Fixed-Point Theorems on Fuzzy Bipolar b -Metric Spaces

: In this manuscript, we establish some ﬁxed-point theorems without continuity by using the triangular property on a fuzzy bipolar b -metric space as a generalized version and expansion of the well-known results. We also provide some examples and applications of the integral equation to the solution for our main results


Introduction and Preliminaries
In 1960, the concept of continuous triangular norm was initiated by Schweizer and Sklar [1].After that, Zadeh [2] initiated the theory of fuzzy sets in 1965.This concept of fuzziness was described by Kramosil and Michalek [3] in 1975 using the fuzzy metric space with the help of the continuous t-norm.Moreover, they were interested the notion of fuzziness to improve the well-known fixed-point results by Grabeic [4] in the sense of such spaces.After that, Gregori and Sapena [5] extended the fuzzy Banach contraction map to the fuzzy metric space.In 1994, the fuzzy metric space definition was modified by George and Veeramani [6].Moreover, Shamas, Rehman, Aydi, Mahmood, and Ameer [7] described more results on the fixed-point results.Mutlu and Gurdal [8] introduced the concept of generalized metric space as a bipolar metric space.Bartwal, Dimri, and Prasad [9] proposed a fuzzy bipolar metric space and proved fixed-point results.Additional articles which relate to the concept can be seen in [10][11][12][13][14].In this paper, we demonstrate some fixed-point results on a fuzzy bipolar b-metric space.Now, we recall some basics Definition 1 ([9]).Let X and Ỹ be two nonempty sets.A quadruple ( X, Ỹ, , * ) is called a fuzzy bipolar metric space (FB-space), where * and are a continuous -norm and fuzzy set on X × Ỹ × (0, ∞), respectively, such that for all , η, δ > 0: Definition 2. Let X and Ỹ be two nonempty sets and s ≥ 1 be a given real number.A five tuple ( X, Ỹ, , s, * ) is called a fuzzy bipolar b-metric space (FBBMS), where * , is a continuous -norm and fuzzy set on X × Ỹ × (0, ∞), respectively, such that for all , η, δ > 0: (A4) A fuzzy bipolar b-metric space is called complete if every Cauchy bisequence is convergent.
Lemma 1.Let ( X, Ỹ, , s, * ) be a fuzzy bipolar b-metric space.If u ∈ X ∩ Ỹ is a limit of a sequence, then it is a unique limit of the sequence.
Hence, the fuzzy bipolar b-metric is b-triangular.
In this manuscript, we prove fixed-point results on FBBMS with some applications.

Main Result
Now, we prove our first result: Theorem 1.Let ( X, Ỹ, , s, * ) be a complete FBBMS with constant s ≥ 1 and the mapping Then, Ř has a UFP (unique fixed point).

Application II
We recall many important definitions from fractional calculus theory [16,17].For a function ∈ C[0, 1], the order δ > 0 of Riemann-Liouville fractional derivative is From ( 2), the right-hand side is pointwise defined on [0, 1], where [δ] and Γ are the integer part of the number δ and the Euler gamma function.
Consider the following FDE (fractional differential equation) where : [0, 1] × R → R is a continuous function and e D represents the Caputo fractional derivative of order , and it is defined by for all ( , x) ∈ X × Ỹ.Then, ( X, Ỹ, , s, ) is a complete FBBMS.Theorem 6.Let us consider the nonlinear FDE (3).Suppose that the following conditions are satisfied: We can find j ∈ (0, Taking the supremum on both sides, we obtain ( Ř , Ř x) ≤ j ( , x).

Definition 4 .
Let ( X, Ỹ, , s, * ) be a fuzzy bipolar b-metric space.The fuzzy bipolar b-metric space is b-triangular