# Φ,Ψ-Weak Contractions in Neutrosophic Cone Metric Spaces via Fixed Point Theorems

Authors: Wadei F. Al-Omeri; Saeid Jafari; Florentin Smarandache
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

In this manuscript, we obtain common fixed point theorems in the neutrosophic cone metric space. Also, notion ofΦ,Ψ-weak contraction is defined in the neutrosophic cone metric space by using the idea of altering distance function. Finally, we review many examples of cone metric spaces to verify some properties.


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## Body

## 1. Introduction

The concept of fuzzy sets was introduced by Zadeh [1]. The study of fuzzy topological spaces was initiated by Chang [2]. The notion of intuitionistic fuzzy sets was introduced by Atanassov [3]. The notion of intuitionistic L-topological spaces was introduced by Atanassov and Stoeva [4] by extending L-topology to intuitionistic L-fuzzy setting. The notion of the intuitionistic fuzzy topological space was introduced by Çoker [5]. The concept of generalized fuzzy closed set was presented by Balasubramanian and Sundaram [6]. Smarandache extended the intuitionistic fuzzy sets to neutrosophic sets [7]. After the introduction of the neutrosophic set concept [8, 9] in 2019 by Smarandache and Shumrani on the nonstandard analysis, the nonstandard neutrosophic topology was developed. In recent years, neutrosophic algebraic structures have been investigated. Neutrosophy has laid the foundation for a whole family of new mathematical theories, generalizing both their classical and fuzzy counterparts, such as a neutrosophic theory in any field, see [10, 11]. Recently, there were introduced neutrosophic mapping and neutrosophic connectedness. The concept of the neutrosophic metric space presented by [12] Al-Omeri et al. is a generalization of the intuitionistic fuzzy metric space due to Veeramani and George [13]. In 2019 and 2020, Smarandache generalized the classical Algebraic Structures to NeutroAlgebraic Structures (or NeutroAlgebras) whose operations and axioms are partially true, partially indeterminate, and partially false as extensions of Partial Algebra and to AntiAlgebraic Structures (or AntiAlgebras) whose operations and axioms are totally false. And in general, he extended any classical structure, in no matter what field of knowledge, to a NeutroStructure and an AntiStructure, see [14, 15]. In 2007, Huang and Zhang [16] introduced the concept of cone metric space and proved some fixed point theorems for contractive mappings. Recently, Öner et al. [17] introduced the concept of the fuzzy cone metric space that generalized the corresponding notions of the fuzzy metric space by George and Veeramani [13] and proved the fuzzy cone Banach contraction theorem. In 2010, Vetro et al. [18] extended the notion of Φ,Ψ-weak contraction to fuzzy metric spaces and proved some common fixed point theorems for four mappings in fuzzy metric spaces by using the idea of an altering distance function. Gupta et al. and Wasfi et al. [19, 20] introduced the notions of E. A and common E. A on the modified intuitionistic generalized fuzzy metric space. They extended the notions of the common limit range property and E. A property for coupled maps on modified intuitionistic fuzzy metric spaces. This paper is devoted to the study of extending and generalizing the Φ,Ψ-weak contraction to the neutrosophic cone metric space and prove some results. In Section 2, we will recall some materials which will be used throughout this paper. In Section 3, we give definitions and present the cone neutrosophic metric space and explain a number of properties. In Section 4, the results obtained from theorems and theoretical application of the neutrosophic fixed point are also presented. The last section contains the conclusions of the paper.


## 2. Preliminaries

Definition 1 (see [21]).
LetΣ be a non-empty fixed set. A neutrosophic set (briefly, NS) R is an object having the form R=t,ξRt,ϱRt,ηRt:t∈Σ, where ξRt,ϱRt, and ηRt which represent the degree of membership function (namely, ξRt), the degree of indeterminacy (namely, ξRt), and the degree of nonmembership (namely, ηRt), respectively, of each element t∈Γ to the set R.
A neutrosophic setH=t,ξHt,ϱHt,ηHt:t∈Γ can be identified to an ordered triple ξHt,ϱHt,ηHt in ⌊0−,1+⌋ on Γ.Remark 1 (see [21]).
By using symbolH=t,ξHt,ϱHt,ηHt for the NS, H=t,ξHt,ϱHt,ηHt:t∈Γ.Definition 2 (see [13]).
LetH=ξHt,ϱHt,ηHt be a NS on Γ. The complement of Hbriefly,CH may be defined as three kinds of complements:(1)
CH=r,1−ξHt,1−ηHt:t∈Γ(2)
CH=r,ηHt,1−ϱHt,ξHt:t∈Γ(3)
CH=r,ηHt,ϱHt,ξHt:t∈Γ
We have the following NSs (see [21]), which will be used in the sequel:(1)
1N=t,1,0,0:t∈Γ or(2)
1N=t,1,0,1:t∈Γ,(3)
1N=t,1,1,0:t∈Γ,(4)
1N=t,1,1,1:t∈Γ.(1)
0N=t,0,1,1:t∈Γ or(2)
0N=t,0,0,1:t∈Γ,(3)
0N=t,0,0,0:t∈Γ,(4)
0N=t,0,1,0:t∈Γ.Definition 3 (see [21]).
LetHj:j∈J be an arbitrary family of NSs in Γ. Then,(1)
∩Hi may be defined as follows:(i)
∩Hi=t,∧i∈∧ξHit,∧i∈∧ϱHit,∨i∈∧ηHit(ii)
∩Hi=t,∧i∈∧ξHit,∨i∈∧ϱHit,∨i∈∧ηHit(2)
∪Hi may be defined as follows:(i)
∪Hi=t,∨i∈∧ξHit,∨i∈∧ϱHit,∧i∈∧ηHit(ii)
∪Hi=t,∨i∈∧ξHit,∧i∈∧ϱHit,∧i∈∧ηHitDefinition 4 (see [21]).
For anyr≠∅, let neutrosophic sets R and Γ be in the form R=r,ξRr,ϱRr,ηRr and Γ=r,ξΓr,ϱΓr,ηΓr. The two possible definitions of R⊆Γare as follows:(1)
R⊆Γ⟺ξRr≤ξΓr,ϱRr≥ϱΓr,andηRr≤ηΓr(2)
R⊆Γ⟺ξRr≤ξΓr,ϱRr≥ϱΓr,andηRr≥ηΓrDefinition 5 (see [22]).
A neutrosophic topology (NT for short) and a nonempty setΓ is a family Ξ of neutrosophic subsets in Γ satisfying the following axioms:(1)
0N,1N∈Ξ(2)
B1∩B2∈Ξ for any B1,B2∈Ξ(3)
∪Bi∈Ξ, ∀Bii∈I⊆Ξ
The elements ofΞ are called open neutrosophic sets. The pair Γ,Ξ is called a neutrosophic topological space, and any neutrosophic set in Ξ is known as the neutrosophic open set (NOS) in Γ. A neutrosophic set B is closed if its complement is neutrosophic-open, denoted by CB. Throughout this paper, we suppose that all cone metrics have nonempty interior.
For anyNTSR in Γ,Ξ [23], we have ClRc=IntRc and IntRc=ClRc.Definition 6.
A subsetμ of Σ is said to be a cone in the following cases:(1)
If boths∈μ and −s∈μ, then s=ϕ(2)
Ifs,r∈S, s,r≥0, and u,v∈μ, then su+rv∈μ(3)
μ is closed, nonempty, and μ≠ϕ
For a given cone, partial ordering (≼) on Σ via μ is defined by u≼v iff v−u∈μ. u≺v will stand for u≼v and u≪v, while u≠v will stand for v−u∈Intμ.
If∃ a constant K>0 such that for all ∅≼u≼v, u,v∈Σ implies u≤Kv, and the least positive number K satisfying this property is called the normal constant of P, where P is the normal.Definition 7.
LetΓ be a nonempty set and s≥1 be a given real number. A function d:Γ×Γ↦Σ is said to be a cone metric on Γ if the following conditions hold:(1)
dm1,m2=dm2,m1 for all m1,m2∈Γ(2)
0≼dm1,m2 for all m1,m2∈Γ(3)
dm1,m3≼sdm1,m2+dm2,m3∀m1,m2,m3∈Γ(4)
dm1,m2=0 iff m1=m2
The pairΓ,d is called a cone metric space (shortly, CMS).Definition 8.
At-norm is continuous for any binary operation ⋇:0,1×0,1⟶0,1 if ⋇ verifies the following statements:(1)
⋇ is continuous(2)
⋇ is commutative and associative(3)
n1⋇n2≤n3⋇n4 whenever n1≤n3 and n2≤n4 for all n1,n2,n3,n4∈0,1(4)
n1⋇1=n1 for all n1∈0,1Definition 9.
LetΓ,d be a CMS. Then, for any d1≫0 and d2≫0, d1,d2∈Σ, ∃d≫0, and d∈Σ such that d≪d1 and d≪d2.Example 1.
n1⋇n2=maxn1,n2 and n1⋇n2=n1n2.Example 2.
n1◇n2=maxn1,n2 and m1◇n2=minn1+n2,1.Definition 10.
At-conorm of a binary operation ◇:0,1×0,1⟶0,1 is continuous if ◇ verifies the following statements:(1)
◇ is continuous(2)
◇ is associative and commutative(3)
q1◇q2≤q3◇q4 whenever q1≤q3 and q2≤q4 for all q1,q2,q3,q4∈0,1(4)
q1◇1=q1 for all q1∈0,1Definition 11 (see [12]).
Γ,ψ,ϕ,⋇,◇ is said to be a neutrosophic cone metric space if μ is NCMS of Σ, Γ is an arbitrary set, ◇ is a N-continuous t-conorm, ⋇ is a N-continuous t-norm, and ψ,ϕ are neutrosophic sets on Γ3×Intμ, which satisfy the following statements: ∀ε1,ε2,ε3∈Γ and n,m∈Intμ (that is, n≫0ϕ and m≫0ϕ):(1)
ψε1,ε2,ε3,m>0ϕ∀ε1,ε2,ε3∈Γ(2)
ψε1,ε2,ε3,m=1 iff ε1=ε2=ε3(3)
ψε1,ε2,ε3,m=ψpε1,ε2,ε3,m, where p is permutation(4)
ψε1,ε2,ε3,m⋇ψε2,ε3,n≤ψε1,ε3,m+n(5)
ψε1,ε2,ε3,.:Intμ⟶ ⌋0−, 1+⌊ is neutrosophic-continuousDefinition 12 (see [12]).
LetΓ,ψ,ϕ,⋇,◇ be a NCMS. For m≫0ϕ, the open ball Γx,s,m with center ε1 and radius s∈0,1 is defined by ε1,s,m=ε2∈Γ:ψε1,ε2,m>1−m,ϕε1,ε2,m<s.Example 3.
LetΣ=R+. Then, μ=p1,p2,p3:p1,p2,p3≥0⊆Σ is a normal cone, and P=1 is a normal constant. Let s⋇t=st, Γ=R, and ψ:Γ3×intμ⟶0,1, defined by ψε1,ε2,ε3,t=1/eε1−ε2+ε2−ε3+ε3−ε1/t∀ε1,ε2,ε3∈Γ and t≫∅.Definition 13 (see [12]).
AnΓ,ψ,ϕ,⋇,◇ neutrosophic cone metric is called complete neutrosophic if any sequence which is Cauchy in NCMSΓ,ψ,ϕ is convergent.Definition 14 (see [12]).
Γ,ψ,ϕ,⋇,◇ is said to be a neutrosophic CMS if μ is a neutrosophic cone metric (shortly, NCMS) of Σ, where Γ is an arbitrary set, ⋇ is a neutrosophic continuous t-norm, ◇ is a neutrosophic continuous t-conorm, and ψ,ϕ are neutrosophic sets on Γ3×Intμ, which satisfy the following statements: ∀ε1,ε2,ε3∈Γ and m,n∈Intμ (that is, n≫0ϕ and m≫0ϕ):(1)
ψε1,ε2,ε3,m=1 iff ε1=ε2=ε3(2)
ψε1,ε2,ε3,m⋇ψε2,ε3,n≤ψε1,ε3,n+m(3)
ψε1,ε2,ε3,m=ψpε1,ε2,ε3,m, where p is permutation(4)
ψε1,ε2,ε3,m+ϕε1,ε2,ε3≤1ϕ(5)
ψε1,ε2,ε3,.:Intμ⟶ ⌋0−, 1+⌊ is neutrosophic-continuous(6)
ϕε1,ε2,ε3,m◇ϕε2,ε3,n≥ϕε1,ε3,m+n(7)
ϕε1,ε2,ε3,.:Intμ◇ ⌋0−, 1+⌊ is neutrosophic-continuous(8)
ϕε1,ε2,ε3,m<0ϕ(9)
ϕε1,ε2,ε3,m=0ϕ if and only if ε1=ε2=ε3(10)
ϕε1,ε2,ε3,m>0ϕ∀ε1,ε2,ε3∈Γ(11)
ϕε1,ε2,ε3,m=ϕpε1,ε2,ε3,m, where p is permutation
Then,ψ,ϕ is called a neutrosophic cone metric on Γ.
The functionsψε1,ε2,m and ϕε1,ε2,m are defined by the degree of non-nearness between ε1 and ε2 with respect to m, respectively.Definition 15 (see [12]).
LetΓ,ψ,ϕ,⋇,◇ be a NCMS, ε1∈Γ, and ε1n be a sequence in Γ. Then, ε1n is said to be convergent to ε1 if for all m≫0ϕ and all s∈0,1, there exists n0∈N such that ψε1n,ε1,m>1−s,ϕε1n,ε1,m≤s for any n≥n0. We defined that limn⟶∞ε1n=ε1 or ε1n⟶ε1 as n⟶∞.Definition 16.
A functionΦ:0,∞⟶0,∞ is an altering distance if Φn is monotone increasing and continuous, and Φn=0 iff n=∅.Definition 17.
LetΓ,d be a metric space and let Σ=R+. Defined μ1◇μ2=minμ1+μ2,1 and μ1⋇μ1=μ1μ2 for any μ1,μ2∈0,1, and let Γ and ψ be fuzzy sets on Γ3×intμ represented by ψε1,ε2,ε3,μ=ktn/ktn+ℒD∗ε1,ε2,ε3 and ϕε1,ε2,ε3,μ=D∗ε1,ε2,ε3/mtn+ℒD∗ε1,ε2,ε3.


## 3. Main Result

Definition 18.
LetΓ,ψ,ϕ,⋇,◇ be a neutrosophic cone metric space (CMS) and T,ℋ:Γ⟶Γ be two mappings. Mapping ℋ is said to be neutrosophic Φ,Ψ-weak contraction if there exists a function Ψ:0,∞⟶0,∞ with Ψs>0 and Ψs=0 for s>0 and an alternating distance function Φ such that(1)Φ1ψℋε1,ℋε2,ℋε3,m−1ϕ≤Φ1ψTε1,Tε2,Tε3,m−1ϕ−Ψ1ψTε1,Tε2,Tε3,m−1ϕ,Φϕℋε1,ℋε2,ℋε3,m≤ΦϕTε1,Tε2,Tε3,m−ΨϕTε1,Tε2,Tε3,m.hold for all ε1,ε2,ε3∈ψ and each m≫0ϕ. If T is the identity map, then ℋ is called a neutrosophic Φ,Ψ-weak contraction mapping.Definition 19.
LetΓ,ψ,ϕ,⋇,◇ be a neutrosophic cone metric space and T,ℋ:Γ⟶Γ be two mappings. Point v is said to be a coincidence point in ψ of T and ℋ if ε3=Tv=ℋv.Definition 20.
LetTi and ℋi be two finite families of self-mappings on ψ. They are called pairwise commuting if(1)
TiTj=TjTi, where i,j∈1,2,…,n(2)
ℋiℋj=ℋjℋi, where i,j∈1,2,…,m(3)
Tiℋj=ℋjTi, where i∈1,2,…,n and j∈1,2,…,mTheorem 1.
LetΓ,ψ,ϕ,⋇,◇ be a neutrosophic cone metric space and ℋ:Γ⟶Γ be a neutrosophic Φ,Ψ-weak contraction with respect to T:Γ⟶Γ. If ℋψ⊆Tψ and Tψ or ℋψ is a complete subset of ψ, then T and ℋ have a unique common fixed point in ψ provided that Ψ is a continuous function.Proof.
Lett0∈ψ be an arbitrary point. Let point t1∈ψ such that ℋt0=Tt1. This can be done since ℋψ⊆Tψ. Continuing this process, we obtain a sequence tn∈ψ such that sn=ℋtn=Ttn+1. We assume that sn≠sn+1 for all n∈ℕ; otherwise, T and ℋ have a coincidence point. Now, we get(2)Φ1ψsn,sn,sn+1,m−1ϕ=Φ1ψℋtn,ℋtn,ℋtn+1,m−1ϕ≤Φ1ψTtn,Ttn,Ttn+1,m−1ϕ−Ψ1ψTtn,Ttn,Ttn+1,m−1ϕ≤Φ1ψsn−1,sn−1,sn,m−1ϕ−Ψ1ψsn−1,sn−1,sn,m−1ϕ≤Φ1ψsn−1,sn−1,sn,m−1ϕ,which suppose that T mapping is nondecreasing; hence, ψsn,sn,sn+1,m>ψsn−1,sn−1,sn,m∀n∈ℕ. Hence, ψsn−1,sn−1,sn,m is an increasing sequence of positive real numbers in 0,1. Let Vm=limn⟶∞ψsn−1,sn−1,sn,m. We prove that Vm=1∀m≫0ϕ. If not, there exists m≫0ϕ such that Vm<1ϕ. Then, from the above inequality on taking n⟶∞, we obtain(3)Φ1Vm−1ϕ≤Φ1Vm−1ϕ−Ψ1Vm−1ϕ,which is a contradiction. Therefore, ψsn,sn,sn+1,m⟶1 as n⟶∞. Now, for each k≥0, by Definition 18, we get(4)ψsn,sn,sn+k,m≥ψsn,sn,sn+1,mk∗ψsn+1,sn+1,sn+2,mk∗⋯∗ψsn+k−1,sn+k−1,sn+k,mk.
It follows thatlimn⟶∞ψsn,sn,sn+k,m≥1∗1∗⋯∗1=1. At the same time, we have(5)Φϕsn,sn,sn+1,m=Φϕℋtn,ℋtn,ℋtn+1,m≤ΦϕTtn,Ttn,Ttn+1,m−ΨϕTtn,Ttn,Ttn+1,m≤Φϕsn−1,sn−1,sn,m−Ψϕsn−1,sn−1,sn,m<Φϕsn−1,sn−1,sn,m.in which considering that the T mapping is nondecreasing, then ϕsn,sn,sn+1,m<ϕsn−1,sn−1,sn,m∀n∈ℕ. Thus, ϕsn−1,sn−1,sn,m is a decreasing sequence of positive real numbers in 0,1. Let Um=limn⟶∞ϕsn−1,sn−1,sn,m. We show that Um=0ϕ for all m≫0ϕ. If this is not the case, there exists m≫0ϕ such that Um>0ϕ. Then, it follows from (5), by taking n⟶∞, that ΦUm≤ΦUm−ΨUm, which is a contraction. Therefore, ϕsn,sn,sn+1,m⟶0ϕ as n⟶∞. Now, for each k≥0, by Definition 14 (9), we have(6)ψsn,sn,sn+k,m+ϕsn,sn,sn+k,m≤1ϕ,limn⟶∞ψsn,sn,sn+k,m+ϕsn,sn,sn+k,m≤1ϕ.
It follows thatlimn⟶∞ϕsn,sn,sn+k,m=0ϕ. Hence, sn is a Cauchy sequence. If Tψ is complete, then there exists k∈Tψ such that sn⟶k as n⟶∞. The same holds if ℋψ is complete with k∈ℋψ. Let k∈ψ and Tk=p. Now, we shall show that k is a coincidence point of T and ℋ. In fact, we have taken(7)Φ1ψℋk,ℋk,Ttn+1,m−1ϕ=Φ1ψℋk,ℋk,ℋtn,m−1ϕ≤Φ1ψTk,Tk,Ttn,m−1ϕ−Ψ1ψTk,Tk,Ttn,m−1ϕ,for every m≫0ϕ, in which by letting n⟶∞,(8)limn⟶∞ψℋk,ℋk,Ttn+1,m=limn⟶∞ψℋk,ℋk,ℋtn,m=ψℋk,ℋk,Tk,m=1.
Therefore,Tk=ℋk=p. Now, we shall prove that Tp=p. If it is not so, then we have(9)Φ1ψTp,Tp,Tp,m−1ϕ=Φ1ψℋp,ℋp,ℋk,m−1ϕ≤Φ1ψTp,Tp,Tk,m−1ϕ−Ψ1ψTp,Tp,Tk,m−1ϕ≤Φ1ψTp,Tp,p,m−1ϕ−Ψ1ψTp,Tp,p,m−1ϕ,which is a contradiction. By inequalities (4) and (5) we prove the uniqueness. The desired equality is obtained.Example 4.
LetΓ,ψ,ϕ,⋇,◇ be a complete neutrosophic cone metric space, Γ=1/n:n∈ℕ∪0ϕ, ◇ be a maximum norm, and ⋇ be a minimum norm. Let ψ,ϕ be defined by(10)ψε1,ε2,ε3,m=mm+t+s+s+r+r+t,ifm>0ϕ,0,ifm=0ϕ,ϕε1,ε2,ε3,m=t+s+s+r+r+tm+t+s+s+r+r+t,ifm>0ϕ,0,ifm=0ϕ.
Also, defineΦ,Ψ:0,∞⟶0,∞ by Φm=m/2 and Ψm=m/8, for all m≫p, Tt=t/2, and ℋt=t/4. Obviously, ℋΓ⊆TΓ, and Ψ is a continuous function. Then, we have(11)Φ1ψTt,Ts,Tr,m−1ϕ−Ψ1ψTt,Ts,Tr,m−1ϕ=3t+s+s+r+r+t16m≥2t+s+s+r+r+t16m=Φ1ψℋt,ℋs,ℋr,m−1ϕ.
From the above inequality and the fact thatϕ=1ϕ−ψ, we conclude that the conditions (1) and (2) in Definition 2.18 are satisfied. Thus, ℋ is a neutrosophic Φ−Ψ-weak contraction with respect to T.Corollary 1.
LetΓ,ψ,ϕ,⋇,◇ be a neutrosophic cone metric space and ℋ:Γ⟶Γ be a neutrosophic Φ,Ψ-weak contraction. If Ψ is continuous, then ℋ has a unique fixed point.Corollary 2.
LetΓ,ψ,ϕ,⋇,◇ be a neutrosophic cone metric space. Then, ℋ:Γ◇Γ is a mapping satisfying(12)Φ1ψℋt,ℋs,ℋr,m−1ϕ≤pΦ1ψt,s,r,m−1ϕ,Φϕℋt,ℋs,ℋr,m≤pΦϕt,s,r,m.for each t,s,r∈Γ, m≫0ϕ, and p∈0,1.Theorem 2.
LetΓ,ψ,ϕ,⋇,◇ be a neutrosophic cone metric space and Tj,ℋi be two finite self-mappings on Γ with T=T1.T2…Tm and ℋ=ℋ1.ℋ2…ℋn such that i∈1,2,…,n and j∈1,2,…,m. Suppose ℋ be a generalized neutrosophic Φ,Ψ-weak contraction which is given with respect to T. If TΓ and ℋΓ⊆TΓ or ℋΓ is a complete subset of Γ, then ℋi,Tj have a common fixed point in which Γ is unique, provided a description of Ψ is a continuous function and the families Tj,ℋi commute pairwise.Proof.
By Theorem1, we obtain that T and ℋ have a common fixed point that is unique, say p. In order to prove that p remains as a fixed point of all self-mappings, let(13)ℋℋjp=ℋ1ℋ2…ℋnℋjp=ℋ1ℋ2…ℋn−1ℋnℋjp=ℋ1ℋ2…ℋn−1ℋjℋnp=…=ℋ1ℋjℋ2ℋ3…ℋnp=ℋjℋ1ℋ2ℋ3…ℋnp=ℋjℋp=ℋjp.
Since the other conditions are similarly proved, we can show thatℋTip=Tiℋp=Tip, TTip=TiTp=Tip, and Tℋjp=ℋjTp=ℋjp, which imply that ∀i,j, ℋjp, and Intip are other fixed points of mapping T,ℋ. For the uniqueness of T and ℋ of self-mappings, we get p=ℋjp=Tip, which shows that p is a common fixed point of Tj and ℋi, ∀i,j.Example 5.
LetΓ,ψ,ϕ,⋇,◇ be a complete neutrosophic cone metric space, k=ℝ+, and Γ=0,∞. Define Φ=Ψ:0,∞⟶0,∞ by Φm=m/2,Ψm=m/4, for all m≫ϕ and two families of self mappings Tj and ℋi where i,j∈1,2,…,n by(14)Tjx=0,ifm=0ϕ,1xn6,ifm>0ϕ,ℋix=0,ifm>0ϕ,1xn2,ifm=0ϕ.
Then, we have(15)Φ1ψTt,Ts,Tr,m−1ϕ−Ψ1ψTt,Ts,Tr,m−1ϕ=3z6t6+s6+t6s6+r6+s6r6+t62mt6s6r6≥z2t2+s2+t2s2+r2+s2r2+t22mt2s2r2=Φ1ψℋt,ℋs,ℋr,m−1ϕ.
From the above and the idea ofϕ=1−ψ, we get that statements (i) and (ii) hold. All statements of Theorem 2 hold; therefore, Tj and ℋi have uniqueness.


## 4. Conclusion

In this paper, the definition of the neutrosophic cone metric space is introduced and studied. Based on this definition, we also stated and proved some fixed point theorems on the neutrosophic CMS. We provided a description of the example and investigated some properties in Section3. We established and extended the definition of the Φ,Ψ-weak contraction in the intuitionistic generalized fuzzy cone metric space.



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