# A Note on Orthogonal Fuzzy Metric Space, Its Properties, and Fixed Point Theorems

Authors: Khalil Javed; Awais Asif; Ekrem Savas
Journal: Journal of Function Spaces
 (2022)

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## Abstract

The article generalizes the notion of orthogonal fuzzy metric space into a broader term, named as orthogonal picture fuzzy metric space. The obtained results improve and extend the idea of the orthogonal fuzzy metric space and its related results. However, this article outstretches the above-mentioned notion further into a newly defined concept, named as orthogonal picture fuzzy metric space. A detailed insight is given into the topic by presenting some fixed point results in the frame of the newly defined structure. To elaborate the results more precisely, some concrete examples are given.


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## Body

## 1. Introduction

In 2013, Cuong [2] proposed a new concept named picture fuzzy sets (PFS), which is an extension of fuzzy sets and intuitionistic fuzzy sets. In a picture fuzzy set, each element is specified by the degree of membership, the degree of non-membership, and degree of neutrality together with the condition that the sum of these grades should be less or equal to 1.In this regard, Phong et al. [7] studied some compositions of picture fuzzy relations. Cuong and Hai [8] investigated main fuzzy logic operators: negations, conjunctions, disjunctions, and implications on picture fuzzy sets, and constructed the main operations for fuzzy inference processes in picture fuzzy systems. Singh [9] studied the correlation coefficients of picture fuzzy sets. Cuong et al. [10] then investigated the classification of representable picture t-norms and picture t-conorms operators for picture fuzzy sets.Eshaghi et al. [4] presented a new generalization of the Banach fixed point theorem (BFPT) by defining the notion of orthogonal sets. The orthogonal set is a non-empty set equipped with a binary relation (called orthogonal relation) having a special structure (see [4]). The metric defined on the orthogonal set is called orthogonal metric space. The orthogonal metric space contains partially ordered metric space and graphical metric space. Hezarjaribi [5] further extended the results of [4] to orthogonal fuzzy metric space. Also, Ishtiaq et al. [6] extended the results of [4] to orthogonal neutrosophic metric space. Some more details about generalized orthogonal metric spaces have been provided by Javed et al. [11], Uddin et al. [12, 13], and Senapati et al. [14].In this paper, we introduce orthogonal picture fuzzy metric space which generalize picture fuzzy metric space and orthogonal fuzzy metric spaces. We show that every picture fuzzy metric space is an orthogonal picture fuzzy metric space but not conversely. We investigate different conditions on the picture fuzzy to show the existence of fixed points in various types of contractions. We also present some examples in support of the obtained results. The authors intend to further widen the interesting idea of orthogonality to the intuitionistic fuzzy metric space and spherical fuzzy metric spaces. Some interesting results on the same two topics can be read in the articles [15, 16] and [17], respectively.


## 2. Preliminaries

Definition 1 (see [1]).
A fuzzy set is a pairⱲ,f, where Ⱳ is a non-empty set, f:Ⱳ⟶0,1 is a membership function and for each I∈Ⱳ,fI is called the grade of membership of I in Ⱳ,f.Definition 2 (see [2]).
A picture fuzzy setA on the universe set Ⱳ is an object of the form where Y∂∈0,1 is called the degree of positive membership of ∂ in A, M∂∈0,1 is called the “degree of neutral membership of∂ in A,” and Ɒ∂∈0,1 is called the degree of negative membership of ∂ in A, and Y∂,M∂,Ɒ∂ satisfy
(1)Y∂+M∂+Ɒ∂≤1,for all ∂∈A. Then,
(2)forall∂∈A,1−Y∂+M∂+Ɒ∂,is called the degree of refusal membership of ∂ in A.Definition 3 (see [3]).
SupposeⱲ≠∅ is an arbitrary set, assume a five tuple Ⱳ,Y,M,∗,Δ where ∗ is a CTN, Δ is a CTCN, andY,Mare FSs onⱲ×Ⱳ×0,∞. If Ⱳ,Y,M,∗,Δ meet the following circumstances for all I,ℏ,∂∈Ⱳandπ,℘0:
(B1)YI,ℏ,℘+MI,ℏ,℘≤1,
(B2)YI,ℏ,℘>0,
(B3)YI,ℏ,℘=1⟺I=ℏ,
(B4)YI,ℏ,℘=Yℏ,I,℘,
(B5)YI,∂,℘+π≥YI,ℏ,℘∗Yℏ,∂,π,
(B6)YI,ℏ,∙ is non decreasing (ND) function of ℝ+ and ℘YI,ℏ,℘=1,
(B7)MI,ℏ,℘>0,
(B8)MI,ℏ,℘=0⟺I=ℏ,
(B9)MI,ℏ,℘=Mℏ,I,℘,
(B10)MI,∂,℘+π≤MI,ℏ,℘ΔMℏ,∂,π,
(B11)MI,ℏ,∙ is non increasing (NI) function of ℝ+ and lim℘⟶∞MI,ℏ,℘=0.
Then,Ⱳ,Y,M,∗,Δ is an IFMS.Definition 4.
SupposeⱲ≠∅, assume five tuples Ⱳ,Y,M,Ɒ,∗,Δ where ∗ is a CTN, Δ is a CTCN, andY,M,Ɒare picture fuzzy set onⱲ×Ⱳ×ℝ+. If Ⱳ,Y,M,Ɒ,∗,Δ meet the following circumstances for all I,ℏ,∂∈Ⱳandπ,℘>0:
(P1)YI,ℏ,℘+MI,ℏ,℘+ⱰI,ℏ,℘≤1,
(P2)0≤YI,ℏ,℘≤1,
(P3)YI,ℏ,℘=1⟺I=ℏ,
(P4)YI,ℏ,℘=Yℏ,I,℘,
(P5)YI,∂,℘+π≥YI,ℏ,℘∗Yℏ,∂,π,
(P6)YI,ℏ,∙ is non decreasing (ND) function of ℝ+ and lim℘⟶∞YI,ℏ,℘=1,
(P7)0≤MI,ℏ,℘≤1,
(P8)MI,ℏ,℘=0⟺I=ℏ,
(P9)MI,ℏ,℘=Mℏ,I,℘,
(P10)MI,∂,℘+π≤MI,ℏ,℘ΔSℏ,∂,π,
(P11)MI,ℏ,∙ is non increasing (NI) function of ℝ+ and lim℘⟶∞MI,ℏ,℘=0,
(P12)0≤ⱰI,ℏ,℘≤1,
(P13)ⱰI,ℏ,℘=0⟺I=ℏ,
(P14)ⱰI,ℏ,℘=Ɒℏ,I,℘,
(P15)ⱰI,∂,℘+π≤ⱰI,ℏ,℘Δvℏ,∂,π,
(P16)ⱰI,ℏ,∙ is non increasing (NI) function of ℝ+ and lim℘⟶∞ⱰI,ℏ,℘=0,
(P17) If℘≤0,then YI,ℏ,℘=0,MI,ℏ,℘=1 and ⱰI,ℏ,℘=1.
Then,Ⱳ,Y,M,Ɒ,∗,Δ is a PFMS.Definition 5 (see [4]).
AssumeⱲ≠⏀ and ˫∈Ⱳ×Ⱳ is a binary relation. Assume there exists I0∈Ⱳ such that I0˫I or I˫I0 for allI∈Ⱳ. Thus, Ⱳ is said to be an OS. Furthermore, we denote OS byⱲ,˫.Definition 6 (see [4]).
Suppose thatⱲ,˫ is an OS. A sequence In for n∈ℕ is called an (OS) if for all n,In˫In+1 or for all n,In+1˫In.


### 2.1. Orthogonal Picture Fuzzy Metric Space

Definition 7.
LetⱲ,Y,M,Ɒ,∗,Δ,˫ be called an OPFMS if Ⱳ is a non-empty OS, ∗ is a CTN, Δ is a CTCN, and Y,M,Ɒ are pfs on Ⱳ×Ⱳ×ℝ+ if the following condition are satisfied for all I,ℏ,∂∈Ⱳ with either (I˫ℏ or ℏ˫I), either (I˫ℏ or δ˫I), and either (I˫ℏ∨ℏ˫I):
P1) YI,ℏ,℘+MI,ℏ,℘+ⱰI,ℏ,℘≤1,
P˫2) 0≤YI,ℏ,℘≤1,
P˫3) YI,ℏ,℘=1 if and only if I=ℏ,
P˫4) YI,ℏ,℘=Yℏ,I,℘,
P˫5) YI,∂,℘+π≥YI,ℏ,λ∗Yℏ,e,π,
P˫6) YI,ℏ,∙:0,∞⟶0,1 is continuous,
P˫7) 0≤MI,ℏ,℘≤1,
P˫8) MI,ℏ,℘=1 if and only if I=ℏ,
P˫9) MI,ℏ,℘=Mℏ,I,℘,
P˫10) MI,∂,℘+π≤MI,ℏ,℘ΔMℏ,∂,π,
P˫11) MI,ℏ,∙:0,∞⟶0,1 is continuous,
P˫12) 0≤ⱰI,ℏ,℘≤1,
P˫13) ⱰI,ℏ,℘=1 if and only if I=ℏ,
P˫14) ⱰI,ℏ,℘=Ɒℏ,I,℘,
P˫15) ⱰI,∂,℘+π≤ⱰI,ℏ,℘ΔⱰℏ,∂,π,
P˫16) ⱰI,ℏ,∙:0,∞⟶0,1 is continuous,
P˫17) If ℘≤0 then YI,ℏ,℘=0,MI,ℏ,℘=1 and ⱰI,ℏ,℘=1,
Then,Ⱳ,Y,M,Ɒ,∗,Δ,˫ is called OPFMS.Remark 8.
Every PFMS is an OPFMS but the converse is not true.Example 1.
LetⱲ=−7,7 and define a CTN asa∗b=ab, CTCN as aΔb=maxa,b and define a binary relation ˫ by I˫ℏ iff I+ℏ≥0. Take
(3)YI,ℏ,℘=1ifI=ℏ,℘℘+maxI,ℏifotherwise,MI,ℏ,℘=0ifI=ℏ,maxI,ℏ℘+maxI,ℏifotherwise,ⱰI,ℏ,℘=0ifI=ℏ,maxI,ℏ℘ifotherwise,
for allI,ℏ∈Ⱳ,℘>0, then it is OPFS, but not an PFMS.It is easy to see that forπ=℘=1,I=−1,ℏ=−1/2,∂=−2. (P˫5), (P˫10), and (P˫15) fails.Remark 9.
The above example is also OPFMS if we take(4)MI,ℏ,℘=0ifI=ℏ,1−℘℘+maxI,ℏifotherwise.Definition 10.
An OSIn in an OPFMS Ⱳ,Y,M,Ɒ,∗,Δ,˫ is said to be orthogonal convergent (O-C) toI∈Ⱳ if
(5)limn⟶∞YIn,I,℘=1,∀℘>0,limn⟶∞MIn,I,℘=0,∀℘>0,limn⟶∞ⱰIn,I,℘=0,∀℘>0.Definition 11.
An OSIn in an OPFMS Ⱳ,Y,M,∗,Δ,˫ is said to be Orthogonal Cauchy (O-CS) if there exists n∈ℕ such that
(6)limn⟶∞YIn,In+p,℘=1,limn⟶∞MIn,In+p,℘=0,limn⟶∞ⱰIn,In+p,℘=0,for all ℘≥0,p≥1.Definition 12.
Ω:Ⱳ⟶Ⱳ is OC at I∈Ⱳ in an OPFMS Ⱳ,Y,M,Ɒ,∗,Δ,˫, whenever for each OS In for all n∈ℕ in Ⱳ if limn⟶∞RIn,I,℘=1, limn⟶∞MIn,I,℘=0, and limn⟶∞ⱰIn,I,℘=0 for all℘>0, then limn⟶∞YΩIn,ΩI,℘=1, limn⟶∞MΩIn,ΩI,℘=0, and limn⟶∞ⱰΩIn,ΩI,℘=0 for all ℘>0.Definition 13.
An OPFMSⱲ,Y,M,Ɒ,∗,Δ,˫ is said to be orthogonally complete (O-complete) if every O-CS is convergent.Example 2.
Assume OPFMS as given in Example1 and define a sequence In in Ⱳby In=1−1/n,∀n∈ℕ such that ∀n;In˫In+1 or ∀n;In+1˫In. Define a CTN asa∗b=ab, CTCN as aΔb=maxa,b, and define a binary relation ˫ by I˫ℏ iff I+ℏ≥0. Take
(7)limn⟶∞YIn,I,℘=limn⟶∞1ifI=ℏ,℘℘+maxIn,Iifotherwise,=1ifI=ℏ,℘℘+maxI,Iifotherwise,limn⟶∞MIn,I,℘=limn⟶∞0ifI=ℏ,maxIn,I℘+maxIn,Iifotherwise,=0ifI=ℏmaxI,I℘+maxI,Iifotherwise,limn⟶∞ⱰIn,I,℘=limn⟶∞0ifI=ℏmaxIj,I℘ifotherwise=0ifI=ℏ,maxI,I℘ifotherwise.Example 3.
From proof of Example2, In=1−1/n,∀n∈ℕ is a O-CS in an OPFMS.
(8)limn⟶∞YIn,In+p,℘=limn⟶∞1ifI=ℏ,℘℘+maxIn,In+pifotherwise,=1ifI=ℏ,℘℘+maxI,Iifotherwise,limn⟶∞MIn,In+p,℘=limn⟶∞0ifI=ℏ,maxIn,In+p℘+maxIn,In+pifotherwise,=0ifI=ℏ,maxI,I℘+maxI,Iifotherwise,limn⟶∞ⱰIn,In+p,℘=limn⟶∞0ifI=ℏ,maxIn,In+p℘ifotherwise=0ifI=ℏ,maxI,I℘ifotherwise,for all ℘≥0,p≥1.Lemma 14.
If for somev∈0,1 and I,ℏ∈Ⱳ,(9)YI,ℏ,℘≥YI,ℏ,℘v,℘>0,MI,ℏ,℘≤MI,ℏ,℘v,℘>0ⱰI,ℏ,℘≤ⱰI,ℏ,℘v,℘>0thenI=ℏ.Definition 15.
LetⱲ,Y,M,Ɒ,∗,Δ,˫ be an OPFMS. A map Ω:Ⱳ⟶Ⱳ is an orthogonal contraction if there exists θ∈0,1 such that for every ℘>0 and I,δ∈Ⱳ with I˫ℏ, we have
(10)YψI,Ωℏ,θλ≥YI,ℏ,℘,(11)MΩI,Ωℏ,θ℘≤MI,ℏ,℘,(12)ⱰΩI,Ωℏ,θ℘≤ⱰI,ℏ,℘.Theorem 16.
LetⱲ,Y,M,Ɒ,∗,Δ,˫ be an O-complete PFMS such that
(13)lim℘⟶∞YI,ℏ,℘=1,lim℘⟶∞MI,ℏ,℘=0,andlim℘⟶∞ⱰI,ℏ,℘=0,∀I,ℏ∈Ⱳ.
LetΩ:Ⱳ⟶Ⱳ be an OC, O-CON and OPR. Thus, Ω has a unique FP, say I∗∈Ⱳ. Furthermore,
(14)limn⟶∞YΩnI,I∗,℘=1,limn⟶∞MΩnI,I∗,℘=0,andlimn⟶∞ⱰΩnI,I∗,℘=0∀I∈Ⱳand℘>0.Proof.
SinceⱲ,Y,M,Ɒ,∗,Δ,˫ is an O-complete PFMS, there exists I0∈Ⱳ such that
(15)I0˫ℏforallℏ∈Ⱳ.
That is,I0˫ΩI0. Take
(16)In=ΩnI0=ΩIn−1foralln∈M.
SinceΩ is OPR, In is an OS. Now, since Ω is an O-CON, we get
(17)YIn+1,In,θ℘=YΩIn,ΩIn−1,θ℘≥YIn,In−1,℘,for all n∈M and ℘>0. Note that Y is nondecreasing on 0,∞. Therefore, by applying the above expression, we can deduce
(18)YIn+1,In,℘≥YIn+1,In,θ℘=YΩIn,ΩIn−1,θ℘≥YIn,In−1,℘=YΩIn−1,ΩIn−2,℘≥YIn−1,In−2,℘θ≥⋯≥YI1,I0,℘θn,for all n∈M and ℘>0. Thus, from (15) and (P˫5), we have
(19)YIn,In+α,℘≥YIn,In+1,℘2∗YIn+1,In+α,℘2≥YIn,In+1,℘2∗YIn+1,In+2,℘22∗YIn+2,In+3,℘23∗⋮∗YIn+α−1,In+α,℘2n+α≥YI1,I0,℘2θn∗YI1,I0,℘22θn∗⋮∗YI1,I0,℘2n+αθn.
We know thatlim℘⟶∞YI,ℏ,℘=1, for all I,ℏ∈Ⱳ and ℘>0. So, from (19) we get,
(20)limn⟶∞YIn,In+α,℘≥1∗1∗⋯∗1=1.MIn+1,In,θ℘=MΩIn,ΩIn−1,θ℘≤MIn,In−1,℘,for all n∈ℏ and ℘>0. Therefore, by applying the above expression, we can deduce
(21)MIn+1,In,℘≤MIn+1,In,θ℘=MΩIn,ΩIn−1,θ℘≤MIn,In−1,℘=MΩIn−1,ψIn−2,℘≤MIn−1,In−2,℘θ≤⋯≤MI1,I0,℘θn,for all n∈ℏ and ℘>0. Thus, from (21) and (P⊥10), we have
(22)MIn,In+α,℘≤MIn,In+1,℘2ΔMIn+1,In+α,℘2≤MIn,In+1,℘2ΔMIn+1,In+2,℘22ΔMIn+2,I,℘23Δ⋯ΔMIn+α−1,In+α,℘2n+α≤MI,I0,℘2θnΔMI1,I0,℘22θnΔ⋯ΔMI1,I0,℘2n+αθn.
We know thatlim℘⟶∞MI,ℏ,℘=0, for all I,ℏ∈Ⱳ and ℘>0. So, from (22) we get,
(23)limn⟶∞MIn,In+α,℘≤0Δ0Δ⋯Δ0=0,ⱰIn+1,In,θ℘=ⱰΩIn,ΩIn−1,θ℘≤ⱰIn,In−1,℘,for all n∈M and ℘>0. Therefore, by applying the above expression, we can deduce
(24)ⱰIn+1,In,℘≤ⱰIn+1,In,θ℘=ⱰΩIn,ΩIn−1,θ℘≤ⱰIn,In−1,℘=ⱰΩIn−1,ΩIn−2,℘≤ⱰIn−1,In−2,℘θ≤⋯≤ⱰI1,I0,℘θn,for all n∈M and ℘>0. Thus, from (24) and (P⊥15), we have
(25)ⱰIn,In+α,℘≤ⱰIn,In+1,℘2ΔⱰIn+1,In+α,℘2≤ⱰIn,In+1,℘2ΔⱰIn+1,In+2,℘22ΔⱰIn+2,In+3,℘23Δ..ΔⱰIn+α−1,In+α,℘2n+α≤ⱰI1,I0,℘2θnΔⱰI1,I0,℘22θnΔ⋯ΔⱰI1,I0,℘2n+αθn,
We know thatlim℘⟶∞ⱰI,ℏ,℘=0, for all I,ℏ∈Ⱳ and ℘>0. So, from (25) we get,
(26)limn⟶∞ⱰIn,In+α,℘≤0Δ0Δ⋯Δ0=0.
So,In is a O-CS. The O-completeness of the PFMS Ⱳ,Y,M,Ɒ,∗,Δ,˫ ensures that there existI∗∈Ⱳ such that YIn,I∗,℘⟶1,MIn,I∗,℘⟶0, and ⱰIn,I∗,℘⟶0 as n⟶+∞ for all ℘>0. Now, since Ω is an OC, YIn+1,ΩI∗,℘=YΩIn,ΩI∗,℘⟶1, MIn+1,ΩI∗,℘=MΩIn,ΩI∗,℘⟶0, and ⱰIn+1,ΩI∗,℘=ⱰΩIn,ΩI∗,℘⟶0 as n⟶+∞. Now, we have
(27)YI∗,ψI∗,℘≥YI∗,In+1,℘2∗YIn+1,ΩI∗,℘2,MI∗,ΩI∗,℘≤MI∗,In+1,℘2ΔMIn+1,ΩI∗,℘2,ⱰI∗,ΩI∗,℘≤ⱰI∗,In+1,℘2ΔⱰIn+1,ΩI∗,℘2.
Taking limit asn⟶+∞, we get YI∗,ΩI∗,℘=1∗1=1,MI∗,ΩI∗,℘=0Δ0=0, and ⱰI∗,ΩI∗,℘=0Δ0=0 and hence ΩI∗=I∗.
Now, we show the uniqueness of the FP of the mappingΩ. Assume that I∗ and ℏ∗ are two FPs of Ω such thatI∗≠δ∗. We can get
(28)I0˫I∗andI0˫ℏ∗.
SinceⱰ is OPR, one writes
(29)ΩnI0˫ΩnI∗andΩnI0˫Ωnℏ∗,
for alln∈M. So from (10), we can derive
(30)YΩnI0,ΩnI∗,℘≥YΩnI0,ΩnI∗,θ℘≥YI0,I∗,℘θn,YΩnI0,Ωnℏ∗,℘≥YΩnI0,Ωnℏ∗,θ℘≥YI0,ℏ∗,℘θn.
Therefore,(31)YI∗,ℏ∗,℘=YΩnI∗,Ωnℏ∗,℘≥YΩnI0,ΩnI∗,℘2∗YΩnI0,Ωnℏ∗,℘2≥YI0,I∗,℘2θn∗YI0,ℏ∗,℘2θn⟶1asn⟶∞.
So from (11), we can derive
(32)MΩnI0,ΩnI∗,℘≤MΩnI0,ΩnI∗,θ℘≤MI0,I∗,℘θn,MΩnI0,Ωnℏ∗,℘≤MΩnI0,Ωnℏ∗,θ℘≤MI0,ℏ∗,℘θn.
Therefore,(33)MI∗,ℏ∗,℘=MΩnI∗,Ωnℏ∗,℘≤MΩnI0,ΩnI∗,℘2ΔMΩnI0,Ωnℏ∗,℘2≤MI0,I∗,℘2θnΔMI0,ℏ∗,℘2θn⟶0asn⟶∞.
Similarly, from (12), we can derive
(34)ⱰΩnI0,ΩnI∗,℘≤ⱰΩnI0,ΩnI∗,θ℘≤ⱰI0,I∗,℘θn,ⱰΩnI0,Ωnℏ∗,℘≤ⱰΩnI0,Ωnℏ∗,θ℘≤ⱰI0,ℏ∗,℘θn.
Therefore,(35)ⱰI∗,ℏ∗,℘=ⱰΩnI∗,Ωnℏ∗,℘≤ⱰΩnI0,ΩnI∗,℘2ΔⱰΩnI0,Ωnℏ∗,℘2≤ⱰI0,I∗,℘2θnΔⱰI0,ℏ∗,℘2θn⟶0asn⟶∞.
So,I∗=ℏ; hence, I∗ is the unique FP.Corollary 17.
AssumeⱲ,Y,M,Ɒ,∗,Δ,˫ be an O-complete PFMS. Assume Ω:Ⱳ⟶Ⱳ be O-CON and OPR and if In is an OS with In⟶I∈Ⱳ, thenI˫In for all n∈ℕ. Then, Ω has a unique FP, say I∗∈Ⱳ.Proof.
We can similarly derive as in the proof of Theorem16 that In is a O-CS and so it converges to I∗∈Ⱳ. Hence, I∗˫In for all n∈ℕ. from (10), we can get
(36)YΩI∗,In+1,℘=YΩI∗,ΩIn,℘≥YΩI∗,ΩIn,℘θ≥YI∗,In,℘,limn⟶∞YΩI∗,In+1,℘=1.
Then, we can write(37)YI∗,ΩI∗,℘≥YI∗,In+1,℘2∗YIn+1,ΩI∗,℘2.
Taking limit asn⟶+∞, we get YI∗,ΩI∗,℘=1∗1=1 and from (11), we can get
(38)MΩI∗,In+1,℘=MΩI∗,ΩIn,℘≤MΩI∗,ΩIn,℘θ≤MI∗,In,℘,limn⟶∞MΩI∗,In+1,℘=0.
Then, we can write(39)MI∗,ΩI∗,℘≤MI∗,In+1,℘2ΔMIn+1,ΩI∗,℘2.
Taking limit asn⟶+∞, we get
(40)MI∗,ΩI∗,℘=0Δ0=0,and from (12), we can get
(41)ⱰΩI∗,In+1,℘=ⱰΩI∗,ΩIn,℘≤ⱰΩI∗,ΩIn,℘θ≤ⱰI∗,In,℘,limn⟶∞ⱰΩI∗,In+1,℘=0.
Then, we can write(42)ⱰI∗,ΩI∗,℘≤ⱰI∗,In+1,℘2ΔⱰIn+1,ΩI∗,℘2.
Taking limit asn⟶+∞, we get
(43)ⱰI∗,ΩI∗,℘=0Δ0=0,
soΩI∗=I∗. Next proof is similar as in Theorem 16.Example 4.
LetⱲ=−3,3. We define a binary relation ˫ by I˫ℏ⟺I+ℏ≥0.
Define an OPFMS as in Example 1 by(44)YI,ℏ,℘=1ifI=ℏ,℘℘+maxI,ℏifotherwise,MI,ℏ,℘=0ifI=ℏ,maxI,ℏ℘+maxI,ℏifotherwise,ⱰI,ℏ,℘=0ifI=ℏ,maxI,ℏ℘ifotherwise,
for allI,δ∈Ⱳ,℘>0, with the CTN a∗b=a∙b and CTCN aΔb=maxa,b. Then, Ⱳ,Y,M,Ɒ,∗,Δ,˫ is an O-complete PFMS. Define Ω:Ⱳ⟶Ⱳ by
(45)ΩI=I4,I∈−3,00,I∈0,3.
Then, the following cases are satisfied:(1)
IfI∈−3,0 andℏ∈0,3, then ΩI=I/4 andψℏ=0(2)
IfI,ℏ∈−3,0, then ΩI=I/4 andΩℏ=ℏ/4(3)
IfI,ℏ∈0,3, then ΩI=0 andΩℏ=0(4)
IfI∈0,3 andℏ∈−3,0, then ΩI=0 andΩℏ=ℏ/4
This clearly implies thatΩI+Ωℏ≥0. Hence, Ω is OPR. We can easily see that if limn⟶∞YIn,I,℘=1, then limn⟶∞YΩIj,ΩI,℘=1,limn⟶∞MIn,I,℘=0, then limn⟶∞MΩIn,ΩI,℘=0 and limn⟶∞ⱰIn,I,℘=0, then limn⟶∞ⱰΩIn,ΩI,℘=0 for allI∈Ⱳ and ℘>0. Hence, Ω is OC.
The above four cases forθ∈1/2,1 satisfies the below contractive conditions:
(46)YΩI,Ωℏ,θ℘≥YI,ℏ,℘,MΩI,Ωℏ,θ℘≤MI,ℏ,℘,ⱰΩI,Ωℏ,θ℘≤ⱰI,ℏ,℘.
All conditions of Theorem16 are satisfied. Also, 0 is FP of Ω.Theorem 18.
LetⱲ,Y,M,Ɒ,∗,Δ,˫ be an O-complete PFMS such that
(47)lim℘⟶∞YI,ℏ,℘=1andlim℘⟶∞MI,ℏ,℘=0,∀I,ℏ∈Ⱳand℘>0.
LetΩ:Ⱳ⟶Ⱳ be OC,O-CON and OPR. Assume that there exist θ∈0,1 and ℘>0 such that
(48)YΩI,Ωℏ,θ℘≥minYΩI,I,℘,YΩℏ,ℏ,℘,MΩI,Ωℏ,θ℘≤minMΩI,I,℘,MΩℏ,ℏ,℘,ⱰΩI,Ωℏ,θ℘≤minⱰΩI,I,℘,ⱰΩℏ,ℏ,℘,for all I,ℏ∈Ⱳ,℘>0. Then, Ω has a unique FP, soI∗∈Ⱳ. Furthermore, limn⟶∞YΩnI,I∗,℘=1, limn⟶∞MΩnI,I∗,℘=0, and limn⟶∞ⱰΩnI,I∗,℘=0 for all I∈Ⱳ and ℘>0.Proof.
SinceⱲ,Y,M,Ɒ,∗,Δ,˫ is an O-complete PFMS, there exists I0∈Ⱳ such that
(49)I0˫ℏ,∀δ∈Ⱳ.
Thus,I˫h.Consider
(50)In=ΩnI0=ΩIn−1,∀n∈S.
SinceΩ is OPR, In is an OS. We can get
(51)YIn+1,In,℘≥YIn+1,In,θ℘=YΩIn,ΩIn−1,θ℘≥minYΩIn,In,℘,YΩIn−1,In−1,℘,MIn+1,In,℘≤MIn+1,In,θ℘=MΩIn,ΩIn−1,θ℘≤minMΩIn,In,℘,MΩIn−1,In−1,℘,ⱰIn+1,In,℘≤ⱰIn+1,In,θ℘=ⱰΩIn,ΩIn−1,θ℘≤minⱰΩIn,In,℘,ⱰΩIn−1,In−1,℘.
Two cases arise:
Case1: IfYIn+1,In,℘≥YΩIn,In,℘, then
(52)YIn+1,In,℘≥YIn+1,In,θ℘≥YΩIn,In,℘=YIn+1,In,℘,MIn+1,In,℘≤MΩIn,In,℘.
Then,(53)MIn+1,In,℘≤MIn+1,In,θ℘≤MΩIn,In,℘=MIn+1,In,℘,ⱰIn+1,In,℘≤ⱰΩIn,In,℘.
Then,(54)ⱰIn+1,In,℘≤ⱰIn+1,In,θ℘≤ⱰΩIn,In,℘=ⱰIn+1,In,℘.
Then by Lemma14, In=In+1 for all n∈ℕ.
Case2: IfYIn+1,In,℘≥YΩIn−1,In−1,℘, then
(55)YIn+1,In,℘≥YIn+1,In,θ℘≥YΩIn−1,In−1,℘≥YIn,In−1,℘,MIn+1,In,℘≤MΩIn−1,In−1,℘.
Then,(56)MIn+1,In,℘≤MIn+1,In,θ℘≤MΩIn−1,In−1,℘≤MIn,In−1,℘,and
(57)ⱰIn+1,In,℘≤ⱰΩIn−1,In−1,℘.
Then,(58)ⱰIn+1,In,℘≤ⱰIn+1,In,θ℘≤ⱰΩIn−1,In−1,℘≤ⱰIn,In−1,℘,for all n∈ℕ and ℘>0. Then by Theorem 16, we have a OCS. By completeness of Ⱳ,Y,M,Ɒ,∗,Δ,˫, there exists I∗∈Ⱳ such that limn⟶∞RIn,I∗,℘=1, limn⟶∞MIn,I∗,℘=0, and limn⟶∞ⱰIn,I∗,℘=0, for all ℘>0.
We know thatΩ is an OC, then
(59)limn⟶∞YIn+1,ΩI∗,℘=limn⟶∞YΩIn,ΩI∗,℘=1,limn⟶∞MIn+1,ΩI∗,℘=limn⟶∞MΩIn,ΩI∗,℘=0,limn⟶∞ⱰIn+1,ΩI∗,℘=limn⟶∞ⱰΩIj,ΩI∗,℘=0.
Now, we prove thatI∗ is a FP of Ω. Let ℘1∈θ,1 and ℘2=1−℘1. Then,
(60)YΩI∗,I∗,℘≥YΩI∗,In+1,℘℘12∗YIn+1,I∗,℘℘22=YΩI∗,ΩIn,℘℘12∗YIn+1,I∗,℘℘22≥minYΩI∗,I∗,℘℘12θ,YψIn,In,℘℘12θ∗YIn+1,I∗,℘℘22=minYΩI∗,I∗,℘℘12θ,YIn+1,In,℘℘12θ∗YIn+1,I∗,℘℘22.
Takingn⟶∞, we get
(61)YΩI∗,I∗,℘≥minYΩI∗,I∗,℘℘12θ,1∗1,YΩI∗,I∗,℘≥YΩI∗,I∗,℘v,℘>0,MΩI∗,I∗,℘≤MΩI∗,In+1,℘℘12ΔMIn+1,I∗,℘℘22=MΩI∗,ΩIn,℘℘12ΔMIn+1,I∗,℘℘22≤minMΩI∗,I∗,℘℘12θ,MΩIn,In,℘℘12θΔMIn+1,I∗,℘℘22=minMΩI∗,I∗,℘℘12θ,MIn+1,In,℘℘12θΔMIn+1,I∗,℘℘22.
Takingn⟶∞, we get
(62)MΩI∗,I∗,℘≤minMΩI∗,I∗,℘℘12θ,0Δ0,MΩI∗,I∗,℘≤MΩI∗,I∗,℘v,℘>0,ⱰΩI∗,I∗,℘≤ⱰΩI∗,In+1,℘℘12ΔⱰIn+1,I∗,℘℘22=ⱰΩI∗,ΩIn,℘℘12ΔⱰIn+1,I∗,℘℘22≤minⱰΩI∗,I∗,℘℘12θ,ⱰΩIn,In,℘℘12θΔⱰIn+1,I,℘℘22=minⱰΩI∗,I∗,℘℘12θ,ⱰIn+1,In,℘℘12θΔⱰIn+1,I∗,℘℘22.
Takingn⟶∞, we get
(63)ⱰΩI∗,I∗,℘≤minⱰΩI∗,I∗,℘℘12θ,0Δ0,ⱰΩI∗,I∗,℘≤ⱰΩI∗,I∗,℘v,℘>0.
Here,v=2θ/℘1∈0,1, from Lemma 14, we haveΩI∗=I∗. Suppose v∗ and ℏ∗ the FPs of Ω. We have
(64)I0˫I∗andI0˫I∗.
BecauseΩ is an OPR, so we can write
(65)ΩnI0˫ΩnI∗andΩnI0˫ΩnI∗foralln∈ℕ.
We can write(66)YΩnI0,ΩnI∗,℘≥YΩnI0,ΩnI∗,θ℘≥minYΩnI0,I0,℘,YΩnI∗,I∗,℘,YΩnI0,Ωnℏ∗,℘≥YΩnI0,Ωnℏ∗,θ℘≥minYΩnI0,I0,℘,YΩnℏ∗,ℏ∗,℘.
Hence, we write that(67)YI∗,ℏ∗,℘=YΩnI∗,Ωnℏ∗,℘≥minYΩnI∗,I∗,℘θ,YΩnℏ∗,ℏ∗,℘θ=min1,1=1,MΩnI0,ΩnI∗,℘≤MΩnI0,ΩnI∗,θ℘≤minMΩnI0,I0,℘,MΩnI∗,I∗,℘,MΩnI0,Ωnℏ∗,℘≤MΩnI0,Ωnℏ∗,θ℘≤minMΩnI0,I0,℘,MΩnℏ∗,ℏ∗,℘.
Hence, we write that(68)MI∗,ℏ∗,℘=MΩnI∗,Ωnℏ∗,℘≤minMΩnI∗,I∗,℘θ,MΩnℏ∗,ℏ∗,℘θ=min0,0=0,ⱰΩnI0,ΩnI∗,℘≤ⱰΩnI0,ΩnI∗,θ℘≤minⱰΩnI0,I0,℘,ⱰΩnI∗,I∗,℘,ⱰΩnI0,Ωnℏ∗,℘≤ⱰΩnI0,Ωnℏ∗,θ℘≤minⱰΩnI0,I0,℘,ⱰΩnℏ∗,ℏ,℘.
Hence, we write that(69)ⱰI∗,ℏ∗,℘=ⱰΩnI∗,Ωnℏ∗,℘≤minⱰΩnI∗,I∗,℘θ,ⱰΩnℏ∗,ℏ∗,℘θ=min0,0=0,for all ℘>0. Hence, I∗=ℏ∗.Corollary 19.
LetⱲ,Y,M,∗,Δ,˫ be an O-complete PFMS and Ω:Ⱳ⟶Ⱳ be an OC and OPR. Then θ∈0,1, we get ℘>0,
(70)YΩI,Ωℏ,θ℘≥minYΩI,I,℘,YΩℏ,ℏ,℘,YI,ℏ,℘,MΩI,Ωℏ,θ℘≤minMΩI,I,℘,MΩℏ,ℏ,℘,MI,ℏ,℘,ⱰΩI,Ωℏ,θ℘≤minⱰΩI,I,℘,ⱰΩℏ,ℏ,℘,ⱰI,ℏ,℘,
Then,Ω has a unique FP.Proof.
It follows from Theorem16 and Theorem 18.Example 5.
LetⱲ=−2,2 and define a binary relation ⊥ by
(71)I˫ℏ⟺I+ℏ≥0.
DefineYandMby
(72)YI,ℏ,℘=1ifI=ℏ℘℘+maxI,ℏotherwise,MI,ℏ,℘=0ifI=ℏmaxI,ℏλ+maxI,ℏotherwise,ⱰI,ℏ,℘=0ifI=ℏ,maxI,ℏλotherwise,
for allI,ℏ∈Ⱳ and ℘>0, with the CTN and CTCN, respectively;a∗b=a.b, aΔb=maxa,b then Ⱳ,Y,M,Ɒ,∗,Δ,˫ is an O-complete PFMS. Note thatlim℘⟶∞YI,ℏ,℘=1, lim℘⟶∞MI,ℏ,℘=0, and lim℘⟶∞ⱰI,ℏ,℘=0 ∀ I,ℏ∈E. Define Ω:Ⱳ⟶Ⱳ by
(73)ΩI=I4,I∈−2,231−I,I∈23,1I−12,I∈1,2.
We have the following cases:(1)
IfI,ℏ∈−2,2/3, then ΩI=I/4 andΩℏ=ℏ/4(2)
IfI,ℏ∈2/3,1, then ΩI=1−I andΩℏ=1−ℏ(3)
IfI,ℏ∈1,2, then ΩI=I−1/2 andΩℏ=ℏ−1/2(4)
IfI∈−2,2/3 andℏ∈2/3,1, then ΩI=I/4 andΩℏ=1−ℏ(5)
IfI∈−2,2/3 and ℏ∈1,2, then ΩI=I/4 andΩℏ=ℏ−1/2(6)
IfI∈2/3,1 and ℏ∈1,2, then ΩI=1−I andΩℏ=ℏ−1/2(7)
IfI∈1,2 and ℏ∈2/3,1, then ΩI=I−1/2 andΩℏ=1−ℏ(8)
IfI∈1,2 and ℏ∈−2,2/3, then ΩI=I−1/2 andΩℏ=ℏ/4(9)
IfI∈2/3,1 and ℏ∈−2,2/3, then ΩI=1−I andΩℏ=ℏ/4
BecauseI˫ℏ⟺I+ℏ≥0, it is clearly implying thatΩI+Ωℏ≥0. Hence, Ω is OPR. Let In be an arbitrary OS in Ⱳ that converges toI∈Ⱳ. We have
(74)limn⟶∞YIn,I,℘=limn⟶∞1ifI=ℏ,℘℘+maxIn,Iotherwise=1,limn⟶∞MIn,I,℘=limn⟶∞1ifI=ℏ,maxIn,I℘+maxIn,Iotherwise=0,limn⟶∞ⱰIn,I,℘=limn⟶∞1ifI=ℏ,maxIn,I℘otherwise=0.
Note that iflimn⟶∞YIn,I,λ=1, limn⟶∞SIn,I,℘=0, and limn⟶∞ⱰIn,I,℘=0, then limn⟶∞YΩIn,ΩI,℘=1, limn⟶∞MΩIn,ΩI,℘=0, and limn⟶∞ⱰΩIn,ΩI,℘=0 for allI∈Ⱳ and ℘>0. Hence, Ω is OC. The case I=ℏ is clear. LetI≠ℏ. We have
(75)YΩI,Ωℏ,θ℘≥minYΩI,I,℘,YΩℏ,ℏ,℘,MΩI,Ωℏ,θ℘≤minMΩI,I,℘,MΩℏ,ℏ,℘,ⱰΩI,Ωℏ,θ℘≤minⱰΩI,I,℘,ⱰΩℏ,ℏ,℘.
Indeed, it is satisfied for all above 9 cases. But,Ωis not a contraction. Assume
(76)minYΩI,I,℘,YΩℏ,ℏ,℘=YΩI,I,℘,minMΩI,I,℘,MΩℏ,ℏ,℘=MΩI,I,℘,minⱰΩI,I,℘,TΩℏ,ℏ,℘=ⱰΩI,I,℘,then for I=−1,ℏ=−2, we have
(77)YΩI,Ωℏ,θ℘=θ℘θ℘+maxI/4,δ/4=4θ℘4θ℘−1≥1,MΩI,Ωℏ,θ℘=maxI/4,ℏ/4θλ+maxI/4,ℏ/4=−14θ℘−1≤0,MΩI,Ωℏ,θ℘=maxI/4,ℏ/4θλ=−14θ℘≤0.
It is a contradiction. Hence, all the conditions of Theorem18 are satisfied and 0 is the unique FP ofψ.Definition 20.
LetⱲ,Y,M,Ɒ,∗,Δ,˫ be an OPFMS. A mapping Ω:Ⱳ⟶Ⱳ is named to be an PF ˫-contractive if ∃ θ∈0,1 so that
(78)1YΩI,Ωℏ,λ−1≤θ1YI,ℏ,λ−1,(79)MΩI,Ωℏ,℘≤θMI,ℏ,℘andⱰΩI,Ωℏ,℘≤θⱰI,ℏ,℘,
for allI,ℏ∈Ⱳ and ℘>0. Here, θ is called the PFS ˫-contractive constant ofΩ.Theorem 21.
LetⱲ,Y,M,Ɒ,∗,Δ,˫ be an O-complete PFMS such that
(80)lim℘⟶∞YI,ℏ,℘=1,lim℘⟶∞MI,ℏ,℘=0,andlim℘⟶∞ⱰI,ℏ,℘=0,∀I,ℏ∈Ⱳ.
LetΩ:Ⱳ⟶Ⱳ be an OC,˫ -contraction and OPR. Thus, Ω has a FP, sayυ∈Ⱳ, Yυ,υ,℘=1,Mυ,υ,℘=0 and Ɒυ,υ,℘=0 for all ℘>0.Proof.
LetⱲ,Y,M,Ɒ,∗,Δ,˫ be an O-complete PFMS. For an arbitraryI0∈Ⱳ,
(81)I0˫ℏ,∀ℏ∈Ⱳ.
That is,I0˫ΩI0. Consider
(82)In=ΩnI0=ΩIn−1foralln∈ℕ.
SinceΩ is OPR, In is an OS. If In=In−1 for some n∈ℕ, then In is a FP ofΩ. We assume that In≠In−1 for all n∈ℕ. For all ℘>0 and n∈ℕ, we get from (12),
(83)1YIn,In+1,℘−1=1YΩIn−1,ΩIn,℘−1≤θ1YIn−1,In,℘−1,MIn,In+1,℘=MΩIn−1,ΩIn,℘≤θMIn−1,In,℘,ⱰIn,In+1,℘=ⱰΩIn−1,ΩIn,℘≤θⱰIn−1,In,℘.
We have∀℘>0(84)1YIn,In+1,℘≤θYIn−1,In,℘+1−θ,
Implying that(85)θYΩIn−2,ΩIn−1,℘+1−θ≤θ2YIn−2,In−1,℘+θ1−θ+1−θ.
Continuing in this way, we get(86)1YIn,In+1,℘≤θnYI0,I1,℘+θn−11−θ+θn−21−θ+⋯+θ1−θ+1−θ≤θnYI0,I1,℘+θn−1+θn−2+⋯+11−θ≤θnYI0,I1,℘+1−θn.
We have(87)1θn/YI0,I1,℘+1−θn≤YIn,In+1,℘,∀℘>0,n∈ℕ,(88)MIn,In+1,℘=MΩIn−1,ΩIn,℘≤θMIn−1,In,℘=θMΩIn−2,ΩIn−1,℘≤θ2MIn−2,In−1,℘≤⋯≤θnMI0,I1,℘.∀℘>0,n∈ℕ,(89)ⱰIn,In+1,℘=ⱰψIn−1,ψIn,℘≤θⱰIn−1,In,℘=θⱰψIn−2,ψIn−1,℘≤θ2Ɒβn−2,In−1,℘≤⋯≤θnⱰI0,I1,℘.∀℘>0,n∈ℕ.
Now, form≥1 and n∈ℕ, we have
(90)YIn,In+m,℘≥YIn,In+1,℘2∗YIn+1,In+m,℘2≥YIn,In+1,℘2∗YIn+1,In+2,℘22∗YIn+2,In+m,℘22.
Again, continuing in this way, we get(91)YIn,In+m,℘≥YIn,In+1,℘2∗YIn+1,In+2,℘22∗⋯∗YIn+m−1,In+m,℘2m−1,MIn,In+p,℘≤MIn,In+1,℘2ΔMIn+1,In+p,℘2≤MIn,In+1,℘2ΔMIn+1,In+2,℘22ΔMIn+2,In+p,℘22.
Continuing in this way, we get(92)MIn,In+p,℘≤MIn,In+1,℘2ΔMIn+1,In+2,℘22Δ⋯ΔMIn+p−1,In+p,℘2p−1,ⱰIn,In+p,℘≤ⱰIn,In+1,℘2ΔⱰIn+1,In+p,℘2≤ⱰIn,In+1,℘2ΔⱰIn+1,In+2,℘22ΔⱰIn+2,In+p,℘22.
Continuing in this way, we get(93)ⱰIn,In+p,℘≤ⱰIn,In+1,℘2ΔⱰIn+1,In+2,℘22Δ⋯ΔⱰIn+p−1,In+p,℘2p−1.
By using (87) in the above inequality, we have
(94)YIn,In+m,℘≥1θn/YI0,I1,℘/2+1−θn∗1θn+1/YI0,I1,℘/22+1−θn+1∗⋯∗1θn+m−1/YI0,I1,℘/2m−1+1−θn+m−1,≥1θn/YI0,I1,℘/2+1∗1θn+1/YI0,I1,℘/22+1∗⋯∗1θn+m−1/YI0,I1,℘/2m−1+1,using (88),
(95)SIn,In+p,℘≤SIn,In+1,℘2ΔSIn+1,In+2,℘22Δ⋯ΔSIn+p−1,In+p,℘2p−1,and using (89)
(96)ⱰIn,In+p,℘≤ⱰIn,In+1,℘2ΔⱰIn+1,In+2,℘22Δ⋯ΔⱰIn+p−1,In+p,℘2p−1,
θ∈0,1 we deduce from the above expression that limn⟶∞YIn,In+m,℘=1,limn⟶∞MIn,In+m,℘=0, and limn⟶∞ⱰIn,In+m,℘=0 for all ℘>0,m≥1.
Therefore,In is a O-CS in Ⱳ,Y,M,Ɒ,∗,Δ,˫. By the completeness of Ⱳ,Y,M,Ɒ,∗,Δ,˫, we know that Ω is an OC and there exists υ∈Ⱳ such that
(97)limn⟶∞YIn+1,υ,℘=limn⟶∞YΩIn,Ωυ,℘=1,∀℘>0,(98)limn⟶∞MIn+1,υ,℘=limn⟶∞MΩIn,Ωυ,℘=0∀℘>0,(99)limn⟶∞ⱰIn+1,υ,℘=limn⟶∞ⱰΩIn,Ωυ,℘=0∀℘>0.
Now, we prove thatυ is a FP of Ω. For this, we obtain from (78) that
(100)1YΩIn,Ωυ,℘−1≤θ1YIn,υ,℘−1=θYIn,υ,℘−θ.
That is,(101)1θ/YIn,υ,℘+1−θ≤YΩIn,Ωυ,℘.
Using the above inequality, we obtain(102)Yυ,Ωυ,℘≥Yυ,In+1,℘2∗YIn+1,Ωυ,℘2=Yυ,In+1,℘2∗YΩIn,Ωυ,℘≥Yυ,In+1,℘2∗1θ/YIn,υ,℘/2+1−θ,Mw,v,℘=MΩw,Ωv,℘≤θMw,v,℘<Mw,v,℘,=Mw,In+1,℘2ΔMΩIn,Ωw,℘≤Mw,In+1,℘2ΔθMIn,w,℘2,Ɒw,v,℘=ⱰΩw,Ωv,℘≤θⱰw,v,℘<Ɒw,v,℘,=Ɒw,In+1,℘2ΔⱰΩIn,Ωw,℘2≤Ɒw,In+1,℘2ΔθⱰIn,w,℘2.
Taking limit asn⟶∞ and using (97), (98), and (99) in the above expression, we get that Yυ,Ωυ,℘=1, Mυ,Ωυ,℘=0, and Ɒυ,Ωυ,℘=0, that is, Ωυ=υ. Therefore, υ is a FP of Ω, Yυ,υ,℘=1, Mυ,υ,℘=0, and Ɒυ,υ,℘=0 for all ℘>0.Corollary 22.
LetⱲ,Y,M,Ɒ,∗,Δ,˫ be a O-complete PFMS and Ω:Ⱳ⟶Ⱳ satisfy
(103)1YΩnI,Ωnℏ,℘−1≤θ1YI,ℏ,℘−1,MΩnI,Ωnℏ,℘≤θMI,ℏ,℘,ⱰΩnI,Ωnℏ,℘≤θⱰI,ℏ,℘,for all n∈ℕ,I,ℏ∈Ⱳ,℘>0, where 0<θ<1. Then, Ω has a FP.Proof.
υ∈Ⱳ is the unique FP of Ωn by using Theorem 21, and Yυ,υ,℘=1,Mυ,υ,℘=0,Ɒυ,υ,℘=0,∀℘>0.Ωυ is also a FP of Ωn as ΩnΩυ=Ωυ. From Theorem 21, Ωυ=υ,υ is a FP since the FP of Ω is also a FP of Ωn.Example 6.
LetⱲ=−1,2 and define a binary relation ˫ by
(104)I˫ℏ⟺I+ℏ≥0.
DefineY,M,Ɒ by
(105)YI,ℏ,℘=1ifI=ℏ,℘℘+maxI,ℏifotherwise,MI,ℏ,℘=0ifI=ℏ,1−℘℘+maxI,ℏifotherwise,ⱰI,ℏ,℘=0ifI=ℏ,maxI,ℏ℘ifotherwise.
With CTNa∗b=a.b and CTCN aΔb=maxa,b then Ⱳ,Y,M,Ɒ,∗,Δ,˫ is an O-complete PFMS. Also observe thatlim℘⟶∞YI,ℏ,℘=1, lim℘⟶∞SI,ℏ,℘=0, andlim℘⟶∞ⱰI,ℏ,℘=0, ∀ I,ℏ∈Ⱳ.
DefineΩ:Ⱳ⟶Ⱳ by
(106)ΩI=2−I,I∈−1,11,I∈1,2.
Therefore, it will satisfy the following cases:(1)
IfI,ℏ∈−1,1, then ΩI=2−I and Ωℏ=2−ℏ(2)
IfI,ℏ∈1,2, then ΩI=Ωℏ=1(3)
IfI∈−1,1 and ℏ∈1,2, then ΩI=2−I and Ωℏ=1(4)
IfI∈1,2 and ℏ∈−1,1, then ΩI=1 and Ωℏ=2−ℏ
BecauseI˫ℏ⟺I+ℏ≥0, it is clearly implying thatΩI+Ωℏ≥0. Hence, Ω is OPR. Let In be an arbitrary OS in Ⱳ that In converges toI∈Ⱳ.
(107)limn⟶∞YIn,I,℘=1,limn⟶∞MIn,I,℘=0,limn⟶∞ⱰIn,I,℘=0,asIn converges to I. We can easily see that if limn⟶∞YIn,I,℘=1, limn⟶∞MIn,I,℘=0, and limn⟶∞ⱰIn,I,℘=0, then clearly limn⟶∞YΩIn,ΩI,℘=1, limn⟶∞MΩIn,ΩI,℘=0, and limn⟶∞ⱰΩIn,ΩI,℘=0 for allI∈Ⱳ and ℘>0. Hence, Ω is OC. Also above all cases satisfied PFS ˫-contractive mapping
(108)1YΩI,Ωℏ,℘−1≤θ1YI,ℏ,℘−1,MΩI,Ωℏ,℘≤θMI,ℏ,℘,ⱰΩI,Ωℏ,℘≤θⱰI,ℏ,℘.
All conditions of Theorem21 are satisfied and 1 is a FP of Ω.


## 2.1. Orthogonal Picture Fuzzy Metric Space

Definition 7.
LetⱲ,Y,M,Ɒ,∗,Δ,˫ be called an OPFMS if Ⱳ is a non-empty OS, ∗ is a CTN, Δ is a CTCN, and Y,M,Ɒ are pfs on Ⱳ×Ⱳ×ℝ+ if the following condition are satisfied for all I,ℏ,∂∈Ⱳ with either (I˫ℏ or ℏ˫I), either (I˫ℏ or δ˫I), and either (I˫ℏ∨ℏ˫I):
P1) YI,ℏ,℘+MI,ℏ,℘+ⱰI,ℏ,℘≤1,
P˫2) 0≤YI,ℏ,℘≤1,
P˫3) YI,ℏ,℘=1 if and only if I=ℏ,
P˫4) YI,ℏ,℘=Yℏ,I,℘,
P˫5) YI,∂,℘+π≥YI,ℏ,λ∗Yℏ,e,π,
P˫6) YI,ℏ,∙:0,∞⟶0,1 is continuous,
P˫7) 0≤MI,ℏ,℘≤1,
P˫8) MI,ℏ,℘=1 if and only if I=ℏ,
P˫9) MI,ℏ,℘=Mℏ,I,℘,
P˫10) MI,∂,℘+π≤MI,ℏ,℘ΔMℏ,∂,π,
P˫11) MI,ℏ,∙:0,∞⟶0,1 is continuous,
P˫12) 0≤ⱰI,ℏ,℘≤1,
P˫13) ⱰI,ℏ,℘=1 if and only if I=ℏ,
P˫14) ⱰI,ℏ,℘=Ɒℏ,I,℘,
P˫15) ⱰI,∂,℘+π≤ⱰI,ℏ,℘ΔⱰℏ,∂,π,
P˫16) ⱰI,ℏ,∙:0,∞⟶0,1 is continuous,
P˫17) If ℘≤0 then YI,ℏ,℘=0,MI,ℏ,℘=1 and ⱰI,ℏ,℘=1,
Then,Ⱳ,Y,M,Ɒ,∗,Δ,˫ is called OPFMS.Remark 8.
Every PFMS is an OPFMS but the converse is not true.Example 1.
LetⱲ=−7,7 and define a CTN asa∗b=ab, CTCN as aΔb=maxa,b and define a binary relation ˫ by I˫ℏ iff I+ℏ≥0. Take
(3)YI,ℏ,℘=1ifI=ℏ,℘℘+maxI,ℏifotherwise,MI,ℏ,℘=0ifI=ℏ,maxI,ℏ℘+maxI,ℏifotherwise,ⱰI,ℏ,℘=0ifI=ℏ,maxI,ℏ℘ifotherwise,
for allI,ℏ∈Ⱳ,℘>0, then it is OPFS, but not an PFMS.It is easy to see that forπ=℘=1,I=−1,ℏ=−1/2,∂=−2. (P˫5), (P˫10), and (P˫15) fails.Remark 9.
The above example is also OPFMS if we take(4)MI,ℏ,℘=0ifI=ℏ,1−℘℘+maxI,ℏifotherwise.Definition 10.
An OSIn in an OPFMS Ⱳ,Y,M,Ɒ,∗,Δ,˫ is said to be orthogonal convergent (O-C) toI∈Ⱳ if
(5)limn⟶∞YIn,I,℘=1,∀℘>0,limn⟶∞MIn,I,℘=0,∀℘>0,limn⟶∞ⱰIn,I,℘=0,∀℘>0.Definition 11.
An OSIn in an OPFMS Ⱳ,Y,M,∗,Δ,˫ is said to be Orthogonal Cauchy (O-CS) if there exists n∈ℕ such that
(6)limn⟶∞YIn,In+p,℘=1,limn⟶∞MIn,In+p,℘=0,limn⟶∞ⱰIn,In+p,℘=0,for all ℘≥0,p≥1.Definition 12.
Ω:Ⱳ⟶Ⱳ is OC at I∈Ⱳ in an OPFMS Ⱳ,Y,M,Ɒ,∗,Δ,˫, whenever for each OS In for all n∈ℕ in Ⱳ if limn⟶∞RIn,I,℘=1, limn⟶∞MIn,I,℘=0, and limn⟶∞ⱰIn,I,℘=0 for all℘>0, then limn⟶∞YΩIn,ΩI,℘=1, limn⟶∞MΩIn,ΩI,℘=0, and limn⟶∞ⱰΩIn,ΩI,℘=0 for all ℘>0.Definition 13.
An OPFMSⱲ,Y,M,Ɒ,∗,Δ,˫ is said to be orthogonally complete (O-complete) if every O-CS is convergent.Example 2.
Assume OPFMS as given in Example1 and define a sequence In in Ⱳby In=1−1/n,∀n∈ℕ such that ∀n;In˫In+1 or ∀n;In+1˫In. Define a CTN asa∗b=ab, CTCN as aΔb=maxa,b, and define a binary relation ˫ by I˫ℏ iff I+ℏ≥0. Take
(7)limn⟶∞YIn,I,℘=limn⟶∞1ifI=ℏ,℘℘+maxIn,Iifotherwise,=1ifI=ℏ,℘℘+maxI,Iifotherwise,limn⟶∞MIn,I,℘=limn⟶∞0ifI=ℏ,maxIn,I℘+maxIn,Iifotherwise,=0ifI=ℏmaxI,I℘+maxI,Iifotherwise,limn⟶∞ⱰIn,I,℘=limn⟶∞0ifI=ℏmaxIj,I℘ifotherwise=0ifI=ℏ,maxI,I℘ifotherwise.Example 3.
From proof of Example2, In=1−1/n,∀n∈ℕ is a O-CS in an OPFMS.
(8)limn⟶∞YIn,In+p,℘=limn⟶∞1ifI=ℏ,℘℘+maxIn,In+pifotherwise,=1ifI=ℏ,℘℘+maxI,Iifotherwise,limn⟶∞MIn,In+p,℘=limn⟶∞0ifI=ℏ,maxIn,In+p℘+maxIn,In+pifotherwise,=0ifI=ℏ,maxI,I℘+maxI,Iifotherwise,limn⟶∞ⱰIn,In+p,℘=limn⟶∞0ifI=ℏ,maxIn,In+p℘ifotherwise=0ifI=ℏ,maxI,I℘ifotherwise,for all ℘≥0,p≥1.Lemma 14.
If for somev∈0,1 and I,ℏ∈Ⱳ,(9)YI,ℏ,℘≥YI,ℏ,℘v,℘>0,MI,ℏ,℘≤MI,ℏ,℘v,℘>0ⱰI,ℏ,℘≤ⱰI,ℏ,℘v,℘>0thenI=ℏ.Definition 15.
LetⱲ,Y,M,Ɒ,∗,Δ,˫ be an OPFMS. A map Ω:Ⱳ⟶Ⱳ is an orthogonal contraction if there exists θ∈0,1 such that for every ℘>0 and I,δ∈Ⱳ with I˫ℏ, we have
(10)YψI,Ωℏ,θλ≥YI,ℏ,℘,(11)MΩI,Ωℏ,θ℘≤MI,ℏ,℘,(12)ⱰΩI,Ωℏ,θ℘≤ⱰI,ℏ,℘.Theorem 16.
LetⱲ,Y,M,Ɒ,∗,Δ,˫ be an O-complete PFMS such that
(13)lim℘⟶∞YI,ℏ,℘=1,lim℘⟶∞MI,ℏ,℘=0,andlim℘⟶∞ⱰI,ℏ,℘=0,∀I,ℏ∈Ⱳ.
LetΩ:Ⱳ⟶Ⱳ be an OC, O-CON and OPR. Thus, Ω has a unique FP, say I∗∈Ⱳ. Furthermore,
(14)limn⟶∞YΩnI,I∗,℘=1,limn⟶∞MΩnI,I∗,℘=0,andlimn⟶∞ⱰΩnI,I∗,℘=0∀I∈Ⱳand℘>0.Proof.
SinceⱲ,Y,M,Ɒ,∗,Δ,˫ is an O-complete PFMS, there exists I0∈Ⱳ such that
(15)I0˫ℏforallℏ∈Ⱳ.
That is,I0˫ΩI0. Take
(16)In=ΩnI0=ΩIn−1foralln∈M.
SinceΩ is OPR, In is an OS. Now, since Ω is an O-CON, we get
(17)YIn+1,In,θ℘=YΩIn,ΩIn−1,θ℘≥YIn,In−1,℘,for all n∈M and ℘>0. Note that Y is nondecreasing on 0,∞. Therefore, by applying the above expression, we can deduce
(18)YIn+1,In,℘≥YIn+1,In,θ℘=YΩIn,ΩIn−1,θ℘≥YIn,In−1,℘=YΩIn−1,ΩIn−2,℘≥YIn−1,In−2,℘θ≥⋯≥YI1,I0,℘θn,for all n∈M and ℘>0. Thus, from (15) and (P˫5), we have
(19)YIn,In+α,℘≥YIn,In+1,℘2∗YIn+1,In+α,℘2≥YIn,In+1,℘2∗YIn+1,In+2,℘22∗YIn+2,In+3,℘23∗⋮∗YIn+α−1,In+α,℘2n+α≥YI1,I0,℘2θn∗YI1,I0,℘22θn∗⋮∗YI1,I0,℘2n+αθn.
We know thatlim℘⟶∞YI,ℏ,℘=1, for all I,ℏ∈Ⱳ and ℘>0. So, from (19) we get,
(20)limn⟶∞YIn,In+α,℘≥1∗1∗⋯∗1=1.MIn+1,In,θ℘=MΩIn,ΩIn−1,θ℘≤MIn,In−1,℘,for all n∈ℏ and ℘>0. Therefore, by applying the above expression, we can deduce
(21)MIn+1,In,℘≤MIn+1,In,θ℘=MΩIn,ΩIn−1,θ℘≤MIn,In−1,℘=MΩIn−1,ψIn−2,℘≤MIn−1,In−2,℘θ≤⋯≤MI1,I0,℘θn,for all n∈ℏ and ℘>0. Thus, from (21) and (P⊥10), we have
(22)MIn,In+α,℘≤MIn,In+1,℘2ΔMIn+1,In+α,℘2≤MIn,In+1,℘2ΔMIn+1,In+2,℘22ΔMIn+2,I,℘23Δ⋯ΔMIn+α−1,In+α,℘2n+α≤MI,I0,℘2θnΔMI1,I0,℘22θnΔ⋯ΔMI1,I0,℘2n+αθn.
We know thatlim℘⟶∞MI,ℏ,℘=0, for all I,ℏ∈Ⱳ and ℘>0. So, from (22) we get,
(23)limn⟶∞MIn,In+α,℘≤0Δ0Δ⋯Δ0=0,ⱰIn+1,In,θ℘=ⱰΩIn,ΩIn−1,θ℘≤ⱰIn,In−1,℘,for all n∈M and ℘>0. Therefore, by applying the above expression, we can deduce
(24)ⱰIn+1,In,℘≤ⱰIn+1,In,θ℘=ⱰΩIn,ΩIn−1,θ℘≤ⱰIn,In−1,℘=ⱰΩIn−1,ΩIn−2,℘≤ⱰIn−1,In−2,℘θ≤⋯≤ⱰI1,I0,℘θn,for all n∈M and ℘>0. Thus, from (24) and (P⊥15), we have
(25)ⱰIn,In+α,℘≤ⱰIn,In+1,℘2ΔⱰIn+1,In+α,℘2≤ⱰIn,In+1,℘2ΔⱰIn+1,In+2,℘22ΔⱰIn+2,In+3,℘23Δ..ΔⱰIn+α−1,In+α,℘2n+α≤ⱰI1,I0,℘2θnΔⱰI1,I0,℘22θnΔ⋯ΔⱰI1,I0,℘2n+αθn,
We know thatlim℘⟶∞ⱰI,ℏ,℘=0, for all I,ℏ∈Ⱳ and ℘>0. So, from (25) we get,
(26)limn⟶∞ⱰIn,In+α,℘≤0Δ0Δ⋯Δ0=0.
So,In is a O-CS. The O-completeness of the PFMS Ⱳ,Y,M,Ɒ,∗,Δ,˫ ensures that there existI∗∈Ⱳ such that YIn,I∗,℘⟶1,MIn,I∗,℘⟶0, and ⱰIn,I∗,℘⟶0 as n⟶+∞ for all ℘>0. Now, since Ω is an OC, YIn+1,ΩI∗,℘=YΩIn,ΩI∗,℘⟶1, MIn+1,ΩI∗,℘=MΩIn,ΩI∗,℘⟶0, and ⱰIn+1,ΩI∗,℘=ⱰΩIn,ΩI∗,℘⟶0 as n⟶+∞. Now, we have
(27)YI∗,ψI∗,℘≥YI∗,In+1,℘2∗YIn+1,ΩI∗,℘2,MI∗,ΩI∗,℘≤MI∗,In+1,℘2ΔMIn+1,ΩI∗,℘2,ⱰI∗,ΩI∗,℘≤ⱰI∗,In+1,℘2ΔⱰIn+1,ΩI∗,℘2.
Taking limit asn⟶+∞, we get YI∗,ΩI∗,℘=1∗1=1,MI∗,ΩI∗,℘=0Δ0=0, and ⱰI∗,ΩI∗,℘=0Δ0=0 and hence ΩI∗=I∗.
Now, we show the uniqueness of the FP of the mappingΩ. Assume that I∗ and ℏ∗ are two FPs of Ω such thatI∗≠δ∗. We can get
(28)I0˫I∗andI0˫ℏ∗.
SinceⱰ is OPR, one writes
(29)ΩnI0˫ΩnI∗andΩnI0˫Ωnℏ∗,
for alln∈M. So from (10), we can derive
(30)YΩnI0,ΩnI∗,℘≥YΩnI0,ΩnI∗,θ℘≥YI0,I∗,℘θn,YΩnI0,Ωnℏ∗,℘≥YΩnI0,Ωnℏ∗,θ℘≥YI0,ℏ∗,℘θn.
Therefore,(31)YI∗,ℏ∗,℘=YΩnI∗,Ωnℏ∗,℘≥YΩnI0,ΩnI∗,℘2∗YΩnI0,Ωnℏ∗,℘2≥YI0,I∗,℘2θn∗YI0,ℏ∗,℘2θn⟶1asn⟶∞.
So from (11), we can derive
(32)MΩnI0,ΩnI∗,℘≤MΩnI0,ΩnI∗,θ℘≤MI0,I∗,℘θn,MΩnI0,Ωnℏ∗,℘≤MΩnI0,Ωnℏ∗,θ℘≤MI0,ℏ∗,℘θn.
Therefore,(33)MI∗,ℏ∗,℘=MΩnI∗,Ωnℏ∗,℘≤MΩnI0,ΩnI∗,℘2ΔMΩnI0,Ωnℏ∗,℘2≤MI0,I∗,℘2θnΔMI0,ℏ∗,℘2θn⟶0asn⟶∞.
Similarly, from (12), we can derive
(34)ⱰΩnI0,ΩnI∗,℘≤ⱰΩnI0,ΩnI∗,θ℘≤ⱰI0,I∗,℘θn,ⱰΩnI0,Ωnℏ∗,℘≤ⱰΩnI0,Ωnℏ∗,θ℘≤ⱰI0,ℏ∗,℘θn.
Therefore,(35)ⱰI∗,ℏ∗,℘=ⱰΩnI∗,Ωnℏ∗,℘≤ⱰΩnI0,ΩnI∗,℘2ΔⱰΩnI0,Ωnℏ∗,℘2≤ⱰI0,I∗,℘2θnΔⱰI0,ℏ∗,℘2θn⟶0asn⟶∞.
So,I∗=ℏ; hence, I∗ is the unique FP.Corollary 17.
AssumeⱲ,Y,M,Ɒ,∗,Δ,˫ be an O-complete PFMS. Assume Ω:Ⱳ⟶Ⱳ be O-CON and OPR and if In is an OS with In⟶I∈Ⱳ, thenI˫In for all n∈ℕ. Then, Ω has a unique FP, say I∗∈Ⱳ.Proof.
We can similarly derive as in the proof of Theorem16 that In is a O-CS and so it converges to I∗∈Ⱳ. Hence, I∗˫In for all n∈ℕ. from (10), we can get
(36)YΩI∗,In+1,℘=YΩI∗,ΩIn,℘≥YΩI∗,ΩIn,℘θ≥YI∗,In,℘,limn⟶∞YΩI∗,In+1,℘=1.
Then, we can write(37)YI∗,ΩI∗,℘≥YI∗,In+1,℘2∗YIn+1,ΩI∗,℘2.
Taking limit asn⟶+∞, we get YI∗,ΩI∗,℘=1∗1=1 and from (11), we can get
(38)MΩI∗,In+1,℘=MΩI∗,ΩIn,℘≤MΩI∗,ΩIn,℘θ≤MI∗,In,℘,limn⟶∞MΩI∗,In+1,℘=0.
Then, we can write(39)MI∗,ΩI∗,℘≤MI∗,In+1,℘2ΔMIn+1,ΩI∗,℘2.
Taking limit asn⟶+∞, we get
(40)MI∗,ΩI∗,℘=0Δ0=0,and from (12), we can get
(41)ⱰΩI∗,In+1,℘=ⱰΩI∗,ΩIn,℘≤ⱰΩI∗,ΩIn,℘θ≤ⱰI∗,In,℘,limn⟶∞ⱰΩI∗,In+1,℘=0.
Then, we can write(42)ⱰI∗,ΩI∗,℘≤ⱰI∗,In+1,℘2ΔⱰIn+1,ΩI∗,℘2.
Taking limit asn⟶+∞, we get
(43)ⱰI∗,ΩI∗,℘=0Δ0=0,
soΩI∗=I∗. Next proof is similar as in Theorem 16.Example 4.
LetⱲ=−3,3. We define a binary relation ˫ by I˫ℏ⟺I+ℏ≥0.
Define an OPFMS as in Example 1 by(44)YI,ℏ,℘=1ifI=ℏ,℘℘+maxI,ℏifotherwise,MI,ℏ,℘=0ifI=ℏ,maxI,ℏ℘+maxI,ℏifotherwise,ⱰI,ℏ,℘=0ifI=ℏ,maxI,ℏ℘ifotherwise,
for allI,δ∈Ⱳ,℘>0, with the CTN a∗b=a∙b and CTCN aΔb=maxa,b. Then, Ⱳ,Y,M,Ɒ,∗,Δ,˫ is an O-complete PFMS. Define Ω:Ⱳ⟶Ⱳ by
(45)ΩI=I4,I∈−3,00,I∈0,3.
Then, the following cases are satisfied:(1)
IfI∈−3,0 andℏ∈0,3, then ΩI=I/4 andψℏ=0(2)
IfI,ℏ∈−3,0, then ΩI=I/4 andΩℏ=ℏ/4(3)
IfI,ℏ∈0,3, then ΩI=0 andΩℏ=0(4)
IfI∈0,3 andℏ∈−3,0, then ΩI=0 andΩℏ=ℏ/4
This clearly implies thatΩI+Ωℏ≥0. Hence, Ω is OPR. We can easily see that if limn⟶∞YIn,I,℘=1, then limn⟶∞YΩIj,ΩI,℘=1,limn⟶∞MIn,I,℘=0, then limn⟶∞MΩIn,ΩI,℘=0 and limn⟶∞ⱰIn,I,℘=0, then limn⟶∞ⱰΩIn,ΩI,℘=0 for allI∈Ⱳ and ℘>0. Hence, Ω is OC.
The above four cases forθ∈1/2,1 satisfies the below contractive conditions:
(46)YΩI,Ωℏ,θ℘≥YI,ℏ,℘,MΩI,Ωℏ,θ℘≤MI,ℏ,℘,ⱰΩI,Ωℏ,θ℘≤ⱰI,ℏ,℘.
All conditions of Theorem16 are satisfied. Also, 0 is FP of Ω.Theorem 18.
LetⱲ,Y,M,Ɒ,∗,Δ,˫ be an O-complete PFMS such that
(47)lim℘⟶∞YI,ℏ,℘=1andlim℘⟶∞MI,ℏ,℘=0,∀I,ℏ∈Ⱳand℘>0.
LetΩ:Ⱳ⟶Ⱳ be OC,O-CON and OPR. Assume that there exist θ∈0,1 and ℘>0 such that
(48)YΩI,Ωℏ,θ℘≥minYΩI,I,℘,YΩℏ,ℏ,℘,MΩI,Ωℏ,θ℘≤minMΩI,I,℘,MΩℏ,ℏ,℘,ⱰΩI,Ωℏ,θ℘≤minⱰΩI,I,℘,ⱰΩℏ,ℏ,℘,for all I,ℏ∈Ⱳ,℘>0. Then, Ω has a unique FP, soI∗∈Ⱳ. Furthermore, limn⟶∞YΩnI,I∗,℘=1, limn⟶∞MΩnI,I∗,℘=0, and limn⟶∞ⱰΩnI,I∗,℘=0 for all I∈Ⱳ and ℘>0.Proof.
SinceⱲ,Y,M,Ɒ,∗,Δ,˫ is an O-complete PFMS, there exists I0∈Ⱳ such that
(49)I0˫ℏ,∀δ∈Ⱳ.
Thus,I˫h.Consider
(50)In=ΩnI0=ΩIn−1,∀n∈S.
SinceΩ is OPR, In is an OS. We can get
(51)YIn+1,In,℘≥YIn+1,In,θ℘=YΩIn,ΩIn−1,θ℘≥minYΩIn,In,℘,YΩIn−1,In−1,℘,MIn+1,In,℘≤MIn+1,In,θ℘=MΩIn,ΩIn−1,θ℘≤minMΩIn,In,℘,MΩIn−1,In−1,℘,ⱰIn+1,In,℘≤ⱰIn+1,In,θ℘=ⱰΩIn,ΩIn−1,θ℘≤minⱰΩIn,In,℘,ⱰΩIn−1,In−1,℘.
Two cases arise:
Case1: IfYIn+1,In,℘≥YΩIn,In,℘, then
(52)YIn+1,In,℘≥YIn+1,In,θ℘≥YΩIn,In,℘=YIn+1,In,℘,MIn+1,In,℘≤MΩIn,In,℘.
Then,(53)MIn+1,In,℘≤MIn+1,In,θ℘≤MΩIn,In,℘=MIn+1,In,℘,ⱰIn+1,In,℘≤ⱰΩIn,In,℘.
Then,(54)ⱰIn+1,In,℘≤ⱰIn+1,In,θ℘≤ⱰΩIn,In,℘=ⱰIn+1,In,℘.
Then by Lemma14, In=In+1 for all n∈ℕ.
Case2: IfYIn+1,In,℘≥YΩIn−1,In−1,℘, then
(55)YIn+1,In,℘≥YIn+1,In,θ℘≥YΩIn−1,In−1,℘≥YIn,In−1,℘,MIn+1,In,℘≤MΩIn−1,In−1,℘.
Then,(56)MIn+1,In,℘≤MIn+1,In,θ℘≤MΩIn−1,In−1,℘≤MIn,In−1,℘,and
(57)ⱰIn+1,In,℘≤ⱰΩIn−1,In−1,℘.
Then,(58)ⱰIn+1,In,℘≤ⱰIn+1,In,θ℘≤ⱰΩIn−1,In−1,℘≤ⱰIn,In−1,℘,for all n∈ℕ and ℘>0. Then by Theorem 16, we have a OCS. By completeness of Ⱳ,Y,M,Ɒ,∗,Δ,˫, there exists I∗∈Ⱳ such that limn⟶∞RIn,I∗,℘=1, limn⟶∞MIn,I∗,℘=0, and limn⟶∞ⱰIn,I∗,℘=0, for all ℘>0.
We know thatΩ is an OC, then
(59)limn⟶∞YIn+1,ΩI∗,℘=limn⟶∞YΩIn,ΩI∗,℘=1,limn⟶∞MIn+1,ΩI∗,℘=limn⟶∞MΩIn,ΩI∗,℘=0,limn⟶∞ⱰIn+1,ΩI∗,℘=limn⟶∞ⱰΩIj,ΩI∗,℘=0.
Now, we prove thatI∗ is a FP of Ω. Let ℘1∈θ,1 and ℘2=1−℘1. Then,
(60)YΩI∗,I∗,℘≥YΩI∗,In+1,℘℘12∗YIn+1,I∗,℘℘22=YΩI∗,ΩIn,℘℘12∗YIn+1,I∗,℘℘22≥minYΩI∗,I∗,℘℘12θ,YψIn,In,℘℘12θ∗YIn+1,I∗,℘℘22=minYΩI∗,I∗,℘℘12θ,YIn+1,In,℘℘12θ∗YIn+1,I∗,℘℘22.
Takingn⟶∞, we get
(61)YΩI∗,I∗,℘≥minYΩI∗,I∗,℘℘12θ,1∗1,YΩI∗,I∗,℘≥YΩI∗,I∗,℘v,℘>0,MΩI∗,I∗,℘≤MΩI∗,In+1,℘℘12ΔMIn+1,I∗,℘℘22=MΩI∗,ΩIn,℘℘12ΔMIn+1,I∗,℘℘22≤minMΩI∗,I∗,℘℘12θ,MΩIn,In,℘℘12θΔMIn+1,I∗,℘℘22=minMΩI∗,I∗,℘℘12θ,MIn+1,In,℘℘12θΔMIn+1,I∗,℘℘22.
Takingn⟶∞, we get
(62)MΩI∗,I∗,℘≤minMΩI∗,I∗,℘℘12θ,0Δ0,MΩI∗,I∗,℘≤MΩI∗,I∗,℘v,℘>0,ⱰΩI∗,I∗,℘≤ⱰΩI∗,In+1,℘℘12ΔⱰIn+1,I∗,℘℘22=ⱰΩI∗,ΩIn,℘℘12ΔⱰIn+1,I∗,℘℘22≤minⱰΩI∗,I∗,℘℘12θ,ⱰΩIn,In,℘℘12θΔⱰIn+1,I,℘℘22=minⱰΩI∗,I∗,℘℘12θ,ⱰIn+1,In,℘℘12θΔⱰIn+1,I∗,℘℘22.
Takingn⟶∞, we get
(63)ⱰΩI∗,I∗,℘≤minⱰΩI∗,I∗,℘℘12θ,0Δ0,ⱰΩI∗,I∗,℘≤ⱰΩI∗,I∗,℘v,℘>0.
Here,v=2θ/℘1∈0,1, from Lemma 14, we haveΩI∗=I∗. Suppose v∗ and ℏ∗ the FPs of Ω. We have
(64)I0˫I∗andI0˫I∗.
BecauseΩ is an OPR, so we can write
(65)ΩnI0˫ΩnI∗andΩnI0˫ΩnI∗foralln∈ℕ.
We can write(66)YΩnI0,ΩnI∗,℘≥YΩnI0,ΩnI∗,θ℘≥minYΩnI0,I0,℘,YΩnI∗,I∗,℘,YΩnI0,Ωnℏ∗,℘≥YΩnI0,Ωnℏ∗,θ℘≥minYΩnI0,I0,℘,YΩnℏ∗,ℏ∗,℘.
Hence, we write that(67)YI∗,ℏ∗,℘=YΩnI∗,Ωnℏ∗,℘≥minYΩnI∗,I∗,℘θ,YΩnℏ∗,ℏ∗,℘θ=min1,1=1,MΩnI0,ΩnI∗,℘≤MΩnI0,ΩnI∗,θ℘≤minMΩnI0,I0,℘,MΩnI∗,I∗,℘,MΩnI0,Ωnℏ∗,℘≤MΩnI0,Ωnℏ∗,θ℘≤minMΩnI0,I0,℘,MΩnℏ∗,ℏ∗,℘.
Hence, we write that(68)MI∗,ℏ∗,℘=MΩnI∗,Ωnℏ∗,℘≤minMΩnI∗,I∗,℘θ,MΩnℏ∗,ℏ∗,℘θ=min0,0=0,ⱰΩnI0,ΩnI∗,℘≤ⱰΩnI0,ΩnI∗,θ℘≤minⱰΩnI0,I0,℘,ⱰΩnI∗,I∗,℘,ⱰΩnI0,Ωnℏ∗,℘≤ⱰΩnI0,Ωnℏ∗,θ℘≤minⱰΩnI0,I0,℘,ⱰΩnℏ∗,ℏ,℘.
Hence, we write that(69)ⱰI∗,ℏ∗,℘=ⱰΩnI∗,Ωnℏ∗,℘≤minⱰΩnI∗,I∗,℘θ,ⱰΩnℏ∗,ℏ∗,℘θ=min0,0=0,for all ℘>0. Hence, I∗=ℏ∗.Corollary 19.
LetⱲ,Y,M,∗,Δ,˫ be an O-complete PFMS and Ω:Ⱳ⟶Ⱳ be an OC and OPR. Then θ∈0,1, we get ℘>0,
(70)YΩI,Ωℏ,θ℘≥minYΩI,I,℘,YΩℏ,ℏ,℘,YI,ℏ,℘,MΩI,Ωℏ,θ℘≤minMΩI,I,℘,MΩℏ,ℏ,℘,MI,ℏ,℘,ⱰΩI,Ωℏ,θ℘≤minⱰΩI,I,℘,ⱰΩℏ,ℏ,℘,ⱰI,ℏ,℘,
Then,Ω has a unique FP.Proof.
It follows from Theorem16 and Theorem 18.Example 5.
LetⱲ=−2,2 and define a binary relation ⊥ by
(71)I˫ℏ⟺I+ℏ≥0.
DefineYandMby
(72)YI,ℏ,℘=1ifI=ℏ℘℘+maxI,ℏotherwise,MI,ℏ,℘=0ifI=ℏmaxI,ℏλ+maxI,ℏotherwise,ⱰI,ℏ,℘=0ifI=ℏ,maxI,ℏλotherwise,
for allI,ℏ∈Ⱳ and ℘>0, with the CTN and CTCN, respectively;a∗b=a.b, aΔb=maxa,b then Ⱳ,Y,M,Ɒ,∗,Δ,˫ is an O-complete PFMS. Note thatlim℘⟶∞YI,ℏ,℘=1, lim℘⟶∞MI,ℏ,℘=0, and lim℘⟶∞ⱰI,ℏ,℘=0 ∀ I,ℏ∈E. Define Ω:Ⱳ⟶Ⱳ by
(73)ΩI=I4,I∈−2,231−I,I∈23,1I−12,I∈1,2.
We have the following cases:(1)
IfI,ℏ∈−2,2/3, then ΩI=I/4 andΩℏ=ℏ/4(2)
IfI,ℏ∈2/3,1, then ΩI=1−I andΩℏ=1−ℏ(3)
IfI,ℏ∈1,2, then ΩI=I−1/2 andΩℏ=ℏ−1/2(4)
IfI∈−2,2/3 andℏ∈2/3,1, then ΩI=I/4 andΩℏ=1−ℏ(5)
IfI∈−2,2/3 and ℏ∈1,2, then ΩI=I/4 andΩℏ=ℏ−1/2(6)
IfI∈2/3,1 and ℏ∈1,2, then ΩI=1−I andΩℏ=ℏ−1/2(7)
IfI∈1,2 and ℏ∈2/3,1, then ΩI=I−1/2 andΩℏ=1−ℏ(8)
IfI∈1,2 and ℏ∈−2,2/3, then ΩI=I−1/2 andΩℏ=ℏ/4(9)
IfI∈2/3,1 and ℏ∈−2,2/3, then ΩI=1−I andΩℏ=ℏ/4
BecauseI˫ℏ⟺I+ℏ≥0, it is clearly implying thatΩI+Ωℏ≥0. Hence, Ω is OPR. Let In be an arbitrary OS in Ⱳ that converges toI∈Ⱳ. We have
(74)limn⟶∞YIn,I,℘=limn⟶∞1ifI=ℏ,℘℘+maxIn,Iotherwise=1,limn⟶∞MIn,I,℘=limn⟶∞1ifI=ℏ,maxIn,I℘+maxIn,Iotherwise=0,limn⟶∞ⱰIn,I,℘=limn⟶∞1ifI=ℏ,maxIn,I℘otherwise=0.
Note that iflimn⟶∞YIn,I,λ=1, limn⟶∞SIn,I,℘=0, and limn⟶∞ⱰIn,I,℘=0, then limn⟶∞YΩIn,ΩI,℘=1, limn⟶∞MΩIn,ΩI,℘=0, and limn⟶∞ⱰΩIn,ΩI,℘=0 for allI∈Ⱳ and ℘>0. Hence, Ω is OC. The case I=ℏ is clear. LetI≠ℏ. We have
(75)YΩI,Ωℏ,θ℘≥minYΩI,I,℘,YΩℏ,ℏ,℘,MΩI,Ωℏ,θ℘≤minMΩI,I,℘,MΩℏ,ℏ,℘,ⱰΩI,Ωℏ,θ℘≤minⱰΩI,I,℘,ⱰΩℏ,ℏ,℘.
Indeed, it is satisfied for all above 9 cases. But,Ωis not a contraction. Assume
(76)minYΩI,I,℘,YΩℏ,ℏ,℘=YΩI,I,℘,minMΩI,I,℘,MΩℏ,ℏ,℘=MΩI,I,℘,minⱰΩI,I,℘,TΩℏ,ℏ,℘=ⱰΩI,I,℘,then for I=−1,ℏ=−2, we have
(77)YΩI,Ωℏ,θ℘=θ℘θ℘+maxI/4,δ/4=4θ℘4θ℘−1≥1,MΩI,Ωℏ,θ℘=maxI/4,ℏ/4θλ+maxI/4,ℏ/4=−14θ℘−1≤0,MΩI,Ωℏ,θ℘=maxI/4,ℏ/4θλ=−14θ℘≤0.
It is a contradiction. Hence, all the conditions of Theorem18 are satisfied and 0 is the unique FP ofψ.Definition 20.
LetⱲ,Y,M,Ɒ,∗,Δ,˫ be an OPFMS. A mapping Ω:Ⱳ⟶Ⱳ is named to be an PF ˫-contractive if ∃ θ∈0,1 so that
(78)1YΩI,Ωℏ,λ−1≤θ1YI,ℏ,λ−1,(79)MΩI,Ωℏ,℘≤θMI,ℏ,℘andⱰΩI,Ωℏ,℘≤θⱰI,ℏ,℘,
for allI,ℏ∈Ⱳ and ℘>0. Here, θ is called the PFS ˫-contractive constant ofΩ.Theorem 21.
LetⱲ,Y,M,Ɒ,∗,Δ,˫ be an O-complete PFMS such that
(80)lim℘⟶∞YI,ℏ,℘=1,lim℘⟶∞MI,ℏ,℘=0,andlim℘⟶∞ⱰI,ℏ,℘=0,∀I,ℏ∈Ⱳ.
LetΩ:Ⱳ⟶Ⱳ be an OC,˫ -contraction and OPR. Thus, Ω has a FP, sayυ∈Ⱳ, Yυ,υ,℘=1,Mυ,υ,℘=0 and Ɒυ,υ,℘=0 for all ℘>0.Proof.
LetⱲ,Y,M,Ɒ,∗,Δ,˫ be an O-complete PFMS. For an arbitraryI0∈Ⱳ,
(81)I0˫ℏ,∀ℏ∈Ⱳ.
That is,I0˫ΩI0. Consider
(82)In=ΩnI0=ΩIn−1foralln∈ℕ.
SinceΩ is OPR, In is an OS. If In=In−1 for some n∈ℕ, then In is a FP ofΩ. We assume that In≠In−1 for all n∈ℕ. For all ℘>0 and n∈ℕ, we get from (12),
(83)1YIn,In+1,℘−1=1YΩIn−1,ΩIn,℘−1≤θ1YIn−1,In,℘−1,MIn,In+1,℘=MΩIn−1,ΩIn,℘≤θMIn−1,In,℘,ⱰIn,In+1,℘=ⱰΩIn−1,ΩIn,℘≤θⱰIn−1,In,℘.
We have∀℘>0(84)1YIn,In+1,℘≤θYIn−1,In,℘+1−θ,
Implying that(85)θYΩIn−2,ΩIn−1,℘+1−θ≤θ2YIn−2,In−1,℘+θ1−θ+1−θ.
Continuing in this way, we get(86)1YIn,In+1,℘≤θnYI0,I1,℘+θn−11−θ+θn−21−θ+⋯+θ1−θ+1−θ≤θnYI0,I1,℘+θn−1+θn−2+⋯+11−θ≤θnYI0,I1,℘+1−θn.
We have(87)1θn/YI0,I1,℘+1−θn≤YIn,In+1,℘,∀℘>0,n∈ℕ,(88)MIn,In+1,℘=MΩIn−1,ΩIn,℘≤θMIn−1,In,℘=θMΩIn−2,ΩIn−1,℘≤θ2MIn−2,In−1,℘≤⋯≤θnMI0,I1,℘.∀℘>0,n∈ℕ,(89)ⱰIn,In+1,℘=ⱰψIn−1,ψIn,℘≤θⱰIn−1,In,℘=θⱰψIn−2,ψIn−1,℘≤θ2Ɒβn−2,In−1,℘≤⋯≤θnⱰI0,I1,℘.∀℘>0,n∈ℕ.
Now, form≥1 and n∈ℕ, we have
(90)YIn,In+m,℘≥YIn,In+1,℘2∗YIn+1,In+m,℘2≥YIn,In+1,℘2∗YIn+1,In+2,℘22∗YIn+2,In+m,℘22.
Again, continuing in this way, we get(91)YIn,In+m,℘≥YIn,In+1,℘2∗YIn+1,In+2,℘22∗⋯∗YIn+m−1,In+m,℘2m−1,MIn,In+p,℘≤MIn,In+1,℘2ΔMIn+1,In+p,℘2≤MIn,In+1,℘2ΔMIn+1,In+2,℘22ΔMIn+2,In+p,℘22.
Continuing in this way, we get(92)MIn,In+p,℘≤MIn,In+1,℘2ΔMIn+1,In+2,℘22Δ⋯ΔMIn+p−1,In+p,℘2p−1,ⱰIn,In+p,℘≤ⱰIn,In+1,℘2ΔⱰIn+1,In+p,℘2≤ⱰIn,In+1,℘2ΔⱰIn+1,In+2,℘22ΔⱰIn+2,In+p,℘22.
Continuing in this way, we get(93)ⱰIn,In+p,℘≤ⱰIn,In+1,℘2ΔⱰIn+1,In+2,℘22Δ⋯ΔⱰIn+p−1,In+p,℘2p−1.
By using (87) in the above inequality, we have
(94)YIn,In+m,℘≥1θn/YI0,I1,℘/2+1−θn∗1θn+1/YI0,I1,℘/22+1−θn+1∗⋯∗1θn+m−1/YI0,I1,℘/2m−1+1−θn+m−1,≥1θn/YI0,I1,℘/2+1∗1θn+1/YI0,I1,℘/22+1∗⋯∗1θn+m−1/YI0,I1,℘/2m−1+1,using (88),
(95)SIn,In+p,℘≤SIn,In+1,℘2ΔSIn+1,In+2,℘22Δ⋯ΔSIn+p−1,In+p,℘2p−1,and using (89)
(96)ⱰIn,In+p,℘≤ⱰIn,In+1,℘2ΔⱰIn+1,In+2,℘22Δ⋯ΔⱰIn+p−1,In+p,℘2p−1,
θ∈0,1 we deduce from the above expression that limn⟶∞YIn,In+m,℘=1,limn⟶∞MIn,In+m,℘=0, and limn⟶∞ⱰIn,In+m,℘=0 for all ℘>0,m≥1.
Therefore,In is a O-CS in Ⱳ,Y,M,Ɒ,∗,Δ,˫. By the completeness of Ⱳ,Y,M,Ɒ,∗,Δ,˫, we know that Ω is an OC and there exists υ∈Ⱳ such that
(97)limn⟶∞YIn+1,υ,℘=limn⟶∞YΩIn,Ωυ,℘=1,∀℘>0,(98)limn⟶∞MIn+1,υ,℘=limn⟶∞MΩIn,Ωυ,℘=0∀℘>0,(99)limn⟶∞ⱰIn+1,υ,℘=limn⟶∞ⱰΩIn,Ωυ,℘=0∀℘>0.
Now, we prove thatυ is a FP of Ω. For this, we obtain from (78) that
(100)1YΩIn,Ωυ,℘−1≤θ1YIn,υ,℘−1=θYIn,υ,℘−θ.
That is,(101)1θ/YIn,υ,℘+1−θ≤YΩIn,Ωυ,℘.
Using the above inequality, we obtain(102)Yυ,Ωυ,℘≥Yυ,In+1,℘2∗YIn+1,Ωυ,℘2=Yυ,In+1,℘2∗YΩIn,Ωυ,℘≥Yυ,In+1,℘2∗1θ/YIn,υ,℘/2+1−θ,Mw,v,℘=MΩw,Ωv,℘≤θMw,v,℘<Mw,v,℘,=Mw,In+1,℘2ΔMΩIn,Ωw,℘≤Mw,In+1,℘2ΔθMIn,w,℘2,Ɒw,v,℘=ⱰΩw,Ωv,℘≤θⱰw,v,℘<Ɒw,v,℘,=Ɒw,In+1,℘2ΔⱰΩIn,Ωw,℘2≤Ɒw,In+1,℘2ΔθⱰIn,w,℘2.
Taking limit asn⟶∞ and using (97), (98), and (99) in the above expression, we get that Yυ,Ωυ,℘=1, Mυ,Ωυ,℘=0, and Ɒυ,Ωυ,℘=0, that is, Ωυ=υ. Therefore, υ is a FP of Ω, Yυ,υ,℘=1, Mυ,υ,℘=0, and Ɒυ,υ,℘=0 for all ℘>0.Corollary 22.
LetⱲ,Y,M,Ɒ,∗,Δ,˫ be a O-complete PFMS and Ω:Ⱳ⟶Ⱳ satisfy
(103)1YΩnI,Ωnℏ,℘−1≤θ1YI,ℏ,℘−1,MΩnI,Ωnℏ,℘≤θMI,ℏ,℘,ⱰΩnI,Ωnℏ,℘≤θⱰI,ℏ,℘,for all n∈ℕ,I,ℏ∈Ⱳ,℘>0, where 0<θ<1. Then, Ω has a FP.Proof.
υ∈Ⱳ is the unique FP of Ωn by using Theorem 21, and Yυ,υ,℘=1,Mυ,υ,℘=0,Ɒυ,υ,℘=0,∀℘>0.Ωυ is also a FP of Ωn as ΩnΩυ=Ωυ. From Theorem 21, Ωυ=υ,υ is a FP since the FP of Ω is also a FP of Ωn.Example 6.
LetⱲ=−1,2 and define a binary relation ˫ by
(104)I˫ℏ⟺I+ℏ≥0.
DefineY,M,Ɒ by
(105)YI,ℏ,℘=1ifI=ℏ,℘℘+maxI,ℏifotherwise,MI,ℏ,℘=0ifI=ℏ,1−℘℘+maxI,ℏifotherwise,ⱰI,ℏ,℘=0ifI=ℏ,maxI,ℏ℘ifotherwise.
With CTNa∗b=a.b and CTCN aΔb=maxa,b then Ⱳ,Y,M,Ɒ,∗,Δ,˫ is an O-complete PFMS. Also observe thatlim℘⟶∞YI,ℏ,℘=1, lim℘⟶∞SI,ℏ,℘=0, andlim℘⟶∞ⱰI,ℏ,℘=0, ∀ I,ℏ∈Ⱳ.
DefineΩ:Ⱳ⟶Ⱳ by
(106)ΩI=2−I,I∈−1,11,I∈1,2.
Therefore, it will satisfy the following cases:(1)
IfI,ℏ∈−1,1, then ΩI=2−I and Ωℏ=2−ℏ(2)
IfI,ℏ∈1,2, then ΩI=Ωℏ=1(3)
IfI∈−1,1 and ℏ∈1,2, then ΩI=2−I and Ωℏ=1(4)
IfI∈1,2 and ℏ∈−1,1, then ΩI=1 and Ωℏ=2−ℏ
BecauseI˫ℏ⟺I+ℏ≥0, it is clearly implying thatΩI+Ωℏ≥0. Hence, Ω is OPR. Let In be an arbitrary OS in Ⱳ that In converges toI∈Ⱳ.
(107)limn⟶∞YIn,I,℘=1,limn⟶∞MIn,I,℘=0,limn⟶∞ⱰIn,I,℘=0,asIn converges to I. We can easily see that if limn⟶∞YIn,I,℘=1, limn⟶∞MIn,I,℘=0, and limn⟶∞ⱰIn,I,℘=0, then clearly limn⟶∞YΩIn,ΩI,℘=1, limn⟶∞MΩIn,ΩI,℘=0, and limn⟶∞ⱰΩIn,ΩI,℘=0 for allI∈Ⱳ and ℘>0. Hence, Ω is OC. Also above all cases satisfied PFS ˫-contractive mapping
(108)1YΩI,Ωℏ,℘−1≤θ1YI,ℏ,℘−1,MΩI,Ωℏ,℘≤θMI,ℏ,℘,ⱰΩI,Ωℏ,℘≤θⱰI,ℏ,℘.
All conditions of Theorem21 are satisfied and 1 is a FP of Ω.


## 3. Conclusions

A picture fuzzy set is more proficient and more capable than an intuitionistic fuzzy set and fuzzy to cope with uncertain and unpredictable information in realistic issues. Herein, we have introduced the notion of orthogonal picture fuzzy metric space and investigated some new type of fixed point theorems in this new setting. Moreover, we have provided non-trivial examples to demonstrate the viability of the proposed results. Since our structure is more general than the class of picture fuzzy metric spaces, our results and notions expand and generalize several previous results. This work can be easily extended in the structure of orthogonal picture fuzzy cone metric spaces, and orthogonal picture fuzzy bipolar metric spaces.



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*Source: 5863328-2022-05-11.xml*