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

Abstract

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.

1. 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 2016, Mutlu and Gürdal [2,3] introduced the concepts of bipolar metric space and they investigated certain basic fixed point and coupled fixed point theorems for co-variant and contra-variant maps under certain contractive conditions. Since then, fixed point results using various contractive conditions in the setting of bipolar metric spaces, have been reported by various researchers [4,5,6,7,8,9,10,11,12,13,14,15].

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.

2. Preliminaries

The following basic definitions are required in the sequel.

Definition 1

([2]). Let $\mathcal{O}$, $\mathcal{L}$ be two nonempty sets and $\mathcal{K}:\mathcal{O}\times \mathcal{L}\to {\mathbb{R}}^{+}$ such that

(a) 

$\mathcal{K}(\kappa ,\mu )=0$ iff $\kappa =\mu$, for all $(\kappa ,\mu )\in \mathcal{O}\times \mathcal{L}$.

(b) 

$\mathcal{K}(\kappa ,\mu )=\mathcal{K}(\mu ,\kappa )$, for all $(\kappa ,\mu )\in \mathcal{O}\cap \mathcal{L}$.

(c) 

$\mathcal{K}(\kappa ,\mu )\le \mathcal{K}(\kappa ,{\rm Y})+\mathcal{K}(\nu ,{\rm Y})+\mathcal{K}(\nu ,\mu )$, for all $\kappa ,\nu \in \mathcal{O}$ and ${\rm Y},\mu \in \mathcal{L}$.

The triplet $(\mathcal{O},\mathcal{L},\mathcal{K})$ is called a bipolar metric space.

Now we define the bipolar-controlled metric space and study some topological properties.

Definition 2.

Let $\mathcal{O}$, $\mathcal{L}$ be two nonempty sets and $\mathcal{K}:\mathcal{O}\times \mathcal{L}\to [0,+\infty )$, ${\rm Y}:\mathcal{O}\times \mathcal{L}\to [0,+\infty )$ such that

(a) 

$\mathcal{K}(\kappa ,\mu )=0$ iff $\kappa =\mu$, for all $(\kappa ,\mu )\in \mathcal{O}\times \mathcal{L}$..

(b) 

$\mathcal{K}(\kappa ,\mu )=\mathcal{K}(\mu ,\kappa )$, for all $(\kappa ,\mu )\in \mathcal{O}\cap \mathcal{L}$.

(c) 

$\mathcal{K}(\kappa ,\mu )\le {\rm Y}(\kappa ,{\rm Y})\mathcal{K}(\kappa ,{\rm Y})+{\rm Y}(\nu ,{\rm Y})\mathcal{K}(\nu ,{\rm Y})+{\rm Y}(\nu ,\mu )\mathcal{K}(\nu ,\mu )$, for all $\kappa ,\nu \in \mathcal{O}$ and ${\rm Y},\mu \in \mathcal{L}$.

Then $(\mathcal{O};\mathcal{L};\mathcal{K};{\rm Y})$ is called a bipolar-controlled metric space.

Let $(\mathcal{O};\mathcal{L};\mathcal{K};{\rm Y})$ be a bipolar-controlled metric space. The set

$$\begin{array}{c}\hfill \mathcal{B}(\kappa ;\epsilon )=\{\eta \in \mathcal{L}:\mathcal{K}(\eta ,\kappa )<\epsilon \}\end{array}$$

is called open ball of radius $\epsilon >0$ and at center $\kappa \in \mathcal{O}$. Similarly, the set

$$\begin{array}{c}\hfill \mathcal{B}[\kappa ;\epsilon ]=\{\eta \in \mathcal{L}:\mathcal{K}(\eta ,\kappa )\le \epsilon \}\end{array}$$

is called closed ball of radius $\epsilon >0$ and at center $\kappa \in \mathcal{O}$. The set of open balls

$$\begin{array}{c}\hfill \mathcal{U}=\left\{\mathcal{B}\right(\kappa ;\epsilon ):\eta \in \mathcal{O},\epsilon >0\},\end{array}$$

form a basis of some topology ${\omega }_2$ on $\mathcal{L}$. The set

$$\begin{array}{c}\hfill \mathfrak{B}(\kappa ;\epsilon )=\{\eta \in \mathcal{O}:\mathcal{K}(\eta ,\kappa )<\epsilon \}\end{array}$$

is called open ball of radius $\epsilon >0$ and at center $\kappa \in \mathcal{L}$. Similarly, the set

$$\begin{array}{c}\hfill \mathfrak{B}[\kappa ;\epsilon ]=\{\eta \in \mathcal{O}:\mathcal{K}(\eta ,\kappa )\le \epsilon \}\end{array}$$

is called closed ball of radius $\epsilon >0$ with center $\kappa \in \mathcal{L}$. The set of open balls

$$\begin{array}{c}\hfill \mathcal{V}=\left\{\mathfrak{B}\right(\kappa ;\epsilon ):\kappa \in \mathcal{L},\epsilon >0\},\end{array}$$

form a basis of some topology ${\omega }_1$ on $\mathcal{O}$.

Let $\mathsf{B}$ denote the family of all subsets of $\mathcal{O}\times \mathcal{L}$ of the form $\mathsf{U}\times \mathsf{V}$, where $\mathsf{U}$ is open in $\mathcal{O}$ and $\mathsf{V}$ open in $\mathcal{L}$. Then $\bigcup \mathsf{B}=\mathcal{O}\times \mathcal{L}$ and the intersection of any two members of $\mathsf{B}$ lies in $\mathsf{B}$. Therefore $\mathsf{B}$ is a base for a topology on $\mathcal{O}\times \mathcal{L}$. This topology is called the product topology.

Remark 1.

Note that, if ${\rm Y}(\kappa ,\eta )=1$, for all $\kappa \in \mathcal{O}$ and $\eta \in \mathcal{L}$, 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.

Example 1.

Let $\mathcal{O}=\{0,\frac{1}{2},1,2,3,4,5\}$ and $\mathcal{L}=\{0,\frac{1}{3},\frac{1}{4},\frac{1}{5},6\}$ be equipped with $\mathcal{K}(\kappa ,\mu )={|\kappa -\mu |}^2$ and ${\rm Y}(\kappa ,\mu )=\kappa \mu +1$ for all $\kappa \in \mathcal{O}$ and $\mu \in \mathcal{L}$. Then, $(\mathcal{O};\mathcal{L};\mathcal{K};{\rm Y})$ is a complete bipolar-controlled metric space. However, we see that

$$\begin{array}{c}\hfill \mathcal{K}(0,6)=36>\mathcal{K}(0,\frac{1}{3})+\mathcal{K}(\frac{1}{2},\frac{1}{3})+\mathcal{K}(\frac{1}{2},6)=30.37.\end{array}$$

But

$$\begin{array}{c}\hfill \mathcal{K}(0,6)=36\le \mathcal{K}(0,\frac{1}{3})+1.17\mathcal{K}(\frac{1}{2},\frac{1}{3})+4\mathcal{K}(\frac{1}{2},6).\end{array}$$

Therefore, bipolar-controlled metric space need not be a bipolar metric space. However, we have the following:

1. 

$\mathcal{B}(\frac{1}{2};1)=\{0,\frac{1}{3},\frac{1}{4},\frac{1}{5}\}$ and there does not exist any open ball with center 0 and contained in $\mathcal{B}(\frac{1}{2};1)$. So $\mathcal{B}(\frac{1}{2};1)$ is not an open set on $\mathcal{L}$. $\mathcal{B}(\frac{1}{3};1)=\{0,\frac{1}{2}\}$ and there does not exist any open ball with center $\frac{1}{2}$ and contained in $\mathcal{B}(\frac{1}{3};1)$. So $\mathcal{B}(\frac{1}{3};1)$ is not an open set on $\mathcal{O}$. Then $\mathcal{B}(\frac{1}{2};1)\times \mathcal{B}(\frac{1}{3};1)$ is not an open in $\mathcal{O}\times \mathcal{L}$.

2. 

There does not exist any ${\epsilon }_1,{\epsilon }_2>0$ such that $\mathcal{B}(\frac{1}{2};{\epsilon }_1)\times \mathcal{B}(\frac{1}{3};{\epsilon }_2)$ and so $(\mathcal{O},\mathcal{L},\mathcal{K})$ is not Hausdorff.

Definition 3.

(A1) 

Let $(\mathcal{O};\mathcal{L};\mathcal{K};{\rm Y})$ be a bipolar-controlled metric space. Then the points of the sets $\mathcal{O}$, $\mathcal{L}$ and $\mathcal{O}\cap \mathcal{L}$ are named as left, right and central points, respectively, and any sequence that consists of only left (or right, or central) points is called a left (or right, or central) sequence on $(\mathcal{O};\mathcal{L};\mathcal{K};{\rm Y})$.

(A2) 

Let $({\mathcal{O}}_1,{\mathcal{L}}_1,{\mathcal{K}}_1,{{\rm Y}}_1)$ and $({\mathcal{O}}_2,{\mathcal{L}}_2,{\mathcal{K}}_2,{{\rm Y}}_2)$ be bipolar-controlled metric spaces and $\top :{\mathcal{O}}_1\cup {\mathcal{L}}_1\to {\mathcal{O}}_2\cup {\mathcal{L}}_2$ be a function. If $\top \left({\mathcal{O}}_1\right)\subseteq {\mathcal{O}}_2$ and $\top \left({\mathcal{L}}_1\right)\subseteq {\mathcal{L}}_2$, then ⊤ is called a covariant map, or a map from $({\mathcal{O}}_1,{\mathcal{L}}_1,{\mathcal{K}}_1,{{\rm Y}}_1)$ to $({\mathcal{O}}_2,{\mathcal{L}}_2,{\mathcal{K}}_2,{{\rm Y}}_2)$ and this is written as $\top :({\mathcal{O}}_1,{\mathcal{L}}_1,{\mathcal{K}}_1,{{\rm Y}}_1)\rightrightarrows ({\mathcal{O}}_2,{\mathcal{L}}_2,{\mathcal{K}}_2,{{\rm Y}}_2)$. If $\top :({\mathcal{O}}_1,{\mathcal{L}}_1,{\mathcal{K}}_1,{{\rm Y}}_1)\rightrightarrows ({\mathcal{L}}_2,{\mathcal{O}}_2,{\overline{\mathcal{K}}}_2,{{\rm Y}}_2)$ where ${\overline{\mathcal{K}}}_2:{\mathcal{L}}_2\times {\mathcal{O}}_2\to {\mathbb{R}}^{+}$, is a map, then ⊤ is called a contravariant map from $({\mathcal{O}}_1,{\mathcal{L}}_1,{\mathcal{K}}_1,{{\rm Y}}_1)$ to $({\mathcal{O}}_2,{\mathcal{L}}_2,{\mathcal{K}}_2,{{\rm Y}}_2)$ and this is denoted as $\top :({\mathcal{O}}_1,{\mathcal{L}}_1,{\mathcal{K}}_1,{{\rm Y}}_1)\rightleftarrows ({\mathcal{O}}_2,{\mathcal{L}}_2,{\mathcal{K}}_2,{{\rm Y}}_2)$.

(A3) 

If a covariant map ⊤ is right and left continuous at each $\kappa \in {\mathcal{O}}_1$ and $\mu \in {\mathcal{L}}_1$, then it is continuous.

Definition 3 implies that a covariant or a contra-variant function ⊤, which is defined from $({\mathcal{O}}_1,{\mathcal{L}}_1,{\mathcal{K}}_1,{{\rm Y}}_1)$ to $({\mathcal{O}}_2,{\mathcal{L}}_2,{\mathcal{K}}_2,{{\rm Y}}_2)$, is continuous, iff $\left\{{\kappa }_{\xi }\right\}\to u$ on $({\mathcal{O}}_1,{\mathcal{L}}_1,{\mathcal{K}}_1,{{\rm Y}}_1)$ implies $\left\{\top \left({\kappa }_{\xi }\right)\right\}\to \top u$ on $({\mathcal{O}}_2,{\mathcal{L}}_2,{\mathcal{K}}_2,{{\rm Y}}_2)$.

Definition 4.

Let $(\mathcal{O};\mathcal{L};\mathcal{K};{\rm Y})$ be a bipolar-controlled metric space.

(B1) 

A left sequence $\left\{{\kappa }_{\xi }\right\}$ converges to μ be a right point iff $\underset{\xi \to \infty }{lim}\mathcal{K}({\kappa }_{\xi },\mu )=0$.

(B2) 

A right seqence $\left\{{\mu }_{\xi }\right\}$ converges to a left point κ iff $\underset{\xi \to \infty }{lim}\mathcal{K}({\mu }_{\xi },\kappa )=0$.

(B3) 

A sequence $(\left\{{\kappa }_{\xi }\right\},\left\{{\mu }_{\xi }\right\})$ on the set $\mathcal{O}\times \mathcal{L}$ is known as a bisequence on $(\mathcal{O};\mathcal{L};\mathcal{K};{\rm Y})$.

(B4) 

If both $\left\{{\kappa }_{\xi }\right\}$ and $\left\{{\mu }_{\xi }\right\}$ converge, then the bisequence $(\left\{{\kappa }_{\xi }\right\},\left\{{\mu }_{\xi }\right\})$ is known as convergent. If $\left\{{\kappa }_{\xi }\right\}$ and $\left\{{\mu }_{\xi }\right\}$ both converge to a point $u\in \mathcal{O}\cap \mathcal{L}$, then this bi-sequence known as bi-convergent.

(B5) 

A bi-sequence $(\left\{{\kappa }_{\xi }\right\},\left\{{\mu }_{\xi }\right\})$ on $(\mathcal{O};\mathcal{L};\mathcal{K};{\rm Y})$ known as a Cauchy bi-sequence, if for every $\epsilon >0$, we can find ${\xi }_0\in \mathbb{N}$, satisfying for all non negative integers $\xi ,\tau \ge {\xi }_0,\mathcal{K}({\kappa }_{\xi },{\mu }_{\tau })<\epsilon$

(B6) 

A Cauchy bisequence is any biconvergent bisequence.

(B7) 

Every biconvergent Cauchy bisequence is convergent.

Definition 5.

A bipolar-controlled metric space is called complete, if every Cauchy bisequence in this space is convergent.

3. Main Results

In this section, we prove fixed point theorems in bipolar-controlled metric spaces.

Theorem 1.

Let $(\mathcal{O},\mathcal{L},\mathcal{K},{\rm Y})$ be a complete bipolar-controlled metric space. Consider the mapping $\top :(\mathcal{O},\mathcal{L},\mathcal{K},{\rm Y})\rightrightarrows (\mathcal{O},\mathcal{L},\mathcal{K},{\rm Y})$ such that

$$\begin{array}{c}\hfill \mathcal{K}(\top \kappa ,\top \mu )\le \rho \mathcal{K}(\kappa ,\mu ),\end{array}$$

for all $\kappa \in \mathcal{O}$ and $\mu \in \mathcal{L}$, where $0<\rho <1$. Suppose that

$$\begin{array}{c}\hfill \underset{\tau \ge 1}{sup}\underset{\varsigma \to \infty }{lim}\frac{{\rm Y}({\kappa }_{\varsigma +1},{\eta }_{\varsigma +2})}{{\rm Y}({\kappa }_{\varsigma },{\eta }_{\varsigma +1})}{\rm Y}({\kappa }_{\varsigma +1},{\eta }_{\tau })<\frac{1}{\rho }\end{array}$$

(1) 

and

$$\begin{array}{c}\hfill \underset{\tau \ge 1}{sup}\underset{\varsigma \to \infty }{lim}\frac{{\rm Y}({\kappa }_{\varsigma +1},{\eta }_{\varsigma +1})}{{\rm Y}({\kappa }_{\varsigma },{\eta }_{\varsigma })}{\rm Y}({\kappa }_{\varsigma +1},{\eta }_{\tau })<\frac{1}{\rho }.\end{array}$$

(2) 

Then the function $\top :\mathcal{O}\cup \mathcal{L}\to \mathcal{O}\cup \mathcal{L}$ has a UFP.

Proof. 

Let ${\kappa }_0$∈$\mathcal{O}$ and ${\mu }_0$∈$\mathcal{L}$. Define $\top \left({\kappa }_{\xi }\right)={\kappa }_{\xi +1}$ and $\top \left({\mu }_{\xi }\right)={\mu }_{\xi +1},\forall \xi \in \mathbb{N}$. Then $(\left\{{\kappa }_{\xi }\right\}$, $\left\{{\mu }_{\xi }\right\})$ is a bisequence on ($\mathcal{O},\mathcal{L},\mathcal{K}$) and

$$\begin{array}{cc}\hfill \mathcal{K}({\kappa }_{\xi },{\mu }_{\xi })& =\mathcal{K}(\top \left({\kappa }_{\xi -1}\right),\top \left({\mu }_{\xi -1}\right))\hfill \\ \hfill & \le \rho \mathcal{K}({\kappa }_{\xi -1},{\mu }_{\xi -1})\hfill \\ \hfill & ⋮\hfill \\ \hfill & \le {\rho }^{\xi }\mathcal{K}({\kappa }_0,{\mu }_0).\hfill \end{array}$$

Furthermore,

$$\begin{array}{cc}\hfill \mathcal{K}({\kappa }_{\xi },{\mu }_{\xi +1})& =\mathcal{K}(\top \left({\kappa }_{\xi -1}\right),\top \left({\mu }_{\xi }\right))\hfill \\ \hfill & \le \rho \mathcal{K}({\kappa }_{\xi -1},{\mu }_{\xi })\hfill \\ \hfill & ⋮\hfill \\ \hfill & \le {\rho }^{\xi }\mathcal{K}({\kappa }_0,{\mu }_1).\hfill \end{array}$$

For all natural numbers $\xi <\tau$, we have

$$\begin{array}{cc}\hfill \mathcal{K}({\kappa }_{\xi },{\mu }_{\tau })& \le {\rm Y}({\kappa }_{\xi },{\mu }_{\xi +1})\mathcal{K}({\kappa }_{\xi },{\mu }_{\xi +1})+{\rm Y}({\kappa }_{\xi +1},{\mu }_{\xi +1})\mathcal{K}({\kappa }_{\xi +1},{\mu }_{\xi +1})\hfill \\ \hfill & +{\rm Y}({\kappa }_{\xi +1},{\mu }_{\tau })\mathcal{K}({\kappa }_{\xi +1},{\mu }_{\tau })\hfill \\ \hfill & \le {\rm Y}({\kappa }_{\xi },{\mu }_{\xi +1})\mathcal{K}({\kappa }_{\xi },{\mu }_{\xi +1})+{\rm Y}({\kappa }_{\xi +1},{\mu }_{\xi +1})\mathcal{K}({\kappa }_{\xi +1},{\mu }_{\xi +1})\hfill \\ \hfill & +{\rm Y}({\kappa }_{\xi +1},{\mu }_{\tau }){\rm Y}({\kappa }_{\xi +1},{\mu }_{\xi +2})\mathcal{K}({\kappa }_{\xi +1},{\mu }_{\xi +2})\hfill \\ \hfill & +{\rm Y}({\kappa }_{\xi +1},{\mu }_{\tau }){\rm Y}({\kappa }_{\xi +2},{\mu }_{\xi +2})\mathcal{K}({\kappa }_{\xi +2},{\mu }_{\xi +2})\hfill \\ \hfill & +{\rm Y}({\kappa }_{\xi +1},{\mu }_{\tau }){\rm Y}({\kappa }_{\xi +2},{\mu }_{\tau })\mathcal{K}({\kappa }_{\xi +2},{\mu }_{\tau })\hfill \\ \hfill & ⋮\hfill \\ \hfill & \le {\rm Y}({\kappa }_{\xi },{\mu }_{\xi +1})\mathcal{K}({\kappa }_{\xi },{\mu }_{\xi +1})+{\rm Y}({\kappa }_{\xi +1},{\mu }_{\xi +1})\mathcal{K}({\kappa }_{\xi +1},{\mu }_{\xi +1})\hfill \\ \hfill & +\sum _{\varsigma =\xi +1}^{\tau -1}\prod _{\mathfrak{j}=\xi +1}^{\varsigma }{\rm Y}({\kappa }_{\mathfrak{j}},{\mu }_{\tau }){\rm Y}({\kappa }_{\varsigma },{\mu }_{\varsigma +1})\mathcal{K}({\kappa }_{\varsigma },{\mu }_{\varsigma +1})\hfill \\ \hfill & +\sum _{\varsigma =\xi +1}^{\tau -1}\prod _{\mathfrak{j}=\xi +1}^{\varsigma }{\rm Y}({\kappa }_{\mathfrak{j}},{\mu }_{\tau }){\rm Y}({\kappa }_{\varsigma },{\mu }_{\varsigma })\mathcal{K}({\kappa }_{\varsigma },{\mu }_{\varsigma })\hfill \\ \hfill & +\prod _{\mathfrak{k}=\xi +1}^{\tau }{\rm Y}({\kappa }_{\mathfrak{k}},{\mu }_{\tau })\mathcal{K}({\kappa }_{\tau },{\mu }_{\tau })\hfill \\ \hfill & \le {\rm Y}({\kappa }_{\xi },{\mu }_{\xi +1}){\rho }^{\xi }\mathcal{K}({\kappa }_0,{\mu }_1)+{\rm Y}({\kappa }_{\xi +1},{\mu }_{\xi +1}){\rho }^{\xi +1}\mathcal{K}({\kappa }_0,{\mu }_0)\hfill \\ \hfill & +\sum _{\varsigma =\xi +1}^{\tau -1}\prod _{\mathfrak{j}=\xi +1}^{\varsigma }{\rm Y}({\kappa }_{\mathfrak{j}},{\mu }_{\tau }){\rm Y}({\kappa }_{\varsigma },{\mu }_{\varsigma +1}){\rho }^{\varsigma }\mathcal{K}({\kappa }_0,{\mu }_1)\hfill \\ \hfill & +\sum _{\varsigma =\xi +1}^{\tau -1}\prod _{\mathfrak{j}=\xi +1}^{\varsigma }{\rm Y}({\kappa }_{\mathfrak{j}},{\mu }_{\tau }){\rm Y}({\kappa }_{\varsigma },{\mu }_{\varsigma }){\rho }^{\varsigma }\mathcal{K}({\kappa }_0,{\mu }_0)\hfill \\ \hfill & +\prod _{\mathfrak{k}=\xi +1}^{\tau }{\rm Y}({\kappa }_{\mathfrak{k}},{\mu }_{\tau }){\rho }^{\tau }\mathcal{K}({\kappa }_0,{\mu }_1)\hfill \\ \hfill & ={\rm Y}({\kappa }_{\xi },{\mu }_{\xi +1}){\rho }^{\xi }\mathcal{K}({\kappa }_0,{\mu }_1)+{\rm Y}({\kappa }_{\xi +1},{\mu }_{\xi +1}){\rho }^{\xi +1}\mathcal{K}({\kappa }_0,{\mu }_0)\hfill \\ \hfill & +\sum _{\varsigma =\xi +1}^{\tau }\prod _{\mathfrak{j}=\xi +1}^{\varsigma }{\rm Y}({\kappa }_{\mathfrak{j}},{\mu }_{\tau }){\rm Y}({\kappa }_{\varsigma },{\mu }_{\varsigma +1}){\rho }^{\varsigma }\mathcal{K}({\kappa }_0,{\mu }_1)\hfill \\ \hfill & +\sum _{\varsigma =\xi +1}^{\tau -1}\prod _{\mathfrak{j}=\xi +1}^{\varsigma }{\rm Y}({\kappa }_{\mathfrak{j}},{\mu }_{\tau }){\rm Y}({\kappa }_{\varsigma },{\mu }_{\varsigma }){\rho }^{\varsigma }\mathcal{K}({\kappa }_0,{\mu }_0)\hfill \\ \hfill & \le {\rm Y}({\kappa }_{\xi },{\mu }_{\xi +1}){\rho }^{\xi }\mathcal{K}({\kappa }_0,{\mu }_1)+{\rm Y}({\kappa }_{\xi +1},{\mu }_{\xi +1}){\rho }^{\xi +1}\mathcal{K}({\kappa }_0,{\mu }_0)\hfill \\ \hfill & +\sum _{\varsigma =\xi +1}^{\tau }\prod _{\mathfrak{j}=0}^{\varsigma }{\rm Y}({\kappa }_{\mathfrak{j}},{\mu }_{\tau }){\rm Y}({\kappa }_{\varsigma },{\mu }_{\varsigma +1}){\rho }^{\varsigma }\mathcal{K}({\kappa }_0,{\mu }_1)\hfill \\ \hfill & +\sum _{\varsigma =\xi +1}^{\tau -1}\prod _{\mathfrak{j}=0}^{\varsigma }{\rm Y}({\kappa }_{\mathfrak{j}},{\mu }_{\tau }){\rm Y}({\kappa }_{\varsigma },{\mu }_{\varsigma }){\rho }^{\varsigma }\mathcal{K}({\kappa }_0,{\mu }_0).\hfill \end{array}$$

Let

$$\begin{array}{c}\hfill {\mathcal{G}}_{\mathfrak{e}}=\sum _{\varsigma =0}^{\mathfrak{e}}\prod _{\mathfrak{j}=0}^{\varsigma }{\rm Y}({\kappa }_{\mathfrak{j}},{\mu }_{\tau }){\rm Y}({\kappa }_{\varsigma },{\mu }_{\varsigma +1}){\rho }^{\varsigma }\end{array}$$

and

$$\begin{array}{c}\hfill {\mathcal{Q}}_{\mathfrak{e}}=\sum _{\varsigma =0}^{\mathfrak{e}}\prod _{\mathfrak{j}=0}^{\varsigma }{\rm Y}({\kappa }_{\mathfrak{j}},{\mu }_{\tau }){\rm Y}({\kappa }_{\varsigma },{\mu }_{\varsigma }){\rho }^{\varsigma }.\end{array}$$

Hence, we have

$$\begin{array}{cc}\hfill \mathcal{K}({\kappa }_{\xi },{\mu }_{\tau })& \le \mathcal{K}({\kappa }_0,{\mu }_1)[{\rm Y}({\kappa }_{\xi },{\mu }_{\xi +1}){\rho }^{\xi }+({\mathcal{G}}_{\tau }-{\mathcal{G}}_{\xi })]\hfill \\ \hfill & +\mathcal{K}({\kappa }_0,{\mu }_0)[{\rm Y}({\kappa }_{\xi +1},{\mu }_{\xi +1}){\rho }^{\xi +1}+({\mathcal{Q}}_{\tau -1}-{\mathcal{Q}}_{\xi })].\hfill \end{array}$$

(3)

Now by conditions (1), (2) and the ratio test, we obtain that $\underset{\xi \to \infty }{lim}{\mathcal{G}}_{\xi }$ and $\underset{\xi \to \infty }{lim}{\mathcal{Q}}_{\xi }$ exists. Therefore $\left\{{\mathcal{G}}_{\xi }\right\}$ and $\left\{{\mathcal{Q}}_{\xi }\right\}$ are Cauchy sequences. As $\xi ,\tau \to \infty$, we get

$$\begin{array}{c}\hfill \underset{\xi ,\tau \to \infty }{lim}\mathcal{K}({\kappa }_{\xi },{\mu }_{\tau })=0.\end{array}$$

(4)

Similarly, we can derive

$$\begin{array}{c}\hfill \underset{\xi ,\tau \to \infty }{lim}\mathcal{K}({\kappa }_{\tau },{\mu }_{\xi })=0.\end{array}$$

(5)

Hence $(\left\{{\kappa }_{\xi }\right\},\left\{{\mu }_{\xi }\right\})$ is a Cauchy bisequence. By completeness property, $\left\{{\kappa }_{\xi }\right\}\to \mathfrak{e},\left\{{\mu }_{\xi }\right\}\to \mathfrak{e}$, where $\mathfrak{e}\in \mathcal{O}\cap \mathcal{L}$ and

$$\begin{array}{c}\hfill \left\{\top \left({\mu }_{\xi }\right)\right\}=\left\{{\mu }_{\xi +1}\right\}\to \mathfrak{e}\in \mathcal{O}\cap \mathcal{L}\end{array}$$

Since ⊤ is continuous $\top \left({\mu }_{\xi }\right)\to \top \left(\mathfrak{e}\right)$, so $\top \left(\mathfrak{e}\right)=\mathfrak{e}$. Hence ⊤ has a fixed point $\mathfrak{e}$. If $\mathfrak{d}$ is another fixed point of ⊤, then $\top \left(\mathfrak{d}\right)=\mathfrak{d}$ implies that $\mathfrak{d}\in \mathcal{O}\cap \mathcal{L}$ and we have

$$\begin{array}{c}\hfill \mathcal{K}(\mathfrak{e},\mathfrak{d})=\mathcal{K}\left(\top \right(\mathfrak{e}),\top (\mathfrak{d})\le \rho \mathcal{K}(\mathfrak{e},\mathfrak{d})\end{array}$$

where $0<\rho <1$, which implies $\mathcal{K}(\mathfrak{e},\mathfrak{d})=0$, and so $\mathfrak{e}=\mathfrak{d}$. □

Example 2.

Let $\mathcal{O}=[0,1]$ and $\mathcal{L}=[1,2]$ be equipped with $\mathcal{K}(\kappa ,\mu )={|\kappa -\mu |}^2$ and ${\rm Y}(\kappa ,\mu )=\kappa \mu +1$ for all $\kappa \in \mathcal{O}$ and $\mu \in \mathcal{L}$. Then, $(\mathcal{O},\mathcal{L},\mathcal{K})$ is a complete bipolar-controlled metric space. Define $\top :\mathcal{O}\cup \mathcal{L}\rightrightarrows \mathcal{O}\cup \mathcal{L}$ by

$$\top \left(\kappa \right)=\left\{\begin{array}{cc}\frac{\kappa }{4},\hfill & if\kappa \in (0,1),\hfill \\ 1,\hfill & if\kappa \in [1,2],\hfill \end{array}\right.$$

∀$\kappa \in \mathcal{O}\cup \mathcal{L}$. Let $\rho =\frac{1}{2}\in (0,1)$ and ${\kappa }_0=\frac{1}{2}\in \mathcal{O}$, ${\mu }_0=2\in \mathcal{L}$, then ${\kappa }_{\xi }=\top \left({\kappa }_{\xi -1}\right)=\frac{1}{4\xi }$ for all $\xi \in \mathbb{N}$ and ${\mu }_{\xi }=\top \left({\mu }_{\xi -1}\right)=1$ for all $\xi \in \mathcal{L}$. We can easily get

$$\begin{array}{c}\hfill \underset{\tau \ge 1}{sup}\underset{\xi \to \infty }{lim}\frac{{\rm Y}({\kappa }_{\xi +1},{\mu }_{\xi +2})}{{\rm Y}({\kappa }_{\xi },{\mu }_{\xi +1})}{\rm Y}({\kappa }_{\xi +1},{\mu }_{\tau })<\frac{1}{\rho }\end{array}$$

and

$$\begin{array}{c}\hfill \underset{\tau \ge 1}{sup}\underset{\xi \to \infty }{lim}\frac{{\rm Y}({\kappa }_{\xi +1},{\mu }_{\xi +1})}{{\rm Y}({\kappa }_{\xi },{\mu }_{\xi })}{\rm Y}({\kappa }_{\xi +1},{\mu }_{\tau })<\frac{1}{\rho }.\end{array}$$

Let $\kappa \in \mathcal{O}$ and $\mu \in \mathcal{L}$, then

$$\begin{array}{cc}\hfill \mathcal{K}(\top \kappa ,\top \mu )& =|\frac{\kappa }{4}{-1|}^2\hfill \\ \hfill & \le \frac{1}{2}{|\kappa -\mu |}^2.\hfill \end{array}$$

Thus, all the criteria of Theorem 1 are satisfied and ⊤ has a UFP $\kappa =1$.

Example 3.

Let $\mathcal{O}={\mathcal{U}}_{\xi }\left(\mathbb{R}\right)$ and $\mathcal{L}={\mathcal{L}}_{\xi }\left(\mathbb{R}\right)$ be the set of all upper and lower triangular matrices of order ξ, respectively. Let $\mathcal{K}:\mathcal{U}\left(\mathbb{R}\right)\times \mathcal{L}\left(\mathbb{R}\right)\to [0,\infty )$ be defined by

$$\begin{array}{c}\hfill \mathcal{K}(\Sigma ,\mathrm{\Gamma })=\sum _{\mathfrak{r},\mathfrak{t}=1}^{\xi }{|{\kappa }_{\mathfrak{r}\mathfrak{t}}-{\mu }_{\mathfrak{r}\mathfrak{t}}|}^2\end{array}$$

(6)

for all $\Sigma ={\left({\kappa }_{\mathfrak{r}\mathfrak{t}}\right)}_{\xi \times \xi }\in \mathcal{O}$ and $\mathrm{\Gamma }={\left({\mu }_{\mathfrak{r}\mathfrak{t}}\right)}_{\xi \times \xi }\in \mathcal{L}$ and ${\rm Y}(\kappa ,\mu )=\kappa \mu +1$ for all $\kappa \in \mathcal{O}$ and $\mu \in \mathcal{L}$. Then, $(\mathcal{O},\mathcal{L},\mathcal{K})$ is a complete bipolar-controlled metric space. Define $\top :\mathcal{O}\cup \mathcal{L}\rightrightarrows \mathcal{O}\cup \mathcal{L}$ by

$$\top {\left({\kappa }_{\mathfrak{r}\mathfrak{t}}\right)}_{\xi \times \xi }={\left(\frac{{\kappa }_{\mathfrak{r}\mathfrak{t}}}{4}\right)}_{\xi \times \xi }$$

∀${\left({\kappa }_{\mathfrak{r}\mathfrak{t}}\right)}_{\xi \times \xi }\in \mathcal{O}\cup \mathcal{L}$. Let $\rho =\frac{1}{2}\in (0,1)$ and ${\kappa }_0={\left(\frac{{\kappa }_{\mathfrak{r}\mathfrak{t}}}{2}\right)}_{\xi \times \xi }\in \mathcal{O}\cup \mathcal{L}$, then ${\left({\kappa }_{\mathfrak{r}\mathfrak{t}}\right)}_{\xi \times \xi }=\top {\left({\kappa }_{\mathfrak{r}\mathfrak{t}-1}\right)}_{\xi \times \xi }={\left(\frac{{\kappa }_{\mathfrak{r}\mathfrak{t}}}{2\left(4^{\xi }\right)}\right)}_{\xi \times \xi }$ for all $\xi \in \mathbb{N}$. We can easily get

$$\begin{array}{c}\hfill \underset{\tau \ge 1}{sup}\underset{\xi \to \infty }{lim}\frac{{\rm Y}({\kappa }_{\xi +1},{\mu }_{\xi +2})}{{\rm Y}({\kappa }_{\xi },{\mu }_{\xi +1})}{\rm Y}({\kappa }_{\xi +1},{\mu }_{\tau })<\frac{1}{\rho }\end{array}$$

and

$$\begin{array}{c}\hfill \underset{\tau \ge 1}{sup}\underset{\xi \to \infty }{lim}\frac{{\rm Y}({\kappa }_{\xi +1},{\mu }_{\xi +1})}{{\rm Y}({\kappa }_{\xi },{\mu }_{\xi })}{\rm Y}({\kappa }_{\xi +1},{\mu }_{\tau })<\frac{1}{\rho }.\end{array}$$

Let $\Sigma \in \mathcal{O}$ and $\mathrm{\Gamma }\in \mathcal{L}$, then

$$\begin{array}{cc}\hfill \mathcal{K}\left(\top \right(\Sigma ),\top (\mathrm{\Gamma }\left)\right)& =\frac{1}{4}\sum _{\mathfrak{r},\mathfrak{t}=1}^{\xi }{|{\kappa }_{\mathfrak{r}\mathfrak{t}}-{\mu }_{\mathfrak{r}\mathfrak{t}}|}^2\hfill \\ \hfill & \le \frac{1}{2}\sum _{\mathfrak{r},\mathfrak{t}=1}^{\xi }{|{\kappa }_{\mathfrak{r}\mathfrak{t}}-{\mu }_{\mathfrak{r}\mathfrak{t}}|}^2\hfill \\ \hfill & =\frac{1}{2}\mathcal{K}(\Sigma ,\mathrm{\Gamma }).\hfill \end{array}$$

Hence Theorem 1 is satisfied and ⊤ has a UFP $O_{\xi \times \xi }$.

Theorem 2.

Let $(\mathcal{O},\mathcal{L},\mathcal{K},{\rm Y})$ be a complete bipolar-controlled metric space. Consider the mapping $\top :(\mathcal{O},\mathcal{L},\mathcal{K},{\rm Y})\leftrightarrows (\mathcal{O},\mathcal{L},\mathcal{K},{\rm Y})$ such that

$$\begin{array}{c}\hfill \mathcal{K}(\top \kappa ,\top \mu )\le \rho \mathcal{K}(\mu ,\kappa ),\end{array}$$

for all $\kappa \in \mathcal{O}$ and $\mu \in \mathcal{L}$, where $0<\rho <1$. Suppose that

$$\begin{array}{c}\hfill \underset{\tau \ge 1}{sup}\underset{\varsigma \to \infty }{lim}\frac{{\rm Y}({\kappa }_{\varsigma +2},{\eta }_{\varsigma +2})}{{\rm Y}({\kappa }_{\varsigma +1},{\eta }_{\varsigma +1})}{\rm Y}({\kappa }_{\tau },{\eta }_{\varsigma +1})<\frac{1}{{\rho }^2}\end{array}$$

(7)

and

$$\begin{array}{c}\hfill \underset{\tau \ge 1}{sup}\underset{\varsigma \to \infty }{lim}\frac{{\rm Y}({\kappa }_{\varsigma +2},{\eta }_{\varsigma +1})}{{\rm Y}({\kappa }_{\varsigma +1},{\eta }_{\varsigma })}{\rm Y}({\kappa }_{\tau },{\eta }_{\varsigma +1})<\frac{1}{{\rho }^2}.\end{array}$$

(8)

Then the function $\top :\mathcal{O}\cup \mathcal{L}\to \mathcal{O}\cup \mathcal{L}$ has a UFP.

Proof. 

Let ${\kappa }_0\in \mathcal{O}$. Define $\top \left({\kappa }_{\xi }\right)={\mu }_{\xi }$ and $\top \left({\mu }_{\xi }\right)={\kappa }_{\xi +1},\forall \xi \in \mathbb{N}$. Then $(\left\{{\kappa }_{\xi }\right\},\left\{{\mu }_{\xi }\right\})$ are bisequence on $(\mathcal{O},\mathcal{L},\mathcal{K},{\rm Y})$. Now,

$$\begin{array}{cc}\hfill \mathcal{K}({\kappa }_{\xi },{\mu }_{\xi })& =\mathcal{K}(\top \left({\mu }_{\xi -1}\right),\top \left({\kappa }_{\xi }\right))\hfill \\ \hfill & \le \mathcal{K}({\kappa }_{\xi },{\mu }_{\xi -1})\hfill \\ \hfill & =\rho .\mathcal{K}(\top \left({\mu }_{\xi -1}\right),\top \left({\kappa }_{\xi -1}\right))\hfill \\ \hfill & \le {\rho }^2\mathcal{K}({\kappa }_{\xi -1},{\mu }_{\xi -1})\hfill \\ \hfill & ⋮\hfill \\ \hfill & \le {\rho }^{2\xi }\mathcal{K}({\kappa }_0,{\mu }_0)\hfill \\ \hfill \mathcal{K}({\kappa }_{\xi +1},{\mu }_{\xi })& =\mathcal{K}(\top \left({\mu }_{\xi }\right),\top \left({\kappa }_{\xi }\right))\hfill \\ \hfill & \le \rho \mathcal{K}({\kappa }_{\xi },{\mu }_{\xi })\hfill \\ \hfill & \le {\rho }^{2\xi +1}\mathcal{K}({\kappa }_0,{\mu }_0).\hfill \end{array}$$

For all natural numbers $\xi <\tau$, we have

$$\begin{array}{cc}\hfill \mathcal{K}({\kappa }_{\tau },{\mu }_{\xi })& \le {\rm Y}({\kappa }_{\tau },{\mu }_{\xi +1})\mathcal{K}({\kappa }_{\tau },{\mu }_{\xi +1})+{\rm Y}({\kappa }_{\xi +1},{\mu }_{\xi +1})\mathcal{K}({\kappa }_{\xi +1},{\mu }_{\xi +1})\hfill \\ \hfill & +{\rm Y}({\kappa }_{\xi +1},{\mu }_{\xi })\mathcal{K}({\kappa }_{\xi +1},{\mu }_{\xi })\hfill \\ \hfill & \le {\rm Y}({\kappa }_{\tau },{\mu }_{\xi +1}){\rm Y}({\kappa }_{\tau },{\mu }_{\xi +2})\mathcal{K}({\kappa }_{\tau },{\mu }_{\xi +2})\hfill \\ & +{\rm Y}({\kappa }_{\tau },{\mu }_{\xi +1}){\rm Y}({\kappa }_{\xi +2},{\mu }_{\xi +2})\mathcal{K}({\kappa }_{\xi +2},{\mu }_{\xi +2})\hfill \\ \hfill & +{\rm Y}({\kappa }_{\tau },{\mu }_{\xi +1}){\rm Y}({\kappa }_{\xi +2},{\mu }_{\xi +1})\mathcal{K}({\kappa }_{\xi +2},{\mu }_{\xi +1})\hfill \\ \hfill & +{\rm Y}({\kappa }_{\xi +1},{\mu }_{\xi +1})\mathcal{K}({\kappa }_{\xi +1},{\mu }_{\xi +1})+{\rm Y}({\kappa }_{\xi +1},{\mu }_{\xi })\mathcal{K}({\kappa }_{\xi +1},{\mu }_{\xi })\hfill \\ \hfill & ⋮\hfill \\ \hfill & \le \prod _{\mathfrak{k}=\xi +1}^{\tau }{\rm Y}({\kappa }_{\tau },{\mu }_{\mathfrak{k}})\mathcal{K}({\kappa }_{\tau },{\mu }_{\tau })\hfill \\ \hfill & +\sum _{\varsigma =\xi +1}^{\tau -1}\prod _{\mathfrak{j}=\xi +1}^{\varsigma }{\rm Y}({\kappa }_{\tau },{\mu }_{\mathfrak{j}}){\rm Y}({\kappa }_{\varsigma },{\mu }_{\varsigma })\mathcal{K}({\kappa }_{\varsigma },{\mu }_{\varsigma })\hfill \\ \hfill & +\sum _{\varsigma =\xi +1}^{\tau -1}\prod _{\mathfrak{j}=\xi +1}^{\varsigma }{\rm Y}({\kappa }_{\tau },{\mu }_{\mathfrak{j}}){\rm Y}({\kappa }_{\varsigma +1},{\mu }_{\varsigma })\mathcal{K}({\kappa }_{\varsigma +1},{\mu }_{\varsigma })\hfill \\ \hfill & +{\rm Y}({\kappa }_{\xi +1},{\mu }_{\xi +1})\mathcal{K}({\kappa }_{\xi +1},{\mu }_{\xi +1})+{\rm Y}({\kappa }_{\xi +1},{\mu }_{\xi })\mathcal{K}({\kappa }_{\xi +1},{\mu }_{\xi })\hfill \\ \hfill & \le \prod _{\mathfrak{k}=\xi +1}^{\tau }{\rm Y}({\kappa }_{\tau },{\mu }_{\mathfrak{k}}){\rho }^{2\tau }\mathcal{K}({\kappa }_0,{\mu }_0)\hfill \\ \hfill & +\sum _{\varsigma =\xi +1}^{\tau -1}\prod _{\mathfrak{j}=\xi +1}^{\varsigma }{\rm Y}({\kappa }_{\tau },{\mu }_{\mathfrak{j}}){\rm Y}({\kappa }_{\varsigma },{\mu }_{\varsigma }){\rho }^{2\varsigma }\mathcal{K}({\kappa }_0,{\mu }_0)\hfill \\ \hfill & +\sum _{\varsigma =\xi +1}^{\tau -1}\prod _{\mathfrak{j}=\xi +1}^{\varsigma }{\rm Y}({\kappa }_{\tau },{\mu }_{\mathfrak{j}}){\rm Y}({\kappa }_{\varsigma +1},{\mu }_{\varsigma }){\rho }^{2\varsigma +1}\mathcal{K}({\kappa }_0,{\mu }_0)\hfill \\ \hfill & +{\rm Y}({\kappa }_{\xi +1},{\mu }_{\xi +1}){\rho }^{2\xi }\mathcal{K}({\kappa }_0,{\mu }_0)+{\rm Y}({\kappa }_{\xi +1},{\mu }_{\xi }){\rho }^{2\xi +1}\mathcal{K}({\kappa }_0,{\mu }_0)\hfill \\ \hfill & =\sum _{\varsigma =\xi +1}^{\tau }\prod _{\mathfrak{j}=\xi +1}^{\varsigma }{\rm Y}({\kappa }_{\tau },{\mu }_{\mathfrak{j}}){\rm Y}({\kappa }_{\varsigma },{\mu }_{\varsigma }){\rho }^{2\varsigma }\mathcal{K}({\kappa }_0,{\mu }_0)\hfill \\ \hfill & +\sum _{\varsigma =\xi +1}^{\tau -1}\prod _{\mathfrak{j}=\xi +1}^{\varsigma }{\rm Y}({\kappa }_{\tau },{\mu }_{\mathfrak{j}}){\rm Y}({\kappa }_{\varsigma +1},{\mu }_{\varsigma }){\rho }^{2\varsigma +1}\mathcal{K}({\kappa }_0,{\mu }_0)\hfill \\ \hfill & +{\rm Y}({\kappa }_{\xi +1},{\mu }_{\xi +1}){\rho }^{2\xi }\mathcal{K}({\kappa }_0,{\mu }_0)+{\rm Y}({\kappa }_{\xi +1},{\mu }_{\xi }){\rho }^{2\xi +1}\mathcal{K}({\kappa }_0,{\mu }_0)\hfill \\ \hfill & \le \sum _{\varsigma =\xi +1}^{\tau }\prod _{\mathfrak{j}=0}^{\varsigma }{\rm Y}({\kappa }_{\tau },{\mu }_{\mathfrak{j}}){\rm Y}({\kappa }_{\varsigma +1},{\mu }_{\varsigma +1}){\rho }^{2\varsigma }\mathcal{K}({\kappa }_0,{\mu }_0)\hfill \\ \hfill & +\sum _{\varsigma =\xi +1}^{\tau -1}\prod _{\mathfrak{j}=0}^{\varsigma }{\rm Y}({\kappa }_{\tau },{\mu }_{\mathfrak{j}}){\rm Y}({\kappa }_{\varsigma +1},{\mu }_{\varsigma }){\rho }^{2\varsigma +1}\mathcal{K}({\kappa }_0,{\mu }_0)\hfill \\ \hfill & +{\rm Y}({\kappa }_{\xi +1},{\mu }_{\xi +1}){\rho }^{2\xi }\mathcal{K}({\kappa }_0,{\mu }_0)+{\rm Y}({\kappa }_{\xi +1},{\mu }_{\xi }){\rho }^{2\xi +1}\mathcal{K}({\kappa }_0,{\mu }_0)\hfill \end{array}$$

Let

$$\begin{array}{c}\hfill {\mathcal{O}}_{\mathfrak{e}}=\sum _{\varsigma =0}^{\mathfrak{e}}\prod _{\mathfrak{j}=0}^{\varsigma }{\rm Y}({\kappa }_{\tau },{\mu }_{\mathfrak{j}}){\rm Y}({\kappa }_{\varsigma +1},{\mu }_{\varsigma +1}){\rho }^{2\varsigma }\end{array}$$

and

$$\begin{array}{c}\hfill {\mathcal{R}}_{\mathfrak{e}}=\sum _{\varsigma =0}^{\mathfrak{e}}\prod _{\mathfrak{j}=0}^{\varsigma }{\rm Y}({\kappa }_{\tau },{\mu }_{\mathfrak{j}}){\rm Y}({\kappa }_{\varsigma +1},{\mu }_{\varsigma }){\rho }^{2\varsigma +1}.\end{array}$$

Hence, we have

$$\begin{array}{cc}\hfill \mathcal{K}({\kappa }_{\tau },{\mu }_{\xi })& \le \mathcal{K}({\kappa }_0,{\mu }_0)[({\mathcal{O}}_{\tau }-{\mathcal{O}}_{\xi })+({\mathcal{R}}_{\tau -1}-{\mathcal{R}}_{\xi })\hfill \\ \hfill & +{\rm Y}({\kappa }_{\xi +1},{\mu }_{\xi +1}){\rho }^{2\xi }+{\rm Y}({\kappa }_{\xi +1},{\mu }_{\xi }){\rho }^{2\xi +1}]\hfill \end{array}$$

Now by conditions (7), (8) and ratio test, we obtain that $\underset{\xi \to \infty }{lim}{\mathcal{O}}_{\xi }$ and $\underset{\xi \to \infty }{lim}{\mathcal{R}}_{\xi }$ exists. Therefore $\left\{{\mathcal{G}}_{\xi }\right\}$ and $\left\{{\mathcal{Q}}_{\xi }\right\}$ are Cauchy sequences. As $\xi ,\tau \to \infty$, we get

$$\begin{array}{c}\hfill \underset{\xi ,\tau \to \infty }{lim}\mathcal{K}({\kappa }_{\tau },{\mu }_{\xi })=0.\end{array}$$

(9)

Similarly, we can derive

$$\begin{array}{c}\hfill \underset{\xi ,\tau \to \infty }{lim}\mathcal{K}({\kappa }_{\xi },{\mu }_{\tau })=0.\end{array}$$

(10)

Therefore $(\left\{{\kappa }_{\xi }\right\},\left\{{\mu }_{\xi }\right\})$ is a Cauchy bisequence. By completeness property, $\left\{{\kappa }_{\xi }\right\}\to \mathfrak{e},\left\{{\mu }_{\xi }\right\}\to \mathfrak{e}$, where $\mathfrak{e}\in \mathcal{O}\cap \mathcal{L}$. Since the contravariant function ⊤ is continuous

$$\begin{array}{c}\hfill \left\{{\kappa }_{\xi }\right\}\to \mathfrak{e},\end{array}$$

which implies that

$$\begin{array}{c}\hfill \left\{{\mu }_{\xi }\right\}=\left\{\top {\kappa }_{\xi }\right)\}\to \top \left(\mathfrak{e}\right)\end{array}$$

and combining this with $\left\{{\mu }_{\xi }\right\}\to \mathfrak{e}$ gives $\top \left(\mathfrak{e}\right)=\mathfrak{e}$. If also $\mathfrak{d}$ is a fixed point of ⊤, then $\top \left(\mathfrak{d}\right)=\mathfrak{d}\Rightarrow \mathfrak{d}\in \mathcal{O}\cap \mathcal{L}$ so that

$$\begin{array}{cc}\hfill \mathcal{K}(\mathfrak{e},\mathfrak{d})& =\mathcal{K}\left(\top \right(\mathfrak{e}),\top (\mathfrak{d}\left)\right)\hfill \\ \hfill & \le \rho \mathcal{K}(\mathfrak{e},\mathfrak{d}),\hfill \end{array}$$

which gives $\mathcal{K}(\mathfrak{e},\mathfrak{d})=0$. Hence $\mathfrak{e}=\mathfrak{d}$. □

Example 4.

Let $\mathcal{O}=\{0,1,2,7\}$ and $\mathcal{L}=\{0,\frac{1}{4},\frac{1}{2},3\}$ be equipped with $\mathcal{K}(\kappa ,\mu )={|\kappa -\mu |}^2$ and ${\rm Y}(\kappa ,\mu )=\kappa \mu +1$ for all $\kappa \in \mathcal{O}$ and $\mu \in \mathcal{L}$. Then, $(\mathcal{O},\mathcal{L},\mathcal{K},{\rm Y})$ is a complete bipolar-controlled metric space. Define $\top :\mathcal{O}\cup \mathcal{L}\rightleftarrows \mathcal{O}\cup \mathcal{L}$ by

$$\top \left(\kappa \right)=\left\{\begin{array}{cc}\frac{1}{2},\hfill & if\kappa \in \{2,7\},\hfill \\ 1,\hfill & if\kappa \in \{0,\frac{1}{4},\frac{1}{2},1,3\},\hfill \end{array}\right.$$

∀$\kappa \in \mathcal{O}\cup \mathcal{L}$. Let $\rho =\frac{1}{2}\in (0,1)$, ${\kappa }_0=7$ and assume $\top \left({\kappa }_{\xi }\right)={\mu }_{\xi }$ and $\top \left({\mu }_{\xi }\right)={\kappa }_{\xi +1}$ for all $\xi \in \mathbb{N}\cup \left\{0\right\}$, then ${\kappa }_{\xi }=\{7,1,1,....\}$ and ${\mu }_{\xi }=\{\frac{1}{2},1,1,....\}$ for all $\xi \in \mathbb{N}\cup \left\{0\right\}$. We can easily get

$$\begin{array}{c}\hfill \underset{\tau \ge 1}{sup}\underset{\xi \to \infty }{lim}\frac{{\rm Y}({\kappa }_{\xi +1},{\mu }_{\xi +1})}{{\rm Y}({\kappa }_{\xi },{\mu }_{\xi })}{\rm Y}({\kappa }_{\xi +1},{\mu }_{\tau })<\frac{1}{{\rho }^2}\end{array}$$

(11)

and

$$\begin{array}{c}\hfill \underset{\tau \ge 1}{sup}\underset{\xi \to \infty }{lim}\frac{{\rm Y}({\kappa }_{\xi +2},{\mu }_{\xi +1})}{{\rm Y}({\kappa }_{\xi +1},{\mu }_{\xi })}{\rm Y}({\kappa }_{\xi +1},{\mu }_{\tau })<\frac{1}{{\rho }^2}.\end{array}$$

(12)

Let $\kappa \in \mathcal{O}$ and $\mu \in \mathcal{L}$, then we can easily get

$$\begin{array}{c}\hfill \mathcal{K}(\top \kappa ,\top \mu )\le \frac{1}{2}\mathcal{K}(\kappa ,\mu ).\end{array}$$

Thus, the conditions of Theorem 2 are satisfied and ⊤ has a UFP $\kappa =1$.

Now we present a theorem based on Kannan’s fixed point theorem [16].

Theorem 3.

Let $(\mathcal{O},\mathcal{L},\mathcal{K},{\rm Y})$ be a complete bipolar-controlled metric space. Consider the mapping $\top :(\mathcal{O},\mathcal{L},\mathcal{K},{\rm Y})\leftrightarrows (\mathcal{O},\mathcal{L},\mathcal{K},{\rm Y})$ such that

$$\begin{array}{c}\hfill \mathcal{K}(\top \mu ,\top \kappa )\le \nu \left(\mathcal{K}\right(\kappa ,\top \kappa )+\mathcal{K}(\top \mu ,\mu \left)\right)\end{array}$$

for all $\kappa \in \mathcal{O}$ and $\mu \in \mathcal{L}$, where $0<\nu <\frac{1}{2}$. Suppose that

$$\begin{array}{c}\hfill \underset{\tau \ge 1}{sup}\underset{\varsigma \to \infty }{lim}\frac{{\rm Y}({\kappa }_{\varsigma +1},{\eta }_{\varsigma +1})}{{\rm Y}({\kappa }_{\varsigma },{\eta }_{\varsigma })}{\rm Y}({\kappa }_{\varsigma +1},{\eta }_{\tau })<\frac{1}{{\rho }^2}\end{array}$$

(13)

and

$$\begin{array}{c}\hfill \underset{\tau \ge 1}{sup}\underset{\varsigma \to \infty }{lim}\frac{{\rm Y}({\kappa }_{\varsigma +2},{\eta }_{\varsigma +1})}{{\rm Y}({\kappa }_{\varsigma +1},{\eta }_{\varsigma })}{\rm Y}({\kappa }_{\varsigma +1},{\eta }_{\tau })<\frac{1}{{\rho }^2},\end{array}$$

(14)

where $\rho :=\frac{\nu }{1-\nu }$. Then the function $\top :\mathcal{O}\cup \mathcal{L}\to \mathcal{O}\cup \mathcal{L}$ has a UFP.

Proof. 

Let ${\kappa }_0\in \mathcal{O}$. Define ${\mu }_{\xi }=\top {\kappa }_{\xi }$ and ${\kappa }_{\xi +1}=\top {\mu }_{\xi }$ for all $\xi \in \mathbb{N}$. Then, we have

$$\begin{array}{cc}\hfill \mathcal{K}({\kappa }_{\xi },{\mu }_{\xi })& =\mathcal{K}(\top {\mu }_{\xi -1},\top {\kappa }_{\xi })\hfill \\ \hfill & \le \nu (\mathcal{K}({\kappa }_{\xi },\top {\kappa }_{\xi })+\mathcal{K}(\top {\mu }_{\xi -1},{\mu }_{\xi -1}))\hfill \\ \hfill & =\nu (\mathcal{K}({\kappa }_{\xi },{\mu }_{\xi })+\mathcal{K}({\kappa }_{\xi },{\mu }_{\xi -1}))\hfill \end{array}$$

for all integers $\xi \ge 1$. Now,

$$\begin{array}{c}\hfill \mathcal{K}({\kappa }_{\xi },{\mu }_{\xi })\le \frac{\nu }{1-\nu }\mathcal{K}({\kappa }_{\xi },{\mu }_{\xi -1}),\end{array}$$

and

$$\begin{array}{cc}\hfill \mathcal{K}({\kappa }_{\xi },{\mu }_{\xi -1})& =\mathcal{K}(\top {\mu }_{\xi -1},\top {\kappa }_{\xi -1})\hfill \\ \hfill & \le \nu (\mathcal{K}({\kappa }_{\xi -1},\top {\kappa }_{\xi -1})+\mathcal{K}(\top {\mu }_{\xi -1},{\mu }_{\xi -1}))\hfill \\ \hfill & =\nu \left(\mathcal{K}({\kappa }_{\xi -1},{\mu }_{\xi -1})\right)+\mathcal{K}({\kappa }_{\xi },{\mu }_{\xi -1})),\hfill \end{array}$$

so that

$$\begin{array}{c}\hfill \mathcal{K}({\kappa }_{\xi },{\mu }_{\xi -1})\le \frac{\nu }{1-\nu }\mathcal{K}({\kappa }_{\xi -1},{\mu }_{\xi -1}).\end{array}$$

If we say $\rho :=\frac{\nu }{1-\nu }$, then we have $\rho \in (0,1)$ since $\nu \in (0,\frac{1}{2})$. Now

$$\begin{array}{cc}\hfill \mathcal{K}({\kappa }_{\xi },{\mu }_{\xi })& \le {\rho }^{2\xi }\mathcal{K}({\kappa }_0,{\mu }_0),\hfill \\ \hfill \mathcal{K}({\kappa }_{\xi },{\mu }_{\xi -1})& \le {\rho }^{2\xi -1}\mathcal{K}({\kappa }_0,{\mu }_0).\hfill \end{array}$$

For all natural numbers $\xi <\tau$, we have

$$\begin{array}{cc}\hfill \mathcal{K}({\kappa }_{\xi },{\mu }_{\tau })& \le {\rm Y}({\kappa }_{\xi },{\mu }_{\xi })\mathcal{K}({\kappa }_{\xi },{\mu }_{\xi })+{\rm Y}({\kappa }_{\xi +1},{\mu }_{\xi })\mathcal{K}({\kappa }_{\xi +1},{\mu }_{\xi })\hfill \\ \hfill & +{\rm Y}({\kappa }_{\xi +1},{\mu }_{\tau })\mathcal{K}({\kappa }_{\xi +1},{\mu }_{\tau })\hfill \\ \hfill & \le {\rm Y}({\kappa }_{\xi },{\mu }_{\xi })\mathcal{K}({\kappa }_{\xi },{\mu }_{\xi })+{\rm Y}({\kappa }_{\xi +1},{\mu }_{\xi })\mathcal{K}({\kappa }_{\xi +1},{\mu }_{\xi })\hfill \\ \hfill & +{\rm Y}({\kappa }_{\xi +1},{\mu }_{\tau }){\rm Y}({\kappa }_{\xi +1},{\mu }_{\xi +1})\mathcal{K}({\kappa }_{\xi +1},{\mu }_{\xi +1})\hfill \\ \hfill & +{\rm Y}({\kappa }_{\xi +1},{\mu }_{\tau }){\rm Y}({\kappa }_{\xi +2},{\mu }_{\xi +1})\mathcal{K}({\kappa }_{\xi +2},{\mu }_{\xi +1})\hfill \\ \hfill & +{\rm Y}({\kappa }_{\xi +1},{\mu }_{\tau }){\rm Y}({\kappa }_{\xi +2},{\mu }_{\tau })\mathcal{K}({\kappa }_{\xi +2},{\mu }_{\tau })\hfill \\ \hfill & ⋮\hfill \\ \hfill & \le {\rm Y}({\kappa }_{\xi },{\mu }_{\xi })\mathcal{K}({\kappa }_{\xi },{\mu }_{\xi })+{\rm Y}({\kappa }_{\xi +1},{\mu }_{\xi })\mathcal{K}({\kappa }_{\xi +1},{\mu }_{\xi })\hfill \\ \hfill & +\sum _{\varsigma =\xi +1}^{\tau -1}\prod _{\mathfrak{j}=\xi +1}^{\varsigma }{\rm Y}({\kappa }_{\mathfrak{j}},{\mu }_{\tau }){\rm Y}({\kappa }_{\varsigma },{\mu }_{\varsigma })\mathcal{K}({\kappa }_{\varsigma },{\mu }_{\varsigma })\hfill \\ \hfill & +\sum _{\varsigma =\xi +1}^{\tau -1}\prod _{\mathfrak{j}=\xi +1}^{\varsigma }{\rm Y}({\kappa }_{\mathfrak{j}},{\mu }_{\tau }){\rm Y}({\kappa }_{\varsigma +1},{\mu }_{\varsigma })\mathcal{K}({\kappa }_{\varsigma +1},{\mu }_{\varsigma })\hfill \\ \hfill & +\prod _{\mathfrak{k}=\xi +1}^{\tau }{\rm Y}({\kappa }_{\mathfrak{k}},{\mu }_{\tau })\mathcal{K}({\kappa }_{\tau },{\mu }_{\tau })\hfill \\ \hfill & \le {\rm Y}({\kappa }_{\xi },{\mu }_{\xi }){\rho }^{2\xi }\mathcal{K}({\kappa }_0,{\mu }_0)+{\rm Y}({\kappa }_{\xi +1},{\mu }_{\xi }){\rho }^{2\xi +1}\mathcal{K}({\kappa }_0,{\mu }_0)\hfill \\ \hfill & +\sum _{\varsigma =\xi +1}^{\tau -1}\prod _{\mathfrak{j}=\xi +1}^{\varsigma }{\rm Y}({\kappa }_{\mathfrak{j}},{\mu }_{\tau }){\rm Y}({\kappa }_{\varsigma },{\mu }_{\varsigma }){\rho }^{2\varsigma }\mathcal{K}({\kappa }_0,{\mu }_0)\hfill \\ \hfill & +\sum _{\varsigma =\xi +1}^{\tau -1}\prod _{\mathfrak{j}=\xi +1}^{\varsigma }{\rm Y}({\kappa }_{\mathfrak{j}},{\mu }_{\tau }){\rm Y}({\kappa }_{\varsigma +1},{\mu }_{\varsigma }){\rho }^{2\varsigma +1}\mathcal{K}({\kappa }_0,{\mu }_0)\hfill \\ \hfill & +\prod _{\mathfrak{k}=\xi +1}^{\tau }{\rm Y}({\kappa }_{\mathfrak{k}},{\mu }_{\tau }){\rho }^{2\tau }\mathcal{K}({\kappa }_0,{\mu }_0)\hfill \\ \hfill & \le {\rm Y}({\kappa }_{\xi },{\mu }_{\xi }){\rho }^{2\xi }\mathcal{K}({\kappa }_0,{\mu }_0)+{\rm Y}({\kappa }_{\xi +1},{\mu }_{\xi }){\rho }^{2\xi +1}\mathcal{K}({\kappa }_0,{\mu }_0)\hfill \\ \hfill & +\sum _{\varsigma =\xi +1}^{\tau -1}\prod _{\mathfrak{j}=\xi +1}^{\varsigma }{\rm Y}({\kappa }_{\mathfrak{j}},{\mu }_{\tau }){\rm Y}({\kappa }_{\varsigma },{\mu }_{\varsigma }){\rho }^{2\varsigma }\mathcal{K}({\kappa }_0,{\mu }_0)\hfill \\ \hfill & +\sum _{\varsigma =\xi +1}^{\tau -1}\prod _{\mathfrak{j}=\xi +1}^{\varsigma }{\rm Y}({\kappa }_{\mathfrak{j}},{\mu }_{\tau }){\rm Y}({\kappa }_{\varsigma +1},{\mu }_{\varsigma }){\rho }^{2\varsigma +1}\mathcal{K}({\kappa }_0,{\mu }_0)\hfill \\ \hfill & +\prod _{\mathfrak{k}=\xi +1}^{\tau }{\rm Y}({\kappa }_{\mathfrak{k}},{\mu }_{\tau }){\rm Y}({\kappa }_{\tau },{\mu }_{\tau }){\rho }^{2\tau }\mathcal{K}({\kappa }_0,{\mu }_0)\hfill \\ \hfill & ={\rm Y}({\kappa }_{\xi },{\mu }_{\xi }){\rho }^{2\xi }\mathcal{K}({\kappa }_0,{\mu }_0)+{\rm Y}({\kappa }_{\xi +1},{\mu }_{\xi }){\rho }^{2\xi +1}\mathcal{K}({\kappa }_0,{\mu }_0)\hfill \\ \hfill & +\sum _{\varsigma =\xi +1}^{\tau }\prod _{\mathfrak{j}=\xi +1}^{\varsigma }{\rm Y}({\kappa }_{\mathfrak{j}},{\mu }_{\tau }){\rm Y}({\kappa }_{\varsigma },{\mu }_{\varsigma }){\rho }^{2\varsigma }\mathcal{K}({\kappa }_0,{\mu }_0)\hfill \\ \hfill & +\sum _{\varsigma =\xi +1}^{\tau -1}\prod _{\mathfrak{j}=\xi +1}^{\varsigma }{\rm Y}({\kappa }_{\mathfrak{j}},{\mu }_{\tau }){\rm Y}({\kappa }_{\varsigma +1},{\mu }_{\varsigma }){\rho }^{2\varsigma +1}\mathcal{K}({\kappa }_0,{\mu }_0)\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le {\rm Y}({\kappa }_{\xi },{\mu }_{\xi }){\rho }^{2\xi }\mathcal{K}({\kappa }_0,{\mu }_0)+{\rm Y}({\kappa }_{\xi +1},{\mu }_{\xi }){\rho }^{2\xi +1}\mathcal{K}({\kappa }_0,{\mu }_0)\hfill \\ \hfill & +\sum _{\varsigma =\xi +1}^{\tau }\prod _{\mathfrak{j}=0}^{\varsigma }{\rm Y}({\kappa }_{\mathfrak{j}},{\mu }_{\tau }){\rm Y}({\kappa }_{\varsigma },{\mu }_{\varsigma }){\rho }^{2\varsigma }\mathcal{K}({\kappa }_0,{\mu }_0)\hfill \\ \hfill & +\sum _{\varsigma =\xi +1}^{\tau -1}\prod _{\mathfrak{j}=0}^{\varsigma }{\rm Y}({\kappa }_{\mathfrak{j}},{\mu }_{\tau }){\rm Y}({\kappa }_{\varsigma +1},{\mu }_{\varsigma }){\rho }^{2\varsigma +1}\mathcal{K}({\kappa }_0,{\mu }_0).\hfill \end{array}$$

Let

$$\begin{array}{c}\hfill {\mathcal{Z}}_{\mathfrak{e}}=\sum _{\varsigma =0}^{\mathfrak{e}}\prod _{\mathfrak{j}=0}^{\varsigma }{\rm Y}({\kappa }_{\mathfrak{j}},{\mu }_{\tau }){\rm Y}({\kappa }_{\varsigma },{\mu }_{\varsigma }){\rho }^{2\varsigma }\end{array}$$

and

$$\begin{array}{c}\hfill {\mathcal{M}}_{\mathfrak{e}}=\sum _{\varsigma =0}^{\mathfrak{e}}\prod _{\mathfrak{j}=0}^{\varsigma }{\rm Y}({\kappa }_{\mathfrak{j}},{\mu }_{\tau }){\rm Y}({\kappa }_{\varsigma +1},{\mu }_{\varsigma }){\rho }^{2\varsigma +1}.\end{array}$$

Hence, we have

$$\begin{array}{cc}\hfill \mathcal{K}({\kappa }_{\xi },{\mu }_{\tau })& \le \mathcal{K}({\kappa }_0,{\mu }_0)[({\mathcal{Z}}_{\tau }-{\mathcal{Z}}_{\xi })+({\mathcal{M}}_{\tau -1}-{\mathcal{M}}_{\xi })\hfill \\ \hfill & +{\rm Y}({\kappa }_{\xi +1},{\mu }_{\xi +1}){\rho }^{2\xi }+{\rm Y}({\kappa }_{\xi +1},{\mu }_{\xi }){\rho }^{2\xi +1}]\hfill \end{array}$$

Now by conditions (13), (14) and the ratio test, we obtain that $\underset{\xi \to \infty }{lim}{\mathcal{Z}}_{\xi }$ and $\underset{\xi \to \infty }{lim}{\mathcal{M}}_{\xi }$ exists. Therefore $\left\{{\mathcal{Z}}_{\xi }\right\}$ and $\left\{{\mathcal{M}}_{\xi }\right\}$ are Cauchy sequences. As $\xi ,\tau \to \infty$, we get

$$\begin{array}{c}\hfill \underset{\xi ,\tau \to \infty }{lim}\mathcal{K}({\kappa }_{\xi },{\mu }_{\tau })=0.\end{array}$$

(15)

Therefore, $(\left\{{\kappa }_{\xi }\right\},\left\{{\mu }_m\right\})$ is a Cauchy bisequence. By completeness property, $\left\{{\kappa }_{\xi }\right\}\to \mathfrak{e},\left\{{\mu }_{\tau }\right\}\to \mathfrak{e}$ where $\mathfrak{e}\in \mathcal{O}\cup \mathcal{L}$. Since

$$\begin{array}{c}\hfill \left\{\top {\kappa }_{\xi }\right\}=\left\{{\mu }_{\xi }\right\}\to \mathfrak{e},\mathcal{K}(\top \mathfrak{e},\top {\kappa }_{\xi })\to \mathcal{K}(\top \mathfrak{e},\mathfrak{e}).\end{array}$$

On the other hand,

$$\begin{array}{c}\hfill \mathcal{K}(\top \mathfrak{e},\top {\kappa }_{\xi })\le \nu (\mathcal{K}({\kappa }_{\xi },\top {\kappa }_{\xi })+\mathcal{K}(\top \mathfrak{e},\mathfrak{e}))=\nu (\mathcal{K}({\kappa }_{\xi },{\mu }_{\xi })+\mathcal{K}(\top \mathfrak{e},\mathfrak{e})).\end{array}$$

Therefore, $\mathcal{K}(\top \mathfrak{e},\mathfrak{e})\le \nu \mathcal{K}(\top \mathfrak{e},\mathfrak{e})$. Hence $\top \mathfrak{e}=\mathfrak{e}$. If $\mathfrak{d}$ is any fixed point of ⊤, then $\top \mathfrak{d}=\mathfrak{d}$, which implies that $\mathfrak{d}\in \mathcal{O}\cap \mathcal{L}$. Then

$$\begin{array}{cc}\hfill \mathcal{K}(\mathfrak{e},\mathfrak{d})=\mathcal{K}(\top \mathfrak{e},\top \mathfrak{d})& \le \nu \left(\mathcal{K}\right(\mathfrak{e},\top \mathfrak{e})+\mathcal{K}(\top \mathfrak{d},\mathfrak{d}\left)\right)\hfill \\ \hfill & =\nu \left(\mathcal{K}\right(\mathfrak{e},\mathfrak{e})+\mathcal{K}(\mathfrak{d},\mathfrak{d}\left)\right)=0.\hfill \end{array}$$

Consequently $\mathfrak{e}=\mathfrak{d}$. □

Now we present a fixed point result based on Reich type fixed point theorem [17].

Theorem 4.

Let $(\mathcal{O},\mathcal{L},\mathcal{K},{\rm Y})$ be a complete bipolar-controlled metric space. Consider the mapping $\top :(\mathcal{O},\mathcal{L},\mathcal{K},{\rm Y})\leftrightarrows (\mathcal{O},\mathcal{L},\mathcal{K},{\rm Y})$ such that

$$\begin{array}{c}\hfill \mathcal{K}(\top \mu ,\top \kappa )\le \alpha \mathcal{K}(\kappa ,\mu )+\mathfrak{p}\mathcal{K}(\kappa ,\top \kappa )+\nu \mathcal{K}(\top \mu ,\mu )\end{array}$$

for all $\kappa \in \mathcal{O}$ and $\mu \in \mathcal{L}$, where $\alpha ,\mathfrak{p},\nu \ge 0$ such that $\alpha +\mathfrak{p}+\nu <1$. Suppose that

$$\begin{array}{c}\hfill \underset{\tau \ge 1}{sup}\underset{\varsigma \to \infty }{lim}\frac{{\rm Y}({\kappa }_{\varsigma +1},{\eta }_{\varsigma +1})}{{\rm Y}({\kappa }_{\varsigma },{\eta }_{\varsigma })}{\rm Y}({\kappa }_{\varsigma +1},{\eta }_{\tau })<\frac{1}{{\mathfrak{f}}^2}\end{array}$$

(16)

and

$$\begin{array}{c}\hfill \underset{\tau \ge 1}{sup}\underset{\varsigma \to \infty }{lim}\frac{{\rm Y}({\kappa }_{\varsigma +2},{\eta }_{\varsigma +1})}{{\rm Y}({\kappa }_{\varsigma +1},{\eta }_{\varsigma })}{\rm Y}({\kappa }_{\varsigma +1},{\eta }_{\tau })<\frac{1}{{\rho }^2},\end{array}$$

(17)

where $\mathfrak{f}:=\frac{\alpha +\nu }{1-\mathfrak{p}}$ and $\rho :=\frac{\alpha +\mathfrak{p}}{1-\nu }$. Then the function $\top :\mathcal{O}\cup \mathcal{L}\to \mathcal{O}\cup \mathcal{L}$ has a UFP.

Proof. 

Let ${\kappa }_0\in \mathcal{O}$. Define ${\mu }_{\xi }=\top {\kappa }_{\xi }$ and ${\kappa }_{\xi +1}=\top {\mu }_{\xi }$ for all $\xi \in \mathbb{N}$. Then, we have

$$\begin{array}{cc}\hfill \mathcal{K}({\kappa }_{\xi },{\mu }_{\xi })& =\mathcal{K}(\top {\mu }_{\xi -1},\top {\kappa }_{\xi })\hfill \\ \hfill & \le \alpha \mathcal{K}({\kappa }_{\xi },{\mu }_{\xi -1})+\mathfrak{p}\mathcal{K}({\kappa }_{\xi },\top {\kappa }_{\xi })+\nu \mathcal{K}(\top {\mu }_{\xi -1},{\mu }_{\xi -1})\hfill \\ \hfill & =(\alpha +\nu )\mathcal{K}({\kappa }_{\xi },{\mu }_{\xi -1}))+\mathfrak{p}\mathcal{K}({\kappa }_{\xi },{\mu }_{\xi })\hfill \end{array}$$

for all integers $\xi \ge 1$. Now,

$$\begin{array}{c}\hfill \mathcal{K}({\kappa }_{\xi },{\mu }_{\xi })\le \left(\frac{\alpha +\nu }{1-\mathfrak{p}}\right)\mathcal{K}({\kappa }_{\xi },{\mu }_{\xi -1}),\end{array}$$

and

$$\begin{array}{cc}\hfill \mathcal{K}({\kappa }_{\xi },{\mu }_{\xi -1})& =\mathcal{K}(\top {\mu }_{\xi -1},\top {\kappa }_{\xi -1})\hfill \\ \hfill & \le \alpha \mathcal{K}({\kappa }_{\xi -1},{\mu }_{\xi -1})+\mathfrak{p}\mathcal{K}({\kappa }_{\xi -1},\top {\kappa }_{\xi -1})+\nu \mathcal{K}(\top {\mu }_{\xi -1},{\mu }_{\xi -1}))\hfill \\ \hfill & =(\alpha +\mathfrak{p})\left(\mathcal{K}({\kappa }_{\xi -1},{\mu }_{\xi -1})\right)+\nu \mathcal{K}({\kappa }_{\xi },{\mu }_{\xi -1})),\hfill \end{array}$$

so that

$$\begin{array}{c}\hfill \mathcal{K}({\kappa }_{\xi },{\mu }_{\xi -1})\le \left(\frac{\alpha +\mathfrak{p}}{1-\nu }\right)\mathcal{K}({\kappa }_{\xi -1},{\mu }_{\xi -1}).\end{array}$$

If we say $\rho :=\frac{\alpha +\mathfrak{p}}{1-\nu }$ and $\mathfrak{f}:=\frac{\alpha +\nu }{1-\mathfrak{p}}$, then we have $\rho ,\mathfrak{f}\in (0,1)$. Now

$$\begin{array}{cc}\hfill \mathcal{K}({\kappa }_{\xi },{\mu }_{\xi })& \le {\mathfrak{f}}^{2\xi }\mathcal{K}({\kappa }_0,{\mu }_0),\hfill \\ \hfill \mathcal{K}({\kappa }_{\xi },{\mu }_{\xi -1})& \le {\rho }^{2\xi -1}\mathcal{K}({\kappa }_0,{\mu }_0).\hfill \end{array}$$

For all natural numbers $\xi <\tau$, we have

$$\begin{array}{cc}\hfill \mathcal{K}({\kappa }_{\xi },{\mu }_{\tau })& \le {\rm Y}({\kappa }_{\xi },{\mu }_{\xi })\mathcal{K}({\kappa }_{\xi },{\mu }_{\xi })+{\rm Y}({\kappa }_{\xi +1},{\mu }_{\xi })\mathcal{K}({\kappa }_{\xi +1},{\mu }_{\xi })\hfill \\ \hfill & +{\rm Y}({\kappa }_{\xi +1},{\mu }_{\tau })\mathcal{K}({\kappa }_{\xi +1},{\mu }_{\tau })\hfill \\ \hfill & \le {\rm Y}({\kappa }_{\xi },{\mu }_{\xi })\mathcal{K}({\kappa }_{\xi },{\mu }_{\xi })+{\rm Y}({\kappa }_{\xi +1},{\mu }_{\xi })\mathcal{K}({\kappa }_{\xi +1},{\mu }_{\xi })\hfill \\ \hfill & +{\rm Y}({\kappa }_{\xi +1},{\mu }_{\tau }){\rm Y}({\kappa }_{\xi +1},{\mu }_{\xi +1})\mathcal{K}({\kappa }_{\xi +1},{\mu }_{\xi +1})\hfill \\ \hfill & +{\rm Y}({\kappa }_{\xi +1},{\mu }_{\tau }){\rm Y}({\kappa }_{\xi +2},{\mu }_{\xi +1})\mathcal{K}({\kappa }_{\xi +2},{\mu }_{\xi +1})\hfill \\ \hfill & +{\rm Y}({\kappa }_{\xi +1},{\mu }_{\tau }){\rm Y}({\kappa }_{\xi +2},{\mu }_{\tau })\mathcal{K}({\kappa }_{\xi +2},{\mu }_{\tau })\hfill \\ \hfill & ⋮\hfill \\ \hfill & \le {\rm Y}({\kappa }_{\xi },{\mu }_{\xi })\mathcal{K}({\kappa }_{\xi },{\mu }_{\xi })+{\rm Y}({\kappa }_{\xi +1},{\mu }_{\xi })\mathcal{K}({\kappa }_{\xi +1},{\mu }_{\xi })\hfill \\ \hfill & +\sum _{\varsigma =\xi +1}^{\tau -1}\prod _{\mathfrak{j}=\xi +1}^{\varsigma }{\rm Y}({\kappa }_{\mathfrak{j}},{\mu }_{\tau }){\rm Y}({\kappa }_{\varsigma },{\mu }_{\varsigma })\mathcal{K}({\kappa }_{\varsigma },{\mu }_{\varsigma })\hfill \\ \hfill & +\sum _{\varsigma =\xi +1}^{\tau -1}\prod _{\mathfrak{j}=\xi +1}^{\varsigma }{\rm Y}({\kappa }_{\mathfrak{j}},{\mu }_{\tau }){\rm Y}({\kappa }_{\varsigma +1},{\mu }_{\varsigma })\mathcal{K}({\kappa }_{\varsigma +1},{\mu }_{\varsigma })\hfill \\ \hfill & +\prod _{\mathfrak{k}=\xi +1}^{\tau }{\rm Y}({\kappa }_{\mathfrak{k}},{\mu }_{\tau })\mathcal{K}({\kappa }_{\tau },{\mu }_{\tau })\hfill \\ \hfill & \le {\rm Y}({\kappa }_{\xi },{\mu }_{\xi }){\mathfrak{f}}^{2\xi }\mathcal{K}({\kappa }_0,{\mu }_0)+{\rm Y}({\kappa }_{\xi +1},{\mu }_{\xi }){\rho }^{2\xi +1}\mathcal{K}({\kappa }_0,{\mu }_0)\hfill \\ \hfill & +\sum _{\varsigma =\xi +1}^{\tau -1}\prod _{\mathfrak{j}=\xi +1}^{\varsigma }{\rm Y}({\kappa }_{\mathfrak{j}},{\mu }_{\tau }){\rm Y}({\kappa }_{\varsigma },{\mu }_{\varsigma }){\mathfrak{f}}^{2\varsigma }\mathcal{K}({\kappa }_0,{\mu }_0)\hfill \\ \hfill & +\sum _{\varsigma =\xi +1}^{\tau -1}\prod _{\mathfrak{j}=\xi +1}^{\varsigma }{\rm Y}({\kappa }_{\mathfrak{j}},{\mu }_{\tau }){\rm Y}({\kappa }_{\varsigma +1},{\mu }_{\varsigma }){\rho }^{2\varsigma +1}\mathcal{K}({\kappa }_0,{\mu }_0)\hfill \\ \hfill & +\prod _{\mathfrak{k}=\xi +1}^{\tau }{\rm Y}({\kappa }_{\mathfrak{k}},{\mu }_{\tau }){\mathfrak{f}}^{2\tau }\mathcal{K}({\kappa }_0,{\mu }_0)\hfill \\ \hfill & \le {\rm Y}({\kappa }_{\xi },{\mu }_{\xi }){\mathfrak{f}}^{2\xi }\mathcal{K}({\kappa }_0,{\mu }_0)+{\rm Y}({\kappa }_{\xi +1},{\mu }_{\xi }){\rho }^{2\xi +1}\mathcal{K}({\kappa }_0,{\mu }_0)\hfill \\ \hfill & +\sum _{\varsigma =\xi +1}^{\tau -1}\prod _{\mathfrak{j}=\xi +1}^{\varsigma }{\rm Y}({\kappa }_{\mathfrak{j}},{\mu }_{\tau }){\rm Y}({\kappa }_{\varsigma },{\mu }_{\varsigma }){\mathfrak{f}}^{2\varsigma }\mathcal{K}({\kappa }_0,{\mu }_0)\hfill \\ \hfill & +\sum _{\varsigma =\xi +1}^{\tau -1}\prod _{\mathfrak{j}=\xi +1}^{\varsigma }{\rm Y}({\kappa }_{\mathfrak{j}},{\mu }_{\tau }){\rm Y}({\kappa }_{\varsigma +1},{\mu }_{\varsigma }){\rho }^{2\varsigma +1}\mathcal{K}({\kappa }_0,{\mu }_0)\hfill \\ \hfill & +\prod _{\mathfrak{k}=\xi +1}^{\tau }{\rm Y}({\kappa }_{\mathfrak{k}},{\mu }_{\tau }){\rm Y}({\kappa }_{\tau },{\mu }_{\tau }){\mathfrak{f}}^{2\tau }\mathcal{K}({\kappa }_0,{\mu }_0)\hfill \\ \hfill & ={\rm Y}({\kappa }_{\xi },{\mu }_{\xi }){\mathfrak{f}}^{2\xi }\mathcal{K}({\kappa }_0,{\mu }_0)+{\rm Y}({\kappa }_{\xi +1},{\mu }_{\xi }){\rho }^{2\xi +1}\mathcal{K}({\kappa }_0,{\mu }_0)\hfill \\ \hfill & +\sum _{\varsigma =\xi +1}^{\tau }\prod _{\mathfrak{j}=\xi +1}^{\varsigma }{\rm Y}({\kappa }_{\mathfrak{j}},{\mu }_{\tau }){\rm Y}({\kappa }_{\varsigma },{\mu }_{\varsigma }){\mathfrak{f}}^{2\varsigma }\mathcal{K}({\kappa }_0,{\mu }_0)\hfill \\ \hfill & +\sum _{\varsigma =\xi +1}^{\tau -1}\prod _{\mathfrak{j}=\xi +1}^{\varsigma }{\rm Y}({\kappa }_{\mathfrak{j}},{\mu }_{\tau }){\rm Y}({\kappa }_{\varsigma +1},{\mu }_{\varsigma }){\rho }^{2\varsigma +1}\mathcal{K}({\kappa }_0,{\mu }_0)\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le {\rm Y}({\kappa }_{\xi },{\mu }_{\xi }){\mathfrak{f}}^{2\xi }\mathcal{K}({\kappa }_0,{\mu }_0)+{\rm Y}({\kappa }_{\xi +1},{\mu }_{\xi }){\rho }^{2\xi +1}\mathcal{K}({\kappa }_0,{\mu }_0)\hfill \\ \hfill & +\sum _{\varsigma =\xi +1}^{\tau }\prod _{\mathfrak{j}=0}^{\varsigma }{\rm Y}({\kappa }_{\mathfrak{j}},{\mu }_{\tau }){\rm Y}({\kappa }_{\varsigma },{\mu }_{\varsigma }){\mathfrak{f}}^{2\varsigma }\mathcal{K}({\kappa }_0,{\mu }_0)\hfill \\ \hfill & +\sum _{\varsigma =\xi +1}^{\tau -1}\prod _{\mathfrak{j}=0}^{\varsigma }{\rm Y}({\kappa }_{\mathfrak{j}},{\mu }_{\tau }){\rm Y}({\kappa }_{\varsigma +1},{\mu }_{\varsigma }){\rho }^{2\varsigma +1}\mathcal{K}({\kappa }_0,{\mu }_0).\hfill \end{array}$$

Let

$$\begin{array}{c}\hfill {\mathcal{Z}}_{\mathfrak{e}}=\sum _{\varsigma =0}^{\mathfrak{e}}\prod _{\mathfrak{j}=0}^{\varsigma }{\rm Y}({\kappa }_{\mathfrak{j}},{\mu }_{\tau }){\rm Y}({\kappa }_{\varsigma },{\mu }_{\varsigma }){\mathfrak{f}}^{2\varsigma }\end{array}$$

and

$$\begin{array}{c}\hfill {\mathcal{M}}_{\mathfrak{e}}=\sum _{\varsigma =0}^{\mathfrak{e}}\prod _{\mathfrak{j}=0}^{\varsigma }{\rm Y}({\kappa }_{\mathfrak{j}},{\mu }_{\tau }){\rm Y}({\kappa }_{\varsigma +1},{\mu }_{\varsigma }){\rho }^{2\varsigma +1}.\end{array}$$

Hence, we have

$$\begin{array}{cc}\hfill \mathcal{K}({\kappa }_{\xi },{\mu }_{\tau })& \le \mathcal{K}({\kappa }_0,{\mu }_0)[({\mathcal{Z}}_{\tau }-{\mathcal{Z}}_{\xi })+({\mathcal{M}}_{\tau -1}-{\mathcal{M}}_{\xi })\hfill \\ \hfill & +{\rm Y}({\kappa }_{\xi +1},{\mu }_{\xi +1}){\mathfrak{f}}^{2\xi }+{\rm Y}({\kappa }_{\xi +1},{\mu }_{\xi }){\rho }^{2\xi +1}]\hfill \end{array}$$

Now by conditions (16), (17) and the ratio test, we obtain that $\underset{\xi \to \infty }{lim}{\mathcal{Z}}_{\xi }$ and $\underset{\xi \to \infty }{lim}{\mathcal{M}}_{\xi }$ exists. Therefore $\left\{{\mathcal{Z}}_{\xi }\right\}$ and $\left\{{\mathcal{M}}_{\xi }\right\}$ are Cauchy sequences. As $\xi ,\tau \to \infty$, we get

$$\begin{array}{c}\hfill \underset{\xi ,\tau \to \infty }{lim}\mathcal{K}({\kappa }_{\xi },{\mu }_{\tau })=0.\end{array}$$

(18)

Therefore, $(\left\{{\kappa }_{\xi }\right\},\left\{{\mu }_m\right\})$ is a Cauchy bisequence. By completeness property, $\left\{{\kappa }_{\xi }\right\}\to \mathfrak{e},\left\{{\mu }_{\tau }\right\}\to \mathfrak{e}$ where $\mathfrak{e}\in \mathcal{O}\cup \mathcal{L}$. Since

$$\begin{array}{c}\hfill \left\{\top {\kappa }_{\xi }\right\}=\left\{{\mu }_{\xi }\right\}\to \mathfrak{e},\mathcal{K}(\top \mathfrak{e},\top {\kappa }_{\xi })\to \mathcal{K}(\top \mathfrak{e},\mathfrak{e}).\end{array}$$

On the other hand,

$$\begin{array}{c}\hfill \mathcal{K}(\top \mathfrak{e},\top {\kappa }_{\xi })\le \alpha \mathcal{K}({\kappa }_{\xi },\mathfrak{e})+\mathfrak{p}\mathcal{K}({\kappa }_{\xi },\top {\kappa }_{\xi })+\nu \mathcal{K}(\top \mathfrak{e},\mathfrak{e})=\alpha \mathcal{K}({\kappa }_{\xi },\mathfrak{e})+\mathfrak{p}\mathcal{K}({\kappa }_{\xi },{\mu }_{\xi })+\nu \mathcal{K}(\top \mathfrak{e},\mathfrak{e}).\end{array}$$

Therefore, $\mathcal{K}(\top \mathfrak{e},\mathfrak{e})\le \nu \mathcal{K}(\top \mathfrak{e},\mathfrak{e})$. Hence $\top \mathfrak{e}=\mathfrak{e}$. If $\mathfrak{d}$ is any fixed point of ⊤, then $\top \mathfrak{d}=\mathfrak{d}$, implies that $\mathfrak{d}\in \mathcal{O}\cap \mathcal{L}$. Then

$$\begin{array}{cc}\hfill \mathcal{K}(\mathfrak{e},\mathfrak{d})=\mathcal{K}(\top \mathfrak{e},\top \mathfrak{d})& \le \alpha \mathcal{K}(\mathfrak{d},\mathfrak{e})+\mathfrak{p}\mathcal{K}(\mathfrak{e},\top \mathfrak{e})+\nu \mathcal{K}(\top \mathfrak{d},\mathfrak{d})\hfill \\ \hfill & =\alpha \mathcal{K}(\mathfrak{d},\mathfrak{e})\hfill \\ \hfill & <\mathcal{K}(\mathfrak{e},\mathfrak{d}).\hfill \end{array}$$

Consequently $\mathfrak{e}=\mathfrak{d}$. □

4. Application

Now, we apply the derived results of Theorem 1 to find an analytical solution to an integral equation.

Theorem 5.

Suppose we have the following equation:

$$\begin{array}{c}\hfill \kappa \left(\phi \right)=\mathfrak{h}\left(\phi \right)+{\int }_{{\chi }_1\cup {\chi }_2}\mathcal{G}(\phi ,\mathfrak{q},\kappa \left(\mathfrak{q}\right))d\mathfrak{q},\kappa \in {\chi }_1\cup {\chi }_2,\end{array}$$

where ${\chi }_1\cup {\chi }_2$ is a Lebesgue measurable set. let us assume that

(T1) 

$\mathcal{G}:({\chi }_1^2\cup {\chi }_2^2)\times {\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ and $b\in L^{\infty }\left({\chi }_1\right)\cup L^{\infty }\left({\chi }_2\right)$,

(T2) 

there is a continuous function $\theta :{\chi }_1^2\cup {\chi }_2^2\to {\mathbb{R}}^{+}$ and $\rho \in (0,1)$ such that

$$\begin{array}{c}\hfill |\mathcal{G}(\phi ,\mathfrak{q},\kappa \left(\mathfrak{q}\right))-\mathcal{G}(\phi ,\mathfrak{q},\mu \left(\mathfrak{q}\right)|\le \sqrt{\rho \theta (\phi ,\mathfrak{q})}\left(\right|\kappa \left(\mathfrak{q}\right)-\mu \left(\mathfrak{q}\right)|,\end{array}$$

for $\kappa ,\mathfrak{q}\in {\chi }_1^2\cup {\chi }_2^2$,

(T3) 

$|{\int }_{{\chi }_1\cup {\chi }_2}\theta (\phi ,\mathfrak{q})d\mathfrak{q}|\le 1$, i.e., ${sup}_{\phi \in {\chi }_1\cup {\chi }_2}{\int }_{{\chi }_1\cup {\chi }_2}\theta (\phi ,\mathfrak{q})d\mathfrak{q}\le 1$.

Then the integral equation has a unique solution in $L^{\infty }\left({\chi }_1\right)\cup L^{\infty }\left({\chi }_2\right)$.

Proof. 

Let $\mathcal{O}=L^{\infty }\left({\chi }_1\right)$ and $\mathcal{L}=L^{\infty }\left({\chi }_2\right)$ be two normed linear spaces, where ${\chi }_1,{\chi }_2$ are Lebesgue measurable sets and $m({\chi }_1\cup {\chi }_2)<\infty$. Consider $\mathcal{K}:\mathcal{O}\times \mathcal{L}\to {\mathbb{R}}^{+}$ to be defined by $\mathcal{K}(\kappa ,\mu )={|\kappa -\mu |}^2$ and ${\rm Y}(\kappa ,\mu )=3$ for all $(\kappa ,\mu )\in \mathcal{O}\times \mathcal{L}$. Then $(\mathcal{O};\mathcal{L};\mathcal{K};{\rm Y})$ is a complete bipolar conrolled metric space. Define the covariant mapping $\top :L^{\infty }\left({\chi }_1\right)\cup L^{\infty }\left({\chi }_2\right)\to L^{\infty }\left({\chi }_1\right)\cup L^{\infty }\left({\chi }_2\right)$ by

$$\begin{array}{c}\hfill \top \left(\kappa \left(\phi \right)\right)=\mathfrak{h}\left(\phi \right)+{\int }_{{\chi }_1\cup {\chi }_2}\mathcal{G}(\phi ,\mathfrak{q},\kappa \left(\mathfrak{q}\right))d\mathfrak{q},\phi \in {\chi }_1\cup {\chi }_2.\end{array}$$

Now, we have

$$\begin{array}{cc}\hfill \mathcal{K}\left(\top \kappa \right(\phi ),\top \mu (\phi \left)\right)& ={|\top \kappa \left(\phi \right)-\top \mu \left(\phi \right)|}^2\hfill \\ \hfill & =|\mathfrak{h}\left(\phi \right)+{\int }_{{\chi }_1\cup {\chi }_2}\mathcal{G}(\phi ,\mathfrak{q},\kappa \left(\mathfrak{q}\right))d\mathfrak{q}-\left(\mathfrak{h}\left(\phi \right)+{\int }_{{\chi }_1\cup {\chi }_2}\mathcal{G}(\phi ,\mathfrak{q},\kappa \left(\mathfrak{q}\right))d\mathfrak{q}\right){|}^2\hfill \\ \hfill & \le {\int }_{{\chi }_1\cup {\chi }_2}{|\mathcal{G}(\phi ,\mathfrak{q},\kappa \left(\mathfrak{q}\right))-\mathcal{G}(\phi ,\mathfrak{q},\mu \left(\mathfrak{q}\right))|}^{2}d\mathfrak{q}\hfill \\ \hfill & \le {\int }_{{\chi }_1\cup {\chi }_2}{\rho \theta (\phi ,\mathfrak{q})\left(\right|\kappa \left(\mathfrak{q}\right)-\mu \left(\mathfrak{q}\right)|}^2)d\mathfrak{q}\hfill \\ \hfill & \le {\rho |\kappa \left(\mathfrak{q}\right)-\mu \left(\mathfrak{q}\right)|}^2\hfill \\ \hfill & =\rho \mathcal{K}(\phi ,\mu )\hfill \end{array}$$

Thus, we have established that the conditions of Theorem 1 are satisfied and hence the integral equation has a unique solution. □

Application to Fractional Differential Equations

Before proceeding to this subsection, let us recall the following: For a function $\mathfrak{v}\in \mathcal{C}[0,1]$, the Reiman–Liouville fractional derivative of order $\delta >0$ is given by

$$\begin{array}{c}\hfill \frac{1}{\mathrm{\Gamma }(\xi -\delta )}\frac{d^{\xi }}{d{\phi }^{\xi }}{\int }_0^{\phi }\frac{\mathfrak{v}\left(\mathfrak{e}\right)d\mathfrak{e}}{{(\phi -\mathfrak{e})}^{\delta -\xi +1}}={\mathcal{D}}^{\delta }\mathfrak{v}\left(\phi \right),\end{array}$$

provided that the right hand side is pointwise defined on $[0,1]$, where $\left[\delta \right]$ is the integer part of the number $\delta ,\mathrm{\Gamma }$ is the Euler gamma function. Consider the following fractional differential equation

$$\begin{array}{cc}\hfill & {}^{\mathfrak{e}}{\mathcal{D}}^{\mu }\mathfrak{v}\left(\phi \right)+\mathfrak{f}(\phi ,\mathfrak{v}\left(\phi \right))=0,1\le \phi \le 0,2\le \mu >1;\hfill \\ \hfill & \mathfrak{v}\left(0\right)=\mathfrak{v}\left(1\right)=0,\hfill \end{array}$$

(19)

where $\mathfrak{f}$ is a continuous function from $[0,1]\times \mathbb{R}$ to $\mathbb{R}$ and ${}^{\mathfrak{e}}{\mathcal{D}}^{\mu }$ represents the Caputo fractional derivative of order $\mu$ and it is defined by

$$\begin{array}{c}\hfill {}^{\mathfrak{e}}{\mathcal{D}}^{\mu }=\frac{1}{\mathrm{\Gamma }(\xi -\mu )}{\int }_0^{\kappa }\frac{{\mathfrak{v}}^{\xi }\left(\mathfrak{e}\right)d\mathfrak{e}}{{(\phi -\mathfrak{e})}^{\mu -\xi +1}}\end{array}$$

Let $\mathcal{O}=\left(\mathcal{C}\right[0,1],[0,\infty \left)\right)$ be the set of all continuous functions defined on $[0,1]$ with values in the interval $[0,\infty )$ and $\mathcal{L}=\left(\mathcal{C}\right[0,1],[0,\infty \left)\right)$ be the set of all continuous functions defined on $[0,1]$ with values in the interval $[0,\infty )$. Consider $\mathcal{K}:\mathcal{O}\times \mathcal{L}\to {\mathbb{R}}^{+}$ to be defined by

$$\begin{array}{c}\hfill \mathcal{K}(\mathfrak{v},{\mathfrak{v}}^{{}^{\prime }})=\underset{\phi \in [0,1]}{sup}{|\mathfrak{v}\left(\phi \right)-{\mathfrak{v}}^{{}^{\prime }}\left(\phi \right)|}^2\end{array}$$

and ${\rm Y}(\mathfrak{v},{\mathfrak{v}}^{{}^{\prime }})=3$ for all $(\mathfrak{v},{\mathfrak{v}}^{{}^{\prime }})\in \mathcal{O}\times \mathcal{L}$. Then $(\mathcal{O};\mathcal{L};\mathcal{K};{\rm Y})$ is a complete bipolar conrolled metric space.

Theorem 6.

Let us consider the fractional differential Equation (FDE) (19). Assuming that the following holds:

(i) 

there exists $\phi \in [0,1]$,$\rho \in (0,1)$ and $(\mathfrak{v},{\mathfrak{v}}^{{}^{\prime }})\in \mathcal{O}\times \mathcal{L}$ such that

$$\begin{array}{c}\hfill |\mathfrak{f}(\phi ,\mathfrak{v})-\mathfrak{f}(\phi ,{\mathfrak{v}}^{{}^{\prime }})|\le \sqrt{\rho }|\mathfrak{v}\left(\phi \right)-{\mathfrak{v}}^{{}^{\prime }}\left(\phi \right)|;\end{array}$$

(ii) 

$$\begin{array}{c}\hfill \underset{\phi \in [0,1]}{sup}{\int }_0^1{\left|\mathcal{G}(\phi ,\mathfrak{e})\right|}^{2}d\mathfrak{q}\le 1.\end{array}$$

Then the FDE (19) has a unique solution in $\mathcal{O}\cup \mathcal{L}$.

Proof. 

Equation (19) is equivalent to

$$\begin{array}{c}\hfill \mathfrak{v}\left(\phi \right)={\int }_0^1\mathcal{G}(\phi ,\mathfrak{e})\mathfrak{f}(\mathfrak{q},\mathfrak{v}\left(\mathfrak{e}\right))d\mathfrak{e},\end{array}$$

where

$$\begin{array}{c}\hfill \mathcal{G}(\phi ,\mathfrak{e})=\left\{\begin{array}{cc}\frac{{\left[\phi (1-\mathfrak{e})\right]}^{\mu -1}-{(\phi -\mathfrak{e})}^{\mu -1}}{\mathrm{\Gamma }\left(\mu \right)},\hfill & 0\le \mathfrak{e}\le \phi \le 1,\hfill \\ \frac{{\left[\phi (1-\mathfrak{e})\right]}^{\mu -1}}{\mathrm{\Gamma }\left(\mu \right)},\hfill & 0\le \phi \le \mathfrak{e}\le 1.\hfill \end{array}\right.\end{array}$$

Define the covariant mapping $\top :\mathcal{O}\cup \mathcal{L}\to \mathcal{O}\cup \mathcal{L}$ defined by

$$\begin{array}{c}\hfill \top \mathfrak{v}\left(\phi \right)={\int }_0^1\mathcal{G}(\phi ,\mathfrak{e})\mathfrak{f}(\mathfrak{q},\mathfrak{v}\left(\mathfrak{e}\right))d\mathfrak{e}.\end{array}$$

It is easy to note that if ${\mathfrak{v}}^{*}\in \top$ is a fixed point of ⊤ then ${\mathfrak{v}}^{*}$ is a solution to the problem (19).

Now

$$\begin{array}{cc}\hfill |\top \mathfrak{v}\left(\phi \right)-\top {\mathfrak{v}}^{{}^{\prime }}{\left(\phi \right)|}^2& =|{\int }_0^1\mathcal{G}(\phi ,\mathfrak{e})\mathfrak{f}(\mathfrak{q},\mathfrak{v}\left(\mathfrak{e}\right))d\mathfrak{e}-{\int }_o^1\mathcal{G}(\phi ,\mathfrak{e})\mathfrak{f}(\mathfrak{q},{\mathfrak{v}}^{{}^{\prime }}\left(\mathfrak{e}\right))d\mathfrak{e}{|}^2\hfill \\ \hfill & \le {\int }_0^1{\left|\mathcal{G}(\phi ,\mathfrak{e})\right|}^{2}d\mathfrak{e}·{\int }_0^1|\mathfrak{f}(\mathfrak{q},\mathfrak{v}\left(\mathfrak{e}\right))-\mathfrak{f}(\mathfrak{q},{\mathfrak{v}}^{{}^{\prime }}\left(\mathfrak{e}\right)){|}^{2}d\mathfrak{e}\hfill \\ \hfill & \le \rho |\mathfrak{v}\left(\phi \right)-{\mathfrak{v}}^{{}^{\prime }}\left(\phi \right){|}^2.\hfill \end{array}$$

Taking the supremum on both sides, we get

$$\begin{array}{c}\hfill \mathcal{K}(\top \mathfrak{v},\top {\mathfrak{v}}^{{}^{\prime }})\le \rho \mathcal{K}(\mathfrak{v},{\mathfrak{v}}^{{}^{\prime }}).\end{array}$$

Thus, the conditions of Theorem 1 are satisfied and hence the fractional differential Equation (19) has a unique solution. □

Example 5.Let us consider the following fractional differential equation :

$$\begin{array}{c}\hfill {\mathcal{D}}^{\mu }\mathfrak{v}\left(\phi \right)+\mathfrak{v}\left(\phi \right)={\displaystyle \frac{2}{\mathrm{\Gamma }(3-\mu )}}{\phi }^{2-\mu }+{\phi }^3,\end{array}$$

(20)

with initial condition: $\mathfrak{v}\left(0\right)=0,{\mathfrak{v}}^{\prime }\left(\mathfrak{0}\right)=0$.

Equation (20) has the exact solution with $\mu =1.9$:

$$\begin{array}{c}\hfill \mathfrak{v}\left(\phi \right)={\phi }^2\end{array}$$

By Equation (19), we can express Equation (20) in the homotopy form;

$$\begin{array}{c}\hfill {\mathcal{D}}^{\mu }\mathfrak{v}\left(\phi \right)+\mathfrak{u}\mathfrak{v}\left(\phi \right)-{\displaystyle \frac{2}{\mathrm{\Gamma }(3-\mu )}}{\phi }^{2-\mu }-{\phi }^3=0,\end{array}$$

(21)

the solution of Equation (20) is:

$$\begin{array}{c}\hfill \mathfrak{v}\left(\phi \right)={\mathfrak{v}}_0\left(\phi \right)+\mathfrak{u}{\mathfrak{v}}_1\left(\phi \right)+{\mathfrak{u}}^2{\mathfrak{v}}_2\left(\phi \right)+\cdots .\end{array}$$

(22)

Substituting Equation (22) in (21) and collecting terms with the power of $\mathfrak{u}$, we get

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\mathfrak{u}}^0:\hfill & {\mathcal{D}}^{\mu }{\mathfrak{v}}_0\left(\phi \right)=0\hfill \\ {\mathfrak{u}}^1:\hfill & {\mathcal{D}}^{\mu }{\mathfrak{v}}_1\left(\phi \right)=-{\mathfrak{v}}_0\left(\phi \right)+\mathfrak{f}\left(\phi \right)\hfill \\ {\mathfrak{u}}^2:\hfill & {\mathcal{D}}^{\mu }{\mathfrak{v}}_2\left(\phi \right)=-{\mathfrak{v}}_1\left(\phi \right)\hfill \\ {\mathfrak{u}}^3:\hfill & {\mathcal{D}}^{\mu }{\mathfrak{v}}_3\left(\phi \right)=-{\mathfrak{v}}_2\left(\phi \right)\hfill \\ ⋮\end{array}\right.\end{array}$$

(23)

Applying ${{\rm Y}}^{\mu }$ and the inverse operation of ${\mathcal{D}}^{\mu }$, on both sides of Equation (23) and fractional integral operation $\left({{\rm Y}}^{\mu }\right)$ of order $\mu >0$, we have

$$\begin{array}{cc}\hfill {\mathfrak{v}}_0\left(\phi \right)& =\sum _{\mathfrak{i}=0}^1{\mathfrak{v}}^{\mathfrak{i}}\left(0\right){\displaystyle \frac{{\phi }^{\mathfrak{i}}}{\mathfrak{i}!}}\hfill \\ & =\mathfrak{v}\left(0\right){\displaystyle \frac{{\phi }^0}{0!}}+{\mathfrak{v}}^{\prime }\left(0\right){\displaystyle \frac{{\phi }^1}{1!}}\hfill \\ \hfill {\mathfrak{v}}_1\left(\phi \right)& =-{{\rm Y}}^{\mu }\left[{\mathfrak{v}}_0\left(\phi \right)+{{\rm Y}}^{\mu }\left[\mathfrak{f}\left(\phi \right)\right]\right]\hfill \\ & ={\phi }^2+{\displaystyle \frac{\mathrm{\Gamma }\left(4\right)}{\mathrm{\Gamma }(4+\mu )}}{\phi }^{3+\mu },\hfill \\ \hfill {\mathfrak{v}}_2\left(\phi \right)& =-{{\rm Y}}^{\mu }\left[{\mathfrak{v}}_1\left(\phi \right)\right]\hfill \\ & ={\displaystyle \frac{2}{\mathrm{\Gamma }(3+\mu )}}{\phi }^{2+\mu }-{\displaystyle \frac{6}{\mathrm{\Gamma }(3+2\mu )}}{\phi }^{3+2\mu },\hfill \\ \hfill {\mathfrak{v}}_3\left(\phi \right)& =-{{\rm Y}}^{\mu }\left[{\mathfrak{v}}_2\left(\phi \right)\right]\hfill \\ & ={\displaystyle \frac{2}{\mathrm{\Gamma }(3+2\mu )}}{\phi }^{2+2\mu }-{\displaystyle \frac{6}{\mathrm{\Gamma }(3+3\mu )}}{\phi }^{3+3\mu }.\hfill \end{array}$$

Hence the solution of Equation (20) is

$$\begin{array}{cc}\hfill \mathfrak{v}\left(\phi \right)& ={\mathfrak{v}}_0\left(\phi \right)+{\mathfrak{v}}_1\left(\phi \right)+{\mathfrak{v}}_2\left(\phi \right)+\cdots \hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill \mathfrak{v}\left(\phi \right)& ={\phi }^2+{\displaystyle \frac{\mathrm{\Gamma }\left(4\right)}{\mathrm{\Gamma }(4+\mu )}}{\phi }^{(3+\mu )}-{\displaystyle \frac{2}{\mathrm{\Gamma }(3+\mu )}}{\phi }^{(2+\mu )}-{\displaystyle \frac{6}{\mathrm{\Gamma }(4+2\mu )}}{\phi }^{(3+2\mu )}+\cdots ,\hfill \end{array}$$

(25)

when $\mu =1.9$

$$\begin{array}{cc}\hfill \mathfrak{v}\left(\phi \right)& ={\phi }^2+{\displaystyle \frac{6}{\mathrm{\Gamma }\left(5.9\right)}}{\phi }^{\left(4.9\right)}-{\displaystyle \frac{2}{\mathrm{\Gamma }\left(4.9\right)}}{\phi }^{\left(3.9\right)}-{\displaystyle \frac{6}{\mathrm{\Gamma }\left(7.8\right)}}{\phi }^{\left(6.8\right)}+\cdots \hfill \\ & ={\phi }^2-\mathrm{small}\mathrm{terms}\hfill \\ & \approx {\phi }^2.\hfill \end{array}$$

For $\mu =1.9$ and $\xi =51$, the results (both numerical and exact) using the matrix approach are presented in

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

$\mathit{\phi }$	$\mathfrak{v}\left(\mathit{\phi }\right)$	${\mathfrak{v}}_{\mathit{\xi }}\left(\mathit{\phi }\right)$	$|\mathfrak{v}\left(\mathit{\phi }\right)-{\mathfrak{v}}_{\mathit{\xi }}\left(\mathit{\phi }\right)|$
0.10000	0.01000	0.00862	0.00138
0.20000	0.04000	0.03769	0.00231
0.30000	0.09000	0.08654	0.00346
0.40000	0.16000	0.15474	0.00526
0.50000	0.25000	0.24193	0.00807
0.60000	0.36000	0.34786	0.01214
0.70000	0.49000	0.47244	0.01756
0.80000	0.64000	0.61581	0.02419
0.90000	0.81000	0.77841	0.03159
1.00000	1.00000	0.96098	0.03902

and the maximum error observed was $\xi =51$ is $0.039016195358901$.

While Figure 1a compares the numerical and exact solutions of the FDE (20), Figure 1b displays the absolute error between them.

5. 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.