Fixed-Point Theorems for Covariant and Contravariant Multivalued Mappings in Bipolar b-Metric Spaces

Abstract

The purpose of this work is to present the notions of the Pompeiu–Hausdorff bipolar b-metric for multivalued covariant and contravariant contraction mappings in bipolar b-metric spaces. We also develop three essential fixed-point theorems, which are backed up by three significant corollaries.

1. Introduction

One of the most powerful and productive tools from nonlinear analysis is fixed-point theory. It is an important mathematical discipline because of its applications in different areas such as differential equations, optimization theory, and variational analysis. In this theory, the concept of metric space and the Banach contraction principle are fundamental. By employing different contraction mappings in various metric spaces, many researchers have developed the Banach contraction principle [1,2,3,4,5].

The concept of b-metric spaces, first introduced in the foundational works [6,7] and further developed in [8], generalizes classical metric spaces by relaxing the triangle inequality. This broader framework extends the applicability of fixed-point results, such as the Banach contraction principle, and has been widely utilized in various mathematical studies. Building on this generalization, recent work by Belhenniche et al. [9] advanced the theory by introducing extended b-metric spaces, where they established a common fixed-point theorem for Ćirić-type operators to tackle nonlinear and dynamic programming equations.

The authors of [10] introduced the notion of bipolar metric spaces as a form of partial distance. In this work, they investigated the relationship between traditional metric spaces and bipolar metric spaces, focusing particularly on the property of completeness. Furthermore, they established several extensions of classical fixed-point theorems within this new setting. After that, many authors proved popular theorems of fixed-point theory in bipolar metric spaces (see [11,12,13,14,15]). In [16] Karapınar and Cvetković investigated the relationship between results obtained in bipolar metric space and analogous fixed-point results in metric space, ultimately reaching equivalence.

On the other hand, Markin [17] initiated the study of fixed points for multivalued contraction mappings using the Hausdorff metric. Later, Multu et al. [18] introduced the concepts of the Pompeiu–Hausdorff bipolar metric and multivalued covariant and contravariant contraction mappings in bipolar metric spaces and recovered well-known classical results.

Very recently, in 2024, Sedghi et al. [19] introduced the concept of bipolar b-metric space. They further clarified the relationship between b-metric spaces and bipolar b-metric spaces and extended established fixed-point theorems, including Banach’s fixed-point theorem.

Motivated by the above results, this work presents the concepts of the Pompeiu–Hausdorff bipolar b-metric and multivalued covariant and contravariant contraction mappings in bipolar b-metric spaces. We also provide three important fixed-point theorems associated with these multivalued mappings, which are backed up by three significant corollaries. Lastly, we provide an example that illustrates how our results can be applied.

Definition 1

([19]). Let ℵ and ${\aleph }^{*}$ be two nonempty sets. Consider a metric, $\varpi :\aleph \times {\aleph }^{*}\to {\mathbb{R}}^{+}$. The function ϖ is called bipolar b-metric on the pair $(\aleph ,{\aleph }^{*})$; it satisfies the following:

1. 

$\varpi (\upsilon ,\zeta )=0$ if and only if $\upsilon =\zeta$;

2. 

$\varpi (\upsilon ,\zeta )=\varpi (\zeta ,\upsilon )$ for all $\upsilon ,\zeta \in \aleph \cap {\aleph }^{*}$;

3. 

$\varpi (\upsilon ,\zeta )\le s\left[\varpi (\upsilon ,{\zeta }_1)+\varpi ({\upsilon }_1,{\zeta }_1)+\varpi ({\upsilon }_1,\zeta )\right]$ for all $\upsilon ,{\upsilon }_1\in \aleph$, $\zeta ,{\zeta }_1\in {\aleph }^{*}$, and $s\ge 1$.

The triple $\left(\aleph ,{\aleph }^{*},\varpi \right)$ is called bipolar b-metric space.

Definition 2

([19]). Let $(\aleph ,{\aleph }^{*},\varpi )$ be a bipolar b-metric space.

1. 

The set ℵ is called the left pole, ${\aleph }^{*}$ is called the right pole, and $\aleph \cap {\aleph }^{*}$ is called the center of $(\aleph ,{\aleph }^{*},\varpi )$. Especially, the points on the left pole are called left points, the points on the right pole are called right points, and the points in the center are called central points.

2. 

The sequence $\left\{{\upsilon }_n\right\}\subseteq \aleph$ is called a left sequence, and the sequence $\left\{{\zeta }_n\right\}\subseteq {\aleph }^{*}$ is called a right sequence.

3. 

The sequence $\left\{{\upsilon }_n\right\}$ is said to be convergent to the point υ, if and only if $\left\{{\upsilon }_n\right\}$ is a left sequence, υ is a right point, and ${lim}_{n\to \infty }\varpi \left({\upsilon }_n,\upsilon \right)=0$ or if $\left\{{\upsilon }_n\right\}$ is a right sequence, υ is a left point, and ${lim}_{n\to \infty }\varpi \left(\upsilon ,{\upsilon }_n\right)=0$.

4. 

The bi-sequence $\left\{\left({\upsilon }_n,{\zeta }_n\right)\right\}$ on $(\aleph ,{\aleph }^{*},\varpi )$ is a sequence on the set $\aleph \times {\aleph }^{*}$. If the sequences $\left\{{\upsilon }_n\right\}$ and $\left\{{\zeta }_n\right\}$ are convergent, then the bi-sequence $\left\{\left({\upsilon }_n,{\zeta }_n\right)\right\}$ is said to be convergent, and if $\left\{{\upsilon }_n\right\}$ and $\left\{{\zeta }_n\right\}$ converge to a common fixed point, then $\left\{\left({\upsilon }_n,{\zeta }_n\right)\right\}$ called biconvergent.

5. 

$\left\{\left({\upsilon }_n,{\zeta }_n\right)\right\}$ is a Cauchy bi-sequence if ${\displaystyle \underset{n,l⟶\infty }{lim}\varpi \left({\upsilon }_n,{\zeta }_l\right)=0}$.

6. 

A bipolar b-metric space is said to be complete if every Cauchy bi-sequence is convergent.

Definition 3

([19]). Consider the following: let $\left({\aleph }_1,{\aleph }_1^{*},{\varpi }_1\right)$ and $\left({\aleph }_2,{\aleph }_2^{*},{\varpi }_2\right)$ be bipolar $b-$metric spaces and $L:{\aleph }_1\cup {\aleph }_1^{*}\to {\aleph }_2\cup {\aleph }_2^{*}$ be a function.

1. 

If $L\left({\aleph }_1\right)\subseteq {\aleph }_2$ and $L\left({\aleph }_1^{*}\right)\subseteq {\aleph }_2^{*}$, then L is called a covariant map, or a map from $\left({\aleph }_1,{\aleph }_1^{*},{\varpi }_1\right)$ to $\left({\aleph }_2,{\aleph }_2^{*},{\varpi }_2\right)$, and this is written as $L:\left({\aleph }_1,{\aleph }_1^{*},{\varpi }_1\right)\rightrightarrows \left({\aleph }_2,{\aleph }_2^{*},{\varpi }_2\right)$.

2. 

If $L\left({\aleph }_1\right)\subseteq {\aleph }_2^{*}$ and $L\left({\aleph }_1^{*}\right)\subseteq {\aleph }_2$, then L is called a contravariant map from $\left({\aleph }_1,{\aleph }_1^{*},{\varpi }_1\right)$ to $\left({\aleph }_2,{\aleph }_2^{*},{\varpi }_2\right)$, and this is denoted as $L:\left({\aleph }_1,{\aleph }_1^{*},{\varpi }_1\right)\leftrightarrows \left({\aleph }_2,{\aleph }_2^{*},{\varpi }_2\right)$.

Definition 4

([19]). Consider the following: $\left({\aleph }_1,{\aleph }_1^{*},{\varpi }_1\right)$ and $\left({\aleph }_2,{\aleph }_2^{*},{\varpi }_2\right)$. Let $b-$metric spaces be bipolar.

1. 

The map $L:\left({\aleph }_1,{\aleph }_1^{*},{\varpi }_1\right)\rightrightarrows \left({\aleph }_2,{\aleph }_2^{*},{\varpi }_2\right)$ is said to be left-continuous at ${\upsilon }_0\in {\aleph }_1$, if

$$\forall \epsilon >0,\exists \delta >0:{\varpi }_1\left({\upsilon }_0,\zeta \right)<\delta \Rightarrow {\varpi }_2\left(L{\upsilon }_0,L\zeta \right)<\epsilon ,\forall \zeta \in {\aleph }_1^{*}.$$

2. 

The map $L:\left({\aleph }_1,{\aleph }_1^{*},{\varpi }_1\right)\rightrightarrows \left({\aleph }_2,{\aleph }_2^{*},{\varpi }_2\right)$ is said to be right-continuous at ${\zeta }_0\in {\aleph }_1^{*}$, if

$$\forall \epsilon >0,\exists \delta >0:{\varpi }_1\left(\upsilon ,{\zeta }_0\right)<\delta \Rightarrow {\varpi }_2\left(L\upsilon ,L{\zeta }_0\right)<\epsilon ,\forall \upsilon \in {\aleph }_1.$$

3. 

The map L is said to be continuous if it is left-continuous at each point $\upsilon \in {\aleph }_1$, and right-continuous at each point $\zeta \in {\aleph }_1^{*}$.

4. 

The contravariant map $L:\left({\aleph }_1,{\aleph }_1^{*}\right)\rightleftarrows \left({\aleph }_2,{\aleph }_2^{*}\right)$ is continuous if and only if it is continuous as the covariant map $L:\left({\aleph }_1,{\aleph }_1^{*}\right)\rightrightarrows \left({\aleph }_2^{*},{\aleph }_2\right)$.

2. Main Result

Definition 5.

Consider $(\aleph ,{\aleph }^{*},\varpi )$ a bipolar b-metric space. The set $A\subseteq \aleph \cup {\aleph }^{*}$ is called closed if every limit of the convergent sequence in A belongs to A.

Definition 6.

Consider $(\aleph ,{\aleph }^{*},\varpi )$ a bipolar b-metric space.

1. 

The set $M\subseteq \aleph$ is called bounded if $\delta \left(M\right)=sup\left\{\varpi \right(m,\zeta ):m\in M\}<\infty$ for all $\zeta \in {\aleph }^{*}$.

2. 

The set $N\subseteq {\aleph }^{*}$ is called bounded if $\delta \left(N\right)=sup\left\{\varpi \right(\upsilon ,n):n\in N\}<\infty$ for all $\upsilon \in \aleph$.

Definition 7.

Consider $(\aleph ,{\aleph }^{*},\varpi )$ a bipolar b-metric space. We denote

$$\begin{array}{cc}\hfill CB(\aleph )& =\{M:M\mathit{is\; a\; nonempty\; closed\; and\; bounded\; subset\; of}\aleph \}\hfill \\ \hfill CB\left({\aleph }^{*}\right)& =\{N:N\mathit{is\; a\; nonempty\; closed\; and\; bounded\; subset\; of}{\aleph }^{*}\}\hfill \\ \hfill \varpi (a,N)& =inf\{\varpi (a,n):n\in N\subset {\aleph }^{*}\},a\in \aleph \hfill \\ \hfill D(M,b)& =inf\{\varpi (m,b):m\in M\subset \aleph \},b\in {\aleph }^{*}\hfill \\ \hfill H(M,N)& =max\{sup\{D(m,N):m\in M\},sup\{D(M,n):n\in N\left\}\right\}\hfill \end{array}$$

for all $A\in CB(\aleph )$ and $B\in CB\left({\aleph }^{*}\right)$. Then H is a bipolar b-metric on $(CB(\aleph ),CB\left({\aleph }^{*}\right))$, called the Pompeiu–Hausdorff bipolar b-metric induced by the bipolar b-metric ϖ.

Lemma 1.

Let $(\aleph ,{\aleph }^{*},\varpi )$ be a bipolar b-metric space, $A\in CB(\aleph )$, $B\in CB\left({\aleph }^{*}\right)$, and $h>1$. Then there exists $b=b\left(a\right)\in B$ for any $a\in A$ (or there exists $a=a\left(b\right)\in A$ for any $b\in B$) such that

$$\varpi (a,b)\le hH(A,B).$$

Lemma 2.

Let $(\aleph ,{\aleph }^{*},\varpi )$ be a bipolar b-metric space with $s\ge 1$, and suppose that $\left\{{\upsilon }_n\right\}\subset \aleph$ and $\left\{{\zeta }_n\right\}\subset {\aleph }^{*}$ are convergent to ζ and υ, respectively, where $\zeta \in {\aleph }^{*}$ and $\upsilon \in \aleph$. Then we have

$${\displaystyle \frac{1}{s}}\varpi (\upsilon ,\zeta )\le \underset{n\to \infty }{lim\; inf}\varpi ({\upsilon }_n,{\zeta }_n)\le \underset{n\to \infty }{lim\; sup}\varpi ({\upsilon }_n,{\zeta }_n)\le s\varpi (\upsilon ,\zeta ).$$

In particular, if $s=1$, then we have

$$\underset{n\to \infty }{lim}\varpi ({\upsilon }_n,{\zeta }_n)=\varpi (\upsilon ,\zeta ).$$

The first main result is the following.

Theorem 1.

Let $(\aleph ,{\aleph }^{*},\varpi )$ be a complete bipolar b-metric space and $Q:(\aleph ,{\aleph }^{*})\rightrightarrows (CB(\aleph ),CB\left({\aleph }^{*}\right))$ be a multivalued covariant mapping, such that there exists $\lambda \in (0,1)$:

$$H(Q\upsilon ,Q\zeta )\le \lambda \varpi (\upsilon ,\zeta ),\forall \upsilon \in \aleph ,\forall \zeta \in {\aleph }^{*}.$$

(1)

Then Q has a fixed point $(p\in Qp)$.

Proof. 

Let ${\upsilon }_0\in \aleph ,{\zeta }_0\in {\aleph }^{*}$, by Lemma 1, for any $h>1$ and $n\in \mathbb{N}$

$$\exists {\zeta }_n\in Q{\zeta }_{n-1}:\varpi ({\upsilon }_n,{\zeta }_n)\le hH(Q{\upsilon }_{n-1},Q{\zeta }_{n-1}),$$

$$\exists {\upsilon }_{n+1}\in Q{\upsilon }_n:\varpi ({\upsilon }_{n+1},{\zeta }_n)\le hH(Q{\upsilon }_n,Q{\zeta }_{n-1}).$$

From (1), we get

$$\varpi ({\upsilon }_n,{\zeta }_n)\le hH(Q{\upsilon }_{n-1},Q{\zeta }_{n-1})\le h\lambda \varpi ({\upsilon }_{n-1},{\zeta }_{n-1}),$$

$$\varpi ({\upsilon }_{n+1},{\zeta }_n)\le hH(Q{\upsilon }_n,Q{\zeta }_{n-1})\le h\lambda \varpi ({\upsilon }_n,{\zeta }_{n-1}).$$

By repeating this process n times, we have

$$\varpi ({\upsilon }_n,{\zeta }_n)\le h^n{\lambda }^n\varpi ({\upsilon }_0,{\zeta }_0),$$

$$\varpi ({\upsilon }_{n+1},{\zeta }_n)\le h^n{\lambda }^n\varpi ({\upsilon }_1,{\zeta }_0).$$

We choose $1<h<{\displaystyle \frac{1}{s\lambda }}$ and let $n\ge 1$ and $p\ge 1$. Then,

$$\begin{array}{cc}\hfill \varpi ({\upsilon }_n,{\zeta }_{n+p})& \le s\left[\varpi ({\upsilon }_n,{\zeta }_n)+\varpi ({\upsilon }_{n+1},{\zeta }_n)+\varpi ({\upsilon }_{n+1},{\zeta }_{n+p})\right]\hfill \\ & \le s\left[\varpi ({\upsilon }_n,{\zeta }_n)+\varpi ({\upsilon }_{n+1},{\zeta }_n)\right]\hfill \\ & +s^2\left[\varpi ({\upsilon }_{n+1},{\zeta }_{n+1})+\varpi ({\upsilon }_{n+2},{\zeta }_{n+1})+\varpi ({\upsilon }_{n+2},{\zeta }_{n+p})\right]\hfill \\ & \le s\left[\varpi ({\upsilon }_n,{\zeta }_n)+\varpi ({\upsilon }_{n+1},{\zeta }_n)\right]\hfill \\ & +s^2\left[\varpi ({\upsilon }_{n+1},{\zeta }_{n+1})+\varpi ({\upsilon }_{n+2},{\zeta }_{n+1})\right]\hfill \\ & +s^3\left[\varpi ({\upsilon }_{n+2},{\zeta }_{n+2})+\varpi ({\upsilon }_{n+3},{\zeta }_{n+2})+\varpi ({\upsilon }_{n+3},{\zeta }_{n+p})\right]\hfill \\ & \le \hfill \\ & ⋮\hfill \\ & \le s\left[\varpi ({\upsilon }_n,{\zeta }_n)+\varpi ({\upsilon }_{n+1},{\zeta }_n)\right]+s^2\left[\varpi ({\upsilon }_{n+1},{\zeta }_{n+1})+\varpi ({\upsilon }_{n+2},{\zeta }_{n+1})\right]\hfill \\ & +s^3\left[\varpi ({\upsilon }_{n+2},{\zeta }_{n+2})+\varpi ({\upsilon }_{n+3},{\zeta }_{n+2})\right]+·+s^p\left[\varpi ({\upsilon }_{n+p-1},{\zeta }_{n+p-1})+\varpi ({\upsilon }_{n+p},{\zeta }_{n+p-1})\right]\hfill \\ & =\left(s{h}^n{\lambda }^n+s^2{h}^{n+1}{\lambda }^{n+1}+·+s^p{h}^{n+p-1}{\lambda }^{n+p-1}\right)\left[\varpi ({\upsilon }_0,{\zeta }_0)+\varpi ({\upsilon }_1,{\zeta }_0)\right]\hfill \\ & =s{h}^n{\lambda }^n\left(1+sh\lambda +·+s^{p-1}{h}^{p-1}{\lambda }^{p-1}\right)\left[\varpi ({\upsilon }_0,{\zeta }_0)+\varpi ({\upsilon }_1,{\zeta }_0)\right].\hfill \end{array}$$

Since $sh\lambda <1$, then $({\upsilon }_n,{\zeta }_n)$ is a Cauchy bi-sequence. Now, $(\aleph ,{\aleph }^{*},\varpi )$ being a complete bipolar b-metric space, the bi-sequence $({\upsilon }_n,{\zeta }_n)$ is biconvergent to some $p\in \aleph \cap {\aleph }^{*}$ such that

$$\underset{n\to \infty }{lim}{\upsilon }_n=p,\mathrm{and}\underset{n\to \infty }{lim}{\zeta }_n=p.$$

On the other hand, we have

$$\begin{array}{cc}\hfill D(p,Qp)& \le s\left[\varpi (p,{\zeta }_{n+1})+\varpi ({\upsilon }_{n+1},{\zeta }_{n+1})+D({\upsilon }_{n+1},Qp)\right]\hfill \\ & \le s\left[\varpi (p,{\zeta }_{n+1})+\varpi ({\upsilon }_{n+1},{\zeta }_{n+1})+H({\upsilon }_{n+1},Qp)\right]\hfill \\ & \le s\left[\varpi (p,{\zeta }_{n+1})+\varpi ({\upsilon }_{n+1},{\zeta }_{n+1})+\lambda \varpi ({\upsilon }_n,p)\right].\hfill \end{array}$$

Then

$$D(p,Qp)\le s\left[\varpi (p,{\zeta }_{n+1})+\varpi ({\upsilon }_{n+1},{\zeta }_{n+1})+\lambda \varpi ({\upsilon }_n,p)\right].$$

(2)

Now, by taking the upper limit when $n\to \infty$ in (2) and using Lemma 2, we get

$$\begin{array}{cc}\hfill \underset{n\to \infty }{lim\; sup}D(p,Qp)& \le \underset{n\to \infty }{lim\; sup}s\left[\varpi (p,{\zeta }_{n+1})+\varpi ({\upsilon }_{n+1},{\zeta }_{n+1})+\lambda \varpi ({\upsilon }_n,p)\right]\hfill \\ & \le s^2\left[\varpi (p,p)+\varpi (p,p)+\lambda \varpi (p,p)\right]\hfill \\ & =0.\hfill \end{array}$$

Then,

$$D(p,Qp)=0.$$

And since $Qp$ is closed, we get $p\in Qp$. Hence, p is a fixed point of Q. □

Example 1.

Let $\aleph =\left\{\right(0,0),(1,0),(2,0\left)\right\}$ and ${\aleph }^{*}=\{(2,0),(3,1),(4,2)\}$. Define $\varpi :\aleph \times {\aleph }^{*}\to {\mathbb{R}}^{+}$ such that

$$\varpi (\upsilon ,\zeta )={(a-e)}^2+\left|f\right|,\forall \upsilon =(a,b)\in \aleph \forall \zeta =(e,f)\in {\aleph }^{*}.$$

Then $(\aleph ,{\aleph }^{*},\varpi )$ is a complete bipolar b-metric space for $s=3$. The covariant mapping

$$Q:(\aleph ,{\aleph }^{*})\rightrightarrows (CB(\aleph ),CB\left({\aleph }^{*}\right))$$

is defined by

$$Q(0,0)=\left\{\right(1,0\left)\right\},Q(1,0)=Q(2,0)=\left\{\right(2,0\left)\right\},Q(3,1)=\left\{\right(2,0),(4,2\left)\right\},Q(4,2)=\left\{\right(3,1\left)\right\}.$$

Note that $Q\upsilon$ and $Q\zeta$ are closed and bounded for all $\upsilon \in \aleph$ and $\zeta \in {\aleph }^{*}$ with respect to the bipolar b-metric space $(\aleph ,{\aleph }^{*},d)$. Then we conclude that the condition

$$H(Q\upsilon ,Q\zeta )\le p\varpi (\upsilon ,\zeta ),$$

for all $\upsilon \in \aleph$ and $\zeta \in {\aleph }^{*}$ is satisfied for the constant $p={\displaystyle \frac{1}{2}}$. From Theorem 1, we say that Q has a fixed point. It is $(2,0)\in \aleph \cap {\aleph }^{*}$.

We next give an example of a multivalued covariant mapping, Q, that does not satisfy the condition in (1).

Example 2.

Let $\aleph =\left\{\right(0,0),(1,0),(2,0\left)\right\}$ and ${\aleph }^{*}=\{(2,0),(1,0),(3,1)\}$. Define $\varpi :\aleph \times {\aleph }^{*}\to {\mathbb{R}}^{+}$ such that

$$\varpi (\upsilon ,\zeta )={(a-e)}^2+\left|f\right|,\forall \upsilon =(a,b)\in \aleph \forall \zeta =(e,f)\in {\aleph }^{*}.$$

The covariant mapping

$$Q:(\aleph ,{\aleph }^{*})\rightrightarrows (CB(\aleph ),CB\left({\aleph }^{*}\right))$$

is defined by

$$Q(2,0)=\left\{\right(1,0\left)\right\},Q(1,0)=Q(0,0)=\left\{\right(2,0\left)\right\},Q(3,1)=\left\{\right(2,0),(1,0\left)\right\}.$$

Note that $Q\upsilon$ and $Q\zeta$ are closed and bounded for all $\upsilon \in \aleph$ and $\zeta \in {\aleph }^{*}$ with respect to the bipolar b-metric space $(\aleph ,{\aleph }^{*},d)$. But Q does not verify the condition (1). Hence, Q does not have a fixed point.

Theorem 2.

Let $(\aleph ,{\aleph }^{*},\varpi )$ be a complete bipolar b-metric space and $Q:(\aleph ,{\aleph }^{*})\rightleftarrows (CB(\aleph ),CB\left({\aleph }^{*}\right))$ be multivalued contravariant mapping, such that there exists $\lambda \in (0,1)$:

$$H(Q\zeta ,Q\upsilon )\le \lambda \varpi (\upsilon ,\zeta ),\forall \upsilon \in \aleph ,\forall \zeta \in {\aleph }^{*}.$$

(3)

Then Q has a fixed point $(p\in Qp)$.

Proof. 

Let ${\upsilon }_0\in \aleph ,{\zeta }_0\in Q{\upsilon }_0$. We choose an ${\upsilon }_1\in Q{\zeta }_0$. Then it follows from Lemma 1 that for any $h>1$ and $n\in \mathbb{N}$,

$$\begin{array}{cc}& \exists {\zeta }_n\in Q{\upsilon }_n;\varpi \left({\upsilon }_n,{\zeta }_n\right)\le hH\left(T{\zeta }_{n-1},Q{\upsilon }_n\right),\hfill \\ & \exists {\upsilon }_{n+1}\in Q{\zeta }_n;\varpi \left({\upsilon }_{n+1},{\zeta }_n\right)\le hH\left(Q{\zeta }_n,Q{\upsilon }_n\right),\hfill \end{array}$$

From (3), we get

$$\varpi \left({\upsilon }_n,{\zeta }_n\right)\le hH\left(Q{\zeta }_{n-1},Q{\upsilon }_n\right)\le h\lambda \varpi ({\upsilon }_n,{\zeta }_{n-1}),$$

$$\varpi ({\upsilon }_{n+1},{\zeta }_n)\le hH\left(Q{\zeta }_n,Q{\upsilon }_n\right)\le h\lambda \varpi ({\upsilon }_n,{\zeta }_n).$$

By repeating this process n times, we have

$$\varpi ({\upsilon }_n,{\zeta }_n)\le h^n{\lambda }^n\varpi ({\upsilon }_1,{\zeta }_0),$$

$$\varpi ({\upsilon }_{n+1},{\zeta }_n)\le h^n{\lambda }^n\varpi ({\upsilon }_0,{\zeta }_0).$$

We choose $1<h<{\displaystyle \frac{1}{s\lambda }}$ and let $n\ge 1$, $p\ge 1$. Then, analogously to the proof of the previous theorem,

$$\varpi ({\upsilon }_n,{\zeta }_{n+p})\le s{h}^n{\lambda }^n\left(1+sh\lambda +·+s^{p-1}{h}^{p-1}{\lambda }^{p-1}\right)\left[\varpi ({\upsilon }_0,{\zeta }_0)+\varpi ({\upsilon }_1,{\zeta }_0)\right].$$

Since $sh\lambda <1$, then $({\upsilon }_n,{\zeta }_n)$ is a Cauchy bi-sequence. Now, $(\aleph ,{\aleph }^{*},\varpi )$ being a complete bipolar b-metric space, the bi-sequence $({\upsilon }_n,{\zeta }_n)$ is biconvergent to some $p\in \aleph \cap {\aleph }^{*}$ such that

$$\underset{n\to \infty }{lim}{\upsilon }_n=p,\mathrm{and}\underset{n\to \infty }{lim}{\zeta }_n=p.$$

On the other hand, we have

$$\begin{array}{cc}\hfill D(p,Qp)& \le s\left[\varpi (p,{\zeta }_{n+1})+\varpi ({\upsilon }_{n+1},{\zeta }_{n+1})+D({\upsilon }_{n+1},Qp)\right]\hfill \\ & \le s\left[\varpi (p,{\zeta }_{n+1})+\varpi ({\upsilon }_{n+1},{\zeta }_{n+1})+H(Q{\zeta }_n,Qp)\right]\hfill \\ & \le s\left[\varpi (p,{\zeta }_{n+1})+\varpi ({\upsilon }_{n+1},{\zeta }_{n+1})+\lambda \varpi ({\zeta }_n,p)\right].\hfill \end{array}$$

Then

$$D(p,Qp)\le s\left[\varpi (p,{\zeta }_{n+1})+\varpi ({\upsilon }_{n+1},{\zeta }_{n+1})+\lambda \varpi ({\zeta }_n,p)\right].$$

(4)

Now, by taking the upper limit when $n\to \infty$ in (4) and using Lemma 2, we get

$$\begin{array}{cc}\hfill \underset{n\to \infty }{lim\; sup}D(p,Qp)& \le \underset{n\to \infty }{lim\; sup}s\left[\varpi (p,{\zeta }_{n+1})+\varpi ({\upsilon }_{n+1},{\zeta }_{n+1})+\lambda \varpi ({\zeta }_n,p)\right]\hfill \\ & \le s^2\left[\varpi (p,p)+\varpi (p,p)+\lambda \varpi (p,p)\right]\hfill \\ & =0.\hfill \end{array}$$

Then,

$$D(p,Qp)=0.$$

And since $Qp$ is closed, we get $p\in Qp$. Hence, p is a fixed point of Q. □

Example 3.

For $M\ge 0$, let

$$\aleph =\left\{\left({\upsilon }_n\right)\in {\mathbb{R}}^{\infty }\mid {\upsilon }_n\le 0\mathit{for}\mathit{each}n\in \mathbb{N}\mathit{and}\sum _{n=1}^{\infty }\sqrt{-{\upsilon }_n}\le M\right\}$$

$${\aleph }^{*}=\left\{\left({\zeta }_n\right)\in {\mathbb{R}}^{\infty }\mid {\zeta }_n\ge 0\mathit{for}\mathit{each}n\in \mathbb{N}\mathit{and}\sum _{n=1}^{\infty }\sqrt{{\zeta }_n}\le M\right\}.$$

Define $\varpi :\aleph \times {\aleph }^{*}\to {\mathbb{R}}^{+}$ such that

$$\varpi (\upsilon ,\zeta )={\left(\sum _{n=1}^{\infty }\sqrt{{\zeta }_n-{\upsilon }_n}\right)}^2,\mathit{where}\upsilon =\left({\upsilon }_n\right)\in \aleph ,\zeta =\left({\zeta }_n\right)\in {\aleph }^{*}.$$

Then $(\aleph ,{\aleph }^{*},\varpi )$ is a complete bipolar b-metric space for $s=3$. The contravariant mapping

$$Q:(\aleph ,{\aleph }^{*})\rightleftarrows (CB(\aleph ),CB\left({\aleph }^{*}\right))$$

is defined by

$$Q\left({\upsilon }_n\right)=\left\{-{\displaystyle \frac{a^n}{2}}\times {\upsilon }_n:a\in [0,1]\right\},n\in \mathbb{N}.$$

Note that $Q\upsilon$ and $Q\zeta$ are closed and bounded for all $\upsilon \in \aleph$ and $\zeta \in {\aleph }^{*}$ with respect to the bipolar b-metric space $(\aleph ,{\aleph }^{*},\varpi )$. Then we conclude that the condition

$$H(Q\zeta ,Q\upsilon )\le p\varpi (\upsilon ,\zeta )$$

for all $\upsilon \in \aleph$ and $\zeta \in {\aleph }^{*}$ is satisfied for the constant $p={\displaystyle \frac{1}{2}}$. From Theorem 2, we say that Q has a fixed point. It is $0=(0,0,\cdots )\in \aleph \cap {\aleph }^{*}$.

Also, we give an example of a multivalued contravariant mapping, Q, that does not satisfy the condition (3).

Example 4.

In the same bipolar b-metric space defined in the previous example $(\aleph ,{\aleph }^{*},\varpi )$, let the contravariant mapping

$$Q:(\aleph ,{\aleph }^{*})\rightleftarrows (CB(\aleph ),CB\left({\aleph }^{*}\right))$$

be defined by

$$Q\left({\upsilon }_n\right)=\left\{-{\upsilon }_n-{(1/2)}^n,-{\upsilon }_n-{(1/3)}^n\right\},n\in \mathbb{N}.$$

Note that $Q\upsilon$ and $Q\zeta$ are closed and bounded for all $\upsilon \in \aleph$ and $\zeta \in {\aleph }^{*}$ with respect to the bipolar b-metric space $(\aleph ,{\aleph }^{*},d)$. But Q does not verify the condition in (3). Hence, Q has not a fixed point.

Corollary 1.

Let $(\aleph ,{\aleph }^{*},\varpi )$ be a complete bipolar b-metric space and

$$Q:(\aleph ,{\aleph }^{*})\rightrightarrows (\aleph ,{\aleph }^{*})$$

be a covariant mapping such that

$$\varpi (Q\upsilon ,Q\zeta )\le \lambda \varpi (\upsilon ,\zeta )$$

for all $\upsilon \in \aleph$ and $\zeta \in {\aleph }^{*}$, where $\lambda \in (0,1)$. Then Q has a fixed point.

Corollary 2.

Let $(\aleph ,{\aleph }^{*},\varpi )$ be a complete bipolar b-metric space and

$$Q:(\aleph ,{\aleph }^{*})\rightleftarrows (\aleph ,{\aleph }^{*})$$

be a contravariant mapping such that

$$\varpi (Q\zeta ,Q\upsilon )\le \lambda \varpi (\upsilon ,\zeta )$$

for all $\upsilon \in \aleph$ and $\zeta \in {\aleph }^{*}$, where $\lambda \in (0,1)$. Then Q has a fixed point.

Theorem 3.

Let $(\aleph ,{\aleph }^{*},\varpi )$ be a complete bipolar b-metric space and $Q:(\aleph ,{\aleph }^{*})\rightleftarrows (CB(\aleph ),CB\left({\aleph }^{*}\right))$ be a multivalued contravariant mapping, such that there exist $\mu ,\nu \in {\mathbb{R}}^{+}$ and $\mu +2\nu <1$:

$$H(Q\zeta ,Q\upsilon )\le \mu \varpi (\upsilon ,\zeta )+\nu \left[D(\upsilon ,Q\upsilon )+D(Q\zeta ,\zeta )\right],\forall \upsilon \in \aleph ,\forall \zeta \in {\aleph }^{*}.$$

(5)

Then Q has a fixed point $(p\in Qp)$.

Proof. 

Let ${\upsilon }_0\in \aleph ,{\zeta }_0\in Q{\upsilon }_0$. We choose an ${\upsilon }_1\in Q{\zeta }_0$. Then it follows from Lemma 1 that for any $h>1$ and $n\in \mathbb{N}$,

$$\begin{array}{cc}& \exists {\zeta }_n\in Q{\upsilon }_n;\varpi \left({\upsilon }_n,{\zeta }_n\right)\le hH\left(Q{\zeta }_{n-1},Q{\upsilon }_n\right),\hfill \\ & \exists {\upsilon }_{n+1}\in Q{\zeta }_n;\varpi \left({\upsilon }_{n+1},{\zeta }_n\right)\le hH\left(Q{\zeta }_n,Q{\upsilon }_n\right),\hfill \end{array}$$

From (5), we get

$$\varpi \left({\upsilon }_n,{\zeta }_n\right)\le hH\left(Q{\zeta }_{n-1},Q{\upsilon }_n\right)\le h\left[\mu \varpi ({\upsilon }_n,{\zeta }_{n-1})+\nu (\varpi ({\upsilon }_n,{\zeta }_n)+\varpi ({\upsilon }_n,{\zeta }_{n-1}))\right],$$

$$\varpi ({\upsilon }_n,{\zeta }_{n-1})\le hH\left(Q{\zeta }_{n-1},Q{\upsilon }_{n-1}\right)\le h\left[\mu \varpi ({\upsilon }_{n-1},{\zeta }_{n-1})+\nu (\varpi ({\upsilon }_{n-1},{\zeta }_{n-1})+\varpi ({\upsilon }_n,{\zeta }_{n-1}))\right].$$

Then,

$$\varpi \left({\upsilon }_n,{\zeta }_n\right)\le h{\displaystyle \frac{\mu +\lambda }{1-\nu }}\varpi ({\upsilon }_n,{\zeta }_{n-1}),$$

$$\varpi ({\upsilon }_n,{\zeta }_{n-1})\le h{\displaystyle \frac{\mu +\nu }{1-\nu }}\varpi ({\upsilon }_{n-1},{\zeta }_{n-1}).$$

Then we conclude that

$$\begin{array}{cc}\hfill \varpi ({\upsilon }_n,{\zeta }_n)& \le {\left(h{\displaystyle \frac{\mu +\nu }{1-\nu }}\right)}^2\varpi ({\upsilon }_{n-1},{\zeta }_{n-1})\hfill \\ & ⋮\hfill \\ & \le {\left(h{\displaystyle \frac{\mu +\nu }{1-\nu }}\right)}^{2n}\varpi ({\upsilon }_0,{\zeta }_0)\hfill \end{array}$$

Let $\lambda ={\displaystyle \frac{\mu +\nu }{1-\nu }}$; then

$$\varpi ({\upsilon }_n,{\zeta }_n)\le h^{2n}{\lambda }^{2n}\varpi ({\upsilon }_0,{\zeta }_0)$$

We choose $1<h<{\displaystyle \frac{1}{s\lambda }}$ and let $n\ge 1$, $p\ge 1$. Then, analogously to the proof of the previous theorem,

$$\varpi ({\upsilon }_n,{\zeta }_{n+p})\le s{h}^{2n}{\lambda }^{2n}\left(1+sh\lambda +·+s^{p-1}{h}^{p-1}{\lambda }^{p-1}\right)\left[\varpi ({\upsilon }_0,{\zeta }_0)+\varpi ({\upsilon }_1,{\zeta }_0)\right]$$

Since $sh\lambda <1$, then $({\upsilon }_n,{\zeta }_n)$ is a Cauchy bi-sequence. Now, $(\aleph ,{\aleph }^{*},\varpi )$ being a complete bipolar b-metric space, the bi-sequence $({\upsilon }_n,{\zeta }_n)$ is biconvergent to some $p\in \aleph \cap {\aleph }^{*}$ such that

$$\underset{n\to \infty }{lim}{\upsilon }_n=p,\mathrm{and}\underset{n\to \infty }{lim}{\zeta }_n=p.$$

On the other hand, we have

$$\begin{array}{cc}\hfill D(p,Qp)& \le s\left[\varpi (p,{\zeta }_{n+1})+\varpi ({\upsilon }_{n+1},{\zeta }_{n+1})+D({\upsilon }_{n+1},Qp)\right]\hfill \\ & \le s\left[\varpi (p,{\zeta }_{n+1})+\varpi ({\upsilon }_{n+1},{\zeta }_{n+1})+H(Q{\zeta }_n,Qp)\right]\hfill \\ & \le s\left[\varpi (p,{\zeta }_{n+1})+\varpi ({\upsilon }_{n+1},{\zeta }_{n+1})+\lambda \varpi ({\zeta }_n,p)\right].\hfill \end{array}$$

Then

$$D(p,Qp)\le s\left[\varpi (p,{\zeta }_{n+1})+\varpi ({\upsilon }_{n+1},{\zeta }_{n+1})+\lambda \varpi ({\zeta }_n,p)\right].$$

(6)

Now, by taking the upper limit when $n\to \infty$ in (6) and using Lemma 2, we get

$$\begin{array}{cc}\hfill \underset{n\to \infty }{lim\; sup}D(p,Qp)& \le \underset{n\to \infty }{lim\; sup}s\left[\varpi (p,{\zeta }_{n+1})+\varpi ({\upsilon }_{n+1},{\zeta }_{n+1})+\lambda \varpi ({\zeta }_n,p)\right]\hfill \\ & \le s^2\left[\varpi (p,p)+\varpi (p,p)+\lambda \varpi (p,p)\right]\hfill \\ & =0.\hfill \end{array}$$

Then,

$$D(p,Qp)=0.$$

And since $Qp$ is closed, we get $p\in Qp$. Hence, p is a fixed point of Q. □

Corollary 3.

Let $(\aleph ,{\aleph }^{*},\varpi )$ be a complete bipolar b-metric space and

$$Q:(\aleph ,{\aleph }^{*})\rightleftarrows (\aleph ,{\aleph }^{*})$$

be a contravariant mapping such that

$$\varpi (Q\zeta ,Q\upsilon )\le \mu \varpi (\upsilon ,\zeta )+\nu \left[\varpi \right(\upsilon ,Q\upsilon )+\varpi (Q\zeta ,\zeta \left)\right]$$

for all $\upsilon \in \aleph$ and $\zeta \in {\aleph }^{*}$, where $\mu ,\nu \ge 0$ and $\mu +2\nu <1$. Then Q has a fixed point.

3. Conclusions

This work establishes fundamental fixed-point theorems for multivalued covariant and contravariant mappings in bipolar b-metric spaces, introducing the Pompeiu–Hausdorff bipolar b-metric and validating results through concrete examples. Future research will extend these methods to fractional systems, coincidence point theory, and equilibrium problems in asymmetric frameworks. These advances provide efficient theoretical tools for complex nonlinear phenomena across applied mathematics.