A Note on Orthogonal Fuzzy Metric Space, Its Properties, and Fixed Point Theorems

The article generalizes the notion of orthogonal fuzzy metric space into a broader term, named as orthogonal picture fuzzy metric space. The obtained results improve and extend the idea of the orthogonal fuzzy metric space and its related results. However, this article outstretches the above-mentioned notion further into a newly de ﬁ ned concept, named as orthogonal picture fuzzy metric space. A detailed insight is given into the topic by presenting some ﬁ xed point results in the frame of the newly de ﬁ ned structure. To elaborate the results more precisely, some concrete examples are given.


Introduction
In 2013, Cuong [2] proposed a new concept named picture fuzzy sets (PFS), which is an extension of fuzzy sets and intuitionistic fuzzy sets. In a picture fuzzy set, each element is specified by the degree of membership, the degree of nonmembership, and degree of neutrality together with the condition that the sum of these grades should be less or equal to 1.
In this regard, Phong et al. [7] studied some compositions of picture fuzzy relations. Cuong and Hai [8] investigated main fuzzy logic operators: negations, conjunctions, disjunctions, and implications on picture fuzzy sets, and constructed the main operations for fuzzy inference processes in picture fuzzy systems. Singh [9] studied the correlation coefficients of picture fuzzy sets. Cuong et al. [10] then investigated the classification of representable picture t-norms and picture tconorms operators for picture fuzzy sets.
Eshaghi et al. [4] presented a new generalization of the Banach fixed point theorem (BFPT) by defining the notion of orthogonal sets. The orthogonal set is a non-empty set equipped with a binary relation (called orthogonal relation) having a special structure (see [4]). The metric defined on the orthogonal set is called orthogonal metric space. The orthogonal metric space contains partially ordered metric space and graphical metric space. Hezarjaribi [5] further extended the results of [4] to orthogonal fuzzy metric space. Also, Ishtiaq et al. [6] extended the results of [4] to orthogonal neutrosophic metric space. Some more details about generalized orthogonal metric spaces have been provided by Javed et al. [11], Uddin et al. [12,13], and Senapati et al. [14].
In this paper, we introduce orthogonal picture fuzzy metric space which generalize picture fuzzy metric space and orthogonal fuzzy metric spaces. We show that every picture fuzzy metric space is an orthogonal picture fuzzy metric space but not conversely. We investigate different conditions on the picture fuzzy to show the existence of fixed points in various types of contractions. We also present some examples in support of the obtained results. The authors intend to further widen the interesting idea of orthogonality to the intuitionistic fuzzy metric space and spherical fuzzy metric spaces. Some interesting results on the same two topics can be read in the articles [15,16] and [17], respectively.

Preliminaries
Definition 1 (see [1]). A fuzzy set is a pair ðⱲ, f Þ, where Ⱳ is a non-empty set, f : Ⱳ ⟶ ½0, 1 is a membership function and for each I ∈ Ⱳ, f ðIÞ is called the grade of membership of I in ðⱲ, f Þ: Definition 2 (see [2]). A picture fuzzy set A on the universe set Ⱳ is an object of the form where Yð∂Þ ∈ ½0, 1 is called the degree of positive membership of ∂ in A, Mð∂Þ ∈ ½0, 1 is called the "degree of neutral membership of ∂ in A," and Ɒð∂Þ ∈ ½0, 1 is called the degree of negative membership of ∂ in A, and Yð∂Þ, Mð∂Þ, Ɒð∂Þ satisfy for all ∂ ∈ A. Then, is called the degree of refusal membership of ∂ in A.
Definition 3 (see [3]  Definition 5 (see [4]). Assume Ⱳ ≠ ⏀ and ˫∈Ⱳ × Ⱳ is a binary relation. Assume there exists I 0 ∈ Ⱳ such that I 0 ˫ I or I˫I 0 for all I ∈ Ⱳ. Thus, Ⱳ is said to be an OS. Furthermore, we denote OS by ðⱲ, ˫Þ.
Definition 6 (see [4]). Suppose that ðⱲ, ˫Þ is an OS. A sequence fI n g for n ∈ ℕ is called an (OS) if for all n, I n ˫ I n+1 or for all n, I n+1 ˫I n .
Remark 9. The above example is also OPFMS if we take for all ℘≥0, p ≥ 1. Example 2. Assume OPFMS as given in Example 1 and define a sequence fI n g in Ⱳ by I n = 1 − 1/n, ∀n ∈ ℕ such that ð∀n ; I n ˫I n+1 Þ or ð∀n ; I n+1 ˫I n Þ. Define a CTN as a * b = ab, CTCN as aΔb = max fa, bg, and define a binary relation ˫ by

Journal of Function Spaces
Example 3. From proof of Example 2, I n = 1 − 1/n, ∀n ∈ ℕ is a O-CS in an OPFMS.

Journal of Function Spaces
Since Ɒ is OPR, one writes Ω n I 0 ˫Ω n I * and Ω n I 0 ˫Ω n ℏ * , ð29Þ for all n ∈ M: So from (10), we can derive Therefore, So from (11), we can derive Therefore, Similarly, from (12), we can derive Therefore, So, I * = ℏ ; hence, I * is the unique FP. Proof. We can similarly derive as in the proof of Theorem 16 that fI n g is a O-CS and so it converges to I * ∈ Ⱳ: Hence, I * ˫I n for all n ∈ ℕ: from (10), we can get Then, we can write Taking limit as n ⟶ +∞, we get YðI * , ΩI * ,℘Þ = 1 * 1 = 1 and from (11), we can get Then, we can write Taking limit as n ⟶ +∞, we get and from (12), we can get Then, we can write Taking limit as n ⟶ +∞, we get so ΩI * = I * : Next proof is similar as in Theorem 16.

Data Availability
No data was used during this research

Conflicts of Interest
The authors declare that they have no competing interests.