# On Geraghty Contractive Mappings and an Application

Authors: S. Batul; F. Mehmood; D-e-Shehwar Sagheer; M. Sohail Ashraf; H. Aydi; Y. U. Gaba
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this article, certain fixed point results for Geraghty-type contractions in the setting ofGr-Complete extended fuzzy b-metric spaces are established. An example is elaborated, and a supporting application is also established. Many known existing results are special cases of our obtained results.


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## Body

## 1. Introduction

Kramosoil and Michalek [1] defined the notion of a fuzzy metric space (FMS) by using the concept of fuzzy sets introduced by Zadeh [2]. Grabiec [3] gave the concept of weak Cauchy sequences, which is called a G-Cauchy sequence and proved the Banach contraction principle (BCP) [4] in the setting of a FMS. George and Veermani [5] modified the definition of a FMS given by Kramosil and Michalek [1] and established some fixed point results. For more works in FMSs, see [6–10].In 1989, Bakhtin [11] introduced the notion of a b-metric space (BMS). Later on, the concept of a BMS was further used by Czerwick [12] to establish different fixed point results on this platform. The study of b-metric space endows an imperative place in fixed point theory with multiple aspects. Many mathematicians (Abdeljawad et al. [13, 14], Akkouchi [15], Chifu and Karapinar [16], Kadelburg and Radenović [17], Chauhan and Gupta [18], Kamran et al. [19] and Gupta [20], etc) led the foundation to improve fixed point theory in BMSs. Another innovative task has been achieved by Kamran et al. [21] in 2017 by introducing the notion of an extended b-metric space (EBMS), which generalizes the notion of a BMS. Some fixed point results are proved in this new setting. See for instance, the works shown in [22, 23].By considering an auxiliary function, Geraghty [24] established a generalisation of the Banach contraction principle in the complete metric spaces. Later on, Gupta et al. [25] proved the fixed point theorems for Ψ,β-Geraghty contraction type maps in ordered metric spaces. For more results using Geraghty contraction type maps in metric spaces can be seen in [26–30].Nǎdǎban [31] generalized the notion of b-metric space (BMS) by introducing the concept of fuzzy b-metric space. The idea of an extended fuzzy b-metric space (EFBMS) was introduced by Mahmood et al. in [32]. In the present article, some fixed point results for Geraghty-type contractions in Gr-complete EFBMS are established. Our results are generalizations of many existing results on FMS. See, for example, [32–35]. At the end, by applying our results, we give a real application.


## 2. Preliminaries

Recently, the concept of an EFBMS in [32] has been introduced as follows:Definition 1 (see [32]).
LetΔ be a non empty set. Given θ:Δ×Δ⟶1,∞ and let ∗ be a continuous t-norm. A fuzzy set τθ in Δ×Δ×0,∞ is called an extended fuzzy b-metric on Δ if for all ϖ1,ϖ2,ϖ3∈Δ, the following conditions hold:FbMθ1:τθϖ1,ϖ2,0=0;FbMθ2:τθϖ1,ϖ2,γ=1,∀γ>0 if and only if ϖ1=ϖ2;FbMθ3:τθϖ1,ϖ2,γ=τθϖ2,ϖ1,γ;FbMθ4:τθϖ1,ϖ3,θϖ1,ϖ3γ+α≥τθϖ1,ϖ2,γ∗τθϖ2,ϖ3,α∀γ,α≥0;FbMθ5:τθϖ1,ϖ2,.:0,∞⟶0,1 is left continuous, and limγ⟶∞τθϖ1,ϖ2,γ=1.
Here,Δ,τθ,∗,θϖ1,ϖ2 is called an EFBMS.Remark 1.
Takingθϖ1,ϖ3=b>1, the notion of a FBMS defined in [31] is obtained and by taking θϖ1,ϖ3=1, the notion of a FMS defined in [1] is obtained.Example 1 (see [32]).
LetΔ=1,2,3 and define db:Δ×Δ⟶ℝ by dbϖ1,ϖ2=ϖ1−ϖ22. Clearly, Δ,db is a BMS. Define the mapping θ:Δ×Δ⟶1,∞ by(1)θϖ1,ϖ2=1+ϖ1+ϖ2,Let τθ:Δ×Δ×0,∞⟶0,1 be defined by(2)τθϖ1,ϖ2,γ=γγ+dbϖ1,ϖ2,ifγ>0,0,ifγ=0,and take the continuous t-norm ∗=∧, that is, t1∗t2=t1∧t2=mint1,t2. Then Δ,τθ,∧,θϖ1,ϖ2 is an EFBMS.
The notions of convergence, Cauchyness and completeness in an EFBMS can be generalized naturally as follows:Definition 2 (see [32]).
LetΔ,τθ,∗ be an EFBMS.(i)
A sequenceϖl in Δ is said to be convergent if there exits ϖ1∈Δ such that(3)liml⟶∞Fθϖl,ϖ1,γ=1,∀γ>0.Grabiec [3] gave the concept of weak Cauchyness which we will denote by Gr-Cauchyness as follows:(ii)
A sequenceϖl in Δ is said to be a Gr-Cauchy sequence if liml⟶∞Fθϖl,ϖm+q,γ=1 for all γ>0 and q>0(iii)
An EFBMS in which everyGr-Cauchy sequence is convergent is called a Gr-complete EFBMS.Lemma 1 (see [34]).
LetΔ,Fb,∗ be a complete FBMS and Fϖ1,ϖ2,kt≥Fϖ1,ϖ2,γ for all ϖ1,ϖ2∈Δ,k∈0,1 and γ>0, then ϖ1=ϖ2.
LetΔ,τb,∗ be a Gr complete FBMS. Throughout this article, let(4)Fβ=β:0,∞⟶0,1b;limsupn⟶∞βtn=1bimplieslimn⟶∞tn=0,where b≥1.


## 3. Main Results

The BCP in the setting ofGr-complete EFBMSs, is established as follows:Theorem 1.
LetΔ,τθ,∗ be a Gr-complete EFBMS with θϖ1,ϖ2≥1. Let T:Δ⟶Δ be a mapping satisfying(5)τθTϖ1,Tϖ2,βτθϖ1,ϖ2,γγ≥τθϖ1,ϖ2,γ,for all ϖ1,ϖ2∈Δ, where β∈Fβ and βτθϖ1,ϖ2,γθϖ1,ϖ2<1. Then T has a unique fixed point.Proof.
Letϖ0∈Δ. Generate a sequence ϖl by the iterative process ϖl=Tlϖ0l∈ℕ. For all l,γ>0, by (5), we have(6)τθϖl,ϖl+1,γ=τθTϖl−1,Tϖl,γ≥τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ≥τθϖl−2,ϖl−1,γβτθϖl−1,ϖl,γ⋅βτθϖl−2,ϖl−1,γ≥τθϖl−3,ϖl−2,γβτθϖl−1,ϖl,γ⋅βτθϖl−2,ϖl−1,γ⋅βτθϖl−3,ϖl−2,γ≥…≥τθϖ0,ϖ1,γβτθϖl−1,ϖl,γ⋅βτθϖl−2,ϖl−1,γ…βτθϖ0,ϖ1,γ,So, we have(7)τθϖl,ϖl+1,γ≥τθϖ0,ϖ1,γβτθϖl−1,ϖl,γ⋅βτθϖl−2,ϖl−1,γ…βτθϖ0,ϖ1,γ.For any q∈ℕ, taking γ=γ/q+γ/q+…+γ/q and using FbMθ4 repeatedly, one can write(8)τθϖl,ϖm+q,γ≥τθϖl,ϖl+1,γqθϖl,ϖm+q∗τθϖl+1,ϖl+2,γqθϖl,ϖm+qθϖl+1,ϖm+q∗τθϖl+2,ϖl+3,γqθϖl,ϖm+qθϖl+1,ϖm+qθϖl+2,ϖm+q∗⋯∗τθϖl+q−1,ϖm+q,γqθϖl,ϖm+qθϖl+1,ϖm+qθϖl+2,ϖm+q…θϖl+q−1,ϖm+q,Using (7) and FbMθ5 , we get(9)τθϖl,ϖm+q,γ≥τθϖ0,ϖ1,γqθϖl,ϖm+qβτθϖl−1,ϖl,γ⋅βτθϖl−2,ϖl−1,γ…βτθϖ0,ϖ1,γ∗τθϖ0,ϖ1,γqθϖl,ϖm+qθϖl+1,ϖm+qβτθϖl,ϖl+1,γ⋅βτθϖl−1,ϖl,γ…βτθϖ0,ϖ1,γ∗⋯∗τθϖ0,ϖ1,γqθϖl,ϖm+q…θϖl+q−1,ϖm+qβτθϖm+q,ϖl+q−1,γ…βτθϖ0,ϖ1,γ,Then(10)τθϖl,ϖm+q,γ≥τθϖ0,ϖ1,bl−1γqθϖl,ϖm+qβτθϖl−1,ϖl,γ∗τθϖ0,ϖ1,bl−1γqθϖl,ϖm+qθϖl+1,ϖm+qβτθϖl,ϖl+1,γ⋅βτθϖl−1,ϖl,γ∗⋯∗τθϖ0,ϖ1,bl−1γqθϖl,ϖm+q…θϖl+q−1,ϖm+qβτθϖm+q,ϖl+q−1,γ…βτθϖl−1,ϖl,γ,Since for all l,q∈ℕ, we have θϖl,ϖm+qβτθϖl−1,ϖl,γ<1, taking limit as l⟶∞, we get(11)liml⟶∞τθϖl,ϖm+q,γ=1∗1∗⋯∗1=1,Hence, ϖl is a Gr-Cauchy sequence. Since Δ,τθ,∗ is a Gr-complete EFBMS, there exists p1∈Δ such that(12)liml⟶∞ϖl=p1.
We want to show thatp1 is a fixed point of T. Consider(13)τθTp1,p1,γ≥τθTp1,Tϖl,γ2θTp1,p1∗τθTϖl,p1,γ2θTp1,p1≥τθp1,ϖl,γ2βp1,ϖl,tθTp1,p1∗τθϖl+1,ϖl,γ2θϖl+1,ϖl⟶1∗1=1.
That is,Tp1=p1 is a fixed point.


### 3.1. Uniqueness

AssumeTp2=p2 for some p2∈Δ. Then(14)τθp2,p1,γ=τθTp2,Tp1,γ≥τθp2,p1,γβτθp2,p1,γ=τθTp2,Tp1,γβτθp2,p1,γ≥τθp2,p1,γβτθp2,p1,γ2≥…≥τθp2,p1,γβτθp2,p1,γn=τθp2,p1,bnγ⟶1asl⟶∞.Hence, the fixed point is unique.Example 2.
LetΔ=0,1,2 and τθϖ1,ϖ2,γ=γ/γ+ϖ1−ϖ22
It is easy to verify thatΔ,τθ,∗ is a Gr-complete EFBMS. Define a mapping T:Δ⟶Δ such that Tϖ1=βτθϖ1,ϖ2,γϖ1/1+ϖ1. Now, for all γ>0 , we have(15)τθTϖ1,Tϖ2,βτθϖ1,ϖ2,γγ=βτθϖ1,ϖ2,γγβτθϖ1,ϖ2,γγ+βτθϖ1,ϖ2,γϖ1/1+ϖ1−βτθϖ1,ϖ2,γϖ2/1+ϖ22=βτθϖ1,ϖ2,γγβτθϖ1,ϖ2,γγ+βτθϖ1,ϖ2,γϖ1/1+ϖ1−ϖ2/1+ϖ22=γγ+ϖ1−ϖ22/1+ϖ121+ϖ12,Recall that(16)ϖ1−ϖ221+ϖ121+ϖ12≤ϖ1−ϖ22γ+ϖ1−ϖ221+ϖ121+ϖ12≤γ+ϖ1−ϖ221γ+ϖ1−ϖ22/1+ϖ121+ϖ12≥1γ+ϖ1−ϖ22γγ+ϖ1−ϖ22/1+ϖ121+ϖ12≥γγ+ϖ1−ϖ22.
This implies that(17)τθTϖ1,Tϖ2,βτθϖ1,ϖ2,γγ≥τθϖ1,ϖ2,γ,Also, ϖ1=0 is the unique fixed point of T.Remark 2.
Takingθϖ1,ϖ2=b, we get Theorem 3.1 of [36].Remark 3.
Takingβτθϖ1,ϖ2,γ=k, the main result of [32] is obtained and for θϖ1,ϖ2=1 and for βτθϖ1,ϖ2,γ=k, the main result of [3] is obtained.Theorem 2.
Consider aGr-complete EFBMS Δ,τθ,∗ with θϖ1,ϖ2≥1. Let T:Δ⟶Δ be a mapping satisfying the condition(18)τθTϖ1,Tϖ2,βτθϖ1,ϖ2,γγ≥minτθTϖ1,Tϖ2,γ,τθϖ1,Tϖ1,γ,τθϖ2,Tϖ2,γ,τθϖ1,ϖ2,γ,for all ϖ1,ϖ2∈Δ, where β∈Fβ and θϖ1,ϖ2βτθϖ1,ϖ2,γ<1. Then T has a unique fixed point.Proof.
Starting by the same way as in Theorem1, we have(19)τθϖl,ϖl+1,γ=τθTϖl−1,Tϖl,γ≥minτθTϖl−1,Tϖl,γβτθϖl−1,ϖl,γ,τθϖl−1,Tϖl−1,γβτθϖl−1,ϖl,γτθϖl,Tϖl,γβτθϖl−1,ϖl,γ,τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ≥minτθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ,τθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ,Hence(20)τθϖl,ϖl+1,γ≥minτθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ,If(21)minτθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ=τθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,then (20) implies(22)τθϖl,ϖl+1,γ≥τθϖl,ϖl+1,γβτθϖl−1,ϖl,γ.
By Lemma1, T has a fixed point.
If(23)minτθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ=τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ,then from (20), we have(24)τθϖl,ϖl+1,γ≥τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ≥…≥τθϖ0,ϖ1,γβτθϖl−1,ϖl,γ⋅βτθϖl−2,ϖl−1,γ…βτθϖ0,ϖ1,γ.
The rest of the proof follows from Theorem1.Remark 4.
Takingθϖ1,ϖ2=b we get Theorem 3.6 of [36].
The following result is an extension of the main result of Gupta at el. [34].Theorem 3.
Consider aGr-complete EFBMS Δ,τθ,∗ with θϖ1,ϖ2≥1. Let T:Δ⟶Δ be a mapping satisfying(25)τθTϖ1,Tϖ2,βτθϖ1,ϖ2,γγ≥minτθϖ2,Tϖ2,γ1+τθϖ1,Tϖ1,γ1+τθϖ1,ϖ2,γ,τθϖ1,ϖ2,γ,for all ϖ1,ϖ2∈Δ, where β∈Fβ and βτθϖ1,ϖ2,γθϖ1,ϖ2<1, then T has a unique fixed point.Proof.
Consider(26)τθϖl,ϖl+1,γ=τθTϖl−1,Tϖl,γ≥minτθϖl,Tϖl,γ/βτθϖl−1,ϖl,γ1+τθϖl−1,Tϖl−1,γ/βτθϖl−1,ϖl,γ1+τθϖl−1,ϖl,γ/βτθϖl−1,ϖl,γτθϖl−1,ϖl,γβτθϖl−1,ϖl,γ≥minτθϖl,ϖl+1,γ/βτθϖl−1,ϖl,γ1+τθϖl−1,ϖl,γ/βτθϖl−1,ϖl,γ1+τθϖl−1,ϖl,γ/βτθϖl−1,ϖl,γ,τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ≥minτθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ.If(27)minτθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ=τθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,then (26) implies(28)τθϖl,ϖl+1,γ≥τθϖl,ϖl+1,γβτθϖl−1,ϖl,γ.
Then the result is trivial by Lemma1.If(29)minτθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ=τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ,then from (26) we have(30)τθϖl,ϖl+1,γ≥τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ.
Continuing in this way, one writes(31)τθϖl,ϖl+1,γ≥τθϖ0,ϖ1,γβτθϖl−1,ϖl,γ⋅βτθϖl−2,ϖl−1γ…βτθϖ0,ϖ1,γ.
Therefore, proceeding as in Theorem1 after inequality (7), the desired result is obtained.Theorem 4.
LetX,τθ,∗ be a Gr-complete EFBMS with θϖ1,ϖ2≥1. Let T:Δ⟶Δ be a mapping satisfying(32)τθTϖ1,Tϖ2,βτθϖ1,ϖ2,γγ≥minτθϖ2,Tϖ2,γ1+τθϖ1,Tϖ1,γ+τθϖ2,Tϖ1,γ2+τθϖ1,ϖ2,γ,τθϖ1,ϖ2,γ,for all ϖ1,ϖ2∈Δ, where β∈Fβ and βτθϖ1,ϖ2,γθϖ1,ϖ2<1, then T has a unique fixed point.Proof.
Forϖ0∈Δ, we choose a sequence ϖl in Δ and start by ϖl+1=Tϖl. For all l,γ>0, we have(33)τθϖl,ϖl+1,γ=τθTϖl−1,Tϖl,γ≥minτθϖl,Tϖl,γ/βτθϖl−1,ϖl,γ1+τθϖl−1,Tϖl−1,γ/βτθϖl−1,ϖl,γ+τθϖl,Tϖl−1,γ/βτθϖl−1,ϖl,γ2+τθϖl−1,ϖl,γ/βτθϖl−1,ϖl,γ,τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ=minτθϖl,ϖl+1,γ/βτθϖl−1,ϖl,γ1+τθϖl−1,ϖl,γ/βτθϖl−1,ϖl,γ+τθϖl,ϖl,γ/βτθϖl−1,ϖl,γ2+τθϖl−1,ϖl,γ/βτθϖl−1,ϖl,γ,τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ=minτθϖl,ϖl+1,γ/βτθϖl−1,ϖl,γ1+τθϖl−1,ϖl,γ/βτθϖl−1,ϖl,γ+12+τθϖl−1,ϖl,γ/βτθϖl−1,ϖl,γ,τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ=minτθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ,So, we have(34)τθϖl,ϖl+1,γ≥minτθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ,If(35)minτθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ=τθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,then from (34)(36)τθϖl,ϖl+1,γ≥τθϖl,ϖl+1,γβτθϖl−1,ϖl,γ.
Then result is trivial by Lemma1. If(37)minτθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ=τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ,then from (34)(38)τθϖl,ϖl+1,t≥τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ,Continuing in this direction, we get(39)τθϖl,ϖl+1,γ≥τθϖ0,ϖ1,γβτθϖl−1,ϖl,γ⋅βτθϖl−2,ϖl−1,γ…βτθϖ0,ϖ1,γ.
The desired result is then established by the same procedure as in Theorem1. The following result is the extension of the main result of Roshan et al. [35].Theorem 5.
Consider aGr-complete EFBMS X,τθ,∗ with θϖ1,ϖ2≥1. Let T:Δ⟶Δ be a mapping satisfying(40)τθTϖ1,Tϖ2,βτθϖ1,ϖ2,γγ≥δϖ1,ϖ2,γ,for all ϖ1,ϖ2∈Δ, where β∈Fβandβτθϖ1,ϖ2,γθϖ1,ϖ2<1, where(41)δϖ1,ϖ2,γ=minτθϖ1,Tϖ1,γ1+τθϖ2,Tϖ2,t1+τθTϖ1,Tϖ2,γ,τθϖ2,Tϖ2,γ1+τθϖ1,Tϖ1,γ1+τθϖ1,ϖ2,γ,τθϖ1,Tϖ1,γ2+τθϖ1,Tϖ2,γ1+τθϖ1,Tϖ2,γ+τθϖ2,Tϖ1,γ,τθϖ1,ϖ2,γ,then T has a unique fixed point.Proof.
Letϖ0∈Δ. Choose a sequence ϖl in Δ by starting an iterative process ϖl+1=Tϖl. For all l,γ>0, we have(42)τθϖl,ϖl+1,γ=τθTϖl−1,Tϖl,γ≥δϖl−1,ϖl,γβτθϖl−1,ϖl,γ,Now(43)δϖl−1,ϖl,γβτθϖl−1,ϖl,γ=minτθϖl−1,Tϖl−1,γ/βτθϖl−1,ϖl,γ1+τθϖl,Tϖl,γ/βτθϖl−1,ϖl,γ1+τθTϖl−1,Tϖl,γ/βτθϖl−1,ϖl,γ,τθϖl,Tϖl,γ/βτθϖl−1,ϖl,γ1+τθϖl−1,Tϖl−1,γ/βτθϖl−1,ϖl,γ1+τθϖl−1,ϖl,γ/βτθϖl−1,ϖl,γ,τθϖl−1,Tϖl−1,γ/βτθϖl−1,ϖl,γ2+τθϖl−1,Tϖl,γ/βτθϖl−1,ϖl,τθϖl−1,ϖl,γ/βτθϖl−1,ϖl,γγ1+τθϖl−1,Tϖl,γ/βτθϖl−1,ϖl,γ+τθϖl,Tϖl−1,γ/βτθϖl−1,ϖl,γτθϖl−1,ϖl,γ/βτθϖl−1,ϖl,γ,That is(44)δϖl−1,ϖl,γβτθϖl−1,ϖl,γ=minτθϖl−1,Tϖl−1,γ/βτθϖl−1,ϖl,γ1+τθϖl,Tϖl,γ/βτθϖl−1,ϖl,γ1+τθTϖl−1,Tϖl,γ/βτθϖl−1,ϖl,γ,τθϖl,Tϖl,γ/βτθϖl−1,ϖl,γ1+τθϖl−1,Tϖl−1,γ/βτθϖl−1,ϖl,γ1+τθϖl−1,ϖl,γ/βτθϖl−1,ϖl,γ,τθϖl−1,Tϖl−1,γ/βτθϖl−1,ϖl,γ2+τθϖl−1,Tϖl,γ/βτθϖl−1,ϖl,τθϖl−1,ϖl,γ/βτθϖl−1,ϖl,γγ1+τθϖl−1,Tϖl,γ/βτθϖl−1,ϖl,γ+τθϖl,Tϖl−1,γ/βτθϖl−1,ϖl,γτθϖl−1,ϖl,γ/βτθϖl−1,ϖl,γ,One writes(45)δϖl−1,ϖl,γβτθϖl−1,ϖl,γ=minτθϖl−1,Tϖl−1,γ/βτθϖl−1,ϖl,γ1+τθϖl,Tϖl,γ/βτθϖl−1,ϖl,γ1+τθTϖl−1,Tϖl,γ/βτθϖl−1,ϖl,γ,τθϖl,Tϖl,γ/βτθϖl−1,ϖl,γ1+τθϖl−1,Tϖl−1,γ/βτθϖl−1,ϖl,γ1+τθϖl−1,ϖl,γ/βτθϖl−1,ϖl,γ,τθϖl−1,Tϖl−1,γ/βτθϖl−1,ϖl,γ2+τθϖl−1,Tϖl,γ/βτθϖl−1,ϖl,τθϖl−1,ϖl,γ/βτθϖl−1,ϖl,γγ1+τθϖl−1,Tϖl,γ/βτθϖl−1,ϖl,γ+τθϖl,Tϖl−1,γ/βτθϖl−1,ϖl,γτθϖl−1,ϖl,γ/βτθϖl−1,ϖl,γ,We obtain(46)δϖl−1,ϖl,γβτθϖl−1,ϖl,γ=minτθϖl−1,ϖl,γβτθϖl−1,ϖl,γ,τθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,If minτθϖl−1,ϖl,γ/βτθϖl−1,ϖl,γ,τθϖl,ϖl+1,γ/βτθϖl−1,ϖl,γ=τθϖl,ϖl+1,γ/βτθϖl−1,ϖl,γ then from (42)(47)τbϖl,ϖl+1,γ≥τθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,Since the range of β is 0,1/b, he result is obvious by Lemma 1. If minτθϖl−1,ϖl,γ/βτθϖl−1,ϖl,γ,τθϖl,ϖl+1,γ/βτθϖl−1,ϖl,γ=τθϖl−1,ϖl,γ/βτθϖl−1,ϖl,γ then from (42)(48)τθϖl,ϖl+1,γ≥τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ,Continuing in this way, we have(49)τθϖl,ϖl+1,γ≥τθϖ0,ϖ1,γβτθϖl−1,ϖl,γ⋅βτθϖl−2,ϖl−1,γ…βτθϖ0,ϖ1,γ.
Then the result is trivial by Lemma1.Theorem 6.
LetΔ,τθ,∗ be a Gr-complete EFBMS with θϖ1,ϖ2≥1. Let T:Δ⟶Δ be a mapping satisfying the condition(50)τθTϖ1,Tϖ2,βτθϖ1,ϖ2,γγ≥αϖ1,ϖ2,γmaxτθϖ1,Tϖ1,γ,τθϖ2,Tϖ2,γ,where(51)αϖ1,ϖ2,γ=minτθTϖ1,Tϖ2,γ⋅τθϖ1,ϖ2,γ,τθϖ1,Tϖ1,t⋅τθϖ2,Tϖ2,γ,for all ϖ1,ϖ2∈Δ, where β∈Fβ and θϖ1,ϖ2βτθϖ1,ϖ2,γ<1, then T has a unique fixed point.Proof.
Consider(52)τθϖl,ϖl+1,γ=τθTϖl−1,Tϖl,γ≥αϖl,ϖl+1,γ/βτθϖl−1,ϖl,γmaxτθϖl−1,Tϖl−1,γ/βτθϖl−1,ϖl,γ,τθϖl,Tϖl,γ/βτθϖl−1,ϖl,γ,Now(53)αϖl,ϖl+1,γβϖl−1,ϖl,γ=minτθTϖl−1,Tϖl,γβτθϖl−1,ϖl,γ⋅τθϖl−1,ϖl,γβτθϖl−1,ϖl,γτθϖl−1,Tϖl−1,γβτθϖl−1,ϖl,γ⋅τθϖl,Tϖl,γβτθϖl−1,ϖl,γ=minτθϖl,ϖl+1,γβτθϖl−1,ϖl,γ⋅τθϖl−1,ϖl,γβτθϖl−1,ϖl,γτθϖl−1,ϖl,γβτθϖl−1,ϖl,γ⋅τθϖl,ϖl+1,γβτθϖl−1,ϖl,γ=τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ⋅τθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,Using (14) in (52) we get(54)τθϖl,ϖl+1,γ≥τθϖl,ϖl+1,γ/βτθϖl−1,ϖl,γ⋅τθϖl−1,ϖl,γ/βτθϖl−1,ϖl,γmaxτθϖl−1,ϖl,γ/βτθϖl−1,ϖl,γ,τθϖl,ϖl+1,γ/βτθϖl−1,ϖl,γ,If(55)maxτθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ=τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ,then (54) implies(56)τθϖl,ϖl+1,kt≥τθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,Then nothing to prove by Lemma 1.If(57)maxτθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ=τθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,then from (54), we have(58)τθϖl,ϖl+1,γ≥τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ≥…≥τθϖ0,ϖ1,γβτθϖl−1,ϖl,γ⋅βτθϖl−2,ϖl−1,γ…βτθϖ0,ϖ1,γ,By using the same method as in Theorem 1, one can complete the proof.Remark 5.
Takingθϖ1,ϖ2=b we get Theorem 3.9 of [36].Theorem 7.
LetΔ,τθ,∗ be a Gr-complete EFBMS with θϖ1,ϖ2≥1 and T:Δ⟶Δ be a mapping satisfying the condition(59)τθTϖ1,Tϖ2,βτθϖ1,ϖ2,γγ≥λϖ1,ϖ2,γ∗μϖ1,ϖ2,γ,where(60)λϖ1,ϖ2,γ=minτθTϖ1,Tϖ2,γ,τθϖ1,Tϖ1,γ,τθϖ2,Tϖ2,t,τθϖ1,ϖ2,γμϖ1,ϖ2,γ=maxτθϖ1,Tϖ2,γ,τθTϖ1,ϖ2,γ,for all ϖ1,ϖ2∈Δ, where β∈Fβ and θϖ1,ϖ2βτθϖ1,ϖ2,γ<1, then T has a unique fixed point, where a∗b=mina,b.Proof.
Start by(61)τθϖl,ϖl+1,γ=τθTϖl−1,Tϖl,γ≥λϖl−1,ϖl,γβϖl−1,ϖl,t∗μϖl−1,ϖl,γβϖl−1,ϖl,γ,Now(62)λϖl−1,ϖl,γβϖl−1,ϖl,γ=minτθTϖl−1,Tϖl,γβτθϖl−1,ϖl,γ,τθϖl−1,Tϖl−1,γβϖl−1,ϖl,γ,τθϖl,Tϖl,γβτθϖl−1,ϖl,γ,τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ=minτθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ,τθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,τθϖl−1,ϖl,γβτθϖl−1,ϖl,γλϖl−1,ϖl−1,γβτθϖl−1,ϖl,γ=minτθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ,(63)μϖl−1,ϖl,γβϖl−1,ϖl,γ=maxτθϖl−1,Tϖl,γβτθϖl−1,ϖl,γ,τθTϖl−1,ϖl,γβτθϖl−1,ϖl,γ=maxτθϖl−1,ϖl+1,γβτθϖl−1,ϖl,γ,τθϖl,ϖl,γβτθϖl−1,ϖl,γ=maxτθϖl−1,ϖl+1,γβτθϖl−1,ϖl,γ,1μϖl−1,ϖl,γβτθϖl−1,ϖl,γ=1,Using (62) and (63) in (61), we have(64)τθϖl,ϖl+1,γ≥minτθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ∗1τθϖl,ϖl+1,γ≥minτθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ,If(65)minτθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ=τθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,then (64) implies(66)τθϖl,ϖl+1,γ≥τθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,then T has a fixed point by Lemma 1. If(67)minτθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,τθϖl−1,ϖl,γβτθϖl−1,ϖl,γτθϖl−1,ϖl,γβτθϖl−1,ϖl,γ,then from (64), we have(68)τθϖl,ϖl+1,γ≥τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ.
Continuing in this way, we will get(69)τθϖl,ϖl+1,γ≥τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ≥…≥τθϖ0,ϖ1,γβτθϖl−1,ϖl,γ⋅βτθϖl−2,ϖl−1,γ…βτθϖ0,ϖ1,γ,Using the method of Theorem 1 after inequality (7), the desired result is established.Remark 6.
Takingθϖ1,ϖ2=b we get Theorem 3.11 of [36].Theorem 8.
LetΔ,τθ,∗ be a Gr-complete EFBMS with θϖ1,ϖ2≥1. Let T:Δ⟶Δ be a mapping satisfying the condition(70)τθTϖ1,Tϖ2,βτθϖ1,ϖ2,γγ≥λϖ1,ϖ2,γ∗μϖ1,ϖ2,γαϖ1,ϖ2,γ,where(71)λϖ1,ϖ2,γ=minτθTϖ1,Tϖ2,γ⋅τθϖ1,ϖ2,γ,τθϖ1,Tϖ1,γ⋅τθϖ2,Tϖ2,γμϖ1,ϖ2,γ=maxτθϖ1,Tϖ1,γ⋅τθϖ1,Tϖ2,γ,τθϖ2,Tϖ1,γ2αϖ1,ϖ2,γ=maxτθϖ1,Tϖ1,γ,τθϖ2,Tϖ2,γ,for all ϖ1,ϖ2∈Δ, where β∈Fβ and θϖ1,ϖ2βτθϖ1,ϖ2,γ<1. Then T has a unique fixed point.Proof.
By the same way as in Theorem1, we have(72)τθϖl,ϖl+1,γ=τθTϖl−1,Tϖl,γ≥λϖl−1,ϖl,γ/βτθϖl−1,ϖl,γ∗μϖl−1,ϖl,γ/βτθϖl−1,ϖl,γαϖ1,ϖ2,γ/βτθϖl−1,ϖl,γ,(73)λϖl−1,ϖl,γβτθϖl−1,ϖl,γ=minτθTϖl−1,Tϖl,γβτθϖl−1,ϖl,γ⋅τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ,τθϖl−1,Tϖl−1,γβτθϖl−1,ϖl,γ⋅τθϖl,Tϖl,γβτθϖl−1,ϖl,γ=minτθϖl,ϖl+1,γβτθϖl−1,ϖl,γ⋅τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ,τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ⋅τθϖl,ϖl+1,γβτθϖl−1,ϖl,γ=τθϖl,ϖl+1,γβτθϖl−1,ϖl,γ⋅τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ,(74)μϖl−1,ϖl,γβτθϖl−1,ϖl,γ=maxτθϖl−1,Tϖl−1,γβτθϖl−1,ϖl,γ⋅τθϖl−1,Tϖl,γβτθϖl−1,ϖl,γ,τθϖl,Tϖl−1,γβτθϖl−1,ϖl,γ2=maxτθϖl−1,ϖl,γβτθϖl−1,ϖl,γ⋅τθϖl−1,ϖl+1,γβτθϖl−1,ϖl,γ,τθϖl,ϖl,γβτθϖl−1,ϖl,γ2=maxτθϖl−1,ϖl,γβτθϖl−1,ϖl,γ⋅τθϖl−1,ϖl+1,γβτθϖl−1,ϖl,γ,1μϖl−1,ϖl,γβτθϖl−1,ϖl,γ=1,(75)αϖl−1,ϖl,γβτθϖl−1,ϖl,γ=maxτθϖl−1,Tϖl−1,γβτθϖl−1,ϖl,γ,τθϖl,Tϖl,γβτθϖl−1,ϖl,γ=maxτθϖl−1,ϖl,γβτθϖl−1,ϖl,γ,τθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,Using (73), (74) and (75) in (72), we have(76)τθϖl,ϖl+1,γ≥τθϖl,ϖl+1,γ/βτθϖl−1,ϖl,γ⋅τθϖl−1,ϖl,γ/βτθϖl−1,ϖl,γmaxτθϖl−1,ϖl,γ/βτθϖl−1,ϖl,γ,τθϖl,ϖl+1,γ/βτθϖl−1,ϖl,γ,If(77)maxτθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ=τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ,then (76) implies(78)τθϖl,ϖl+1,γ≥τθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,then T has a fixed point by Lemma 1. If(79)maxτθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ=τθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,then from (76), we have(80)τθϖl,ϖl+1,γ≥τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ.
Continuing in this way, we will get(81)τθϖl,ϖl+1,γ≥τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ≥…≥τθϖ0,ϖ1,γβτθϖl−1,ϖl,γ⋅βτθϖl−2,ϖl−1,γ…βτθϖ0,ϖ1,γ.
By adopting the same procedure used in Theorem1, the required result is obtained.Remark 7.
By consideringθϖ1,ϖ2=b, Theorem 3.11 of [36] can be obtained.


## 3.1. Uniqueness

AssumeTp2=p2 for some p2∈Δ. Then(14)τθp2,p1,γ=τθTp2,Tp1,γ≥τθp2,p1,γβτθp2,p1,γ=τθTp2,Tp1,γβτθp2,p1,γ≥τθp2,p1,γβτθp2,p1,γ2≥…≥τθp2,p1,γβτθp2,p1,γn=τθp2,p1,bnγ⟶1asl⟶∞.Hence, the fixed point is unique.Example 2.
LetΔ=0,1,2 and τθϖ1,ϖ2,γ=γ/γ+ϖ1−ϖ22
It is easy to verify thatΔ,τθ,∗ is a Gr-complete EFBMS. Define a mapping T:Δ⟶Δ such that Tϖ1=βτθϖ1,ϖ2,γϖ1/1+ϖ1. Now, for all γ>0 , we have(15)τθTϖ1,Tϖ2,βτθϖ1,ϖ2,γγ=βτθϖ1,ϖ2,γγβτθϖ1,ϖ2,γγ+βτθϖ1,ϖ2,γϖ1/1+ϖ1−βτθϖ1,ϖ2,γϖ2/1+ϖ22=βτθϖ1,ϖ2,γγβτθϖ1,ϖ2,γγ+βτθϖ1,ϖ2,γϖ1/1+ϖ1−ϖ2/1+ϖ22=γγ+ϖ1−ϖ22/1+ϖ121+ϖ12,Recall that(16)ϖ1−ϖ221+ϖ121+ϖ12≤ϖ1−ϖ22γ+ϖ1−ϖ221+ϖ121+ϖ12≤γ+ϖ1−ϖ221γ+ϖ1−ϖ22/1+ϖ121+ϖ12≥1γ+ϖ1−ϖ22γγ+ϖ1−ϖ22/1+ϖ121+ϖ12≥γγ+ϖ1−ϖ22.
This implies that(17)τθTϖ1,Tϖ2,βτθϖ1,ϖ2,γγ≥τθϖ1,ϖ2,γ,Also, ϖ1=0 is the unique fixed point of T.Remark 2.
Takingθϖ1,ϖ2=b, we get Theorem 3.1 of [36].Remark 3.
Takingβτθϖ1,ϖ2,γ=k, the main result of [32] is obtained and for θϖ1,ϖ2=1 and for βτθϖ1,ϖ2,γ=k, the main result of [3] is obtained.Theorem 2.
Consider aGr-complete EFBMS Δ,τθ,∗ with θϖ1,ϖ2≥1. Let T:Δ⟶Δ be a mapping satisfying the condition(18)τθTϖ1,Tϖ2,βτθϖ1,ϖ2,γγ≥minτθTϖ1,Tϖ2,γ,τθϖ1,Tϖ1,γ,τθϖ2,Tϖ2,γ,τθϖ1,ϖ2,γ,for all ϖ1,ϖ2∈Δ, where β∈Fβ and θϖ1,ϖ2βτθϖ1,ϖ2,γ<1. Then T has a unique fixed point.Proof.
Starting by the same way as in Theorem1, we have(19)τθϖl,ϖl+1,γ=τθTϖl−1,Tϖl,γ≥minτθTϖl−1,Tϖl,γβτθϖl−1,ϖl,γ,τθϖl−1,Tϖl−1,γβτθϖl−1,ϖl,γτθϖl,Tϖl,γβτθϖl−1,ϖl,γ,τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ≥minτθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ,τθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ,Hence(20)τθϖl,ϖl+1,γ≥minτθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ,If(21)minτθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ=τθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,then (20) implies(22)τθϖl,ϖl+1,γ≥τθϖl,ϖl+1,γβτθϖl−1,ϖl,γ.
By Lemma1, T has a fixed point.
If(23)minτθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ=τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ,then from (20), we have(24)τθϖl,ϖl+1,γ≥τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ≥…≥τθϖ0,ϖ1,γβτθϖl−1,ϖl,γ⋅βτθϖl−2,ϖl−1,γ…βτθϖ0,ϖ1,γ.
The rest of the proof follows from Theorem1.Remark 4.
Takingθϖ1,ϖ2=b we get Theorem 3.6 of [36].
The following result is an extension of the main result of Gupta at el. [34].Theorem 3.
Consider aGr-complete EFBMS Δ,τθ,∗ with θϖ1,ϖ2≥1. Let T:Δ⟶Δ be a mapping satisfying(25)τθTϖ1,Tϖ2,βτθϖ1,ϖ2,γγ≥minτθϖ2,Tϖ2,γ1+τθϖ1,Tϖ1,γ1+τθϖ1,ϖ2,γ,τθϖ1,ϖ2,γ,for all ϖ1,ϖ2∈Δ, where β∈Fβ and βτθϖ1,ϖ2,γθϖ1,ϖ2<1, then T has a unique fixed point.Proof.
Consider(26)τθϖl,ϖl+1,γ=τθTϖl−1,Tϖl,γ≥minτθϖl,Tϖl,γ/βτθϖl−1,ϖl,γ1+τθϖl−1,Tϖl−1,γ/βτθϖl−1,ϖl,γ1+τθϖl−1,ϖl,γ/βτθϖl−1,ϖl,γτθϖl−1,ϖl,γβτθϖl−1,ϖl,γ≥minτθϖl,ϖl+1,γ/βτθϖl−1,ϖl,γ1+τθϖl−1,ϖl,γ/βτθϖl−1,ϖl,γ1+τθϖl−1,ϖl,γ/βτθϖl−1,ϖl,γ,τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ≥minτθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ.If(27)minτθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ=τθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,then (26) implies(28)τθϖl,ϖl+1,γ≥τθϖl,ϖl+1,γβτθϖl−1,ϖl,γ.
Then the result is trivial by Lemma1.If(29)minτθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ=τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ,then from (26) we have(30)τθϖl,ϖl+1,γ≥τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ.
Continuing in this way, one writes(31)τθϖl,ϖl+1,γ≥τθϖ0,ϖ1,γβτθϖl−1,ϖl,γ⋅βτθϖl−2,ϖl−1γ…βτθϖ0,ϖ1,γ.
Therefore, proceeding as in Theorem1 after inequality (7), the desired result is obtained.Theorem 4.
LetX,τθ,∗ be a Gr-complete EFBMS with θϖ1,ϖ2≥1. Let T:Δ⟶Δ be a mapping satisfying(32)τθTϖ1,Tϖ2,βτθϖ1,ϖ2,γγ≥minτθϖ2,Tϖ2,γ1+τθϖ1,Tϖ1,γ+τθϖ2,Tϖ1,γ2+τθϖ1,ϖ2,γ,τθϖ1,ϖ2,γ,for all ϖ1,ϖ2∈Δ, where β∈Fβ and βτθϖ1,ϖ2,γθϖ1,ϖ2<1, then T has a unique fixed point.Proof.
Forϖ0∈Δ, we choose a sequence ϖl in Δ and start by ϖl+1=Tϖl. For all l,γ>0, we have(33)τθϖl,ϖl+1,γ=τθTϖl−1,Tϖl,γ≥minτθϖl,Tϖl,γ/βτθϖl−1,ϖl,γ1+τθϖl−1,Tϖl−1,γ/βτθϖl−1,ϖl,γ+τθϖl,Tϖl−1,γ/βτθϖl−1,ϖl,γ2+τθϖl−1,ϖl,γ/βτθϖl−1,ϖl,γ,τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ=minτθϖl,ϖl+1,γ/βτθϖl−1,ϖl,γ1+τθϖl−1,ϖl,γ/βτθϖl−1,ϖl,γ+τθϖl,ϖl,γ/βτθϖl−1,ϖl,γ2+τθϖl−1,ϖl,γ/βτθϖl−1,ϖl,γ,τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ=minτθϖl,ϖl+1,γ/βτθϖl−1,ϖl,γ1+τθϖl−1,ϖl,γ/βτθϖl−1,ϖl,γ+12+τθϖl−1,ϖl,γ/βτθϖl−1,ϖl,γ,τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ=minτθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ,So, we have(34)τθϖl,ϖl+1,γ≥minτθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ,If(35)minτθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ=τθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,then from (34)(36)τθϖl,ϖl+1,γ≥τθϖl,ϖl+1,γβτθϖl−1,ϖl,γ.
Then result is trivial by Lemma1. If(37)minτθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ=τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ,then from (34)(38)τθϖl,ϖl+1,t≥τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ,Continuing in this direction, we get(39)τθϖl,ϖl+1,γ≥τθϖ0,ϖ1,γβτθϖl−1,ϖl,γ⋅βτθϖl−2,ϖl−1,γ…βτθϖ0,ϖ1,γ.
The desired result is then established by the same procedure as in Theorem1. The following result is the extension of the main result of Roshan et al. [35].Theorem 5.
Consider aGr-complete EFBMS X,τθ,∗ with θϖ1,ϖ2≥1. Let T:Δ⟶Δ be a mapping satisfying(40)τθTϖ1,Tϖ2,βτθϖ1,ϖ2,γγ≥δϖ1,ϖ2,γ,for all ϖ1,ϖ2∈Δ, where β∈Fβandβτθϖ1,ϖ2,γθϖ1,ϖ2<1, where(41)δϖ1,ϖ2,γ=minτθϖ1,Tϖ1,γ1+τθϖ2,Tϖ2,t1+τθTϖ1,Tϖ2,γ,τθϖ2,Tϖ2,γ1+τθϖ1,Tϖ1,γ1+τθϖ1,ϖ2,γ,τθϖ1,Tϖ1,γ2+τθϖ1,Tϖ2,γ1+τθϖ1,Tϖ2,γ+τθϖ2,Tϖ1,γ,τθϖ1,ϖ2,γ,then T has a unique fixed point.Proof.
Letϖ0∈Δ. Choose a sequence ϖl in Δ by starting an iterative process ϖl+1=Tϖl. For all l,γ>0, we have(42)τθϖl,ϖl+1,γ=τθTϖl−1,Tϖl,γ≥δϖl−1,ϖl,γβτθϖl−1,ϖl,γ,Now(43)δϖl−1,ϖl,γβτθϖl−1,ϖl,γ=minτθϖl−1,Tϖl−1,γ/βτθϖl−1,ϖl,γ1+τθϖl,Tϖl,γ/βτθϖl−1,ϖl,γ1+τθTϖl−1,Tϖl,γ/βτθϖl−1,ϖl,γ,τθϖl,Tϖl,γ/βτθϖl−1,ϖl,γ1+τθϖl−1,Tϖl−1,γ/βτθϖl−1,ϖl,γ1+τθϖl−1,ϖl,γ/βτθϖl−1,ϖl,γ,τθϖl−1,Tϖl−1,γ/βτθϖl−1,ϖl,γ2+τθϖl−1,Tϖl,γ/βτθϖl−1,ϖl,τθϖl−1,ϖl,γ/βτθϖl−1,ϖl,γγ1+τθϖl−1,Tϖl,γ/βτθϖl−1,ϖl,γ+τθϖl,Tϖl−1,γ/βτθϖl−1,ϖl,γτθϖl−1,ϖl,γ/βτθϖl−1,ϖl,γ,That is(44)δϖl−1,ϖl,γβτθϖl−1,ϖl,γ=minτθϖl−1,Tϖl−1,γ/βτθϖl−1,ϖl,γ1+τθϖl,Tϖl,γ/βτθϖl−1,ϖl,γ1+τθTϖl−1,Tϖl,γ/βτθϖl−1,ϖl,γ,τθϖl,Tϖl,γ/βτθϖl−1,ϖl,γ1+τθϖl−1,Tϖl−1,γ/βτθϖl−1,ϖl,γ1+τθϖl−1,ϖl,γ/βτθϖl−1,ϖl,γ,τθϖl−1,Tϖl−1,γ/βτθϖl−1,ϖl,γ2+τθϖl−1,Tϖl,γ/βτθϖl−1,ϖl,τθϖl−1,ϖl,γ/βτθϖl−1,ϖl,γγ1+τθϖl−1,Tϖl,γ/βτθϖl−1,ϖl,γ+τθϖl,Tϖl−1,γ/βτθϖl−1,ϖl,γτθϖl−1,ϖl,γ/βτθϖl−1,ϖl,γ,One writes(45)δϖl−1,ϖl,γβτθϖl−1,ϖl,γ=minτθϖl−1,Tϖl−1,γ/βτθϖl−1,ϖl,γ1+τθϖl,Tϖl,γ/βτθϖl−1,ϖl,γ1+τθTϖl−1,Tϖl,γ/βτθϖl−1,ϖl,γ,τθϖl,Tϖl,γ/βτθϖl−1,ϖl,γ1+τθϖl−1,Tϖl−1,γ/βτθϖl−1,ϖl,γ1+τθϖl−1,ϖl,γ/βτθϖl−1,ϖl,γ,τθϖl−1,Tϖl−1,γ/βτθϖl−1,ϖl,γ2+τθϖl−1,Tϖl,γ/βτθϖl−1,ϖl,τθϖl−1,ϖl,γ/βτθϖl−1,ϖl,γγ1+τθϖl−1,Tϖl,γ/βτθϖl−1,ϖl,γ+τθϖl,Tϖl−1,γ/βτθϖl−1,ϖl,γτθϖl−1,ϖl,γ/βτθϖl−1,ϖl,γ,We obtain(46)δϖl−1,ϖl,γβτθϖl−1,ϖl,γ=minτθϖl−1,ϖl,γβτθϖl−1,ϖl,γ,τθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,If minτθϖl−1,ϖl,γ/βτθϖl−1,ϖl,γ,τθϖl,ϖl+1,γ/βτθϖl−1,ϖl,γ=τθϖl,ϖl+1,γ/βτθϖl−1,ϖl,γ then from (42)(47)τbϖl,ϖl+1,γ≥τθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,Since the range of β is 0,1/b, he result is obvious by Lemma 1. If minτθϖl−1,ϖl,γ/βτθϖl−1,ϖl,γ,τθϖl,ϖl+1,γ/βτθϖl−1,ϖl,γ=τθϖl−1,ϖl,γ/βτθϖl−1,ϖl,γ then from (42)(48)τθϖl,ϖl+1,γ≥τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ,Continuing in this way, we have(49)τθϖl,ϖl+1,γ≥τθϖ0,ϖ1,γβτθϖl−1,ϖl,γ⋅βτθϖl−2,ϖl−1,γ…βτθϖ0,ϖ1,γ.
Then the result is trivial by Lemma1.Theorem 6.
LetΔ,τθ,∗ be a Gr-complete EFBMS with θϖ1,ϖ2≥1. Let T:Δ⟶Δ be a mapping satisfying the condition(50)τθTϖ1,Tϖ2,βτθϖ1,ϖ2,γγ≥αϖ1,ϖ2,γmaxτθϖ1,Tϖ1,γ,τθϖ2,Tϖ2,γ,where(51)αϖ1,ϖ2,γ=minτθTϖ1,Tϖ2,γ⋅τθϖ1,ϖ2,γ,τθϖ1,Tϖ1,t⋅τθϖ2,Tϖ2,γ,for all ϖ1,ϖ2∈Δ, where β∈Fβ and θϖ1,ϖ2βτθϖ1,ϖ2,γ<1, then T has a unique fixed point.Proof.
Consider(52)τθϖl,ϖl+1,γ=τθTϖl−1,Tϖl,γ≥αϖl,ϖl+1,γ/βτθϖl−1,ϖl,γmaxτθϖl−1,Tϖl−1,γ/βτθϖl−1,ϖl,γ,τθϖl,Tϖl,γ/βτθϖl−1,ϖl,γ,Now(53)αϖl,ϖl+1,γβϖl−1,ϖl,γ=minτθTϖl−1,Tϖl,γβτθϖl−1,ϖl,γ⋅τθϖl−1,ϖl,γβτθϖl−1,ϖl,γτθϖl−1,Tϖl−1,γβτθϖl−1,ϖl,γ⋅τθϖl,Tϖl,γβτθϖl−1,ϖl,γ=minτθϖl,ϖl+1,γβτθϖl−1,ϖl,γ⋅τθϖl−1,ϖl,γβτθϖl−1,ϖl,γτθϖl−1,ϖl,γβτθϖl−1,ϖl,γ⋅τθϖl,ϖl+1,γβτθϖl−1,ϖl,γ=τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ⋅τθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,Using (14) in (52) we get(54)τθϖl,ϖl+1,γ≥τθϖl,ϖl+1,γ/βτθϖl−1,ϖl,γ⋅τθϖl−1,ϖl,γ/βτθϖl−1,ϖl,γmaxτθϖl−1,ϖl,γ/βτθϖl−1,ϖl,γ,τθϖl,ϖl+1,γ/βτθϖl−1,ϖl,γ,If(55)maxτθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ=τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ,then (54) implies(56)τθϖl,ϖl+1,kt≥τθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,Then nothing to prove by Lemma 1.If(57)maxτθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ=τθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,then from (54), we have(58)τθϖl,ϖl+1,γ≥τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ≥…≥τθϖ0,ϖ1,γβτθϖl−1,ϖl,γ⋅βτθϖl−2,ϖl−1,γ…βτθϖ0,ϖ1,γ,By using the same method as in Theorem 1, one can complete the proof.Remark 5.
Takingθϖ1,ϖ2=b we get Theorem 3.9 of [36].Theorem 7.
LetΔ,τθ,∗ be a Gr-complete EFBMS with θϖ1,ϖ2≥1 and T:Δ⟶Δ be a mapping satisfying the condition(59)τθTϖ1,Tϖ2,βτθϖ1,ϖ2,γγ≥λϖ1,ϖ2,γ∗μϖ1,ϖ2,γ,where(60)λϖ1,ϖ2,γ=minτθTϖ1,Tϖ2,γ,τθϖ1,Tϖ1,γ,τθϖ2,Tϖ2,t,τθϖ1,ϖ2,γμϖ1,ϖ2,γ=maxτθϖ1,Tϖ2,γ,τθTϖ1,ϖ2,γ,for all ϖ1,ϖ2∈Δ, where β∈Fβ and θϖ1,ϖ2βτθϖ1,ϖ2,γ<1, then T has a unique fixed point, where a∗b=mina,b.Proof.
Start by(61)τθϖl,ϖl+1,γ=τθTϖl−1,Tϖl,γ≥λϖl−1,ϖl,γβϖl−1,ϖl,t∗μϖl−1,ϖl,γβϖl−1,ϖl,γ,Now(62)λϖl−1,ϖl,γβϖl−1,ϖl,γ=minτθTϖl−1,Tϖl,γβτθϖl−1,ϖl,γ,τθϖl−1,Tϖl−1,γβϖl−1,ϖl,γ,τθϖl,Tϖl,γβτθϖl−1,ϖl,γ,τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ=minτθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ,τθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,τθϖl−1,ϖl,γβτθϖl−1,ϖl,γλϖl−1,ϖl−1,γβτθϖl−1,ϖl,γ=minτθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ,(63)μϖl−1,ϖl,γβϖl−1,ϖl,γ=maxτθϖl−1,Tϖl,γβτθϖl−1,ϖl,γ,τθTϖl−1,ϖl,γβτθϖl−1,ϖl,γ=maxτθϖl−1,ϖl+1,γβτθϖl−1,ϖl,γ,τθϖl,ϖl,γβτθϖl−1,ϖl,γ=maxτθϖl−1,ϖl+1,γβτθϖl−1,ϖl,γ,1μϖl−1,ϖl,γβτθϖl−1,ϖl,γ=1,Using (62) and (63) in (61), we have(64)τθϖl,ϖl+1,γ≥minτθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ∗1τθϖl,ϖl+1,γ≥minτθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ,If(65)minτθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ=τθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,then (64) implies(66)τθϖl,ϖl+1,γ≥τθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,then T has a fixed point by Lemma 1. If(67)minτθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,τθϖl−1,ϖl,γβτθϖl−1,ϖl,γτθϖl−1,ϖl,γβτθϖl−1,ϖl,γ,then from (64), we have(68)τθϖl,ϖl+1,γ≥τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ.
Continuing in this way, we will get(69)τθϖl,ϖl+1,γ≥τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ≥…≥τθϖ0,ϖ1,γβτθϖl−1,ϖl,γ⋅βτθϖl−2,ϖl−1,γ…βτθϖ0,ϖ1,γ,Using the method of Theorem 1 after inequality (7), the desired result is established.Remark 6.
Takingθϖ1,ϖ2=b we get Theorem 3.11 of [36].Theorem 8.
LetΔ,τθ,∗ be a Gr-complete EFBMS with θϖ1,ϖ2≥1. Let T:Δ⟶Δ be a mapping satisfying the condition(70)τθTϖ1,Tϖ2,βτθϖ1,ϖ2,γγ≥λϖ1,ϖ2,γ∗μϖ1,ϖ2,γαϖ1,ϖ2,γ,where(71)λϖ1,ϖ2,γ=minτθTϖ1,Tϖ2,γ⋅τθϖ1,ϖ2,γ,τθϖ1,Tϖ1,γ⋅τθϖ2,Tϖ2,γμϖ1,ϖ2,γ=maxτθϖ1,Tϖ1,γ⋅τθϖ1,Tϖ2,γ,τθϖ2,Tϖ1,γ2αϖ1,ϖ2,γ=maxτθϖ1,Tϖ1,γ,τθϖ2,Tϖ2,γ,for all ϖ1,ϖ2∈Δ, where β∈Fβ and θϖ1,ϖ2βτθϖ1,ϖ2,γ<1. Then T has a unique fixed point.Proof.
By the same way as in Theorem1, we have(72)τθϖl,ϖl+1,γ=τθTϖl−1,Tϖl,γ≥λϖl−1,ϖl,γ/βτθϖl−1,ϖl,γ∗μϖl−1,ϖl,γ/βτθϖl−1,ϖl,γαϖ1,ϖ2,γ/βτθϖl−1,ϖl,γ,(73)λϖl−1,ϖl,γβτθϖl−1,ϖl,γ=minτθTϖl−1,Tϖl,γβτθϖl−1,ϖl,γ⋅τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ,τθϖl−1,Tϖl−1,γβτθϖl−1,ϖl,γ⋅τθϖl,Tϖl,γβτθϖl−1,ϖl,γ=minτθϖl,ϖl+1,γβτθϖl−1,ϖl,γ⋅τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ,τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ⋅τθϖl,ϖl+1,γβτθϖl−1,ϖl,γ=τθϖl,ϖl+1,γβτθϖl−1,ϖl,γ⋅τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ,(74)μϖl−1,ϖl,γβτθϖl−1,ϖl,γ=maxτθϖl−1,Tϖl−1,γβτθϖl−1,ϖl,γ⋅τθϖl−1,Tϖl,γβτθϖl−1,ϖl,γ,τθϖl,Tϖl−1,γβτθϖl−1,ϖl,γ2=maxτθϖl−1,ϖl,γβτθϖl−1,ϖl,γ⋅τθϖl−1,ϖl+1,γβτθϖl−1,ϖl,γ,τθϖl,ϖl,γβτθϖl−1,ϖl,γ2=maxτθϖl−1,ϖl,γβτθϖl−1,ϖl,γ⋅τθϖl−1,ϖl+1,γβτθϖl−1,ϖl,γ,1μϖl−1,ϖl,γβτθϖl−1,ϖl,γ=1,(75)αϖl−1,ϖl,γβτθϖl−1,ϖl,γ=maxτθϖl−1,Tϖl−1,γβτθϖl−1,ϖl,γ,τθϖl,Tϖl,γβτθϖl−1,ϖl,γ=maxτθϖl−1,ϖl,γβτθϖl−1,ϖl,γ,τθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,Using (73), (74) and (75) in (72), we have(76)τθϖl,ϖl+1,γ≥τθϖl,ϖl+1,γ/βτθϖl−1,ϖl,γ⋅τθϖl−1,ϖl,γ/βτθϖl−1,ϖl,γmaxτθϖl−1,ϖl,γ/βτθϖl−1,ϖl,γ,τθϖl,ϖl+1,γ/βτθϖl−1,ϖl,γ,If(77)maxτθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ=τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ,then (76) implies(78)τθϖl,ϖl+1,γ≥τθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,then T has a fixed point by Lemma 1. If(79)maxτθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ=τθϖl,ϖl+1,γβτθϖl−1,ϖl,γ,then from (76), we have(80)τθϖl,ϖl+1,γ≥τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ.
Continuing in this way, we will get(81)τθϖl,ϖl+1,γ≥τθϖl−1,ϖl,γβτθϖl−1,ϖl,γ≥…≥τθϖ0,ϖ1,γβτθϖl−1,ϖl,γ⋅βτθϖl−2,ϖl−1,γ…βτθϖ0,ϖ1,γ.
By adopting the same procedure used in Theorem1, the required result is obtained.Remark 7.
By consideringθϖ1,ϖ2=b, Theorem 3.11 of [36] can be obtained.


## 4. Application

The application of our main result stated in Theorem1 is established here. Let Δ=C0,I be the set of real valued continuous functions on 0,I and define a Gr-complete EFBMS Fα:Δ×Δ×0,∞⟶0,1 by(82)τθϖ1,ϖ2,γ=e−supr∈0,Iϖ1r−ϖ2r2γ,ifγ>0,0,ifγ>0,with(83)θϖ1,ϖ2=1+ϖ1+ϖ2.Consider the integral equation(84)ϖ1s=fs+∫0Ihs,rHs,r,ϖ1rdr,where I>0, and f:0,I⟶ℝ, h:0,I×0,I⟶ℝ, and H:0,I×0,I×ℝ⟶ℝ all are continuous functions.Theorem 9.
Suppose that the following conditions hold:(i)
for alls,r∈0,I, ϖ1,ϖ2∈Δ and β∈Fb, we have(85)Hs,r,ϖ1r−Hs,r,ϖ2r<βτθϖ1,ϖ2,γϖ1r−ϖ2r.(ii)
for alls,r∈0,I ,(86)supr∈0,I∫0Ihs,r2dr≤1I.
Then the integral equation (84) has a solution in Δ.Proof.
LetT:Δ⟶Δ be the integral operator defined by(87)Tϖ1s=fs+∫0Ihs,rHs,r,ϖ1rdr,for all ϖ1∈Δ,ands,r∈0,I. Now, for all ϖ1,ϖ2∈Δ and by using Conditions (i) and (ii), we have(88)τθTϖ1,Tϖ2,βτθϖ1,ϖ2,γγ=e−supr∈0,ITϖ1r−Tϖ2r2/βτθϖ1,ϖ2,γγ=e−supr∈0,I∫0Ih∫0Is,rHs,r,ϖ1rdr−∫0Ihs,rHs,r,ϖ2rdr/βτθϖ1,ϖ2,γγ=e−supr∈0,I∫0Ihs,rHs,r,ϖ1r−Hs,r,ϖ2rdr2/βτθϖ1,ϖ2,γγ≥e−supr∈0,I∫0Ihs,r2dr∫0IFs,r,ϖ1r−Fs,r,ϖ2r2/βτθϖ1,ϖ2,γγ≥e−supr∈0,I1/I∫0Iβτθϖ1,ϖ2,γϖ1r−ϖ2r2/βτθϖ1,ϖ2,γγ≥e−supr∈0,Iβτθϖ1,ϖ2,γϖ1r−ϖ2r2/βτθϖ1,ϖ2,γγ,That is(89)τθTϖ1,Tϖ2,βτθϖ1,ϖ2,γγ≥e−supr∈0,Iϖ1r−ϖ2r2/γ=τθϖ1,ϖ2,γ.
Consequently(90)τθTϖ1,Tϖ2,βτθϖ1,ϖ2,γγ≥τθϖ1,ϖ2,γ.
Since all the conditions of Theorem1 are satisfied, the mapping T has a fixed point. That is, the integral equation (84) has a solution.



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