, respectively. Thus, we can remark that α-admissible and its various forms can be extended to G-metric as well as S-metric spaces and further to Gb-metric and Sb-metric spaces. With this idea, we introduce various forms of α-admissible mappings in the context of G-metric space and present following definitions. For notation, we use αG for α-admissible mappings in G-metric space.Definition 22.
LetR:U⟶U and αG:U×U×U⟶0,+∞, then R is said to be αG-admissible if ξ,θ,ϕ∈U, αGξ,θ,ϕ≥1 implies αGRξ,Rθ,Rϕ≥1.Definition 23.
LetR,S:U⟶U and αG:U×U×U⟶0,+∞. We say that the pair R,S is αG−admissible if ξ,θ,ϕ∈U such that αGξ,θ,ϕ≥1, then we have αGRξ,Sθ,Sϕ≥1 and αGSξ,Rθ,Rϕ≥1.Definition 24.
LetR:U⟶U and αG:U×U×U⟶0,+∞. We say that R is triangular αG−admissible mapping if the following holds:(i)
αGξ,θ,ϕ≥1 implies αGRξ,Rθ,Rϕ≥1,ξ,θ,ϕ∈U.(ii)
αGξ,t,t≥1 and αGt,θ,ϕ≥1 implies αGξ,θ,ϕ≥1,ξ,θ,ϕ,t∈U.Definition 25.
LetR:U⟶U and let αG,ηG:U×U×U⟶0,+∞ be functions. We say that R is αG−admissible mapping with respect to ηG if ξ,θ,ϕ∈U,(8)αGξ,θ,ϕ≥ηGξ,θ,ϕimpliesαGRξ,Rθ,Rϕ≥ηGRξ,Rθ,Rϕ.
Note that if we takeηGξ,θ,ϕ=1, then this definition becomes Definition 22. Also, if we take αGξ,θ,ϕ=1, then it is said that R is an ηG−subadmissible mapping.Definition 26.
LetR,S:U⟶U and αG,ηG:U×U×U⟶0,+∞. We say that the pair R,S is αG−admissible mapping with respect to ηG if ξ,θ,ϕ∈U such that αGξ,θ,ϕ≥ηGξ,θ,ϕ, then we have αGRξ,Sθ,Sϕ≥ηGRξ,Sθ,Sϕ and αGSξ,Rθ,Rϕ≥ηGSξ,Rθ,Rϕ.Lemma 4.
LetR,S:U⟶U are triangular αG-admissible mappings. Suppose that there exists ξ0∈U such that αGξ0,Rξ0,Rξ0≥1. Define sequences(9)ξ2i+1=Rξ2i,ξ2i+2=Sξ2i+1,wherei=0,1,2,….
Then, we haveαGξn,ξm,ξm≥1, m,n∈ℕ∪0, n<m.


## 2. Main Results

Let us takeG as the collection of functions g:0,+∞⟶0,1 such that gtn⟶1 gives tn⟶0, where tn is a bounded sequence of positive real numbers.We start our results with the following definitions.Definition 27.
In aG−metric space U,G, let αG:U×U×U⟶0,+∞ be a function. We say that mappings R,S:U⟶U is a pair of generalized rational αG−Geraghty contraction mappings of type-I if for all ξ,θ,ϕ∈U and g∈G,(10)αGξ,θ,ϕGRξ,Sθ,Sϕ≤g∇1ξ,θ,ϕ∇1ξ,θ,ϕ,where(11)∇1ξ,θ,ϕ=maxGξ,θ,ϕ,GRξ,Sθ,Sϕ,Gξ,Rξ,RξGθ,Sθ,Sθ1+Gξ,θ,ϕ+GRξ,Sθ,Sϕ,Gθ,Sθ,SθGϕ,Sϕ,Sϕ1+Gξ,θ,ϕ+GRξ,Sθ,Sϕ,Gϕ,Sϕ,SϕGξ,Rξ,Rξ1+Gξ,θ,ϕ+GRξ,Sθ,Sϕ.Definition 28.
In aG-metric space U,G, let αG:U×U×U⟶0,+∞ be a function. We say that the mappings R,S:U⟶U are a pair of generalized rational αG−Geraghty contraction mappings of type-II if for all ξ,θ∈U and g∈G,(12)αGξ,θ,θGRξ,Sθ,Sθ≤g∇2ξ,θ,θ∇2ξ,θ,θ,where(13)∇2ξ,θ,θ=maxGξ,θ,θ,GRξ,Sθ,Sθ,Gξ,Rξ,RξGθ,Sθ,Sθ1+Gξ,θ,θ+GRξ,Sθ,Sθ,Gθ,Sθ,SθGθ,Sθ,Sθ1+Gξ,θ,θ+GRξ,Sθ,Sθ.
IfR=S, then we have the following.Definition 29.
In aG−metric space U,G, let αG:U×U×U⟶0,+∞ be a function. We say that mapping R:U⟶U is a generalized rational αG−Geraghty contraction mappings of type-I if there exists g∈G such that for all ξ,θ,ϕ∈U,(14)αGξ,θ,ϕGRξ,Rθ,Rϕ≤g∇1ξ,θ,ϕ∇1ξ,θ,ϕ,where(15)∇1ξ,θ,ϕ=maxGξ,θ,ϕ,GRξ,Rθ,Rϕ,Gξ,Rξ,RξGθ,Rθ,Rθ1+Gξ,θ,ϕ+GRξ,Rθ,Rϕ,Gθ,Rθ,RθGϕ,Rϕ,Rϕ1+Gξ,θ,ϕ+GRξ,Rθ,Rϕ,Gϕ,Rϕ,RϕGξ,Rξ,Rξ1+Gξ,θ,ϕ+GRξ,Rθ,Rϕ.Definition 30.
In aG-metric space U,G, let αG:U×U×U⟶0,+∞ be a function. We say that the mapping R:U⟶U is a generalized rational αG−Geraghty contraction mappings of type-II if there exists g∈G such that for all ξ,θ∈U,(16)αGξ,θ,θGRξ,Rθ,Rθ≤g∇2ξ,θ,θ∇2ξ,θ,θ,where(17)∇2ξ,θ,θ=maxGξ,θ,θ,GRξ,Rθ,Rθ,Gξ,Rξ,RξGθ,Rθ,Rθ1+Gξ,θ,θ+GRξ,Rθ,Rθ,Gθ,Rθ,RθGθ,Rθ,Rθ1+Gξ,θ,θ+GRξ,Rθ,Rθ.Theorem 1.
In a completeG-metric space U,G, let αG:U×U×U⟶0,+∞ be a function. Let R,S:U⟶U be two mappings satisfying the following:(i)
R,S is pair of generalized rational αG-Geraghty contraction mappings of type-I.(ii)
R,S is a pair of triangular αG-admissible mappings.(iii)
There existsξ0∈U such that αGξ0,Rξ0,Rξ0≥1.(iv)
R and S are continuous.
Then, a common fixed point exists for the pairR,S.Proof.
Letξ1∈U be such that ξ1=Rξ0 and ξ2=Sξ1. Inductively, we construct a sequence ξn in U as follows:(18)ξ2i+1=Rξ2i,ξ2i+2=Sξ2i+1,where i=0,1,2,3,….
By assumptionαGξ0,ξ1,ξ1≥1 and the pair R,S is αG-admissible, by Lemma 4, we have(19)αGξn,ξn+1,ξn+1≥1,for alln∈ℕ∪0.
Then,(20)Gξ2i+1,ξ2i+2,ξ2i+2=GRξ2i,Sξ2i+1,Sξ2i+1≤αGξ2i,ξ2i+1,ξ2i+1GRξ2i,Sξ2i+1,Sξ2i+1≤g∇1ξ2i,ξ2i+1,ξ2i+1∇1ξ2i,ξ2i+1,ξ2i+1,for all i∈ℕ∪0.
Now,(21)∇1ξ2i,ξ2i+1,ξ2i+1=maxGξ2i,ξ2i+1,ξ2i+1,GRξ2i,Sξ2i+1,Sξ2i+1,Gξ2i,Rξ2i,Rξ2iGξ2i+1,Sξ2i+1,Sξ2i+11+Gξ2i,ξ2i+1,ξ2i+1+GRξ2i,Sξ2i+1,Sξ2i+1,Gξ2i+1,Sξ2i+1,Sξ2i+1Gξ2i+1,Sξ2i+1,Sξ2i+11+Gξ2i,ξ2i+1,ξ2i+1+GRξ2i,Sξ2i+1,Sξ2i+1,Gξ2i+1,Sξ2i+1,Sξ2i+1Gξ2i,Rξ2i,Rξ2i1+Gξ2i,ξ2i+1,ξ2i+1+GRξ2i,Sξ2i+1,Sξ2i+1=maxGξ2i,ξ2i+1,ξ2i+1,Gξ2i+1,ξ2i+2,ξ2i+2,Gξ2i,ξ2i+1,ξ2i+1Gξ2i,ξ2i+1,ξ2i+11+Gξ2i,ξ2i+1,ξ2i+1+Gξ2i+1,ξ2i+2,ξ2i+2,Gξ2i+1,ξ2i+2,ξ2i+2Gξ2i+1,ξ2i+2,ξ2i+21+Gξ2i,ξ2i+1,ξ2i+1+Gξ2i+1,ξ2i+2,ξ2i+2,Gξ2i+1,ξ2i+2,ξ2i+2Gξ2i,ξ2i+1,ξ2i+11+Gξ2i,ξ2i+1,ξ2i+1+Gξ2i+1,ξ2i+2,ξ2i+2=maxGξ2i,ξ2i+1,ξ2i+1,Gξ2i+1,ξ2i+2,ξ2i+2.
IfmaxGξ2i,ξ2i+1,ξ2i+1,Gξ2i+1,ξ2i+2,ξ2i+2=Gξ2i+1,ξ2i+2,ξ2i+2, then(22)Gξ2i+1,ξ2i+2,ξ2i+2≤gGξ2i+1,ξ2i+2,ξ2i+2Gξ2i+1,ξ2i+2,ξ2i+2<Gξ2i+1,ξ2i+2,ξ2i+2,which is a contradiction. Hence,(23)Gξ2i+1,ξ2i+2,ξ2i+2<Gξ2i,ξ2i+1,ξ2i+1.
This implies that(24)Gξn+1,ξn+2,ξn+2<Gξn,ξn+1,ξn+1,for all n∈ℕ∪0.
So the sequenceGξn,ξn+1,ξn+1 is nonnegative and nonincreasing. Now we prove that Gξn,ξn+1,ξn+1⟶0. It is clear that Gξn,ξn+1,ξn+1 is a decreasing sequence. So, for some r>0, we have limn⟶+∞Gξn,ξn+1,ξn+1=r.
From (23),(25)Gξn+1,ξn+2,ξn+2Gξn,ξn+1,ξn+1≤gGξn,ξn+1,ξn+1≤1.
Now, by taking limit asn⟶+∞, we have(26)1≤limn⟶+∞gGξn,ξn+1,ξn+1<1,that is,(27)limn⟶+∞gGξn,ξn+1,ξn+1=1.
By the property ofg, we have(28)limn⟶+∞Gξn,ξn+1,ξn+1=0.
We have to show thatξn is a Cauchy sequence. If possible, let ξn is not a Cauchy sequence. Then, there exist ε>0 and sequences ξmk and ξnk such that, for all positive integers k, we get mk>nk>k:(29)Gξnk,ξmk,ξmk≥ε,Gξnk,ξmk−1,ξmk−1<ε.
Therefore,(30)ε≤Gξnk,ξmk,ξmk≤Gξnk,ξmk−1,ξmk−1+Gξmk−1,ξmk,ξmk<ε+Gξmk−1,ξmk,ξmk.
Takingk⟶+∞,(31)ε≤limk⟶+∞supGξnk,ξmk,ξmk<ε.
Therefore,(32)limk⟶+∞Gξnk,ξmk,ξmk=ε.
Also, from the triangular inequality, we have(33)Gξnk,ξmk,ξmk≤Gξnk,ξnk+1,ξnk+1+Gξnk+1,ξmk,ξmk,Gξnk+1,ξmk,ξmk≤Gξnk+1,ξnk,ξnk+Gξnk,ξmk,ξmk≤Gξnk+1,ξnk+1,ξnk+Gξnk+1,ξnk+1,ξnk+Gξnk,ξmk,ξmk.
Taking upper limit ask⟶+∞ above, we obtain(34)ε≤limk⟶+∞supGξnk,ξmk,ξmk≤limk⟶+∞supGξnk+1,ξmk,ξmk,limk⟶+∞supGξnk+1,ξmk,ξmk≤limk⟶+∞Gξnk,ξmk,ξmk≤ε.
Thus,(35)limk⟶+∞supGξnk+1,ξmk,ξmk=ε,limk⟶+∞supGξnk,ξmk+1,ξmk+1=ε.
By triangle inequality, we have(36)Gξnk+1,ξmk,ξmk≤Gξnk+1,ξmk+1,ξmk+1+Gξmk+1,ξmk,ξmk.
Taking limit ask⟶+∞, we have(37)ε≤limk⟶+∞supGξnk+1,ξmk+1,ξmk+1.
Following the above process, we have(38)limk⟶+∞supGξnk+1,ξmk+1,ξmk+1≤ε.
Combining, we have(39)limk⟶+∞Gξnk+1,ξmk+1,ξmk+1=ε.
By Lemma4, αGξnk,ξmk,ξmk≥1, we have(40)Gξnk+1,ξmk+1,ξmk+1=GRξnk,Sξmk,Sξmk≤αGξnk,ξmk,ξmkGRξnk,Sξmk,Sξmk≤g∇1ξnk,ξmk,ξmk∇1ξnk,ξmk,ξmk.
Finally, we conclude that(41)Gξnk+1,ξmk+1,ξmk+1∇1ξnk,ξmk,ξmk≤g∇1ξnk,ξmk,ξmk.
Applyingk⟶+∞, we obtain(42)limk⟶+∞gGξnk,ξmk,ξmk=1.
So,(43)limk⟶+∞Gξnk,ξmk,ξmk=0<ε,a contradiction. Thus, ξn is a Cauchy sequence. By completeness of U, there exists a∈U such that ξn⟶a implies that ξ2i+1⟶a and ξ2i+2⟶a. As R and S are continuous, we get Sξ2i+1⟶Sa and Rξ2i+2⟶Ra. Thus, a=Sa. Similarly, a=Ra, and we have Ra=Sa=a. Then, R,S have common fixed point.
In the next theorem, we dropped the continuity condition.Theorem 2.
In a completeG-metric space U,G, let αG:U×U×U⟶ℝ be a function. Let R,S:U⟶U be two mappings satisfying the following:(i)
R,S is a pair of generalized rational αG-Geraghty contraction mappings of type-I.(ii)
R,S is a pair of triangular αG-admissible mappings.(iii)
There existsξ0∈U such that αGξ0,Rξ0,Rξ0≥1.(iv)
Ifξn is a sequence in U such that αGξn,ξn+1,ξn+1≥1 for all n∈ℕ∪0 and ξn⟶a∈U as n⟶+∞, then a subsequence ξnk of ξn exists satisfying αGξnk,a,a≥1 for all k.
Then,R,S have a common fixed point.Proof.
It follows the similar lines of Theorem1. Define a sequence ξ2i+1=Rξ2i and ξ2i+2=Sξ2i+1, where i=0,1,2,… converges to a∈U. By the hypothesis of iv, a subsequence ξnk of ξn exists satisfying αGξ2nk,a,a≥1 for all k. Now, we have(44)Gξ2nk+1,Sa,Sa=GRξ2nk,Sa,Sa≤αGξ2nk,a,aGRξ2nk,Sa,Sa≤g∇1ξ2nk,a,a∇1ξ2nk,a,a,so that(45)Gξ2nk+1,Sa,Sa≤g∇1ξ2nk,a,a∇1ξ2nk,a,a.
On the contrary, we obtain(46)∇1ξ2nk,a,a=maxGξ2nk,a,a,GRξ2nk,Sa,Sa,Gξ2nk,Rξ2nk,Rξ2nkGa,Sa,Sa1+Gξ2nk,a,a+GRξ2nk,Sa,Sa,Ga,Sa,SaGa,Sa,Sa1+Gξ2nk,a,a+GRξ2nk,Sa,Sa,Ga,Sa,SaGξ2nk,Rξ2nk,Rξ2nk1+Gξ2nk,a,a+GRξ2nk,Sa,Sa=maxGξ2nk,a,a,Gξ2nk+1,Sa,Sa,Gξ2nk,ξ2nk+1,ξ2nk+1Ga,Sa,Sa1+Gξ2nk,a,a+Gξ2nk+1,Sa,Sa,Ga,Sa,SaGa,Sa,Sa1+Gξ2nk,a,a+Gξ2nk+1,Sa,Sa,Ga,Sa,SaGξ2nk,ξ2nk+1,ξ2nk+11+Gξ2nk,a,a+Gξ2nk+1,Sa,Sa.
Lettingk⟶+∞, then we have(47)limk⟶+∞∇1ξ2nk,a,a=Ga,Sa,Sa.
Suppose thatGa,Sa,Sa>0. From (47), for a large k, we have ∇1ξ2nk,a,a>0, which implies that(48)g∇1ξ2nk,a,a<1.
Then, we have(49)Gξ2nk+1,Sa,Sa<∇1ξ2nk,a,a.
Lettingk⟶+∞ in (49), we claim that(50)Ga,Sa,Sa<Ga,Sa,Sa,which is a contradiction. Thus, we find that Ga,Sa,Sa=0 implies a=Sa.
Alsoa=Ra showing that a in U is a common fixed point of R and S.


## 3. Consequences

If(51)∇1ξ,θ,ϕ=maxGξ,θ,ϕ,GRξ,Sθ,Sϕ,Gξ,Rξ,RξGθ,Sθ,Sθ1+Gξ,θ,ϕ+GRξ,Sθ,Sϕ,Gθ,Sθ,SθGϕ,Sϕ,Sϕ1+Gξ,θ,ϕ+GRξ,Sθ,Sϕ,Gϕ,Sϕ,SϕGξ,Rξ,Rξ1+Gξ,θ,ϕ+GRξ,Sθ,Sϕ,and R=S in Theorems 1 and 2, we have the following corollaries.Corollary 1.
In a completeG-metric space U,G, let R be αG-admissible mapping satisfying the following:(i)
R is generalized rational αG-Geraghty contraction mappings of type-I.(ii)
R is triangular αG-admissible.(iii)
There existsξ0∈U such that αGξ0,Rξ0,Rξ0≥1.(iv)
R is continuous.Then,R has a fixed point a∈U, and R is a Picard operator; that is, Rnξ0 converges to a.Corollary 2.
In a completeG-metric space U,G, let R be αG-admissible mapping satisfying the following:(i)
R is a generalized rational αG-Geraghty contraction mappings of type-I.(ii)
R is triangular αG-admissible.(iii)
There existsξ0∈U such that αGξ0,Rξ0,Rξ0≥1.(iv)
Ifξn is a sequence in U such that αξn,ξn+1,ξn+1≥1 for all n∈ℕ∪0 and ξn⟶a∈U as n⟶+∞, then a subsequence ξnk of ξn exists satisfying αξnk,a,a≥1 for all k.Then,R has a fixed point a∈U and R is a Picard operator; that is, Rnξ0 converges to a.If(52)∇1ξ,θ,ϕ=maxGξ,θ,ϕ,Gξ,Rξ,Rξ,Gθ,Sθ,Sθ,Gϕ,Sϕ,Sϕin Theorems 1 and 2, we can have another result.LetU,G be a G-metric space and let αG,ηG:U×U×U⟶0,+∞ be functions. Mappings R,S:U⟶U are called a pair of generalized rational αG-Geraghty contraction-type mappings with respect to ηG if there exists g∈G such that for all ξ,θ,ϕ∈U:(53)αGξ,θ,ϕ≥ηGξ,θ,ϕ⟹GRξ,Sθ,Sϕ≤g∇1ξ,θ,ϕ∇1ξ,θ,ϕ,where(54)∇1ξ,θ,ϕ=maxGξ,θ,ϕ,GRξ,Sθ,Sϕ,Gξ,Rξ,RξGθ,Sθ,Sθ1+Gξ,θ,ϕ+GRξ,Sθ,Sϕ,Gθ,Sθ,SθGϕ,Sϕ,Sϕ1+Gξ,θ,ϕ+GRξ,Sθ,Sϕ,Gϕ,Sϕ,SϕGξ,Rξ,Rξ1+Gξ,θ,ϕ+GRξ,Sθ,Sϕ.Theorem 3.
In a completeG-metric space U,G, let R be αG-admissible mapping with respect to ηG satisfying the following:(i)
R,S is a pair of a generalized rational αG-Geraghty contraction type mapping.(ii)
R,S is a pair of triangular αG-admissible mappings.(iii)
There existsξ0∈U such that αGξ0,Rξ0,Rξ0≥ηGξ0,Rξ0,Rξ0.(iv)
R and S are continuous.Then,R,S have common fixed point.Proof.
Letξ1∈U be such that ξ1=Rξ0 and ξ2=Sξ1. Inductively, we form a sequence ξn in U as follows:(55)ξ2i+1=Rξ2i,ξ2i+2=Sξ2i+1,where i=0,1,2,3,….
By assumptionαGξ0,ξ1,ξ1≥ηGξ0,ξ1,ξ1 and the pair R,S is αG-admissible with respect to ηG, we have αGRξ0,Sξ1,Sξ1≥ηGRξ0,Sξ1,Sξ1 from which we deduce that αGξ1,ξ2,ξ2≥ηGξ1,ξ2,ξ2 which also implies that αGSξ1,Rξ2,Rξ2≥ηGSξ1,Rξ2,Rξ2. Continuing in this way, we obtain αGξn,ξn+1,ξn+1≥ηGξn,ξn+1,ξn+1 for all n∈ℕ∪0:(56)Gξ2i+1,ξ2i+2,ξ2i+2=GRξ2i,Sξ2i+1,Sξ2i+1≤g∇1ξ2i,ξ2i+1,ξ2i+1∇1ξ2i,ξ2i+1,ξ2i+1.
Therefore,(57)Gξ2i+1,ξ2i+2,ξ2i+2≤αGξ2i,ξ2i+1,ξ2i+1GRξ2i,Sξ2i+1,Sξ2i+1,for all i∈ℕ∪0.
Now,(58)∇1ξ2i,ξ2i+1,ξ2i+1=maxGξ2i,ξ2i+1,ξ2i+1,GRξ2i,Sξ2i+1,Sξ2i+1,Gξ2i,Rξ2i,Rξ2iGξ2i+1,Sξ2i+1,Sξ2i+11+Gξ2i,ξ2i+1,ξ2i+1+GRξ2i,Sξ2i+1,Sξ2i+1,Gξ2i+1,Sξ2i+1,Sξ2i+1Gξ2i+1,Sξ2i+1,Sξ2i+11+Gξ2i,ξ2i+1,ξ2i+1+GRξ2i,Sξ2i+1,Sξ2i+1,Gξ2i+1,Sξ2i+1,Sξ2i+1Gξ2i,Rξ2i,Rξ2i1+Gξ2i,ξ2i+1,ξ2i+1+GRξ2i,Sξ2i+1,Sξ2i+1=maxGξ2i,ξ2i+1,ξ2i+1,Gξ2i+1,ξ2i+2,ξ2i+2,Gξ2i,ξ2i+1,ξ2i+1Gξ2i,ξ2i+1,ξ2i+11+Gξ2i,ξ2i+1,ξ2i+1+Gξ2i+1,ξ2i+2,ξ2i+2,Gξ2i,ξ2i+1,ξ2i+1Gξ2i+1,ξ2i+2,ξ2i+21+Gξ2i,ξ2i+1,ξ2i+1+Gξ2i+1,ξ2i+2,ξ2i+2,Gξ2i+1,ξ2i+2,ξ2i+2Gξ2i,ξ2i+1,ξ2i+11+Gξ2i,ξ2i+1,ξ2i+1+Gξ2i+1,ξ2i+2,ξ2i+2=maxGξ2i,ξ2i+1,ξ2i+1,Gξ2i+1,ξ2i+2,ξ2i+2.
From the definition ofg, the case ∇1ξ2i,ξ2i+1,ξ2i+1=Gξ2i+1,ξ2i+2,ξ2i+2 is impossible.(59)Gξ2i+1,ξ2i+2,ξ2i+2≤g∇1ξ2i,ξ2i+1,ξ2i+1∇1ξ2i,ξ2i+1,ξ2i+1≤gGξ2i+1,ξ2i+2,ξ2i+2Gξ2i+1,ξ2i+2,ξ2i+2<Gξ2i+1,ξ2i+2,ξ2i+2,which is a contradiction. Otherwise, in other case,(60)Gξ2i+1,ξ2i+2,ξ2i+2≤g∇1ξ2i,ξ2i+1,ξ2i+1∇1ξ2i,ξ2i+1,ξ2i+1≤gGξ2i,ξ2i+1,ξ2i+1Gξ2i,ξ2i+1,ξ2i+1<Gξ2i,ξ2i+1,ξ2i+1.
This implies that(61)Gξn+1,ξn+2,ξn+2<Gξn,ξn+1,ξn+1,for all n∈ℕ∪0.
Following the similar lines of the Theorem1, we can prove that R and S have a common fixed point.Theorem 4.
In a completeG-metric space U,G, let R,S be a pair of αG-admissible mappings with respect to ηG satisfying the following:(i)
The pairR,S is a generalized rational αG-Geraghty contraction type mappings.(ii)
The pairR,S is triangular αG-admissible.(iii)
There existsξ0∈U such that αGξ0,Rξ0,Rξ0≥ηGξ0,Rξ0,Rξ0.(iv)
Ifξn is a sequence in U such that αGξn,ξn+1,ξn+1≥ηGξn,ξn+1,ξn+1 for all n∈ℕ∪0 and ξn⟶a∈U as n⟶+∞, then a subsequence ξnk of ξn exists satisfying αGξnk,a,a≥ηGξnk,a,a for all k.Then,R and S have common fixed point.Proof.
It follows the similar line of Theorem2.
If(62)∇1ξ,θ,ϕ=maxGξ,θ,ϕ,GRξ,Rθ,Rϕ,Gξ,Rξ,RξGθ,Rθ,Rθ1+Gξ,θ,ϕ+GRξ,Rθ,Rϕ,Gθ,Rθ,RθGϕ,Rϕ,Rϕ1+Gξ,θ,ϕ+GRξ,Rθ,Rϕ,Gϕ,Rϕ,RϕGξ,Rξ,Rξ1+Gξ,θ,ϕ+GRξ,Rθ,Rϕ,and R=S in Theorems 3 and 4, we get the following corollaries.Corollary 3.
In a completeG-metric space U,G, let R be αG-admissible mappings with respect to ηG satisfying the following:(i)
R is a generalized rational αG-Geraghty contraction type mapping.(ii)
R is triangular αG-admissible.(iii)
There existsξ0∈U such that αGξ0,Rξ0,Rξ0≥ηGξ0,Rξ0,Rξ0.(iv)
R is continuous.Then,R has a fixed point a∈U and R be a Picard operator; that is, Rnξ0 converges to a.Corollary 4.
In a completeG-metric space U,G, let R be αG-admissible mapping with respect to ηG satisfying the following:(i)
R is a generalized rational αG-Geraghty contraction type mapping.(ii)
R is triangular αG-admissible.(iii)
There existsξ0∈U such that αGξ0,Rξ0,Rξ0≥ηGξ0,Rξ0,Rξ0.(iv)
There existsξ0∈U such that αGξn,Rξn+1,Rξn+1≥ηGξn,Rξn+1,Rξn+1 for all n∈ℕ∪0 and ξn⟶a∈U as n⟶+∞, then a subsequence ξnk of ξn exists satisfying αGξnk,a,a≥ηGξnk,a,a for all k.Then,R has a fixed point a∈U, and R is a Picard operator; that is, Rnξ0 converges to a.Example 3.
LetU=1,2,3 with G-metric, then G1,3,3=G3,1,1=5/7, G1,1,1=G2,2,2=G3,3,3=0, G1,2,2=G2,1,1=1, G2,3,3=G3,2,2=4/7, and(63)αGξ,θ,ϕ=1,ifξ,θ,ϕ∈U,0,otherwise.
Define the mappingsR,S:U⟶U as follows Rξ=1 for each ξ∈U, S1=S3=1, S2=3, and g:0,+∞⟶0,1, then(64)αGξ,θ,ϕGSξ,Sθ,Sϕ≤g∇1ξ,θ,ϕ∇1ξ,θ,ϕ.
Letξ=2, θ=3, ϕ=3, then condition (i) is not satisfied by the mapping S as GS2,S3,S3=G3,1,1=5/7, where(65)∇1ξ,θ,ϕ=maxG2,3,3,GS2,S3,S3,G2,S2,S2G3,S3,S31+G2,3,3+GS2,S3,S3,G3,S3,S3G3,S3,S31+G2,3,3+GS2,S3,S3=maxG2,3,3,G3,1,1,G2,3,3G3,1,11+G2,3,3+G3,1,1,G3,1,1G3,1,11+G2,3,3+G3,1,1=max47,57,528,25112=57.
Thus,αGξ,θ,ϕGSξ,Sθ,Sϕ≤g∇1ξ,θ,ϕ∇1ξ,θ,ϕ is not true.
We prove that Theorem1 can be applied to R and S. Let ξ,θ,ϕ∈U; clearly, R,S is αG-admissible such that αGξ,θ,ϕ≥1. Let ξ,θ,ϕ∈U so that Rξ,Sθ,Sϕ∈U and αGRξ,Sθ,Sϕ=1. Hence, R,S is αG-admissible. We know that condition (i) of Theorem 1 is satisfied.
Ifξ,θ,ϕ∈U, then αGξ,θ,ϕ=1, and we have(66)αGξ,θ,ϕGRξ,Sθ,Sϕ≤g∇1ξ,θ,ϕ∇1ξ,θ,ϕ,where(67)∇1ξ,θ,ϕ=maxG2,3,3,GR2,S3,S3,G2,R2,R2G3,S3,S31+G2,3,3+GR2,S3,S3,G3,S3,S3G3,S3,S31+G2,3,3+GR2,S3,S3=maxG2,3,3,G1,1,1,G2,1,1G3,1,11+G2,3,3+G1,1,1,G3,1,1G3,1,11+G2,3,3+G1,1,1=max47,0,511,2577=47,and GR2,S3,S3=G1,1,1=0.(68)αGξ,θ,ϕGRξ,Sθ,Sϕ≤g∇1ξ,θ,ϕ∇1ξ,θ,ϕ.
Hence, the conditions of Theorem1 are satisfied. So, R and S have a common fixed point.



---

*Source: 6661045-2021-03-09.xml*