# Single and Multivalued Maps on Parametric Metric Spaces Endowed with an Equivalence Relation

Authors: Muhammad Usman Ali; Hassen Aydi; Asma Batool; Vahid Parvaneh; Naeem Saleem
Journal: Advances in Mathematical Physics
 (2022)

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## Abstract

This article presents theE-parametric metric space, which is a generalized concept of parametric metric space. After that, the discussion is concerned with the existence of fixed points of single and multivalued maps on E-parametric metric spaces satisfying some contractive inequalities defined by an auxiliary function.


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## Body

## 1. Introduction and Preliminaries

Metric fixed point theory is a very fruitful area of research belonging to nonlinear analysis and operator theory. It has much usage in other fields of mathematics. Numerous generalizations of Banach contraction principle have flourished this field in several aspects through using generalized forms of metric spaces or contraction conditions. It is not wrong to say that the contraction condition given by Salimi et al. in [1] is the most renowned condition of this decade, whereas on the account of metric spaces, we have many abstract forms like b-metric spaces [2–5], partial metric spaces [6, 7], and partial b-metric spaces [8, 9]. Recently, the notions of R-metric [10] and parametric metric spaces [11] have been initiated.In the following, we recall the concept of parametric metric spaces [11] and the convergence of sequences.A mapPm:H×H×0,∞⟶0,∞ (where H≠∅) is said to be a parametric metric on H if it satisfies the following axioms:
(i)
Pmℓ,ℓ′,⋎=0 for all ⋎>0⟺ℓ=ℓ′(ii)
Pmℓ,ℓ′,⋎=Pℓ′,ℓ,⋎ for all ⋎>0(iii)
Pmℓ,ℓ′,⋎≤Pmℓ,ℓ′′,⋎+Pmℓ′′,ℓ′,⋎ for all ℓ,ℓ′,ℓ′′∈H and for all ⋎>0Then,H,Pm is a parametric metric space. If H,Pm be a parametric metric space and ℓn⊆H, then
(1)
ℓn⟶ℓ∈H⟺limn⟶∞Pmℓn,ℓ,⋎=0 for all ⋎>0(2)
ℓnisCauchyiflimm,n⟶∞Pmℓm,ℓn,⋎=0 for all ⋎>0Note thatH,Pm is complete if each Cauchy sequence in H converges in H.One of the ways to generalize the contraction condition is to define it via some auxiliary functions. For example, an auxiliary type function was given in [12], where Qξ represents the collection of functions Q:ℝ+4⟶ℝ+=0,∞ satisfying the below axioms:
(i)
Q is nondecreasing and continuous with respect to its each coordinate(ii)
Ifq1,q2∈ℝ+ with q1<q2 and q1≤Qq2,q2,q1,q2, then q1≤ξq2;(iii)
Ifq1,q2∈ℝ+ with q1≥q2 and q1≤Qq1,q2,q1,q1, then q1=0(iv)
Ifq∈ℝ+ with q≤Q0,0,q,1/2q or q≤Q0,q,0,1/2q or q≤Qq,0,0,q, then q=0and ξ:ℝ+⟶ℝ+ is a nondecreasing map with ∑n=1∞ξnt<∞ for all t≥0 and ξt<t for all t>0.As examples ofQ∈Qξ, we cite the following:
(i)
LetQ1q1,q2,q3,q4=kmaxq1,q2,q3,q4 with ξt=kt, where k∈0,1.(ii)
LetQ2q1,q2,q3,q4=kq4 with ξt=kt, where k∈0,1.(iii)
LetQ3q1,q2,q3,q4=a∗q1+b∗q2+c∗q4 with ξt=a∗+b∗+c∗t, where a∗,b∗,c∗≥0 and a∗+b∗+c∗∈0,1.Khalehoghli et al. [10] took a binary relation E on H and a simple metric d on H to define an R-metric space, denoted as H,d,BR. They also said that a self map T on H is a BR-contraction, if
(1)dTℓ,Tℓ′≤kdℓ,ℓ′,for all ℓ,ℓ′∈H with ℓBRℓ′, where 0<k<1. Along with this definition, the authors also defined the concepts of BR-continuity and BR-preserving property in order to extend the result of Banach on R-metric spaces.


## 2. Main Results

LetH≠∅ and E be an equivalence relation on H. We introduce the following: a function Pm:H×H×0,∞⟶0,∞ is an E-parametric-metric on H, provided that the following axioms hold:
(i)
Pmℓ,ℓ′,⋎=0⟺ℓ=ℓ′forallℓ~ℓ′andforall⋎>0(ii)
Pmℓ,ℓ′,⋎=Pmℓ′,ℓ,⋎forall⋎>0andforallℓ~ℓ′(iii)
Pmℓ,ℓ′,⋎≤Pmℓ,ℓ′′,⋎+Pmℓ′′,ℓ′,⋎forallℓ,ℓ′,ℓ′′∈H,forall⋎>0andforallℓ~ℓ′~ℓ′′Then,H,E,Pm is an E-parametric metric space.Example 1.
TakeH as the collection of all functions g:0,∞⟶R. Consider an equivalence relation E on H defined by gEf if tft−gt for all t∈0,∞ (note that ∣ means that t divides ft−gt). Define Pm:H×H×0,∞⟶0,∞ by
(2)Pmg,f,⋎=g⋎−f⋎,forallg,f∈Handforall⋎>0.Then,H,E,Pm is an E-parametric metric space.Below,H,E,Pm stands for an E-parametric metric space.Definition 1.
Letκm be an E-sequence in H,E,Pm, that is, κm~κm+1 for each m∈ℕ. Then, we say that
(i)κm is a convergent sequence in H and converges to κ∈H if limm⟶∞Pmκm,κ,⋎=0 for all ⋎>0 and κm~κ for each m≥k (for some value of k)
(ii)κm is Cauchy if limm,n⟶∞Pmκm,κn,⋎=0 for all ⋎>0.Definition 2.
H,E,Pm is E-complete if each Cauchy E-sequence in H is convergent in H.Definition 3.
H,E,Pm is called regular whenever if Pmℓ,ℓ′,⋎0=0 for some ⋎0>0, then we have Pmℓ,ℓ′,⋎=0 for all ⋎.We now present the first main theorem of the section.Theorem 4.
LetH,E,Pm be an E-complete regular E-parametric metric space. Let T:H⟶H and α:H×H⟶0,∞ be two maps such that
(3)αℓ,ℓ′PmTℓ,Tℓ′,⋎≤QPmℓ,ℓ′,⋎,Pmℓ,Tℓ,⋎,Pmℓ′,Tℓ′,⋎,Pmℓ′,Tℓ,⋎+Pmℓ,Tℓ′,⋎2,for all ⋎>0 and for all ℓ,ℓ′∈H with ℓ~ℓ′, where Q∈Qξ. Also, consider the following assertions:
(a)
T is E-preserving, that is, ℓ~ℓ′ implies Tℓ~Tℓ′(b)
There isℓ0∈H such that ℓ0~Tℓ0 and αℓ0,Tℓ0≥1(c)
For everyℓ,ℓ′∈H such that ℓ~ℓ′ and αℓ,ℓ′≥1, we have αTℓ,Tℓ′≥1(d)
For eachE-sequence ℓm in H with αℓm,ℓm+1≥1 for all m∈ℕ and ℓm⟶ℓ, we have αℓm,ℓ≥1 for all m∈ℕ
Then,T has a fixed point.Proof.
By (b), there must beκ0∈H of the form κ0~Tκ0 and ακ0,Tκ0≥1. Since T is E-preserving, we have Tκ0~T2κ0, and by (c), αTκ0,T2κ0≥1. The repetition of these steps yields that Tmκ0~Tm+1κ0 and αTmκ0,Tm+1κ0≥1 for all m∈ℕ. Take a sequence κm with the terms as κm=Tmκ0 for all m∈ℕ. Thus, κm~κm+1 and ακm,κm+1≥1 for all m∈W=ℕ∪0. Before doing the next part of the proof, we take κm≠κm+1 for all m∈W. From (3), for each m∈W, we get
(4)PmTκm,Tκm+1,⋎≤ακm,κm+1PmTκm,Tκm+1,⋎≤QPmκm,κm+1,⋎,Pmκm,Tκm,⋎,Pmκm+1,Tκm+1,⋎,12Pmκm+1,Tκm,⋎+Pmκm,Tκm+1,⋎=QPmκm,κm+1,⋎,Pmκm,κm+1,⋎,Pmκm+1,κm+2,⋎,12Pmκm+1,κm+1,⋎+Pmκm,κm+2,⋎≤QPmκm,κm+1,⋎,Pmκm,κm+1,⋎,Pmκm+1,κm+2,⋎,12Pmκm,κm+1,⋎+Pmκm+1,κm+2,⋎,for all ⋎>0.☐Here, our claim is, for each⋎>0, Pmκm,κm+1,⋎m∈W is a strictly decreasing sequence. Suppose this is not true in general, then we have at least one m0∈W such that Pmκm0,κm0+1,⋎0≤Pmκm0+1,κm0+2,⋎0 for some ⋎0>0. Then, from (4) and property (i) of Qξ, we get
(5)Pmκm0+1,κm0+2,⋎0=PmTκm0,Tκm0+1,⋎0≤QPmκm0,κm0+1,⋎0,Pmκm0,κm0+1,⋎0,Pmκm0+1,κm0+2,⋎0,12Pmκm0,κm0+1,⋎0+Pmκm0+1,κm0+2,⋎0≤QPmκm0+1,κm0+2,⋎0,Pmκm0,κm0+1,⋎0,Pmκm0+1,κm0+2,⋎0,Pmκm0+1,κm0+2,⋎0.Property (iii) ofQξ together with the above inequality implies that
(6)Pmκm0+1,κm0+2,⋎0=0,for⋎0>0.This fact and regularity condition give us(7)Pmκm0+1,κm0+2,⋎=0,forall⋎>0,which is not possible under the assumption. Thus, Pmκm,κm+1,⋎>Pmκm+1,κm+2,⋎ for all ⋎>0 and m∈W. Again, we use (4) and property (i) of Qξ to obtain
(8)Pmκm+1,κm+2,⋎=PmTκm,Tκm+1,⋎≤QPmκm,κm+1,⋎,Pmκm,κm+1,⋎,Pmκm+1,κm+2,⋎,Pmκm,κm+1,⋎,for all ⋎>0 and m∈W.Property (ii) ofQξ together with (8) implies
(9)Pmκm+1,κm+2,⋎≤ξPmκm,κm+1,⋎,forall⋎>0andm∈W.After a few simplification, we get(10)Pmκm,κm+1,⋎≤ξmPmκ0,κ1,⋎,forall⋎>0andm∈ℕ.Hence,κm is a sequence in H with κm~κm+1 and Pmκm,κm+1,⋎≤ξmPmκ0,κ1,⋎ for all ⋎>0 and m∈ℕ. Now, we will show that this E-sequence is also Cauchy. For each n>m∈ℕ, we can obtain
(11)Pmκm,κn,⋎≤∑j=mn−1Pmκj,κj+1,⋎≤∑j=m∞ξjPmκ0,κ1,⋎,forall⋎>0.The convergence of∑j=1∞ξjPmκ0,κ1,⋎ for each ⋎>0 implies that κm is a Cauchy E-sequence in H,E,Pm. The E-completeness of H,E,Pm leads to the existence of a point κ∗∈H with limm⟶∞Pmκm,κ∗,⋎=0 for all ⋎>0 and κm~κ∗ for each m≥k, for some k. By (d), we also conclude that ακm,κ∗≥1 for each m≥k. Now, from (3), we get
(12)PmTκm,Tκ∗,⋎≤ακm,κ∗PmTκm,Tκ∗,⋎≤QPmκm,κ∗,⋎,Pmκm,Tκm,⋎,Pmκ∗,Tκ∗,⋎,12Pmκ∗,Tκm,⋎+Pmκm,Tκ∗,⋎=QPmκm,κ∗,⋎,Pmκm,κm+1,⋎,Pmκ∗,Tκ∗,⋎,12Pmκm+1,κ∗,⋎+Pmκm,Tκ∗,⋎,for all ⋎>0.By triangle property and (12), we get
(13)Pmκ∗,Tκ∗,⋎≤Pmκ∗,κm+1,⋎+Pmκm+1,Tκ∗,⋎=Pmκ∗,κm+1,⋎+PmTκm,Tκ∗,⋎≤Pmκ∗,κm+1,⋎+QPmκm,κ∗,⋎,Pmκm,κm+1,⋎,Pmκ∗,Tκ∗,⋎,12Pmκm+1,κ∗,⋎+Pmκm,Tκ∗,⋎≤Pmκ∗,κm+1,⋎+QPmκm,κ∗,⋎,Pmκm,κm+1,⋎,Pmκ∗,Tκ∗,⋎,12Pmκm+1,κ∗,⋎+Pmκm,κ∗,⋎+Pmκ∗,Tκ∗,⋎,for all ⋎>0.Applying the limit whenm⟶∞ in (13), we get
(14)Pmκ∗,Tκ∗,⋎≤Q0,0,Pmκ∗,Tκ∗,⋎,12Pmκ∗,Tκ∗,⋎,for all ⋎>0.Thus, by property (iv) ofQξ, we get Pmκ∗,Tκ∗,⋎=0 for all ⋎>0. Therefore, κ∗=Tκ∗.In the following result, (15) is a special case of (3), but with (15) we do not need the condition of the regularity of the E-parametric metric space.Theorem 5.
LetH,E,Pm be an E-complete E-parametric metric space. Let T:H⟶H and α:H×H⟶0,∞ be two maps such that
(15)αℓ,ℓ′PmTℓ,Tℓ′,⋎≤ξmaxPmℓ,ℓ′,⋎,Pmℓ,Tℓ,⋎,Pmℓ′,Tℓ′,⋎,Pmℓ′,Tℓ,⋎+Pmℓ,Tℓ′,⋎2,for all ⋎>0 and for all ℓ,ℓ′∈H with ℓ~ℓ′, where ξ:ℝ+⟶ℝ+ is a continuous and nondecreasing map with ∑n=1∞ξnt<∞ for all t≥0 and ξt<t for all t>0. Also, consider the following assertions:
(a)
T is E-preserving, that is, ℓ~ℓ′ implies Tℓ~Tℓ′(b)
There isℓ0∈H such that ℓ0~Tℓ0 and αℓ0,Tℓ0≥1(c)
For everyℓ,ℓ′∈H such that ℓ~ℓ′ and αℓ,ℓ′≥1, we have αTℓ,Tℓ′≥1(d)
For eachE-sequence ℓm in H with αℓm,ℓm+1≥1 for all m∈ℕ and ℓm⟶ℓ, we have αℓm,ℓ≥1 for all m∈ℕ
Then,T has a fixed point.Proof.
By (b), we know that there isκ0∈H of the form κ0~Tκ0 and ακ0,Tκ0≥1. In view of the fact that T has E-preserving nature, thus Tκ0~T2κ0 and by (c), αTκ0,T2κ0≥1. Continuing this process, we get Tmκ0~Tm+1κ0 and αTmκ0,Tm+1κ0≥1 for all m∈ℕ. Take a sequence κm with the terms as κm=Tmκ0 for all m∈ℕ. Thus, κm~κm+1 and ακm,κm+1≥1 for all m∈W. Now, we take κm≠κm+1 for all m∈W. From (15), one writes for each m∈W,
(16)Pmκm+1,κm+2,⋎=PmTκm,Tκm+1,⋎≤ακm,κm+1PmTκm,Tκm+1,⋎≤ξmaxPmκm,κm+1,⋎,Pmκm,Tκm,⋎,Pmκm+1,Tκm+1,⋎,12Pmκm+1,Tκm,⋎+Pmκm,Tκm+1,⋎=ξmaxPmκm,κm+1,⋎,Pmκm,κm+1,⋎,Pmκm+1,κm+2,⋎,12Pmκm+1,κm+1,⋎+Pmκm,κm+2,⋎≤ξmaxPmκm,κm+1,⋎,Pmκm+1,κm+2,⋎,for all ⋎>0.Sinceκm≠κm+1 for all m∈W, (16) implies that
(17)Pmκm+1,κm+2,⋎≤ξPmκm,κm+1,⋎,forall⋎>0andm∈W.Therefore,(18)Pmκm,κm+1,⋎≤ξmPmκ0,κ1,⋎,forall⋎>0andm∈ℕ.Now, by following the proof of the above result, we may ensure thatκm is a Cauchy E-sequence in H,E,Pm, and there must exists κ∗∈H with limm⟶∞Pmκm,κ∗,⋎=0 for all ⋎>0 and κm~κ∗ for each m≥k, for some k. Also, using (d), we conclude that ακm,κ∗≥1 for each m≥k. Now, from (15), we get
(19)PmTκm,Tκ∗,⋎≤ακm,κ∗PmTκm,Tκ∗,⋎≤ξmaxPmκm,κ∗,⋎,Pmκm,Tκm,⋎,Pmκ∗,Tκ∗,⋎,12Pmκ∗,Tκm,⋎+Pmκm,Tκ∗,⋎=ξmaxPmκm,κ∗,⋎,Pmκm,κm+1,⋎,Pmκ∗,Tκ∗,⋎,12Pmκm+1,κ∗,⋎+Pmκm,Tκ∗,⋎,for all ⋎>0 and for all m≥k.By triangle property and (19), we get
(20)Pmκ∗,Tκ∗,⋎≤Pmκ∗,κm+1,⋎+Pmκm+1,Tκ∗,⋎=Pmκ∗,κm+1,⋎+PmTκm,Tκ∗,⋎≤Pmκ∗,κm+1,⋎+ξmaxPmκm,κ∗,⋎,Pmκm,κm+1,⋎,Pmκ∗,Tκ∗,⋎,12Pmκm+1,κ∗,⋎+Pmκm,Tκ∗,⋎≤Pmκ∗,κm+1,⋎+ξmaxPmκm,κ∗,⋎,Pmκm,κm+1,⋎,Pmκ∗,Tκ∗,⋎,12Pmκm+1,κ∗,⋎+Pmκm,κ∗,⋎+Pmκ∗,Tκ∗,⋎,for all ⋎>0 and for all m≥k.Applying the limit whenm⟶∞ in (20) and using the continuity of ξ, we get
(21)Pmκ∗,Tκ∗,⋎≤ξmaxPmκ∗,Tκ∗,⋎,12Pmκ∗,Tκ∗,⋎,for all ⋎>0.The above inequality is only possible whenPmκ∗,Tκ∗,⋎=0 for all ⋎>0. Therefore, κ∗=Tκ∗.Example 6.
TakeH=C1,∞ with equivalence relation E on H defined as hg~hf if t∣hgt−hft. Define Pm:H×H×0,∞⟶0,∞ by
(22)Pmhg,hf,⋎=hg⋎−hf⋎,forallhg,hf∈Handforall⋎≥1,0,forallhg,hf∈Handforall⋎∈0,1.Define the mapsT:H⟶H and α:H×H⟶0,∞ by
(23)Thgt=hgt+12,ifhgt≥0,−hgt2,otherwise,αhg,hf=1,ifhg,hf≥0,0,otherwise.Here, one can verify that all the axioms of Theorem5 are valid, and there is a fixed point of T in H.LetH,E,Pm be an E-parametric metric space. Let NH,E denotes the collection of all nonempty subsets of H having the following properties:
(i)
For anyAH∈NH,E, supPmκ,ω,⋎:κ,ω∈AH exists for each ⋎>0(ii)
Ifκn⊂AH and κn⟶κ, then κ∈AHThe HausdroffE-parametric distance HPm is a mapping from NH,E×NH,E×0,∞ into 0,∞ defined by
(24)HPmAH,BH,⋎=maxλAH,BH,⋎,λBH,AH,⋎,for all ⋎>0 where
(25)λAH,BH,⋎=supinfPmκ,ω,⋎:ω∈BH:κ∈AH,λBH,AH,⋎=supinfPmω,κ,⋎:κ∈AH:ω∈BH.Here, everyone should know that for eachAH∈NH,E, ω∈H,E,Pm and for each value of ⋎ there must exists κq∈AH with Pmκq,ω,⋎≤qinfPmκ,ω,⋎:κ∈AH, where q>1. Subsequently, we take infPmω,κ,⋎:κ∈AH=Pmω,AH,⋎.In the following theorem, we will discuss the case of the existence of fixed points for multivalued maps.Theorem 7.
LetH,E,Pm be an E-complete regular E-parametric metric space. Let T:NH,E⟶NH,E and α:H×H⟶0,∞ be two maps such that
(26)αcTℓ,Tℓ′HPmTℓ,Tℓ′,⋎≤QPmℓ,ℓ′,⋎,Pmℓ,Tℓ,⋎,Pmℓ′,Tℓ′,⋎,Pmℓ′,Tℓ,⋎+Pmℓ,Tℓ′,⋎2,for all ⋎>0 and for all ℓ,ℓ′∈H with ℓ~ℓ′, where Q∈Qξ and αcTℓ,Tℓ′=infαj,j′:j∈Tℓ,j′∈Tℓ′. Also, consider the following assertions:
(a)
T is E-preserving, that is, ℓ~ℓ′ implies j~j′ for each j∈Tℓ and j′∈Tℓ′(b)
There isκ0∈H and κ1∈Tκ0 such that κ0~κ1 and ακ0,κ1>1(c)
For everyℓ,ℓ′∈H such that ℓ~ℓ′ and αℓ,ℓ′>1, we have αcTℓ,Tℓ′>1(d)
For eachE-sequence ℓm in H with αℓm,ℓm+1>1 for all m∈ℕ and ℓm⟶ℓ, we have αℓm,ℓ>1 for all m∈ℕ(e)
For eachq>1, Tκ∈NH,E and ω∈H there is κq∈Tκ with Pmκq,ω,⋎≤qPmω,Tκ,⋎ for all ⋎>0, where infPmω,κ∗,⋎:κ∗∈Tκ=Pmω,Tκ,⋎.
Then,T has a fixed point.Proof.
By (b) and (c), we observe thatαcTκ0,Tκ1>1, since κ0∈H and κ1∈Tκ0 are of the form κ0~κ1 and ακ0,κ1>1. Here, we take κ0≠κ1. Otherwise, κ0∈Tκ0. By (26), we get
(27)αcTκ0,Tκ1Pmκ1,Tκ1,⋎≤αcTκ0,Tκ1HPmTκ0,Tκ1,⋎≤QPmκ0,κ1,⋎,Pmκ0,Tκ0,⋎,Pmκ1,Tκ1,⋎,12Pmκ1,Tκ0,⋎+Pmκ0,Tκ1,⋎,for all ⋎>0.SinceαcTκ0,Tκ1>1, by (e), there is κ2∈Tκ1 such that
(28)Pmκ1,κ2,⋎≤αcTκ0,Tκ1Pmκ1,Tκ1,⋎,for all ⋎>0. Here, again, we take κ1≠κ2. Otherwise, κ1∈Tκ1. Thus, by using it in (27), we get
(29)Pmκ1,κ2,⋎≤αcTκ0,Tκ1Pmκ1,Tκ1,⋎≤QPmκ0,κ1,⋎,Pmκ0,Tκ0,⋎,Pmκ1,Tκ1,⋎,12Pmκ1,Tκ0,⋎+Pmκ0,Tκ1,⋎≤QPmκ0,κ1,⋎,Pmκ0,κ1,⋎,Pmκ1,κ2,⋎,12Pmκ0,κ1,⋎+Pmκ1,κ2,⋎,for all ⋎>0.Here, our claim isPmκ0,κ1,⋎>Pmκ1,κ2,⋎∀⋎>0. Suppose this is not true. Then, Pmκ0,κ1,⋎0≤Pmκ1,κ2,⋎0 for some ⋎0>0. Then, from (29) and by property (i) of Qξ, for ⋎0>0, we get
(30)Pmκ1,κ2,⋎0≤QPmκ1,κ2,⋎0,Pmκ0,κ1,⋎0,Pmκ1,κ2,⋎0,Pmκ1,κ2,⋎0.From property (iii) ofQξ and the above inequality, we get
(31)Pmκ1,κ2,⋎0=0,forsome⋎0>0.This fact together with regularity condition implies(32)Pmκ1,κ2,⋎=0,forall⋎>0,and it is not possible under the assumption. Thus, Pmκ0,κ1,⋎>Pmκ1,κ2,⋎ for all ⋎>0. Again, we use (29) and property (i) of Qξ to obtain
(33)Pmκ1,κ2,⋎≤QPmκ0,κ1,⋎,Pmκ0,κ1,⋎,Pmκ1,κ2,⋎,Pmκ0,κ1,⋎,forall⋎>0.The property (ii) ofQξ and (33) imply
(34)Pmκ1,κ2,⋎≤ξPmκ0,κ1,⋎,forall⋎>0.Clearly,ακ1,κ2>1 due to the fact that κ1∈Tκ0, κ2∈Tκ1 and αcTκ0,Tκ1>1. Also, the E-preserving characteristic of T yields κ1~κ2. Thus, by (c), we get αcTκ1,Tκ2>1. The repeated application of the above steps yields the sequence κm with κm∈Tκm−1, κm−1~κm, ακm−1,κm>1 and Pmκm−1,κm,⋎>Pmκm,κm+1,⋎ for all ⋎>0 and
(35)Pmκm,κm+1,⋎≤ξmPmκ0,κ1,⋎,for all ⋎>0 and for all m∈ℕ.The steps given below will show thatκm is a Cauchy E-sequence. For each n>m∈ℕ, the triangle inequality and (35) imply that
(36)Pmκm,κn,⋎≤∑j=mn−1Pmκj,κj+1,⋎≤∑j=m∞ξjPmκ0,κ1,⋎,for all ⋎>0.The convergence of∑j=1∞ξjPmκ0,κ1,⋎ together with the above inequality confirm that κm is a Cauchy E-sequence in H,E,Pm. The E-completeness of H,E,Pm ensures the existence of a point κ∗∈H with limm⟶∞Pmκm,κ∗,⋎=0 for all ⋎>0 and κm~κ∗ for each m≥k, for some k. We now able to conclude that, by (d), ακm,κ∗>1 for each m∈ℕ. Since κm~κ∗ and ακm,κ∗>1 for each m≥k, using (c), we have αTκm,Tκ∗>1. Thus, by (26), we get
(37)HPmTκm,Tκ∗,⋎<αTκm,Tκ∗HPmTκm,Tκ∗,⋎≤QPmκm,κ∗,⋎,Pmκm,Tκm,⋎,Pmκ∗,Tκ∗,⋎,12Pmκ∗,Tκm,⋎+Pmκm,Tκ∗,⋎=QPmκm,κ∗,⋎,Pmκm,κm+1,⋎,Pmκ∗,Tκ∗,⋎,12Pmκm+1,κ∗,⋎+Pmκm,Tκ∗,⋎,for all ⋎>0.By triangle property and (37), we get
(38)Pmκ∗,Tκ∗,⋎≤Pmκ∗,κm+1,⋎+Pmκm+1,Tκ∗,⋎≤Pmκ∗,κm+1,⋎+HPmTκm,Tκ∗,⋎<Pmκ∗,κm+1,⋎+QPmκm,κ∗,⋎,Pmκm,κm+1,⋎,Pmκ∗,Tκ∗,⋎,12Pmκm+1,κ∗,⋎+Pmκm,Tκ∗,⋎≤Pmκ∗,κm+1,⋎+QPmκm,κ∗,⋎,Pmκm,κm+1,⋎,Pmκ∗,Tκ∗,⋎,12Pmκm+1,κ∗,⋎+Pmκm,κ∗,⋎+Pmκ∗,Tκ∗,⋎,for all ⋎>0.Applying the limit whenm⟶∞ in (38), we get
(39)Pmκ∗,Tκ∗,⋎≤Q0,0,Pmκ∗,Tκ∗,⋎,12Pmκ∗,Tκ∗,⋎,forall⋎>0.Thus, by property (iv) ofQξ, we get Pmκ∗,Tκ∗,⋎=0 for all ⋎>0. Therefore, κ∗∈Tκ∗.In the following theorem, (40) is a special case of (26), but with (40), we do not need the condition of regularity of the E-parametric metric space.Theorem 8.
LetH,E,Pm be an E-complete E-parametric metric space. Let T:NH,E⟶NH,E and α:H×H⟶0,∞ be two maps such that
(40)αcTℓ,Tℓ′HPmTℓ,Tℓ′,⋎≤ξmaxPmℓ,ℓ′,⋎,Pmℓ,Tℓ,⋎,Pmℓ′,Tℓ′,⋎,Pmℓ′,Tℓ,⋎+Pmℓ,Tℓ′,⋎2,for all ⋎>0 and for all ℓ,ℓ′∈H with ℓ~ℓ′, where ξ:ℝ+⟶ℝ+ is a continuous and nondecreasing mapping with ∑n=1∞ξnt<∞ for all t≥0 and ξt<t for all t>0, and αcTℓ,Tℓ′=infαj,j′:j∈Tℓ,j′∈Tℓ′. Also, consider the following assertions:
(a)
T is E-preserving, that is, ℓ~ℓ′ implies j~j′ for each j∈Tℓ and j′∈Tℓ′(b)
There isκ0∈H and κ1∈Tκ0 such that κ0~κ1 and ακ0,κ1>1(c)
For everyℓ,ℓ′∈H such that ℓ~ℓ′ and αℓ,ℓ′>1, we have αcTℓ,Tℓ′>1(d)
For eachE-sequence ℓm in H with αℓm,ℓm+1>1 for all m∈ℕ and ℓm⟶ℓ, we have αℓm,ℓ>1(e)
For eachq>1, Tκ∈NH,E and ω∈H there exists κq∈Tκ with Pmκq,ω,⋎≤qPmω,Tκ,⋎ for all ⋎>0, where infPmω,κ∗,⋎:κ∗∈Tκ=Pmω,Tκ,⋎.
Then,T has a fixed point.The proof of this theorem will be done by following the techniques of Theorems5 and 6.


## 3. Application to Existence of Solutions of Integral Equations

LetX=C0,T,ℝ be the set of real continuous functions defined on 0,T and Pm:X×X×0,∞⟶0,+∞ be defined by
(41)Pmx,y,α=supt∈0,T11+αt2xt−yt,for all x,y∈X and all t∈0,T. Then, X,Pm is a complete parametric metric space. Let ~ be the equivalence relation on X defined by x~y if and only if t∣xt−yt. Then, X,Pm,~ is a complete E-parametric metric space. Consider the following integral equation:
(42)xt=pt+∫0TSt,sfs,xsds,where
(A)
f:0,T×ℝ⟶ℝ is continuous(B)
p:0,T⟶ℝ is continuous(C)
S:0,T×0,T⟶0,+∞ is continuous and(43)supt∈0,T∫0TSt,sds≤1,(D)
There existsk∈0,1 such that(44)0≤fs,ys−fs,xs≤k1+αs2maxxs−ys,xs−Hxs,ys−Hys,xs−Hys+ys−Hxs2,for all x,y∈X with x~y and s∈0,T where
(45)Hxt=pt+∫0TSt,sfs,xsds,t∈0,T,forallx∈X.(E)
There existsx0∈X such that(46)x0t≤pt+∫0TSt,sfs,x0sds,tx0t−pt+∫0TSt,sfs,x0sds.(F)
s∣∫0TSt,sfs,ys−fs,xsds for all x,y∈X with x~y and s∈0,TWe have the following result on the existence of solutions for integral equations.Theorem 9.
Under assumptionsA−F, the integral Equation (42) has a unique solution in X=C0,T,R.Proof.
LetH:X⟶X be defined by
(47)Hxt=pt+∫0TSt,sfs,xsds,t∈0,T,forallx∈X.First, we will prove thatH is E-preserving. Let x~y. Then, we have s∣∫0TSt,sfs,ysds−∫0TSt,sfs,xsds for all s∈0,T. On the other hand, by the definition of H, we have
(48)Hy−Hx=∫0TSt,sfs,ys−fs,xsds,forallt∈0,T.So,Hx~Hy.Now, suppose thatx,y∈X with x~y. Then, by C, D and the definition of H, we get
(49)PmHx,Hy,α=supt∈0,T11+αt2∣Hxt−Hyt∣≤supt∈0,T11+αt2∫0TSt,s∣fs,xs−fs,ys∣ds≤k11+αt2maxxs−ys,xs−Hxs,ys−Hys,xs−Hys+ys−Hxs2≤k11+αt2maxsups∈0,Txs−ys,sups∈0,Txs−Hxs,sups∈0,Tys−Hys,sups∈0,Txs−Hys+sups∈0,Tys−Hxs2≤kmaxPmx,y,α,Pmx,Hx,α,Pmy,Hy,α,Pmx,Hy,α+Pmy,Hx,α2.Now, byE, there exists x0∈X such that x0~Hx0. Then, the conditions of Theorem 4 are satisfied and hence the integral Equation (42) has a unique solution in X=C0,T,ℝ. Note that α≡1 when ℓ~ℓ′ and α≡0 when ℓ≁ℓ′.


## 4. Conclusion

This article presents fixed point theorems for mappings satisfying the contraction type conditions defined with auxiliary functions overE-parametric metric spaces. The readers will get many fixed point results as the outcome of this work. These results have been obtained in the following ways:
(1)
Defining an auxiliary function through a particular form(2)
Equipping theE-parametric metric space H,E,Pm with a directed graph and defining αx,y=1, whenever x,y is an edge in a directed graph defined on H and αx,y=0 otherwise. Similarly, we may consider a partial order on H instead of a directed graph(3)
Defining an equivalence relation onH by edge/path of a graph, a partial order, a preorder, etc.(4)
TakingPmℓ,ℓ′,⋎=dℓ,ℓ′ for all ⋎, where d is used for a metric on H



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*Source: 6188108-2022-01-28.xml*