# Fixed Point Results in Orthogonal Neutrosophic Metric Spaces

Authors: Umar Ishtiaq; Khalil Javed; Fahim Uddin; Manuel de la Sen; Khalil Ahmed; Muhammad Usman Ali
Journal: Complexity
 (2021)

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## Abstract

Neutrosophy deals with neutrosophic logic, probability, and sets. Actually, the neutrosophic set is a generalization of the classical set, fuzzy set, and intuitionistic fuzzy set. A neutrosophic set is a mathematical notion serving issues containing inconsistent, indeterminate, and imprecise data. The notion of intuitionistic fuzzy metric space is useful in modelling some phenomena, where it is necessary to study the relationship between two probability functions. In this study, the concept of an orthogonal neutrosophic metric space is initiated. It is a generalization of the neutrosophic metric space. Some fixed point results are investigated in this setting. For the validity of the obtained results, some nontrivial examples are given.


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## Body

## 1. Introduction

Zadeh [1] was the first to put forward the concept of fuzzy sets (FSs), and this idea has deeply influenced many scientific fields since its inception. Using the concepts of probabilistic metric space and fuzzy sets, fuzzy metric space (FMS) was introduced in [2]. Kaleva and Seikkala [3] defined the FMS as a distance between two points to be nonnegative fuzzy numbers. Afterward, the utility of FMS appeared in applied sciences such as fixed point theory, image and signal processing, medical imaging, and decision-making. This classical result was generalized in different spaces, and different structures were attained using this topic, and one may recall the existing notions, ordered theoretic results in fuzzy b metric spaces [4], fuzzy b-metric-like spaces [5], orthogonal control fuzzy metric spaces [6], fuzzy generalized prioritized weighted average operator [7], fuzzy C-means clustering with locality preservation [8], the measure of divergence with fuzzy [9], and many more.In 1986, Zadeh [1] introduced the concept of intuitionistic fuzzy sets (IFSs), which is considered as a new version of the idea of the classical set, defined by Hezarjaribi [10]. Park [11] utilized the idea of IFSs and initiated the concept of intuitionistic fuzzy metric spaces (IFMSs). Later, Kirişci and Simsek [12] introduced the notion of neutrosophic metric spaces (NMSs) which deal with membership, nonmembership, and naturalness functions. Simsek and Kirişci [13] and Sowndrarajan et al. [14] proved some fixed point results in the setting of NMSs.On the other hand, Eshaghi et al. [15] introduced the approach of an orthogonal set (OS) and generalized the Banach fixed point (FP) theorem. Furthermore, fixed point results on orthogonal (generalized) metric spaces have been provided by Senapati et al. [16], Javed et al. [17], and Uddin et al. [18]. Using the fuzzy structure, Hezarjaribi [10] initiated the notion of an orthogonal fuzzy metric space (OFMS) and established the Banach contraction principle.In this study, we have introduced the notion of orthogonal neutrosophic metric space, which is a generalization of FMS, IFMS, and NMS, and we have provided nontrivial examples to justify our claim that our results would be more efficient in dealing with many scientific problems than those in the literature.The main objectives of this study are as follows.(1)
To introduce the concept of orthogonal neutrosophic metric space (ONMSs)(2)
To prove some fixed point results in the sense of orthogonal neutrosophic metric space(3)
To enhance existing literature of fuzzy metric spaces and fuzzy fixed point theoryThis study is organized with some rudimentary concepts of IFMS, NMS, OSs, and O-sequences. The concept of ONMSs is discussed in detail, and some fixed point results with nontrivial examples are imparted. A conclusion is provided for the obtained results.In the end, some basic definitions which are important for the understanding of this study and proofs of some new results (Theorem2 and Corollary 1) are accommodated in the Appendix section to avoid obscurity and vagueness.Throughout this study, CTN is used for a continuous triangular norm and CTCN for a continuous triangular conorm.


## 2. Preliminaries

In this section, some basic definitions are given that are helpful to understand the main results.Park introduced the concept of intuitionistic fuzzy metric spaces (IFMSs) and utilized this idea to investigate fixed point results. Park defined the notion of IFMSs as follows.Definition 1.
(See [11]). Suppose E≠∅ is an arbitrary set, assume a five-tuple E,R,S,∗,Δ, where ∗ is a CTN, Δ is a CTCN, and R,S are the FSs on E×E×0,∞. If E,R,S,∗,Δ meet the following circumstances for all β,δ,∂∈Eandπ,λ>0,(B1)
Rβ,δ,λ+Sβ,δ,λ≤1(B2)
Rβ,δ,λ>0(B3)
Rβ,δ,λ=1⟺β=δ(B4)
Rβ,δ,λ=Rδ,β,λ(B5)
Rβ,∂,λ+π≥Rβ,δ,λ∗Rδ,∂,π(B6)
Rβ,δ,Δ is a nondecreasing (ND) function of ℝ+ and limλ⟶∞Rβ,δ,λ=1(B7)
Sβ,δ,λ>0(B8)
Sβ,δ,λ=0⟺β=δ(B9)
Sβ,δ,λ=Sδ,β,λ(B10)
Sβ,∂,λ+π≤Sβ,δ,λ⟺Sδ,∂,π(B11)
Sβ,δ,Δ is a nonincreasing (NI) function of ℝ+ and limλ⟶∞Sβ,δ,λ=0
Then,E,R,S,∗,Δ is an IFMS.
The concept of neutrosophic metric spaces was discussed by Kirişci and Simsek, and they defined the said concept as follows:Definition 2.
(See [11]). Suppose E≠∅, assume a five-tuple E,R,S,T,∗,Δ, where ∗ is a CTN, Δ is a CTCN, and R,S,T are the neutrosophic sets (NSs) on E×E×0,∞. If E,R,S,T,∗,Δ meet the following circumstances for all β,δ,∂∈Eandπ,λ>0,(N1)
Rβ,δ,λ+Sβ,δ,λ+Tβ,δ,λ≤3(N2)
Rβ,δ,λ>0(N3)
Rβ,δ,λ=1⟺β=δ(N4)
Rβ,δ,λ=Rδ,β,λ(N5)
Rβ,∂,λ+π≥Rβ,δ,λ∗Rδ,∂,π(N6)
Rβ,δ,Δ is a nondecreasing (ND) function of ℝ+ and limλ⟶∞Rβ,δ,λ=1(N7)
Sβ,δ,λ<1(N8)
Sβ,δ,λ=0⟺β=δ(N9)
Sβ,δ,λ=Sδ,β,λ(N10)
Sβ,∂,λ+π≤Sβ,δ,λΔSδ,∂,π(N11)
Sβ,δ,Δ is a nonincreasing (NI) function of ℝ+ and limλ⟶∞Sβ,δ,λ=0(N12)
Tβ,δ,λ<1(N13)
Tβ,δ,λ=0⟺β=δ(N14)
Tβ,δ,λ=Tδ,β,λ(N15)
Tβ,∂,λ+π≤Tβ,δ,λΔTδ,∂,π(N16)
Tβ,δ,Δ is a nonincreasing (NI) function of ℝ+ and limλ⟶∞Tβ,δ,λ=0(N17)
Ifλ≤0, then Rβ,δ,λ=0,Sβ,δ,λ=1 and Tβ,δ,λ=1
Then,E,R,S,T,∗,Δ is a NMS.
Eshaghi et al. introduced the concept of orthogonal sets using a binary relation and defined it as:Definition 3 (See [15]).
AssumeE≠Δ and ⊥∈E×E is a binary relation. Assume there exists β0∈E, such that β0⊥β or β⊥β0 for all β∈E. Thus, E is said to be an OS. Furthermore, we denote OS by E,⊥.Definition 4 (See [15]).
Suppose thatE,⊥ is an OS. A sequence βn for n∈ℕ is called an O-sequence if ∀n;βn⊥βn+1 or ∀n;βn+1⊥βn.
More definitions and basic results related to these topics are available in the Appendix section at the end of this study.


## 3. Orthogonal Neutrosophic Metric Spaces

Now, we introduce the notion of ONMSs and utilize this concept to investigate some fixed point results.Definition 5.
LetE,R,S,T,∗,Δ,⊥ is called an ONMS if E is a (nonempty) OS, where ∗ is a CTN, Δ is a CTCN, and R,S,T are the NSs on E×E×0,∞ meeting the following circumstances:B⊥(1) Rβ,δ,λ+Sβ,δ,λ+Tβ,δ,λ≤3for allβ,δ∈E,λ>0,such thatβ⊥δandδ⊥βB⊥(2) Rβ,δ,λ>0, for all β,δ∈E,λ>0, such that β⊥δ and δ⊥βB⊥(3) Rβ,δ,λ=1 if and only if β=δ, for all β,δ∈E,λ>0, such that β⊥δ and δ⊥βB⊥(4) Rβ,δ,λ=Rδ,β,λ, for all β,δ∈E,λ>0, such that β⊥δ and δ⊥βB⊥(5) Rβ,∂,λ+π≥Rβ,δ,λ∗Rδ,e,π, for all β,δ,∂∈E,λ,π>0, such that β⊥δ, δ⊥∂ and β⊥∂B⊥(6) Rβ,δ,Δ:0,∞⟶0,1 is continuous, for all β,δ∈E, such that β⊥δ and δ⊥βB⊥(7) Sβ,δ,λ>0, for all β,δ∈E,λ>0, such that β⊥δ and δ⊥βB⊥(8) Sβ,δ,λ=1 if and only if β=δ, for all β,δ∈E,λ>0, such that β⊥δ and δ⊥βB⊥(9) Sβ,δ,λ=Sδ,β,λ, for all β,δ∈E,λ>0, such that β⊥δ and δ⊥βB⊥(10) Sβ,∂,λ+π≤Sβ,δ,λΔSδ,∂,π, for all β,δ,∂∈E,λ,π>0, such that β⊥δ, δ⊥∂, and β⊥∂B⊥(11) Sβ,δ,Δ:0,∞⟶0,1 is continuous, for all β,δ∈E, such that β⊥δ and δ⊥βB⊥(12) Tβ,δ,λ>0, for all β,δ∈E,λ>0, such that β⊥δ and δ⊥βB⊥(13) Tβ,δ,λ=1 if and only if β=δ, for all β,δ∈E,λ>0, such that β⊥δ and δ⊥βB⊥(14) Tβ,δ,λ=Tδ,β,λ, for all β,δ∈E,λ>0, such that β⊥δ and δ⊥βB⊥(15) Tβ,∂,λ+π≤Tβ,δ,λΔTδ,∂,π for all β,δ,∂∈E,λ,π>0, such that β⊥δ, δ⊥∂, and β⊥∂B⊥(16) Tβ,δ,Δ:0,∞⟶0,1 is continuous, for all β,δ∈E, such that β⊥δ and δ⊥βB⊥(17) If λ≤0 then Rβ,δ,λ=0,Sβ,δ,λ=1 and Tβ,δ,λ=1, for all β,δ∈E,λ>0, such that β⊥δ and δ⊥β.
Then,E,R,S,T,∗,Δ,⊥ is called ONMS.Remark 1.
Every NMS is an ONMS, but the converse is not necessarily true.Example 1.
LetE=−5,5 and define a CTN as a∗b=ab, CTCN as aΔb=maxa,b, and a binary relation ⊥ by β⊥δ iff β+δ≥0. Take(1)Rβ,δ,λ=1,ifβ=δ,λλ+maxβ,δ,if otherwise,Sβ,δ,λ=0,ifβ=δ,maxβ,δλ+maxβ,δ,if otherwise,Tβ,δ,λ=0,ifβ=δ,maxβ,δλ,if otherwise,for all β,δ∈E,λ>0, then it is ONMS, but not an NMS.
Easy to see, forπ=λ=1,β=−1,δ=−1/2,∂=−2. (B5), (B10), and (B15) fail.Remark 2.
The above example is also true for CTNa∗b=mina,b and CTCN aΔb=maxa,b.Remark 3.
The above example is also ONMS if we take(2)Sβ,δ,λ=0,ifβ=δ,1−λλ+maxβ,δ,if otherwise.Definition 6.
An O-sequenceβn in an ONMS E,R,S,T,∗,Δ,⊥ is called to be orthogonal convergent (O-convergent) to β∈E, if(3)limn⟶∞Rβn,β,λ=1,∀λ>0,limn⟶∞Sβn,β,λ=0,∀λ>0,limn⟶∞Tβn,β,λ=0,∀λ>0.Definition 7.
An O-sequenceβn in an ONMS E,R,S,∗,Δ,⊥ is named to be orthogonal Cauchy (O-Cauchy) if there exists n∈ℕ, such that(4)limn⟶∞Rβn,βn+p,λ=1,limn⟶∞Sβn,βn+p,λ=0,limn⟶∞Tβn,βn+p,λ=0,for allλ≥0,p≥1.Definition 8.
ψ:E⟶E is ⊥-continuous at β∈E in an ONMS E,R,S,T,∗,Δ,⊥, whenever for each O-sequence βn for all n∈ℕ in E; if limn⟶∞Rβn,β,λ=1, limn⟶∞Sβn,β,λ=0, and limn⟶∞Tβn,β,λ=0 for all λ>0, then limn⟶∞Rψβn,ψβ,λ=1, limn⟶∞Sψβn,ψβ,λ=0, and limn⟶∞Tψβn,ψβ,λ=0 for all λ>0. Furthermore, ψ is ⊥-continuous on E if ψ is ⊥-continuous at each β∈E. Also, ψ is ⊥-preserving if ψβ⊥ψδ when β⊥δ.Definition 9.
An ONMSE,R,S,T,∗,Δ,⊥ is said to orthogonally complete (O-complete) if every O-Cauchy sequence is convergent.Remark 4.
It is necessary for the limit of a convergent O-sequence to be unique in an ONMS.Example 2.
Assume ONMS given in an Example1 and define a sequence βn in E by βn=1−1/n,∀n∈ℕ, such that ∀n;βn⊥βn+1 or ∀n;βn+1⊥βn. Define a CTN as a∗b=ab, CTCN as aΔb=maxa,b, and a binary relation ⊥ by β⊥δ iff β+δ≥0. Take(5)limn⟶∞Rβn,β,λ=limn⟶∞1,ifβ=δ,λλ+maxβn,β,if otherwise=1,ifβ=δ,λλ+maxβ,β,if otherwise=1,limn⟶∞Sβn,β,λ=limn⟶∞0,ifβ=δ,maxβn,βλ+maxβn,β,if otherwise=0,ifβ=δ,maxβ,βλ+maxβ,β,if otherwise=0,limn⟶∞Tβn,β,λ=limn⟶∞0,ifβ=δ,maxβn,βλ,if otherwise=0,ifβ=δ,maxβ,βλ,if otherwise=0.
Similarly, for any convergent O-sequence, limit will be unique in an ONMS.Remark 5.
It is necessary that a convergent O-sequence is O-Cauchy in an ONMS.Example 3.
From proof of above Example2βn=1−1/n,∀n∈ℕ is a convergent O-sequence in an ONMS.(6)limn⟶∞Rβn,βn+p,λ=limn⟶∞1,ifβ=δ,λλ+maxβn,βn+p,if otherwise=1,ifβ=δ,λλ+maxβ,β,if otherwise=1,limn⟶∞Sβn,βn+p,λ=limn⟶∞0,ifβ=δ,maxβn,βn+pλ+maxβn,βn+p,if otherwise=0,ifβ=δ,maxβ,βλ+maxβ,β,if otherwise=0,limn⟶∞Tβn,βn+p,λ=limn⟶∞0,ifβ=δ,maxβn,βn+pλ,if otherwise=0,ifβ=δ,maxβ,βλ,if otherwise=0,for all λ≥0,p≥1. Similarly, every convergent O-sequence is an O-Cauchy sequence in an ONMS.Lemma 1.
If for somev∈0,1 and β,δ∈E,(7)Rβ,δ,λ≥Rβ,δ,λv,λ>0,Sβ,δ,λ≤Sβ,δ,λv,λ>0,Tβ,δ,λ≤Tβ,δ,λv,λ>0,then β=δ.Proof.
Easy to show on the lines of [24].
Now, we introduce the notion of Banach contraction in the sense of ONMSs.Definition 10.
LetE,R,S,T,∗,Δ,⊥ be an ONMS. A map ψ:E⟶E is an orthogonal contraction if there exists θ∈0,1, such that for every λ>0 and β,δ∈E with β⊥δ, we have(8)Rψβ,ψδ,θλ≥Rβ,δ,λ,(9)Sψβ,ψδ,θλ≤Sβ,δ,λ,(10)Tψβ,ψδ,θλ≤Tβ,δ,λ.Theorem 1.
LetE,R,S,T,∗,Δ,⊥ be an O-complete NMS, such that(11)limλ⟶∞Rβ,δ,λ=1,limλ⟶∞Sβ,δ,λ=0,limλ⟶∞Tβ,δ,λ=0,∀β,δ∈E.
Letψ:E⟶E be an ⊥-continuous, ⊥-contraction, and ⊥-preserving mapping. Thus, ψ has a unique FP, say β∗∈E. Furthermore,(12)limn⟶∞Rψnβ,β∗,λ=1,limn⟶∞Sψnβ,β∗,λ=0,limn⟶∞Tψnβ,β∗,λ=0,for allβ∈Eandλ>0.Proof.
SinceE,R,S,T,∗,Δ,⊥ is an O-complete NMS, there exists β0∈E, such that(13)β0⊥δ,for allδ∈E.
That is,β0⊥ψβ0. Take(14)βn=ψnβ0=ψβn−1,for alln∈S.
Sinceψ is ⊥-preserving, βn is an O-sequence. Now, since ψ is an ⊥-contraction, we can get(15)Rβn+1,βn,θλ=Rψβn,ψβn−1,θλ≥Rβn,βn−1,λ,for all n∈S and λ>0. Note that R is nondecreasing on 0,∞. Therefore, by applying the above expression, we can deduce(16)Rβn+1,βn,λ≥Rβn+1,βn,θλ=Rψβn,ψβn−1,θλ≥Rβn,βn−1,λ=Rψβn−1,ψβn−2,λ≥Rβn−1,βn−2,λθ≥⋯≥Rβ1,β0,λθn,for all n∈S and λ>0. Thus, from (13) and (B4), we have(17)Rβn,βn+α,λ≥Rβn,βn+1,λ2∗Rβn+1,βn+α,λ2≥Rβn,βn+1,λ2∗Rβn+1,βn+2,λ22∗Rβn+2,βn+3,λ23∗⋯∗Rβn+α−1,βn+α,λ2n+α≥Rβ1,β0,λ2θn∗Rβ1,β0,λ22θn∗⋯∗Rβ1,β0,λ2n+αθn.
We know thatlimλ⟶∞Rβ,δ,λ=1, for all β,δ∈E and λ>0. So, from (17), we get(18)limn⟶∞Rβn,βn+α,λ≥1∗1∗⋯∗1=1,Sβn+1,βn,θλ=Sψβn,ψβn−1,θλ≤Sβn,βn−1,λ,for all n∈S and λ>0. Therefore, by applying the above expression, we can deduce(19)Sβn+1,βn,λ≤Sβn+1,βn,θλ=Sψβn,ψβn−1,θλ≤Sβn,βn−1,λ=Sψβn−1,ψβn−2,λ≤Sβn−1,βn−2,λθ≤⋯≤Sβ1,β0,λθn,for all n∈S and λ>0. Thus, from (19) and (B10), we have(20)Sβn,βn+α,λ≤Sβn,βn+1,λ2ΔSβn+1,βn+α,λ2≤Sβn,βn+1,λ2ΔSβn+1,βn+2,λ22ΔSβn+2,βn+3,λ23Δ⋯ΔSβn+α−1,βn+α,λ2n+α≤Sβ1,β0,λ2θnΔSβ1,β0,λ22θnΔ⋯ΔSβ1,β0,λ2n+αθn.
We know thatlimλ⟶∞Sβ,δ,λ=0 for all β,δ∈E and λ>0. So, from (20), we get(21)limn⟶∞Sβn,βn+α,λ≤0Δ0Δ⋯Δ0=0,Tβn+1,βn,θλ=Tψβn,ψβn−1,θλ≤Tβn,βn−1,λ,for all n∈S and λ>0. Therefore, by applying the above expression, we can deduce(22)Tβn+1,βn,λ≤Tβn+1,βn,θλ=Tψβn,ψβn−1,θλ≤Tβn,βn−1,λ=Tψβn−1,ψβn−2,λ≤Tβn−1,βn−2,λθ≤⋯≤Tβ1,β0,λθn,for all n∈S and λ>0. Thus, from (22) and (B15), we have(23)Tβn,βn+α,λ≤Tβn,βn+1,λ2ΔTβn+1,βn+α,λ2≤Tβn,βn+1,λ2ΔTβn+1,βn+2,λ22ΔTβn+2,βn+3,λ23Δ⋯ΔTβn+α−1,βn+α,λ2n+α≤Tβ1,β0,λ2θnΔTβ1,β0,λ22θnΔ⋯ΔTβ1,β0,λ2n+αθn.
We know thatlimλ⟶∞Tβ,δ,λ=0, for all β,δ∈E, and λ>0. So, from (23), we get(24)limn⟶∞Tβn,βn+α,λ≤0Δ0Δ⋯Δ0=0.
So,βn is a Cauchy O-sequence. The O-completeness of the NMS E,R,S,T,∗,Δ,⊥ ensures that there exists β∗∈E, such that Rβn,β∗,λ⟶1, Sβn,β∗,λ⟶0, and Tβn,β∗,λ⟶0 as n⟶+∞ for all λ>0. Now, since ψ is an ⊥-continuous mapping, Rβn+1,ψβ∗,λ=Rψβn,ψβ∗,λ⟶1, Sβn+1,ψβ∗,λ=Sψβn,ψβ∗,λ⟶0, and Tβn+1,ψβ∗,λ=Tψβn,ψβ∗,λ⟶0 as n⟶+∞. Now, we have(25)Rβ∗,ψβ∗,λ≥Rβ∗,βn+1,λ2∗Rβn+1,ψβ∗,λ2,Sβ∗,ψβ∗,λ≤Sβ∗,βn+1,λ2ΔSβn+1,ψβ∗,λ2,Tβ∗,ψβ∗,λ≤Tβ∗,βn+1,λ2ΔTβn+1,ψβ∗,λ2.
Taking limit asn⟶+∞, we get Rβ∗,ψβ∗,λ=1∗1=1, Sβ∗,ψβ∗,λ=0Δ0=0, and Tβ∗,ψβ∗,λ=0Δ0=0, and hence, ψβ∗=β∗.
Now, we show the uniqueness of the FP of the mappingψ. Assume that β∗ and δ∗ are two FPs of ψ, such that β∗≠δ∗. We can get(26)β0⊥β∗,β0⊥δ∗.
SinceT is ⊥-preserving, one writes(27)ψnβ0⊥ψnβ∗,ψnβ0⊥ψnδ∗,for all n∈S. So from (8), we can derive(28)Rψnβ0,ψnβ∗,λ≥Rψnβ0,ψnβ∗,θλ≥Rβ0,β∗,λθn,Rψnβ0,ψnδ∗,λ≥Rψnβ0,ψnδ∗,θλ≥Rβ0,δ∗,λθn.
Therefore,(29)Rβ∗,δ∗,λ=Rψnβ∗,ψnδ∗,λ≥Rψnβ0,ψnβ∗,λ2∗Rψnβ0,ψnδ∗,λ2≥Rβ0,β∗,λ2θn∗Rβ0,δ∗,λ2θn⟶1,asn⟶∞.
So from (9), we can derive(30)Sψnβ0,ψnβ∗,λ≤Sψnβ0,ψnβ∗,θλ≤Sβ0,β∗,λθn,Sψnβ0,ψnδ∗,λ≤Sψnβ0,ψnδ∗,θλ≤Sβ0,δ∗,λθn.
Therefore,(31)Sβ∗,δ∗,λ=Sψnβ∗,ψnδ∗,λ≤Sψnβ0,ψnβ∗,λ2ΔSψnβ0,ψnδ∗,λ2≤Sβ0,β∗,λ2θnΔSβ0,δ∗,λ2θn⟶0,asn⟶∞.
Similarly, from (10), we can derive(32)Tψnβ0,ψnβ∗,λ≤Tψnβ0,ψnβ∗,θλ≤Tβ0,β∗,λθn,Tψnβ0,ψnδ∗,λ≤Tψnβ0,ψnδ∗,θλ≤Tβ0,δ∗,λθn.
Therefore,(33)Tβ∗,δ∗,λ=Tψnβ∗,ψnδ∗,λ≤Tψnβ0,ψnβ∗,λ2ΔTψnβ0,ψnδ∗,λ2≤Tβ0,β∗,λ2θnΔTβ0,δ∗,λ2θn⟶0,asn⟶∞.
So,β∗=δ∗; hence, β∗ is the unique FP.Corollary 1.
LetE,R,S,T,∗,Δ,⊥ be an O-complete NMS. Let ψ:E⟶E be ⊥-contraction and ⊥-preserving. Suppose that if βn is an O-sequence with βn⟶β∈E, then β⊥βn for all n∈ℕ. Then, ψ has a unique FP, say β∗∈E. Furthermore, limn⟶∞Rψnβ,β∗,λ=1, limn⟶∞Sψnβ,β∗,λ=0, and limn⟶∞Tψnβ,β∗,λ=0, for all β∈E and λ>0.Proof.
See Appendix.Example 4.
LetE=−2,2. We define a binary relation ⊥ by(34)β⊥δ⟺β+δ≥0.
Define an ONMS as in Example2 by(35)Rβ,δ,λ=1,ifβ=δ,λλ+maxβ,δ,if otherwise,Sβ,δ,λ=0,ifβ=δ,maxβ,δλ+maxβ,δ,if otherwise,Tβ,δ,λ=0,ifβ=δ,maxβ,δλ,if otherwise,for all β,δ∈E,λ>0, with the CTN a∗b=aΔb and CTCN aΔb=maxa,b. Then, E,R,S,T,∗,Δ,⊥ is an O-complete NMS. Define ψ:E⟶E by(36)ψβ=β4,β∈−2,0,0,β∈0,2.
Then, the following cases are satisfied.(1)
Ifβ∈−2,0 and δ∈0,2, then ψβ=β/4 and ψδ=0(2)
Ifβ,δ∈−2,0, then ψβ=β/4 and ψδ=δ/4(3)
Ifβ,δ∈0,2, then ψβ=0 and ψδ=0(4)
Ifβ∈0,2 and δ∈−2,0, then ψβ=0 and ψδ=δ/4
This clearly implies thatψβ+ψδ≥0. Hence, ψ is ⊥-preserving. We can easily see that if limn⟶∞Rβn,β,λ=1, then limn⟶∞Rψβn,ψβ,λ=1, limn⟶∞Sβn,β,λ=0, then limn⟶∞Sψβn,ψβ,λ=0 and limn⟶∞Tβn,β,λ=0, then limn⟶∞Tψβn,ψβ,λ=0 for all β∈E and λ>0. Hence, ψ is ⊥-continuous.
The above cases 1–4 forθ∈1/2,1 are satisfied contractive conditions:(37)Rψβ,ψδ,θλ≥Rβ,δ,λ,Sψβ,ψδ,θλ≤Sβ,δ,λ,Tψβ,ψδ,θλ≤Tβ,δ,λ.
All conditions of Theorem1 are satisfied. Also, 0 is the unique FP of ψ.Theorem 2.
LetE,R,S,T,∗,Δ,⊥ be an O-complete NMS, such that(38)limλ⟶∞Rβ,δ,λ=1,limλ⟶∞Sβ,δ,λ=0,∀β,δ∈Eandλ>0.
Letψ:E⟶E be ⊥-continuous, ⊥-contraction, and ⊥-preserving. Assume that there exist θ∈0,1 and λ>0, such that(39)Rψβ,ψδ,θλ≥minRψβ,β,λ,Rψδ,δ,λ,Sψβ,ψδ,θλ≤minSψβ,β,λ,Sψδ,δ,λ,Tψβ,ψδ,θλ≤minTψβ,β,λ,Tψδ,δ,λ,for all β,δ∈E,λ>0. Then, ψ has a unique FP, so β∗∈E. Furthermore, limn⟶∞Rψnβ,β∗,λ=1, limn⟶∞Sψnβ,β∗,λ=0, and limn⟶∞Tψnβ,β∗,λ=0 for all β∈E and λ>0.Proof.
See Appendix.Corollary 2.
LetE,R,S,∗,Δ,⊥ be an O-complete IFMS and ψ:E⟶E be an ⊥-continuous and ⊥-preserving mapping. Assume that there exists θ∈0,1, so that for all λ>0,(40)Rψβ,ψδ,θλ≥minRψβ,β,λ,Rψδ,δ,λ,Rβ,δ,λ,Sψβ,ψδ,θλ≤minSψβ,β,λ,Sψδ,δ,λ,Sβ,δ,λ,Tψβ,ψδ,θλ≤minTψβ,β,λ,Tψδ,δ,λ,Tβ,δ,λ.Then,ψ has a unique FP.Proof.
It follows from Theorems1 and 2.Example 5.
LetE=−2,2 and define a binary relation ⊥ by(41)β⊥δ⟺β+δ≥0.
DefineRandS by(42)Rβ,δ,λ=1,ifβ=δ,λλ+maxβ,δ,otherwise,Sβ,δ,λ=0,ifβ=δ,maxβ,δλ+maxβ,δ,otherwise,Tβ,δ,λ=0,ifβ=δ,maxβ,δλ,otherwise,for all β,δ∈E, and λ>0, with the CTN and CTCN, respectively: a∗b=a⋅b, aΔb=maxa,b, then E,R,S,T,∗,Δ,⊥ is an O-complete NMS. Note that limλ⟶∞Rβ,δ,λ=1, limλ⟶∞Sβ,δ,λ=0, and limλ⟶∞Tβ,δ,λ=0 ∀ β,δ∈E. Define ψ:E⟶E by(43)ψβ=β4,β∈−2,23,1−β,β∈23,1,β−12,β∈1,2.
We have the following cases:(1)
Ifβ,δ∈−2,2/3, then ψβ=β/4 and ψδ=δ/4(2)
Ifβ,δ∈2/3,1, then ψβ=1−β and ψδ=1−δ(3)
Ifβ,δ∈1,2, then ψβ=β−1/2 and ψδ=δ−1/2(4)
Ifβ∈−2,2/3 and δ∈2/3,1, then ψβ=β/4 and ψδ=1−δ(5)
Ifβ∈−2,2/3 and δ∈1,2, then ψβ=β/4 and ψδ=δ−1/2(6)
Ifβ∈2/3,1 and δ∈1,2, then ψβ=1−β and ψδ=δ−1/2(7)
Ifβ∈1,2 and δ∈2/3,1, then ψβ=β−1/2 and ψδ=1−δ(8)
Ifβ∈1,2 and δ∈−2,2/3, then ψβ=β−1/2 and ψδ=δ/4(9)
Ifβ∈2/3,1 and δ∈−2,2/3, then ψβ=1−β and ψδ=δ/4
Becauseβ⊥δ⟺β+δ≥0, it clearly implies that ψβ+ψδ≥0. Hence, ψ is ⊥-preserving. Let βn be an arbitrary O-sequence in E that converges to β∈E. We have(44)limn⟶∞Rβn,β,λ=limn⟶∞1,ifβ=δ,λλ+maxβn,β,otherwise=1,limn⟶∞Sβn,β,λ=limn⟶∞1,ifβ=δ,maxβn,βλ+maxβn,β,otherwise=0,limn⟶∞Tβn,β,λ=limn⟶∞1,ifβ=δ,maxβn,βλ,otherwise=0.
Note that iflimn⟶∞Rβn,β,λ=1, limn⟶∞Sβn,β,λ=0, and limn⟶∞Tβn,β,λ=0, then limn⟶∞Rψβn,ψβ,λ=1, limn⟶∞Sψβn,ψβ,λ=0, and limn⟶∞Tψβn,ψβ,λ=0 for all β∈E and λ>0. Hence, ψ is orthogonal continuous. The case β=δ is clear. Let β≠δ. We have(45)Rψβ,ψδ,θλ≥minRψβ,β,λ,Rψδ,δ,λ,Sψβ,ψδ,θλ≤minSψβ,β,λ,Sψδ,δ,λ,Tψβ,ψδ,θλ≤minTψβ,β,λ,Tψδ,δ,λ.
Indeed, it is satisfied for all above 9 cases. But,ψ is not a contraction. Assume(46)minRψβ,β,λ,Rψδ,δ,λ=Rψβ,β,λ,minSψβ,β,λ,Sψδ,δ,λ=Sψβ,β,λ,minTψβ,β,λ,Tψδ,δ,λ=Tψβ,β,λ,then for β=−1,δ=−2, we have(47)Rψβ,ψδ,θλ=θλθλ+maxβ/4,δ/4=4θλ4θλ−1≥1,Sψβ,ψδ,θλ=maxβ/4,δ/4θλ+maxβ/4,δ/4=−14θλ−1≤0,Sψβ,ψδ,θλ=maxβ/4,δ/4θλ=−14θλ≤0.
It is a contradiction. Hence, all the conditions of Theorem2 are satisfied and 0 is the unique FP of ψ.Definition 11.
LetE,R,S,T,∗,Δ,⊥ be an ONMS. A mapping ψ:E⟶E is named to be an NS ⊥-contractive if ∃ θ∈0,1, so that(48)1Rψβ,ψδ,λ−1≤θ1Rβ,δ,λ−1,Sψβ,ψδ,λ≤θSβ,δ,λ,Tψβ,ψδ,λ≤θTβ,δ,λ,for all β,δ∈E and λ>0. Here, θ is called the NS ⊥-contractive constant of ψ.Theorem 3.
LetE,R,S,T,∗,Δ,⊥ be an O-complete NMS, such that(49)limλ⟶∞Rβ,δ,λ=1,limλ⟶∞Sβ,δ,λ=0,limλ⟶∞Tβ,δ,λ=0,∀β,δ∈E.
Letψ:E⟶E be an ⊥-continuous, NS⊥-contraction, and ⊥-preserving mapping. Thus, ψ has a FP, say υ∈E, Rυ,υ,λ=1,Sυ,υ,λ=0, and Tυ,υ,λ=0 for all λ>0.Proof.
LetE,R,S,T,∗,Δ,⊥ be an O-complete NMS. For an arbitrary β0∈E,(50)β0⊥δ,∀δ∈E.
That is,β0⊥ψβ0. Consider(51)βn=ψnβ0=ψβn−1,for alln∈ℕ.
Sinceψ is ⊥-preserving, βn is an O-sequence. If βn=βn−1 for some n∈ℕ, then βn is a FP of ψ. We assume that βn≠βn−1 for all n∈ℕ. For all λ>0 and n∈ℕ, we get from (48),(52)1Rβn,βn+1,λ−1=1Rψβn−1,ψβn,λ−1≤θ1Rβn−1,βn,λ−1,Sβn,βn+1,λ=Sψβn−1,ψβn,λ≤θSβn−1,βn,λ,Tβn,βn+1,λ=Tψβn−1,ψβn,λ≤θTβn−1,βn,λ.
We have(53)1Rβn,βn+1,λ≤θRβn−1,βn,λ+1−θ,∀λ>0=θRψβn−2,ψβn−1,λ+1−θ≤θ2Rβn−2,βn−1,λ+θ1−θ+1−θ.
Continuing in this way, we get(54)1Rβn,βn+1,λ≤θnRβ0,β1,λ+θn−11−θ+θn−21−θ+⋯+θ1−θ+1−θ≤θnRβ0,β1,λ+θn−1+θn−2+⋯+11−θ≤θnRβ0,β1,λ+1−θn.
We have(55)1θn/Rβ0,β1,λ+1−θn≤Rβn,βn+1,λ,∀λ>0,n∈ℕ,(56)Sβn,βn+1,λ=Sψβn−1,ψβn,λ≤θSβn−1,βn,λ=θSψβn−2,ψβn−1,λ≤θ2Sβn−2,βn−1,λ≤⋯≤θnSβ0,β1,λ,∀λ>0,n∈ℕ,(57)Tβn,βn+1,λ=Tψβn−1,ψβn,λ≤θTβn−1,βn,λ=θTψβn−2,ψβn−1,λ≤θ2Tβn−2,βn−1,λ≤⋯≤θnTβ0,β1,λ,∀λ>0,n∈ℕ.
Now, form≥1 and n∈ℕ, we have(58)Rβn,βn+m,λ≥Rβn,βn+1,λ2∗Rβn+1,βn+m,λ2≥Rβn,βn+1,λ2∗Rβn+1,βn+2,λ22∗Rβn+2,βn+m,λ22.
Again, continuing in this way, we get(59)Rβn,βn+m,λ≥Rβn,βn+1,λ2∗Rβn+1,βn+2,λ22∗⋯∗Rβn+m−1,βn+m,λ2m−1,Sβn,βn+p,λ≤Sβn,βn+1,λ2ΔSβn+1,βn+p,λ2≤Sβn,βn+1,λ2ΔSβn+1,βn+2,λ22ΔSβn+2,βn+p,λ22.
Continuing in this way, we get(60)Sβn,βn+p,λ≤Sβn,βn+1,λ2ΔSβn+1,βn+2,λ22Δ⋯ΔSβn+p−1,βn+p,λ2p−1Tβn,βn+p,λ≤Tβn,βn+1,λ2ΔTβn+1,βn+p,λ2≤Tβn,βn+1,λ2ΔTβn+1,βn+2,λ22ΔTβn+2,βn+p,λ22.
Continuing in this way, we get(61)Tβn,βn+p,λ≤Tβn,βn+1,λ2ΔTβn+1,βn+2,λ22Δ⋯ΔTβn+p−1,βn+p,λ2p−1.
By using (55) in the above inequality, we have(62)Rβn,βn+m,λ≥1θn/Rβ0,β1,λ/2+1−θn∗1θn+1/Rβ0,β1,λ/22+1−θn+1∗⋯∗1θn+m−1/Rβ0,β1,λ/2m−1+1−θn+m−1,≥1θn/Rβ0,β1,λ/2+1∗1θn+1/Rβ0,β1,λ/22+1∗⋯∗1θn+m−1/Rβ0,β1,λ/2m−1+1,using (56),(63)Sβn,βn+p,λ≤Sβn,βn+1,λ2ΔSβn+1,βn+2,λ22Δ⋯ΔSβn+p−1,βn+p,λ2p−1,and using (57), (64)Tβn,βn+p,λ≤Tβn,βn+1,λ2ΔTβn+1,βn+2,λ22Δ⋯ΔTβn+p−1,βn+p,λ2p−1,θ∈0,1, we deduce from the above expression that(65)limn⟶∞Rβn,βn+m,λ=1,limn⟶∞Sβn,βn+m,λ=0,limn⟶∞Tβn,βn+m,λ=0,for allλ>0,m≥1.
Therefore,βn is a Cauchy O-sequence in E,R,S,T,∗,Δ,⊥. By the completeness of E,R,S,T,∗,Δ,⊥, we know that ψ is an ⊥-continuous mapping and there exists υ∈E, such that(66)limn⟶∞Rβn+1,υ,λ=limn⟶∞Rψβn,ψυ,λ=1,∀λ>0,(67)limn⟶∞Sβn+1,υ,λ=limn⟶∞Sψβn,ψυ,λ=0,∀λ>0,(68)limn⟶∞Tβn+1,υ,λ=limn⟶∞Tψβn,ψυ,λ=0,∀λ>0.
Now, we prove thatυ is a FP of ψ. For this, we obtain from (48) that(69)1Rψβn,ψυ,λ−1≤θ1Rβn,υ,λ−1=θRβn,υ,λ−θ.
That is,(70)1θ/Rβn,υ,λ+1−θ≤Rψβn,ψυ,λ.
Using the above inequality, we obtain(71)Rυ,ψυ,λ≥Rυ,βn+1,λ2∗Rβn+1,ψυ,λ2=Rυ,βn+1,λ2∗Rψβn,ψυ,λ2≥Rυ,βn+1,λ2∗1θ/Rβn,υ,λ/2+1−θ,Sw,v,λ=Sψw,ψv,λ≤θSw,v,λ<Sw,v,λ,=Sw,βn+1,λ2ΔSψβn,ψw,λ2≤Sw,βn+1,λ2ΔθSβn,w,λ2,Tw,v,λ=Tψw,ψv,λ≤θTw,v,λ<Tw,v,λ=Tw,βn+1,λ2ΔTψβn,ψw,λ2≤Tw,βn+1,λ2ΔθTβn,w,λ2.
Taking limit asn⟶∞ and using (66)–(68) in the above expression, we get that Rυ,ψυ,λ=1,Sυ,ψυ,λ=0 and Tυ,ψυ,λ=0, that is, ψυ=υ. Therefore, υ is a FP of ψ, Rυ,υ,λ=1, Sυ,υ,λ=0, and Tυ,υ,λ=0 for all λ>0.Corollary 3.
LetE,R,S,T,∗,Δ,⊥ be a O-complete NMS and ψ:E⟶E satisfy(72)1Rψnβ,ψnδ,λ−1≤θ1Rβ,δ,λ−1,Sψnβ,ψnδ,λ≤θSβ,δ,λ,Tψnβ,ψnδ,λ≤θTβ,δ,λ,for all n∈ℕ,β,δ∈E,λ>0, where 0<θ<1. Then, ψ has a FP.Proof.
υ∈E is the unique FP of ψn by using Theorem 3, and Rυ,υ,λ=1,Sυ,υ,λ=0,Tυ,υ,λ,∀λ>0, ψυ is also a FP of ψn as ψnψυ=ψυ. From Theorem 3, ψυ=υ, where υ is a FP since the FP of ψ is also a FP of ψn.Example 6.
LetE=−1,2 and define a binary relation ⊥ by(73)β⊥δ⟺β+δ≥0.
DefineR,S,T by(74)Rβ,δ,λ=1,ifβ=δ,λλ+maxβ,δ,if otherwise,Sβ,δ,λ=0,ifβ=δ,1−λλ+maxβ,δ,if otherwise,Tβ,δ,λ=0,ifβ=δ,maxβ,δλ,if otherwise.
With CTNa∗b=a⋅b and CTCN aΔb=maxa,b, E,R,S,T,∗,Δ,⊥ is an O-complete NMS. Also, observe that limλ⟶∞Rβ,δ,λ=1, limλ⟶∞Sβ,δ,λ=0, and limλ⟶∞Tβ,δ,λ=0, ∀ β,δ∈E.
Defineψ:E⟶E by(75)ψβ=2−β,β∈−1,1,1,β∈1,2.
Therefore, it will satisfy the following cases:(1)
Ifβ,δ∈−1,1, then ψβ=2−β and ψδ=2−δ(2)
Ifβ,δ∈1,2, then ψβ=ψδ=1(3)
Ifβ∈−1,1 and δ∈1,2, then ψβ=2−β and ψδ=1(4)
Ifβ∈1,2 and δ∈−1,1, then ψβ=1 and ψδ=2−δ
Becauseβ⊥δ⟺β+δ≥0, it clearly implies that ψβ+ψδ≥0. Hence, ψ is ⊥-preserving. Let βn be an arbitrary O-sequence in E that βn converges to β∈E.(76)limn⟶∞Rβn,β,λ=1,limn⟶∞Sβn,β,λ=0,limn⟶∞Tβn,β,λ=0,as βn converges to β. We can easily see that if limn⟶∞Rβn,β,λ=1,limn⟶∞Sβn,β,λ=0, and limn⟶∞Tβn,β,λ=0, then clearly limn⟶∞Rψβn,ψβ,λ=1, limn⟶∞Sψβn,ψβ,λ=0, and limn⟶∞Tψβn,ψβ,λ=0 for all β∈E and λ>0. Hence, ψ is orthogonal continuous. Also, above all cases satisfied NS ⊥-contractive mapping(77)1Rψβ,ψδ,λ−1≤θ1Rβ,δ,λ−1,Sψβ,ψδ,λ≤θSβ,δ,λ,Tψβ,ψδ,λ≤θTβ,δ,λ.
All conditions of Theorem3 are satisfied and 1 is a FP of ψ.


## 4. Conclusion

Herein, we have introduced the notion of orthogonal neutrosophic metric space and investigated some new type fixed point theorems in this new setting. Moreover, we have provided nontrivial examples to demonstrate the viability of the proposed methods. Since our structure is more general than the class of neutrosophic metric spaces, our results and notions expand and generalize a number of previously published results. This study can be easily extended in the structure of orthogonal neutrosophic cone metric spaces, orthogonal neutrosophic triple partial metric spaces, orthogonal neutrosophic triple V-generalized metric spaces, orthogonal neutrosophic triple partial bipolar metric spaces, and orthogonal neutrosophic triple partial g-metric spaces.



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