# On Pentagonal Controlled Fuzzy Metric Spaces with an Application to Dynamic Market Equilibrium

Authors: Aftab Hussain; Umar Ishtiaq; Khaleel Ahmed; Hamed Al-Sulami
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this manuscript, we coined pentagonal controlled fuzzy metric spaces and fuzzy controlled hexagonal metric space as generalizations of fuzzy triple controlled metric spaces and fuzzy extended hexagonal b-metric spaces. We use a control function in fuzzy controlled hexagonal metric space and introduce five noncomparable control functions in pentagonal controlled fuzzy metric spaces. In the scenario of pentagonal controlled fuzzy metric spaces, we prove the Banach fixed point theorem, which generalizes the Banach fixed point theorem for the aforementioned spaces. An example is offered to support our main point. We also presented an application to dynamic market equilibrium.


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## Body

## 1. Introduction

Fuzzy notions are used to describe the degrees of possession of a certain property. The ability of fuzzy set (FS) theory to address circumstances that fixed point theory finds problematic originates from its attractiveness in tackling control problems. FSs are used to govern ill-defined, convoluted, and nonlinear systems The fabulous idea of FSs was presented by Zadeh [1] in his research paper. A FS extends the concept of a crisp set by associating all elements with membership values in the range of [0,1]. The FS theory has been widely employed in mathematics since then. Schweizer and Sklar [2] presented continuous t-norms (CTNs). Itoh [3] proved fixed point theorems with an application to random differential equations in Banach spaces. Kramosil and Michálek [4] presented the fuzzy metric space (FMS) approach. George and Veeramani [5] modify the notion of FMS and presented the Hausdorff topology in FMS. Grabiec [6] proved the Banach contraction theorem in fuzzy version, and also, he proved the Edelstein theorem in FMS. Han [7] proved the Banach fixed point theorem from the view point of digital topology. Uddin et al. [8] gave a solution of the Fredholm integral inclusions via Suzuki-type fuzzy contractions. Kamran et al. [9] presented the approach of extended metric space and proved several fixed point results for contraction mappings. Mehmood et al. [10] presented fuzzy rectangular b-metric spaces and proved fixed point theorems. Saleem et al. [11] coined the notion of fuzzy double controlled metric spaces and proved several fixed point results. Badshah-e-Rome and Sarwar [12] presented the approach of extended fuzzy rectangular b-metric spaces and proved fixed point results for contraction mappings via α-admissibility. Furqan et al. [13] presented the notion of fuzzy triple controlled metric spaces (FTCMSs) as a generalization of various spaces. Zubair et al. [14] presented fuzzy extended hexagonal b-metric spaces (FEHBMSs) and proved several fixed point results.In this manuscript, we generalized the ideas of FTCMSs and FEHBMSs and present the approaches of pentagonal controlled fuzzy metric spaces (PCFMSs) and fuzzy controlled hexagonal metric spaces (FCHMSs). We extend the Banach contraction principle in the setting of FCHMSs. At the end, an application to dynamic market equilibrium is given to validate the main result.


## 2. Preliminaries

This section contains some important definitions that aid comprehension of the main section.Definition 1 (see [2]).
A binary operation∗:0,1×0,1⟶0,1 is a CTN if
(1)
τ∗θ=θ∗τ,∀τ,θ∈0,1(2)
∗ is continuous(3)
τ∗1=τ,∀τ∈0,1(4)
τ∗θ∗ρ=τ∗θ∗ρ,∀τ,θ,ρ∈0,1(5)
Ifτ≤ρ and θ≤σ, with τ,θ,ρ,σ∈0,1, then τ∗θ≤ρ∗σDefinition 2 (see [13]).
LetX be a nonempty set. A 3-tuple X,N,∗ is named a FTCMS if ∗ is a CTN, N is a FS on X×X×0,∞, and Q,W,E:X×X⟶1,∞ are noncomparable and fulfill the following assertions for all z,d,e,f∈X,z≠e,e≠f,f≠d, and α,β,γ≥0; the following circumstances are fulfilled:
(S1)
Nz,d,0=0(S2)
Nz,d,α=1 implies z=d(S3)
Nz,d,α=Md,z,α(S4)
Nz,d,α+β+γ≥Nz,e,α/Qz,e∗Ne,f,β/We,f∗Nf,g,γ/Ef,d(S5)
Nz,d,.:0,∞⟶0,1 is left continuous and limα⟶∞Nz,d,α=1Definition 3 (see [14]).
LetX be a nonempty set. A 4-tuple X,L,∗ is a FEHBMS, if ∗ is a CTN, L is a FS on X×X×0,∞, and Q:X×X⟶1,∞ fulfills the following assertions for all z,d,e,f,g,k∈X,z≠e,e≠f,f≠g,g≠k,k≠d, and α,β,γ,δ,w≥0; the following circumstances are fulfilled:
(F1)
Lz,d,0=0(F2)
Lz,d,α=1 implies z=d(F3)
Lz,d,α=Ld,z,α(F4)
Lz,d,Qz,dα+β+γ+δ+w≥Lz,e,α∗Le,f,β∗Lf,g,γ∗Lg,k,δ∗Lk,d,w(F5)
Lz,d,.:0,∞⟶0,1 is left continuous


## 3. Main Results

The definitions of FCHMS and PCFMS are presented in this section, as well as proofs of fixed point findings.Definition 4.
LetX be a nonempty set. A 4-tuple X,H,∗,Q is a FCHMS if ∗ is a CTN, H is a FS on X×X×0,∞, and Q:X×X⟶1,∞ fulfills the following assertions for all z,d,e,f,g,k∈X,z≠e,e≠f,f≠g,g≠k,k≠d, and α,β,γ,δ,w≥0; the following circumstances are fulfilled:
(T1)
Hz,d,0=0(T2)
Hz,d,α=1 implies z=d(T3)
Hz,d,α=Md,z,α(T4)
Hz,d,α+β+γ+δ+w≥Hz,e,α/Qz,e∗He,f,β/Qe,f∗Hf,g,γ/Qf,g∗Hg,k,δ/Qg,k∗Hk,d,w/Qk,d(T5)
Hz,d,.:0,∞⟶0,1 is left continuous and limα⟶∞Hz,d,α=1Example 5.
LetX=1,2,3,4,5,6. Define H:X×X×0,∞⟶0,1 as
(1)Hz,d,α=ez−d2/α−1forallα>0,with the CTN ∗ such that α1∗α2=α1α2. Then, X,M,∗ is FCHMS with control functions Qz,d=1+z+d.Definition 6.
LetX be a nonempty set. A triplet X,M,∗ is a PCFMS if ∗ is a CTN, M is a fuzzy set on X×X×0,∞, and Q,W,E,R,T:X×X⟶1,∞ are five noncomparable functions that fulfill the following assertions for all z,d,e,f,g,k∈X,z≠e,e≠f,f≠g,g≠k,k≠d, and α,β,γ,δ,w≥0; the following circumstances are fulfilled:
(A1)
Mz,d,0=0(A2)
Mz,d,α=1 implies z=d(A3)
Mz,d,α=Md,z,α(A4)
Mz,d,α+β+γ+δ+w≥Mz,e,α/Qz,e∗Me,f,β/We,f∗Mf,g,γ/Ef,g∗Mg,k,δ/Rg,k∗Mk,d,w/Tk,d(A5)
Mz,d,.:0,∞⟶0,1 is left continuous and limα⟶∞Mz,d,α=1Example 7.
LetX=1,2,3,4,5,6. Define M:X×X×0,∞⟶0,1 as
(2)Mz,d,α=αα+z−d6forallα>0,with the CTN ∗ such that α1∗α2=α1α2. Then, X,M,∗ is a PCFMS with noncomparable control functions Qz,d=1+z+d,(3)Wz,d=1+z2+d2,Ez,d=1+zd,Rz,d=1+dz,Tz,d=1+z2+d.Remark 8.
From the definition of PCFMS,(i)
If we takeWe,f=Qe,f,Ef,g=Qf,g,Rg,k=Qg,k,Tk,d=Qk,d, then it will becomes the definition of FCHMS(ii)
If we takeQz,e=We,f=Ef,g=Rg,k=Tk,d=bz,d, then it will becomes the definition of FEHBMS in [14](iii)
Ifg=k=d and γ=δ=w=r′in (A4), then it will becomes FTCMS in [13]Definition 9.
LetX,M,∗ be a PCFMS and zn be a sequence in X, thenzn is named to be
(i)
a convergent, if there existsz∈X such that
(4)limn⟶∞Mzn,z,α=1forallα>0(ii)
a Cauchy, if and only if for eachω>0,α>0, there exists n0∈ℕ such that
(5)Mzn,zm,α≥1−ω,foralln,m≥n0
If every Cauchy sequence is convergent inX, then X,M,∗ is a complete PCFMS.Definition 10.
LetX,M,∗ be a PCFMS, then we define an open ball Bz,r,α with centre z, radius r,0<r<1, and α>0 as follows:
(6)Bz,r,α=d∈X:Mz,d,α>1−r,and the topology that corresponds to it is defined as
(7)τpz=D⊂X:Bz,r,α⊂D.Theorem 11.
LetX,M,∗ be a complete PCFMS and Q,W,E,R,T:X×X⟶1,∞ such that
(8)limα⟶∞Mz,d,α=1,forallα>0,z,d∈X.
LetF:X⟶X be a mapping satisfying
(9)MFz,Fd,qα≥Mz,d,α,forallα>0,z,d∈X,where 0<p<1. Furthermore, if, for z0∈Xandn,q∈1,2,3,⋯, it holds bzn,zn+q<1/p where zn=Fnz0, then F has a unique fixed point.Proof.
Assumez0∈X and construct a sequence zn by
(10)zn=Fzn−1foralln∈1,2,3,⋯.
Without restricting generality, suppose thatzn≠zn+1foralln∈0,1,2,3,⋯. With the help of (9), we deduce
(11)Mzn,zn+1,α=MFzn−1,Fzn,α≥Mzn−1,zn,αp⋮≥Mz0,z1,αpn.
Continuing in this way, we obtain(12)Mzn,zn+2,α≥Mz0,z2,αpn,(13)Mzn,zn+3,α≥Mz0,z3,αpn,(14)Mzn,zn+4,α≥Mz0,z4,αpn.
It implies, ifm=1,2,3,⋯,(15)Mzn,zn+4m+1,α≥Mz0,z4m+1,αpn,(16)Mzn,zn+4m+2,α≥Mz0,z4m+2,αpn,(17)Mzn,zn+4m+3,α≥Mz0,z4m+3,αpn,(18)Mzn,zn+4m+4,α≥Mz0,z4m+4,αpn.
Expressingα=α/5+α/5+α/5+α/5+α/5 and by using (11) and (A1), we obtain
(19)Mz0,z5,α≥Mz0,z1,α5Qz0,z1∗Mz1,z2,α5Wz1,z2∗Mz2,z3,α5Ez2,z3∗Mz3,z4,α5Rbz3,z4∗Mz4,z5,α5Tbz4,z5≥Mz0,z1,α5Qz0,z1∗Mz0,z1,α5pWz1,z2∗Mz0,z1,α5p2Ez2,z3∗Mz0,z1,α5p3Rz3,z4∗Mz0,z1,α5p4Tz4,z5.
In similar manner, we can deduce(20)Mz0,z9,α≥Mz0,z1,α5Qz0,z1∗Mz1,z2,α5Wz1,z2∗Mz2,z3,α5Ez2,z3∗Mz3,z4,α5Rz3,z4∗Mz4,z9,α5Tz4,z9≥Mz0,z1,α5Qz0,z1∗Mz1,z2,α5Wz1,z2∗Mz2,z3,α5Ez2,z3∗Mz3,z4,α5Rz3,z4∗Mz4,z5,α52Tz4,z9Qz4,z5∗Mz5,z6,α52Tz4,z9Wz5,z6∗Mz6,z7,α52Tz4,z9Ez6,z7∗Mz7,z8,α52Tz4,z9Rz7,z8∗Mz8,z9,α52Tz4,z9Tz8,z9≥Mz0,z1,α5Qz0,z1∗Mz0,z1,α5pWz1,z2∗Mz0,z1,α5p2Ez2,z3∗Mz0,z1,α5p3Rz3,z4∗Mz0,z1,α52p4Tz4,z9Qz4,z5∗Mz0,z1,α52p5Tz4,z9Wz5,z6∗Mz0,z1,α52p6Tz4,z9Ez6,z7∗Mz0,z1,α52p7Tz4,z9Rz7,z8∗Mz0,z1,α52p8Tz4,z9Tz8,z9.
We obtain for eachm=1,2,3,⋯,(21)Mz0,z4m+1,α≥Mz0,z1,α5Qz0,z1∗Mz1,z2,α5Wz1,z2∗Mz2,z3,α5Ez2,z3∗Mz3,z4,α5Rz3,z4∗Mz4,z4m+1,α5Tz4,z4m+1≥Mz0,z1,α5Qz0,z1∗Mz1,z2,α5Wz1,z2∗Mz2,z3,α5Ez2,z3∗Mz3,z4,α5Rz3,z4∗Mz4,z5,α52Tz4,z4m+1Qz4,z5∗Mz5,z6,α52Tz4,z4m+1Wz5,z6∗Mz6,z7,α52Tz4,z4m+1Ez6,z7∗Mz7,z8,α52Tz4,z4m+1Rz7,z8∗Mz8,z4m+1,α52Tz4,z4m+1Tz8,z4m+1≥Mz0,z1,α5Qz0,z1∗Mz1,z2,α5Wz1,z2∗Mz2,z3,α5Ez2,z3∗Mz3,z4,α5Rz3,z4∗Mz4,z5,α52Tz4,z4m+1Qz4,z5∗Mz5,z6,α52Tz4,z4m+1Wz5,z6∗Mz6,z7,α52Tz4,z4m+1Ez6,z7∗Mz7,z8,α52Tz4,z4m+1Rz7,z8∗Mz8,z9,α53Tz4,z4m+1Tz8,z4m+1Qz8,z9∗Mz9,z10,α53Tz4,z4m+1Tz8,z4m+1Wz9,z10∗Mz10,z11,α53Tz4,z4m+1Tz8,z4m+1Ez10,z11∗Mz11,z12,α53Tz4,z4m+1Tz8,z4m+1Rz11,z12∗Mz12,z4m+1,α53Tz4,z4m+1Tz8,z4m+1Tz12,z4m+1≥Mz0,z1,α5Qz0,z1∗Mz1,z2,α5Wz1,z2∗Mz2,z3,α5Ez2,z3∗Mz3,z4,α5Rz3,z4∗Mz4,z5,α52Tz4,z4m+1Qz4,z5∗Mz5,z6,α52Tz4,z4m+1Wz5,z6∗Mz6,z7,α52Tz4,z4m+1Ez6,z7∗Mz7,z8,α52Tz4,z4m+1Rz7,z8∗Mz8,z9,α53Tz4,z4m+1Tz8,z4m+1Qz8,z9∗Mz9,z10,α53Tz4,z4m+1Tz8,z4m+1Wz9,z10∗Mz10,z11,α53Tz4,z4m+1Tz8,z4m+1Ez10,z11∗Mz11,z12,α53Tz4,z4m+1Tz8,z4m+1Rz11,z12∗Mz12,z13,α54Tz4,z4m+1Tz8,z4m+1Tz12,z4m+1Qz12,z13∗⋮∗Mz4m,z4m+1,α5mTz4,z4m+1Tz8,z4m+1Tz12,z4m+1⋯Tz4m−4,z4m+1Tz4m,z4m+1≥Mz0,z1,α5Qz0,z1∗Mz0,z1,α5pWz1,z2∗Mz0,z1,α5p2Ez2,z3∗Mz0,z1,α5p3Rz3,z4∗Mz0,z1,α52p4Tz4,z4m+1Qz4,z5∗Mz0,z1,α52p5Tz4,z4m+1Wz5,z6∗Mz0,z1,α52p6Tz4,z4m+1Ez6,z7∗Mz0,z1,α52p7Tz4,z4m+1Rz7,z8∗Mz0,z1,α53p8Tz4,z4m+1Tz8,z4m+1Qz8,z9∗Mz0,z1,α53p9Tz4,z4m+1Tz8,z4m+1Wz9,z10∗Mz0,z1,α53p10Tz4,z4m+1Tz8,z4m+1Ez10,z11∗Mz0,z1,α53p11Tz4,z4m+1Tz8,z4m+1Rz11,z12∗Mz0,z1,α54p12Tz4,z4m+1Tz8,z4m+1Tz12,z4m+1Qz12,z13∗⋮∗Mz0,z1,α5mp4mTz4,z4m+1Tz8,z4m+1Tz12,z4m+1⋯Tz4m−4,z4m+1Tz4m,z4m+1.
Now, using (15), we deduce that
(22)Mzn,zn+4m+1,α≥Mz0,z4m+1,αpn≥Mz0,z1,α5pnQz0,z1∗Mz0,z1,α5pn+1Wz1,z2∗Mz0,z1,α5pn+2Ez2,z3∗Mz0,z1,α5pn+3Rz3,z4∗Mz0,z1,α52pn+4Tz4,z4m+1Qz4,z5∗Mz0,z1,α52pn+5Tz4,z4m+1Wz5,z6∗Mz0,z1,α52pn+6Tz4,z4m+1Ez6,z7∗Mz0,z1,α52pn+7Tz4,z4m+1Rz7,z8∗Mz0,z1,α53pn+8Tz4,z4m+1Tz8,z4m+1Qz8,z9∗Mz0,z1,α53pn+9Tz4,z4m+1Tz8,z4m+1Wz9,z10∗Mz0,z1,α53pn+10Tz4,z4m+1Tz8,z4m+1Ez10,z11∗Mz0,z1,α53pn+11Tz4,z4m+1Tz8,z4m+1Rz11,z12∗Mz0,z1,α54pn+12Tz4,z4m+1Tz8,z4m+1Tz12,z4m+1Qz12,z13∗⋮∗Mz0,z1,α5mpn+4mTz4,z4m+1Tz8,z4m+1Tz12,z4m+1⋯Tz4m−4,z4m+1Tz4m,z4m+1.
Furthermore, from (11) and (12), we can obtain
(23)Mz0,z6,α≥Mz0,z1,α5Qz0,z1∗Mz1,z2,α5Wz1,z2∗Mz2,z3,α5Ez2,z3∗Mz3,z4,α5Rz3,z4∗Mz4,z6,α5Tz4,z6≥Mz0,z1,α5Qz0,z1∗Mz0,z1,α5pWz1,z2∗Mz0,z1,α5p2Ez2,z3∗Mz0,z1,α5p3Rz3,z4∗Mz0,z2,α5p4Tz4,z6.
In similar manner, we can deduce(24)Mz0,z10,α≥Mz0,z1,α5Qz0,z1∗Mz1,z2,α5Wz1,z2∗Mz2,z3,α5Ez2,z3∗Mz3,z4,α5Rz3,z4∗Mz4,z10,α5Tz4,z10≥Mz0,z1,α5Qz0,z1∗Mz1,z2,α5Wz1,z2∗Mz2,z3,α5Ez2,z3∗Mz3,z4,α5Rz3,z4∗Mz4,z5,α52Tz4,z10Qz4,z5∗Mz5,z6,α52Tz4,z10Wz5,z6∗Mz6,z7,α52Tz4,z10Ez6,z7∗Mz7,z8,α52Tz4,z10Rz7,z8∗Mz8,z10,α52Tz4,z9Tz8,z10≥Mz0,z1,α5Qz0,z1∗Mz0,z1,α5pWz1,z2∗Mz0,z1,α5p2Ez2,z3∗Mz0,z1,α5p3Rz3,z4∗Mz0,z1,α52p4Tz4,z9Qz4,z5∗Mz0,z1,α52p5Tz4,z9Wz5,z6∗Mz0,z1,α52p6Tz4,z9Ez6,z7∗Mz0,z1,α52p7Tz4,z9Rz7,z8∗Mz0,z2,α52p8Tz4,z9Tz8,z10.
We obtain for eachm=1,2,3,⋯,(25)Mz0,z4m+2,α≥Mz0,z1,α5Qz0,z1∗Mz1,z2,α5Wz1,z2∗Mz2,z3,α5Ez2,z3∗Mz3,z4,α5Rz3,z4∗Mz4,z4m+2,α5Tz4,z4m+2≥Mz0,z1,α5Qz0,z1∗Mz1,z2,α5Wz1,z2∗Mz2,z3,α5Ez2,z3∗Mz3,z4,α5Rz3,z4∗Mz4,z5,α52Tz4,z4m+2Qz4,z5∗Mz5,z6,α52Tz4,z4m+2Wz5,z6∗Mz6,z7,α52Tz4,z4m+2Ez6,z7∗Mz7,z8,α52Tz4,z4m+2Rz7,z8∗Mz8,z4m+2,α52Tz4,z4m+2Tz8,z4m+2≥Mz0,z1,α5Qz0,z1∗Mz1,z2,α5Wz1,z2∗Mz2,z3,α5Ez2,z3∗Mz3,z4,α5Rz3,z4∗Mz4,z5,α52Tz4,z4m+2Qz4,z5∗Mz5,z6,α52Tz4,z4m+2Wz5,z6∗Mz6,z7,α52Tz4,z4m+2Ez6,z7∗Mz7,z8,α52Tz4,z4m+2Rz7,z8∗Mz8,z9,α53Tz4,z4m+2Tz8,z4m+2Qz8,z9∗Mz9,z10,α53Tz4,z4m+2Tz8,z4m+2Wz9,z10∗Mz10,z11,α53Tz4,z4m+2Tz8,z4m+2Ez10,z11∗Mz11,z12,α53Tz4,z4m+2Tz8,z4m+2Rz11,z12∗Mz12,z4m+2,α53Tz4,z4m+2Tz8,z4m+2Tz12,z4m+2≥Mz0,z1,α5Qz0,z1∗Mz1,z2,α5Wz1,z2∗Mz2,z3,α5Ez2,z3∗Mz3,z4,α5Rz3,z4∗Mz4,z5,α52Tz4,z4m+2Qz4,z5∗Mz5,z6,α52Tz4,z4m+2Wz5,z6∗Mz6,z7,α52Tz4,z4m+2Ez6,z7∗Mz7,z8,α52Tz4,z4m+2Rz7,z8∗Mz8,z9,α53Tz4,z4m+2Tz8,z4m+2Qz8,z9∗Mz9,z10,α53Tz4,z4m+2Tz8,z4m+2Wz9,z10∗Mz10,z11,α53Tz4,z4m+2Tz8,z4m+2Ez10,z11∗Mz11,z12,α53Tz4,z4m+2Tz8,z4m+2Rz11,z12∗Mz12,z13,α54Tz4,z4m+2Tz8,z4m+2Tz12,z4m+2Qz12,z13∗⋮∗Mz4m,z4m+2,α5mTz4,z4m+2Tz8,z4m+2Tz12,z4m+2⋯Tz4m−4,z4m+2Tz4m,z4m+2≥Mz0,z1,α5Qz0,z1∗Mz0,z1,α5pWz1,z2∗Mz0,z1,α5p2Ez2,z3∗Mz0,z1,α5p3Rz3,z4∗Mz0,z1,α52p4Tz4,z4m+2Qz4,z5∗Mz0,z1,α52p5Tz4,z4m+2Wz5,z6∗Mz0,z1,α52p6Tz4,z4m+2Ez6,z7∗Mz0,z1,α52p7Tz4,z4m+2Rz7,z8∗Mz0,z1,α53p8Tz4,z4m+2Tz8,z4m+2Qz8,z9∗Mz0,z1,α53p9Tz4,z4m+2Tz8,z4m+2Wz9,z10∗Mz0,z1,α53p10Tz4,z4m+2Tz8,z4m+2Ez10,z11∗Mz0,z1,α53p11Tz4,z4m+2Tz8,z4m+2Rz11,z12∗Mz0,z1,α54p12Tz4,z4m+2Tz8,z4m+2Tz12,z4m+2Qz12,z13∗⋮∗Mz0,z2,α5mp4mTz4,z4m+2Tz8,z4m+2Tz12,z4m+2⋯Tz4m−4,z4m+2Tz4m,z4m+2.
Now, using (16), we deduce that
(26)Mzn,zn+4m+2,α≥Mz0,z4m+2,αpn≥Mz0,z1,α5pnQz0,z1∗Mz0,z1,α5pn+1Wz1,z2∗Mz0,z1,α5pn+2Ez2,z3∗Mz0,z1,α5pn+3Rz3,z4∗Mz0,z1,α52pn+4Tz4,z4m+2Qz4,z5∗Mz0,z1,α52pn+5Tz4,z4m+2Wz5,z6∗Mz0,z1,α52pn+6Tz4,z4m+2Ez6,z7∗Mz0,z1,α52pn+7Tz4,z4m+2Rz7,z8∗Mz0,z1,α53pn+8Tz4,z4m+2Tz8,z4m+2Qz8,z9∗Mz0,z1,α53pn+9Tz4,z4m+2Tz8,z4m+2Wz9,z10∗Mz0,z1,α53pn+10Tz4,z4m+2Tz8,z4m+2Ez10,z11∗Mz0,z1,α53pn+11Tz4,z4m+2Tz8,z4m+2Rz11,z12∗Mz0,z1,α54pn+12Tz4,z4m+2Tz8,z4m+2Tz12,z4m+2Qz12,z13∗⋮∗Mz0,z2,α5mpn+4mTz4,z4m+2Tz8,z4m+2Tz12,z4m+2⋯Tz4m−4,z4m+2Tz4m,z4m+2.
Accordingly, from (27) and (28) can be deduced as
(27)Mzn,zn+4m+3,α≥Mz0,z4m+3,αpn≥Mz0,z1,α5pnQz0,z1∗Mz0,z1,α5pn+1Wz1,z2∗Mz0,z1,α5pn+2Ez2,z3∗Mz0,z1,α5pn+3Rz3,z4∗Mz0,z1,α52pn+4Tz4,z4m+3Qz4,z5∗Mz0,z1,α52pn+5Tz4,z4m+3Wz5,z6∗Mz0,z1,α52pn+6Tz4,z4m+3Ez6,z7∗Mz0,z1,α52pn+7Tz4,z4m+3Rz7,z8∗Mz0,z1,α53pn+8Tz4,z4m+3Tz8,z4m+3Qz8,z9∗Mz0,z1,α53pn+9Tz4,z4m+3Tz8,z4m+3Wz9,z10∗Mz0,z1,α53pn+10Tz4,z4m+3Tz8,z4m+3Ez10,z11∗Mz0,z1,α53pn+11Tz4,z4m+3Tz8,z4m+3Rz11,z12∗Mz0,z1,α54pn+12Tz4,z4m+3Tz8,z4m+3Tz12,z4m+3Qz12,z13∗⋮∗Mz0,z3,α5mpn+4mTz4,z4m+3Tz8,z4m+3Tz12,z4m+3⋯Tz4m−4,z4m+3Tz4m,z4m+3,(28)Mzn,zn+4m+4,α≥Mz0,z4m+4,αpn≥Mz0,z1,α5pnQz0,z1∗Mz0,z1,α5pn+1Wz1,z2∗Mz0,z1,α5pn+2Ez2,z3∗Mz0,z1,α5pn+3Rz3,z4∗Mz0,z1,α52pn+4Tz4,z4m+4Qz4,z5∗Mz0,z1,α52pn+5Tz4,z4m+4Wz5,z6∗Mz0,z1,α52pn+6Tz4,z4m+4Ez6,z7∗Mz0,z1,α52pn+7Tz4,z4m+4Rz7,z8∗Mz0,z1,α53pn+8Tz4,z4m+4Tz8,z4m+4Qz8,z9∗Mz0,z1,α53pn+9Tz4,z4m+4Tz8,z4m+4Wz9,z10∗Mz0,z1,α53pn+10Tz4,z4m+4Tz8,z4m+4Ez10,z11∗Mz0,z1,α53pn+11Tz4,z4m+4Tz8,z4m+4Rz11,z12∗Mz0,z1,α54pn+12Tz4,z4m+4Tz8,z4m+4Tz12,z4m+4Qz12,z13∗⋮∗Mz0,z4,α5mpn+4mTz4,z4m+4Tz8,z4m+4Tz12,z4m+4⋯Tz4m−4,z4m+4Tz4m,z4m+4.
Therefore, for eachq and from inequalities (22)–(28), we obtain
(29)limn⟶∞Mzn,zn+q,α=1∗1∗1∗⋯∗1=1,as bzn,zn+q<1/pforalln,q∈ℕandp∈0,1,i.e.,zn is a Cauchy sequence in X. Since X,M,∗ is complete, there exists z∈Xsuchthatzn⟶zasn⟶∞. We assert that z is the fixed point of F. From equation (29) and condition (A4), we obtain
(30)Mz,Fz,α≥Mz,zn,α5Qz,zn∗MFzn−1,Fzn,α5Wzn,zn+1∗MFzn,Fzn+1,α5Ezn+1,zn+2∗MFzn+1,Fzn+2,α5Rzn+2,zn+3∗MFzn+2,Fz,α5Tzn+3,Fz≥Mz,zn,α5Qz,zn∗Mzn−1,zn,α5pWzn,zn+1∗Mzn,zn+1,α5pEzn+1,zn+2∗Mzn+1,zn+2,α5pRzn+2,zn+3∗Mzn+2,z,α5pTzn+3,Fz.
Lettingn⟶∞ in the above inequality, we get Fz=z,i.e.,z is a fixed point of F. By using the inequality (9), we can easily examine that z is a unique fixed point of F.Theorem 12.
LetX,M,∗,Q be a complete FCHMS and Q:X×X⟶1,∞ such that
(31)limα⟶∞Mz,d,α=1,forallα>0,z,d∈X.
LetF:X⟶X be a mapping satisfying
(32)MFz,Fd,qα≥Mz,d,α,forallα>0,z,d∈X,where 0<p<1. Furthermore, if, for z0∈Xandn,q∈1,2,3,⋯, it holds bzn,zn+q<1/p where zn=Fnz0, then F has a unique fixed point.Proof.
It is easy to prove on the lines of Theorem11.Example 13.
LetX=0,1. Define M:X×X×0,∞⟶0,1 as
(33)Mz,d,α=e−z−dk/αforallα>0,k≥1,with the CTN ∗ such that α1∗α2=α1α2. Then, X,M,∗ is a complete PCFMS with noncomparable control functions Qz,d=1+z+d,Wz,d=1+z2+d2,Ez,d=1+z/d,Rz,d=1+d/z,andTz,d=1+z2+d.
DefineF:X⟶X by Fz=z/6. Consider
(34)MFz,Fd,qα=Mz6,d6,qα=e−z/6−d/6k/α=e−z−dk/6kα≥e−z−dk/α=Mz,d,α.
Hence, by Theorem11, F has unique fixed point, which is 0.


## 4. Application

Now, we show how our established result can be used to find the unique solution to an integral equation in dynamic market equilibrium economics. For supplyQβ and demand Qd, in many markets, current prices and pricing trends (whether prices are rising or dropping and whether they are rising or falling at an increasing or decreasing rate) have an impact. The economist, therefore, wants to know what the current price is Pα, the first derivative dPα/dα, and the second derivative d2Pα/dα2. Assume
(35)Qβ=g1+γ1Pα+e1dPαdα+z1d2Pαdα2,Qd=g2+γ2Pα+e2dPαdα+z2d2Pαdα2.g1,g2,γ1,γ2,e1,ande2 are constants. If pricing clears the market at each point in time, we comment on the dynamic stability of the market. In equilibrium,Qβ=Qd. So,
(36)g1+γ1Pα+e1dPαdα+z1d2Pαdα2=g2+γ2Pα+e2dPαdα+z2d2Pαdα2.Since(37)z1−z2d2Pαdα2+e1−e2ddPαdα+γ1−γ2Pα=−g1−g2.Lettingz=z1−z2,e=e1−e2,γ=γ1−γ2,andg=g1−g2 in above, we have
(38)zd2Pαdα2+edPαdα+γPα=−g.Dividing through byz,Pα is governed by the following initial value problem
(39)P′′+ezP′′+γzPα=−gz,P0=0,P′0=0,wheree2/z=4γ/zandγ/e=μ is a continuous function. It is easy to show that problem (39) is equivalent to the integral equation:
(40)Pα=∫0Tξα,rFα,r,Prdr,whereξα,r is Green’s function given by
(41)ξα,r=reμ/2α−rif0≤r≤α≤T,αeμ/2r−αif0≤α≤r≤α≤T.We will show the existence of a solution to the integral equation:(42)Pα=∫0TGα,r,Prdr.LetX=C0,T set of real continuous functions defined on0,Tforα>0, we define
(43)Mh,z,α=0,ifα=0,supαϵ0,Tminh,z+αmaxh,z+α,otherwise.For allh,zϵX with the CTN “ ∗” such that α1∗α2=α1α2, define Q,W,E,R,T:X×X⟶1,∞as
(44)Qz,d=1+z+d,Wz,d=1+z2+d2,Ez,d=1+zd,Rz,d=1+dz,Tz,d=1+z2+d.It is easy to prove thatX,M,∗ is a complete PCFMS, and F:X⟶X is defined by
(45)FPα=∫0TGα,r,Prdr.Theorem 14.
Consider equation (42) and suppose that
(i)
G:0,T×0,T⟶R+ is a continuous function(ii)
There exists a continuous functionξ:0,T×0,T⟶R+ such that supα∈0,T∫0Tξα,rdr≥1(iii)
maxGα,r,hr−Gα,r,zr≥ξα,rmaxhr,zrandminGα,r,hr−Gα,r,zr≥ξα,rminhr,zr
Then, the integral equation (42) has a unique solution.Proof.
Forh,z∈X, by use of assumptions (i) to (iii), we have
(46)MFh,Fz,α=supαϵ0,Tmin∫0TGα,r,hrdr,∫0TGα,r,zrdr+αmax∫0TGα,r,hrdr,∫0TGα,r,zrdr+α=supαϵ0,T∫0TminGα,r,hr,Gα,r,zrdr+α∫0TmaxGα,r,hr,Gα,r,zrdr+α≥supαϵ0,T∫0Tξα,rminhr,zrdr+α∫0Tξα,rmaxhr,zrdr+α≥supαϵ0,Tminhr,zr∫0Tξα,rdr+αmaxhr,zr∫0Tξα,rdr+α≥minhr,zr+αmaxhr,zr+α=Mh,z,α.
Thus,MFh,Fz,qα≥Mh,z,αforallh,z∈X, and all conditions of Theorem 11 are satisfied. Therefore, equation (42) has a unique fixed point.


## 5. Conclusion

The aim of this study is to present the notions of FEHMSs and PCFMS and prove Banach fixed point theorems in these spaces; nontrivial examples and an application to integral equation are also given to support our results. Due to a diverse range of applications of metric fixed point theory in mathematics, science, and economics, it is researched widely. Different types of fixed point results for single- and multivalued mappings can be proven in the sense of the above defined notions in this manuscript.



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*Source: 5301293-2022-01-11.xml*