Application of Fixed Points in Bipolar Controlled Metric Space to Solve Fractional Differential Equation

: Fixed point results and metric ﬁxed point theory play a vital role to ﬁnd the unique solution to differential and integral equations. Likewise, fractal calculus has vast physical applications. In this article, we introduce the concept of bipolar-controlled metric space and prove ﬁxed point theorems. The derived results expand and extend certain well-known results from the research literature and are supported with a non-trivial example. We have applied the ﬁxed point result to ﬁnd the analytical solution to the integral equation and fractional differential equation. The analytical solution has been supplemented with numerical simulation.


Introduction
The Banach Fixed Point Theorem [1] is being studied not only by mathematicians but also by researchers in other branches of science to examine its applicability to real-world situations and accordingly very popular among scientists across various domains such as computer science, physics, applied mathematics and even finance.Even medical experts examine its applicability in their field in real-life situations.
In addition, metric fixed point theory has a wide range of applications such as dynamic programming, variational inequalities, fractal dynamics, dynamical systems of mathematics, as well as the deployment of satellites in their appropriate orbits in space science, etc.It also ensures that patients receive the most appropriate diagnosis and examines the intensity of the spread of contagious diseases in various cities.In mathematics, new discoveries of space and their properties are always of interest to researchers.The notion of probabilistic metric spaces, in which the probabilistic distance between two points is examined, has provided a new dimension to the subject and interest in learning more about stars in the cosmos.
Recently, popular topics in fixed point theory are addressing the existence of fixed points of contraction mappings in bipolar metric spaces, which can be considered as generalizations of the Banach contraction principle.
In this paper, we introduce the notion of Bipolar controlled metric space and prove fixed point theorems on bipolar-controlled metric space.We supplement our results with suitable examples.We also present an application of the derived result to find analytical and numerical solutions to a fractional differential equation.
The rest of the paper is organized as follows: In Section 2, we define Bipolar controlled metric space and study some topological properties.In Section 3, we present our main results by establishing fixed point results in the setting of Bipolar controlled metric spaces.In Section 4, we present an application to find the solution to the fractional differential equation.We conclude this paper with some open problems for further research.

Preliminaries
The following basic definitions are required in the sequel.

Definition 1 ([2]
).Let O, L be two nonempty sets and The triplet (O, L, K) is called a bipolar metric space.
Now we define the bipolar-controlled metric space and study some topological properties.

Definition 2.
Let O, L be two nonempty sets and Then (O; L; K; Υ) is called a bipolar-controlled metric space.
Let (O; L; K; Υ) be a bipolar-controlled metric space.The set form a basis of some topology ω 2 on L. The set is called open ball of radius ε > 0 and at center κ ∈ L. Similarly, the set Therefore B is a base for a topology on O × L. This topology is called the product topology.
Remark 1.Note that, if Υ(κ, η) = 1, for all κ ∈ O and η ∈ L, then bipolar-controlled metric space reduces to a bipolar metric space.That is to say, bipolar-controlled metric space is a generalization of bipolar metric space.
Remark 2. Every bipolar metric space is bipolar-controlled metric space but the converse is not always true and also bipolar-controlled metric space is not Hausdorff.The following example illustrates this.
B( 1 2 ; 1) = {0, ) and this is written as ) and this is denoted as If a covariant map is right and left continuous at each κ ∈ O 1 and µ ∈ L 1 , then it is continuous.
Definition 5. A bipolar-controlled metric space is called complete, if every Cauchy bisequence in this space is convergent.

Main Results
In this section, we prove fixed point theorems in bipolar-controlled metric spaces.
Theorem 1.Let (O, L, K, Υ) be a complete bipolar-controlled metric space.Consider the mapping for all κ ∈ O and µ ∈ L, where 0 < ρ < 1. Suppose that and Then the function For all natural numbers ξ < τ, we have and Hence, we have Now by conditions (1), ( 2) and the ratio test, we obtain that lim ξ→∞ G ξ and lim ξ→∞ Q ξ exists.
Then, (O, L, K) is a complete bipolar-controlled metric space.Define for all ξ ∈ N. We can easily get Hence Theorem 1 is satisfied and has a UFP O ξ×ξ .
Theorem 2. Let (O, L, K, Υ) be a complete bipolar-controlled metric space.Consider the mapping for all κ ∈ O and µ ∈ L, where 0 < ρ < 1. Suppose that and Then the function . . .
Now we present a theorem based on Kannan's fixed point theorem [16].

Application to Fractional Differential Equations
Before proceeding to this subsection, let us recall the following: For a function v ∈ C[0, 1], the Reiman-Liouville fractional derivative of order δ > 0 is given by 1 Γ(ξ − δ) provided that the right hand side is pointwise defined on [0, 1], where [δ] is the integer part of the number δ, Γ is the Euler gamma function.Consider the following fractional differential equation where f is a continuous function from [0, 1] × R to R and e D µ represents the Caputo fractional derivative of order µ and it is defined by Then the FDE (19) has a unique solution in O ∪ L.
Proof.Equation ( 19) is equivalent to Define the covariant mapping : Taking the supremum on both sides, we get Thus, the conditions of Theorem 1 are satisfied and hence the fractional differential Equation (19) has a unique solution.

Conclusions
The notion of bipolar-controlled metric space was introduced and its basic topological properties were discussed in this article.We have established fixed-point results in these spaces.The derived results have been applied to find the analytical solution to the integral equation and fractional differential equation.The analytical solution has been supplemented with numerical computations.It is an open problem to investigate the existence of fixed points using various types of contractions such as Presi'c, Meir-Keeler, etc., and apply the results to find solutions to problems involving differential and integral equations.
is called closed ball of radius ε > 0 with center κ ∈ L. The set of open balls V = {B(κ; ε) : κ ∈ L, ε > 0}, form a basis of some topology ω 1 on O. Let B denote the family of all subsets of O × L of the form U × V, where U is open in O and V open in L. Then B = O × L and the intersection of any two members of B lies in B.

Figure 1 .
Figure 1.For Example 5: (a) compares the numerical and exact solutions of the FDE; (b) displays the absolute error between them.

Table 1 .
The numerical and exact solution using the matrix approach method where ∼ = 51.