Orthogonal neutrosophic 2-metric spaces

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.


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 fixedpoint results in weak and rational (αψ)-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.
Therefore, Λr = r.Thus, r = Υ r = Λr = Θr = Γ r.Hence, r is a common fixed point of Υ , Λ, Θ, and Γ .Let p be another common fixed point of Υ , Λ, Θ, and Γ for all ς ∈ Φ with ς = r, ς = p, and ℘ > 0, we have Since, Γ is ⊥-preserving, one writes 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 .

Application
In this section, we given an application to the Fredholm integral equation as below: Hence, all the conditions of Theorem 3.1 are satisfied.Hence, Υ and Λ have a unique common solution.

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.

Definition 2 . 9
called the open ball with center ν and radius h with respect to ℘. Suppose (Φ, Q, F, G, * , +, ⊥) is a ON2MS.Then, an open set of U ⊂ Φ of its points is the center of a open ball contained in U .The open set in a N2MS