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Orthogonal neutrosophic 2-metric spaces
Journal of Inequalities and Applications volume 2023, Article number: 112 (2023)
Abstract
In this study, we introduce the notion of an orthogonal neutrosophic 2-metric space and prove the common fixed-point theorem on an orthogonal neutrosophic 2-metric space. From the obtained results, we give an example to support our results.
1 Introduction
Nowadays, a fuzzy concept has become the subject of several research works. Finding the fuzzy equivalents of the classical set theory is one of the advancements made to the basic theory of fuzzy sets provided by Zadeh [1]. Following that, the use of a fuzzy metric space in applied sciences including fixed-point theory, image and signal processing, medical imaging, and decision making occurred. The concept of intuitionistic fuzzy metric spaces was first proposed by Park [2]. The domains of population dynamics [3] computer programming [4], chaos control [5], nonlinear dynamical system [6], and medicine [7] are only a few examples of the scientific and technological fields that have utilized it. Gahler [8] presented a study on a 2-metric space. Schweizer and Sklar [9] explored the statistical metric spaces. The concept of intuitionistic fuzzy sets was presented by Atanassov [10] and Çoker [11] and the concept of intuitionistic fuzzy topological was discussed in [12]. In [13] the authors introduced the concepts of intuitionistic fuzzy 2-normed spaces and in [14] intuitionistic fuzzy 2-metric spaces.
Bera and Mahapatra [15] established the neutrosophic soft linear space. The neutrosophic normed linear space was established by Bera and Mahapatra [16]. The concept of an orthogonal neutrosophic metric space was introduced by Ishtiaq et al. [17] who proved several fixed-point results in the context of an orthogonal neutrosophic metric space. The contraction mapping was used to prove common fixed-point results in the context of a neutrosophic metric space established by Jeyaraman and Sowndrarajan [18]. Several fixed-point results in weak and rational \((\alpha -\psi )\)-contractions in an ordered 2-metric space were established by Fathollahi et al. [19]. Many authors like Salama and Alblowi [20] worked on neutrosophic topological spaces and Al-omeri et al. [21] worked on a neutrosophic cone metric space, etc. Mursaleen and Lohani [22] introduced the idea of an intuitionistic 2-normed space and an intuitionistic 2-metric space. Ali Asghar and Aftab Hussain [23] established the basic properties of N2MSs and demonstrated some fixed-point findings. Umar Ishtiaq [24] introduced the notion of ONMSs and investigated some fixed-point results. The idea of orthogonality has several applications in mathematics. The notion of orthogonality in a metric space was established by Eshagi Gordji, Ramezani, De la Sen and Cho [25] and also expanded the findings in the setting of a metric space with new orthogonality and proved fixed-point theorems.
The main objectives of this study are as follows:
-
(i)
To introduce the concept of an orthogonal neutrosophic 2-metric space (ON2MS).
-
(ii)
To prove common fixed-point results on the orthogonal neutrosophic 2-metric space.
-
(iii)
To enhance the literature of an intuitionistic fuzzy 2-metric space and a neutrosophic metric space.
-
(iv)
To prove the uniqueness of the solution of integral equations.
Now, we provide some basic definitions to help to understand the main section.
2 Preliminaries
Here, “con-t-nm” means continuous triangular-norm, “con-t-conm” means continuous-triangular-conorm, “NMS” means neutrosophic metric space, “N2MS” means neutrosophic 2-metric space, “ON2MS” means orthogonal neutrosophic 2-metric space. Some basic definitions are given below:
Definition 2.1
[26] Let \(\ast \colon [0, 1]\times [0, 1]\rightarrow [0, 1]\) be a con-t-nm on a binary operation, then:
-
(I)
∗ is associative and commutative;
-
(II)
∗ is continuous;
-
(III)
\(\mu \ast 1=\mu \) for all \(\mu \in [0, 1]\);
-
(IV)
\(\mu \ast \alpha \leq \eta \ast \gamma \), when \(\mu \leq \eta \) and \(\alpha \leq \gamma \) for all \(\mu , \alpha , \eta , \gamma \in [0, 1]\).
Definition 2.2
Let \(+\colon [0, 1]\times [0, 1]\rightarrow [0, 1]\) be a con-t-conm on a binary operation, then it satisfies (I), (II), (IV), and
-
(III)
\(\mu +0=\mu \) for all \(\mu \in [0, 1]\).
Definition 2.3
Let Φ be the universe. A neutrosophic set (NS) \(\mathcal{A}\) in Φ is characterized by a truth membership function \(\mathcal{Q}_{\mathcal{A}}\), an indeterminacy membership function \(\mathcal{F}_{\mathcal{A}}\), and a falsity membership function \(\mathcal{G}_{\mathcal{A}}\), where \(\mathcal{Q}_{\mathcal{A}}\), \(\mathcal{F}_{\mathcal{A}}\), and \(\mathcal{G}_{\mathcal{A}}\) are real standard elements of \([0, 1]\). This can be written as:
There is no restriction on the sum of \(\mathcal{Q}_{\mathcal{A}}(\nu )\), \(\mathcal{F}_{\mathcal{A}}(\nu )\), and \(\mathcal{G}_{\mathcal{A}}(\nu )\) and so \(0^{-}\leq \mathcal{Q}_{\mathcal{A}}(\nu )+ \mathcal{F}_{\mathcal{A}}( \nu )+ \mathcal{G}_{\mathcal{A}}(\nu )\leq 3^{+}\).
Definition 2.4
[27] Let \(\varPhi \neq \emptyset \). A 6-tuple \((\varPhi , \mathcal{Q}, \mathcal{F}, \mathcal{G}, \ast , +)\), where ∗ is a con-t-nm, + is a con-t-conm, \(\mathcal{Q}\), \(\mathcal{F}\), and \(\mathcal{G}\) are neutrosophic sets on \(\varPhi \times \varPhi \times (0, \infty )\). If \((\varPhi , \mathcal{Q}, \mathcal{F}, \mathcal{G}, \ast , +)\), satisfies the conditions below for all \(\nu , \varrho , \mathfrak{z}\in \varPhi \), and \(\wp , \mathfrak{s}>0\):
-
(N1)
\(\mathcal{Q}(\nu , \varrho , \wp )+\mathcal{F}(\nu , \varrho , \wp )+ \mathcal{G}(\nu , \varrho , \wp )\leq 3\);
-
(N2)
\(0\leq \mathcal{Q}(\nu , \varrho , \wp )\leq 1\);
-
(N3)
\(\mathcal{Q}(\nu , \varrho , \wp )=1\) if and only if \(\nu =\varrho \);
-
(N4)
\(\mathcal{Q}(\nu , \varrho , \wp )=\mathcal{Q}(\varrho , \nu , \wp )\);
-
(N5)
\(\mathcal{Q}(\nu , \varsigma , \wp +\mathfrak{s})\geq \mathcal{Q}( \nu , \varrho , \wp )\ast \mathcal{Q}(\varrho , \varsigma , \mathfrak{s})\);
-
(N6)
\(\mathcal{Q}(\nu , \varrho , \cdot )\colon [0, \infty )\rightarrow [0, 1]\) is continuous;
-
(N7)
\(\lim_{\wp \rightarrow \infty}\mathcal{Q}(\nu , \varrho , \wp )=1\);
-
(N8)
\(0\leq \mathcal{F}(\nu , \varrho , \wp )\leq 1\);
-
(N9)
\(\mathcal{F}(\nu , \varrho , \wp )=0\) if and only if \(\nu =\varrho \);
-
(N10)
\(\mathcal{F}(\nu , \varrho , \wp )=\mathcal{F}(\varrho , \nu , \wp )\);
-
(N11)
\(\mathcal{F}(\nu , \varsigma , \wp +\mathfrak{s})\leq \mathcal{F}( \nu , \varrho , \wp )+\mathcal{F}(\varrho , \varsigma , \mathfrak{s})\);
-
(N12)
\(\mathcal{F}(\nu , \varrho , \cdot )\colon [0, \infty )\rightarrow [0, 1]\) is continuous;
-
(N13)
\(\lim_{\wp \rightarrow \infty}\mathcal{F}(\nu , \varrho , \wp )=0\);
-
(N14)
\(0\leq \mathcal{G}(\nu , \varrho , \wp )\leq 1\);
-
(N15)
\(\mathcal{F}(\nu , \varrho , \wp )=0\) if and only if \(\nu =\varrho \);
-
(N16)
\(\mathcal{G}(\nu , \varrho , \wp )=\mathcal{G}(\varrho , \nu , \wp )\);
-
(N17)
\(\mathcal{G}(\nu , \varsigma , \wp +\mathfrak{s})\leq \mathcal{G}( \nu , \varrho , \wp )+\mathcal{G}(\varrho , \varsigma , \mathfrak{s})\);
-
(N18)
\(\mathcal{G}(\nu , \varrho , \cdot )\colon [0, \infty )\rightarrow [0, 1]\) is continuous;
-
(N19)
\(\lim_{\wp \rightarrow \infty}\mathcal{G}(\nu , \varrho , \wp )=0\);
-
(N20)
if \(\wp \leq 0\), then \(\mathcal{Q}(\nu , \varrho , \wp )=0\), \(\mathcal{F}(\nu , \varrho , \wp )=1\), \(\mathcal{G}(\nu , \varrho , \wp )=1\).
Then, \((\mathcal{Q}, \mathcal{F}, \mathcal{G})\) is a neutrosophic metric and \((\varPhi , \mathcal{Q}, \mathcal{F}, \mathcal{G}, \ast , +)\) is a NMS.
Definition 2.5
[28] The 5-tuple \((\varPhi , \mathcal{Q}, \mathcal{F}, \ast , +)\) is called an intuitionistic fuzzy 2-metric space if Φ is any nonvoid set, ∗ is a con-t-nm, + is a con-t-conm, and \(\mathcal{Q}\), \(\mathcal{F}\) are fuzzy sets on \(\varPhi \times \varPhi \times \varPhi \times (0, \infty )\), then it satisfies for all \(\nu , \varrho , \varsigma , \mathfrak{w}\in \varPhi \), and \(\mathfrak{s}, \wp >0\):
-
(a)
\(\mathcal{Q}(\nu , \varrho , \varsigma , \wp )+\mathcal{F}(\nu , \varrho , \varsigma , \wp )\leq 1\);
-
(b)
Let ν, ϱ of Φ, there exists an element ς of Φ such that \(0\leq \mathcal{Q}(\nu , \varrho , \varsigma , \wp )\leq 1\);
-
(c)
\(\mathcal{Q}(\nu , \varrho , \varsigma , \wp )=1\) if at least two of ν, ϱ, ς are equal;
-
(d)
\(\mathcal{Q}(\nu , \varrho , \varsigma , \wp )=\mathcal{Q}(\nu , \varsigma , \varrho , \wp )=\mathcal{Q}(\varrho , \varsigma , \nu , \wp )\) for all ν, ϱ, ς in Φ;
-
(e)
\(\mathcal{Q}(\nu , \varrho , \mathfrak{w}, \wp )\ast \mathcal{Q}(\nu , \mathfrak{w}, \varsigma , \mathfrak{s})\ast \mathcal{Q}(\mathfrak{w}, \varrho , \varsigma , \mathfrak{h})\leq \mathcal{Q}(\nu , \varrho , \varsigma , \wp +\mathfrak{s}+\mathfrak{h})\) for all \(\nu , \varrho , \varsigma , \mathfrak{w}\in \varPhi \);
-
(f)
\(\mathcal{Q}(\nu , \varrho , \varsigma , \cdot )\colon (0, \infty ) \rightarrow (0, 1]\) is continuous;
-
(g)
\(\mathcal{F}(\nu , \varrho , \varsigma , \wp )<1\);
-
(h)
\(\mathcal{F}(\nu , \varrho , \varsigma , \wp )=0\) if at least two of ν, ϱ, ς are equal;
-
(i)
\(\mathcal{F}(\nu , \varrho , \varsigma , \wp )=\mathcal{F}(\nu , \varsigma , \varrho , \wp )=\mathcal{F}(\varrho , \varsigma , \nu , \wp )\) for all ν, ϱ, ς in Φ;
-
(j)
\(\mathcal{F}(\nu , \varrho , \mathfrak{w}, \wp )+\mathcal{F}(\nu , \mathfrak{w}, \varsigma , \mathfrak{s})+\mathcal{F}(\mathfrak{w}, \varrho , \varsigma , \mathfrak{h})\geq \mathcal{F}(\nu , \varrho , \varsigma , \wp +\mathfrak{s}+\mathfrak{h})\);
-
(k)
\(\mathcal{F}(\nu , \varrho , \varsigma , \cdot )\colon (0, \infty ) \rightarrow (0, 1]\) is continuous.
Definition 2.6
The 6-tuple \((\varPhi , \mathcal{Q}, \mathcal{F}, \mathcal{G}, \ast , +)\) is said to be a N2MS if Φ is any nonempty set, ∗ is a con-t-nm, + is a con-t-conm, and \(\mathcal{Q}\), \(\mathcal{F}\), \(\mathcal{G}\) are neutrosophic sets on \(\varPhi \times \varPhi \times \varPhi \times (0, \infty )\), then it satisfies for all \(\nu , \varrho , \varsigma , \mathfrak{w}\in \varPhi \), and \(\mathfrak{s}, \wp >0\);
-
(N2MS1)
\(\mathcal{Q}(\nu , \varrho , \varsigma , \wp )+\mathcal{F}(\nu , \varrho , \varsigma , \wp )+\mathcal{G}(\nu , \varrho , \varsigma , \wp )\leq 3\);
-
(N2MS2)
Let ν, ϱ of Φ, there exists an element ς of Φ such that \(0\leq \mathcal{Q}(\nu , \varrho , \varsigma , \wp )\leq 1\);
-
(N2MS3)
\(\mathcal{Q}(\nu , \varrho , \varsigma , \wp )=1\) if at least two of ν, ϱ, ς are equal;
-
(N2MS4)
\(\mathcal{Q}(\nu , \varrho , \varsigma , \wp )=\mathcal{Q}(\nu , \varsigma , \varrho , \wp )=\mathcal{Q}(\varrho , \varsigma , \nu , \wp )\);
-
(N2MS5)
\(\mathcal{Q}(\nu , \varrho , \mathfrak{w}, \wp )\ast \mathcal{Q}(\nu , \mathfrak{w}, \varsigma , \mathfrak{s})\ast \mathcal{Q}(\mathfrak{w}, \varrho , \varsigma , \mathfrak{h})\leq \mathcal{Q}(\nu , \varrho , \varsigma , \wp +\mathfrak{s}+\mathfrak{h})\);
-
(N2MS6)
\(\mathcal{Q}(\nu , \varrho , \varsigma , \cdot )\colon (0, \infty ) \rightarrow (0, 1]\) is continuous for all \(\nu , \varrho , \varsigma \in \varPhi \) such that \(\nu \perp \varrho \perp \varsigma \);
-
(N2MS7)
\(\mathcal{F}(\nu , \varrho , \varsigma , \wp )\leq 1\);
-
(N2MS8)
\(\mathcal{F}(\nu , \varrho , \varsigma , \wp )=0\) if at least two of ν, ϱ, ς are equal;
-
(N2MS9)
\(\mathcal{F}(\nu , \varrho , \varsigma , \wp )=\mathcal{F}(\nu , \varsigma , \varrho , \wp )=\mathcal{F}(\varrho , \varsigma , \nu , \wp )\);
-
(N2MS10)
\(\mathcal{F}(\nu , \varrho , \mathfrak{w}, \wp )+\mathcal{F}(\nu , \mathfrak{w}, \varsigma , \mathfrak{s})+\mathcal{F}(\mathfrak{w}, \varrho , \varsigma , \mathfrak{h})\geq \mathcal{F}(\nu , \varrho , \varsigma , \wp +\mathfrak{s}+\mathfrak{h})\);
-
(N2MS11)
\(\mathcal{F}(\nu , \varrho , \varsigma , \cdot )\colon (0, \infty ) \rightarrow (0, 1]\) is continuous;
-
(N2MS12)
\(\mathcal{G}(\nu , \varrho , \varsigma , \wp )\leq 1\);
-
(N2MS13)
\(\mathcal{G}(\nu , \varrho , \varsigma , \wp )=0\) if at least two of ν, ϱ, ς are equal;
-
(N2MS14)
\(\mathcal{G}(\nu , \varrho , \varsigma , \wp )=\mathcal{G}(\nu , \varsigma , \varrho , \wp )=\mathcal{G}(\varrho , \varsigma , \nu , \wp )\);
-
(N2MS15)
\(\mathcal{G}(\nu , \varrho , \mathfrak{w}, \wp )+\mathcal{G}(\nu , \mathfrak{w}, \varsigma , \mathfrak{s})+\mathcal{G}(\mathfrak{w}, \varrho , \varsigma , \mathfrak{h})\geq \mathcal{G}(\nu , \varrho , \varsigma , \wp +\mathfrak{s}+\mathfrak{h})\);
-
(N2MS16)
\(\mathcal{G}(\nu , \varrho , \varsigma , \cdot )\colon (0, \infty ) \rightarrow (0, 1]\) is continuous.
Here, the functions \(\mathcal{Q}(\nu , \varrho , \varsigma , \wp )\), \(\mathcal{F}(\nu , \varrho , \varsigma , \wp )\), and \(\mathcal{G}(\nu , \varrho , \varsigma , \wp )\) denotes the degree of nearness, the degree of nonnearness, and the degree of naturalness between ν, ϱ, and ς with respect to ℘, respectively.
Now, we define the notion of ON2MS
Definition 2.7
The 6-tuple \((\varPhi , \mathcal{Q}, \mathcal{F}, \mathcal{G}, \ast , +, \perp )\) is said to be a ON2MS if Φ is any nonempty set, ∗ is a con-t-nm, + is a con-t-conm, and \(\mathcal{Q}\), \(\mathcal{F}\), \(\mathcal{G}\) are neutrosophic sets on \(\varPhi \times \varPhi \times \varPhi \times (0, \infty )\), then it satisfies for all \(\nu , \varrho , \varsigma , \mathfrak{w}\in \varPhi \) and \(\mathfrak{s}, \wp >0\);
-
(ON2MS1)
\(\mathcal{Q}(\nu , \varrho , \varsigma , \wp )+\mathcal{F}(\nu , \varrho , \varsigma , \wp )+\mathcal{G}(\nu , \varrho , \varsigma , \wp )\leq 3\) for all \(\nu , \varrho , \varsigma \in \varPhi \), \(\wp >0\) such that \(\nu \perp \varrho \perp \varsigma \);
-
(ON2MS2)
Let ν, ϱ of Φ, there exists an element ς of Φ such that \(0\leq \mathcal{Q}(\nu , \varrho , \varsigma , \wp )\leq 1\) such that \(\nu \perp \varrho \perp \varsigma \);
-
(ON2MS3)
\(\mathcal{Q}(\nu , \varrho , \varsigma , \wp )=1\) if at least two of ν, ϱ, ς are equal such that \(\nu \perp \varrho \perp \varsigma \);
-
(ON2MS4)
\(\mathcal{Q}(\nu , \varrho , \varsigma , \wp )=\mathcal{Q}(\nu , \varsigma , \varrho , \wp )=\mathcal{Q}(\varrho , \varsigma , \nu , \wp )\) for all ν, ϱ, ς in Φ such that \(\nu \perp \varrho \perp \varsigma \);
-
(ON2MS5)
\(\mathcal{Q}(\nu , \varrho , \mathfrak{w}, \wp )\ast \mathcal{Q}(\nu , \mathfrak{w}, \varsigma , \mathfrak{s})\ast \mathcal{Q}(\mathfrak{w}, \varrho , \varsigma , \mathfrak{h})\leq \mathcal{Q}(\nu , \varrho , \varsigma , \wp +\mathfrak{s}+\mathfrak{h})\) for all \(\nu , \varrho , \varsigma , \mathfrak{w}\in \varPhi \) such that \(\nu \perp \varrho \perp \varsigma \);
-
(ON2MS6)
\(\mathcal{Q}(\nu , \varrho , \varsigma , \cdot )\colon (0, \infty ) \rightarrow (0, 1]\) is continuous for all \(\nu , \varrho , \varsigma \in \varPhi \) such that \(\nu \perp \varrho \perp \varsigma \);
-
(ON2MS7)
\(\mathcal{F}(\nu , \varrho , \varsigma , \wp )\leq 1\), for all \(\nu , \varrho , \varsigma \in \varPhi \) such that \(\nu \perp \varrho \perp \varsigma \);
-
(ON2MS8)
\(\mathcal{F}(\nu , \varrho , \varsigma , \wp )=0\) if at least two of ν, ϱ, ς are equal for all \(\nu , \varrho , \varsigma \in \varPhi \) such that \(\nu \perp \varrho \perp \varsigma \);
-
(ON2MS9)
\(\mathcal{F}(\nu , \varrho , \varsigma , \wp )=\mathcal{F}(\nu , \varsigma , \varrho , \wp )=\mathcal{F}(\varrho , \varsigma , \nu , \wp )\) for all \(\nu , \varrho , \varsigma \in \varPhi \) such that \(\nu \perp \varrho \perp \varsigma \);
-
(ON2MS10)
\(\mathcal{F}(\nu , \varrho , \mathfrak{w}, \wp )+\mathcal{F}(\nu , \mathfrak{w}, \varsigma , \mathfrak{s})+\mathcal{F}(\mathfrak{w}, \varrho , \varsigma , \mathfrak{h})\geq \mathcal{F}(\nu , \varrho , \varsigma , \wp +\mathfrak{s}+\mathfrak{h})\), for all \(\nu , \varrho , \varsigma \in \varPhi \) such that \(\nu \perp \varrho \perp \varsigma \);
-
(ON2MS11)
\(\mathcal{F}(\nu , \varrho , \varsigma , \cdot )\colon (0, \infty ) \rightarrow (0, 1]\) is continuous for all \(\nu , \varrho , \varsigma \in \varPhi \) such that \(\nu \perp \varrho \perp \varsigma \);
-
(ON2MS12)
\(\mathcal{G}(\nu , \varrho , \varsigma , \wp )\leq 1\), for all \(\nu , \varrho , \varsigma \in \varPhi \) such that \(\nu \perp \varrho \perp \varsigma \);
-
(ON2MS13)
\(\mathcal{G}(\nu , \varrho , \varsigma , \wp )=0\) if at least two of ν, ϱ, ς are equal for all \(\nu , \varrho , \varsigma \in \varPhi \) such that \(\nu \perp \varrho \perp \varsigma \);
-
(ON2MS14)
\(\mathcal{G}(\nu , \varrho , \varsigma , \wp )=\mathcal{G}(\nu , \varsigma , \varrho , \wp )=\mathcal{G}(\varrho , \varsigma , \nu , \wp )\) for all \(\nu , \varrho , \varsigma \in \varPhi \) such that \(\nu \perp \varrho \perp \varsigma \);
-
(ON2MS15)
\(\mathcal{G}(\nu , \varrho , \mathfrak{w}, \wp )+\mathcal{G}(\nu , \mathfrak{w}, \varsigma , \mathfrak{s})+\mathcal{G}(\mathfrak{w}, \varrho , \varsigma , \mathfrak{h})\geq \mathcal{G}(\nu , \varrho , \varsigma , \wp +\mathfrak{s}+\mathfrak{h})\), for all \(\nu , \varrho , \varsigma \in \varPhi \) such that \(\nu \perp \varrho \perp \varsigma \);
-
(ON2MS16)
\(\mathcal{G}(\nu , \varrho , \varsigma , \cdot )\colon (0, \infty ) \rightarrow (0, 1]\) is continuous for all \(\nu , \varrho , \varsigma \in \varPhi \) such that \(\nu \perp \varrho \perp \varsigma \).
Definition 2.8
Suppose \((\varPhi , \mathcal{Q}, \mathcal{F}, \mathcal{G}, \ast , +, \perp )\) is a ON2MS. Suppose \(\mathfrak{h}\in (0, 1)\), \(\wp >0\) and \(\nu \in \varPhi \). The set \(\mathbb{B}(\nu , \mathfrak{h}, \wp )=\{\varrho \in \varPhi \colon \mathcal{Q}(\nu , \varrho , \varsigma , \wp )>1-\mathfrak{h}, \mathcal{F}(\nu , \varrho , \varsigma , \wp )<\mathfrak{h}\text{ and } \mathcal{G}(\nu , \varrho , \varsigma , \wp )<\mathfrak{h}\text{ for all } \varsigma \in \varPhi \}\) is called the open ball with center ν and radius \(\mathfrak{h}\) with respect to ℘.
Definition 2.9
Suppose \((\varPhi , \mathcal{Q}, \mathcal{F}, \mathcal{G}, \ast , +, \perp )\) is a ON2MS. Then, an open set of \(\mathcal{U}\subset \varPhi \) of its points is the center of a open ball contained in \(\mathcal{U}\). The open set in a N2MS \((\varPhi , \mathcal{Q}, \mathcal{F}, \mathcal{G}, \ast , +, \perp )\) is represented by \(\mathbb{U}\).
Definition 2.10
Assume \((\varPhi , \mathcal{Q}, \mathcal{F}, \mathcal{G}, \ast , +, \perp )\) is a ON2MS. A sequence \((\nu _{\mathfrak{n}})\) in Φ is a Cauchy one if for each \(\epsilon >0\) and each \(\wp >0\), there exist \(\mathfrak{n}^{*}\in \mathbb{N}\) such that \(\mathcal{Q}(\nu _{\mathfrak{n}}, \nu _{\mathfrak{m}}, \mathfrak{h}, \wp )>1-\mathfrak{h}\), \(\mathcal{F}(\nu _{\mathfrak{n}}, \nu _{ \mathfrak{m}}, \mathfrak{h}, \wp )<\mathfrak{h}\) and \(\mathcal{G}(\nu _{\mathfrak{n}}, \nu _{\mathfrak{m}}, \mathfrak{h}, \wp )<\mathfrak{h}\) for all \(\mathfrak{n}, \mathfrak{m}\geq \mathfrak{n}^{*}\) for all \(\mathfrak{h}\in \varPhi \).
Definition 2.11
Suppose \((\varPhi , \mathcal{Q}, \mathcal{F}, \mathcal{G}, \ast , +, \perp )\) is a ON2MS. A sequence \(\nu =(\nu _{\imath})\) is convergent to \(\mathfrak{l}\in \varPhi \), with respect to the ON2MS if, for every \(\epsilon >0\) and \(\wp >0\), there exist \(\imath _{0}\in \mathbb{N}\) such that \(\mathcal{Q}(\nu _{\imath}, \mathfrak{l}, \mathfrak{h}, \wp )>1- \epsilon \), \(\mathcal{F}(\nu _{\imath}, \mathfrak{l}, \mathfrak{h}, \wp )<\epsilon \), and \(\mathcal{G}(\nu _{\imath}, \mathfrak{l}, \mathfrak{h}, \wp ) \epsilon \) for all \(\imath \geq \imath _{0}\) and for all \(\mathfrak{h}\in \varPhi \). In this case, we write \((\mathcal{Q}, \mathcal{F}, \mathcal{G})_{2}-\lim \nu =\mathfrak{l}\) (or) \(\nu _{\imath} \xrightarrow{(\mathcal{Q}, \mathcal{F}, \mathcal{G})_{2}}\mathfrak{l}\) as \(\imath \rightarrow \infty \).
Definition 2.12
Let \((\varPhi , \mathcal{Q}, \mathcal{F}, \mathcal{G}, \ast , +, \perp )\) be a ON2MS. Define \(\tau _{(\mathcal{Q}, \mathcal{F}, \mathcal{G})_{2}}=\varUpsilon \subset \varPhi \colon \) for each \(\nu \in \varPhi \), there exist \(\wp >0\) and \(\mathfrak{h}\in (0, 1)\) such that \(\mathbb{B}(\nu , \mathfrak{h}, \wp )\subset \varPhi \}\). Then, \(\tau _{(\mathcal{Q}, \mathcal{F}, \mathcal{G})_{2}}\) is a topology on \((\varPhi , \mathcal{Q}, \mathcal{F}, \mathcal{G}, \ast , +, \perp )\).
Definition 2.13
Let \((\varPhi , \mathcal{Q}, \mathcal{F}, \mathcal{G}, \ast , +, \perp )\) be a ON2MS. If each Cauchy sequence converges with respect to \(\varsigma (\mathcal{Q}, \mathcal{F}, \mathcal{G})_{2}\) then it is called complete.
Theorem 2.1
Every open ball \(\mathbb{B}(\nu , \mathfrak{h}, \wp )\) in ON2MS is an open set.
Proof
Consider \(\mathbb{B}(\nu , \mathfrak{h}, \wp )\) to be an open ball with center ν and radius \(\mathfrak{h}\). Assume \(\varrho \in \mathbb{B}(\nu , \mathfrak{h}, \wp )\). Therefore, \(\mathcal{Q}(\nu , \varrho , \varsigma , \wp )>1-\mathfrak{h}\), \(\mathcal{F}(\nu , \varrho , \varsigma , \wp )<\mathfrak{h}\), and \(\mathcal{G}(\nu , \varrho , \varsigma , \wp )<\mathfrak{h}\) for each \(\varsigma \in \Xi \). There exists \(\frac{\wp}{3}\in (0, \wp )\) such that \(\mathcal{Q}(\nu , \varrho , \mathfrak{p}, \frac{\wp}{3})>1- \mathfrak{h}\), \(\mathcal{F}(\nu , \varrho , \mathfrak{p}, \frac{\wp}{3})< \mathfrak{h}\), and \(\mathcal{G}(\nu , \varrho , \mathfrak{p}, \frac{\wp}{3})< \mathfrak{h}\), due to \(\mathcal{Q}(\nu , \varrho , \varsigma , \wp )>1-\mathfrak{h}\). If we take \(\mathfrak{h}_{0}=\mathcal{Q}(\nu , \varrho , \mathfrak{p}, \frac{\wp}{3})\), then for \(\mathfrak{h}_{0}>1-\mathfrak{h}\), \(\epsilon \in (0, 1)\) will exist such that \(\mathfrak{h}_{0}>1-\epsilon >1-\mathfrak{h}\). Given \(\mathfrak{h}_{0}\) and ϵ such that \(\mathfrak{h}_{0}>1-\epsilon \), then \(\{\mathfrak{h}_{\mathfrak{i}}\}^{6}_{\mathfrak{i}=1}\in (0, 1)\) such that \(\mathfrak{h}_{0}\ast \mathfrak{h}_{1}\ast \mathfrak{h}_{2}>1- \epsilon \), \((1-\mathfrak{h}_{0})+(1-\mathfrak{h}_{3})+(1-\mathfrak{h}_{4}) \leq \epsilon \), and \((1-\mathfrak{h}_{0})+(1-\mathfrak{h}_{5})+(1-\mathfrak{h}_{6})\leq \epsilon \). Choose \(\mathfrak{h}_{7}=\max \{\mathfrak{h}_{\mathfrak{i}}\}^{6}_{ \mathfrak{i}=1}\). Consider \(\mathbb{B}(\varrho , 1-\mathfrak{h}_{7}, \frac{\wp}{3})\). To show that \(\mathbb{B}(\varrho , 1-\mathfrak{h}_{7}, \frac{\wp}{3})\subset \mathbb{B}(\nu , \mathfrak{h}, \wp )\), consider \(\mathfrak{v}\in \mathbb{B}(\varrho , 1-\mathfrak{h}_{7}, \frac{\wp}{3})\), then \(\mathcal{Q}(\nu , \mathfrak{p}, \varsigma , \frac{\wp}{3})> \mathfrak{h}_{7}\), \(\mathcal{F}(\nu , \mathfrak{p}, \varsigma , \frac{\wp}{3})<\mathfrak{h}_{7}\), and \(\mathcal{Q}(\nu , \mathfrak{p}, \varsigma , \frac{\wp}{3})< \mathfrak{h}_{7}\) and \(\mathcal{F}(\mathfrak{p}, \varrho , \varsigma , \frac{\wp}{3})> \mathfrak{h}_{7}\), \(\mathcal{F}(\mathfrak{p}, \varrho , \varsigma , \frac{\wp}{3})<\mathfrak{h}_{7}\), and \(\mathcal{G}(\mathfrak{p}, \varrho , \varsigma , \frac{\wp}{3})< \mathfrak{h}_{7}\). Then,
We obtain \(\mathfrak{v}\in \mathbb{B}(\nu , \mathfrak{h}, \wp )\) and \(\mathbb{B}(\varrho , 1-\mathfrak{h}_{7}, \frac{\wp}{3})\subset \mathbb{B}(\nu , \mathfrak{h}, \wp )\). □
Theorem 2.2
Every ON2MS is Hausdorff.
Proof
Let \((\varPhi , \mathcal{Q}, \mathcal{F}, \mathcal{G}, \ast , +)\) be a N2MS. Let ν and ϱ be points in Φ. Then, \(0<\mathcal{Q}(\nu , \varrho , \varsigma , \wp )<1\), \(0<\mathcal{F}( \nu , \varrho , \varsigma , \wp )<1\), and \(0<\mathcal{G}(\nu , \varrho , \varsigma , \wp )<1\) for every \(\varsigma \in \varPhi \). Put \(\mathfrak{h}_{1}=\mathcal{Q}(\nu , \varrho , \varsigma _{1}, \wp )\), \(1- \mathfrak{h}_{2}=\mathcal{F}(\nu , \varrho , \varsigma _{1}, \wp )\), and \(1-\mathfrak{h}_{3}=\mathcal{G}(\nu , \varrho , \varsigma _{1}, \wp )\), \(\mathfrak{h}_{4}=\mathcal{Q}(\nu , \varrho , \mathfrak{p}, \frac{\wp}{3})\), \(1-\mathfrak{h}_{5}=\mathcal{F}(\nu , \varrho , \mathfrak{p}, \frac{\wp}{3})\), \(1-\mathfrak{h}_{6}=\mathcal{G}(\nu , \varrho , \mathfrak{p}, \frac{\wp}{3})\) and \(\mathfrak{h}=\max \{\mathfrak{h}_{1}, 1-\mathfrak{h}_{2}, 1- \mathfrak{h}_{3}, \mathfrak{h}_{4}, 1-\mathfrak{h}_{5}, 1- \mathfrak{h}_{6}\}\). For each \(\mathfrak{h}_{0}\in (\mathfrak{h}, 1)\) there exist \(\mathfrak{h}_{7}\) and \(\mathfrak{h}_{8}\) such that \(\mathfrak{h}_{4}\ast \mathfrak{h}_{7}\ast \mathfrak{h}_{7}\geq \mathfrak{h}_{0}\) and \((1-\mathfrak{h}_{5})\ast (1-\mathfrak{h}_{8})\ast (1-\mathfrak{h}_{8}) \leq 1-\mathfrak{h}_{0}\). Put \(\mathfrak{h}_{9}=\max \{\mathfrak{h}_{7}, \mathfrak{h}_{8}\}\) and consider the open balls \(\mathbb{B}(\nu , 1-\mathfrak{h}_{9}, \frac{\wp}{3})\) and \(\mathbb{B}(\varrho , 1-\mathfrak{h}_{9}, \frac{\wp}{3})\). Then, clearly
If there is \(\mathfrak{p}\in \mathbb{B}(\nu , 1-\mathfrak{h}_{9}, \frac{\wp}{3}) \cap \mathbb{B}(\varrho , 1-\mathfrak{h}_{9}, \frac{\wp}{3})= \emptyset \), then
and similarly, \(1-\mathfrak{h}_{2}<1-\mathfrak{h}_{2}\), is its contrary. Hence, \((\varPhi , \mathcal{Q}, \mathcal{F}, \mathcal{G}, \ast , +)\) is Hausdorff. □
3 Main results
Lemma 1
If \((\varPhi , \mathcal{Q}, \mathcal{F}, \mathcal{G}, \ast , +, \perp )\) is a N2MS. Then, \(\mathcal{Q}(\nu , \varrho , \varsigma , \wp )\) is nondecreasing, \(\mathcal{F}(\nu , \varrho , \varsigma , \wp )\) is nonincreasing, and \(\mathcal{G}(\nu , \varrho , \varsigma , \wp )\) is nonincreasing for all \(\nu , \varrho , \varsigma \in \varPhi \).
Proof
Let \(\mathfrak{s}, \wp >0\) be any points such that \(\wp >\mathfrak{s}\cdot \wp =\mathfrak{s}+\frac{\wp -\mathfrak{s}}{2}+ \frac{\wp -\mathfrak{s}}{2}\). Hence, we have
and
Similarly, \(\mathcal{Q}(\nu , \varrho , \varsigma , \wp )>\mathcal{Q}(\nu , \varrho , \varsigma , \mathfrak{s})\). □
From Lemma 1, let \((\varPhi , \mathcal{Q}, \mathcal{F}, \mathcal{G}, \ast , +, \perp )\) be a ON2MS with the following conditions:
Lemma 2
Let \((\varPhi , \mathcal{Q}, \mathcal{F}, \mathcal{G}, \ast , +, \perp )\) be a ON2MS. If there exists \(\ell \in (0, 1)\) such that \(\mathcal{Q}(\nu , \varrho , \varsigma , \ell \wp +0)\geq \mathcal{Q}( \nu , \varrho , \varsigma , \wp )\), \(\mathcal{F}(\nu , \varrho , \varsigma , \ell \wp +0)\leq \mathcal{F}(\nu , \varrho , \varsigma , \wp )\), and \(\mathcal{G}(\nu , \varrho , \varsigma , \ell \wp +0)\leq \mathcal{G}( \nu , \varrho , \varsigma , \wp )\) for all \(\nu , \varrho , \varsigma \in \varPhi \) with \(\varsigma \neq \nu \), \(\varsigma \neq \varrho \), and \(\wp >0\), then \(\nu =\varrho \).
Proof
Since
and
for all \(\wp >0\), \(\mathcal{Q}(\nu , \varrho , \varsigma , \cdot )\), \(\mathcal{F}( \nu , \varrho , \varsigma , \cdot )\), and \(\mathcal{G}(\nu , \varrho , \varsigma , \cdot )\) are constant. Since \(\lim_{\wp \rightarrow \infty}\mathcal{Q}(\nu , \varrho , \varsigma , \wp )=1\), \(\lim_{\wp \rightarrow \infty}\mathcal{F}(\nu , \varrho , \varsigma , \wp )=0\) and \(\lim_{\wp \rightarrow \infty}\mathcal{G}(\nu , \varrho , \varsigma , \wp )=0\), then \(\mathcal{Q}(\nu , \varrho , \varsigma , \wp )=1\), \(\mathcal{F}(\nu , \varrho , \varsigma , \wp )=0\) and \(\mathcal{G}(\nu , \varrho , \varsigma , \wp )=0\). Consequently, for all \(\wp >0\). Hence, \(\nu =0\) because \(\varsigma \neq \nu \), \(\varsigma \neq \varrho \). □
Lemma 3
Let \((\varPhi , \mathcal{Q}, \mathcal{F}, \mathcal{G}, \ast , +, \perp )\) be a ON2MS and let \(\lim_{\wp \rightarrow \infty}\nu _{\mathfrak{n}}=\nu \), \(\lim_{\wp \rightarrow \infty}\varrho _{\mathfrak{n}}=\varrho \). Then, it satisfies for all \(\tau \in \varPhi \) and \(\wp \geq 0\):
-
(1)
$$\begin{aligned}& \lim_{\mathfrak{n}\rightarrow \infty}\inf \mathcal{Q}(\nu _{ \mathfrak{n}}, \varrho _{\mathfrak{n}}, \tau , \wp )\geq \mathcal{Q}( \nu , \varrho , \tau , \wp ), \\& \lim_{\mathfrak{n}\rightarrow \infty} \sup \mathcal{F}(\nu _{\mathfrak{n}}, \varrho _{\mathfrak{n}}, \tau , \wp )\leq \mathcal{F}(\nu , \varrho , \tau , \wp ) \end{aligned}$$
and
$$\begin{aligned} \lim_{\mathfrak{n}\rightarrow \infty}\sup \mathcal{G}(\nu _{ \mathfrak{n}}, \varrho _{\mathfrak{n}}, \tau , \wp )\leq \mathcal{G}( \nu , \varrho , \tau , \wp ). \end{aligned}$$ -
(2)
$$\begin{aligned}& \mathcal{Q}(\nu , \varrho , \tau , \wp )\geq \lim_{\mathfrak{n} \rightarrow \infty}\sup \mathcal{Q}(\nu _{\mathfrak{n}}, \varrho _{ \mathfrak{n}}, \tau , \wp ), \\& \mathcal{F}(\nu , \varrho , \tau , \wp +0)\leq \lim_{\mathfrak{n}\rightarrow \infty} \inf \mathcal{F}(\nu _{ \mathfrak{n}}, \varrho _{\mathfrak{n}}, \tau , \wp ) \end{aligned}$$
and
$$\begin{aligned} \mathcal{G}(\nu , \varrho , \tau , \wp +0)\leq \lim_{\mathfrak{n} \rightarrow \infty} \inf \mathcal{G}(\nu _{\mathfrak{n}}, \varrho _{ \mathfrak{n}}, \tau , \wp ). \end{aligned}$$
Proof
For all \(\tau \in \varPhi \) and \(\wp \geq 0\), we have
which implies \(\lim_{\mathfrak{n}\rightarrow \infty}\mathcal{Q}(\nu _{\mathfrak{n}}, \varrho _{\mathfrak{n}}, \tau , \wp )\geq 1\ast 1\ast 1\ast \mathcal{Q}(\nu , \varrho , \tau , \wp )\ast 1=\mathcal{Q}(\nu , \varrho , \tau , \wp )\), also
which implies
and
which implies \(\lim_{\mathfrak{n}\rightarrow \infty}\sup \mathcal{G}(\nu _{ \mathfrak{n}}, \varrho _{\mathfrak{n}}, \tau , \wp )\leq 0+0+0+ \mathcal{G}(\nu , \varrho , \tau , \wp )+0=\mathcal{G}(\nu , \varrho , \tau , \wp )\).
Let \(\epsilon >0\) be given. For all \(\tau \in \nu \) and \(\wp >0\), we have
Consequently,
Letting \(\epsilon \rightarrow 0\), we have
Also, we have
Consequently,
Letting \(\epsilon \rightarrow 0\), we have
and
Consequently,
Letting \(\epsilon \rightarrow 0\), we have
□
Lemma 4
Let \((\varPhi , \mathcal{Q}, \mathcal{F}, \mathcal{G}, \ast , +, \perp )\) be a N2MS. Let ϒ and Λ be a continuous self-map on Φ and \([\varUpsilon , \varLambda ]\) are compatible. Let \(\nu _{\mathfrak{n}}\) be a sequence in Φ such that \(\varUpsilon \nu _{\mathfrak{n}}\rightarrow \omega \) and \(\varLambda \nu _{\mathfrak{n}}\rightarrow \omega \). Then, \(\varUpsilon \varLambda \nu _{\mathfrak{n}}\rightarrow \varLambda \omega \).
Proof
Since ϒ, Λ are compatible maps, \(\varUpsilon \varLambda \nu _{\mathfrak{n}}\rightarrow \varUpsilon \omega \), \(\varLambda \varUpsilon \nu _{\mathfrak{n}}\rightarrow \varLambda \omega \) and so, \(\mathcal{Q}(\varUpsilon \varLambda \nu _{\mathfrak{n}}, \varUpsilon \omega , \tau , \frac{\wp}{3})\rightarrow 1\), \(\mathcal{F}(\varLambda \varUpsilon \nu _{\mathfrak{n}}, \varLambda \omega , \tau , \frac{\wp}{3})\rightarrow 0\) and \(\mathcal{G}(\varLambda \varUpsilon \nu _{\mathfrak{n}}, \varLambda \omega , \tau , \frac{\wp}{3})\rightarrow 0\) for all \(\tau \in \varPhi \) and \(\wp >0\),
Also, we have
For all \(\tau \in \Xi \) and \(\wp >0\), and
For all \(\tau \in \varPhi \) and \(\wp >0\). Hence, \(\varUpsilon \varLambda \nu _{\mathfrak{n}}\rightarrow \varLambda \omega \). □
Theorem 3.1
Let \((\varPhi , \mathcal{Q}, \mathcal{F}, \mathcal{G}, \ast , +, \perp )\) be an orthogonal complete neutrosophic 2-metric space with “∗” as con-t-nm and “+” as con-t-conm. Let Θ and Γ be continuous self-mappings on Φ. Then, Θ and Γ have a unique common fixed point in Φ if and only if there exist two self-mappings ϒ, Λ of Φ satisfying:
-
(1)
\(\varUpsilon \varPhi \subset \varGamma \varPhi \), \(\varLambda \varPhi \subset \varTheta \varPhi \);
-
(2)
The pair \(\{\varUpsilon , \varTheta \}\) and \(\{\varLambda , \varGamma \}\) are compatible;
-
(3)
ϒ, Λ, Θ, Γ be ⊥-preserving;
-
(4)
There exists \(\ell \in (0, 1)\) such that for every \(\nu , \varrho , \varsigma \in \varPhi \) and \(\wp >0\),
$$\begin{aligned}& \begin{aligned} \mathcal{Q}(\varUpsilon \nu , \varLambda \varrho , \varsigma , \ell \wp )\geq{}& \min \bigl\{ \mathcal{Q}(\varTheta \nu , \varGamma \nu , \varsigma , \wp ), \mathcal{Q}(\varUpsilon \nu , \varTheta \nu , \varsigma , \wp ), \\ &{} \mathcal{Q}( \varLambda \varrho , \varGamma \varrho , \varsigma , \wp ), \mathcal{Q}( \varUpsilon \nu , \varLambda \varrho , \varsigma , \ell \wp ) \bigr\} , \end{aligned} \\& \begin{aligned} \mathcal{F}(\varUpsilon \nu , \varLambda \varrho , \varsigma , \ell \wp )\leq{}& \max \bigl\{ \mathcal{F}(\varTheta \nu , \varGamma \nu , \varsigma , \wp ), \mathcal{F}(\varUpsilon \nu , \varTheta \nu , \varsigma , \wp ), \\ &{}\mathcal{F}( \varLambda \varrho , \varGamma \varrho , \varsigma , \wp ), \mathcal{F}( \varUpsilon \nu , \varLambda \varrho , \varsigma , \ell \wp ) \bigr\} , \end{aligned} \\& \begin{aligned} \mathcal{G}(\varUpsilon \nu , \varLambda \varrho , \varsigma , \ell \wp )\leq{}& \max \bigl\{ \mathcal{F}(\varTheta \nu , \varGamma \nu , \varsigma , \wp ), \mathcal{F}(\varUpsilon \nu , \varTheta \nu , \varsigma , \wp ), \\ &{}\mathcal{F}( \varLambda \varrho , \varGamma \varrho , \varsigma , \wp ), \mathcal{F}( \varUpsilon \nu , \varLambda \varrho , \varsigma , \ell \wp ) \bigr\} . \end{aligned} \end{aligned}$$
Then ϒ, Λ, Θ, and Γ have a unique common fixed point in Φ.
Proof
Suppose Θ and Γ have a unique common fixed point, say \(\mathfrak{r}\in \varPhi \). Define \(\varUpsilon \colon \varPhi \rightarrow \varPhi \) by \(\varUpsilon \nu =\mathfrak{r}\) for all \(\nu \in \varPhi \) and \(\varLambda \colon \varPhi \rightarrow \varPhi \) by \(\varLambda \nu =\mathfrak{r}\) for all \(\nu \in \varPhi \). Then, it satisfies(1)–(4)
Conversely, if there exist two self-mappings ϒ, Λ of ν this satisfies (1)–(4). From (1) if two sequences are \(\nu _{\mathfrak{n}}\) and \(\varrho _{\mathfrak{n}}\) of Φ such that \(\varrho _{2\mathfrak{n}-1}=\varGamma \nu _{2\mathfrak{n}-1}\) and \(\nu _{2\mathfrak{n}-1}=\varTheta \nu _{2\mathfrak{n}}=\varLambda \nu _{2\mathfrak{n}-1}\) for \(\mathfrak{n}=1, 2, 3\). Putting \(\nu =\nu _{2\mathfrak{n}}\) and \(\nu =\nu _{2\mathfrak{n}+1}\) in condition (4), for all \(\varsigma \in \varPhi \) and \(\wp >0\).
Since \((\varPhi , \mathcal{Q}, \mathcal{F}, \mathcal{G}, \ast , +, \perp )\) is an orthogonal complete neutrosophic 2-metric space there exists \(\nu _{0}\in \varPhi \), such that
Since ϒ is ⊥-preserving, \(\{\nu _{\mathfrak{n}}\}\) is an O-sequence. Now, since ϒ is an ⊥-contraction, we can obtain
and
which implies that
and
By using Lemma 1 and letting \(\nu =\nu _{2\mathfrak{n}+1}\) and \(\varrho =\nu _{2\mathfrak{n}+1}\) in condition (4), we have
and
for all \(\varsigma \in \varPhi \) and \(\wp >0\).
In general, we obtain that for all \(\varsigma \in \varPhi \) and \(\wp >0\) and \(\mathfrak{n}=1, 2, 3, \ldots\) .
and
Thus, for all \(\varsigma \in \varPhi \) and \(\wp >0\), and \(\mathfrak{n}=1, 2, 3, \ldots \)
To show that \(\{\varrho _{\mathfrak{n}}\}\) is a Cauchy sequence in Φ, let \(\mathfrak{m}>\mathfrak{n}\). Then, for all \(\varsigma \in \varPhi \) and \(\wp >\nu \), we obtain
and
Letting \(\mathfrak{m}, \mathfrak{n}\rightarrow \infty \), we have
Thus, \(\{\varrho _{\mathfrak{n}}\}\) is a Cauchy sequence in Φ. By completeness of Φ there exist \(\mathfrak{r}\in \varPhi \) such that
and
From Lemma 4, we have
Meanwhile, for all \(\varsigma \in \varPhi \) with \(\varsigma \neq \varTheta \mathfrak{r}\) and \(\varsigma \neq \varGamma \mathfrak{r}\) and \(\wp >0\), we have
and
Taking the limit as \(\mathfrak{n}\rightarrow \infty \) and using (4), we have for all \(\varsigma \in \varPhi \) with \(\varsigma \neq \varTheta \mathfrak{r}\) and \(\varsigma \neq \varGamma \mathfrak{r}\) and \(\wp >0\),
and
By Lemma 2, we have
From condition (4), we obtain for all \(\varsigma \in \varPhi \) with \(\varsigma \neq \varUpsilon \mathfrak{r}\), \(\varsigma \neq \varGamma \mathfrak{r}\) and \(\wp >0\),
and
Let \(\mathfrak{n}\rightarrow \infty \), by condition (4), and Lemma 3, for all \(\varsigma \in \varPhi \)
and
By Lemma 2, we have
For all \(\varsigma \in \varPhi \) with \(\varsigma \neq \varUpsilon \mathfrak{r}\) and \(\varsigma \neq \varLambda \mathfrak{r}\) and \(\wp >0\), we have
and
By Lemma 2,
It follows that \(\varUpsilon \mathfrak{r}=\varLambda \mathfrak{r}=\varTheta \mathfrak{r}=\varGamma \mathfrak{r}\). For all \(\varsigma \in \varPhi \) with \(\varsigma \neq \varLambda \mathfrak{r}\) and \(\varsigma \neq \mathfrak{r}\), and \(\wp >0\),
and
Taking the limit as \(\mathfrak{n}\rightarrow \infty \) and using (4) and Lemma 3, we have for all \(\varsigma \in \varPhi \) we \(\varsigma \neq \varLambda \mathfrak{r}\), \(\varsigma \neq \mathfrak{r}\) and \(\wp >0\)
and
Hence, we have
and
Therefore, \(\varLambda \mathfrak{r}=\mathfrak{r}\). Thus, \(\mathfrak{r}=\varUpsilon \mathfrak{r}=\varLambda \mathfrak{r}= \varTheta \mathfrak{r}=\varGamma \mathfrak{r}\). Hence, \(\mathfrak{r}\) is a common fixed point of ϒ, Λ, Θ, and Γ.
Let \(\mathfrak{p}\) be another common fixed point of ϒ, Λ, Θ, and Γ for all \(\varsigma \in \varPhi \) with \(\varsigma \neq \mathfrak{r}\), \(\varsigma \neq \mathfrak{p}\), and \(\wp >0\), we have
Since, Γ is ⊥-preserving, one writes
Now,
and
which implies that
and
Hence, \(\mathfrak{r}=\mathfrak{p}\). □
Example 3.2
Let \(\varPhi =[-1, 2]\) and define a binary relation ⊥ by \(\nu \perp \varrho \perp \varsigma \Longleftrightarrow \nu +\varrho + \varsigma \geq 0\). Define \(\mathcal{Q}\), \(\mathcal{F}\), \(\mathcal{G}\) by,
With CTN \(\mu \ast \alpha =\mu \cdot \alpha \) and CTCN \(\mu +\alpha =\max \{\mu +\alpha \}\), \((\varPhi , \mathcal{Q}, \mathcal{F}, \mathcal{G}, \ast , +, \perp )\) is an O-complete N2MS. Also, observe that \(\lim_{\mathfrak{n}\rightarrow \infty}\mathcal{Q}(\varUpsilon \nu , \varLambda \varrho , \varsigma , \ell \wp )=1\), \(\lim_{\mathfrak{n} \rightarrow \infty}\mathcal{F}(\varUpsilon \nu , \varLambda \varrho , \varsigma , \ell \wp )=0\) and \(\lim_{\mathfrak{n}\rightarrow \infty}\mathcal{G}(\varUpsilon \nu , \varLambda \varrho , \varsigma , \ell \wp )=0\) \(\forall \nu , \varrho \), \(\varsigma \in \varPhi \).
Define \(\varUpsilon , \varLambda , \varTheta , \varGamma \colon \varPhi \rightarrow \varPhi \)
From this, we obtain
This implies
All the conditions of the above theorem are satisfied and 1 is a common fixed point of ϒ, Λ, Θ, and Γ.
4 Application
In this section, we given an application to the Fredholm integral equation as below:
Suppose \(\mathcal{I}=\mathcal{C}([\rho , \pi ], \mathbb{R})\) is the set of real-valued continuous functions defined on \([\rho , \pi ]\).
Consider the integral equation,
where \(\delta >0\), \(\mathfrak{f}(\varpi )\) is a neutrosophic function of \(\varpi \colon \varpi \in [\rho , \pi ]\) and \(\mathcal{U}_{1}, \mathcal{U}_{2}\colon \mathcal{C}([\rho , \pi ] \times \mathbb{R})\rightarrow \mathbb{R}^{+}\). Define the binary relation ⊥ on \(\mathcal{X}\) by \(\mathfrak{x}\perp \mathfrak{y}\perp \mathfrak{z}\) iff \(\mathfrak{x}+\mathfrak{y}+\mathfrak{z}\geq 0\) and define \(\mathcal{Q}\), \(\mathcal{F}\) and \(\mathcal{G}\) by
With con-t-nm and con-t-conm defined by \(\rho \ast \pi =\rho \cdot \pi \) and \(\rho +\pi =\max \{\rho , \pi \}\), then \((\Phi , \mathcal{Q}, \mathcal{F}, \mathcal{G}, \ast , +)\) is a O-complete N2MS. Consider \(\int _{\rho}^{\pi}\mathit{d}\theta \leq \ell \wp <1\). Then, the neutrosophic integral equations (8) and (9) have a unique common solution.
Proof
Define \(\varUpsilon , \varLambda \colon \Phi \rightarrow \Phi \) by
The survival of a fixed of the operator \(\mathcal{U}\) has come to the survival of solution of a neutrosophic integral equation,
and
Hence, all the conditions of Theorem 3.1 are satisfied. Hence, ϒ and Λ have a unique common solution. □
5 Conclusion
We introduced the notion of a neutrosophic metric space to an orthogonal neutrosophic 2-metric space that deals with greater ambiguity and uncertainty in engineering and research studies. Finally, we obtained the common fixed-point theorem in an orthogonal neutrosophic 2-metric space.
Availability of data and materials
Not applicable.
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The authors extend their appreciation to Prince Sattam bin Abdulaziz University, Alkharj, Saudi Arabia for funding this research work through the project number (PSAU/2023/01/33030).
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Janardhanan, G., Mani, G., Ege, O. et al. Orthogonal neutrosophic 2-metric spaces. J Inequal Appl 2023, 112 (2023). https://doi.org/10.1186/s13660-023-03024-x
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DOI: https://doi.org/10.1186/s13660-023-03024-x