Common fixed point theorems on complex valued extended $b$-metric spaces for rational contractions with application

1.   Introduction

The notion of complex valued metric space is introduced by Azam et al. ^[1] in 2011 and established common fixed points of self mappings satisfying rational contractions. Later on, Rouzkard et al. ^[2] gave generalized contraction and extended the leading theorem of Azam et al. ^[1]. Subsequently, Sintunavarat et al. ^[3] replaced the constants involved in the contraction with control functions of one variable and generalized the results of Azam et al. ^[1] and Rouzkard et al. ^[2]. Sitthikul et al. ^[4] used control functions of two vaiables in the contraction and established common fixed point theorems in context of complex valued metric space. Although many researchers ^{[5,6,7,8,9,10]} worked in this space and proved different generalized results. Mukheimer ^[11] gave the notion of complex valued $b$-metric space (CV$b$MS) by involving a constant $\pi \geq 1$ in the triangle inequality and generalized the concept of complex valued metric space (CVMS). Kumar ^[12] and Rao et al. ^[13] proved common fixed point results in CV$b$MS for generalized contractions. Naimatullah et al. ^[14] replaced contant with a control function and extended the concept of complex valued $b$-metric space (CV$b$MS) to complex valued extended $b$-metric space (CVE$b$MS). They proved fixed points of multivalued mappings for contractions involving rational expressions in CVE$b$MS. For more details in this direction, we refer the readers to ^{[15,16,17,18,19,20,21,22]}.

In this paper, we obtain common fixed point theorems in complex valued extended $b$-metric spaces (CVE$b$MS) for rational contractions with contractiveness on a closed ball. We also provide a significant example to show the originality of obtained results.

2.   Preliminaries

Azam et al. ^[1] gave the notion of complex valued metric space (CVMS) in this way.

Definition 1. (See ^[1])Let $\omega _{1}, \omega _{2}\in \mathbb{C}$ (set of complex numbers). A partial order $\precsim \ $on $\mathbb{C}$ is defined as follows

It follows that

if one of these assertions is satisfied:

Definition 2. (See ^[1]) Let $\mathcal{W}\not = \emptyset$ and $d:\mathcal{W}\times \mathcal{W}\rightarrow \mathbb{C}$ satisfy

(i) $0\precsim d(\omega, \varrho)\ $and $d(\omega, \varrho) = 0$ if and only if $\omega = \varrho$;

(ii) $d(\omega, \varrho) = d(\varrho, \omega)$;

(iii) $d(\omega, \varrho)\precsim d(\omega, \nu)+d(\nu, \varrho)$;

for all $\omega, \varrho, \nu \in \mathcal{W},$ then $(\mathcal{W}, d)\ $is said to be complex valued metric space (CVMS).

Example 3. (See ^[1]) Let $\mathcal{W} = [0, 1]$. Define $d:\mathcal{W}\times \mathcal{W}\rightarrow \mathbb{C}$ by

for all $\omega, \varrho \in \mathcal{W}\mathfrak{, }$ then $(\mathcal{W}, d)$ is CVMS.

Mukheimer ^[11] gave the conception of complex valued $b$-metric space (CV$b$MS) in this way.

Definition 4. (See ^[11]) Let $\mathcal{W}\not = \emptyset$ and $\pi \geq 1$ be a real number. If a mapping $d:\mathcal{W}\times \mathcal{W}\rightarrow \mathbb{C}$ satisfy

(i) $0\precsim d(\omega, \varrho)\ $and $d(\omega, \varrho) = 0$ if and only if $\omega = \varrho$;

(ii) $d(\omega, \varrho) = d(\varrho, \omega)$;

(iii) $d(\omega, \varrho)\precsim \pi \left[d(\omega, \nu)+d(\nu, \varrho)\right]$;

for all $\omega, \varrho, \nu \in \mathcal{W},$ then $(\mathcal{W}, d)\ $is called a complex valued $b$- metric space (CV$b$MS).

Example 5. (See ^[11]) Let $\mathcal{W} = [0, 1].$ Define $d:\mathcal{W}\times \mathcal{W}\rightarrow \mathbb{C}$ by

for all $\omega, \varrho \in \mathcal{W},$ then $(\mathcal{W}, d)$ is CV$b$MS with $\pi = 2$.

Recently, Naimatullah et al. ^[14] defined the notion of complex valued extended $b$-metric space (CVE$b$MS) in this way.

Definition 6. (See ^[14]) Let $\mathcal{W}\not = \emptyset$ and $\varphi :\mathcal{W} \times \mathcal{W}\rightarrow \lbrack 1, \infty)$. If a mapping $d:\mathcal{W}\times \mathcal{W}\rightarrow \mathbb{C}$ satisfy

(i) $0\precsim d(\omega, \varrho)\ $and $d(\omega, \varrho) = 0$ if and only if $\omega = \varrho$;

(ii) $d(\omega, \varrho) = d(\varrho, \omega)$;

(iii) $d(\omega, \varrho)\precsim \varphi (\omega, \varrho)\left[d(\omega, \nu)+d(\nu, \varrho)\right]$;

for all $\omega, \varrho, \nu \in \mathcal{W},$ then $(\mathcal{W}, d)$ is called CVE$b$MS.

Example 7. (See ^[14]) Let $\mathcal{W}\not = \varnothing$ and $\varphi :\mathcal{W}\times \mathcal{W}\rightarrow \lbrack 1, \infty)$ be defined by

and $d:\mathcal{W}\times \mathcal{W}\rightarrow \mathbb{C}$ by

(i) $d(\omega, \varrho) = \frac{i}{\omega \varrho },$ for all $0 < \omega, \varrho \leq 1;$

(ii) $d(\omega, \varrho) = 0$ if and only if $\omega = \varrho,$ for all $0\leq \omega, \varrho \leq 1;$

(iii) $d(\omega, 0) = d(0, \omega) = \frac{i}{\omega },$ for all $0 < \omega \leq 1.$

Then $(\mathcal{W}, d)$ is a CVE$b$MS.

Example 8. Let $\mathcal{W} = [0, \infty)$ and $\varphi :\mathcal{W}\times \mathcal{W} \rightarrow \lbrack 1, \infty)$ be a function defined by $\varphi (\omega, \varrho) = 1+\omega +\varrho$ and $d:\mathcal{W}\times \mathcal{W} \rightarrow \mathbb{C}$ by

Then $(\mathcal{W}, d)$ is a CVE$b$MS.

Lemma 9. (See ^[14]) Let $(\mathcal{W}, d)\ $be a CVE$b$MS and $\left \{ \omega _{n}\right \}$ $\subseteq \mathcal{W}$, then $\left \{ \omega _{n}\right \}$ converges to $\omega$ if and only if $\left \vert d(\omega _{n}, \omega)\right \vert \rightarrow 0,$ as $n\rightarrow \infty.$

Lemma 10. (See ^[14]) Let $(\mathcal{W}, d)\ $be a CVEbMS and $\left \{ \omega _{n}\right \}$ $\subseteq \mathcal{W}$, then $\left \{ \omega _{n}\right \}$ is a Cauchy sequence if and only if $\ \left \vert d(\omega _{n}, \omega _{m})\right \vert \rightarrow 0\ $as$\ n, m\rightarrow \infty.$

3.   Results

Now we state our main result in this way.

Theorem 11. Let $(\mathcal{W}, d)$ be a complete CVE$b$MS with $\varphi : \mathcal{W}\times \mathcal{W}\rightarrow \lbrack 1, \infty)$ and $\beth _{1}, \beth _{2}:\mathcal{W}\rightarrow \mathcal{W}$. Suppose that there exist $\aleph _{1}, \aleph _{2}, \aleph _{3}, \aleph _{4}, \aleph _{5}\in \lbrack 0, 1)$ with $\aleph _{1}+\aleph _{2}+\aleph _{3}+2\aleph _{4}+2\aleph _{5} < 1$ such that

for all $\omega _{0}, \omega, \varrho \in \overline{B(\omega _{0}, r)}$, $0\prec r\in \mathbb{C}$ and

where $\lambda = \max \{(\frac{\aleph _{1}+\aleph _{4}}{1-\aleph _{2}-\aleph _{4}}), (\frac{\aleph _{1}+\aleph _{5}}{1-\aleph _{2}-\aleph _{5}})\}$. And for each $\omega _{0}\in \overline{B(\omega _{0}, r)}$ and $\lim_{n, m\rightarrow +\infty }\varphi \left(\omega _{n}, \omega _{m}\right) \lambda < 1$, then there exists a unique point $\omega ^{\ast }\in \overline{B(\omega _{0}, r)\mathit{\text{}}}$ such that $\omega ^{\ast } = \beth _{1}\omega ^{\ast } = \beth _{2}\omega ^{\ast }$.

Proof. Let $\omega _{0}$ $\in \mathcal{W}$ and define

for all $n = 0, 1, 2, \ldots.$ Now we show that $\omega _{n}\in \overline{ B(\omega _{0}, r)},$ for all $n\in \mathbb{N}$. By the fact that $\lambda = \max \{(\frac{\aleph _{1}+\aleph _{4}}{ 1-\aleph _{2}-\aleph _{4}}), (\frac{\aleph _{1}+\aleph _{5}}{1-\aleph _{2}-\aleph _{5}})\} < 1$ and inequality (3.2), we have

It implies that $\omega _{1}\in \overline{B(\omega _{0}, r)}.$ Let $\omega _{2}, ..., \omega _{j}\in \overline{B(\omega _{0}, r)}$ for some $j\in N$. If $j = 2n+1$, where $n = 0, 1, 2, \ldots \frac{j-1}{2}$ or $j = 2n+2,$ where $n = 0, 1, 2, \ldots, \frac{j-2}{2}$. By (3.1), we have$\ $

Now $\omega _{2n+1} = \beth _{1}\omega _{2n}$ implies that $d\left(\omega _{2n+1}, \beth _{1}\omega _{2n}\right) = 0,$ so we have

This implies that

Since $|1+d\left(\omega _{2n}, \omega _{2n+1}\right) | > |d\left(\omega _{2n}, \omega _{2n+1}\right) |,$ so we have

Which implies that by triangular inequality

Similarly, we get$\ $

Now $\omega _{2n+2} = \beth _{2}\omega _{2n+1}$ implies that $d\left(\omega _{2n+2}, \beth _{2}\omega _{2n+1}\right) = 0,$ so we have

This implies that

Since $|1+d\left(\omega _{2n+2}, \omega _{2n+1}\right) | > |d\left(\omega _{2n+2}, \omega _{2n+1}\right) |,$ so we have

Which implies that by triangular inequality

Putting $\lambda = \max \{(\frac{\aleph _{1}+\aleph _{4}}{1-\aleph _{2}-\aleph _{4}}), (\frac{\aleph _{1}+\aleph _{5}}{1-\aleph _{2}-\aleph _{5}})\},$ we obtain that

Now

gives $\omega _{j+1}\in \overline{B(\omega _{0}, r)}.$ Hence $\omega _{n}\in \overline{B(\omega _{0}, r)}$ for all $n\in \mathbb{N}.$ One can easily prove that

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

Since $\lim_{n, m\rightarrow +\infty }\varphi \left(\omega _{n}, \omega _{m}\right) \lambda < 1,$ so the series $\sum \limits_{n = 1}^{\infty }\lambda ^{n}\prod \limits_{i = 1}^{p}\varphi \left(\omega _{i}, \omega _{m}\right)$ converges by ratio test for each $m\in \mathbb{N}.$ Let

Thus, for $m > n$, the above inequality can be written as

Now, by taking the limit as $n, m\rightarrow +\infty$, we get

By lemma (10), we conclude that$\ $the sequence $\{ \omega _{n}\}$ is a Cauchy sequence in $\overline{B(\omega _{0}, r)}.$ Consequently there exists $\omega ^{\ast }\in \overline{B(\omega _{0}, r)}$ such that $\lim \limits_{n\rightarrow +\infty }\omega _{n} = \omega ^{\ast }$. $\ $It follows that $\omega ^{\ast } = \beth _{1}\omega ^{\ast }, \ $otherwise $d(\omega ^{\ast }, \beth _{1}\omega ^{\ast }) = \upsilon \succ 0$ and we would then have

which implies that

That is $\left \vert \upsilon \right \vert = 0,$ which is a contradiction. Thus $\omega ^{\ast } = \beth _{1}\omega ^{\ast }.$Similarly, we can prove that $\omega ^{\ast } = \beth _{2}\omega ^{\ast }$.

Now we show uniqueness of common fixed point. We suppose $\omega ^{/}$ in $\mathcal{W}$ is another common fixed point of $\beth _{1}$ and $\beth _{2}$ that is $\omega ^{/} = \beth _{1}\omega ^{/} = \beth _{2}\omega ^{/}$ which is distinct from $\omega ^{\ast }$ that is $\omega ^{\ast }\not = \omega ^{/}.$ Now by (3.1), we have

so that

Since $|1+d(\omega ^{\ast }, \omega ^{/})| > |d(\omega ^{\ast }, \omega ^{/})|,$ so we have

This is contradiction to $\aleph _{1}+\aleph _{3} < 1.$ Hence, $\omega ^{/} = \omega ^{\ast }.$ Therefore $\omega ^{\ast }$ is a unique common fixed point of $\beth _{1}$ and $\beth _{2}.$

Corollary 12. Let $(\mathcal{W}, d)$ be a complete CVE$b$MS with $\varphi : \mathcal{W}\times \mathcal{W}\rightarrow \lbrack 1, \infty)$ and $\beth : \mathcal{W}\rightarrow \mathcal{W}.$ Suppose that there exist $\aleph _{1}, \aleph _{2}, \aleph _{3}, \aleph _{4}$, $\aleph _{5}\in \lbrack 0, 1)$ with $\aleph _{1}+\aleph _{2}+\aleph _{3}+2\aleph _{4}+2\aleph _{5} < 1$ such that

for all $\omega _{0}, \omega, \varrho \in \overline{B(\omega _{0}, r)}$, $0\prec r\in \mathbb{C}$ and

where $\lambda = \max \{(\frac{\aleph _{1}+\aleph _{4}}{1-\aleph _{2}-\aleph _{4}}), (\frac{\aleph _{1}+\aleph _{5}}{1-\aleph _{2}-\aleph _{5}})\}$. And for each $\omega _{0}\in \overline{B(\omega _{0}, r)}$ and $\lim_{n, m\rightarrow +\infty }\varphi \left(\omega _{n}, \omega _{m}\right) \lambda < 1$, there exists a unique point $\omega ^{\ast }\in \overline{B(\omega _{0}, r)\mathit{\text{}}}$ such that $\omega ^{\ast } = \beth \omega ^{\ast }$.

Proof. Taking $\beth _{1} = \beth _{2} = \beth$ in Theorem 11.

Example 13. Suppose

and $\mathcal{W} = \mathcal{W}_{1}\cup \mathcal{W}_{2}.$ Consider complex valued extended $b$-metric $d:\mathcal{W}\times \mathcal{W}\longrightarrow \mathbb{C} \ \ $as follows:

and $\varphi :\mathcal{W}\times \mathcal{W}\rightarrow \lbrack 1, \infty)$ by $\varphi (\omega, \varrho) = 2$. Then $(\mathcal{W}, d)$ is complete CVE$b$MS. Take $\upsilon _{0} = \frac{1}{2}+0i$ and $r = \frac{1}{3}+\frac{1}{4}i.$ Then,

Define $\beth _{1}, \beth _{2}:\mathcal{W\rightarrow W}$ as

Then with $\aleph _{1} = \frac{1}{6}, \aleph _{2} = \frac{1}{24}, \aleph _{3} = \frac{1}{2}, \aleph _{4} = \frac{1}{25}$ and $\aleph _{5} = \frac{1}{26},$ all the assumptions of Theorem 11 are satisfied and hence $0+0i\in \overline{B(\upsilon _{0}, r)}$ is a unique common fixed point $\beth _{1}$ and $\beth _{2}$.

Corollary 14. Let $(\mathcal{W}, d)$ be a complete CVE$b$MS with $\varphi : \mathcal{W}\times \mathcal{W}\rightarrow \lbrack 1, \infty)$ and $\beth _{1}, \beth _{2}:\mathcal{W}\rightarrow \mathcal{W}$. Suppose that there exist $\aleph _{1}, \aleph _{2}, \aleph _{3}, \aleph _{4}\in \lbrack 0, 1)$ with $\aleph _{1}+\aleph _{2}+\aleph _{3}+2\aleph _{4} < 1$ such that

for all $\omega _{0}, \omega, \varrho \in \overline{B(\omega _{0}, r)}$, $0\prec r\in \mathbb{C}$ and

where $\lambda = \max \{ \frac{\aleph _{1}+\aleph _{4}}{1-\aleph _{2}-\aleph _{4}}, \frac{\aleph _{1}}{1-\aleph _{2}}\}$. And for each $\omega _{0}\in \overline{B(\omega _{0}, r)}$, $\lim_{n, m\rightarrow +\infty }\varphi \left(\omega _{n}, \omega _{m}\right) \lambda < 1$, then there exists a unique point $\omega ^{\ast }\in \overline{B(\omega _{0}, r)\mathit{\text{}}}$ such that $\omega ^{\ast } = \beth _{1}\omega ^{\ast } = \beth _{2}\omega ^{\ast }$.

Proof. Taking $\aleph _{5} = 0$ in Theorem 11.

Corollary 15. Let $(\mathcal{W}, d)$ be a complete CVE$b$MS with $\varphi : \mathcal{W}\times \mathcal{W}\rightarrow \lbrack 1, \infty)$ and $\beth : \mathcal{W}\rightarrow \mathcal{W}$. Suppose that there exist $\aleph _{1}, \aleph _{2}, \aleph _{3}, \aleph _{4}\in \lbrack 0, 1)$ with $\aleph _{1}+\aleph _{2}+\aleph _{3}+2\aleph _{4} < 1$ such that

for all $\omega _{0}, \omega, \varrho \in \overline{B(\omega _{0}, r)}$, $0\prec r\in \mathbb{C}$ and

where $\lambda = \max \{ \frac{\aleph _{1}+\aleph _{4}}{1-\aleph _{2}-\aleph _{4}}, \frac{\aleph _{1}}{1-\aleph _{2}}\}$. And for each $\omega _{0}\in \overline{B(\omega _{0}, r)}$ and $\lim_{n, m\rightarrow +\infty }\varphi \left(\omega _{n}, \omega _{m}\right) \lambda < 1$, then there exists a unique point $\omega ^{\ast }\in \overline{B(\omega _{0}, r)\mathit{\text{}}}$ such that $\omega ^{\ast } = \beth \omega ^{\ast }$.

Proof. Taking $\beth _{1} = \beth _{2} = \beth$ in Corollary 14.

Corollary 16. Let $(\mathcal{W}, d)$ be a complete CVE$b$MS with $\varphi : \mathcal{W}\times \mathcal{W}\rightarrow \lbrack 1, \infty)$ and $\beth _{1}, \beth _{2}:\mathcal{W}\rightarrow \mathcal{W}$. Suppose that there exist $\aleph _{1}, \aleph _{2}, \aleph _{3}, \aleph _{5}\in \lbrack 0, 1)$ with $\aleph _{1}+\aleph _{2}+\aleph _{3}+2\aleph _{5} < 1$ such that

for all $\omega _{0}, \omega, \varrho \in \overline{B(\omega _{0}, r)}$, $0\prec r\in \mathbb{C}$ and

where $\lambda = \max \{(\frac{\aleph _{1}}{1-\aleph _{2}}), (\frac{\aleph _{1}+\aleph _{5}}{1-\aleph _{2}-\aleph _{5}})\}$. And for each $\omega _{0}\in \overline{B(\omega _{0}, r)}$ and $\lim_{n, m\rightarrow +\infty }\varphi \left(\omega _{n}, \omega _{m}\right) \lambda < 1$, then there exists a unique point $\omega ^{\ast }\in \overline{B(\omega _{0}, r)\mathit{\text{}}}$ such that $\omega ^{\ast } = \beth _{1}\omega ^{\ast } = \beth _{2}\omega ^{\ast }$.

Proof. Taking $\aleph _{4} = 0$ in Theorem 11.

Corollary 17. Let $(\mathcal{W}, d)$ be a complete CVE$b$MS with $\varphi :\mathcal{W} \times \mathcal{W}\rightarrow \lbrack 1, \infty)$ and $\beth :\mathcal{W} \rightarrow \mathcal{W}$. Suppose that there exist $\aleph _{1}, \aleph _{2}, \aleph _{3}, \aleph _{5}\in \lbrack 0, 1)$ with $\aleph _{1}+\aleph _{2}+\aleph _{3}+2\aleph _{5} < 1$ such that

for all $\omega _{0}, \omega, \varrho \in \overline{B(\omega _{0}, r)}$, $0\prec r\in \mathbb{C}$ and

where $\lambda = \max \{(\frac{\aleph _{1}}{1-\aleph _{2}}), (\frac{\aleph _{1}+\aleph _{5}}{1-\aleph _{2}-\aleph _{5}})\}$. And for each $\omega _{0}\in \overline{B(\omega _{0}, r)}$ and $\lim_{n, m\rightarrow +\infty }\varphi \left(\omega _{n}, \omega _{m}\right) \lambda < 1$, then there exists a unique point $\omega ^{\ast }\in \overline{B(\omega _{0}, r)\mathit{\text{}}}$ such that $\omega ^{\ast } = \beth \omega ^{\ast }$.

Proof. By setting $\beth _{1} = \beth _{2} = \beth$ in Corollary 16.

Corollary 18. Let $(\mathcal{W}, d)$ be a complete CVE$b$MS with $\varphi : \mathcal{W}\times \mathcal{W}\rightarrow \lbrack 1, \infty)$ and $\beth _{1}, \beth _{2}:\mathcal{W}\rightarrow \mathcal{W}$. Suppose that there exist $\aleph _{1}, \aleph _{2}, \aleph _{3}\in \lbrack 0, 1)$ with $\aleph _{1}+\aleph _{2}+\aleph _{3} < 1$ such that

for all $\omega _{0}, \omega, \varrho \in \overline{B(\omega _{0}, r)}$, $0\prec r\in \mathbb{C}$ and

where $\lambda = \frac{\aleph _{1}}{1-\aleph _{2}}$. And for each $\omega _{0}\in \overline{B(\omega _{0}, r)}$ and $\lim_{n, m\rightarrow +\infty }\varphi \left(\omega _{n}, \omega _{m}\right) \lambda < 1$, then there exists a unique point $\omega ^{\ast }\in \overline{B(\omega _{0}, r)\mathit{\text{}}}$ such that $\omega ^{\ast } = \beth _{1}\omega ^{\ast } = \beth _{2}\omega ^{\ast }$.

Proof. By choosing $\aleph _{4} = \aleph _{5} = 0$ in Theorem 11.

Corollary 19. Let $(\mathcal{W}, d)$ be a complete CVE$b$MS with $\varphi : \mathcal{W}\times \mathcal{W}\rightarrow \lbrack 1, \infty)$ and $\beth : \mathcal{W}\rightarrow \mathcal{W}$. Suppose that there exist $\aleph _{1}, \aleph _{2}, \aleph _{3}\in \lbrack 0, 1)$ with $\aleph _{1}+\aleph _{2}+\aleph _{3} < 1$ such that

for all $\omega _{0}, \omega, \varrho \in \overline{B(\omega _{0}, r)}$, $0\prec r\in \mathbb{C}$ and

where $\lambda = \frac{\aleph _{1}}{1-\aleph _{2}}$. And for each $\omega _{0}\in \overline{B(\omega _{0}, r)}$ and $\lim_{n, m\rightarrow +\infty }\varphi \left(\omega _{n}, \omega _{m}\right) \lambda < 1$, then there exists a unique point $\omega ^{\ast }\in \overline{B(\omega _{0}, r)\mathit{\text{}}}$ such that $\omega ^{\ast } = \beth \omega ^{\ast }.$

Proof. Taking $\beth _{1} = \beth _{2} = \beth$ in Corollary 18.

Corollary 20. Let $(\mathcal{W}, d)$ be a complete CVE$b$MS with $\varphi : \mathcal{W}\times \mathcal{W}\rightarrow \lbrack 1, \infty)$ and $\beth _{1}, \beth _{2}:\mathcal{W}\rightarrow \mathcal{W}$. Suppose that there exist $\aleph _{1}, \aleph _{2}\in \lbrack 0, 1)$ with $\aleph _{1}+\aleph _{2} < 1$ such that

for all $\omega _{0}, \omega, \varrho \in \overline{B(\omega _{0}, r)}$, $0\prec r\in \mathbb{C}$ and

where $\lambda = \frac{\aleph _{1}}{1-\aleph _{2}}$. And for each $\omega _{0}\in \overline{B(\omega _{0}, r)}$ and $\lim_{n, m\rightarrow +\infty }\varphi \left(\omega _{n}, \omega _{m}\right) \lambda < 1$, then there exists a unique point $\omega ^{\ast }\in \overline{B(\omega _{0}, r)\mathit{\text{}}}$ such that $\omega ^{\ast } = \beth _{1}\omega ^{\ast } = \beth _{2}\omega ^{\ast }$.

Proof. Taking $\aleph _{3} = \aleph _{4} = \aleph _{5} = 0$ in Theorem 11.

Corollary 21. Let $(\mathcal{W}, d)$ be a complete CVE$b$MS with $\varphi : \mathcal{W}\times \mathcal{W}\rightarrow \lbrack 1, \infty)$ and $\beth : \mathcal{W}\rightarrow \mathcal{W}$. Suppose that there exist $\aleph _{1}, \aleph _{2}\in \lbrack 0, 1)$ with $\aleph _{1}+\aleph _{2} < 1$ such that

for all $\omega _{0}, \omega, \varrho \in \overline{B(\omega _{0}, r)}$, $0\prec r\in \mathbb{C}$ and

where $\lambda = \frac{\aleph _{1}}{1-\aleph _{2}}$. And for each $\omega _{0}\in \overline{B(\omega _{0}, r)}$ and $\lim_{n, m\rightarrow +\infty }\varphi \left(\omega _{n}, \omega _{m}\right) \lambda < 1$, then there exists a unique point $\omega ^{\ast }\in \overline{B(\omega _{0}, r)\mathit{\text{}}}$ such that $\omega ^{\ast } = \beth \omega ^{\ast }.$

Proof. Taking $\beth _{1} = \beth _{2} = \beth$ in Corollary 20.

Now we we establish the following result for two finite families of mappings as an application of Theorem 11.

Theorem 22. If $\{ \aleph _{i}\}_{1}^{m}$ and $\{ \Re _{i}\}_{1}^{n}$ are two finite pairwise commuting finite families of self-mapping defined on a complex valued extended $b$-metric space with $\varphi :\mathcal{W}\times \mathcal{W} \rightarrow \lbrack 1, \infty)$ such that the mappings $\Re$ and $\Im$ (with $\Im = \aleph _{1}\aleph _{2}\cdot \cdot \cdot \aleph _{m}$ and $\Re = \Re _{1}\Re _{2}\cdot \cdot \cdot \Re _{n})$ satisfy (3.1) and (3.2) then the component mappings of these $\{ \aleph _{i}\}_{1}^{m}$ and $\{ \Re _{i}\}_{1}^{n}$ have a unique common fixed point.

Proof. By Theorem 11, one can get $\Im \omega ^{\ast } = \Re \omega ^{\ast } = \omega ^{\ast },$ which is unique. Now by pairwise commutativity of $\{ \aleph _{i}\}_{1}^{m}$ and $\{ \Re _{i}\}_{1}^{n}$, (for every $1\leq k\leq m)$ one can write $\aleph _{k}\omega ^{\ast } = \aleph _{k}\aleph \omega ^{\ast } = \aleph \aleph _{k}\omega ^{\ast }$ and $\aleph _{k}\omega ^{\ast } = \aleph _{k}\Re \omega ^{\ast } = \Re \aleph _{k}\omega ^{\ast }$ which manifest that $\aleph _{k}\omega ^{\ast }$, for all $k,$ is also a common fixed point of $\Im$ and $\Re$. Now utilizing the uniqueness, one can write $\Im _{k}\omega ^{\ast } = \omega ^{\ast }$ (for every $k$) which shows that $\omega ^{\ast }$ is a common fixed point of $\{ \Im _{i}\}_{1}^{m}.$ By doing the same strategy, we can prove that $\Re _{k}\omega ^{\ast } = \omega ^{\ast }$ ($1\leq k\leq n)$. Hence $\{ \aleph _{i}\}_{1}^{m}$ and $\{ \Re _{i}\}_{1}^{n}$ have a unique common fixed point.

Corollary 23. Let $(\mathcal{W}, d)$ be a complete CVE$b$MS with $\varphi : \mathcal{W}\times \mathcal{W}\rightarrow \lbrack 1, \infty)$ and $F, G: \mathcal{W}\rightarrow \mathcal{W}$. Suppose that there exist $\aleph _{1}, \aleph _{2}, \aleph _{3}, \aleph _{4}$, $\aleph _{5}\in \lbrack 0, 1)$ with $\aleph _{1}+\aleph _{2}+\aleph _{3}+2\aleph _{4}+2\aleph _{5} < 1$ such that

for all $\omega _{0}, \omega, \varrho \in \overline{B(\omega _{0}, r)}$, $0\prec r\in \mathbb{C}$ and

where $\lambda = \max \{(\frac{\aleph _{1}+\aleph _{4}}{1-\aleph _{2}-\aleph _{4}}), (\frac{\aleph _{1}+\aleph _{5}}{1-\aleph _{2}-\aleph _{5}})\}$. And for each $\omega _{0}\in \overline{B(\omega _{0}, r)}$ and $\lim_{n, m\rightarrow +\infty }\varphi \left(\omega _{n}, \omega _{m}\right) \lambda < 1$, then there exists a unique point $\omega ^{\ast }\in \overline{ B(\omega _{0}, r)\mathit{\text{}}}$ such that $\omega ^{\ast } = F\omega ^{\ast } = G\omega ^{\ast }$.

Proof. Taking $\aleph _{1} = \aleph _{2} = \cdot \cdot \cdot = \aleph _{m} = F$ and $\Re _{1} = \Re _{2} = \cdot \cdot \cdot = \Re _{n} = G,$ in Theorem 18.

Corollary 24. Let $(\mathcal{W}, d)$ be a complete CVE$b$MS with $\varphi :\mathcal{W} \times \mathcal{W}\rightarrow \lbrack 1, \infty)$ and $\beth :\mathcal{W} \rightarrow \mathcal{W}.$ Suppose that there exist $\aleph _{1}, \aleph _{2}, \aleph _{3}, \aleph _{4}$, $\aleph _{5}\in \lbrack 0, 1)$ with $\aleph _{1}+\aleph _{2}+\aleph _{3}+2\aleph _{4}+2\aleph _{5} < 1$ such that

for all $\omega _{0}, \omega, \varrho \in \overline{B(\omega _{0}, r)}$, $0\prec r\in \mathbb{C}$ and

where $\lambda = \max \{(\frac{\aleph _{1}+\aleph _{4}}{1-\aleph _{2}-\aleph _{4}}), (\frac{\aleph _{1}+\aleph _{5}}{1-\aleph _{2}-\aleph _{5}})\}$. And for each $\omega _{0}\in \overline{B(\omega _{0}, r)}$ and $\lim_{n, m\rightarrow +\infty }\varphi \left(\omega _{n}, \omega _{m}\right) \lambda < 1$, then there exists a unique point $\omega ^{\ast }\in \overline{ B(\omega _{0}, r)\mathit{\text{}}}$ such that $\omega ^{\ast } = \beth \omega ^{\ast }$.

Taking $m = n$ and $F = G = \beth$ in Corollary 23.

4.   Applications

Theorem 25. Let $\mathcal{W} = C([a, b], \mathbb{R}^{n})$, $a > 0\ $and$\ d:\mathcal{W}\times \mathcal{W}\rightarrow \mathbb{C}$ be defined in this way

and $\varphi :\mathcal{W}\times \mathcal{W}\rightarrow \lbrack 1, \infty)$ be defined by $\varphi (\omega, \varrho) = 2.$ Then ($\mathcal{W}, d$) is complete CVE$b$MS. Consider the Urysohn integral equations

for all $t\in \lbrack a, b]\subset \mathbb{R}$, $\omega, \phi, \psi \in \mathcal{W}$.

Assume that $K_{1}, K_{2}:[a, b]\times \lbrack a, b]\times \mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$ are such that $F_{\omega }, G_{\omega }\in \mathcal{W}\ \ $for each$\ \omega \in \mathcal{W}, \ $where,

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

If there exist$\ \aleph _{1}\mathfrak{, }\aleph _{2}\in \lbrack 0, 1)$ with $\aleph _{1}\mathfrak{+}\aleph _{2} < 1$ such that for every $\omega, \varrho \in \mathcal{W}$

where

then Urysohn integral equations (4.1) and (4.2) have a unique common solution.

Proof. Define $\Im _{1}, \Im _{2}:\mathcal{W}\rightarrow \mathcal{W}$ by

Then

It is easily seen that

for every $\omega, \varrho \in \mathcal{W}$. By Theorem 11 with $\aleph _{3} = \aleph _{4} = \aleph _{5} = 0$, the Urysohn integral equations (4.1) and (4.2) have a unique common solution.

5.   Conclusions

In this article, we have utilized the notion of complex valued extended $b$ -metric space (CVE$b$MS) and secured common fixed point results for rational contractions on a closed ball. We have derived common fixed points and fixed points of single valued mappings for contractions on a closed ball. We expect that the obtained consequences in this article will form up to date relations for researchers who are employing in CVE$b$MS.

The future work in this way will target on studying the common fixed points of single valued and multivalued mappings in the setting of CVE$b$MS. Differential and integral equations can be solved as applications of these results.

Acknowledgments

The author would like to thank the anonymous reviewers for their insightful suggestions and careful reading of the manuscript.

Conflict of interest

The authors declare that they have no conflicts of interest.