On common fixed point results in bicomplex valued metric spaces with application

1.   Introduction

Metric fixed-point theory has newly emerging applications to study the internet topology ^[1] and modelling the cyberspace as a digital ecosystem ^[2]. Moreover, new researches in fixed-point theory determine the significance to find the solution of real-world problems. A routing problem, for example, can be solved using functional equations and iterative procedures. The capacitated vehicle routing problem (CVRP) ^[3] outlines a method for determining the best plan to meet the demand of a globally dispersed network of clients while distributing cohesive products from a pickup point that used a large number of (the same) automobiles with a specific adaptive capacity. Meanwhile, fixed point theory is used as a problem-solving tool in communication engineering. Other real-world applications include the solution of chemical equations, genetics, algorithm testing, and control theory. Such results offer delightful circumstances in the study of mathematical analysis to approximating the solutions of linear and nonlinear differential and integral equations ^[4]. Because the theory of fixed-point is an odd synthesis of analysis ^[5,6] and geometry ^[7,8,9,10]. Therefore, it has emerged as a powerful and crucial instrument for the investigation of nonlinear problems ^{[11,12,13,14]}. More recently, Işık and collaborators have discussed such results by using rational ^[15] as well as generalized Wardowski type contractive multi-valued mappings ^[16] and also investigated the common solutions to integral and functional equations ^[17,18]. The aim of this article is to obtain common fixed point results in a bicomplex valued metric space. Therefore, we first define the basic preliminaries involving bicomplex numbers and review further developments related to them in the following paragraphs.

The emergence of complex numbers was established in the 17$^{th}$ century by Sir Carl Fredrich Gauss, but his work was not on record. Later, in the year 1840 Augustin Louis Cauchy started doing analysis of complex numbers, who is known to be an effective founder of complex analysis. The theory of complex numbers has its source in the fact that the solution of the quadratic equation $ax^{2}+bx+c = 0$ was not worthwhile 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 ^[19] which provides a commutative substitute to the skew field of quaternions. These numbers generalize complex numbers more precisely to quaternions. We refer readers to ^[20] for a more in-depth examination of bicomplex numbers. In 2011, Azam et al. ^[21] gave the concept of a complex valued metric space (CVMS) as a special case of cone metric space. Since the concept to introduce complex valued metric spaces is designed to define rational expressions that cannot be defined in cone metric spaces and therefore several results of fixed point theory cannot be proved to cone metric spaces, so complex valued metric space form a special class of cone metric space. Actually, the definition of a cone metric space banks on the underlying Banach space which is not a division ring. However, we can study generalizations of many results of fixed point theory involving divisions in complex valued metric spaces. Moreover, this idea is also used to define complex valued Banach spaces ^[22] which offer a lot of scope for further investigation. In 2017, Choi et al. ^[23] combined the concepts of bicomplex numbers and CVMS and introduced the notion of bicomplex valued metric spaces (bi CVMS) and established common fixed point results for weakly compatible mappings. Later on, Jebril et al. ^[24], utilized this notion of newly introduced space and obtained common fixed point results under rational contractions for a pair of mappings in the background of bi CVMS. More specifically, CVMS ^[25,26] and bi CVMS ^[27,28] has been remained a focus point of recent and past researches. By taking motivation from these facts, we establish some common fixed point theorems in bi CVMS for rational contractions involving control functions of two variables. As an application, we investigate the solutions of integral equations.

2.   Preliminaries

We represent $\mathbb{C} _{0},$ $\mathbb{C} _{1}$ and $\mathbb{C} _{2}$ as the set of real numbers, complex numbers and bicomplex numbers respectively. Segre ^[19] defined the notion of 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 $i_{1}i_{2} = i_{2}i_{1, }$ and $\mathbb{C} _{2}$ is defined 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 $\varrho = z_{1}+i_{2}z_{2}$ and $\hbar = \omega _{1}+i_{2}\omega _{2},$ then the sum is

and the product is

There are four idempotent members 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 demonstrated as the mixture of $e_{1}$and $e_{2}$, namely

This characterization of $\varrho$ is familiar as the idempotent characterization of $\varrho$ and the complex coefficients $\varrho _{1}$ $= \left(z_{1}-i_{1}z_{2}\right)$ and $\varrho _{2} =$ $\left(z_{1}+i_{1}z_{2}\right)$ are called as idempotent components of $\varrho$.

An element $\varrho = z_{1}+i_{2}z_{2}\in \mathbb{C} _{2}$ is called invertible if there exists $\hbar \in \mathbb{C} _{2}$ such that $\varrho \hbar = 1$ and $\hbar$ is called the inverse (multiplicative) of $\varrho.$ Therefore $\varrho$ is called the inverse of $\hbar.$ An element $\varrho = z_{1}+i_{2}z_{2}\in \mathbb{C} _{2}$ is nonsingular iff $\left \vert z_{1}^{2}+z_{2}^{2}\right \vert \not = 0$ and singular iff $\left \vert z_{1}^{2}+z_{2}^{2}\right \vert = 0.$ The inverse of $\varrho$ is defined as

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

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

is degenerated. In this way $\varrho ^{-1}$ exists and it is degenerated too and $\left \Vert \cdot \right \Vert : \mathbb{C} _{2}\rightarrow \mathbb{C} _{0}^{+}$ is defined as

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

The space $\mathbb{C} _{2}$ with respect to the norm given above is a Banach space. If $\varrho, \hbar \in \mathbb{C} _{2},$ then

holds instead of

Therefore, $\mathbb{C} _{2}$ is not a Banach algebra. Let $\varrho = z_{1}+i_{2}z_{2},$ $\hbar = \omega _{1}+i_{2}\omega _{2}\in \mathbb{C} _{2},$ then we define

It implies

if one of these assertions hold:

Specifically, $\varrho \precnsim _{i_{2}}\hbar$ if $\varrho \preceq _{i_{2}}\hbar$ and $\varrho \not = \hbar,$ that is, one of (i), (ii) and (iii) holds. Also $\varrho \prec _{i_{2}}\hbar$ if only condition (iii) is satisfied. For $\varrho,$ $\hbar \in \mathbb{C} _{2},$ we can prove the followings:

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

(ii) $\left \Vert \varrho +\hbar \right \Vert \leq \left \Vert \varrho \right \Vert +\left \Vert \hbar \right \Vert,$

(iii) $\left \Vert a\varrho \right \Vert \leq a\left \Vert \hbar \right \Vert,$ where $a$ is a non negative real number,

(iv) $\left \Vert \varrho \hbar \right \Vert \leq \sqrt{2}\left \Vert \varrho \right \Vert \left \Vert \hbar \right \Vert,$

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

(vi) $\left \Vert \frac{\varrho }{\hbar }\right \Vert = \frac{\left \Vert \varrho \right \Vert }{\left \Vert \hbar \right \Vert },$ if $\hbar$ is a degenerated bicomplex number.

Azam et al. ^[21] gave the conception of CVMS in this way:

Definition 1. (^[21]) Let $\mathfrak{L} \not = \emptyset$, $\preceq$ is a partial order on $\mathbb{C}$ and $\mathfrak{\varsigma }:\mathfrak{L} \times \mathfrak{L} \rightarrow \mathbb{C} _{1}$ be a mapping satisfying

(i) $0\preceq \mathfrak{\varsigma }(\varrho, \hbar), \ $ for all $\varrho, \hbar \in \mathfrak{L} \ $and $\mathfrak{\varsigma }(\varrho, \hbar) = 0$ if and only if $\varrho = \hbar$;

(ii) $\mathfrak{\varsigma }(\varrho, \hbar) = \mathfrak{\varsigma } (\hbar, \varrho)\ $ for all $\varrho, \hbar \in \mathfrak{L};$

(iii) $\mathfrak{\varsigma }(\varrho, \hbar)\preceq \mathfrak{ \varsigma }(\varrho, \nu)+\mathfrak{\varsigma }(\nu, \hbar),$ for all $\varrho, \hbar, \nu \in \mathfrak{L},$

then $(\mathfrak{L}, \mathfrak{\varsigma })$ is a CVMS.

Choi et al. ^[23] defined the bi CVMS as follows:

Definition 2. (^[23]) Let $\mathfrak{L} \not = \emptyset,$ $\preceq _{i_{2}}$ is a partial order on $\mathbb{C} _{2}$ and $\mathfrak{\varsigma }:\mathfrak{L} \times \mathfrak{L} \rightarrow \mathbb{C} _{2}$ be a mapping satisfying

(i) $0\preceq _{i_{2}}\mathfrak{\varsigma }(\varrho, \hbar), \ $for all $\varrho, \hbar \in \mathfrak{L} \ $and $\mathfrak{\varsigma }(\varrho, \hbar) = 0$ if and only if $\varrho = \hbar$;

(ii) $\mathfrak{\varsigma }(\varrho, \hbar) = \mathfrak{\varsigma } (\hbar, \varrho)\ $for all $\varrho, \hbar \in \mathfrak{L};$

(iii) $\mathfrak{\varsigma }(\varrho, \hbar)\preceq _{i_{2}} \mathfrak{\varsigma }(\varrho, \nu)+\mathfrak{\varsigma }(\nu, \hbar),$ for all $\varrho, \hbar, \nu \in \mathfrak{L},$

then $(\mathfrak{L}, \mathfrak{\varsigma })$ is a bi CVMS.

Example 1. (^[29]) Let $\mathfrak{L} = \mathbb{C} _{2}$ and $\varrho, \hbar \in \mathfrak{L}.$ Define $\mathfrak{\varsigma } :\mathfrak{L} \times \mathfrak{L} \rightarrow \mathbb{C} _{2}$ by

where $\varrho = z_{1}+i_{2}z_{2}$ and $\hbar = \omega _{1}+i_{2}\omega _{2}\in \mathbb{C} _{2}.$ Then $(\mathfrak{L}, \mathfrak{\varsigma })$ is a bi CVMS.

Lemma 1. (^[29]) Let $(\mathfrak{L}, \mathfrak{\varsigma })\ $ be a bi CVMS and let $\left \{ \varrho _{r}\right \}$ $\subseteq \mathfrak{L}$. Then $\left \{ \varrho _{r}\right \}$ converges to $\varrho$ if and only if $\left \Vert \mathfrak{\varsigma }(\varrho _{r}, \varrho)\right \Vert \rightarrow 0\ $as$\ r\rightarrow \infty.$

Lemma 2. (^[29]) Let $(\mathfrak{L}, \mathfrak{\varsigma })\ $ be a bi CVMS and let $\left \{ \varrho _{r}\right \}$ $\subseteq \mathfrak{L}$.\ Then $\left \{ \varrho _{r}\right \}$ is a Cauchy sequence if and only if $\ \left \Vert \mathfrak{\varsigma }(\varrho _{r}, \varrho _{r+m})\right \Vert \rightarrow 0\ $as$\ r\rightarrow \infty, \ $where $m\in \mathbb{N}.$

3.   Main result

We state and prove the following proposition which is required in the sequel.

Proposition 1. Let $(\mathfrak{L}, \mathfrak{\varsigma })\ $be a bi CVMS and $\Im _{1}, \Im _{2}:(\mathfrak{L}, \mathfrak{\varsigma })\rightarrow (\mathfrak{ L}, \mathfrak{\varsigma }).$ Let $\varrho _{0}\in$ $\mathfrak{L}.$ Define the sequence $\{\varrho _{r}\}$ by

for all $r = 0, 1, 2, ...$

Assume that there exist $\rho :\mathfrak{L\times L\rightarrow }[0, 1)$ satisfying

for all $\varrho, \hbar \in \mathfrak{L}.$ Then

for all $\varrho, \hbar \in \mathfrak{L}$ and $r = 0, 1, 2, ...$

Proof. Let $\varrho, \hbar \in \mathfrak{L}$ and $r = 0, 1, 2, ...$ Then we have

Similarly, we have

Lemma 3. Let $\rho, \kappa :\mathfrak{L\times L\rightarrow }[0, 1)$ and $\varrho, \hbar \in$ $\mathfrak{L}.$ If $\Im _{1}, \Im _{2}:\mathfrak{L}$ $\rightarrow \mathfrak{L}$ satisfy

and

then

and

Proof. We can write

Similarly, we have

Theorem 1. Let $\left(\mathfrak{L}, \mathfrak{\varsigma }\right)$ be a complete bi CVMS and $\Im _{1}, \Im _{2}$: $\mathfrak{L}$ $\rightarrow \mathfrak{L}$. If there exist mappings $\rho, \kappa, \varpi :\mathfrak{ L\times L\rightarrow }[0, 1)$ such that for all $\varrho, \hbar \in$ $\mathfrak{L, }$

(a) $\rho \left(\Im _{2}\Im _{1}\varrho, \hbar \right) \leq \rho \left(\varrho, \hbar \right)$ and $\rho \left(\varrho, \Im _{1}\Im _{2}\hbar \right) \leq \rho \left(\varrho, \hbar \right),$ $\ \ \ \ \ \kappa \left(\Im _{2}\Im _{1}\varrho, \hbar \right) \leq \kappa \left(\varrho, \hbar \right)$ and $\kappa \left(\varrho, \Im _{1}\Im _{2}\hbar \right) \leq \kappa \left(\varrho, \hbar \right),$ $\ \ \ \ \ \varpi \left(\Im _{2}\Im _{1}\varrho, \hbar \right) \leq \varpi \left(\varrho, \hbar \right)$ and $\varpi \left(\varrho, \Im _{1}\Im _{2}\hbar \right) \leq \varpi \left(\varrho, \hbar \right),$

(b) $\rho \left(\varrho, \hbar \right) +\sqrt{2}\kappa \left(\varrho, \hbar \right) +\sqrt{2}\varpi \left(\varrho, \hbar \right) < 1$,

(c)

then $\Im _{1}$ and $\Im _{2}$ have a unique common fixed point.

Proof. Let $\varrho, \hbar \in \mathfrak{L.}$ From (3.2), we have

By Lemma (3), we get

Similarly, we have

By Lemma (3), we get

Let $\varrho _{0}$ $\in \mathfrak{L}$ and the sequence $\{\varrho _{r}\}$ be defined by (3.1). From Proposition (1) and inequalities (3.3) and (3.4), we have

for all $r = 0, 1, 2, ..$. This implies that

Similarly, we have

which implies that

Let $\lambda =$ $\frac{\rho \left(\varrho _{0}, \varrho _{1}\right) }{1- \sqrt{2}\kappa \left(\varrho _{0}, \varrho _{1}\right) } < 1.$ Then from (3.5) and (3.6), we have

for all $r\in \mathbb{N}.$ Inductively, we can construct a sequence $\{ \varrho _{r}\}$ in $\mathfrak{ L}$ such that

for all $r\in \mathbb{N}.$ Now for $m > r$, we get

Now, by taking $r, m\rightarrow \infty$, we get

By Lemma 2, $\left \{ \varrho _{r}\right \}$ is a Cauchy sequence. As $\mathfrak{L}$ is complete, so there exists $\varrho ^{\ast }\in \mathfrak{L}$ such that $\varrho _{r}\rightarrow \varrho ^{\ast }$ as $r\rightarrow \infty.$

Now, we show that $\varrho ^{\ast }$ is a fixed point of $\Im _{1}.$ From (3.2), we have

This implies that

Letting $r\rightarrow \infty,$ we have $\left \Vert \mathfrak{\varsigma } \left(\varrho ^{\ast }, \Im _{1}\varrho ^{\ast }\right) \right \Vert = 0.$ Thus $\varrho ^{\ast } = \Im _{1}\varrho ^{\ast }.$ Now we prove that $\varrho ^{\ast }$ is a fixed point of $\Im _{2}.$ By (3.2), we have

This implies that

Letting $r\rightarrow \infty,$ we have $\left \Vert \mathfrak{\varsigma } \left(\varrho ^{\ast }, \Im _{2}\varrho ^{\ast }\right) \right \Vert = 0.$ Thus $\varrho ^{\ast } = \Im _{2}\varrho ^{\ast }.$ Thus $\varrho ^{\ast }$ is a common fixed point of $\Im _{1}$ and $\Im _{2}.$ Now we prove that $\varrho ^{\ast }$ is unique. We suppose that

but $\varrho ^{\ast }\not = \varrho ^{/}.$ Now from (3.2), we have

This implies that

As $\rho \left(\varrho ^{\ast }, \varrho ^{/}\right) +\sqrt{2}\varpi \left(\varrho ^{\ast }, \varrho ^{/}\right) < 1,$ we have

Thus $\varrho ^{\ast } = \varrho ^{/}.$

Corollary 1. Let $\left(\mathfrak{L}, \mathfrak{\varsigma }\right)$ be a complete bi CVMS and $\Im _{1}, \Im _{2}:\mathfrak{L}$ $\rightarrow \mathfrak{L}$. If there exist mappings $\rho, \kappa :\mathfrak{L\times L\rightarrow }[0, 1)$ such that

(a) $\rho \left(\Im _{2}\Im _{1}\varrho, \hbar \right) \leq \rho \left(\varrho, \hbar \right)$ and $\rho \left(\varrho, \Im _{1}\Im _{2}\hbar \right) \leq \rho \left(\varrho, \hbar \right),$ $\kappa \left(\Im _{2}\Im _{1}\varrho, \hbar \right) \leq \kappa \left(\varrho, \hbar \right)$ and $\kappa \left(\varrho, \Im _{1}\Im _{2}\hbar \right) \leq \kappa \left(\varrho, \hbar \right),$

(b) $\rho \left(\varrho, \hbar \right) +\kappa \left(\varrho, \hbar \right) < 1$,

(c) $\mathfrak{\varsigma }\left(\Im _{1}\varrho, \Im _{2}\hbar \right) \preceq _{i_{2}}\rho \left(\varrho, \hbar \right) \mathfrak{\varsigma }\left(\varrho, \hbar \right) +\kappa \left(\varrho, \hbar \right) \frac{\mathfrak{ \varsigma }\left(\varrho, \Im _{1}\varrho \right) \mathfrak{\varsigma } \left(\hbar, \Im _{2}\hbar \right) }{1+\mathfrak{\varsigma }\left(\varrho, \hbar \right) }, \text{ }$

for all $\varrho, \hbar \in$ $\mathfrak{L}$, then $\Im _{1}$ and $\Im _{2}$ have a unique common fixed point.

Proof. Setting $\varpi :\mathfrak{L\times L\rightarrow }[0, 1)$ by $\varpi \left(\varrho, \hbar \right) = 0$ in Theorem 1.

Corollary 2. Let $\left(\mathfrak{L}, \mathfrak{\varsigma }\right)$ be a complete bi CVMS and $\Im _{1}, \Im _{2}:\mathfrak{L}\rightarrow \mathfrak{L}$. If there exist mappings $\rho, \varpi :\mathfrak{L\times L\rightarrow }[0, 1)$ such that for all $\varrho, \hbar \in$ $\mathfrak{L, }$

(a) $\rho \left(\Im _{2}\Im _{1}\varrho, \hbar \right) \leq \rho \left(\varrho, \hbar \right)$ and $\rho \left(\varrho, \Im _{1}\Im _{2}\hbar \right) \leq \rho \left(\varrho, \hbar \right),$ $\varpi \left(\Im _{2}\Im _{1}\varrho, \hbar \right) \leq \varpi \left(\varrho, \hbar \right)$ and $\varpi \left(\varrho, \Im _{1}\Im _{2}\hbar \right) \leq \varpi \left(\varrho, \hbar \right),$

(b) $\rho \left(\varrho, \hbar \right) +\varpi \left(\varrho, \hbar \right) < 1,$

(c) $\mathfrak{\varsigma}\left(\Im _{1}\varrho, \Im _{2}\hbar \right) \preceq _{i_{2}}\rho \left(\varrho, \hbar \right) \mathfrak{\varsigma }\left(\varrho, \hbar \right) +\varpi \left(\varrho, \hbar \right) \frac{\mathfrak{ \varsigma }\left(\hbar, \Im _{1}\varrho \right) \mathfrak{\varsigma }\left(\varrho, \Im _{2}\hbar \right) }{1+\mathfrak{\varsigma }\left(\varrho, \hbar \right) }\text{, }$

then $\Im _{1}$ and $\Im _{2}$ have a unique common fixed point.

Proof. Setting $\kappa :\mathfrak{L\times L\rightarrow }[0, 1)$ by $\kappa \left(\varrho, \hbar \right) = 0$ in Theorem 1.

Corollary 3. Let $\left(\mathfrak{L}, \mathfrak{\varsigma }\right)$ be a complete bi CVMS and $\Im _{1}, \Im _{2}$: $\mathfrak{L}$ $\rightarrow \mathfrak{L}$. If there exists mapping $\rho :\mathfrak{L\times L\rightarrow }[0, 1)$ such that

(a) $\rho \left(\Im _{2}\Im _{1}\varrho, \hbar \right) \leq \rho \left(\varrho, \hbar \right)$ and $\rho \left(\varrho, \Im _{1}\Im _{2}\hbar \right) \leq \rho \left(\varrho, \hbar \right),$

(b) $\mathfrak{\varsigma }\left(\Im _{1}\varrho, \Im _{2}\hbar \right) \preceq _{i_{2}}\rho \left(\varrho, \hbar \right) \mathfrak{\varsigma }\left(\varrho, \hbar \right) \text{, }$

for all $\varrho, \hbar \in$ $\mathfrak{L, }$ then $\Im _{1}$ and $\Im _{2}$ have a unique common fixed point.

Proof. Setting $\kappa, \varpi :\mathfrak{L\times L\rightarrow }[0, 1)$ by $\kappa \left(\varrho, \hbar \right) = \varpi \left(\varrho, \hbar \right) = 0$ in Theorem 1.

Corollary 4. Let $\left(\mathfrak{L}, \mathfrak{\varsigma }\right)$ be a complete bi CVMS and $\Im :\mathfrak{L}$ $\rightarrow \mathfrak{L}$. If there exist mappings $\rho, \kappa, \varpi :\mathfrak{L\times L\rightarrow } [0, 1)$ such that

(a) $\rho \left(\Im \varrho, \hbar \right) \leq \rho \left(\varrho, \hbar \right)$ and $\rho \left(\varrho, \Im \hbar \right) \leq \rho \left(\varrho, \hbar \right),$ $\kappa \left(\Im \varrho, \hbar \right) \leq \kappa \left(\varrho, \hbar \right)$ and $\kappa \left(\varrho, \Im \hbar \right) \leq \kappa \left(\varrho, \hbar \right),$ $\varpi \left(\Im \varrho, \hbar \right) \leq \varpi \left(\varrho, \hbar \right)$ and $\varpi \left(\varrho, \Im \hbar \right) \leq \varpi \left(\varrho, \hbar \right),$

(b) $\rho \left(\varrho, \hbar \right) +\kappa \left(\varrho, \hbar \right) +\varpi \left(\varrho, \hbar \right) < 1$,

(c) $\mathfrak{\varsigma }\left(\Im \varrho, \Im \hbar \right) \preceq _{i_{2}}\rho \left(\varrho, \hbar \right) \mathfrak{\varsigma }\left(\varrho, \hbar \right) +\kappa \left(\varrho, \hbar \right) \frac{\mathfrak{ \varsigma }\left(\varrho, \Im \varrho \right) \mathfrak{\varsigma }\left(\hbar, \Im \hbar \right) }{1+\mathfrak{\varsigma }\left(\varrho, \hbar \right) }+\varpi \left(\varrho, \hbar \right) \frac{\mathfrak{\varsigma } \left(\hbar, \Im \varrho \right) \mathfrak{\varsigma }\left(\varrho, \Im \hbar \right) }{1+\mathfrak{\varsigma }\left(\varrho, \hbar \right) }\text{}$,

for all $\varrho, \hbar \in$ $\mathfrak{L}$, then $\Im$ has a unique fixed point.

Proof. Setting $\Im _{1} = \Im _{2} = \Im$ in Theorem 1.

Example 2. Let $\mathfrak{L} = [0, 1]$ and $\mathfrak{\varsigma }:\mathfrak{L}\times \mathfrak{L}\rightarrow \mathbb{C}$ defined by

for all $\varrho, \hbar \in$ $\mathfrak{L}.$ Then $(\mathfrak{L}, \mathfrak{\varsigma })$ is a complete bi CVMS. Define $\Im _{1}, \Im _{2}:\mathfrak{L}$ $\rightarrow \mathfrak{L}$ by

Consider

by

and

and

Then evidently,

Now

and

Also,

and

and

and

Now

Then it is very simple to prove that all the conditions of Theorem 1 are satisfied and $0$ is a common fixed point of mappings $\Im _{1}$ and $\Im _{2}.$

Corollary 5. Let $\left(\mathfrak{L}, \mathfrak{\varsigma }\right)$ be a complete bi CVMS and let $\Im :\mathfrak{L}$ $\rightarrow \mathfrak{L}$. If there exist $\rho, \kappa, \varpi :\mathfrak{L\times L\rightarrow }[0, 1)$ such that

(a) $\rho \left(\Im \varrho, \hbar \right) \leq \rho \left(\varrho, \hbar \right)$ and $\rho \left(\varrho, \Im \hbar \right) \leq \rho \left(\varrho, \hbar \right),$ $\kappa \left(\Im \varrho, \hbar \right) \leq \kappa \left(\varrho, \hbar \right)$ and $\kappa \left(\varrho, \Im \hbar \right) \leq \kappa \left(\varrho, \hbar \right),$ $\varpi \left(\Im \varrho, \hbar \right) \leq \varpi \left(\varrho, \hbar \right)$ and $\varpi \left(\varrho, \Im \hbar \right) \leq \varpi \left(\varrho, \hbar \right),$

(b) $\rho \left(\varrho, \hbar \right) +\kappa \left(\varrho, \hbar \right) +\varpi \left(\varrho, \hbar \right) < 1$,

for all $\varrho, \hbar \in$ $\mathfrak{L}$, then $\Im$ has a unique fixed point.

Proof. From the Corollary (4), we have $\varrho \in \mathfrak{L}$ such that $\Im ^{n}\varrho = \varrho.$ Now from

which implies that

which is possible only whenever $\left \vert \mathfrak{\varsigma }\left(\Im \varrho, \varrho \right) \right \vert = 0.$ Thus $\Im \varrho = \varrho.$

4.   Deduced results

Corollary 6. Let $\left(\mathfrak{L}, \mathfrak{\varsigma }\right)$ be a complete bi CVMS and let $\Im _{1}, \Im _{2}:\mathfrak{L}$ $\rightarrow \mathfrak{L}$. If there exist $\rho, \kappa, \varpi :\mathfrak{L\rightarrow } [0, 1)$ such that for all $\varrho, \hbar \in$ $\mathfrak{L, }$

(a) $\rho \left(\Im _{2}\Im _{1}\varrho \right) \leq \rho \left(\varrho \right),$ $\kappa \left(\Im _{2}\Im _{1}\varrho \right) \leq \kappa \left(\varrho \right),$ $\varpi \left(\Im _{2}\Im _{1}\varrho \right) \leq \varpi \left(\varrho \right),$ (b) $\rho \left(\varrho \right) +\kappa \left(\varrho \right) +\varpi \left(\varrho \right) < 1$,

(c) $\mathfrak{\varsigma }\left(\Im _{1}\varrho, \Im _{2}\hbar \right) \preceq _{i_{2}}\rho \left(\varrho \right) \mathfrak{\varsigma }\left(\varrho, \hbar \right) +\kappa \left(\varrho \right) \frac{\mathfrak{\varsigma } \left(\varrho, \Im _{1}\varrho \right) \mathfrak{\varsigma }\left(\hbar, \Im _{2}\hbar \right) }{1+\mathfrak{\varsigma }\left(\varrho, \hbar \right) }+\varpi \left(\varrho \right) \frac{\mathfrak{\varsigma }\left(\hbar, \Im _{1}\varrho \right) \mathfrak{\varsigma }\left(\varrho, \Im _{2}\hbar \right) }{1+\mathfrak{\varsigma }\left(\varrho, \hbar \right) } \text{, }$

Corollary 7. Let $\left(\mathfrak{L}, \mathfrak{\varsigma }\right)$ be a complete bi CVMS and let $\Im _{1}, \Im _{2}:\mathfrak{L}$ $\rightarrow \mathfrak{L}$. If there exist $\rho, \kappa, \varpi :\mathfrak{L\rightarrow } [0, 1)$ such that for all $\varrho, \hbar \in$ $\mathfrak{L, }$

(a) $\rho \left(\Im _{2}\Im _{1}\varrho \right) \leq \rho \left(\varrho \right),$ $\kappa \left(\Im _{2}\Im _{1}\varrho \right) \leq \kappa \left(\varrho \right),$ $\varpi \left(\Im _{2}\Im _{1}\varrho \right) \leq \varpi \left(\varrho \right),$

(b) $\rho \left(\varrho \right) +\kappa \left(\varrho \right) +\varpi \left(\varrho \right) < 1$,

(c) $\mathfrak{\varsigma }\left(\Im _{1}\varrho, \Im _{2}\hbar \right) \preceq _{i_{2}}\rho \left(\varrho \right) \mathfrak{\varsigma }\left(\varrho, \hbar \right) +\kappa \left(\varrho \right) \frac{\mathfrak{\varsigma } \left(\varrho, \Im _{1}\varrho \right) \mathfrak{\varsigma }\left(\hbar, \Im _{2}\hbar \right) }{1+\mathfrak{\varsigma }\left(\varrho, \hbar \right) }+\varpi \left(\varrho \right) \frac{\mathfrak{\varsigma }\left(\hbar, \Im _{1}\varrho \right) \mathfrak{\varsigma }\left(\varrho, \Im _{2}\hbar \right) }{1+\mathfrak{\varsigma }\left(\varrho, \hbar \right) } \text{, }$

then $\Im _{1}$ and $\Im _{2}$ have a unique common fixed point.

Proof. Define $\rho, \kappa, \varpi :\mathfrak{L\times L\rightarrow }[0, 1)$ by

for all $\varrho, \hbar \in \mathfrak{L.}$ Then for all $\varrho, \hbar \in \mathfrak{L, }$ we have

(a) $\rho \left(\Im _{2}\Im _{1}\varrho, \hbar \right) = \rho \left(\Im _{2}\Im _{1}\varrho \right) \leq \rho \left(\varrho \right) = \rho \left(\varrho, \hbar \right)~~ and~~ \rho \left(\varrho, \Im _{1}\Im _{2}\hbar \right) = \rho \left(\varrho \right) = \rho \left(\varrho, \hbar \right)$,

$\kappa \left(\Im _{2}\Im _{1}\varrho, \hbar \right) = \kappa \left(\Im _{2}\Im _{1}\varrho \right) \leq \kappa \left(\varrho \right) = \kappa \left(\varrho, \hbar \right) ~~and~~ \kappa \left(\varrho, \Im _{1}\Im _{2}\hbar \right) = \kappa \left(\varrho \right) = \kappa \left(\varrho, \hbar \right)$,

$\varpi \left(\Im _{2}\Im _{1}\varrho, \hbar \right) = \varpi \left(\Im _{2}\Im _{1}\varrho \right) \leq \varpi \left(\varrho \right) = \varpi \left(\varrho, \hbar \right)$ and $\varpi \left(\varrho, \Im _{1}\Im _{2}\hbar \right) = \varpi \left(\varrho \right) = \varpi \left(\varrho, \hbar \right)$,

(b) $\ \rho \left(\varrho, \hbar \right) +\kappa \left(\varrho, \hbar \right) +\varpi \left(\varrho, \hbar \right) = \rho \left(\varrho \right) +\kappa \left(\varrho \right) +\varpi \left(\varrho \right) < 1$,

(c) $\mathfrak{\varsigma}\left(\Im _{1}\varrho, \Im _{2}\hbar \right)$ $\preceq _{i_{2}}\rho \left(\varrho \right) \mathfrak{\varsigma }\left(\varrho, \hbar \right) +\kappa \left(\varrho \right) \frac{\mathfrak{\varsigma } \left(\varrho, \Im _{1}\varrho \right) \mathfrak{\varsigma }\left(\hbar, \Im _{2}\hbar \right) }{1+\mathfrak{\varsigma }\left(\varrho, \hbar \right) }+\varpi \left(\varrho \right) \frac{\mathfrak{\varsigma }\left(\hbar, \Im _{1}\varrho \right) \mathfrak{\varsigma }\left(\varrho, \Im _{2}\hbar \right) }{1+\mathfrak{\varsigma }\left(\varrho, \hbar \right) } \text{ }$ = $\rho \left(\varrho, \hbar \right) \mathfrak{\varsigma }\left(\varrho, \hbar \right) +\kappa \left(\varrho, \hbar \right) \frac{\mathfrak{ \varsigma }\left(\varrho, \Im _{1}\varrho \right) \mathfrak{\varsigma } \left(\hbar, \Im _{2}\hbar \right) }{1+\mathfrak{\varsigma }\left(\varrho, \hbar \right) }+\varpi \left(\varrho, \hbar \right) \frac{\mathfrak{ \varsigma }\left(\hbar, \Im _{1}\varrho \right) \mathfrak{\varsigma }\left(\varrho, \Im _{2}\hbar \right) }{1+\mathfrak{\varsigma }\left(\varrho, \hbar \right) }$,

(d) $\lambda = \frac{\rho \left(\varrho _{0}, \varrho _{1}\right) }{1-\kappa \left(\varrho _{0}, \varrho _{1}\right) } = \frac{\rho \left(\varrho _{0}\right) }{1-\kappa \left(\varrho _{0}\right) } < 1.$

By Theorem $1$, $\Im _{1}$ and $\Im _{2}$ have a unique common fixed point.

Corollary 8. Let $\left(\mathfrak{L}, \mathfrak{\varsigma }\right)$ be a complete bi CVMS and let $\Im _{1}, \Im _{2}:\mathfrak{L}$ $\rightarrow \mathfrak{L}$. If there exist $\rho, \kappa, \varpi \in \lbrack 0, 1)$ with $\rho +\kappa +\varpi < 1$ such that

for all $\varrho, \hbar \in$ $\mathfrak{L}$, then $\Im _{1}$ and $\Im _{2}$ have a unique common fixed point.

Proof. Taking $\rho \left(\cdot \right) = \rho,$ $\kappa \left(\cdot \right) = \kappa$ and $\varpi \left(\cdot \right) = \varpi$ in Corollary (6).

Corollary 9. Let $\left(\mathfrak{L}, \mathfrak{\varsigma }\right)$ be a complete bi CVMS and let $\Im _{1}, \Im _{2}:\mathfrak{L}$ $\rightarrow \mathfrak{L}$. If there exist $\rho, \kappa \in \lbrack 0, 1)$ with $\rho +\kappa < 1$ such that

for all $\varrho, \hbar \in$ $\mathfrak{L}$, then $\Im _{1}$ and $\Im _{2}$ have a unique common fixed point.

5.   Applications

Let $\mathfrak{L} = C([a, b], \mathbb{R})$, $(a > 0)$ where $C[a, b]$ denotes the set of all real continuous functions defined on the closed interval $[a, b]\ $and$\ d:\mathfrak{L}\times \mathfrak{L}\rightarrow \mathbb{C} _{2}$ be defined in this way

for all $\varrho, \hbar \in$ $\mathfrak{L}$ and $t\in \lbrack a, b],$ where $\left \vert \cdot \right \vert$ is the usual real modulus$.$ Then ($\mathfrak{ L}, d$) is complete bi CVMS. Consider the integral equations of Urysohn type

where $g:$ $[a, b]\rightarrow \mathbb{R}$ and $K_{1}, K_{2}:[a, b]\times \lbrack a, b]\times \mathbb{R}\rightarrow \mathbb{R}$ are continuous for $t\in \lbrack a, b]$. We define partial order $\preceq _{i_{2}}$ in $\mathbb{C} _{2}$ as follows $\varrho \left(t\right) \preceq _{i_{2}}\hbar \left(t\right)$ if and only if $\varrho \leq \hbar.$

Theorem 2. Suppose the following condition

holds, for all $\varrho, \hbar \in \mathfrak{L}$ with $\varrho \not = \hbar$ and for some control function $\rho :\mathfrak{L\times L\rightarrow }[0, 1)$, then the integral operators defined by (5.1) and (5.2) have a unique common solution.

Proof. Define continuous mappings $\Im _{1}, \Im _{2}$: $\mathfrak{L}$ $\rightarrow \mathfrak{L}$ by

for all$\ t\in \lbrack a, b].$ Consider

Thus

Now with $\kappa, \varpi :\mathfrak{L\times L\rightarrow }[0, 1)$ defined by

for every $\varrho, \hbar \in \mathfrak{L}$, all the assumptions of Theorem (1) are satisfied and the integral equations (5.1) and (5.2) have a unique common solution.

6.   Conclusions

In this article, we have utilized the notion of bicomplex valued metric space (bi CVMS) and obtained common fixed point results for rational contractions involving control functions of two variables. We have derived common fixed points and fixed points of single valued mappings for contractions involving control functions of one variable and constants. We anticipate that the obtained theorems in this article will establish new relationships for those who use bi CVMS. Still there are some open problems that can be addressed in future work. For example:

1) Can the notion of bi complex valued metric space be extended to hypercomplex valued metric space?

2) Can the results proved in this article be extended to multivalued mappings and fuzzy set valued mappings ^[30]?

3) Can differential and integral inclusions can be solved as applications of fixed point results for multivalued mappings in the setting of bi complex valued metric space?

Acknowledgments

The authors extend their appreciation to the deputyship for Research and Innovation, Ministry of Education in Saudi Arabia for funding this research work through project number (IFP-2020-106).

Conflict of interest

The authors declare that they have no conflicts of interest.