Fixed-Point Results of F-Contractions in Bipolar p-Metric Spaces

Abstract

In this paper, we present new findings on F-contraction in bipolar p-metric spaces. We establish a covariant Banach-type fixed-point theorem and a contravariant Reich-type fixed-point theorem based on F-contraction in these spaces. Additionally, we include an example that demonstrates the applicability of our results. Our results non-trivially extend this covariant Banach-type fixed-point theorem and contravariant Reich type theorem via the concept of F-contraction.

1. Introduction

The Banach fixed-point theorem, a key result in mathematics, was established in 1922. Following this, significant advancements occurred in fixed-point theory. In 1989, Bakhtin [1] and, in 1993, Czerwik [2] introduced various contraction conditions in b-metric spaces, extending the concept of metric spaces. In 2016, Mutlu and Gurdal [3,4] defined bipolar metric spaces and developed numerous theorems based on different contractive conditions. For a comprehensive study on the comparison of various definitions of contractive mappings, we refer to the famous work of Rhoades [5]. Wardowski [6] introduced the notion of F-contraction in 2012. For some recent works in F-contractions, we refer to [7,8,9,10]. In 2020, Roy and Saha [11] presented the concept of bipolar cone b-metric spaces. Paul et al. [12] proved some common fixed points in bipolar metric spaces. The idea of p-metric spaces was proposed by Parvaneh et al. [13] in 2017. Concurrent developments in b-metric spaces and Branciari distance were presented in [14,15]. Some historial notes, surveys and non-trivial generalizations of metric spaces and different versions of the Banach’s fixed-point theorem may be found in [16,17,18,19,20,21] and in the references therein.

In this paper, we introduce the concept of F-contraction in bipolar p-metric spaces and explore covariant and contravariant fixed-point theorems within this new framework. Additionally, we present an example to illustrate and validate one of the results.

For some very recent interesting covariant and contravariant fixed-point theorems on bipolar and bipolar-p-metric spaces, we refer to the works of Mutlu et al. [22,23] and Roy et al. [24], respectively. In this paper, in particular, we non-trivially extend Theorems 3.2 and 3.4 of [24] using Wardowski’s F-contraction [6].

2. Preliminaries

Some important results that are related to the present work are listed below:

Definition 1

([1,2]). Suppose $\mathfrak{M}$ is a non-empty set and ${\mathfrak{d}}_{\mathfrak{b}}:\mathfrak{M}\times \mathfrak{M}\to [0,\infty )$ is a mapping. If ${\mathfrak{d}}_{\mathfrak{b}}$ satisfies the following conditions:

(1) 

${\mathfrak{d}}_{\mathfrak{b}}({\varsigma }_1,{\varsigma }_2)=0$ if and only if ${\varsigma }_1={\varsigma }_2$;

(2) 

${\mathfrak{d}}_{\mathfrak{b}}({\varsigma }_1,{\varsigma }_2)={\mathfrak{d}}_{\mathfrak{b}}({\varsigma }_2,{\varsigma }_1)$ for all ${\varsigma }_1,{\varsigma }_2\in \mathfrak{M}$;

(3) 

There exists a real number $s\ge 1$ such that ${\mathfrak{d}}_{\mathfrak{b}}({\varsigma }_1,{\varsigma }_3)\le s[{\mathfrak{d}}_{\mathfrak{b}}({\varsigma }_1,{\varsigma }_2)+{\mathfrak{d}}_{\mathfrak{b}}({\varsigma }_2,{\varsigma }_3)]$ for all ${\varsigma }_1,{\varsigma }_2,{\varsigma }_3\in \mathfrak{M}$, then ${\mathfrak{d}}_{\mathfrak{b}}$ is known as a b-metric on $\mathfrak{M}$ and $(\mathfrak{M},{\mathfrak{d}}_{\mathfrak{b}})$ is a b-metric space.

Definition 2

([25]). Suppose $(\mathfrak{M},{\mathfrak{d}}_{\mathfrak{b}})$ is a b-metric space and $\left\{u_n\right\}$ is a sequence in $\mathfrak{M}$. Then,

(a) 

$\left\{u_n\right\}$ is called a convergent sequence in $(\mathfrak{M},{\mathfrak{d}}_{\mathfrak{b}})$, if for every $\epsilon >0$, ∃ $n_0\in \mathbb{N}$, such that ${\mathfrak{d}}_{\mathfrak{b}}(u_n,u)<\epsilon$∀$n>n_0$. It is denoted by $\underset{n\to \infty }{lim}{u}_n=u$ or $u_n\to u$ as $n\to \infty .$

(b) 

$\left\{u_n\right\}$ is called a Cauchy sequence in $(\mathfrak{M},{\mathfrak{d}}_{\mathfrak{b}})$ if for every $\epsilon >0$ ∃ $n_0\in \mathbb{N}$, such that ${\mathfrak{d}}_{\mathfrak{b}}(u_n,u_{n+p})<\epsilon$∀$n>n_0,p>0$.

(c) 

$(\mathfrak{M},{\mathfrak{d}}_{\mathfrak{b}})$ is called a complete b-metric space if every Cauchy sequence in $\mathfrak{M}$ converges to some $u\in \mathfrak{M}$.

Definition 3

([6]). Suppose the function $\mathfrak{F}:(0,\infty )\to (-\infty ,+\infty )$ satisfies the following conditions:

(F1) 

$\mathfrak{F}$ is strictly increasing;

(F2) 

For every sequence ${\left\{t_n\right\}}_{n\in \mathbb{N}}\subset (0,\infty )$, ${lim}_{n\to \infty }{t}_n=0$ iff ${lim}_{n\to \infty }\mathfrak{F}(t_n)=-\infty$;

(F3) 

There exist $s\in (0,1)$, such that ${lim}_{t\to 0}{t}^s\mathfrak{F}(t)=0.$

Let $\mathcal{F}$ be the collection of all functions $\mathfrak{F}$ and let $(\mathfrak{M},d)$ be a metric space. Then, a mapping $S:\mathfrak{M}\to \mathfrak{M}$ is known as an $\mathfrak{F}$-contraction if ∃$\tau >0$, $\mathfrak{F}\in \mathcal{F}$, such that ∀ $\mathfrak{P},\mathfrak{q}\in \mathfrak{M}$, and we have

$$d(S(\mathfrak{P}),S(\mathfrak{q}))>0\Rightarrow \tau +\mathfrak{F}(d(S(\mathfrak{P}),S(\mathfrak{q})))\le \mathfrak{F}(d(\mathfrak{P},\mathfrak{q})).$$

Definition 4

([13]). Let $\mathfrak{X}\ne \varphi$. A mapping ${\mathfrak{d}}_{\mathfrak{p}}:\mathfrak{X}\times \mathfrak{X}⟶[0,\infty )$ is called an extended b-metric or p-metric if ∃ a strictly increasing continuous mapping $\mathrm{\Omega }:[0,\infty )⟶[0,\infty )$ with ${\mathrm{\Omega }}^{-1}(t)\le t\le \mathrm{\Omega }(t)$, ∀ $t\ge 0$ and ${\mathrm{\Omega }}^{-1}(0)=0=\mathrm{\Omega }(0)$, such that ∀ $x,y,z\in \mathfrak{X}$, and the following conditions hold:

(i) 

${\mathfrak{d}}_{\mathfrak{p}}({\varsigma }_1,{\varsigma }_2)=0$ iff ${\varsigma }_1={\varsigma }_2$;

(ii) 

${\mathfrak{d}}_{\mathfrak{p}}({\varsigma }_1,{\varsigma }_2)={\mathfrak{d}}_{\mathfrak{p}}({\varsigma }_2,{\varsigma }_1)$, for all ${\varsigma }_1,{\varsigma }_2\in \mathfrak{X}\bigcap \mathfrak{Y}$;

(iii) 

${\mathfrak{d}}_{\mathfrak{p}}({\varsigma }_1,{\varsigma }_3)\le \mathrm{\Omega }({\mathfrak{d}}_{\mathfrak{p}}({\varsigma }_1,{\varsigma }_2)+{\mathfrak{d}}_{\mathfrak{p}}({\varsigma }_2,{\varsigma }_3))$.

Then, $(\mathfrak{X},{\mathfrak{d}}_{\mathfrak{p}})$ is known as a p-metric space.

Definition 5

([3]). Consider two non-empty sets $\mathfrak{X}$ and $\mathfrak{Y}$. A mapping ${\mathfrak{d}}_{\mathfrak{bi}}:\mathfrak{X}\times \mathfrak{Y}⟶[0,\infty )$ is called bipolar-metric on $(\mathfrak{X},\mathfrak{Y})$ if it satisfies the following conditions:

(i) 

${\mathfrak{d}}_{\mathfrak{bi}}({\varsigma }_1,{\varsigma }_2)=0$ iff ${\varsigma }_1={\varsigma }_2$;

(ii) 

${\mathfrak{d}}_{\mathfrak{bi}}({\varsigma }_1,{\varsigma }_2)={\mathfrak{d}}_{\mathfrak{bi}}({\varsigma }_2,{\varsigma }_1)$, for all ${\varsigma }_1,{\varsigma }_2\in \mathfrak{X}\bigcap \mathfrak{Y}$;

(iii) 

${\mathfrak{d}}_{\mathfrak{bi}}({\varsigma }_1,{\varsigma }_3)\le {\mathfrak{d}}_{\mathfrak{bi}}({\varsigma }_1,{\varsigma }_2)+{\mathfrak{d}}_{\mathfrak{bi}}(x_1,{\varsigma }_2)+{\mathfrak{d}}_{\mathfrak{bi}}(x_1,{\varsigma }_3)$, for all $({\varsigma }_1,{\varsigma }_2),(x_1,{\varsigma }_3)\in \mathfrak{X}\times \mathfrak{Y}$.

Then, $(\mathfrak{X},\mathfrak{Y},{\mathfrak{d}}_{\mathfrak{bi}})$ is known as a bipolar-metric space.

Definition 6

([24]). Suppose Ω is a strictly increasing continuous function. Consider the two non-empty sets of mappings:

$$\begin{array}{l}\psi =\{\mathrm{\Omega }:[0,\infty )⟶[0,\infty ):{\mathrm{\Omega }}^{-1}(t)\le t\le \mathrm{\Omega }(t),\forall t\ge 0\} and\\ {\psi }^{*}=\{\mathrm{\Omega }\in \psi :{\mathrm{\Omega }}^{-1}(t_1+t_2)\le {\mathrm{\Omega }}^{-1}(t_1)+{\mathrm{\Omega }}^{-1}(t_2),\forall t_1,t_2\ge 0\}.\end{array}$$

Let $\mathfrak{X}$ and $\mathfrak{Y}$ be two non-empty sets. A mapping $p:\mathfrak{X}\times \mathfrak{Y}⟶[0,\infty )$ is known as a bipolar p-metric on $(\mathfrak{X},\mathfrak{Y})$ if it satisfies the following three conditions for a function $\mathrm{\Omega }\in \psi$:

(i) 

$p({\varsigma }_1,{\varsigma }_2)=0$ iff ${\varsigma }_1={\varsigma }_2$;

(ii) 

$p({\varsigma }_1,{\varsigma }_2)=p({\varsigma }_2,{\varsigma }_1)$, for all $({\varsigma }_1,{\varsigma }_2)\in {(\mathfrak{X}\bigcap \mathfrak{Y})}^2$;

(iii) 

$p({\varsigma }_1,{\varsigma }_3)\le \mathrm{\Omega }[p({\varsigma }_1,{\varsigma }_2)+p(x_1,{\varsigma }_2)+p(x_1,{\varsigma }_3)]$, for all $({\varsigma }_1,{\varsigma }_2),(x_1,{\varsigma }_3)\in \mathfrak{X}\times \mathfrak{Y}$.

And $(\mathfrak{X},\mathfrak{Y},p)$ is called a bipolar p-metric space.

Remark 1. 

The definitions of sequence, Cauchy sequence, convergent sequence etc., in a bipolar p-metric space are exactly the same as in the case of a usual metric space or b-metric space. Hence, we omit their exact definitions to avoid repetition.

Remark 2. 

Any metric space, b-metric space, p-metric space, bipolar metric space, and bipolar b-metric space is also a bipolar p-metric space. As such, the results established in the current paper are also true in the aforementioned less general metric spaces.

Definition 7

([24]). Consider two pairs of sets $({\mathfrak{X}}_{\mathfrak{1}},{\mathfrak{Y}}_{\mathfrak{1}})$ and $({\mathfrak{X}}_{\mathfrak{2}},{\mathfrak{Y}}_{\mathfrak{2}})$. The function $f:{\mathfrak{X}}_{\mathfrak{1}}\bigcup {\mathfrak{Y}}_{\mathfrak{1}}\to {\mathfrak{X}}_{\mathfrak{2}}\bigcup {\mathfrak{Y}}_{\mathfrak{2}}$ is known as covariant mapping if $f({\mathfrak{X}}_{\mathfrak{1}})\subset {\mathfrak{X}}_{\mathfrak{2}}$ and $f({\mathfrak{Y}}_{\mathfrak{1}})\subset {\mathfrak{Y}}_{\mathfrak{2}}$ and it is denoted by $f:({\mathfrak{X}}_{\mathfrak{1}},{\mathfrak{Y}}_{\mathfrak{1}})\stackrel{⟶}{\to }({\mathfrak{X}}_{\mathfrak{2}},{\mathfrak{Y}}_{\mathfrak{2}})$.

Definition 8

([24]). Suppose $({\mathfrak{X}}_{\mathfrak{1}},{\mathfrak{Y}}_{\mathfrak{1}})$ and $({\mathfrak{X}}_{\mathfrak{2}},{\mathfrak{Y}}_{\mathfrak{2}})$ are two pairs of sets. The function $f:{\mathfrak{X}}_{\mathfrak{1}}\bigcup {\mathfrak{Y}}_{\mathfrak{1}}\to {\mathfrak{X}}_{\mathfrak{2}}\bigcup {\mathfrak{Y}}_{\mathfrak{2}}$ is known as contravariant mapping if $f({\mathfrak{X}}_{\mathfrak{1}})\subset {\mathfrak{Y}}_{\mathfrak{2}}$ and $f({\mathfrak{Y}}_{\mathfrak{1}})\subset {\mathfrak{X}}_{\mathfrak{2}}$ and it is denoted by $f:({\mathfrak{X}}_{\mathfrak{1}},{\mathfrak{Y}}_{\mathfrak{1}})\rightleftharpoons ({\mathfrak{X}}_{\mathfrak{2}},{\mathfrak{Y}}_{\mathfrak{2}})$.

3. Extended Interpolative $\mathfrak{F}$-Contraction

In this section, we present covariant-type and contravariant-type fixed-point theorems.

Theorem 1. 

Consider a complete bipolar p-MS $(\mathfrak{X},\mathfrak{Y},p)$ for some $\mathrm{\Omega }\in {\psi }^{*}$ and a covariant mapping $f:(\mathfrak{X},\mathfrak{Y},p)\stackrel{⟶}{\to }(\mathfrak{X},\mathfrak{Y},p)$ such that

$$\tau +\mathfrak{F}(p(f({\varsigma }_1),f({\varsigma }_2)))\le c_1\mathfrak{F}(p({\varsigma }_1,{\varsigma }_2))$$

holds for $({\varsigma }_1,{\varsigma }_2)\in \mathfrak{X}\times \mathfrak{Y}$, $\tau >0$, $c_1\in {\Delta }_{\mathrm{\Omega }}$ and for any $p(f({\varsigma }_1),f({\varsigma }_2))>0$.

Then, the function $f:\mathfrak{X}\bigcup \mathfrak{Y}\to \mathfrak{X}\bigcup \mathfrak{Y}$ has a unique fixed point.

Proof. 

Consider $({\varsigma }_0,{\zeta }_0)\in \mathfrak{X}\times \mathfrak{Y}$. Let us consider the iterative sequences ${\varsigma }_n\subset \mathfrak{X}$ and ${\zeta }_n\subset \mathfrak{Y}$ such that ${\varsigma }_n=f({\varsigma }_{n-1})=f^n({\varsigma }_0)$ and ${\zeta }_n=f({\zeta }_{n-1})=f^n({\zeta }_0)$, for all $n\in \mathbb{N}$. Then, $(\left\{{\varsigma }_n\right\},\left\{{\zeta }_n\right\})$ is a bisequence on $(\mathfrak{X},\mathfrak{Y},p)$ and ${\varsigma }_n\ne {\zeta }_n$.

The term bisequence means that the sequence $(\left\{{\varsigma }_n\right\},\left\{{\zeta }_n\right\})$ is a subset of the Cartesian product of $\mathfrak{X}$ and $\mathfrak{Y}$. The concept of convergence of a bisequence is a natural extension of the concept of convergence of a sequence.

We then have

$$\begin{array}{cc}\hfill \tau +\mathfrak{F}(p({\varsigma }_n,{\zeta }_n))& =\tau +\mathfrak{F}(p(f({\varsigma }_{n-1}),f({\zeta }_{n-1})))\hfill \\ \hfill & \le c_1\mathfrak{F}(p({\varsigma }_{n-1},{\zeta }_{n-1})).\hfill \end{array}$$

Therefore,

$$\begin{array}{cc}\hfill \mathfrak{F}(p({\varsigma }_n,{\zeta }_n))& \le c_1\mathfrak{F}(p(f({\varsigma }_{n-1}),f({\zeta }_{n-1})))-\tau \hfill \\ \hfill & \le c_1^2\mathfrak{F}(p({\varsigma }_{n-2},{\zeta }_{n-2}))-2\tau .\hfill \end{array}$$

Proceeding in this way, we have $\forall n\ge 1$,

$$\mathfrak{F}(p({\varsigma }_n,{\zeta }_n))\le c_1^n\mathfrak{F}(p({\varsigma }_0,{\zeta }_0))-n\tau .$$

Taking the limit as $n\to \infty$,

$$\underset{n\to \infty }{lim}\mathfrak{F}(p({\varsigma }_n,{\zeta }_n))=-\infty .$$

Then, from the second property of $\mathfrak{F}$-contraction, we have

$$\underset{n\to \infty }{lim}p({\varsigma }_n,{\zeta }_n)=0.$$

Hence, from the third property of the $\mathfrak{F}$-contraction, $\forall n\in \mathbb{N}$∃$t\in (0,1)$, such that

$$\underset{n\to \infty }{lim}p{({\varsigma }_n,{\zeta }_n)}^t\mathfrak{F}(p({\varsigma }_n,{\zeta }_n))=0.$$

For all $n\in \mathbb{N}$, we have

$$\begin{array}{cc}\hfill p{({\varsigma }_n,{\zeta }_n)}^t\mathfrak{F}(p({\varsigma }_n,{\zeta }_n))& -p{({\varsigma }_n,{\zeta }_n)}^t\mathfrak{F}(p({\varsigma }_0,{\zeta }_0))\hfill \\ \hfill & \le p{({\varsigma }_n,{\zeta }_n)}^t(c_1^n\mathfrak{F}(p({\varsigma }_0,{\zeta }_0))-n\tau )-p{({\varsigma }_n,{\zeta }_n)}^t\mathfrak{F}(p({\varsigma }_0,{\zeta }_0))\hfill \\ \hfill & =-p{({\varsigma }_n,{\zeta }_n)}^{t}n\tau \hfill \\ \hfill & \le 0.\hfill \end{array}$$

From the third property of the $\mathfrak{F}$-contraction and taking the limit as $n\to \infty$, we have

$$\underset{n\to \infty }{lim}np{({\varsigma }_n,{\zeta }_n)}^t=0$$

Hence, there exist $n_1\in \mathbb{N}$ such that $np{({\varsigma }_n,{\zeta }_n)}^t\le 1$ for all $n\ge n_1$.

Therefore, $p({\varsigma }_n,{\zeta }_n)\le {\displaystyle \frac{1}{n^{1/t}}}$ for all $n\ge n_1$.

Again,

$$\begin{array}{cc}\hfill \mathfrak{F}(p({\varsigma }_n,{\zeta }_{n+1}))& =\mathfrak{F}(p(f({\varsigma }_{n-1}),f({\zeta }_n)))\hfill \\ \hfill & \le c_1\mathfrak{F}(p({\varsigma }_{n-1},{\zeta }_n))-\tau \hfill \\ \hfill ⋮\\ \hfill & =c_1^n\mathfrak{F}(p({\varsigma }_0,{\zeta }_1))-n\tau \hfill \end{array}$$

Taking the limit as $n\to \infty$,

$$\underset{n\to \infty }{lim}\mathfrak{F}(p({\varsigma }_n,{\zeta }_{n+1}))=-\infty$$

From the second property of the F-contraction, we have

$$\underset{n\to \infty }{lim}p({\varsigma }_n,{\zeta }_{n+1})=0$$

Hence, from the third property of the F-contraction for all $n\in \mathbb{N}$, there exist $t\in (0,1)$, such that

$$\underset{n\to \infty }{lim}p{({\varsigma }_n,{\zeta }_{n+1})}^t\mathfrak{F}(p({\varsigma }_n,{\zeta }_{n+1}))=0$$

For all $n\in \mathbb{N}$, we have

$$\begin{array}{cc}\hfill p{({\varsigma }_n,{\zeta }_{n+1})}^t\mathfrak{F}(p({\varsigma }_n,{\zeta }_{n+1}))& -p{({\varsigma }_n,{\zeta }_{n+1})}^t\mathfrak{F}(p({\varsigma }_0,{\zeta }_1))\hfill \\ \hfill & \le p{({\varsigma }_n,{\zeta }_{n+1})}^t(c_1^n\mathfrak{F}(p({\varsigma }_0,{\zeta }_1))-n\tau )-p{({\varsigma }_n,{\zeta }_{n+1})}^t\mathfrak{F}(p({\varsigma }_0,{\zeta }_1))\hfill \\ \hfill & =-p{({\varsigma }_n,{\zeta }_{n+1})}^{t}n\tau \hfill \\ \hfill & \le 0.\hfill \end{array}$$

From the third property of the $\mathfrak{F}$-contraction and taking the limit as $n\to \infty$, we have

$$\underset{n\to \infty }{lim}np{({\varsigma }_n,{\zeta }_{n+1})}^t=0$$

Hence, there exist $n_2\in \mathbb{N}$, such that $np{({\varsigma }_n,{\zeta }_{n+1})}^t\le 1$ for all $n\ge n_2$.

Therefore, $p({\varsigma }_n,{\zeta }_{n+1})\le {\displaystyle \frac{1}{n^{1/t}}}$ for all $n\ge n_2$.

Consider $n_0=max\{n_1,n_2\}$.

For some $1\le n<m$, we have

$$\begin{array}{cc}\hfill p({\varsigma }_n,{\zeta }_m)& \le \mathrm{\Omega }[p({\varsigma }_n,{\zeta }_n)+p({\varsigma }_{n+1},{\zeta }_n)+p({\varsigma }_{n+1},{\zeta }_m)]\hfill \\ \hfill \Rightarrow {\mathrm{\Omega }}^{-1}(p({\varsigma }_n,{\zeta }_m))& \le {\displaystyle \frac{1}{n^{\frac{1}{t}}}}+{\displaystyle \frac{1}{n^{\frac{1}{t}}}}+p({\varsigma }_{n+1},{\zeta }_m), \mathrm{for} \mathrm{all} n\ge n_0\hfill \\ \hfill & ={\displaystyle \frac{1}{n^{\frac{1}{t}}}}+{\displaystyle \frac{1}{n^{\frac{1}{t}}}}+\mathrm{\Omega }\left[p({\varsigma }_{n+1},{\zeta }_{n+1})+p({\varsigma }_{n+2},{\zeta }_{n+1})+p({\varsigma }_{n+2},{\zeta }_m)\right]\hfill \\ \hfill \Rightarrow {\mathrm{\Omega }}^{-2}(p({\varsigma }_n,{\zeta }_m))& \le {\mathrm{\Omega }}^{-1}\left[{\displaystyle \frac{1}{n^{\frac{1}{t}}}}+{\displaystyle \frac{1}{n^{\frac{1}{t}}}}\right]+{\displaystyle \frac{1}{{(n+1)}^{\frac{1}{t}}}}+{\displaystyle \frac{1}{{(n+1)}^{\frac{1}{t}}}}+p({\varsigma }_{n+2},{\zeta }_m)\hfill \end{array}$$

Proceeding in a similar way, we have

$$\begin{array}{cc}\hfill {\mathrm{\Omega }}^{-(m-n+1)}(p({\varsigma }_n,{\zeta }_m))& \le {\mathrm{\Omega }}^{-(m-n)}\left[{\displaystyle \frac{1}{n^{\frac{1}{t}}}}+{\displaystyle \frac{1}{n^{\frac{1}{t}}}}\right]+{\mathrm{\Omega }}^{-(m-n)}\left[{\displaystyle \frac{1}{{(n+1)}^{\frac{1}{t}}}}+{\displaystyle \frac{1}{{(n+1)}^{\frac{1}{t}}}}\right]\hfill \\ \hfill & +\dots +{\mathrm{\Omega }}^{-1}\left[{\displaystyle \frac{1}{{(m-1)}^{\frac{1}{t}}}}+{\displaystyle \frac{1}{{(m-1)}^{\frac{1}{t}}}}\right]+p({\varsigma }_{m+1},{\zeta }_m)\hfill \\ \hfill & \le \sum _{i=n}^m{\mathrm{\Omega }}^{-(m-i)}\left[{\displaystyle \frac{1}{i^{\frac{1}{t}}}}+{\displaystyle \frac{1}{i^{\frac{1}{t}}}}\right], \mathrm{for} \mathrm{all} n\ge n_0\hfill \end{array}$$

Hence,

$$p({\varsigma }_n,{\zeta }_m)\le {\mathrm{\Omega }}^{(m-n+1)}\left(\sum _{i=n}^m{\mathrm{\Omega }}^{-(m-i)}\left[{\displaystyle \frac{1}{i^{\frac{1}{t}}}}+{\displaystyle \frac{1}{i^{\frac{1}{t}}}}\right]\right)$$

(1)

Similarly, for any $1\le m<n$, we can show that

$$p({\varsigma }_n,{\zeta }_m)\le {\mathrm{\Omega }}^{(m-n+1)}\left(\sum _{i=m}^n{\mathrm{\Omega }}^{-(n-i)}\left[{\displaystyle \frac{1}{i^{\frac{1}{t}}}}+{\displaystyle \frac{1}{i^{\frac{1}{t}}}}\right]\right)$$

(2)

Since $t\in (0,1)$, the right hand sides of Equations (1) and (2) trend toward 0 as $m,n\to \infty$.

Hence, the series is bi-convergent and $\{{\varsigma }_n,{\zeta }_n\}$ is a Cauchy bisequence in $(\mathfrak{X},\mathfrak{Y})$. Let the $\{{\varsigma }_n,{\zeta }_n\}$ biconverge to some $u\in \mathfrak{X}\bigcap \mathfrak{Y}$. Then, $\left\{{\varsigma }_n\right\}\to u$ and $\left\{{\zeta }_n\right\}\to u$, where $u\in \mathfrak{X}\bigcap \mathfrak{Y}$ and $\left\{f({\varsigma }_n)\right\}=\left\{{\varsigma }_{n+1}\right\}$ where $u\in \mathfrak{X}\bigcap \mathfrak{Y}$. Again, since f is continuous, $f({\varsigma }_n)\to f(u)$. Therefore, $f(u)=u$. Hence, u is a fixed point of f. If possible, let v is another fixed point of f. Then, we have $f(v)=v$, for some $v\in \mathfrak{X}\bigcap \mathfrak{Y}$.

Then,

$$\tau \le \mathfrak{F}(p(f(u),f(v)))-\mathfrak{F}(p(u,v))=0$$

which is a contradiction. Hence, $u=v$. Therefore, f has a unique fixed point in $(\mathfrak{X},\mathfrak{Y},p)$. □

Theorem 2. 

Consider a complete bipolar p-MS $(\mathfrak{X},\mathfrak{Y},p)$ for some $\mathrm{\Omega }\in {\psi }^{*}$ and a function $f:(\mathfrak{X},\mathfrak{Y},p)\rightleftharpoons (\mathfrak{X},\mathfrak{Y},p)$, which is contravariant such that

$$\tau +\mathfrak{F}(p(f(\zeta ),f(\varsigma )))\le c_1\mathfrak{F}(p(\varsigma ,\zeta ))+c_2\mathfrak{F}(p(\varsigma ,f(\varsigma )))+c_3\mathfrak{F}(p(f(\zeta ),\zeta )),$$

for $(\varsigma ,y)\in X\times Y$ and $\tau >0$, where $c_1,c_2,c_3\ge 0$, such that

$c_1+c_2+c_3<1$ and $\left({\displaystyle \frac{c_1+c_3}{1-c_2}}\right)\left({\displaystyle \frac{c_1+c_2}{1-c_2}}\right)\in {\Delta }_{\mathrm{\Omega }}$.

Then, the function $f:\mathfrak{X}\bigcup \mathfrak{Y}\to \mathfrak{X}\bigcup \mathfrak{Y}$ ∀ $t>0$, $c_{3}t<{\mathrm{\Omega }}^{-1}(t)$ has a unique fixed point.

Proof. 

Consider ${\varsigma }_0\in \mathfrak{X}$. Let us construct two iterative sequences ${\varsigma }_n\subset \mathfrak{X}$ and ${\zeta }_n\subset \mathfrak{Y}$ such that for some $n\ge 0$, we construct ${\zeta }_n=f({\varsigma }_n)$ and ${\varsigma }_{n+1}=f({\zeta }_n)$, for all $n\in \mathbb{N}$.

Then, we have

$$\begin{array}{cc}\hfill \tau +\mathfrak{F}(p({\varsigma }_n,{\zeta }_n))& =\tau +\mathfrak{F}(p(f({\zeta }_{n-1}),f({\varsigma }_n)))\hfill \\ \hfill & \le c_1\mathfrak{F}(p({\varsigma }_n,{\zeta }_{n-1}))+c_2\mathfrak{F}(p({\varsigma }_n,f({\varsigma }_n)))\hfill \\ \hfill & +c_3\mathfrak{F}(p(f({\zeta }_{n-1}),{\zeta }_{n-1}))\hfill \\ \hfill & =(c_1+c_3)\mathfrak{F}(p({\varsigma }_n,{\zeta }_{n-1}))+c_2\mathfrak{F}(p({\varsigma }_n,{\zeta }_n)), \mathrm{for} \mathrm{all} n\ge 1\hfill \\ \hfill \Rightarrow \mathfrak{F}(p({\varsigma }_n,{\zeta }_n))-c_2\mathfrak{F}(p({\varsigma }_n,{\zeta }_n))& \le (c_1+c_3)\mathfrak{F}(p({\varsigma }_n,{\zeta }_{n-1}))-\tau \hfill \\ \hfill \Rightarrow \mathfrak{F}(p({\varsigma }_n,{\zeta }_n))& \le \left({\displaystyle \frac{c_1+c_3}{1-c_2}}\right)\mathfrak{F}(p({\varsigma }_n,{\zeta }_{n-1}))-{\displaystyle \frac{1}{1-c_2}}\tau .\hfill \end{array}$$

Again,

$$\begin{array}{cc}\hfill \tau +\mathfrak{F}(p({\varsigma }_n,{\zeta }_{n-1}))& =\tau +\mathfrak{F}(p(f({\zeta }_{n-1}),f({\varsigma }_{n-1})))\hfill \\ \hfill & \le c_1\mathfrak{F}(p({\varsigma }_{n-1},{\zeta }_{n-1}))+c_2\mathfrak{F}(p({\varsigma }_{n-1},f({\varsigma }_{n-1})))\hfill \\ \hfill & +c_3\mathfrak{F}(p(f({\zeta }_{n-1}),{\zeta }_{n-1}))\hfill \\ \hfill & =(c_1+c_2)\mathfrak{F}(p({\varsigma }_{n-1},{\zeta }_{n-1}))+c_3\mathfrak{F}(p({\varsigma }_n,{\zeta }_{n-1}))\hfill \\ \hfill \Rightarrow (1-c_3)\mathfrak{F}(p({\varsigma }_n,{\zeta }_{n-1}))& \le (c_1+c_2)\mathfrak{F}(p({\varsigma }_{n-1},{\zeta }_{n-1}))-\tau \hfill \\ \hfill \Rightarrow \mathfrak{F}(p({\varsigma }_n,{\zeta }_{n-1}))& \le \left({\displaystyle \frac{c_1+c_2}{1-c_3}}\right)\mathfrak{F}(p({\varsigma }_{n-1},{\zeta }_{n-1}))-{\displaystyle \frac{1}{1-c_3}}\tau , \mathrm{for} \mathrm{all} n\ge 1.\hfill \end{array}$$

Therefore we have,

$$\mathfrak{F}(p({\varsigma }_n,{\zeta }_n))\le \left({\displaystyle \frac{c_1+c_3}{1-c_2}}\right)\left({\displaystyle \frac{c_1+c_2}{1-c_3}}\right)\mathfrak{F}(p({\varsigma }_{n-1},{\zeta }_{n-1}))-\left({\displaystyle \frac{c_1+c_3}{1-c_2}}\right)\left({\displaystyle \frac{1}{1-c_3}}\right)\tau -{\displaystyle \frac{1}{1-c_2}}\tau$$

Let $\lambda =\left({\displaystyle \frac{c_1+c_3}{1-c_2}}\right)\left({\displaystyle \frac{c_1+c_2}{1-c_3}}\right)$, therefore

$$\mathfrak{F}(p({\varsigma }_n,{\zeta }_n))\le \lambda \mathfrak{F}(p({\varsigma }_{n-1},{\zeta }_{n-1}))-\left({\displaystyle \frac{c_1+c_3}{1-c_3}}-1\right)\left({\displaystyle \frac{1}{1-c_2}}\right)\tau$$

Proceeding in this way, we have

$$\mathfrak{F}(p({\varsigma }_n,{\zeta }_n))\le {\lambda }^n\mathfrak{F}(p({\varsigma }_0,{\zeta }_0))-n\left({\displaystyle \frac{c_1+c_3}{1-c_3}}-1\right)\left({\displaystyle \frac{1}{1-c_2}}\right)\tau$$

Taking the limit as $n\to \infty$,

$$\underset{n\to \infty }{lim}\mathfrak{F}(p({\varsigma }_n,{\zeta }_n))=-\infty$$

Thus, from the second property of the $\mathfrak{F}$-contraction,

$$\underset{n\to \infty }{lim}p({\varsigma }_n,{\zeta }_n)=0.$$

Hence, from the third property of the $\mathfrak{F}$-contraction, $\forall n\in \mathbb{N}$, ∃ $t\in (0,1)$ such that

$$\underset{n\to \infty }{lim}p{({\varsigma }_n,{\zeta }_n)}^t\mathfrak{F}(p({\varsigma }_n,{\zeta }_n))=0.$$

For all $n\in \mathbb{N}$, we have

$$\begin{array}{cc}\hfill p{({\varsigma }_n,{\zeta }_n)}^t\mathfrak{F}(p({\varsigma }_n,{\zeta }_n))& -p{({\varsigma }_n,{\zeta }_n)}^t\mathfrak{F}(p({\varsigma }_0,{\zeta }_0))\hfill \\ \hfill & \le p{({\varsigma }_n,{\zeta }_n)}^t({\lambda }^n\mathfrak{F}(p({\varsigma }_0,{\zeta }_0))-n\left({\displaystyle \frac{c_1+c_3}{1-c_3}}-1\right)\left({\displaystyle \frac{1}{1-c_2}}\right)\tau )\hfill \\ \hfill & -p{({\varsigma }_n,{\zeta }_n)}^t\mathfrak{F}(p({\varsigma }_0,{\zeta }_0))\hfill \\ \hfill & =-p{({\varsigma }_n,{\zeta }_n)}^{t}n\left({\displaystyle \frac{c_1+c_3}{1-c_3}}-1\right)\left({\displaystyle \frac{1}{1-c_2}}\right)\tau \hfill \\ \hfill & \le 0.\hfill \end{array}$$

From the third property of the $\mathfrak{F}$-contraction and considering the limit as $n\to \infty$, we have

$$\underset{n\to \infty }{lim}n\left({\displaystyle \frac{c_1+c_3}{1-c_3}}-1\right)\left({\displaystyle \frac{1}{1-c_2}}\right)p{({\varsigma }_n,{\zeta }_n)}^t=0$$

Hence, there exist $n_1\in \mathbb{N}$ such that $np{({\varsigma }_n,{\zeta }_n)}^t\le 1$ for all $n\ge n_1$.

Therefore, $p({\varsigma }_n,{\zeta }_n)\le {\displaystyle \frac{1}{n^{1/t}}}$ for all $n\ge n_1$.

Again,

$$\begin{array}{cc}\hfill \mathfrak{F}(p({\varsigma }_{n+1},{\zeta }_n))& =\mathfrak{F}(p(f({\zeta }_n),f({\varsigma }_n)))\hfill \\ \hfill & \le \left({\displaystyle \frac{c_1+c_2}{1-c_2}}\right)\mathfrak{F}(p({\varsigma }_{n+1},{\zeta }_n))-\left({\displaystyle \frac{c_1+c_3}{1-c_3}}-1\right)\left({\displaystyle \frac{1}{1-c_2}}\right)\tau \hfill \\ \hfill ⋮\\ \hfill & \le {\lambda }^n\mathfrak{F}(p({\varsigma }_1,{\zeta }_0))-n\left({\displaystyle \frac{c_1+c_3}{1-c_3}}-1\right)\left({\displaystyle \frac{1}{1-c_2}}\right)\tau \hfill \\ \hfill & \le {\lambda }^n\left({\displaystyle \frac{c_1+c_2}{1-c_3}}\right)\mathfrak{F}(p({\varsigma }_0,{\zeta }_0))-n\left({\displaystyle \frac{c_1+c_2}{1-c_3}}\right)\left({\displaystyle \frac{c_1+c_3}{1-c_3}}-1\right)\left({\displaystyle \frac{1}{1-c_2}}\right)\tau \hfill \end{array}$$

Taking the limit as $n\to \infty$,

$$\underset{n\to \infty }{lim}\mathfrak{F}(p({\varsigma }_{n+1},{\zeta }_n))=-\infty$$

From the second property of the $\mathfrak{F}$-contraction, we have

$$\underset{n\to \infty }{lim}p({\varsigma }_{n+1},{\zeta }_n)=0$$

Hence, from the third property of the $\mathfrak{F}$-contraction for all $n\in \mathbb{N}$, there exist $t\in (0,1)$, such that

$$\underset{n\to \infty }{lim}p{({\varsigma }_{n+1},{\zeta }_n)}^t\mathfrak{F}(p({\varsigma }_{n+1},{\zeta }_n))=0$$

For all $n\in \mathbb{N}$, we have

$$\begin{array}{cc}\hfill p{({\varsigma }_{n+1},{\zeta }_n)}^t\mathfrak{F}(p({\varsigma }_{n+1},{\zeta }_n))& -p{({\varsigma }_{n+1},{\zeta }_n)}^t\mathfrak{F}(p({\varsigma }_1,{\zeta }_0))\hfill \\ \hfill & \le p{({\varsigma }_{n+1},{\zeta }_n)}^t\left[{\lambda }^n\mathfrak{F}(p({\varsigma }_1,{\zeta }_0))-n\left({\displaystyle \frac{c_1+c_3}{1-c_3}}-1\right)\left({\displaystyle \frac{1}{1-c_2}}\right)\tau \right]\hfill \\ \hfill & -p{({\varsigma }_{n+1},{\zeta }_n)}^t\mathfrak{F}(p({\varsigma }_1,{\zeta }_0))\hfill \\ \hfill & =-p{({\varsigma }_{n+1},{\zeta }_n)}^{t}n\left({\displaystyle \frac{c_1+c_3}{1-c_3}}-1\right)\left({\displaystyle \frac{1}{1-c_2}}\right)\tau \hfill \\ \hfill & \le 0.\hfill \end{array}$$

From the third property of the $\mathfrak{F}$-contraction and taking the limit as $n\to \infty$, we have

$$\underset{n\to \infty }{lim}np{({\varsigma }_{n+1},{\zeta }_n)}^t=0$$

Hence, there exist $n_2\in \mathbb{N}$, such that $np{({\varsigma }_n,{\zeta }_{n+1})}^t\le 1$ for all $n\ge n_2$.

Therefore, $p({\varsigma }_{n+1},{\zeta }_n)\le {\displaystyle \frac{1}{n^{1/t}}}$ for all $n\ge n_2$.

Consider $n_0=max\{n_1,n_2\}$.

For any $1\le n<m$, we have

$$\begin{array}{cc}\hfill p({\varsigma }_n,{\zeta }_m)& \le \mathrm{\Omega }\left[p({\varsigma }_n,{\zeta }_n)+p({\varsigma }_{n+1},{\zeta }_n)+p({\varsigma }_{n+1},{\zeta }_m)\right]\hfill \\ \hfill \Rightarrow {\mathrm{\Omega }}^{-1}(p({\varsigma }_n,{\zeta }_m))& \le {\displaystyle \frac{1}{n^{\frac{1}{t}}}}+{\displaystyle \frac{1}{n^{\frac{1}{t}}}}+p({\varsigma }_{n+1},{\zeta }_m), \mathrm{for} \mathrm{all} n\ge n_0\hfill \\ \hfill & ={\displaystyle \frac{1}{n^{\frac{1}{t}}}}+{\displaystyle \frac{1}{n^{\frac{1}{t}}}}+\mathrm{\Omega }\left[p({\varsigma }_{n+1},{\zeta }_{n+1})+p({\varsigma }_{n+2},{\zeta }_{n+1})+p({\varsigma }_{n+2},{\zeta }_m)\right]\hfill \\ \hfill \Rightarrow {\mathrm{\Omega }}^{-2}(p({\varsigma }_n,{\zeta }_m))& \le {\mathrm{\Omega }}^{-1}\left[{\displaystyle \frac{1}{n^{\frac{1}{t}}}}+{\displaystyle \frac{1}{n^{\frac{1}{t}}}}\right]+{\displaystyle \frac{1}{{(n+1)}^{\frac{1}{t}}}}+{\displaystyle \frac{1}{{(n+1)}^{\frac{1}{t}}}}+p({\varsigma }_{n+2},{\zeta }_m)\hfill \end{array}$$

Proceeding in a similar way, we have

$$\begin{array}{cc}\hfill {\mathrm{\Omega }}^{-(m-n+1)}(p({\varsigma }_n,{\zeta }_m))& \le {\mathrm{\Omega }}^{-(m-n)}\left[{\displaystyle \frac{1}{n^{\frac{1}{t}}}}+{\displaystyle \frac{1}{n^{\frac{1}{t}}}}\right]+{\mathrm{\Omega }}^{-(m-n)}\left[{\displaystyle \frac{1}{{(n+1)}^{\frac{1}{t}}}}+{\displaystyle \frac{1}{{(n+1)}^{\frac{1}{t}}}}\right]\hfill \\ \hfill & +\dots +{\mathrm{\Omega }}^{-1}\left[{\displaystyle \frac{1}{{(m-1)}^{\frac{1}{t}}}}+{\displaystyle \frac{1}{{(m-1)}^{\frac{1}{t}}}}\right]+p({\varsigma }_{m+1},{\zeta }_m)\hfill \\ \hfill & \le \sum _{i=n}^m{\mathrm{\Omega }}^{-(m-i)}\left[{\displaystyle \frac{1}{i^{\frac{1}{t}}}}+{\displaystyle \frac{1}{i^{\frac{1}{t}}}}\right], \mathrm{for} \mathrm{all} n\ge n_0\hfill \end{array}$$

Hence,

$$p({\varsigma }_n,{\zeta }_m)\le {\mathrm{\Omega }}^{(m-n+1)}\left(\sum _{i=n}^m{\mathrm{\Omega }}^{-(m-i)}\left[{\displaystyle \frac{1}{i^{\frac{1}{t}}}}+{\displaystyle \frac{1}{i^{\frac{1}{t}}}}\right]\right)$$

(3)

Similarly, for any $1\le m<n$, we can show that

$$p({\varsigma }_n,{\zeta }_m)\le {\mathrm{\Omega }}^{(m-n+1)}\left(\sum _{i=m}^n{\mathrm{\Omega }}^{-(n-i)}\left[{\displaystyle \frac{1}{i^{\frac{1}{t}}}}+{\displaystyle \frac{1}{i^{\frac{1}{t}}}}\right]\right)$$

(4)

Since $t\in (0,1)$, the right hand sides of Equations (3) and (4) trend to 0 as $m,n\to \infty$. Hence the series is bi-convergent.

Therefore, $\{{\varsigma }_n,{\zeta }_n\}$ is a Cauchy bisequence in $(\mathfrak{X},\mathfrak{Y})$.

Let the $\{{\varsigma }_n,{\zeta }_n\}$ biconverge to some $\mathfrak{Z}\in \mathfrak{X}\bigcap \mathfrak{Y}$.

Then, we have

$$\tau +\mathfrak{F}(p(f(\mathfrak{Z}),f({\varsigma }_n)))\le c_1\mathfrak{F}(p(\mathfrak{Z},{\zeta }_n))+c_2\mathfrak{F}(p(\mathfrak{Z},f(\mathfrak{Z})))+c_3\mathfrak{F}(p(f({\zeta }_n),{\zeta }_n))$$

Moreover,

$$\begin{array}{cc}\hfill p(f(\mathfrak{Z}),\mathfrak{Z})& \le \mathrm{\Omega }\left[p(f(\mathfrak{Z}),f({\varsigma }_n))+p({\zeta }_n,f({\zeta }_n))+p({\zeta }_n,\mathfrak{Z})\right]\hfill \\ \hfill & \le \mathrm{\Omega }\left[c_1\mathfrak{F}(p(\mathfrak{Z},{\zeta }_n))+c_2\mathfrak{F}(p(\mathfrak{Z},f(\mathfrak{Z})))+c_3\mathfrak{F}(p(f({\zeta }_n),{\zeta }_n))+p({\zeta }_n,f({\zeta }_n))+p({\zeta }_n,\mathfrak{Z})\right]\hfill \end{array}$$

Taking the limit as $n\to \infty$, we obtain

$$p(f(\mathfrak{Z}),\mathfrak{Z})\le \mathrm{\Omega }\left[c_2\mathfrak{F}(p(\mathfrak{Z},f(\mathfrak{Z})))\right]$$

If $f(\mathfrak{Z})\ne \mathfrak{Z}$, then

$$p(f(\mathfrak{Z}),\mathfrak{Z})\le \mathrm{\Omega }\left[c_2\mathfrak{F}(p(\mathfrak{Z},f(\mathfrak{Z})))\right]<p(f(\mathfrak{Z}),\mathfrak{Z})$$

which is a contradiction.

Hence, $\mathfrak{Z}$ is a fixed point of f.

If possible, let ${\mathfrak{Z}}_2$ is another fixed point of f.

Then, we have for $\mathfrak{Z},{\mathfrak{Z}}_2\in \mathfrak{X}\bigcap \mathfrak{Y}$.

$$\begin{array}{cc}\hfill \tau +p(\mathfrak{Z},{\mathfrak{Z}}_2)& =\tau +p(f(\mathfrak{Z}),f({\mathfrak{Z}}_2))\hfill \\ \hfill & \le c_1\mathfrak{F}(p(\mathfrak{Z},{\mathfrak{Z}}_2))+c_2\mathfrak{F}(p(\mathfrak{Z},f(\mathfrak{Z})))+c_3\mathfrak{F}(p(f({\mathfrak{Z}}_2),{\mathfrak{Z}}_2))\hfill \\ \hfill & <\tau +p(\mathfrak{Z},{\mathfrak{Z}}_2)\hfill \end{array}$$

Therefore, $p(\mathfrak{Z},{\mathfrak{Z}}_2)=0\Rightarrow \mathfrak{Z}={\mathfrak{Z}}_2$.

Therefore f has a unique fixed point in $(\mathfrak{X},\mathfrak{Y},p)$.

□

4. Example

This section includes an example to validate Theorem 2.

Example 1. 

Consider $\mathfrak{X}=[0,2]$ and $\mathfrak{Y}=[2,5]$ and a mapping $p:\mathfrak{X}\times \mathfrak{Y}\to [0,\infty )$ defined as $p(\mathfrak{A},\mathfrak{B})={|\mathfrak{A}-\mathfrak{B}|}^2$ for all $(\mathfrak{A},\mathfrak{B})\in \mathfrak{X}\times \mathfrak{Y}$. Then for the function $\mathrm{\Omega }(t)=3t$ for all $t\ge 0$, $(\mathfrak{X},\mathfrak{Y},p)$ is a complete bipolar p-MS such that ∀ $\mathfrak{v}\in \mathfrak{X}\bigcup \mathfrak{Y}$:

$$f:(\mathfrak{X},\mathfrak{Y},p)\overrightarrow{\to }(\mathfrak{X},\mathfrak{Y},p) \mathit{defined} \mathit{as},$$

$$f(\mathfrak{V})={\displaystyle \frac{(\sqrt{2}+1)-\mathfrak{V}}{\sqrt{2}}}$$

Then, for $c_1={\displaystyle \frac{1}{2}}$, $c_2=0$, $c_3=0$, the map f is a contravariant map.

Next, consider $\tau =ln{\displaystyle \frac{1}{2}}$ and $\mathfrak{F}(\alpha )=ln(\alpha )$.

Now,

$$\begin{array}{cc}\hfill \tau +\mathfrak{F}(p(f(\varsigma ),f(\zeta )))=& ln{\displaystyle \frac{1}{2}}+\mathfrak{F}\left(p\left({\displaystyle \frac{(\sqrt{2}+1)-\varsigma }{\sqrt{2}}},{\displaystyle \frac{(\sqrt{2}+1)-\zeta }{\sqrt{2}}}\right)\right)\hfill \\ \hfill & =ln{\displaystyle \frac{1}{2}}+\mathfrak{F}\left({\left|{\displaystyle \frac{(\sqrt{2}+1)}{\sqrt{2}}}-{\displaystyle \frac{\varsigma }{\sqrt{2}}}-{\displaystyle \frac{(\sqrt{2}+1)}{\sqrt{2}}}+{\displaystyle \frac{\zeta }{\sqrt{2}}}\right|}^2\right)\hfill \\ \hfill & =ln{\displaystyle \frac{1}{2}}+\mathfrak{F}\left({\left|-{\displaystyle \frac{\varsigma }{\sqrt{2}}}+{\displaystyle \frac{\zeta }{\sqrt{2}}}\right|}^2\right)\hfill \\ \hfill & =ln{\displaystyle \frac{1}{2}}+ln{\left({\displaystyle \frac{\varsigma -\zeta }{\sqrt{2}}}\right)}^2\hfill \\ \hfill & =ln{\displaystyle \frac{1}{2}}+2\left[ln(\varsigma -\zeta )-ln\sqrt{2}\right]\hfill \\ \hfill & =2[ln(\varsigma -\zeta )]+\left(ln{\displaystyle \frac{1}{2}}-2ln\sqrt{2}\right)\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill c_1\mathfrak{F}(p(\varsigma ,\zeta ))& ={\displaystyle \frac{1}{2}}{\mathfrak{F}(|\varsigma -\zeta |}^2)\hfill \\ \hfill & ={\displaystyle \frac{1}{2}}{ln(|\varsigma -\zeta |}^2)\hfill \\ \hfill & =ln|\varsigma -\zeta |.\hfill \end{array}$$

Therefore,

$$\tau +\mathfrak{F}(p(f(\zeta ),f(\varsigma )))\le c_1\mathfrak{F}(p(\varsigma ,\zeta ))+c_2\mathfrak{F}(p(\varsigma ,f(\varsigma )))+c_3\mathfrak{F}(p(f(\zeta ),\zeta )),$$

for $(\varsigma ,\zeta )\in \mathfrak{X}\times \mathfrak{Y}$, where $c_1,c_2,c_3\ge 0$, such that $c_1+c_2+c_3<1$ and ${\displaystyle \frac{1}{4}}\in {\Delta }_{\mathrm{\Omega }}$.

Thus, we observe that all conditions of Theorem 2 are satisfied by f.

Hence, f has a unique fixed point in $(\mathfrak{X},\mathfrak{Y},p)$.

5. Conclusions

In this paper, we established new extended versions of a covariant Banach-type fixed-point theorem and a contravariant Rich-type fixed-point theorem in a complete bipolar p-metric space using the concept of F-contraction. As a results of this work, several existing results in the literature on Banach- and Reich-type fixed-point theorems (such as Theorems 3.2 and 3.4 in [24]) may be thought of as special cases of Theorem 1 and Theorem 2, respectively. This work can be extended in future to investigate some new results of the fixed points under different types of contractions as indicated in [5] in bipolar p-metric space. Further, multivalued versions of our results may be investigated, as achieved in the recent interesting paper [22]. Common fixed-point results of such covariant and contravariant mappings may also be studied following the directions of [23].