Common fixed point theorems for multi-valued mappings in bicomplex valued metric spaces with application

1.   Introduction

Gauss gave the theory of complex numbers in 17$^{th}$ century, but his research work was not on record. Later on, Cauchy initiated an in-depth review of complex numbers in the year 1840, and he is familiar as a successful originator of complex analysis. The study of complex numbers had its beginning because the solution of the quadratic equation $ax^{2}+bx+c = 0$ did not exist for $b^{2}-4ac < 0$ in the set of real numbers. Under this background, Euler was the first mathematician who presented the symbol $i$ for $\sqrt{-1},$ with the property, $i^{2} = -1$.

On the other hand, the beginning of bicomplex numbers was set up by Segre ^[1], and they provide a commutative replacement to the skew field of quaternions. These numbers extrapolated complex numbers more firmly and precisely to quaternions. For a better extensive study of investigation in bicomplex numbers, we refer the readers to ^[2]. In 2007, Huang et al. ^[3] presented the notion of a cone metric space (CMS) as an expansion of a traditional metric space (MS) and determined fixed point results for contractive mappings. Later on, Azam et al. ^[4] introduced the concept of a complex valued metric space (CVMS) as a particular case of a CMS. Mebawondu et al. ^[5] investigated the existence of solutions of differential equations by fixed point results in complex valued $b$-metric spaces. Vairaperumal et al. ^[6] established some common fixed point results for rational contractions in complex valued extended $b$-metric spaces. Okeke et al. ^[7] introduced the notion of complex valued convex metric spaces and proved certain fixed point results. In 2017, Choi et al. ^[8] introduced the notion of bicomplex valued metric spaces (bi-CVMS) by combining bicomplex numbers and CVMS. They proved some common fixed point theorems for weakly compatible mappings. Subsequently, Jebril et al. ^[9] used the idea of this novel space and presented theorems for two self mappings in the framework of bi-CVMS. In 2021, Beg et al. ^[10] reinforced the conception of bi-CVMS and proved extrapolated fixed point results. Afterward, Gnanaprakasam et al. ^[11] presented results for a contractive type condition in the framework of bi-CVMSs and explored the solution of linear equations. Later on, Tassaddiq et al. ^[12] involved control functions in the contractive inequality and established common fixed point results. Recently, Albargi et al. ^[13] obtained common fixed points of six self mapping in the setting of bi-CVMS. Mlaiki et al. ^[14] introduced locally contractive mappings in bi-CVMS and proved common fixed point theorems. For more details on CVMS and bi-CVMS, we refer the readers to ^{[15,16,17,18,19,20,21,22,23,24,25,26,27,28]}.

In this research work, we introduce a generalized Hausdorff distance function in the framework of bi-CVMS and obtain common fixed point theorems for generalized contractions. We also furnish a significant example to illustrate the originality of the obtained results.

2.   Preliminaries

We represent by $\mathbb{C} _{0},$ $\mathbb{C} _{1}$ and $\mathbb{C} _{2}$ the set of real numbers, the set of complex numbers and the set of bicomplex numbers, respectively. Segre ^[1] defined the idea of a bicomplex number as follows:

where $a_{1}, a_{2}, a_{3}, a_{4}\in \mathbb{C} _{0},$ and the independent units $i_{1}, i_{2}$ are such that $i_{1}^{2} = i_{2}^{2} = -1$ and $i_{1}i_{2} = i_{2}i_{1}.$ We define the set of bicomplex numbers $\mathbb{C} _{2}$ as

that is,

where $z_{1} = a_{1}+a_{2}i_{1}\in \mathbb{C} _{1}$ and $z_{2} = a_{3}+a_{4}i_{1}\in \mathbb{C} _{1}.$ If $\ell = z_{1}+i_{2}z_{2}$, $\rho = \omega _{1}+i_{2}\omega _{2}$ $\in \mathbb{C} _{2},$ then the sum is

and the product is

There are four idempotent elements in $\mathbb{C} _{2},$ which are, $0, 1, e_{1} = \frac{1+i_{1}i_{2}}{2}$ and $e_{2} = \frac{ 1-i_{1}i_{2}}{2},$ out of which $e_{1}$ and $e_{2}$ are nontrivial such that $e_{1}+e_{2} = 1$ and $e_{1}e_{2} = 0.$ Every bicomplex number $z_{1}+i_{2}z_{2}$ can uniquely be expressed as a combination of $e_{1}$and $e_{2}$, namely,

This description of $\ell$ is familiar as the idempotent representation of $\ell,$ and the complex coefficients $\ell _{1}$ $= \left(z_{1}-i_{1}z_{2}\right)$ and $\ell _{2} =$ $\left(z_{1}+i_{1}z_{2}\right)$ are known as idempotent components of the bicomplex number $\ell$.

An element $\ell = z_{1}+i_{2}z_{2}\in \mathbb{C} _{2}$ is invertible if there exists $\rho \in \mathbb{C} _{2}$ such that $\ell \rho = 1.$ In this way, the element $\rho$ is the multiplicative inverse of $\ell.$ As a consequence, $\ell$ is the multiplicative inverse of $\rho.$

An element $\ell = z_{1}+i_{2}z_{2}\in \mathbb{C} _{2}$ is non-singular if and only if $\left \vert z_{1}^{2}+z_{2}^{2}\right \vert \not = 0$ and singular if and only if $\left \vert z_{1}^{2}+z_{2}^{2}\right \vert = 0.$ The inverse of $\ell$ is defined as

Zero is the only member in $\mathbb{C} _{0}$ that does not possess a multiplicative inverse, and in $\mathbb{C} _{1}$, $0 = 0+i0$ is the only member that does not possess a multiplicative inverse. We represent the sets of singular members of $\mathbb{C} _{0}$ and $\mathbb{C} _{1}$ by $\mathcal{W}_{0}$ and $\mathcal{W}_{1},$ respectively. There are many members in $\mathbb{C} _{2}$ that do not have multiplicative inverse. We represent this set by $\mathcal{W}_{2},$ and evidently $\mathcal{W}_{0}$ $= \mathcal{W}_{1}\subset \mathcal{W}_{2}.$

A bicomplex number $\ell = a_{1}+a_{2}i_{1}+a_{3}i_{2}+a_{4}i_{1}i_{2}\in \mathbb{C} _{2}$ is said to be degenerated if the matrix

is degenerated. In that case, $\ell ^{-1}$ exists, and it is also degenerated.

The norm $\left \Vert \cdot \right \Vert$ of $\mathbb{C} _{2}$ is a positive real valued function, and $\left \Vert \cdot \right \Vert : \mathbb{C} _{2}\rightarrow \mathbb{C} _{0}^{+}$ is defined by

where $\ell = a_{1}+a_{2}i_{1}+a_{3}i_{2}+a_{4}i_{1}i_{2} = z_{1}+i_{2}z_{2}\in \mathbb{C} _{2}.$

A linear space $\mathbb{C} _{2}$ with regard to norm $\left \Vert \cdot \right \Vert$ is a normed linear space, and since $\mathbb{C} _{2}$ is complete, thus $\mathbb{C} _{2}$ is the Banach space. If $\ell, \rho \in \mathbb{C} _{2},$ then

holds instead of

Therefore $\mathbb{C} _{2}$ is not the Banach algebra. The partial order relation $\preceq _{i_{2}}$ on $\mathbb{C} _{2}$ is defined as follows:

Let $\mathbb{C} _{2}$ be the set of bicomplex numbers and $\ell = z_{1}+i_{2}z_{2},$ $\rho = \omega _{1}+i_{2}\omega _{2}\in \mathbb{C} _{2}.$ Then

It follows that

if one of these assertions is satisfied:

In particular, we can write $\ell \precnsim _{i_{2}}\rho$ if $\ell \preceq _{i_{2}}\rho$ and $\ell \not = \rho,$ that is, one of (a), (b) and (c) is satisfied, and we will write $\ell = \rho$ if only (d) is satisfied. For any two bicomplex numbers $\ell,$ $\rho \in \mathbb{C} _{2},$ we can verify the followings:

(ⅰ) $\ell \preceq _{i_{2}}\rho \Longrightarrow \left \Vert \ell \right \Vert \leq \left \Vert \rho \right \Vert,$

(ⅱ) $\left \Vert \ell +\rho \right \Vert \leq \left \Vert \ell \right \Vert +\left \Vert \rho \right \Vert,$

(ⅲ) $\left \Vert a\ell \right \Vert \leq a\left \Vert \rho \right \Vert,$ where $a$ is a non-negative real number,

(ⅳ) $\left \Vert \ell \rho \right \Vert \leq \sqrt{2}\left \Vert \ell \right \Vert \left \Vert \rho \right \Vert,$

(ⅴ) $\left \Vert \ell ^{-1}\right \Vert = \left \Vert \ell \right \Vert ^{-1},$

(ⅵ) $\left \Vert \frac{\ell }{\rho }\right \Vert = \frac{\left \Vert \ell \right \Vert }{\left \Vert \rho \right \Vert }.$

Choi et al. ^[8] defined the notion of a bi-CVMS as follows.

Definition 2.1. ^[8] Let $\mathcal{O}\not = \emptyset$ and $\kappa :\mathcal{O}\times \mathcal{O}\rightarrow \mathbb{C} _{2}$ be a function satisfying

$({\rm{i}})$ $0\preceq _{i_{2}}\kappa (\sigma, \varrho)\ $and $\kappa (\sigma, \varrho) = 0$ $\Leftrightarrow \sigma = \varrho$,

$({\rm{ii}})$ $\kappa (\sigma, \varrho) = \kappa (\rho, \varrho),$

$({\rm{iii}})$ $\kappa (\sigma, \varrho)\preceq _{i_{2}}\kappa (\sigma, \nu)+\kappa (\nu, \varrho),$

for all $\sigma, \varrho, \nu \in \mathcal{O}$. Then $(\mathcal{O}, \kappa)$ is a bi-CVMS.

Example 2.1. ^[10] Let $\mathcal{O} = \mathbb{C} _{2}$ and $\sigma, \varrho \in \mathcal{O}.$ Define $\kappa :\mathcal{O} \times \mathcal{O}\rightarrow \mathbb{C} _{2}$ by

where $\sigma = z_{1}+i_{2}z_{2},$ $\varrho = \omega _{1}+i_{2}\omega _{2}\in \mathbb{C} _{2}.$ Then, $(\mathcal{O}, \kappa)$ is a bi-CVMS.

Lemma 2.1. ^[10] Let $\left \{ \sigma _{n}\right \}$ $\subseteq (\mathcal{O}, \kappa)$. Then, $\left \{ \sigma _{n}\right \}$ converges to $\ell$ if and only if $\left \Vert \kappa (\sigma _{n}, \sigma)\right \Vert \rightarrow 0$ as $n\rightarrow \infty.$

Lemma 2.2. ^[10] Let $\left \{ \sigma _{n}\right \}$ $\subseteq (\mathcal{O}, \kappa)$. Then, $\left \{ \sigma _{n}\right \}$ is a Cauchy sequence if and only if $\ \left \Vert \kappa (\sigma _{n}, \sigma _{n+m})\right \Vert \rightarrow 0$ as $n\rightarrow \infty,$ where $m\in \mathbb{N}.$

Let $(\mathcal{O}, \kappa)\ $be a bi-CVMS. We denote by $N(\mathcal{O})$ (resp. $CB(\mathcal{O}))$ the collection of nonempty (resp. the collection of nonempty, closed and bounded) subsets of $(\mathcal{O}, \kappa)$. Now, we denote generalized Hausdorff distance function as $\wp$ and define

for $\ell \in$ $\mathbb{C} _{2},$ and

for $\sigma \in$ $\mathcal{O}$ and $B\in CB\left(\mathcal{O}\right).$ For $A, B\in CB\left(\mathcal{O}\right),$ we denote

Remark 2.1. Let $(\mathcal{O}, \kappa)\ $be a bi-CVMS. If we take $a_{2} = a_{3} = a_{4},$ then $(\mathcal{O}, \kappa)\ $is a metric space. Furthermore, for $A, B\in CB\left(\mathcal{O}\right),$

is the Hausdorff distance induced by $\kappa$.

Let $\beth :\mathcal{O}\rightarrow CB(\mathcal{O})$ be a multi-valued mapping. For $\sigma \in \mathcal{O},$ and $A\in CB(\mathcal{O})$, define

Thus, for $\sigma, \varrho \in \mathcal{O}$

Definition 2.2. Let $(\mathcal{O}, \kappa)$ be a bi-CVMS. A nonempty subset $A$ of $\mathcal{O}$ is called bounded from below if there exists $\ell \in \mathbb{C} _{2},$ such that $\ell \preceq _{i_{2}}\rho,$ for all $\rho \in A.$

Definition 2.3. Let $(\mathcal{O}, \kappa)$ be a bi-CVMS. A mapping $\mathcal{L}:\mathcal{O} \rightarrow 2^{\mathbb{ \mathbb{C} }_{2}}$ is said to be bounded from below if for $\sigma \in \mathcal{O}$, if there exists $\ell _{\sigma }\in \mathbb{ \mathbb{C} }$ such that

Definition 2.4. Let $(\mathcal{O}, \kappa)$ be a bi-CVMS. A mapping $\beth :\mathcal{O} \rightarrow CB(\mathcal{O})$ is said to have lower bound (l.b) property on $(\mathcal{O}, \kappa)$, if for any $\sigma \in \mathcal{O}$, the multi-valued mapping $\mathcal{L}_{\sigma }:\mathcal{O}\rightarrow 2^{\mathbb{ \mathbb{C} }_{2}}$ defined by,

is bounded from below, that is, for $\sigma, \varrho \in \mathcal{O}$, there exists $l_{\sigma }(\beth \varrho)\in \mathbb{C}$ such that;

for all $\rho \in W_{\sigma }(\beth \varrho),$ where $l_{\sigma }(\beth \varrho)$ is called the lower bound of $\beth$ associated with $(\sigma, \varrho).$

Definition 2.5. Let $(\mathcal{O}, \kappa)$ be a bi-CVMS. A mapping $\beth :\mathcal{O} \rightarrow CB(\mathcal{O})$ is said to satisfy greatest lower bound (g.l.b) property on $(\mathcal{O}, \kappa)$ if a greatest lower bound of $W_{\sigma }(\beth \varrho)$ exists in $\mathbb{C} _{2},$ for all $\sigma, \varrho \in \mathcal{O}$. We represent by $\kappa (\sigma, \beth \varrho)$ the g.l.b. of $W_{\sigma }(\beth \varrho),$ that is,

3.   Main result

Throughout this section, we consider $(\mathcal{O}, \kappa)$ as a complete bi-CVMS, and the mappings $\beth _{1}, \beth _{2}:\mathcal{O}\rightarrow CB(\mathcal{O})$ satisfy the g.l.b. property.

Theorem 3.1. Let $(\mathcal{O}, \kappa)$ be a complete bi-CVMS, and let $\beth _{1}, \beth _{2}:(\mathcal{O}, \kappa)\rightarrow CB(\mathcal{O})$ be such that

for all $\sigma, \varrho \in \mathcal{O}$ and $\aleph _{1}, \aleph _{2}, \aleph _{3}\in \lbrack 0, 1)$ with $\aleph _{1}+\sqrt{2}\aleph _{2}+ \sqrt{2}\aleph _{3} < 1.$ Then, there exists $\varpi \in \mathcal{O}$ such that $\varpi \in \beth _{1}\varpi \cap \beth _{2}\varpi$.

Proof. Let $\sigma _{0}$ $\in \mathcal{O}$ be an arbitrary point and $\sigma _{1}\in \beth _{1}\sigma _{0}.$ From (3.1), we have

This implies that

for $\sigma \in \beth _{1}\sigma _{0}.$ Since $\sigma _{1}\in \beth _{1}\sigma _{0},$ we have

This implies that

So, there exists $\sigma _{2}\in \beth _{2}\sigma _{1}$ and we have

Therefore,

Since the pair ($\beth _{1}, \beth _{2}$) satisfies g.l.b. property, we get

This implies

because $\frac{\left \Vert \kappa \left(\sigma _{0}, \sigma _{1}\right) \right \Vert }{\left \Vert 1+\kappa \left(\sigma _{0}, \sigma _{1}\right) \right \Vert } < 1.$ This yields

Similarly, for $\sigma _{2}\in \beth _{2}\sigma _{1}$ and from (3.1), we have

This implies that

for $\sigma \in \beth _{2}\sigma _{1}.$ Since $\sigma _{2}\in \beth _{2}\sigma _{1},$ we have

This implies that

So, there exists $\sigma _{3}\in \beth _{1}\sigma _{2},$ and we have

Therefore,

Since the pair ($\beth _{1}, \beth _{2}$) satisfies g.l.b. property, we get

This implies

since $\frac{\left \Vert \kappa \left(\sigma _{1}, \sigma _{2}\right) \right \Vert }{\left \Vert 1+\kappa \left(\sigma _{1}, \sigma _{2}\right) \right \Vert } < 1.$ This yields

Let $\frac{\aleph _{1}}{1-\sqrt{2}\aleph _{2}} = \aleph < 1.$ Then, from (3.2) and (3.3), we have

Thus, we can generate a sequence {$\sigma _{n}$} in $\mathcal{O}$ such that

and

for $n\in \mathbb{N}.$ Now, for $m > n$ and by the triangle inequality, we have

Now, by taking $n\rightarrow \infty$, we get

Thus, $\{ \sigma _{n}\}$ is a Cauchy sequence in $\mathcal{O}$ by Lemma 2.2. Therefore there exists $\varpi \in \mathcal{O}$ such that $\lim \limits_{n\rightarrow \infty }\sigma _{n} = \varpi$. Then, also, $\lim \limits_{n\rightarrow \infty }\sigma _{2n} = \varpi,$ and $\lim \limits_{n\rightarrow \infty }\sigma _{2n+1} = \varpi.$ Now, we show that $\varpi \in \beth _{1}\varpi$ and $\varpi \in \beth _{2}\varpi.$ From (3.1), we have

which implies that

for $\sigma \in \beth _{1}\sigma _{2n}.$ Since $\sigma _{2n+1}\in \beth _{1}\sigma _{2n},$ we have

By definition

There exists $\sigma _{n}\in \beth _{2}\varpi$ such that

By definition

Since the pair ($\beth _{1}, \beth _{2}$) satisfies g.l.b. property, we get

By the triangle inequality, we have

Now, using (3.4), we have

which implies

Taking the limit as $n\rightarrow \infty,$ we have $\left \Vert \kappa (\varpi, \varpi _{n})\right \Vert \rightarrow 0$ as $n\rightarrow \infty$. Thus, $\varpi _{n}\rightarrow \varpi$ as $n\rightarrow \infty.$ Since $\beth _{2}\varpi$ is closed, we have $\varpi \in \beth _{2}\varpi.$ Similarly, we can prove that $\varpi \in \beth _{1}\varpi$. Therefore, $\varpi$ is a common fixed point of $\beth _{1}$ and $\beth _{2}.$ □

Corollary 3.1. Let $(\mathcal{O}, \kappa)$ be a complete bi-CVMS and let $\beth :(\mathcal{O}, \kappa)\rightarrow CB(\mathcal{O})$ be a multi-valued mapping with g.l.b. property such that

for all $\sigma, \varrho \in \mathcal{O}$ and $\aleph _{1}, \aleph _{2}, \aleph _{3}\in \lbrack 0, 1)$ with $\aleph _{1}+\sqrt{2}\aleph _{2}+ \sqrt{2}\aleph _{3} < 1.$ Then, there exists $\varpi \in \mathcal{O}$ such that $\varpi \in \beth \varpi$.

Proof. Take $\beth _{1} = \beth _{2} = \beth$ in Theorem 3.1. □

Corollary 3.2. Let $(\mathcal{O}, \kappa)$ be a complete bi-CVMS and let $\beth _{1}, \beth _{2}:(\mathcal{O}, \kappa)\rightarrow CB(\mathcal{O})$ be such that

for all $\sigma, \varrho \in \mathcal{O}$ and $\aleph _{1}, \aleph _{2}\in \lbrack 0, 1)$ with $\aleph _{1}+\sqrt{2}\aleph _{2} < 1.$ Then, there exists $\varpi \in \mathcal{O}$ such that $\varpi \in \beth _{1}\varpi \cap \beth _{2}\varpi$.

Proof. Take $\aleph _{3} = 0$ in Theorem 3.1. □

Corollary 3.3. Let $(\mathcal{O}, \kappa)$ be a complete bi-CVMS and $\beth :(\mathcal{O}, \kappa)\rightarrow CB(\mathcal{O})$ be such that

for all $\sigma, \varrho \in \mathcal{O}$ and $\aleph _{1}, \aleph _{2}\in \lbrack 0, 1)$ with $\aleph _{1}+\sqrt{2}\aleph _{2} < 1.$ Then, there exists $\varpi \in \mathcal{O}$ such that $\varpi \in \beth \varpi$.

Proof. Take $\beth _{1} = \beth _{2} = \beth$ in Corollary 3.2. □

Corollary 3.4. Let $(\mathcal{O}, \kappa)$ be a complete bi-CVMS and $\beth _{1}, \beth _{2}:\mathcal{O}\rightarrow CB(\mathcal{O})$ be such that

for all $\sigma, \varrho \in \mathcal{O}$ and $\aleph _{1}, \aleph _{3}\in \lbrack 0, 1)$ with $\aleph _{1}+\sqrt{2}\aleph _{3} < 1.$ Then, there exists $\varpi \in \mathcal{O}$ such that $\varpi \in \beth _{1}\varpi \cap \beth _{2}\varpi$.

Proof. Take $\aleph _{2} = 0$ in Theorem 3.1. □

Corollary 3.5. Let $(\mathcal{O}, \kappa)$ be a complete bi-CVMS and let $\beth :(\mathcal{O}, \kappa)\rightarrow CB(\mathcal{O})$ be such that

for all $\sigma, \varrho \in \mathcal{O}$ and $\aleph _{1}, \aleph _{3}\in \lbrack 0, 1)$ with $\aleph _{1}+\sqrt{2}\aleph _{3} < 1.$ Then, there exists $\varpi \in \mathcal{O}$ such that $\varpi \in \beth \varpi$.

Proof. Set $\beth _{1} = \beth _{2} = \beth$ in Corollary 3.4. □

Corollary 3.6. Let $(\mathcal{O}, \kappa)$ be a complete bi-CVMS and let $\beth _{1}, \beth _{2}:(\mathcal{O}, \kappa)\rightarrow CB(\mathcal{O})$ be such that

for all $\sigma, \varrho \in \mathcal{O}$ and $\aleph _{1}\in \lbrack 0, 1)$ with $\aleph _{1} < 1.$ Then, there exists $\varpi \in \mathcal{O}$ such that $\varpi \in \beth _{1}\varpi \cap \beth _{2}\varpi$.

Proof. Choose $\aleph _{2} = \aleph _{3} = 0$ in Theorem 3.1. □

Corollary 3.7. Let $(\mathcal{O}, \kappa)$ be a complete bi-CVMS and let $\beth :(\mathcal{O}, \kappa)\rightarrow CB(\mathcal{O})$ be such that

for all $\sigma, \varrho \in \mathcal{O}$ and $\aleph _{1}\in \lbrack 0, 1).$ Then, there exists $\varpi \in \mathcal{O}$ such that $\varpi \in \beth \varpi$.

Proof. Take $\beth _{1} = \beth _{2} = \beth$ in Corollary 3.6. □

Example 3.1. Let $\mathcal{O} = [0, 1]$.$\ $Define $\kappa :\mathcal{O}\times \mathcal{O} \rightarrow \mathbb{C} _{2}$ by

Then, $(\mathcal{O}, \kappa)$ is a complete bi-CVMS$.$ Consider the mappings $\beth _{1}, \beth _{2}:\mathcal{O}\rightarrow CB(\mathcal{O})$ defined by

If $\sigma = \varrho = 0,$ then obviously the contractive condition is satisfied. Now, we assume that $\sigma < \varrho$ Then, we have

and

Consider,

Then, for any values of $\aleph _{2}$ and $\aleph _{3}$ and $\aleph _{1} = \frac{ 1}{6}$, we have

Hence,

Thus, all the axioms of Theorem 3.1 hold, and the pair ($\beth _{1}, \beth _{2})$ has a fixed point $0$.

Now, if we consider $\beth _{1}(\sigma) = \left \{ \sigma \right \}$ and $\beth _{2}(\varrho) = \{ \varrho \}$ in Theorem 3.1, then we can derive the key result of Gnanaprakasam et al. ^[11] in this manner.

Corollary 3.8. ^[11] Let $(\mathcal{O}, \kappa)$ be a complete bi-CVMS and let $\beth _{1}, \beth _{2}:(\mathcal{O}, \kappa)\rightarrow (\mathcal{O}, \kappa)$ be such that

for all $\sigma, \varrho \in \mathcal{O}$ and $\aleph _{1}, \aleph _{2}, \aleph _{3}\in \lbrack 0, 1)$ with $\aleph _{1}+\sqrt{2}\aleph _{2}+ \sqrt{2}\aleph _{3} < 1.$ Then, there exists $\varpi \in \mathcal{O}$ such that $\varpi = \beth _{1}\varpi = \beth _{2}\varpi$.

Corollary 3.9. Let $(\mathcal{O}, \kappa)$ be a complete bi-CVMS and let $\beth :(\mathcal{O}, \kappa)\rightarrow (\mathcal{O}, \kappa)$ be such that

for all $\sigma, \varrho \in \mathcal{O}$ and $\aleph _{1}, \aleph _{2}, \aleph _{3}\in \lbrack 0, 1)$ with $\aleph _{1}+\sqrt{2}\aleph _{2}+ \sqrt{2}\aleph _{3} < 1.$ Then, there exists $\varpi \in \mathcal{O}$ such that $\varpi = \beth \varpi$.

Proof. Take $\beth _{1} = \beth _{2} = \beth$ in Corollary 3.8. □

Corollary 3.10. Let $(\mathcal{O}, \kappa)$ be a complete bi-CVMS and let $\beth _{1}, \beth _{2}:(\mathcal{O}, \kappa)\rightarrow (\mathcal{O}, \kappa)$ be such that

for all $\sigma, \varrho \in \mathcal{O}$ and $\aleph _{1}, \aleph _{2}\in \lbrack 0, 1)$ with $\aleph _{1}+\sqrt{2}\aleph _{2} < 1.$ Then, there exists $\varpi \in \mathcal{O}$ such that $\varpi = \beth _{1}\varpi = \beth _{2}\varpi$.

Proof. Take $\aleph _{3} = 0$ in Corollary 3.8. □

Corollary 3.11. ^[10] Let $(\mathcal{O}, \kappa)$ be a complete bi-CVMS and let $\beth :(\mathcal{O}, \kappa)\rightarrow (\mathcal{O}, \kappa)$ be such that

for all $\sigma, \varrho \in \mathcal{O}$ and $\aleph _{1}, \aleph _{2}\in \lbrack 0, 1)$ with $\aleph _{1}+\sqrt{2}\aleph _{2} < 1.$ Then, there exists $\varpi \in \mathcal{O}$ such that $\varpi = \beth \varpi$.

Proof. Take $\beth _{1} = \beth _{2} = \beth$ in the above Corollary. □

4.   Conclusions

In this paper, we have introduced a generalized Hausdorff distance function in the setting of bi-CVMS and obtained common fixed point results for rational contractions. We hope that the established theorems in this paper will form contemporary associations for researchers who are working in bi-CVMS. As an application of our main results, we have derived some results for self mappings in the context of bi-CVMS, including the leading results of [Demonstr. Math., 54 (2021), 474–487] and [Int. J. Nonlinear Anal. Appl., 12 (2021), 717–727].

Use of AI tools declaration

The author declares she has not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work was funded by the University of Jeddah, Jeddah, Saudi Arabia, under grant No. (UJ-21-DR-68). The author, therefore, thanks the University of Jeddah for its technical and financial support.

Conflict of interest

The author declares that she has no conflict of interest.