Orthogonal neutrosophic 2-metric spaces

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:

$$\begin{aligned} \mathcal{A}= \bigl\{ \bigl\langle \nu , \bigl(\mathcal{Q}_{\mathcal{A}}(\nu ), \mathcal{F}_{\mathcal{A}}(\nu ), \mathcal{G}_{\mathcal{A}}(\nu ) \bigr) \bigr\rangle :\nu \in \varPhi , \mathcal{Q}_{\mathcal{A}}, \mathcal{F}_{\mathcal{A}}, \mathcal{G}_{\mathcal{A}}\in \,]^{-}0, 1^{+}[ \bigr\} . \end{aligned}$$

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,

$$\begin{aligned}& \begin{aligned} \mathcal{Q}(\nu , \varrho , \varsigma , \wp )&\geq \mathcal{Q} \biggl(\nu , \varrho , \mathfrak{p}, \frac{\wp}{3} \biggr)\ast \mathcal{Q} \biggl(\nu , \mathfrak{p}, \varsigma , \frac{\wp}{3} \biggr)\ast \mathcal{Q} \biggl( \mathfrak{p}, \varrho , \varsigma , \frac{\wp}{3} \biggr) \\ &\geq \mathfrak{h}_{0}\ast \mathfrak{h}_{7}\ast \mathfrak{h}_{7}\geq \mathfrak{h}_{0}\ast \mathfrak{h}_{1}\ast \mathfrak{h}_{2}\geq 1- \epsilon >1- \mathfrak{h}, \end{aligned} \\& \begin{aligned} \mathcal{F}(\nu , \varrho , \varsigma , \wp )&\geq \mathcal{F} \biggl(\nu , \varrho , \mathfrak{p}, \frac{\wp}{3} \biggr)\ast \mathcal{F} \biggl(\nu , \mathfrak{p}, \varsigma , \frac{\wp}{3} \biggr)\ast \mathcal{F} \biggl( \mathfrak{p}, \varrho , \varsigma , \frac{\wp}{3} \biggr) \\ &\geq (1-\mathfrak{h}_{0})+(1-\mathfrak{h}_{7})+(1- \mathfrak{h}_{7}) \\ &\geq (1-\mathfrak{h}_{0})+(1-\mathfrak{h}_{1})+(1- \mathfrak{h}_{2}) \leq \epsilon < \mathfrak{h}, \end{aligned} \\& \begin{aligned} \mathcal{G}(\nu , \varrho , \varsigma , \wp )&\geq \mathcal{Q} \biggl(\nu , \varrho , \mathfrak{p}, \frac{\wp}{3} \biggr)\ast \mathcal{Q} \biggl(\nu , \mathfrak{p}, \varsigma , \frac{\wp}{3} \biggr)\ast \mathcal{Q} \biggl( \mathfrak{p}, \varrho , \varsigma , \frac{\wp}{3} \biggr) \\ &\leq (1-\mathfrak{h}_{0})+(1-\mathfrak{h}_{7})+(1- \mathfrak{h}_{7}) \\ &\leq (1-\mathfrak{h}_{0})+(1-\mathfrak{h}_{1})+(1- \mathfrak{h}_{2}) \leq \epsilon < \mathfrak{h}. \end{aligned} \end{aligned}$$

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

$$\begin{aligned} \varrho \mathbb{B} \biggl(\nu , 1-\mathfrak{h}_{9}, \frac{\wp}{3} \biggr)\cap \mathbb{B} \biggl(\varrho , 1- \mathfrak{h}_{9}, \frac{\wp}{3} \biggr)=\emptyset . \end{aligned}$$

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

$$\begin{aligned} \mathfrak{h}_{1}&=\mathcal{Q}(\nu , \varrho , \varsigma _{1}, \wp ) \geq \mathcal{Q} \biggl(\nu , \mathfrak{p}, \varsigma _{1}, \frac{\wp}{3} \biggr) \ast \mathcal{Q} \biggl( \mathfrak{p}, \varrho , \varsigma _{1}, \frac{\wp}{3} \biggr) \ast \mathcal{Q} \biggl(\nu , \varrho , \mathfrak{p}, \frac{\wp}{3} \biggr) \\ &\geq \mathfrak{h}_{4}\ast \mathfrak{h}_{9}\ast \mathfrak{h}_{9}\geq \mathfrak{h}_{4}\ast \mathfrak{h}_{7}\ast \mathfrak{h}_{7}\geq \mathfrak{h}_{0}>\mathfrak{h}_{1} \end{aligned}$$

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

$$\begin{aligned} \mathcal{Y}(\nu , \varrho , \varsigma , \wp )&=\mathcal{Y} \biggl(\nu , \varrho , \varsigma , \mathfrak{s}+\frac{\wp -\mathfrak{s}}{2}+ \frac{\wp -\mathfrak{s}}{2} \biggr), \\ &\leq \mathcal{Y}(\nu , \varrho , \varsigma , \mathfrak{s})+ \mathcal{Y} \biggl( \nu , \varsigma , \varsigma , \frac{\wp -\mathfrak{s}}{2} \biggr)+ \mathcal{Y} \biggl( \varsigma , \varrho , \varsigma , \frac{\wp -\mathfrak{s}}{2} \biggr)=\mathcal{Y}(\nu , \varrho , \varsigma , \mathfrak{s}) \end{aligned}$$

and

$$\begin{aligned} \mathcal{G}(\nu , \varrho , \varsigma , \wp )&=\psi \biggl(\nu , \varrho , \varsigma , \mathfrak{s}+\frac{\wp -\mathfrak{s}}{2}+ \frac{\wp -\mathfrak{s}}{2} \biggr), \\ &\leq \mathcal{G}(\nu , \varrho , \varsigma , \mathfrak{s})+\psi \biggl( \nu , \varsigma , \varsigma , \frac{\wp -\mathfrak{s}}{2} \biggr)+ \mathcal{G} \biggl( \varsigma , \varrho , \varsigma , \frac{\wp -\mathfrak{s}}{2} \biggr)=\mathcal{G}(\nu , \varrho , \varsigma , \mathfrak{s}). \end{aligned}$$

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:

$$\begin{aligned} \lim_{\wp \rightarrow \infty}\mathcal{Q}(\nu , \varrho , \varsigma , \wp )=1, \qquad \lim_{\wp \rightarrow \infty}\mathcal{F}(\nu , \varrho , \varsigma , \wp )=0 \quad \text{and}\quad \lim_{\wp \rightarrow \infty}\mathcal{G}(\nu , \varrho , \varsigma , \wp )=0. \end{aligned}$$

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

$$\begin{aligned}& \mathcal{Q}(\nu , \varrho , \varsigma , \wp )\geq \mathcal{Q}(\nu , \varrho , \varsigma , \ell \wp +0)\geq \mathcal{Q}(\nu ,\varrho , \varsigma , \wp ), \\& \mathcal{F}(\nu , \varrho , \varsigma , \wp )\leq \mathcal{F}(\nu , \varrho , \varsigma , \ell \wp +0)\leq \mathcal{F}(\nu , \varrho , \varsigma , \wp ) \end{aligned}$$

and

$$\begin{aligned} \mathcal{G}(\nu , \varrho , \varsigma , \wp )\leq \mathcal{G}(\nu , \varrho , \varsigma , \ell \wp +0)\leq \mathcal{G}(\nu , \varrho , \varsigma , \wp ), \end{aligned}$$

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

$$\begin{aligned} \begin{aligned} \mathcal{Q}(\nu _{\mathfrak{n}}, \varrho _{\mathfrak{n}}, \tau , \wp )\geq{}& \mathcal{Q}(\nu _{\mathfrak{n}}, \varrho _{\mathfrak{n}}, \nu , \wp _{1})\ast \mathcal{Q}(\nu _{\mathfrak{n}}, \nu , \tau , \wp _{2}) \ast \mathcal{Q}(\nu , \varrho _{\mathfrak{n}}, \tau , \wp ), \quad \wp _{1}+\wp _{2}=0 \\ \geq{}& \mathcal{Q}(\nu _{\mathfrak{n}}, \varrho _{\mathfrak{n}}, \nu , \wp _{1})\ast \mathcal{Q}(\nu _{\mathfrak{n}}, \nu , \tau , \wp _{2}) \ast \mathcal{Q}(\nu , \varrho _{\mathfrak{n}}, \varrho , \wp _{3}) \\ &{}\ast \mathcal{Q}(\nu , \varrho , \tau , \wp _{4})\ast \mathcal{Q}( \varrho , \varrho _{\mathfrak{n}}, \tau , \wp ), \quad \wp _{3}+\wp _{4}=0, \end{aligned} \end{aligned}$$

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

$$\begin{aligned} \mathcal{F}(\nu _{\mathfrak{n}}, \varrho _{\mathfrak{n}}, \tau , \wp )\leq{}& \mathcal{F}(\nu _{\mathfrak{n}}, \varrho _{\mathfrak{n}}, \nu , \wp _{1})+\mathcal{F}(\nu _{\mathfrak{n}}, \nu , \tau , \wp _{2})+ \mathcal{F}(\nu , \varrho _{\mathfrak{n}}, \tau , \wp ), \quad \wp _{1}+ \wp _{2}=0 \\ \leq{}& \mathcal{F}(\nu _{\mathfrak{n}}, \varrho _{\mathfrak{n}}, \nu , \wp _{1})+\mathcal{F}(\nu _{\mathfrak{n}}, \nu , \tau , \wp _{2})+ \mathcal{F}(\nu , \varrho _{\mathfrak{n}}, \varrho , \wp _{3}) \\ &{}+\mathcal{F}(\nu , \varrho , \tau , \wp _{4})+\mathcal{F}(\varrho , \varrho _{\mathfrak{n}}, \tau , \wp ), \quad \wp _{3}+\wp _{4}=0, \end{aligned}$$

which implies

$$\begin{aligned} \lim_{\mathfrak{n}\rightarrow \infty}\sup \mathcal{F}(\nu _{ \mathfrak{n}}, \varrho _{\mathfrak{n}}, \tau , \wp )\leq 0+0+0+ \mathcal{F}(\nu , \varrho , \tau , \wp )+0=\mathcal{F}(\nu , \varrho , \tau , \wp ) \end{aligned}$$

and

$$\begin{aligned} \mathcal{G}(\nu _{\mathfrak{n}}, \varrho _{\mathfrak{n}}, \tau , \wp )\leq{}& \mathcal{G}(\nu _{\mathfrak{n}}, \varrho _{\mathfrak{n}}, \nu , \wp _{1})+\mathcal{G}(\nu _{\mathfrak{n}}, \nu , \tau , \wp _{2})+ \mathcal{G}(\nu , \varrho _{\mathfrak{n}}, \tau , \wp ), \quad \wp _{1}+ \wp _{2}=0 \\ \leq{}& \mathcal{G}(\nu _{\mathfrak{n}}, \varrho _{\mathfrak{n}}, \nu , \wp _{1})+\mathcal{G}(\nu _{\mathfrak{n}}, \nu , \tau , \wp _{2})+ \mathcal{G}(\nu , \varrho _{\mathfrak{n}}, \varrho , \wp _{3}) \\ &{}+\mathcal{G}(\nu , \varrho , \tau , \wp _{4})+\mathcal{G}(\varrho , \varrho _{\mathfrak{n}}, \tau , \wp _{4}), \quad \wp _{3}+\wp _{4}=0, \end{aligned}$$

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

$$\begin{aligned} \mathcal{Q}(\nu , \varrho , \tau , \wp +2\epsilon )\geq{}& \mathcal{Q} \biggl( \nu , \varrho , \nu _{\mathfrak{n}}, \frac{\epsilon}{2} \biggr)\ast \mathcal{Q} \biggl(\nu , \nu _{\mathfrak{n}}, \tau , \frac{\epsilon}{2} \biggr) \ast \mathcal{Q}(\nu _{\mathfrak{n}}, \varrho , \tau , \wp +\epsilon ) \\ \geq{}& \mathcal{Q} \biggl(\nu , \varrho , \nu _{\mathfrak{n}}, \frac{\epsilon}{2} \biggr)\ast \mathcal{Q} \biggl(\nu , \nu _{\mathfrak{n}}, \tau , \frac{\epsilon}{2} \biggr)\ast \mathcal{Q} \biggl(\nu _{\mathfrak{n}}, \varrho , \varrho _{\mathfrak{n}}, \frac{\epsilon}{2} \biggr) \\ &{}\ast \mathcal{Q}( \nu _{ \mathfrak{n}}, \varrho _{\mathfrak{n}}, \tau , \wp )\ast \mathcal{Q} \biggl( \varrho _{\mathfrak{n}}, \varrho , \tau , \frac{\epsilon}{2} \biggr). \end{aligned}$$

Consequently,

$$\begin{aligned} \mathcal{Q}(\nu , \varrho , \tau , \wp +2\epsilon )\geq \lim _{ \mathfrak{n}\rightarrow \infty}\sup \mathcal{Q}(\nu _{\mathfrak{n}}, \varrho _{\mathfrak{n}}, \tau , \wp ). \end{aligned}$$

Letting $\epsilon \rightarrow 0$, we have

$$\begin{aligned} \mathcal{Q}(\nu , \varrho , \tau , \wp +0)\geq \lim_{\mathfrak{n} \rightarrow \infty} \sup \mathcal{Q}(\nu _{\mathfrak{n}}, \varrho _{ \mathfrak{n}}, \tau , \wp ). \end{aligned}$$

Also, we have

$$\begin{aligned} \mathcal{F}(\nu , \varrho , \tau , \wp +2\epsilon )\leq{}& \mathcal{F} \biggl( \nu , \varrho , \nu _{\mathfrak{n}}, \frac{\epsilon}{2} \biggr)\diamond \mathcal{F} \biggl(\nu , \nu _{\mathfrak{n}}, \tau , \frac{\epsilon}{2} \biggr) \diamond \mathcal{F}(\nu _{\mathfrak{n}}, \varrho , \tau , \wp + \epsilon ) \\ \geq{}& \mathcal{F} \biggl(\nu , \varrho , \nu _{\mathfrak{n}}, \frac{\epsilon}{2} \biggr)\diamond \mathcal{F} \biggl(\nu , \nu _{\mathfrak{n}}, \tau , \frac{\epsilon}{2} \biggr)\diamond \mathcal{F}(\nu _{\mathfrak{n}}, \varrho _{\mathfrak{n}}, \tau , \wp ) \\ &{}\diamond \mathcal{F} \biggl(\varrho _{\mathfrak{n}}, \varrho , \tau , \frac{\epsilon}{2} \biggr)\diamond \mathcal{F}(\nu _{\mathfrak{n}}, \varrho _{ \mathfrak{n}}, \tau , \wp )\diamond \mathcal{F} \biggl(\varrho _{ \mathfrak{n}}, \varrho , \tau , \frac{\epsilon}{2} \biggr). \end{aligned}$$

Consequently,

$$\begin{aligned} \mathcal{F}(\nu , \varrho , \tau , \wp +2\epsilon )\leq \lim _{ \mathfrak{n}\rightarrow \infty}\inf \mathcal{F}(\nu _{\mathfrak{n}}, \varrho _{\mathfrak{n}}, \tau , \wp ). \end{aligned}$$

Letting $\epsilon \rightarrow 0$, we have

$$\begin{aligned} \mathcal{F}(\nu , \varrho , \tau , \wp +0)\leq \lim_{\mathfrak{n} \rightarrow \infty} \inf \mathcal{F}(\nu _{\mathfrak{n}}, \varrho _{ \mathfrak{n}}, \tau , \wp ) \end{aligned}$$

and

$$\begin{aligned} \mathcal{G}(\nu , \varrho , \tau , \wp +2\epsilon )\leq{}& \mathcal{G} \biggl( \nu , \varrho , \nu _{\mathfrak{n}}, \frac{\epsilon}{2} \biggr)\diamond \mathcal{G} \biggl(\nu , \nu _{\mathfrak{n}}, \tau , \frac{\epsilon}{2} \biggr) \diamond \mathcal{G}(\nu _{\mathfrak{n}}, \varrho , \tau , \wp + \epsilon ) \\ \geq{}& \mathcal{G} \biggl(\nu , \varrho , \nu _{\mathfrak{n}}, \frac{\epsilon}{2} \biggr)\diamond \mathcal{G} \biggl(\nu , \nu _{\mathfrak{n}}, \tau , \frac{\epsilon}{2} \biggr)\diamond \mathcal{G} \biggl(\nu _{\mathfrak{n}}, \varrho , \varrho _{\mathfrak{n}}, \frac{\epsilon}{2} \biggr)\diamond \mathcal{G}(\nu _{\mathfrak{n}}, \varrho _{\mathfrak{n}}, \tau , \wp ) \\ &{}\diamond \mathcal{G} \biggl(\varrho _{\mathfrak{n}}, \varrho , \tau , \frac{\epsilon}{2} \biggr)\diamond \mathcal{G}(\nu _{\mathfrak{n}}, \varrho _{ \mathfrak{n}}, \tau , \wp )\diamond \mathcal{G} \biggl(\varrho _{ \mathfrak{n}}, \varrho , \tau , \frac{\epsilon}{2} \biggr). \end{aligned}$$

Consequently,

$$\begin{aligned} \mathcal{G}(\nu , \varrho , \tau , \wp +2\epsilon )\leq \lim _{ \mathfrak{n}\rightarrow \infty}\inf \mathcal{G}(\nu _{\mathfrak{n}}, \varrho _{\mathfrak{n}}, \tau , \wp ). \end{aligned}$$

Letting $\epsilon \rightarrow 0$, we have

$$\begin{aligned} \mathcal{G}(\nu , \varrho , \tau , \wp +0)\leq \lim_{\mathfrak{n} \rightarrow \infty} \inf \mathcal{G}(\nu _{\mathfrak{n}}, \varrho _{ \mathfrak{n}}, \tau , \wp ). \end{aligned}$$

 □

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 $\wp >0$,

$$\begin{aligned} \mathcal{Q}(\varUpsilon \varLambda \nu _{\mathfrak{n}}, \varLambda \omega , \tau , \wp )\geq{}& \mathcal{Q} \biggl(\varUpsilon \varLambda \nu _{ \mathfrak{n}}, \varLambda \omega , \varLambda \varUpsilon \nu _{ \mathfrak{n}}, \frac{\wp}{3} \biggr)\ast \mathcal{Q} \biggl( \varUpsilon \varLambda \nu _{\mathfrak{n}}, \varLambda \varUpsilon \nu _{\mathfrak{n}}, \tau , \frac{\wp}{3} \biggr) \\ &{}\ast \mathcal{Q} \biggl(\varLambda \varUpsilon \nu _{ \mathfrak{n}}, \varLambda \omega , \tau , \frac{\wp}{3} \biggr) \\ \geq{}& \mathcal{Q} \biggl(\varLambda \varUpsilon \nu _{\mathfrak{n}}, \varLambda \omega , \varUpsilon \varLambda \nu _{\mathfrak{n}}, \frac{\wp}{3} \biggr) \ast \mathcal{Q} \biggl(\varLambda \varUpsilon \nu _{ \mathfrak{n}}, \omega \varLambda \nu _{\mathfrak{n}}, \tau , \frac{\wp}{3} \biggr) \\ &{}\ast \mathcal{Q} \biggl(\varLambda \varUpsilon \nu _{ \mathfrak{n}}, \varLambda \omega , \tau , \frac{\wp}{3} \biggr)\rightarrow 1. \end{aligned}$$

Also, we have

$$\begin{aligned} \mathcal{F}(\varUpsilon \varLambda \nu _{\mathfrak{n}}, \varLambda \omega , \tau , \wp )\leq{}& \mathcal{F} \biggl(\varUpsilon \varLambda \nu _{ \mathfrak{n}}, \varLambda \omega , \varLambda \varUpsilon \nu _{ \mathfrak{n}}, \frac{\wp}{3} \biggr)+d\mathcal{F} \biggl(\varUpsilon \varLambda \nu _{\mathfrak{n}}, \varLambda \varUpsilon \nu _{\mathfrak{n}}, \tau , \frac{\wp}{3} \biggr) \\ &{}+\mathcal{F} \biggl(\varLambda \varUpsilon \nu _{ \mathfrak{n}}, \varLambda \omega , \tau , \frac{\wp}{3} \biggr) \\ \leq{}& \mathcal{F} \biggl(\varLambda \varUpsilon \nu _{\mathfrak{n}}, \varLambda \omega , \varUpsilon \varLambda \nu _{\mathfrak{n}}, \frac{\wp}{3} \biggr)+\mathcal{F} \biggl(\varLambda \varUpsilon \nu _{\mathfrak{n}}, \omega \varLambda \nu _{\mathfrak{n}}, \tau , \frac{\wp}{3} \biggr) \\ &{}+ \mathcal{F} \biggl(\varLambda \Omega \nu _{\mathfrak{n}}, \varLambda \omega , \tau , \frac{\wp}{3} \biggr)\rightarrow 0. \end{aligned}$$

For all $\tau \in \Xi$ and $\wp >0$, and

$$\begin{aligned} \mathcal{G}(\varUpsilon \varLambda \nu _{\mathfrak{n}}, \varLambda \omega , \tau , \wp )\leq{}& \mathcal{G} \biggl(\varUpsilon \varLambda \nu _{ \mathfrak{n}}, \varLambda \omega , \varLambda \varUpsilon \nu _{ \mathfrak{n}}, \frac{\wp}{3} \biggr)+\mathcal{G} \biggl(\varUpsilon \varLambda \nu _{ \mathfrak{n}}, \varLambda \varUpsilon \nu _{\mathfrak{n}}, \tau , \frac{\wp}{3} \biggr) \\ &{}+\mathcal{G} \biggl(\varLambda \varUpsilon \nu _{\mathfrak{n}}, \varLambda \omega , \tau , \frac{\wp}{3} \biggr) \\ \leq{}& \mathcal{G} \biggl(\varLambda \varUpsilon \nu _{\mathfrak{n}}, \varLambda \omega , \varUpsilon \varLambda \nu _{\mathfrak{n}}, \frac{\wp}{3} \biggr)+\mathcal{G} \biggl(\varLambda \varUpsilon \nu _{\mathfrak{n}}, \omega \varLambda \nu _{\mathfrak{n}}, \tau , \frac{\wp}{3} \biggr) \\ &{}+ \mathcal{G} \biggl(\varLambda \varUpsilon \nu _{\mathfrak{n}}, \varLambda \omega , \tau , \frac{\wp}{3} \biggr)\rightarrow 0. \end{aligned}$$

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

$$\begin{aligned}& \nu _{0}\perp \varrho , \quad \text{for all } \varrho \in \varPhi , \\& \text{i.e.},\ \nu _{0}\perp \varUpsilon \nu _{0} \quad \text{take} \\& \nu _{\mathfrak{n}}=\varUpsilon ^{\mathfrak{n}}\nu _{0}= \varUpsilon \nu _{\mathfrak{n}-1}, \quad \text{for all } \mathfrak{n}\in \mathcal{F}. \end{aligned}$$

Since ϒ is ⊥-preserving, $\{\nu _{\mathfrak{n}}\}$ is an O-sequence. Now, since ϒ is an ⊥-contraction, we can obtain

$$\begin{aligned}& \begin{aligned} \mathcal{Q}(\varrho \nu _{2\mathfrak{n}+1}, \varrho \nu _{2 \mathfrak{n}+2}, \varsigma , \ell \wp )={}&\mathcal{Q}(\varUpsilon \nu _{2 \mathfrak{n}}, \varLambda \nu _{2\mathfrak{n}+1}, \varsigma , \ell \wp ) \\ \geq {}&\min \bigl\{ \mathcal{Q}(\varTheta \nu _{2\mathfrak{n}}, \varGamma \nu _{2\mathfrak{n}+1}, \varsigma , \wp ), \mathcal{Q}(\varUpsilon \nu _{2\mathfrak{n}}, \varTheta \nu _{2\mathfrak{n}}, \varsigma , \wp ), \\ &{}\mathcal{Q}(\varLambda \varrho _{2\mathfrak{n}+1}, \varGamma \nu _{2 \mathfrak{n}+1}, \varsigma , \wp ), \mathcal{Q}(\varUpsilon \nu _{2 \mathfrak{n}}, \varGamma \nu _{2\mathfrak{n}+1}, \varsigma , \wp ) \bigr\} \\ \geq {}&\min \bigl\{ \mathcal{Q}(\varrho \nu _{2\mathfrak{n}}, \varrho \nu _{2 \mathfrak{n}+1}, \varsigma , \ell \wp ), \mathcal{Q}(\varrho \nu _{2 \mathfrak{n}+1}, \varrho \nu _{2\mathfrak{n}+1}, \varsigma , \ell \wp ) \bigr\} , \end{aligned} \\& \begin{aligned} \mathcal{F}(\varrho \nu _{2\mathfrak{n}+1}, \varrho \nu _{2 \mathfrak{n}+2}, \varsigma , \ell \wp )={}&\mathcal{F}(\varUpsilon \nu _{2 \mathfrak{n}}, \varLambda \nu _{2\mathfrak{n}+1}, \varsigma , \ell \wp ) \\ \leq {}&\max \bigl\{ \mathcal{F}(\varTheta \nu _{2\mathfrak{n}}, \varGamma \nu _{2\mathfrak{n}+1}, \varsigma , \wp ), \mathcal{F}(\varUpsilon \nu _{2\mathfrak{n}}, \varTheta \nu _{2\mathfrak{n}}, \varsigma , \wp ), \\ &{}\mathcal{F}(\varLambda \varrho _{2\mathfrak{n}+1}, \varGamma \nu _{2 \mathfrak{n}+1}, \varsigma , \wp ), \mathcal{F}(\varUpsilon \nu _{2 \mathfrak{n}}, \varGamma \nu _{2\mathfrak{n}+1}, \varsigma , \wp ) \bigr\} \\ \leq {}&\max \bigl\{ \mathcal{F}(\varrho \nu _{2\mathfrak{n}}, \varrho \nu _{2 \mathfrak{n}+1}, \varsigma , \ell \wp ), \mathcal{F}(\varrho \nu _{2 \mathfrak{n}+1}, \varrho \nu _{2\mathfrak{n}+1}, \varsigma , \ell \wp ) \bigr\} \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \mathcal{G}(\varrho \nu _{2\mathfrak{n}+1}, \varrho \nu _{2 \mathfrak{n}+2}, \varsigma , \ell \wp )={}&\mathcal{G}(\varUpsilon \nu _{2 \mathfrak{n}}, \varLambda \nu _{2\mathfrak{n}+1}, \varsigma , \ell \wp ) \\ \leq {}&\max \bigl\{ \mathcal{G}(\varTheta \nu _{2\mathfrak{n}}, \varGamma \nu _{2\mathfrak{n}+1}, \varsigma , \wp ), \mathcal{G}(\varUpsilon \nu _{2\mathfrak{n}}, \varTheta \nu _{2\mathfrak{n}}, \varsigma , \wp ), \\ &{}\mathcal{G}(\varLambda \varrho _{2\mathfrak{n}+1}, \varGamma \nu _{2 \mathfrak{n}+1}, \varsigma , \wp ), \mathcal{G}(\varUpsilon \nu _{2 \mathfrak{n}}, \varGamma \nu _{2\mathfrak{n}+1}, \varsigma , \wp ) \bigr\} \\ \leq {}&\max \bigl\{ \mathcal{G}(\varrho \nu _{2\mathfrak{n}}, \varrho \nu _{2 \mathfrak{n}+1}, \varsigma , \ell \wp ), \mathcal{G}(\varrho \nu _{2 \mathfrak{n}+1}, \varrho \nu _{2\mathfrak{n}+1}, \varsigma , \ell \wp ) \bigr\} , \end{aligned}$$

which implies that

$$\begin{aligned}& \mathcal{Q}(\varrho \nu _{2\mathfrak{n}+1}, \varrho \nu _{2 \mathfrak{n}+2}, \varsigma , \ell \wp )\geq \mathcal{Q}(\varrho \nu _{2 \mathfrak{n}+1}, \varrho \nu _{2\mathfrak{n}+1}, \varsigma , \ell \wp ), \\& \mathcal{F}(\varrho \nu _{2\mathfrak{n}+1}, \varrho \nu _{2 \mathfrak{n}+2}, \varsigma , \ell \wp )\leq \mathcal{F}(\varrho \nu _{2 \mathfrak{n}+1}, \varrho \nu _{2\mathfrak{n}+1}, \varsigma , \ell \wp ) \end{aligned}$$

and

$$\begin{aligned} \mathcal{G}(\varrho \nu _{2\mathfrak{n}+1}, \varrho \nu _{2 \mathfrak{n}+2}, \varsigma , \ell \wp )&\leq \mathcal{G}(\varrho \nu _{2 \mathfrak{n}+1}, \varrho \nu _{2\mathfrak{n}+1}, \varsigma , \ell \wp ). \end{aligned}$$

By using Lemma 1 and letting $\nu =\nu _{2\mathfrak{n}+1}$ and $\varrho =\nu _{2\mathfrak{n}+1}$ in condition (4), we have

$$\begin{aligned}& \mathcal{Q}(\varrho _{2\mathfrak{n}+2}, \varrho _{2\mathfrak{n}+3}, \varsigma , \ell \wp )\geq \mathcal{Q}(\varrho _{2\mathfrak{n}+1}, \varrho _{2\mathfrak{n}+1}, \varsigma , \wp ), \\& \mathcal{F}(\varrho _{2\mathfrak{n}+2}, \varrho _{2\mathfrak{n}+3}, \varsigma , \wp )\leq \mathcal{F}(\varrho _{2\mathfrak{n}+1}, \varrho _{2\mathfrak{n}+1}, \varsigma , \wp ) \end{aligned}$$

and

$$\begin{aligned} \mathcal{G}(\varrho _{2\mathfrak{n}+2}, \varrho _{2\mathfrak{n}+3}, \varsigma , \wp )&\leq \mathcal{G}(\varrho _{2\mathfrak{n}+1}, \varrho _{2\mathfrak{n}+1}, \varsigma , \wp ), \end{aligned}$$

for all $\varsigma \in \varPhi$ and $\wp >0$.

In general, we obtain that for all $\varsigma \in \varPhi$ and $\wp >0$ and $\mathfrak{n}=1, 2, 3, \ldots$ .

$$\begin{aligned}& \begin{aligned} &\mathcal{Q}(\varrho _{\mathfrak{n}}, \varrho _{\mathfrak{n}+1}, \varsigma , \ell \wp )\geq \mathcal{Q}(\varrho _{\mathfrak{n}-1}, \varrho _{\mathfrak{n}}, \varsigma , \wp ), \\ &\mathcal{F}(\varrho _{\mathfrak{n}}, \varrho _{\mathfrak{n}+1}, \varsigma , \ell \wp )\leq \mathcal{Q}(\varrho _{\mathfrak{n}-1}, \varrho _{\mathfrak{n}}, \varsigma , \wp ) \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \mathcal{G}(\varrho _{\mathfrak{n}}, \varrho _{\mathfrak{n}+1}, \varsigma , \ell \wp )\leq \mathcal{G}(\varrho _{\mathfrak{n}-1}, \varrho _{\mathfrak{n}}, \varsigma , \wp ). \end{aligned}$$

Thus, for all $\varsigma \in \varPhi$ and $\wp >0$, and $\mathfrak{n}=1, 2, 3, \ldots$

$$\begin{aligned}& \mathcal{Q}(\varrho _{\mathfrak{n}}, \varrho _{\mathfrak{n}+1}, \varsigma , \ell \wp )\geq \mathcal{Q} \biggl(\varrho _{0}, \varrho _{1}, \varsigma , \frac{\wp}{\ell ^{\mathfrak{n}}} \biggr), \end{aligned}$$

(1)

$$\begin{aligned}& \mathcal{F}(\varrho _{\mathfrak{n}}, \varrho _{\mathfrak{n}+1}, \varsigma , \ell \wp )\leq \mathcal{F} \biggl(\varrho _{0}, \varrho _{1}, \varsigma , \frac{\wp}{\ell ^{\mathfrak{n}}} \biggr), \end{aligned}$$

(2)

$$\begin{aligned}& \mathcal{G}(\varrho _{\mathfrak{n}}, \varrho _{\mathfrak{n}+1}, \varsigma , \ell \wp )\leq \mathcal{G} \biggl(\varrho _{0}, \varrho _{1}, \varsigma , \frac{\wp}{\ell ^{\mathfrak{n}}} \biggr). \end{aligned}$$

(3)

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

$$\begin{aligned} \begin{aligned} \mathcal{Q}(\varrho _{\mathfrak{m}}, \varrho _{\mathfrak{n}}, \varsigma , \wp )\geq{}& \mathcal{Q} \biggl(\varrho _{\mathfrak{m}}, \varrho _{ \mathfrak{n}}, \varrho _{\mathfrak{n}+1}, \frac{\wp}{3} \biggr)\ast \mathcal{Q} \biggl( \varrho _{\mathfrak{n}+1}, \varrho _{\mathfrak{n}}, \varsigma , \frac{\wp}{3} \biggr)\ast \mathcal{Q} \biggl(\varrho _{\mathfrak{m}}, \varrho _{\mathfrak{n}+1}, \varsigma , \frac{\wp}{3} \biggr) \\ \geq{}& \mathcal{Q} \biggl(\varrho _{\mathfrak{m}}, \varrho _{\mathfrak{n}}, \varrho _{\mathfrak{n}+1}, \frac{\wp}{3} \biggr)\ast \mathcal{Q} \biggl( \varrho _{ \mathfrak{n}+1}, \varrho _{\mathfrak{n}}, \varsigma , \frac{\wp}{3} \biggr) \ast \mathcal{Q} \biggl(\varrho _{\mathfrak{m}}, \varrho _{\mathfrak{n}+1}, \varrho _{\mathfrak{n}+1}, \frac{\wp}{3} \biggr) \\ &{}\ast \mathcal{Q} \biggl(\varrho _{\mathfrak{n}+2}, \varrho _{\mathfrak{n}+1}, \varsigma , \frac{\wp}{3^{2}} \biggr)\ast \mathcal{Q} \biggl(\varrho _{\mathfrak{m}}, \varrho _{\mathfrak{n}+2}, \varsigma , \frac{\wp}{3^{2}} \biggr)\ast \cdots \\ &{}\ast \mathcal{Q} \biggl(\varrho _{\mathfrak{m}}, \varrho _{\mathfrak{m}-1}, \varsigma , \frac{\wp}{3^{\mathfrak{m}-\mathfrak{n}}} \biggr), \end{aligned} \\ \begin{aligned} \mathcal{F}(\varrho _{\mathfrak{m}}, \varrho _{\mathfrak{n}}, \varsigma , \wp )\leq{}& \mathcal{F} \biggl(\varrho _{\mathfrak{m}}, \varrho _{ \mathfrak{n}}, \varrho _{\mathfrak{n}+1}, \frac{\wp}{3} \biggr)\ast \mathcal{F} \biggl( \varrho _{\mathfrak{n}+1}, \varrho _{\mathfrak{n}}, \varsigma , \frac{\wp}{3} \biggr)\ast \mathcal{F} \biggl(\varrho _{\mathfrak{m}}, \varrho _{\mathfrak{n}+1}, \varsigma , \frac{\wp}{3} \biggr) \\ \leq{}& \mathcal{F} \biggl(\varrho _{\mathfrak{m}}, \varrho _{\mathfrak{n}}, \varrho _{\mathfrak{n}+1}, \frac{\wp}{3} \biggr)\ast \mathcal{F} \biggl( \varrho _{ \mathfrak{n}+1}, \varrho _{\mathfrak{n}}, \varsigma , \frac{\wp}{3} \biggr) \ast \mathcal{F} \biggl(\varrho _{\mathfrak{m}}, \varrho _{\mathfrak{n}+1}, \varrho _{\mathfrak{n}+1}, \frac{\wp}{3} \biggr) \\ &{}\ast \mathcal{F} \biggl(\varrho _{\mathfrak{n}+2}, \varrho _{\mathfrak{n}+1}, \varsigma , \frac{\wp}{3^{2}} \biggr)\ast \mathcal{F} \biggl(\varrho _{\mathfrak{m}}, \varrho _{\mathfrak{n}+2}, \varsigma , \frac{\wp}{3^{2}} \biggr)\ast \cdots \\ &{}\ast \mathcal{F} \biggl(\varrho _{\mathfrak{m}}, \varrho _{\mathfrak{m}-1}, \varsigma , \frac{\wp}{3^{\mathfrak{m}-\mathfrak{n}}} \biggr) \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \mathcal{G}(\varrho _{\mathfrak{m}}, \varrho _{\mathfrak{n}}, \varsigma , \wp )\leq{}& \mathcal{G} \biggl(\varrho _{\mathfrak{m}}, \varrho _{ \mathfrak{n}}, \varrho _{\mathfrak{n}+1}, \frac{\wp}{3} \biggr)\ast \mathcal{G} \biggl( \varrho _{\mathfrak{n}+1}, \varrho _{\mathfrak{n}}, \varsigma , \frac{\wp}{3} \biggr)\ast \mathcal{G} \biggl(\varrho _{\mathfrak{m}}, \varrho _{\mathfrak{n}+1}, \varsigma , \frac{\wp}{3} \biggr) \\ \leq{}& \mathcal{G} \biggl(\varrho _{\mathfrak{m}}, \varrho _{\mathfrak{n}}, \varrho _{\mathfrak{n}+1}, \frac{\wp}{3} \biggr)\ast \mathcal{G} \biggl( \varrho _{ \mathfrak{n}+1}, \varrho _{\mathfrak{n}}, \varsigma , \frac{\wp}{3} \biggr) \ast \mathcal{G} \biggl(\varrho _{\mathfrak{m}}, \varrho _{\mathfrak{n}+1}, \varrho _{\mathfrak{n}+1}, \frac{\wp}{3} \biggr) \\ &{}\ast \mathcal{G} \biggl(\varrho _{\mathfrak{n}+2}, \varrho _{\mathfrak{n}+1}, \varsigma , \frac{\wp}{3^{2}} \biggr)\ast \mathcal{G} \biggl(\varrho _{\mathfrak{m}}, \varrho _{\mathfrak{n}+2}, \varsigma , \frac{\wp}{3^{2}} \biggr)\ast \cdots \\ &{}\ast \mathcal{G} \biggl(\varrho _{\mathfrak{m}}, \varrho _{\mathfrak{m}-1}, \varsigma , \frac{\wp}{3^{\mathfrak{m}-\mathfrak{n}}} \biggr). \end{aligned}$$

Letting $\mathfrak{m}, \mathfrak{n}\rightarrow \infty$, we have

$$\begin{aligned}& \begin{aligned} &\lim_{\mathfrak{n}\rightarrow \infty}\mathcal{Q}(\varrho _{ \mathfrak{m}}, \varrho _{\mathfrak{n}}, \varsigma , \wp )=1,\qquad \lim_{ \mathfrak{n}\rightarrow \infty} \mathcal{F}( \varrho _{\mathfrak{m}}, \varrho _{\mathfrak{n}}, \varsigma , \wp )=0 \quad \text{and} \\ &\lim_{\mathfrak{n}\rightarrow \infty}\mathcal{G}(\varrho _{ \mathfrak{m}}, \varrho _{\mathfrak{n}}, \varsigma , \wp )=0. \end{aligned} \end{aligned}$$

Thus, $\{\varrho _{\mathfrak{n}}\}$ is a Cauchy sequence in Φ. By completeness of Φ there exist $\mathfrak{r}\in \varPhi$ such that

$$\begin{aligned} \lim_{\mathfrak{n}\rightarrow \infty}\varrho _{\mathfrak{n}}= \mathfrak{r}, \qquad \lim_{\mathfrak{n}\rightarrow \infty}\varrho _{2 \mathfrak{n}-1}=\lim_{\mathfrak{n}\rightarrow \infty} \nu _{2 \mathfrak{n}-1}=\lim_{\mathfrak{n}\rightarrow \infty}\varUpsilon \nu _{2\mathfrak{n}-2}=\mathfrak{r} \end{aligned}$$

and

$$\begin{aligned} \lim_{\mathfrak{n}\rightarrow \infty}\varrho _{2\mathfrak{n}}=\lim _{ \mathfrak{n}\rightarrow \infty}\varTheta \nu _{2\mathfrak{n}}=\lim _{ \mathfrak{n}\rightarrow \infty}\varLambda \nu _{2\mathfrak{n}-1}= \mathfrak{r}. \end{aligned}$$

From Lemma 4, we have

$$\begin{aligned} \varUpsilon \varTheta \nu _{2\mathfrak{n}+1}=\varTheta \mathfrak{r} \quad \text{and}\quad \varLambda \varGamma \nu _{2\mathfrak{n}+1}= \varGamma \mathfrak{r}. \end{aligned}$$

(4)

Meanwhile, for all $\varsigma \in \varPhi$ with $\varsigma \neq \varTheta \mathfrak{r}$ and $\varsigma \neq \varGamma \mathfrak{r}$ and $\wp >0$, we have

$$\begin{aligned}& \begin{aligned} &\mathcal{Q}(\varUpsilon \varTheta \nu _{2\mathfrak{n}+1}, \varLambda \varGamma \nu _{2\mathfrak{n}+1}, \varsigma , \ell \wp ) \\ &\quad \geq \min \bigl\{ \mathcal{Q}( \varTheta \varTheta \nu _{2\mathfrak{n}+1}, \varGamma \varGamma \nu _{2\mathfrak{n}+1}, \varsigma , \wp ), \mathcal{Q}( \varUpsilon \varTheta \nu _{2\mathfrak{n}+1}, \varTheta \varTheta \nu _{2\mathfrak{n}+1}, \varsigma , \wp ), \\ &\qquad {}\mathcal{Q}( \varLambda \varGamma \nu _{2\mathfrak{n}+1}, \varGamma \varGamma \nu _{2 \mathfrak{n}+1}, \ell \varsigma , \wp ), \mathcal{Q}(\varUpsilon \varTheta \nu _{2\mathfrak{n}+1}, \varGamma \varGamma \nu _{2 \mathfrak{n}+1}, \varsigma , \wp ) \bigr\} , \end{aligned} \\& \begin{aligned} &\mathcal{F}(\varUpsilon \varTheta \nu _{2\mathfrak{n}+1}, \varLambda \varGamma \nu _{2\mathfrak{n}+1}, \varsigma , \ell \wp ) \\ &\quad \leq\max \bigl\{ \mathcal{F}( \varTheta \varTheta \nu _{2\mathfrak{n}+1}, \varGamma \varGamma \nu _{2\mathfrak{n}+1}, \varsigma , \wp ), \mathcal{F}( \varUpsilon \varTheta \nu _{2\mathfrak{n}+1}, \varTheta \varTheta \nu _{2\mathfrak{n}+1}, \varsigma , \wp ), \\ &\qquad {}\mathcal{F}( \varLambda \varGamma \nu _{2\mathfrak{n}+1}, \varGamma \varGamma \nu _{2 \mathfrak{n}+1}, \ell \varsigma , \wp ), \mathcal{F}(\varUpsilon \varTheta \nu _{2\mathfrak{n}+1}, \varGamma \varGamma \nu _{2 \mathfrak{n}+1}, \varsigma , \wp ) \bigr\} \end{aligned} \end{aligned}$$

and

$$\begin{aligned} &\mathcal{G}(\varUpsilon \varTheta \nu _{2\mathfrak{n}+1}, \varLambda \varGamma \nu _{2\mathfrak{n}+1}, \varsigma , \ell \wp ) \\ &\quad \leq \max \bigl\{ \mathcal{G}( \varTheta \varTheta \nu _{2\mathfrak{n}+1}, \varGamma \varGamma \nu _{2\mathfrak{n}+1}, \varsigma , \wp ), \mathcal{G}(\varUpsilon \varTheta \nu _{2\mathfrak{n}+1}, \varTheta \varTheta \nu _{2\mathfrak{n}+1}, \varsigma , \wp ), \\ &\qquad {}\mathcal{G}( \varLambda \varGamma \nu _{2\mathfrak{n}+1}, \varGamma \varGamma \nu _{2 \mathfrak{n}+1}, \ell \varsigma , \wp ), \mathcal{G}(\varUpsilon \varTheta \nu _{2\mathfrak{n}+1}, \varGamma \varGamma \nu _{2 \mathfrak{n}+1}, \varsigma , \wp ) \bigr\} . \end{aligned}$$

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 $\wp >0$,

$$\begin{aligned}& \begin{aligned} &\mathcal{Q}(\varTheta \mathfrak{r}, \varGamma \mathfrak{r}, \varsigma , \ell \wp +0) \\ &\quad \geq \min \bigl\{ \mathcal{Q}(\varTheta \mathfrak{r}, \varGamma \mathfrak{r}, \varsigma , \wp ), \mathcal{Q}( \varTheta \mathfrak{r}, \varTheta \mathfrak{r}, \varsigma , \wp ), \mathcal{Q}(\varGamma \mathfrak{r}, \varGamma \mathfrak{r}, \varsigma , \wp ), \mathcal{Q}( \varTheta \mathfrak{r}, \varTheta \mathfrak{r}, \varsigma , \wp ) \bigr\} \\ &\quad =\mathcal{Q}(\varTheta \mathfrak{r}, \varGamma \mathfrak{r}, \varsigma , \wp ), \end{aligned} \\& \begin{aligned} &\mathcal{F}(\varTheta \mathfrak{r}, \varGamma \mathfrak{r}, \varsigma , \ell \wp +0) \\ &\quad \leq \max \bigl\{ \mathcal{F}(\varTheta \mathfrak{r}, \varGamma \mathfrak{r}, \varsigma , \wp ), \mathcal{F}( \varTheta \mathfrak{r}, \varTheta \mathfrak{r}, \varsigma , \wp ), \mathcal{F}(\varGamma \mathfrak{r}, \varGamma \mathfrak{r}, \varsigma , \wp ), \mathcal{F}( \varTheta \mathfrak{r}, \varTheta \mathfrak{r}, \varsigma , \wp ) \bigr\} \\ &\quad =\mathcal{F}(\varTheta \mathfrak{r}, \varGamma \mathfrak{r}, \varsigma , \wp ) \end{aligned} \end{aligned}$$

and

$$\begin{aligned} &\mathcal{G}(\varTheta \mathfrak{r}, \varGamma \mathfrak{r}, \varsigma , \ell \wp +0) \\ &\quad \leq \max \bigl\{ \mathcal{G}(\varTheta \mathfrak{r}, \varGamma \mathfrak{r}, \varsigma , \wp ), \mathcal{G}( \varTheta \mathfrak{r}, \varTheta \mathfrak{r}, \varsigma , \wp ), \mathcal{G}(\varGamma \mathfrak{r}, \varGamma \mathfrak{r}, \varsigma , \wp ), \mathcal{G}(\varTheta \mathfrak{r}, \varTheta \mathfrak{r}, \varsigma , \wp ) \bigr\} \\ &\quad =\mathcal{G}(\varTheta \mathfrak{r}, \varGamma \mathfrak{r}, \varsigma , \wp ). \end{aligned}$$

By Lemma 2, we have

$$\begin{aligned} \varTheta \mathfrak{r}=\varGamma \mathfrak{r}. \end{aligned}$$

(5)

From condition (4), we obtain for all $\varsigma \in \varPhi$ with $\varsigma \neq \varUpsilon \mathfrak{r}$, $\varsigma \neq \varGamma \mathfrak{r}$ and $\wp >0$,

$$\begin{aligned}& \begin{aligned} \mathcal{Q}(\varUpsilon \mathfrak{r}, \varLambda \varGamma \nu _{2 \mathfrak{n}+1}, \varsigma , \ell \wp )\geq{} &\min \bigl\{ \mathcal{Q}( \varTheta \mathfrak{r}, \varGamma \varGamma \nu _{2\mathfrak{n}+1}, \varsigma , \wp ), \mathcal{Q}(\varUpsilon _{\mathfrak{r}}, \varTheta _{\mathfrak{r}}, \varsigma , \wp ) \\ &{}\mathcal{Q}(\varLambda \varGamma \nu _{2\mathfrak{n}+1}, \varGamma \varGamma \nu _{2\mathfrak{n}+1}, \varsigma , \wp ), \mathcal{Q}( \varUpsilon \mathfrak{r}, \varGamma \varGamma \nu _{2\mathfrak{n}+1}, \varsigma , \wp ) \bigr\} , \end{aligned} \\& \begin{aligned} \mathcal{F}(\varUpsilon \mathfrak{r}, \varLambda \varGamma \nu _{2 \mathfrak{n}+1}, \varsigma , \ell \wp )\leq {}&\max \bigl\{ \mathcal{F}( \varTheta \mathfrak{r}, \varGamma \varGamma \nu _{2\mathfrak{n}+1}, \varsigma , \wp ), \mathcal{F}(\varUpsilon _{\mathfrak{r}}, \varTheta _{\mathfrak{r}}, \varsigma , \wp ) \\ &{}\mathcal{F}(\varLambda \varGamma \nu _{2\mathfrak{n}+1}, \varGamma \varGamma \nu _{2\mathfrak{n}+1}, \varsigma , \wp ), \mathcal{F}( \varUpsilon \mathfrak{r}, \varGamma \varGamma \nu _{2\mathfrak{n}+1}, \varsigma , \wp ) \bigr\} \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \mathcal{G}(\varUpsilon \mathfrak{r}, \varLambda \varGamma \nu _{2 \mathfrak{n}+1}, \varsigma , \ell \wp )\leq{} &\max \bigl\{ \mathcal{G}( \varTheta \mathfrak{r}, \varGamma \varGamma \nu _{2\mathfrak{n}+1}, \varsigma , \wp ), \mathcal{G}(\varUpsilon _{\mathfrak{r}}, \varTheta _{\mathfrak{r}}, \varsigma , \wp ) \\ &{}\mathcal{G}(\varLambda \varGamma \nu _{2\mathfrak{n}+1}, \varGamma \varGamma \nu _{2\mathfrak{n}+1}, \varsigma , \wp ), \mathcal{G}( \varUpsilon \mathfrak{r}, \varGamma \varGamma \nu _{2\mathfrak{n}+1}, \varsigma , \wp ) \bigr\} . \end{aligned}$$

Let $\mathfrak{n}\rightarrow \infty$, by condition (4), and Lemma 3, for all $\varsigma \in \varPhi$

$$\begin{aligned}& \begin{aligned} \mathcal{Q}(\varUpsilon \mathfrak{r}, \varGamma \mathfrak{r}, \varsigma , \ell \wp +0)\geq{}& \min \bigl\{ \mathcal{Q}(\varTheta \mathfrak{r}, \varGamma \mathfrak{r}, \varsigma , \wp ), \mathcal{Q}( \varUpsilon \mathfrak{r}, \varTheta \mathfrak{r}, \varsigma , \wp ), \\ &{}\mathcal{Q}(\varGamma \mathfrak{r}, \varGamma \mathfrak{r}, \varsigma , \wp ), \mathcal{Q}(\varUpsilon \mathfrak{r}, \varGamma \mathfrak{r}, \varsigma , \wp ) \bigr\} \\ ={}&\mathcal{Q}(\varUpsilon \mathfrak{r}, \varGamma \mathfrak{r}, \varsigma , \wp ), \end{aligned} \\& \begin{aligned} \mathcal{F}(\varUpsilon \mathfrak{r}, \varGamma \mathfrak{r}, \varsigma , \ell \wp +0)\leq{}& \max \bigl\{ \mathcal{F}(\varTheta \mathfrak{r}, \varGamma \mathfrak{r}, \varsigma , \wp ), \mathcal{F}( \varUpsilon \mathfrak{r}, \varTheta \mathfrak{r}, \varsigma , \wp ), \\ &{}\mathcal{F}(\varGamma \mathfrak{r}, \varGamma \mathfrak{r}, \varsigma , \wp ), \mathcal{F}(\varUpsilon \mathfrak{r}, \varGamma \mathfrak{r}, \varsigma , \wp ) \bigr\} \\ ={}&\mathcal{F}(\varUpsilon \mathfrak{r}, \varGamma \mathfrak{r}, \varsigma , \wp ) \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \mathcal{G}(\varUpsilon \mathfrak{r}, \varGamma \mathfrak{r}, \varsigma , \ell \wp +0)\leq{}& \max \bigl\{ \mathcal{G}(\varTheta \mathfrak{r}, \varGamma \mathfrak{r}, \varsigma , \wp ), \mathcal{G}( \varUpsilon \mathfrak{r}, \varTheta \mathfrak{r}, \varsigma , \wp ), \\ &{}\mathcal{G}(\varGamma \mathfrak{r}, \varGamma \mathfrak{r}, \varsigma , \wp ), \mathcal{G}(\varUpsilon \mathfrak{r}, \varGamma \mathfrak{r}, \varsigma , \wp ) \bigr\} \\ ={}&\mathcal{G}(\varUpsilon \mathfrak{r}, \varGamma \mathfrak{r}, \varsigma , \wp ). \end{aligned}$$

By Lemma 2, we have

$$\begin{aligned} \varUpsilon \mathfrak{r}=\varGamma \mathfrak{r}. \end{aligned}$$

(6)

For all $\varsigma \in \varPhi$ with $\varsigma \neq \varUpsilon \mathfrak{r}$ and $\varsigma \neq \varLambda \mathfrak{r}$ and $\wp >0$, we have

$$\begin{aligned} \begin{aligned} &\mathcal{Q}(\varUpsilon \mathfrak{r}, \varLambda \mathfrak{r}, \varsigma , \ell \wp ) \\ &\quad \geq \min \bigl\{ \mathcal{Q}(\varTheta \mathfrak{r}, \varGamma \mathfrak{r}, \varsigma , \wp ), \mathcal{Q}(\varUpsilon \mathfrak{r}, \varGamma \mathfrak{r},\varsigma , \wp ), \mathcal{Q}( \varLambda \mathfrak{r}, \varGamma \mathfrak{r}, \varsigma , \wp ), \mathcal{Q}(\varUpsilon \mathfrak{r}, \varGamma \mathfrak{r}, \varsigma , \wp ) \bigr\} \\ &\quad \geq \min \bigl\{ \mathcal{Q}(\varGamma \mathfrak{r}, \varGamma \mathfrak{r}, \varsigma , \wp ), \mathcal{Q}(\varUpsilon \mathfrak{r}, \varGamma \mathfrak{r},\varsigma , \wp ), \mathcal{Q}(\varLambda \mathfrak{r}, \varUpsilon \mathfrak{r}, \varsigma , \wp ), \mathcal{Q}(\varGamma \mathfrak{r}, \varGamma \mathfrak{r}, \varsigma , \wp ) \bigr\} \\ &\quad =\mathcal{Q}(\varUpsilon \mathfrak{r}, \varLambda \mathfrak{r}, \varsigma , \wp ), \end{aligned} \\ \begin{aligned} &\mathcal{F}(\varUpsilon \mathfrak{r}, \varLambda \mathfrak{r}, \varsigma , \ell \wp ) \\ &\quad \leq \max \bigl\{ \mathcal{F}(\varTheta \mathfrak{r}, \varGamma \mathfrak{r}, \varsigma , \wp ), \mathcal{F}(\varUpsilon \mathfrak{r}, \varGamma \mathfrak{r},\varsigma , \wp ), \mathcal{F}( \varLambda \mathfrak{r}, \varGamma \mathfrak{r}, \varsigma , \wp ), \mathcal{F}(\varUpsilon \mathfrak{r}, \varGamma \mathfrak{r}, \varsigma , \wp ) \bigr\} \\ &\quad \leq \max \bigl\{ \mathcal{F}(\varGamma \mathfrak{r}, \varGamma \mathfrak{r}, \varsigma , \wp ), \mathcal{F}(\varUpsilon \mathfrak{r}, \varGamma \mathfrak{r},\varsigma , \wp ), \mathcal{F}(\varLambda \mathfrak{r}, \varUpsilon \mathfrak{r}, \varsigma , \wp ), \mathcal{F}(\varGamma \mathfrak{r}, \varGamma \mathfrak{r}, \varsigma , \wp ) \bigr\} \\ &\quad =\mathcal{F}(\varUpsilon \mathfrak{r}, \varLambda \mathfrak{r}, \varsigma , \wp ) \end{aligned} \end{aligned}$$

and

$$\begin{aligned} &\mathcal{G}(\varUpsilon \mathfrak{r}, \varLambda \mathfrak{r}, \varsigma , \ell \wp ) \\ &\quad \leq \max \bigl\{ \mathcal{G}(\varTheta \mathfrak{r}, \varGamma \mathfrak{r}, \varsigma , \wp ), \mathcal{G}(\varUpsilon \mathfrak{r}, \varGamma \mathfrak{r}, \varsigma , \wp ), \mathcal{G}( \varLambda \mathfrak{r}, \varGamma \mathfrak{r}, \varsigma , \wp ), \mathcal{G}(\varUpsilon \mathfrak{r}, \varGamma \mathfrak{r}, \varsigma , \wp ) \bigr\} , \\ &\quad \leq \max \bigl\{ \mathcal{G}(\varGamma \mathfrak{r}, \varGamma \mathfrak{r}, \varsigma , \wp ), \mathcal{G}(\varUpsilon \mathfrak{r}, \varGamma \mathfrak{r}, \varsigma , \wp ), \mathcal{G}(\varLambda \mathfrak{r}, \varUpsilon \mathfrak{r}, \varsigma , \wp ), \mathcal{G}(\varGamma \mathfrak{r}, \varGamma \mathfrak{r}, \varsigma , \wp ) \bigr\} \\ &\quad =\mathcal{G}(\varUpsilon \mathfrak{r}, \varLambda \mathfrak{r}, \varsigma , \wp ). \end{aligned}$$

By Lemma 2,

$$\begin{aligned} \varUpsilon \mathfrak{r}=\varLambda \mathfrak{r}. \end{aligned}$$

(7)

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 $\wp >0$,

$$\begin{aligned} \mathcal{Q}(\varUpsilon \nu _{2\mathfrak{n}}, \varLambda \mathfrak{r}, \varsigma , \ell \wp )\geq{}& \min \bigl\{ \mathcal{Q}(\varTheta \nu _{2 \mathfrak{n}}, \varGamma \mathfrak{r}, \varsigma , \wp ), \mathcal{Q}( \varUpsilon \nu _{2\mathfrak{n}}, \varTheta \nu _{2\mathfrak{n}}, \varsigma , \wp ) \\ &{}\mathcal{Q}(\varLambda \mathfrak{r}, \varGamma \mathfrak{r}, \varsigma , \wp ), \mathcal{Q}(\varUpsilon \nu _{2\mathfrak{n}}, \varGamma _{\mathfrak{r}}, \varsigma , \wp ) \bigr\} , \\ \mathcal{F}(\varUpsilon \nu _{2\mathfrak{n}}, \varLambda \mathfrak{r}, \varsigma , \ell \wp )\leq{}& \max \bigl\{ \mathcal{F}(\varTheta \nu _{2 \mathfrak{n}}, \varGamma \mathfrak{r}, \varsigma , \wp ), \mathcal{F}( \varUpsilon \nu _{2\mathfrak{n}}, \varTheta \nu _{2\mathfrak{n}}, \varsigma , \wp ) \\ &{}\mathcal{F}(\varLambda \mathfrak{r}, \varGamma \mathfrak{r}, \varsigma , \wp ), \mathcal{F}(\varUpsilon \nu _{2\mathfrak{n}}, \varGamma _{\mathfrak{r}}, \varsigma , \wp ) \bigr\} \end{aligned}$$

and

$$\begin{aligned} \mathcal{G}(\varUpsilon \nu _{2\mathfrak{n}}, \varLambda \mathfrak{r}, \varsigma , \ell \wp )\leq{}& \max \bigl\{ \mathcal{G}(\varTheta \nu _{2 \mathfrak{n}}, \varGamma \mathfrak{r}, \varsigma , \wp ), \mathcal{G}( \varUpsilon \nu _{2\mathfrak{n}}, \varTheta \nu _{2\mathfrak{n}}, \varsigma , \wp ) \\ &{}\mathcal{G}(\varLambda \mathfrak{r}, \varGamma \mathfrak{r}, \varsigma , \wp ), \mathcal{G}(\varUpsilon \nu _{2\mathfrak{n}}, \varGamma _{\mathfrak{r}}, \varsigma , \wp ) \bigr\} . \end{aligned}$$

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 $\wp >0$

$$\begin{aligned} \begin{aligned} \mathcal{Q}(\mathfrak{r}, \varLambda \mathfrak{r}, \varsigma , \ell \wp +0)&\geq \min \bigl\{ \mathcal{Q}(\mathfrak{r}, \varGamma \mathfrak{r}, \varsigma , \wp ), \mathcal{Q}(\mathfrak{r}, \mathfrak{r}, \mathfrak{r}, \wp ), \mathcal{Q}( \varLambda \mathfrak{r}, \varLambda \mathfrak{r}, \varsigma , \wp ), \mathcal{Q}( \mathfrak{r}, \varGamma \mathfrak{r}, \varsigma , \wp ) \bigr\} \\ &\geq \mathcal{Q}(\mathfrak{r}, \varGamma \mathfrak{r}, \varsigma , \wp )\geq \mathcal{Q}(\mathfrak{r}, \varLambda \mathfrak{r}, \varsigma , \wp ), \end{aligned} \\ \begin{aligned} \mathcal{F}(\mathfrak{r}, \varLambda \mathfrak{r}, \varsigma , \ell \wp +0)&\leq \max \bigl\{ \mathcal{F}(\mathfrak{r}, \varGamma \mathfrak{r}, \varsigma , \wp ), \mathcal{F}(\mathfrak{r}, \mathfrak{r}, \mathfrak{r}, \wp ), \mathcal{F}( \varLambda \mathfrak{r}, \varLambda \mathfrak{r}, \varsigma , \wp ), \mathcal{F}( \mathfrak{r}, \varGamma \mathfrak{r}, \varsigma , \wp ) \bigr\} \\ &\leq \mathcal{F}(\mathfrak{r}, \varGamma \mathfrak{r}, \varsigma , \wp )\leq \mathcal{F}(\mathfrak{r}, \varLambda \mathfrak{r}, \varsigma , \wp ) \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \mathcal{G}(\mathfrak{r}, \varLambda \mathfrak{r}, \varsigma , \ell \wp +0)&\leq \max \bigl\{ \mathcal{G}(\mathfrak{r}, \varGamma \mathfrak{r}, \varsigma , \wp ), \mathcal{G}(\mathfrak{r}, \mathfrak{r}, \mathfrak{r}, \wp ), \mathcal{G}( \varLambda \mathfrak{r}, \varLambda \mathfrak{r}, \varsigma , \wp ), \mathcal{G}( \mathfrak{r}, \varGamma \mathfrak{r}, \varsigma , \wp ) \bigr\} \\ &\leq \mathcal{G}(\mathfrak{r}, \varGamma \mathfrak{r}, \varsigma , \wp )\leq \mathcal{G}(\mathfrak{r}, \varLambda \mathfrak{r}, \varsigma , \wp ). \end{aligned}$$

Hence, we have

$$\begin{aligned} \mathcal{Q}(\mathfrak{r}, \varLambda \mathfrak{r}, \varsigma , \ell \wp )\geq \mathcal{Q}(\mathfrak{r}, \varLambda \mathfrak{r}, \varsigma , \wp ), \qquad \mathcal{F}(\mathfrak{r}, \varLambda \mathfrak{r}, \varsigma , \ell \wp )\leq \mathcal{F}(\mathfrak{r}, \varLambda \mathfrak{r}, \varsigma , \wp ) \end{aligned}$$

and

$$\begin{aligned} \mathcal{G}(\mathfrak{r}, \varLambda \mathfrak{r}, \varsigma , \ell \wp )\leq \mathcal{G}(\mathfrak{r}, \varLambda \mathfrak{r}, \varsigma , \wp ). \end{aligned}$$

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

$$\begin{aligned}& \nu _{0}\perp \nu ^{*}, \\& \nu _{0}\perp \varrho ^{*}. \end{aligned}$$

Since, Γ is ⊥-preserving, one writes

$$\begin{aligned}& \varUpsilon ^{\mathfrak{n}}\nu _{0}\perp \varUpsilon ^{\mathfrak{n}} \nu ^{*}, \\& \varUpsilon ^{\mathfrak{n}}\nu _{0}\perp \varUpsilon ^{\mathfrak{n}} \varrho ^{*}. \end{aligned}$$

Now,

$$\begin{aligned}& \begin{aligned} \mathcal{Q}(\mathfrak{r}, \mathfrak{p}, \varsigma , \ell \wp )={}& \mathcal{Q}( \varUpsilon \mathfrak{r}, \varLambda \mathfrak{p}, \varsigma , \ell \wp ) \\ \geq{}& \min \bigl\{ \mathcal{Q}(\varTheta \mathfrak{r}, \varGamma \mathfrak{p}, \varsigma , \wp ), \mathcal{Q}(\varUpsilon \mathfrak{r}, \varTheta \mathfrak{r}, \varsigma , \wp ), \mathcal{Q}(\varLambda \mathfrak{p}, \varGamma \mathfrak{p}, \varsigma , \wp ), \\ &{}\mathcal{Q}( \varUpsilon \mathfrak{r}, \varGamma \mathfrak{p}, \varsigma , \wp ) \bigr\} \\ \geq{}& \min \bigl\{ \mathcal{Q}(\mathfrak{r}, \mathfrak{p}, \varsigma , \wp ), \mathcal{Q}(\mathfrak{r}, \mathfrak{p}, \varsigma , \wp ), \mathcal{Q}( \mathfrak{p}, \mathfrak{p}, \varsigma , \wp ), \mathcal{Q}(\mathfrak{r}, \mathfrak{p}, \varsigma , \wp ) \bigr\} \\ \geq{}& \mathcal{Q}(\mathfrak{r}, \mathfrak{p}, \varsigma , \wp ), \end{aligned} \\& \begin{aligned} \mathcal{F}(\mathfrak{r}, \mathfrak{p}, \varsigma , \ell \wp )={}& \mathcal{F}( \varUpsilon \mathfrak{r}, \varLambda \mathfrak{p}, \varsigma , \ell \wp ) \\ \leq{}& \max \bigl\{ \mathcal{F}(\varTheta \mathfrak{r}, \varGamma \mathfrak{p}, \varsigma , \wp ), \mathcal{F}(\varUpsilon \mathfrak{r}, \varTheta \mathfrak{r}, \varsigma , \wp ), \mathcal{F}(\varLambda \mathfrak{p}, \varGamma \mathfrak{p}, \varsigma , \wp ), \\ &{} \mathcal{F}( \varUpsilon \mathfrak{r}, \varGamma \mathfrak{p}, \varsigma , \wp ) \bigr\} \\ \leq{}& \max \bigl\{ \mathcal{F}(\mathfrak{r}, \mathfrak{p}, \varsigma , \wp ), \mathcal{F}(\mathfrak{r}, \mathfrak{p}, \varsigma , \wp ), \mathcal{F}( \mathfrak{p}, \mathfrak{p}, \varsigma , \wp ), \mathcal{F}(\mathfrak{r}, \mathfrak{p}, \varsigma , \wp ) \bigr\} \\ \leq{}& \mathcal{F}(\mathfrak{r}, \mathfrak{p}, \varsigma , \wp ) \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \mathcal{G}(\mathfrak{r}, \mathfrak{p}, \varsigma , \ell \wp )={}& \mathcal{G}( \varUpsilon \mathfrak{r}, \varLambda \mathfrak{p}, \varsigma , \ell \wp ) \\ \leq{}& \max \bigl\{ \mathcal{G}(\varTheta \mathfrak{r}, \varGamma \mathfrak{p}, \varsigma , \wp ), \mathcal{G}(\varUpsilon \mathfrak{r}, \varTheta \mathfrak{r}, \varsigma , \wp ), \mathcal{G}(\varLambda \mathfrak{p}, \varGamma \mathfrak{p}, \varsigma , \wp ), \\ &{}\mathcal{G}( \varUpsilon \mathfrak{r}, \varGamma \mathfrak{p}, \varsigma , \wp ) \bigr\} \\ \leq{}& \max \bigl\{ \mathcal{G}(\mathfrak{r}, \mathfrak{p}, \varsigma , \wp ), \mathcal{G}(\mathfrak{r}, \mathfrak{p}, \varsigma , \wp ), \mathcal{G}( \mathfrak{p}, \mathfrak{p}, \varsigma , \wp ), \mathcal{G}(\mathfrak{r}, \mathfrak{p}, \varsigma , \wp ) \bigr\} \\ \leq{}& \mathcal{G}(\mathfrak{r}, \mathfrak{p}, \varsigma , \wp ), \end{aligned}$$

which implies that

$$\begin{aligned}& \mathcal{Q}(\mathfrak{r}, \mathfrak{p}, \varsigma , \ell \wp )\geq \mathcal{Q}( \mathfrak{r}, \mathfrak{p}, \varsigma , \ell \wp ), \\& \mathcal{F}(\mathfrak{r}, \mathfrak{p}, \varsigma , \ell \wp )\leq \mathcal{F}( \mathfrak{r}, \mathfrak{p}, \varsigma , \ell \wp ) \end{aligned}$$

and

$$\begin{aligned} \mathcal{G}(\mathfrak{r}, \mathfrak{p}, \varsigma , \ell \wp )\leq \mathcal{G}( \mathfrak{r}, \mathfrak{p}, \varsigma , \ell \wp ). \end{aligned}$$

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,

$$\begin{aligned}& \mathcal{Q}(\varUpsilon \nu , \varLambda \varrho , \varsigma , \ell \wp )= \textstyle\begin{cases} 1 ,&\text{if } \nu =\varrho =\varsigma , \\ \frac{\ell \wp}{\ell \wp +\min \{\varUpsilon \nu , \varLambda \varrho , \varsigma , \ell \wp \}} ,&\text{if otherwise}, \end{cases}\displaystyle \\& \mathcal{F}(\varUpsilon \nu , \varLambda \varrho , \varsigma , \ell \wp )= \textstyle\begin{cases} 0 ,&\text{if } \nu =\varrho =\varsigma , \\ 1- \frac{\ell \wp}{\ell \wp +\max \{\varUpsilon \nu , \varLambda \varrho , \varsigma , \ell \wp \}} ,&\text{if otherwise}, \end{cases}\displaystyle \\& \mathcal{G}(\varUpsilon \nu , \varLambda \varrho , \varsigma , \ell \wp )= \textstyle\begin{cases} 0 ,&\text{if } \nu =\varrho =\varsigma , \\ \frac{\ell \wp +\max \{\varUpsilon \nu , \varLambda \varrho , \varsigma , \ell \wp \}}{\ell \wp} ,&\text{if otherwise}. \end{cases}\displaystyle \end{aligned}$$

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 $\varUpsilon , \varLambda , \varTheta , \varGamma \colon \varPhi \rightarrow \varPhi$

$$\begin{aligned} \varUpsilon \nu =\nu ^{2},\qquad \varLambda \nu =\nu ,\qquad \varTheta \nu =4\nu ^{4}-3,\qquad \varGamma \nu =4\nu ^{2}-3. \end{aligned}$$

From this, we obtain

$$\begin{aligned}& \mathcal{Q}(\varrho _{\mathfrak{n}}, \varrho _{\mathfrak{n}+1}, \varsigma , \ell \wp )\geq \mathcal{Q}(\varrho _{\mathfrak{n}-1}, \varrho _{\mathfrak{n}}, \varsigma , \wp ), \\& \mathcal{F}(\varrho _{\mathfrak{n}}, \varrho _{\mathfrak{n}+1}, \varsigma , \ell \wp )\leq \mathcal{Q}(\varrho _{\mathfrak{n}-1}, \varrho _{\mathfrak{n}}, \varsigma , \wp ), \\& \mathcal{G}(\varrho _{\mathfrak{n}}, \varrho _{\mathfrak{n}+1}, \varsigma , \ell \wp )\geq \mathcal{Q}(\varrho _{\mathfrak{n}-1}, \varrho _{\mathfrak{n}}, \varsigma , \wp ). \end{aligned}$$

This implies

$$\begin{aligned}& \lim_{\mathfrak{n}\rightarrow \infty}\mathcal{Q}(\varrho _{ \mathfrak{m}}, \varrho _{\mathfrak{n}}, \varsigma , \wp )=1, \qquad \lim_{\mathfrak{n}\rightarrow \infty} \mathcal{F}(\varrho _{ \mathfrak{m}}, \varrho _{\mathfrak{n}}, \varsigma , \wp )=0 \quad \text{and} \\& \lim_{\mathfrak{n}\rightarrow \infty}\mathcal{G}(\varrho _{ \mathfrak{m}}, \varrho _{\mathfrak{n}}, \varsigma , \wp )=0 \quad \text{for all } \nu , \varrho , \varsigma \in \varPhi . \end{aligned}$$

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,

$$\begin{aligned}& \mathit{d}(\varpi )=\mathfrak{f}(\varpi )+\delta \int _{\rho}^{\pi} \mathcal{U}_{1}( \varpi , \theta )\mathfrak{k}(\varpi )\,\mathit{d} \theta \quad \text{for all } \theta , \varpi \in [\rho , \pi ], \end{aligned}$$

(8)

$$\begin{aligned}& \mathit{d}(\varpi )=\mathfrak{f}(\varpi )+\delta \int _{\rho}^{\pi} \mathcal{U}_{1}( \varpi , \theta )\mathfrak{k}(\varpi )\,\mathit{d} \theta \quad \text{for all } \theta , \varpi \in [\rho , \pi ], \end{aligned}$$

(9)

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

$$\begin{aligned}& \mathcal{Q}(\varUpsilon \nu , \varLambda \varrho , \varsigma , \ell \wp )=\min \biggl\{ \frac{\ell \wp}{\ell \wp +\min \{\varUpsilon \nu , \varLambda \varrho , \varsigma , \ell \wp \}} \biggr\} \quad \forall \nu , \varrho , \varsigma \in \Phi \text{ and } \wp >0, \\& \mathcal{F}(\varUpsilon \nu , \varLambda \varrho , \varsigma , \ell \wp )=\max \biggl\{ 1- \frac{\ell \wp}{\ell \wp +\max \{\varUpsilon \nu , \varLambda \varrho , \varsigma , \ell \wp \}} \biggr\} \quad \forall \nu , \varrho , \varsigma \in \Phi \text{ and } \wp >0, \\& \mathcal{G}(\varUpsilon \nu , \varLambda \varrho , \varsigma , \ell \wp )=\max \biggl\{ \frac{\max \{\varUpsilon \nu , \varLambda \varrho , \varsigma , \ell \wp \}}{\ell \wp} \biggr\} \quad \forall \nu , \varrho , \varsigma \in \Phi \text{ and } \wp >0. \end{aligned}$$

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

$$\begin{aligned}& \mathit{d}(\varpi )=\mathfrak{f}(\varpi )+\delta \int _{\rho}^{\pi} \mathcal{U}_{1}( \varpi , \theta )\mathfrak{k}(\varpi )\,\mathit{d} \theta \quad \text{for all } \theta , \varpi \in [\rho , \pi ], \end{aligned}$$

(10)

$$\begin{aligned}& \mathit{d}(\varpi )=\mathfrak{f}(\varpi )+\delta \int _{\rho}^{\pi} \mathcal{U}_{1}( \varpi , \theta )\mathfrak{k}(\varpi )\,\mathit{d} \theta \quad \text{for all } \theta , \varpi \in [\rho , \pi ]. \end{aligned}$$

(11)

The survival of a fixed of the operator $\mathcal{U}$ has come to the survival of solution of a neutrosophic integral equation,

$$\begin{aligned} &\mathcal{Q} \bigl(\varUpsilon \nu (\varpi ), \varLambda \varrho (\varpi ), \varsigma \varpi , \ell \wp \bigr) \\ &\quad=\sup_{\varpi \in [\rho , \pi ]} \frac{\ell \wp}{\ell \wp +\min (\varUpsilon \nu (\varpi ), \varLambda \varrho (\varpi ), \varsigma \varpi , \ell \wp )} \\ &\quad=\sup_{\varpi \in [\rho , \pi ]} {\ell \wp} \big/\biggl(\ell \wp + \biggl\vert \mathfrak{f}(\varpi )+\delta \int _{\rho}^{\pi}\mathcal{U}_{1}( \varpi , \theta )\varUpsilon \nu (\varpi )-\varsigma \varpi -\wp -\mathfrak{f}( \varpi ) \\ &\qquad{} -\delta \int _{\rho}^{\pi}\mathcal{U}_{2}(\varpi , \theta )\varLambda \varrho (\varpi )-\varsigma \varpi -\wp \biggr\vert \biggr) \\ &\quad\geq \sup_{\varpi \in [\rho , \pi ]} \frac{\ell \wp}{\ell \wp + \vert \varUpsilon \nu (\varpi )-\varLambda \varrho (\varpi )- \varsigma \varpi -\wp ) \vert } \\ &\quad\geq \mathcal{Q}(\varUpsilon \nu , \varLambda \varrho , \varsigma , \wp ), \\ &\mathcal{F} \bigl(\varUpsilon \nu (\varpi ), \varLambda \varrho (\varpi ), \varsigma \varpi , \ell \wp \bigr) \\ &\quad=1-\sup_{\varpi \in [\rho , \pi ]} \frac{\ell \wp}{\ell \wp +\max (\varUpsilon \nu (\varpi ), \varLambda \varrho (\varpi ), \varsigma \varpi , \ell \wp )} \\ &\quad=1-\sup_{\varpi \in [\rho , \pi ]} {\ell \wp} \big/\biggl(\ell \wp + \biggl\vert \mathfrak{f}(\varpi )+\delta \int _{\rho}^{\pi}\mathcal{U}_{1}( \varpi , \theta )\varUpsilon \nu (\varpi )-\varsigma \varpi -\wp -\mathfrak{f}( \varpi ) \\ &\qquad{} -\delta \int _{\rho}^{\pi}\mathcal{U}_{2}(\varpi , \theta )\varLambda \varrho (\varpi )-\varsigma \varpi -\wp \biggr\vert \biggr) \\ &\quad\leq 1-\sup_{\varpi \in [\rho , \pi ]} \frac{\ell \wp}{\ell \wp + \vert \varUpsilon \nu (\varpi )-\varLambda \varrho (\varpi )- \varsigma \varpi -\wp ) \vert } \\ &\quad\leq \mathcal{F}(\varUpsilon \nu , \varLambda \varrho , \varsigma , \wp ) \end{aligned}$$

and

$$\begin{aligned} &\mathcal{G} \bigl(\varUpsilon \nu (\varpi ), \varLambda \varrho (\varpi ), \varsigma \varpi , \ell \wp \bigr) \\ &\quad=\sup_{\varpi \in [\rho , \pi ]} \frac{\ell \wp}{\ell \wp +\max (\varUpsilon \nu (\varpi ), \varLambda \varrho (\varpi ), \varsigma \varpi , \ell \wp )} \\ &\quad=\sup_{\varpi \in [\rho , \pi ]} {\ell \wp} / \biggl(\ell \wp + \biggl\vert \mathfrak{f}(\varpi )+\delta \int _{\rho}^{\pi}\mathcal{U}_{1}( \varpi , \theta )\varUpsilon \nu (\varpi )-\varsigma \varpi -\wp -\mathfrak{f}( \varpi ) \\ &\qquad{} -\delta \int _{\rho}^{\pi}\mathcal{U}_{2}(\varpi , \theta )\varLambda \varrho (\varpi )-\varsigma \varpi -\wp \biggr\vert \biggr) \\ &\quad\leq \sup_{\varpi \in [\rho , \pi ]} \frac{\ell \wp}{\ell \wp + \vert \varUpsilon \nu (\varpi )-\varLambda \varrho (\varpi )- \varsigma \varpi -\wp ) \vert } \\ &\quad\leq \mathcal{F}(\varUpsilon \nu , \varLambda \varrho , \varsigma , \wp ). \end{aligned}$$

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.