Application of Fixed Point Result in Complex Valued Extended b-Metric Space

Abstract

The aim of the present research work is to investigate the solution of Urysohn integral equation by common fixed point result in the setting of complex valued b-metric space. To obtain the objective, we used a generalized rational contraction involving control functions and a pair of self-mappings. In this way, we generalize some well-known results of literature. Some non-trivial examples are also flourished to demonstrate the innovation of our principal result.

1. Introduction

In nonlinear analysis, the study of metric spaces has played a significant and essential role. One can find various constructive and magnificent applications of metric spaces in different fields of sciences such as mathematics, computer science, physics, chemistry and biology. Due to its abundant applications, various researchers extended and generalized this notion in different ways. In 2011, Azam et al. [1] generalized this notion by putting the set of complex numbers on the place of set of real numbers in the range of it and gave the idea of complex valued metric space (CVMS). Azam et al. [1] achieved common fixed points of two single-valued mappings under generalized contractions involving certain rational expressions. After a while, Rouzkard et al. [2] extended the leading result of Azam et al. [1] by including a rational term in the contraction of [1]. Thereafter, Sintunavarat et al. [3] supplied a variety of control functions depending on one variable and generalized different results in the literature. Furthermore, Sitthikul et al. [4] extended these control functions by increasing the numbers of variables and demonstrated some results in the backdrop of CVMS. Later on, Rao et al. [5] involved a number $s\ge 1$ in the triangle inequality of CVMS and introduced the idea of complex valued b-metric space (CVbMs). They proved a result for common fixed points of two self mappings with a generalized contraction and derived some of the prime results in the literature. In due course, Mukheimer [6] utilized this new notion to establish certain common fixed point theorems. As a special case, Mukheimer [6] obtained the supreme result of Azam et al. [1]. Thereafter, Kumar et al. [7] broadened the contractive inequality given by Mukheimer [6] and set up some branded common fixed point theorems. In modern days, Ullah et al. [8,9] developed the idea of complex valued extended b-metric space (CVEbMS) as a expansion of CVbMS and presented some contemporary results for contractive-type single-valued and multi-valued mappings. Carmel Pushpa Raj et al. [10] employed this new notion and provided different common fixed point theorems. For further understanding in the direction of CVMS, CVbMS and CVEbMS, we suggest the readers see [11,12,13,14,15].

In the present research work, we introduce certain control functions of one variable in the generalized rational contractions to obtain common fixed point theorems in the foundation of CVEbMS. Some non-trivial and convincing examples are also flourished to express the authenticity of our leading theorem. As outcomes, we obtain the prime results of Azam et al. [1], Rouzkard et al. [2], Sintunavarat et al. [3], Sitthikul et al. [4], Kumar et al. [7] and Carmel Pushpa Raj et al. [10]. To confirm the applicability of the obtained results, we investigate the solution of Urysohn integral equation as an application.

2. Preliminaries

The concept of complex valued metric space was initiated by Azam et al. [1] as an extension of metric space in 2011. They put a set of complex numbers $\mathbb{C}$ in the place of a set of real numbers $\mathbb{R}$ in the range of metric space.

Definition 1 

([1]). Let ${\xi }_1,{\xi }_2\in \mathbb{C}$. A partial order $\precsim$ on $\mathbb{C}$ is defined as follows:

$${\xi }_1\precsim {\xi }_2\iff Re\left({\xi }_1\right)⩽Re\left({\xi }_2\right),Im\left({\xi }_1\right)⩽Im\left({\xi }_2\right).$$

It follows that

$${\xi }_1\precsim {\xi }_2$$

if one of the following conditions is fulfilled:

$$\begin{array}{ccc}\hfill \mathit{\left(}a\mathit{\right)}Re\left({\xi }_1\right)& =& Re\left({\xi }_2\right),Im\left({\xi }_1\right)<Im\left({\xi }_2\right),\hfill \\ \hfill \mathit{\left(}b\mathit{\right)}Re\left({\xi }_1\right)& <& Re\left({\xi }_2\right),Im\left({\xi }_1\right)=Im\left({\xi }_2\right),\hfill \\ \hfill \mathit{\left(}c\mathit{\right)}Re\left({\xi }_1\right)& <& Re\left({\xi }_2\right),Im\left({\xi }_1\right)<Im\left({\xi }_2\right),\hfill \\ \hfill \mathit{\left(}d\mathit{\right)}Re\left({\xi }_1\right)& =& Re\left({\xi }_2\right),Im\left({\xi }_1\right)=Im\left({\xi }_2\right).\hfill \end{array}$$

Definition 2 

([1]). Let $X\ne ⌀$ and $d_{cv}:X\times X\to \mathbb{C}$ be a function satisfying:

(i) 

$0\precsim d_{cv}(\xi ,\varsigma ),$ and $d_{cv}(\xi ,\varsigma )=0$ if and only if $\xi =\varsigma$;

(ii) 

$d_{cv}(\xi ,\varsigma )=d_{cv}(\varsigma ,\xi );$

(iii) 

$d_{cv}(\xi ,\varsigma )\precsim d_{cv}(\xi ,\nu )+d_{cv}(\nu ,\varsigma );$

for all $\xi ,\varsigma ,\nu \in X,$ then $(X,d_{cv})$ is called a CVMS.

Example 1 

([1]). Let $X=[0,1]$ and $\xi ,\varsigma \in X.$ Define $d_{cv}:X\times X\to \mathbb{C}$ by

$$d_{cv}(\xi ,\varsigma )=\left\{\begin{array}{c}0,\mathrm{if}\xi =\varsigma ,\\ \frac{i}{2},\mathrm{if}\xi \ne \varsigma .\end{array}\right.$$

Then, $(X,d_{cv})$ is CVMS.

Rao et al. [5] furnished the concept of complex valued b-metric space (CVbMS) in such a way.

Definition 3 

([5]). Let $X\ne ⌀$, $s\ge 1$ and $d_{cb}:X\times X\to \mathbb{C}$ be a function fulfilling:

(i) 

$0\precsim d_{cb}(\xi ,\varsigma ),$ and $d_{cb}(\xi ,\varsigma )=0$ if and only if $\xi =\varsigma$;

(ii) 

$d_{cb}(\xi ,\varsigma )=d_{cb}(\varsigma ,\xi );$

(iii) 

$d_{cb}(\xi ,\varsigma )\precsim s\left[d_{cb}(\xi ,\nu )+d_{cb}(\nu ,\varsigma )\right];$

for all $\xi ,\varsigma ,\nu \in X,$ then $(X,d_{cb},s)$ is claimed as a CVbMS.

Example 2 

([5]). Let $X=[0,1].$ Define $d_{cb}:X\times X\to \mathbb{C}$ by

$$d_{cb}(\xi ,\varsigma )={|\xi -\varsigma |}^2+i{|\xi -\varsigma |}^2$$

for all $\xi ,\varsigma \in X.$ Then, $(X,d_{cb},2)$ is a CVbMS.

Ullah et al. [8] defined the idea of complex valued extended b-metric space (CVEbMS) as follows.

Definition 4 

([8]). Let  $X\ne ⌀$, $\phi :X\times X\to [1,\infty )$ and $d_{\mathfrak{E}}:X\times X\to \mathbb{C}$ be a function fulfilling:

(i) 

$0\precsim d_{\mathfrak{E}}(\xi ,\varsigma )$ and $d_{\mathfrak{E}}(\xi ,\varsigma )=0$ if and only if $\xi =\varsigma$;

(ii) 

$d_{\mathfrak{E}}(\xi ,\varsigma )=d_{\mathfrak{E}}(\varsigma ,\xi );$

(iii) 

$d_{\mathfrak{E}}(\xi ,\varsigma )\precsim \phi (\xi ,\varsigma )\left[d_{\mathfrak{E}}(\xi ,\nu )+d_{\mathfrak{E}}(\nu ,\varsigma )\right];$

for all $\xi ,\varsigma ,\nu \in X,$ then $(X,d_{\mathfrak{E}},\phi )$ is said to be CVEbMS.

Example 3 

([8]). Let $X\ne ⌀$ and $\phi :X\times X\to [1,\infty )$ be defined by

$$\phi (\xi ,\varsigma )=\frac{1+\xi +\varsigma }{\xi +\varsigma }$$

and $d_{\mathfrak{E}}:X\times X\to \mathbb{C}$ by:

(i) 

$d_{\mathfrak{E}}(\xi ,\varsigma )=\frac{i}{\xi \varsigma },$ for all $0<\xi ,\varsigma \le 1;$

(ii) 

$d_{\mathfrak{E}}(\xi ,\varsigma )=0$ if and only if $\xi =\varsigma ,$ for all $0\le \xi ,\varsigma \le 1;$

(iii) 

$d_{\mathfrak{E}}(\xi ,0)=d_{\mathfrak{E}}(0,\xi )=\frac{i}{\xi },$ for all $0<\xi \le 1.$

Then, the triple $(X,d_{\mathfrak{E}},\phi )$ is CVEbMS.

Example 4.

Let $X=[0,\infty )$ and $\phi :X\times X\to [1,\infty )$ be a function defined by $\phi (\xi ,\varsigma )=1+\xi +\varsigma$ and $d_{\mathfrak{E}}:X\times X\to \mathbb{C}$ by

$$d_{\mathfrak{E}}(\xi ,\varsigma )=\left\{\begin{array}{c}0,\mathrm{if}\xi =\varsigma \\ i,\mathrm{if}\xi \ne \varsigma .\end{array}\right.$$

Then $(X,d_{\mathfrak{E}},\phi )$ is a CVEbMS.

Lemma 1 

([8]). Let $(X,d_{\mathfrak{E}},\phi )$ be a CVEbMS and let $\left\{{\xi }_n\right\}$$\subseteq X$. Then, $\left\{{\xi }_n\right\}$ converges to ξ if and only if $\left|d_{\mathfrak{E}}({\xi }_n,\xi )\right|\to 0$ as $n\to \infty$.

Lemma 2 

([8]). Let $(X,d_{\mathfrak{E}},\phi )$ be a CVEbMS and let $\left\{{\xi }_n\right\}$$\subseteq X$. Then, $\left\{{\xi }_n\right\}$ is a Cauchy sequence if and only if $\left|d_{\mathfrak{E}}({\xi }_n,{\xi }_{n+k})\right|\to 0$ as $n\to \infty ,$ where $k\in \mathbb{N}$.

3. Main Result

Throughout the section, we will consider $\left(X,d_{\mathfrak{E}},\phi \right)$ as complete CVEbMS with $\phi :X\times X\to [1,\infty )$. We state and prove our main result in this way.

Theorem 1.

Let ${\mathcal{F}}_1,{\mathcal{F}}_2:\left({X,d}_{\mathfrak{E}},\phi \right)\to \left({X,d}_{\mathfrak{E}},\phi \right).$ If there exist the mappings $\ell ,\mathfrak{m},\mathfrak{r},\hslash :X\to [0,1)$ such that the following conditions are satisfied:

(a) 

$\ell \left({\mathcal{F}}_1\xi \right)\le \ell \left(\xi \right)$ and $\ell \left({\mathcal{F}}_2\xi \right)\le \ell \left(\xi \right)$

$\mathfrak{m}\left({\mathcal{F}}_1\xi \right)\le \mathfrak{m}\left(\xi \right)$ and $\mathfrak{m}\left({\mathcal{F}}_2\xi \right)\le \mathfrak{m}\left(\xi \right),$

$\mathfrak{r}\left({\mathcal{F}}_1\xi \right)\le \mathfrak{r}\left(\xi \right)$ and $\mathfrak{r}\left({\mathcal{F}}_2\xi \right)\le \mathfrak{r}\left(\xi \right),$

$\hslash \left({\mathcal{F}}_1\xi \right)\le \hslash \left(\xi \right)$ and $\hslash \left({\mathcal{F}}_2\xi \right)\le \hslash \left(\xi \right);$

(b) 

$\ell \left(\xi \right)+2\mathfrak{m}\left(\xi \right)+\mathfrak{r}\left(\xi \right)+\hslash \left(\xi \right)<1;$

(c) 

$$\begin{array}{ccc}\hfill d_{\mathfrak{E}}\left({\mathcal{F}}_1\xi ,{\mathcal{F}}_2\varsigma \right)& ⪯& \ell \left(\xi \right)d_{\mathfrak{E}}\left(\xi ,\varsigma \right)+\mathfrak{m}\left(\xi \right)\left(d_{\mathfrak{E}}\left(\xi ,{\mathcal{F}}_1\xi \right)+d_{\mathfrak{E}}\left(\varsigma ,{\mathcal{F}}_2\varsigma \right)\right)\hfill \\ & & +\mathfrak{r}\left(\xi \right)\frac{d_{\mathfrak{E}}\left(\xi ,{\mathcal{F}}_1\xi \right)d_{\mathfrak{E}}\left(\varsigma ,{\mathcal{F}}_2\varsigma \right)}{1+d_{\mathfrak{E}}\left(\xi ,\varsigma \right)}\hfill \\ & & +\hslash \left(\xi \right)\frac{d_{\mathfrak{E}}\left(\xi ,{\mathcal{F}}_2\varsigma \right)d_{\mathfrak{E}}\left(\varsigma ,{\mathcal{F}}_1\xi \right)}{1+d_{\mathfrak{E}}\left(\xi ,\varsigma \right)}\hfill \end{array}$$

(1)

for all $\xi ,\varsigma \in$$X;$

(d) 

for each ${\xi }_0\in X,$$\lambda =\frac{\ell \left({\xi }_0\right)+\mathfrak{m}\left({\xi }_0\right)}{1-\mathfrak{m}\left({\xi }_0\right)-\mathfrak{r}\left({\xi }_0\right)}<1$ and ${lim}_{n,k\to \infty }\phi \left({\xi }_n,{\xi }_k\right)\lambda <1$ hold, then ${\mathcal{F}}_1$ and ${\mathcal{F}}_2$ have a unique common fixed point.

Proof. 

Let ${\xi }_0$$\in X$ be an arbitrary point and the sequence {${\xi }_n$} be defined by

$${\xi }_{2n+1}={\mathcal{F}}_1{\xi }_{2n}\mathit{and}{\xi }_{2n+2}={\mathcal{F}}_2{\xi }_{2n+1}.$$

(2)

From the Equation (2), we have

$$\begin{array}{ccc}\hfill d_{\mathfrak{E}}\left({\xi }_{2n+1},{\xi }_{2n+2}\right)& =& d_{\mathfrak{E}}\left({\mathcal{F}}_1{\xi }_{2n},{\mathcal{F}}_2{\xi }_{2n+1}\right)⪯\ell \left({\xi }_{2n}\right)d_{\mathfrak{E}}\left({\xi }_{2n},{\xi }_{2n+1}\right)\hfill \\ & & +\mathfrak{m}\left({\xi }_{2n}\right)\left(d_{\mathfrak{E}}\left({\xi }_{2n},{\mathcal{F}}_1{\xi }_{2n}\right)+d_{\mathfrak{E}}\left({\xi }_{2n+1},{\mathcal{F}}_2{\xi }_{2n+1}\right)\right)\hfill \\ & & +\mathfrak{r}\left({\xi }_{2n}\right)\frac{d_{\mathfrak{E}}\left({\xi }_{2n},{\mathcal{F}}_1{\xi }_{2n}\right)d_{\mathfrak{E}}\left({\xi }_{2n+1},{\mathcal{F}}_2{\xi }_{2n+1}\right)}{1+d_{\mathfrak{E}}\left({\xi }_{2n},{\xi }_{2n+1}\right)}\hfill \\ & & +\hslash \left({\xi }_{2n}\right)\frac{d_{\mathfrak{E}}\left({\xi }_{2n},{\mathcal{F}}_2{\xi }_{2n+1}\right)d_{\mathfrak{E}}\left({\xi }_{2n+1},{\mathcal{F}}_1{\xi }_{2n}\right)}{1+d_{\mathfrak{E}}\left({\xi }_{2n},{\xi }_{2n+1}\right)}\hfill \\ & =& \ell \left({\xi }_{2n}\right)d_{\mathfrak{E}}\left({\xi }_{2n},{\xi }_{2n+1}\right)\hfill \\ & & +\mathfrak{m}\left({\xi }_{2n}\right)\left(d_{\mathfrak{E}}\left({\xi }_{2n},{\xi }_{2n+1}\right)+d_{\mathfrak{E}}\left({\xi }_{2n+1},{\xi }_{2n+2}\right)\right)\hfill \\ & & +\mathfrak{r}\left({\xi }_{2n}\right)\frac{d_{\mathfrak{E}}\left({\xi }_{2n},{\xi }_{2n+1}\right)d_{\mathfrak{E}}\left({\xi }_{2n+1},{\xi }_{2n+2}\right)}{1+d_{\mathfrak{E}}\left({\xi }_{2n},{\xi }_{2n+1}\right)}\hfill \\ & & +\hslash \left({\xi }_{2n}\right)\frac{d_{\mathfrak{E}}\left({\xi }_{2n},{\xi }_{2n+2}\right)d_{\mathfrak{E}}\left({\xi }_{2n+1},{\xi }_{2n+1}\right)}{1+d_{\mathfrak{E}}\left({\xi }_{2n},{\xi }_{2n+1}\right)}\hfill \end{array}$$

that is,

$$\begin{array}{ccc}\hfill d_{\mathfrak{E}}\left({\xi }_{2n+1},{\xi }_{2n+2}\right)& ⪯& \ell \left({\xi }_{2n}\right)d_{\mathfrak{E}}\left({\xi }_{2n},{\xi }_{2n+1}\right)\hfill \\ & & +\mathfrak{m}\left({\xi }_{2n}\right)d_{\mathfrak{E}}\left({\xi }_{2n},{\xi }_{2n+1}\right)+\mathfrak{m}\left({\xi }_{2n}\right)d_{\mathfrak{E}}\left({\xi }_{2n+1},{\xi }_{2n+2}\right)\hfill \\ & & +\mathfrak{r}\left({\xi }_{2n}\right)\frac{d_{\mathfrak{E}}\left({\xi }_{2n},{\xi }_{2n+1}\right)d_{\mathfrak{E}}\left({\xi }_{2n+1},{\xi }_{2n+2}\right)}{1+d_{\mathfrak{E}}\left({\xi }_{2n},{\xi }_{2n+1}\right)}\hfill \end{array}$$

which implies that

$$\begin{array}{ccc}\hfill \left|d_{\mathfrak{E}}\left({\xi }_{2n+1},{\xi }_{2n+2}\right)\right|& \le & \ell \left({\xi }_{2n}\right)\left|d_{\mathfrak{E}}\left({\xi }_{2n},{\xi }_{2n+1}\right)\right|\hfill \\ & & +\mathfrak{m}\left({\xi }_{2n}\right)\left|d_{\mathfrak{E}}\left({\xi }_{2n},{\xi }_{2n+1}\right)\right|+\mathfrak{m}\left({\xi }_{2n}\right)\left|d_{\mathfrak{E}}\left({\xi }_{2n+1},{\xi }_{2n+2}\right)\right|\hfill \\ & & +\mathfrak{r}\left({\xi }_{2n}\right)\left|\frac{d_{\mathfrak{E}}\left({\xi }_{2n},{\xi }_{2n+1}\right)}{1+d_{\mathfrak{E}}\left({\xi }_{2n},{\xi }_{2n+1}\right)}\right|\left|d_{\mathfrak{E}}\left({\xi }_{2n+1},{\xi }_{2n+2}\right)\right|\hfill \end{array}$$

since $\left|\frac{d_{\mathfrak{E}}\left({\xi }_{2n},{\xi }_{2n+1}\right)}{1+d_{\mathfrak{E}}\left({\xi }_{2n},{\xi }_{2n+1}\right)}\right|<1,$ so we have

$$\begin{array}{ccc}\hfill \left|d_{\mathfrak{E}}\left({\xi }_{2n+1},{\xi }_{2n+2}\right)\right|& \le & \ell \left({\xi }_{2n}\right)\left|d_{\mathfrak{E}}\left({\xi }_{2n},{\xi }_{2n+1}\right)\right|\hfill \\ & & +\mathfrak{m}\left({\xi }_{2n}\right)\left|d_{\mathfrak{E}}\left({\xi }_{2n},{\xi }_{2n+1}\right)\right|+\mathfrak{m}\left({\xi }_{2n}\right)\left|d_{\mathfrak{E}}\left({\xi }_{2n+1},{\xi }_{2n+2}\right)\right|\hfill \\ & & +\mathfrak{r}\left({\xi }_{2n}\right)\left|d_{\mathfrak{E}}\left({\xi }_{2n+1},{\xi }_{2n+2}\right)\right|\hfill \\ & =& \ell \left({\mathcal{F}}_2{\xi }_{2n-1}\right)\left|d_{\mathfrak{E}}\left({\xi }_{2n},{\xi }_{2n+1}\right)\right|\hfill \\ & & +\mathfrak{m}\left({\mathcal{F}}_2{\xi }_{2n-1}\right)\left|d_{\mathfrak{E}}\left({\xi }_{2n},{\xi }_{2n+1}\right)\right|+\mathfrak{m}\left({\mathcal{F}}_2{\xi }_{2n-1}\right)\left|d_{\mathfrak{E}}\left({\xi }_{2n+1},{\xi }_{2n+2}\right)\right|\hfill \\ & & +\mathfrak{r}\left({\mathcal{F}}_2{\xi }_{2n-1}\right)\left|d_{\mathfrak{E}}\left({\xi }_{2n+1},{\xi }_{2n+2}\right)\right|.\hfill \end{array}$$

By using condition (a), we have

$$\begin{array}{ccc}\hfill \left|d_{\mathfrak{E}}\left({\xi }_{2n+1},{\xi }_{2n+2}\right)\right|& \le & \ell \left({\mathcal{F}}_2{\xi }_{2n-1}\right)\left|d_{\mathfrak{E}}\left({\xi }_{2n},{\xi }_{2n+1}\right)\right|\hfill \\ & & +\mathfrak{m}\left({\mathcal{F}}_2{\xi }_{2n-1}\right)\left|d_{\mathfrak{E}}\left({\xi }_{2n},{\xi }_{2n+1}\right)\right|\hfill \\ & & +\mathfrak{m}\left({\mathcal{F}}_2{\xi }_{2n-1}\right)\left|d_{\mathfrak{E}}\left({\xi }_{2n+1},{\xi }_{2n+2}\right)\right|\hfill \\ & & +\mathfrak{r}\left({\mathcal{F}}_2{\xi }_{2n-1}\right)\left|d_{\mathfrak{E}}\left({\xi }_{2n+1},{\xi }_{2n+2}\right)\right|\hfill \\ & \le & \ell \left({\xi }_{2n-1}\right)\left|d_{\mathfrak{E}}\left({\xi }_{2n},{\xi }_{2n+1}\right)\right|\hfill \\ & & +\mathfrak{m}\left({\xi }_{2n-1}\right)\left|d_{\mathfrak{E}}\left({\xi }_{2n},{\xi }_{2n+1}\right)\right|+\mathfrak{m}\left({\xi }_{2n-1}\right)\left|d_{\mathfrak{E}}\left({\xi }_{2n+1},{\xi }_{2n+2}\right)\right|\hfill \\ & & +\mathfrak{r}\left({\xi }_{2n-1}\right)\left|d_{\mathfrak{E}}\left({\xi }_{2n+1},{\xi }_{2n+2}\right)\right|\hfill \\ & =& \ell \left({\mathcal{F}}_1{\xi }_{2n-2}\right)\left|d_{\mathfrak{E}}\left({\xi }_{2n},{\xi }_{2n+1}\right)\right|\hfill \\ & & +\mathfrak{m}\left({\mathcal{F}}_1{\xi }_{2n-2}\right)\left|d_{\mathfrak{E}}\left({\xi }_{2n},{\xi }_{2n+1}\right)\right|+\mathfrak{m}\left({\mathcal{F}}_1{\xi }_{2n-2}\right)\left|d_{\mathfrak{E}}\left({\xi }_{2n+1},{\xi }_{2n+2}\right)\right|\hfill \\ & & +\mathfrak{r}\left({\mathcal{F}}_1{\xi }_{2n-2}\right)\left|d_{\mathfrak{E}}\left({\xi }_{2n+1},{\xi }_{2n+2}\right)\right|.\hfill \end{array}$$

It yields

$$\begin{array}{ccc}\hfill \left|d_{\mathfrak{E}}\left({\xi }_{2n+1},{\xi }_{2n+2}\right)\right|& \le & \ell \left({\xi }_{2n-2}\right)\left|d_{\mathfrak{E}}\left({\xi }_{2n},{\xi }_{2n+1}\right)\right|\hfill \\ & & +\mathfrak{m}\left({\xi }_{2n-2}\right)\left|d_{\mathfrak{E}}\left({\xi }_{2n},{\xi }_{2n+1}\right)\right|+\mathfrak{m}\left({\xi }_{2n-2}\right)\left|d_{\mathfrak{E}}\left({\xi }_{2n+1},{\xi }_{2n+2}\right)\right|\hfill \\ & & +\mathfrak{r}\left({\xi }_{2n-2}\right)\left|d_{\mathfrak{E}}\left({\xi }_{2n+1},{\xi }_{2n+2}\right)\right|\hfill \\ & \le & ···\le \ell \left({\xi }_0\right)\left|d_{\mathfrak{E}}\left({\xi }_{2n},{\xi }_{2n+1}\right)\right|\hfill \\ & & +\mathfrak{m}\left({\xi }_0\right)\left|d_{\mathfrak{E}}\left({\xi }_{2n},{\xi }_{2n+1}\right)\right|+\mathfrak{m}\left({\xi }_0\right)\left|d_{\mathfrak{E}}\left({\xi }_{2n+1},{\xi }_{2n+2}\right)\right|\hfill \\ & & +\mathfrak{r}\left({\xi }_0\right)\left|d_{\mathfrak{E}}\left({\xi }_{2n+1},{\xi }_{2n+2}\right)\right|.\hfill \end{array}$$

This implies

$$\left|d_{\mathfrak{E}}\left({\xi }_{2n+1},{\xi }_{2n+2}\right)\right|\le \frac{\ell \left({\xi }_0\right)+\mathfrak{m}\left({\xi }_0\right)}{1-\mathfrak{m}\left({\xi }_0\right)-\hslash \left({\xi }_0\right)}\left|d_{\mathfrak{E}}\left({\xi }_{2n},{\xi }_{2n+1}\right)\right|$$

(3)

for all $n\ge 0.$ Similarly, we have

$$\begin{array}{ccc}\hfill d_{\mathfrak{E}}\left({\xi }_{2n+2},{\xi }_{2n+3}\right)& =& d_{\mathfrak{E}}\left({\xi }_{2n+3},{\xi }_{2n+2}\right)=d_{\mathfrak{E}}\left({\mathcal{F}}_1{\xi }_{2n+2},{\mathcal{F}}_2{\xi }_{2n+1}\right)⪯\ell \left({\xi }_{2n+2}\right)d_{\mathfrak{E}}\left({\xi }_{2n+2},{\xi }_{2n+1}\right)\hfill \\ & & +\mathfrak{m}\left({\xi }_{2n+2}\right)\left(d_{\mathfrak{E}}\left({\xi }_{2n+2},{\mathcal{F}}_1{\xi }_{2n+2}\right)+d_{\mathfrak{E}}\left({\xi }_{2n+1},{\mathcal{F}}_2{\xi }_{2n+1}\right)\right)\hfill \\ & & +\mathfrak{r}\left({\xi }_{2n+2}\right)\frac{d_{\mathfrak{E}}\left({\xi }_{2n+2},{\mathcal{F}}_1{\xi }_{2n+2}\right)d_{\mathfrak{E}}\left({\xi }_{2n+1},{\mathcal{F}}_2{\xi }_{2n+1}\right)}{1+d_{\mathfrak{E}}\left({\xi }_{2n+2},{\xi }_{2n+1}\right)}\hfill \\ & & +\hslash \left(\xi \right)\frac{d_{\mathfrak{E}}\left({\xi }_{2n+2},{\mathcal{F}}_2{\xi }_{2n+1}\right)d_{\mathfrak{E}}\left({\xi }_{2n+1},{\mathcal{F}}_1{\xi }_{2n+2}\right)}{1+d_{\mathfrak{E}}\left({\xi }_{2n+2},{\xi }_{2n+1}\right)}\hfill \\ & =& \ell \left({\xi }_{2n+2}\right)d_{\mathfrak{E}}\left({\xi }_{2n+2},{\xi }_{2n+1}\right)\hfill \\ & & +\mathfrak{m}\left({\xi }_{2n+2}\right)\left(d_{\mathfrak{E}}\left({\xi }_{2n+2},{\xi }_{2n+3}\right)+d_{\mathfrak{E}}\left({\xi }_{2n+1},{\xi }_{2n+2}\right)\right)\hfill \\ & & +\mathfrak{r}\left({\xi }_{2n+2}\right)\frac{d_{\mathfrak{E}}\left({\xi }_{2n+2},{\xi }_{2n+3}\right)d_{\mathfrak{E}}\left({\xi }_{2n+1},{\xi }_{2n+2}\right)}{1+d_{\mathfrak{E}}\left({\xi }_{2n+2},{\xi }_{2n+1}\right)}\hfill \\ & & +\hslash \left(\xi \right)\frac{d_{\mathfrak{E}}\left({\xi }_{2n+2},{\xi }_{2n+2}\right)d_{\mathfrak{E}}\left({\xi }_{2n+1},{\xi }_{2n+3}\right)}{1+d_{\mathfrak{E}}\left({\xi }_{2n+2},{\xi }_{2n+1}\right)}\hfill \end{array}$$

which implies

$$\begin{array}{ccc}\hfill \left|d_{\mathfrak{E}}\left({\xi }_{2n+2},{\xi }_{2n+3}\right)\right|& \le & \ell \left({\xi }_{2n+2}\right)\left|d_{\mathfrak{E}}\left({\xi }_{2n+2},{\xi }_{2n+1}\right)\right|\hfill \\ & & +\mathfrak{m}\left({\xi }_{2n+2}\right)\left|d_{\mathfrak{E}}\left({\xi }_{2n+2},{\xi }_{2n+3}\right)\right|+\mathfrak{m}\left({\xi }_{2n+2}\right)\left|d_{\mathfrak{E}}\left({\xi }_{2n+1},{\xi }_{2n+2}\right)\right|\hfill \\ & & +\mathfrak{r}\left({\xi }_{2n+2}\right)\left|d_{\mathfrak{E}}\left({\xi }_{2n+2},{\xi }_{2n+3}\right)\right|\left|\frac{d_{\mathfrak{E}}\left({\xi }_{2n+1},{\xi }_{2n+2}\right)}{1+d_{\mathfrak{E}}\left({\xi }_{2n+2},{\xi }_{2n+1}\right)}\right|\hfill \end{array}$$

since $\left|\frac{d_{\mathfrak{E}}\left({\xi }_{2n+1},{\xi }_{2n+2}\right)}{1+d_{\mathfrak{E}}\left({\xi }_{2n+2},{\xi }_{2n+1}\right)}\right|<1,$ so we have

$$\begin{array}{ccc}\hfill \left|d_{\mathfrak{E}}\left({\xi }_{2n+2},{\xi }_{2n+3}\right)\right|& \le & \ell \left({\xi }_{2n+2}\right)\left|d_{\mathfrak{E}}\left({\xi }_{2n+1},{\xi }_{2n+2}\right)\right|\hfill \\ & & +\mathfrak{m}\left({\xi }_{2n+2}\right)\left|d_{\mathfrak{E}}\left({\xi }_{2n+2},{\xi }_{2n+3}\right)\right|+\mathfrak{m}\left({\xi }_{2n+2}\right)\left|d_{\mathfrak{E}}\left({\xi }_{2n+1},{\xi }_{2n+2}\right)\right|\hfill \\ & & +\mathfrak{r}\left({\xi }_{2n+2}\right)\left|d_{\mathfrak{E}}\left({\xi }_{2n+2},{\xi }_{2n+3}\right)\right|\hfill \end{array}$$

By using the assumption (a), we have

$$\begin{array}{ccc}\hfill \left|d_{\mathfrak{E}}\left({\xi }_{2n+2},{\xi }_{2n+3}\right)\right|& \le & \ell \left({\xi }_{2n+2}\right)\left|d_{\mathfrak{E}}\left({\xi }_{2n+1},{\xi }_{2n+2}\right)\right|\hfill \\ & & +\mathfrak{m}\left({\xi }_{2n+2}\right)\left|d_{\mathfrak{E}}\left({\xi }_{2n+2},{\xi }_{2n+3}\right)\right|+\mathfrak{m}\left({\xi }_{2n+2}\right)\left|d_{\mathfrak{E}}\left({\xi }_{2n+1},{\xi }_{2n+2}\right)\right|\hfill \\ & & +\mathfrak{r}\left({\xi }_{2n+2}\right)\left|d_{\mathfrak{E}}\left({\xi }_{2n+2},{\xi }_{2n+3}\right)\right|\hfill \\ & =& \ell \left({\mathcal{F}}_2{\xi }_{2n+1}\right)\left|d_{\mathfrak{E}}\left({\xi }_{2n+1},{\xi }_{2n+2}\right)\right|\hfill \\ & & +\mathfrak{m}\left({\mathcal{F}}_2{\xi }_{2n+1}\right)\left|d_{\mathfrak{E}}\left({\xi }_{2n+2},{\xi }_{2n+3}\right)\right|+\mathfrak{m}\left({\mathcal{F}}_2{\xi }_{2n+1}\right)\left|d_{\mathfrak{E}}\left({\xi }_{2n+1},{\xi }_{2n+2}\right)\right|\hfill \\ & & +\mathfrak{r}\left({\mathcal{F}}_2{\xi }_{2n+1}\right)\left|d_{\mathfrak{E}}\left({\xi }_{2n+2},{\xi }_{2n+3}\right)\right|\hfill \\ & \le & \ell \left({\xi }_{2n+1}\right)\left|d_{\mathfrak{E}}\left({\xi }_{2n+1},{\xi }_{2n+2}\right)\right|\hfill \\ & & +\mathfrak{m}\left({\xi }_{2n+1}\right)\left|d_{\mathfrak{E}}\left({\xi }_{2n+2},{\xi }_{2n+3}\right)\right|+\mathfrak{m}\left({\xi }_{2n+1}\right)\left|d_{\mathfrak{E}}\left({\xi }_{2n+1},{\xi }_{2n+2}\right)\right|\hfill \\ & & +\mathfrak{r}\left({\xi }_{2n+1}\right)\left|d_{\mathfrak{E}}\left({\xi }_{2n+2},{\xi }_{2n+3}\right)\right|\hfill \\ & =& \ell \left({\mathcal{F}}_1{\xi }_{2n}\right)\left|d_{\mathfrak{E}}\left({\xi }_{2n+1},{\xi }_{2n+2}\right)\right|\hfill \\ & & +\mathfrak{m}\left({\mathcal{F}}_1{\xi }_{2n}\right)\left|d_{\mathfrak{E}}\left({\xi }_{2n+2},{\xi }_{2n+3}\right)\right|+\mathfrak{m}\left({\mathcal{F}}_1{\xi }_{2n}\right)\left|d_{\mathfrak{E}}\left({\xi }_{2n+1},{\xi }_{2n+2}\right)\right|\hfill \\ & & +\mathfrak{r}\left({\mathcal{F}}_1{\xi }_{2n}\right)\left|d_{\mathfrak{E}}\left({\xi }_{2n+2},{\xi }_{2n+3}\right)\right|.\hfill \end{array}$$

It yields

$$\begin{array}{ccc}\hfill \left|d_{\mathfrak{E}}\left({\xi }_{2n+2},{\xi }_{2n+3}\right)\right|& \le & \ell \left({\xi }_{2n}\right)\left|d_{\mathfrak{E}}\left({\xi }_{2n+1},{\xi }_{2n+2}\right)\right|\hfill \\ & & +\mathfrak{m}\left({\xi }_{2n}\right)\left|d_{\mathfrak{E}}\left({\xi }_{2n+2},{\xi }_{2n+3}\right)\right|+\mathfrak{m}\left({\xi }_{2n}\right)\left|d_{\mathfrak{E}}\left({\xi }_{2n+1},{\xi }_{2n+2}\right)\right|\hfill \\ & & +\mathfrak{r}\left({\xi }_{2n}\right)\left|d_{\mathfrak{E}}\left({\xi }_{2n+2},{\xi }_{2n+3}\right)\right|\hfill \\ & \le & ···\le \ell \left({\xi }_0\right)\left|d_{\mathfrak{E}}\left({\xi }_{2n+1},{\xi }_{2n+2}\right)\right|\hfill \\ & & +\mathfrak{m}\left({\xi }_0\right)\left|d_{\mathfrak{E}}\left({\xi }_{2n+2},{\xi }_{2n+3}\right)\right|+\mathfrak{m}\left({\xi }_0\right)\left|d_{\mathfrak{E}}\left({\xi }_{2n+1},{\xi }_{2n+2}\right)\right|\hfill \\ & & +\mathfrak{r}\left({\xi }_0\right)\left|d_{\mathfrak{E}}\left({\xi }_{2n+2},{\xi }_{2n+3}\right)\right|.\hfill \end{array}$$

This implies that

$$\left|d_{\mathfrak{E}}\left({\xi }_{2n+2},{\xi }_{2n+3}\right)\right|\le \frac{\ell \left({\xi }_0\right)+\mathfrak{m}\left({\xi }_0\right)}{1-\mathfrak{m}\left({\xi }_0\right)-\mathfrak{r}\left({\xi }_0\right)}\left|d_{\mathfrak{E}}\left({\xi }_{2n+1},{\xi }_{2n+2}\right)\right|$$

(4)

for all $n\ge 0.$ Let $\lambda =$$\frac{\ell \left({\xi }_0\right)+\mathfrak{m}\left({\xi }_0\right)}{1-\mathfrak{m}\left({\xi }_0\right)-\mathfrak{r}\left({\xi }_0\right)}<1.$ Then from (3) and (4), we have

$$\left|d_{\mathfrak{E}}\left({\xi }_n,{\xi }_{n+1}\right)\right|\le \lambda \left|d_{\mathfrak{E}}\left({\xi }_{n-1},{\xi }_n\right)\right|$$

(5)

for all $n\in \mathbb{N}.$ Thus, we can generate a sequence $\left\{{\xi }_n\right\}$ in X such that

$$\begin{array}{ccc}\hfill \left|d_{\mathfrak{E}}\left({\xi }_n,{\xi }_{n+1}\right)\right|& \le & \lambda \left|d_{\mathfrak{E}}\left({\xi }_{n-1},{\xi }_n\right)\right|\hfill \\ \hfill \left|d_{\mathfrak{E}}\left({\xi }_{n+1},{\xi }_n\right)\right|& \le & {\lambda }^2\left|d_{\mathfrak{E}}\left({\xi }_{n-2},{\xi }_{n-1}\right)\right|\hfill \\ & \le & \\ & & ·\hfill \\ & & ·\hfill \\ & & ·\hfill \\ & \le & {\lambda }^n\left|d_{\mathfrak{E}}\left({\xi }_0,{\xi }_1\right)\right|\hfill \end{array}$$

for all $n\in \mathbb{N}.$ Now for $k>n$, we obtain

$$\begin{array}{ccc}\hfill \left|d_{\mathfrak{E}}\left({\xi }_n,{\xi }_k\right)\right|& \le & \phi \left({\xi }_n,{\xi }_k\right){\lambda }^n\left|d_{\mathfrak{E}}\left({\xi }_0,{\xi }_1\right)\right|\hfill \\ & & +\phi \left({\xi }_n,{\xi }_k\right)\phi \left({\xi }_{n+1},{\xi }_k\right){\lambda }^{n+1}\left|d_{\mathfrak{E}}\left({\xi }_0,{\xi }_1\right)\right|\hfill \\ & & +···+\hfill \\ & & \phi \left({\xi }_n,{\xi }_k\right)\phi \left({\xi }_{n+1},{\xi }_k\right)···\phi \left({\xi }_{k-2},{\xi }_k\right)\phi \left({\xi }_{k-1},{\xi }_k\right){\lambda }^{k-1}\left|d_{\mathfrak{E}}\left({\xi }_0,{\xi }_1\right)\right|\hfill \\ & \le & \left|d_{\mathfrak{E}}\left({\xi }_0,{\xi }_1\right)\right|\left[\begin{array}{c}\phi \left({\xi }_n,{\xi }_k\right){\lambda }^n\\ +\phi \left({\xi }_n,{\xi }_k\right)\phi \left({\xi }_{n+1},{\xi }_k\right){\lambda }^{n+1}+···+\\ \phi \left({\xi }_n,{\xi }_k\right)\phi \left({\xi }_{n+1},{\xi }_k\right)···\phi \left({\xi }_{k-2},{\xi }_k\right)\phi \left({\xi }_{k-1},{\xi }_k\right){\lambda }^{k-1}\end{array}\right].\hfill \end{array}$$

Since ${lim}_{n,k\to \infty }\phi \left({\xi }_n,{\xi }_k\right)\lambda <1,$ so the series ${\displaystyle \sum _{n=1}^{\infty }}{\lambda }^n{\displaystyle \prod _{i=1}^p}\phi \left({\xi }_i,{\xi }_k\right)$ converges by ratio test for each $k\in \mathbb{N}.$ Let

$$S=\sum _{n=1}^{\infty }{\lambda }^n\prod _{i=1}^p\phi \left({\xi }_i,{\xi }_k\right),S_n=\sum _{j=1}^n{\lambda }^j\prod _{i=1}^p\phi \left({\xi }_i,{\xi }_k\right).$$

Hence, for $k>n$, the above inequality can be written as

$$\left|d_{\mathfrak{E}}\left({\xi }_n,{\xi }_k\right)\right|\le \left|d_{\mathfrak{E}}\left({\xi }_0,{\xi }_1\right)\right|\left[S_{k-1}-S_n\right].$$

Now, by taking $n\to \infty$, we obtain

$$\left|d_{\mathfrak{E}}\left({\xi }_n,{\xi }_k\right)\right|\to 0.$$

From Lemma 2, $\left\{{\xi }_n\right\}$ is a Cauchy sequence. As X is complete, so ${\xi }^{*}\in X$ such that ${\xi }_n\to {\xi }^{*}$ as $n\to \infty .$

Now, we show that ${\xi }^{*}$ is fixed point of ${\mathcal{F}}_1.$ From (2), we have

$$\begin{array}{ccc}\hfill d_{\mathfrak{E}}\left({\xi }^{*},{\mathcal{F}}_1{\xi }^{*}\right)& ⪯& \phi \left({\xi }^{*},{\mathcal{F}}_1{\xi }^{*}\right)\left(d_{\mathfrak{E}}\left({\xi }^{*},{\mathcal{F}}_2{\xi }_{2n+1}\right)+d_{\mathfrak{E}}\left({\mathcal{F}}_2{\xi }_{2n+1},{\mathcal{F}}_1{\xi }^{*}\right)\right)\hfill \\ & =& \phi \left({\xi }^{*},{\mathcal{F}}_1{\xi }^{*}\right)\left(d_{\mathfrak{E}}\left({\xi }^{*},{\mathcal{F}}_2{\xi }_{2n+1}\right)+d_{\mathfrak{E}}\left({\mathcal{F}}_1{\xi }^{*},{\mathcal{F}}_2{\xi }_{2n+1}\right)\right)\hfill \\ & ⪯& \phi \left({\xi }^{*},{\mathcal{F}}_1{\xi }^{*}\right)\left(\begin{array}{c}{d}_{\mathfrak{E}}\left({\xi }^{*},{\xi }_{2n+2}\right)\\ +\ell \left({\xi }^{*}\right)d_{\mathfrak{E}}\left({\xi }^{*},{\xi }_{2n+1}\right)\\ +\mathfrak{m}\left({\xi }^{*}\right)\left(d_{\mathfrak{E}}\left({\xi }^{*},{\mathcal{F}}_1{\xi }^{*}\right)+d_{\mathfrak{E}}\left({\xi }_{2n+1},{\mathcal{F}}_2{\xi }_{2n+1}\right)\right)\\ +\mathfrak{r}\left({\xi }^{*}\right)\frac{d_{\mathfrak{E}}\left({\xi }^{*},{\mathcal{F}}_1{\xi }^{*}\right)d_{\mathfrak{E}}\left({\xi }_{2n+1},{\mathcal{F}}_2{\xi }_{2n+1}\right)}{1+d_{\mathfrak{E}}\left({\xi }^{*},{\xi }_{2n+1}\right)}\\ +\hslash \left({\xi }^{*}\right)\frac{d_{\mathfrak{E}}\left({\xi }^{*},{\mathcal{F}}_2{\xi }_{2n+1}\right)d_{\mathfrak{E}}\left({\xi }_{2n+1},{\mathcal{F}}_1{\xi }^{*}\right)}{1+d_{\mathfrak{E}}\left({\xi }^{*},{\xi }_{2n+1}\right)}\end{array}\right)\hfill \\ & ⪯& \phi \left({\xi }^{*},{\mathcal{F}}_1{\xi }^{*}\right)\left(\begin{array}{c}{d}_{\mathfrak{E}}\left({\xi }^{*},{\xi }_{2n+2}\right)+\ell \left({\xi }^{*}\right)d_{\mathfrak{E}}\left({\xi }^{*},{\xi }_{2n+1}\right)\\ +\mathfrak{m}\left({\xi }^{*}\right)\left(d_{\mathfrak{E}}\left({\xi }^{*},{\mathcal{F}}_1{\xi }^{*}\right)+d_{\mathfrak{E}}\left({\xi }_{2n+1},{\xi }_{2n+2}\right)\right)\\ +\mathfrak{r}\left({\xi }^{*}\right)\frac{d_{\mathfrak{E}}\left({\xi }^{*},{\mathcal{F}}_1{\xi }^{*}\right)d_{\mathfrak{E}}\left({\xi }_{2n+1},{\xi }_{2n+2}\right)}{1+d_{\mathfrak{E}}\left({\xi }^{*},{\xi }_{2n+1}\right)}\\ +\hslash \left({\xi }^{*}\right)\frac{d_{\mathfrak{E}}\left({\xi }^{*},{\xi }_{2n+2}\right)d_{\mathfrak{E}}\left({\xi }_{2n+1},{\mathcal{F}}_1{\xi }^{*}\right)}{1+d_{\mathfrak{E}}\left({\xi }^{*},{\xi }_{2n+1}\right)}\end{array}\right).\hfill \end{array}$$

This implies that

$$\left|d_{\mathfrak{E}}\left({\xi }^{*},{\mathcal{F}}_1{\xi }^{*}\right)\right|\le \phi \left({\xi }^{*},{\mathcal{F}}_1{\xi }^{*}\right)\left(\begin{array}{c}\left|d_{\mathfrak{E}}\left({\xi }^{*},{\xi }_{2n+2}\right)\right|+\ell \left({\xi }^{*}\right)\left|d_{\mathfrak{E}}\left({\xi }^{*},{\xi }_{2n+1}\right)\right|\\ +\mathfrak{m}\left({\xi }^{*}\right)\left(\left|d_{\mathfrak{E}}\left({\xi }^{*},{\mathcal{F}}_1{\xi }^{*}\right)\right|+\left|d_{\mathfrak{E}}\left({\xi }_{2n+1},{\xi }_{2n+2}\right)\right|\right)\\ +\mathfrak{r}\left({\xi }^{*}\right)\frac{\left|d_{\mathfrak{E}}\left({\xi }^{*},{\mathcal{F}}_1{\xi }^{*}\right)\right|\left|d_{\mathfrak{E}}\left({\xi }_{2n+1},{\xi }_{2n+2}\right)\right|}{\left|1+d_{\mathfrak{E}}\left({\xi }^{*},{\xi }_{2n+1}\right)\right|}\\ +\hslash \left({\xi }^{*}\right)\frac{\left|d_{\mathfrak{E}}\left({\xi }^{*},{\xi }_{2n+2}\right)\right|\left|d_{\mathfrak{E}}\left({\xi }_{2n+1},{\mathcal{F}}_1{\xi }^{*}\right)\right|}{\left|1+d_{\mathfrak{E}}\left({\xi }^{*},{\xi }_{2n+1}\right)\right|}\end{array}\right)$$

Letting $n\to \infty ,$ we have

$$(1-\phi \left({\xi }^{*},{\mathcal{F}}_1{\xi }^{*}\right)\mathfrak{m}\left({\xi }^{*}\right))\left|d_{\mathfrak{E}}\left({\xi }^{*},{\mathcal{F}}_1{\xi }^{*}\right)\right|=0.$$

Since

$$(1-\phi \left({\xi }^{*},{\mathcal{F}}_1{\xi }^{*}\right)\mathfrak{m}\left({\xi }^{*}\right))\ne 0.$$

Thus ${\xi }^{*}={\mathcal{F}}_1{\xi }^{*}.$ Similarly, one can prove that ${\xi }^{*}$ is a fixed point of ${\mathcal{F}}_2.$ Now we discuss the uniqueness of fixed point. We suppose, on the contrary, that ${\xi }^{/}={\mathcal{F}}_1{\xi }^{/}={\mathcal{F}}_2{\xi }^{/}$ but ${\xi }^{*}\ne {\xi }^{/}.$ Now from (2), we have

$$\begin{array}{ccc}\hfill d_{\mathfrak{E}}\left({\xi }^{*},{\xi }^{/}\right)& =& d_{\mathfrak{E}}\left({\mathcal{F}}_1{\xi }^{*},{\mathcal{F}}_2{\xi }^{/}\right)\hfill \\ & ⪯& \ell \left({\xi }^{*}\right)d_{\mathfrak{E}}\left({\xi }^{*},{\xi }^{/}\right)+\mathfrak{m}\left({\xi }^{*}\right)\left(d_{\mathfrak{E}}\left({\xi }^{*},{\mathcal{F}}_1{\xi }^{*}\right)+d_{\mathfrak{E}}\left({\xi }^{/},{\mathcal{F}}_2{\xi }^{/}\right)\right)\hfill \\ & & +\mathfrak{r}\left({\xi }^{*}\right)\frac{d_{\mathfrak{E}}\left({\xi }^{*},{\mathcal{F}}_1{\xi }^{*}\right)d_{\mathfrak{E}}\left({\xi }^{/},{\mathcal{F}}_2{\xi }^{/}\right)}{1+d_{\mathfrak{E}}\left({\xi }^{*},{\xi }^{/}\right)}\hfill \\ & & +\hslash \left({\xi }^{*}\right)\frac{d_{\mathfrak{E}}\left({\xi }^{/},{\mathcal{F}}_2{\xi }^{*}\right)d_{\mathfrak{E}}\left({\xi }^{*},{\mathcal{F}}_1{\xi }^{/}\right)}{1+d_{\mathfrak{E}}\left({\xi }^{*},{\xi }^{/}\right)}\hfill \\ & =& \ell \left({\xi }^{*}\right)d_{\mathfrak{E}}\left({\xi }^{*},{\xi }^{/}\right)+\hslash \left({\xi }^{*}\right)\frac{d_{\mathfrak{E}}\left({\xi }^{/},{\xi }^{*}\right)d_{\mathfrak{E}}\left({\xi }^{*},{\xi }^{/}\right)}{1+d_{\mathfrak{E}}\left({\xi }^{*},{\xi }^{/}\right)}\hfill \end{array}$$

This implies that we have

$$\begin{array}{ccc}\hfill \left|d_{\mathfrak{E}}\left({\xi }^{*},{\xi }^{/}\right)\right|& \le & \ell \left({\xi }^{*}\right)\left|d_{\mathfrak{E}}\left({\xi }^{*},{\xi }^{/}\right)\right|+\hslash \left({\xi }^{*}\right)\left|\frac{d_{\mathfrak{E}}\left({\xi }^{/},{\xi }^{*}\right)}{1+d_{\mathfrak{E}}\left({\xi }^{*},{\xi }^{/}\right)}\right|\left|d_{\mathfrak{E}}\left({\xi }^{*},{\xi }^{/}\right)\right|\hfill \\ & \le & \ell \left({\xi }^{*}\right)\left|d_{\mathfrak{E}}\left({\xi }^{*},{\xi }^{/}\right)\right|+\hslash \left({\xi }^{*}\right)\left|d_{\mathfrak{E}}\left({\xi }^{*},{\xi }^{/}\right)\right|\hfill \end{array}$$

that is,

$$(1-\ell \left({\xi }^{*}\right)-\hslash \left({\xi }^{*}\right))\left|d_{\mathfrak{E}}\left({\xi }^{*},{\xi }^{/}\right)\right|\le 0$$

which hold only when $\left|d_{\mathfrak{E}}\left({\xi }^{*},{\xi }^{/}\right)\right|=0.$ Thus, ${\xi }^{*}={\xi }^{/}.$ □

Example 5.

Let $X=[0,1]$ and $\phi :X\times X\to [1,\infty )$ be a function defined by

$$\phi (\xi ,\varsigma )=\frac{2+\xi +\varsigma }{1+\xi +\varsigma }$$

and $d_{\mathfrak{E}}:X\times X\to \mathbb{C}$ by

$$d_{\mathfrak{E}}(\xi ,\varsigma )={\left|\xi -\varsigma \right|}^2+i{\left|\xi -\varsigma \right|}^2$$

for all $\xi ,\varsigma \in$ X. Then, $(X,d_{\mathfrak{E}})$ is complete CVEbMS. Define a self mapping $\mathcal{F}:X$$\to X$ by

$${\mathcal{F}}_1\xi =\frac{\xi }{6}$$

and

$${\mathcal{F}}_2\xi =\frac{\xi }{12}$$

Consider

$$\ell ,\mathfrak{m},\mathfrak{r},\hslash :X\to [0,1)$$

by

$$\ell \left(\xi \right)=\frac{\xi }{2},\mathfrak{m}\left(\xi \right)=\frac{\xi }{9},\mathfrak{r}\left(\xi \right)=\frac{\xi }{12}\mathit{and}\hslash \left(\xi \right)=\frac{\xi }{6}.$$

Then,

$$\ell \left({\mathcal{F}}_1\xi \right)=\frac{\xi }{12}\le \frac{\xi }{2}=\ell \left(\xi \right)\mathit{and}\ell \left({\mathcal{F}}_2\xi \right)=\frac{\xi }{24}\le \frac{\xi }{2}=\ell \left(\xi \right)$$

$$\mathfrak{m}\left({\mathcal{F}}_1\xi \right)=\frac{\xi }{54}\le \frac{\xi }{9}=\mathfrak{m}\left(\xi \right)\mathit{and}\mathfrak{m}\left({\mathcal{F}}_2\xi \right)=\frac{\xi }{108}\le \frac{\xi }{9}=\mathfrak{m}\left(\xi \right)$$

$$\mathfrak{r}\left({\mathcal{F}}_1\xi \right)=\frac{\xi }{72}\le \frac{\xi }{12}=\mathfrak{r}\left(\xi \right)\mathit{and}\mathfrak{r}\left({\mathcal{F}}_2\xi \right)=\frac{\xi }{144}\le \frac{\xi }{12}=\mathfrak{r}\left(\xi \right)$$

$$\hslash \left({\mathcal{F}}_1\xi \right)=\frac{\xi }{36}\le \frac{\xi }{6}=\hslash \left(\xi \right)\mathit{and}\hslash \left({\mathcal{F}}_2\xi \right)=\frac{\xi }{72}\le \frac{\xi }{6}=\hslash \left(\xi \right).$$

Also,

$$\ell \left(\xi \right)+2\mathfrak{m}\left(\xi \right)+\mathfrak{r}\left(\xi \right)+\hslash \left(\xi \right)=\frac{31\xi }{36}<1.$$

If $\xi =\varsigma =0$, conditions of Theorem 1 hold trivially. Suppose ξ and ς are non-zero with $\xi <\varsigma$. Then

$$d_{\mathfrak{E}}\left(\xi ,{\mathcal{F}}_1\xi \right)={\left|\xi -\frac{\xi }{6}\right|}^2+i{\left|\xi -\frac{\xi }{6}\right|}^2$$

$$d_{\mathfrak{E}}\left(\varsigma ,{\mathcal{F}}_2\varsigma \right)={\left|\varsigma -\frac{\varsigma }{12}\right|}^2+i{\left|\varsigma -\frac{\varsigma }{12}\right|}^2$$

$$d_{\mathfrak{E}}\left(\varsigma ,{\mathcal{F}}_1\xi \right)={\left|\varsigma -\frac{\xi }{6}\right|}^2+i{\left|\varsigma -\frac{\xi }{6}\right|}^2$$

$$d_{\mathfrak{E}}\left(\xi ,{\mathcal{F}}_2\varsigma \right)={\left|\xi -\frac{\varsigma }{12}\right|}^2+i{\left|\xi -\frac{\varsigma }{12}\right|}^2$$

$$d_{\mathfrak{E}}\left({\mathcal{F}}_1\xi ,{\mathcal{F}}_2\varsigma \right)={\left|\frac{\xi }{6}-\frac{\varsigma }{12}\right|}^2+i{\left|\frac{\xi }{6}-\frac{\varsigma }{12}\right|}^2.$$

Then, it is very simple to prove that all the assumptions of Theorem 1 are satisfied and 0 is unique common fixed point of mappings ${\mathcal{F}}_1$ and ${\mathcal{F}}_2.$

Corollary 1.

Let $\mathcal{F}:\left(X,d_{\mathfrak{E}},\phi \right)\to \left(X,d_{\mathfrak{E}},\phi \right).$ If there exist the mappings $\ell ,\mathfrak{m},\mathfrak{r},\hslash :X\to [0,1)$ such that the following conditions are satisfied:

(a) 

$\ell \left(\mathcal{F}\xi \right)\le \ell \left(\xi \right)$,

$\mathfrak{m}\left(\mathcal{F}\xi \right)\le \mathfrak{m}\left(\xi \right)$,

$\mathfrak{r}\left(\mathcal{F}\xi \right)\le \mathfrak{r}\left(\xi \right),$

$\hslash \left(\mathcal{F}\xi \right)\le \hslash \left(\xi \right)$;

(b) 

$\ell \left(\xi \right)+2\mathfrak{m}\left(\xi \right)+\mathfrak{r}\left(\xi \right)+\hslash \left(\xi \right)<1;$

(c) 

$$\begin{array}{ccc}\hfill d_{\mathfrak{E}}\left(\mathcal{F}\xi ,\mathcal{F}\varsigma \right)& ⪯& \ell \left(\xi \right)d_{\mathfrak{E}}\left(\xi ,\varsigma \right)+\mathfrak{m}\left(\xi \right)\left(d_{\mathfrak{E}}\left(\xi ,\mathcal{F}\xi \right)+d_{\mathfrak{E}}\left(\varsigma ,\mathcal{F}\varsigma \right)\right)\hfill \\ & & +\mathfrak{r}\left(\xi \right)\frac{d_{\mathfrak{E}}\left(\xi ,\mathcal{F}\xi \right)d_{\mathfrak{E}}\left(\varsigma ,\mathcal{F}\varsigma \right)}{1+d_{\mathfrak{E}}\left(\xi ,\varsigma \right)}\hfill \\ & & +\hslash \left(\xi \right)\frac{d_{\mathfrak{E}}\left(\xi ,\mathcal{F}\varsigma \right)d_{\mathfrak{E}}\left(\varsigma ,\mathcal{F}\xi \right)}{1+d_{\mathfrak{E}}\left(\xi ,\varsigma \right)}\hfill \end{array}$$

for all $\xi ,\varsigma \in$$X;$

(d) 

for each ${\xi }_0\in X,$$\lambda =\frac{\ell \left({\xi }_0\right)+\mathfrak{m}\left({\xi }_0\right)}{1-\mathfrak{m}\left({\xi }_0\right)-\mathfrak{r}\left({\xi }_0\right)}<1$ and ${lim}_{n,k\to \infty }\phi \left({\xi }_n,{\xi }_k\right)\lambda <1$ hold, then $\mathcal{F}$ has a unique fixed point.

Proof. 

Setting ${\mathcal{F}}_1={\mathcal{F}}_2=\mathcal{F}$ in Theorem 1. □

Corollary 2.

Let ${\mathcal{F}}_1,{\mathcal{F}}_2:\left({X,d}_{\mathfrak{E}},\phi \right)\to \left({X,d}_{\mathfrak{E}},\phi \right)$. If there exist the mappings $\ell ,\mathfrak{m},\mathfrak{r}:X\to [0,1)$ such that the following conditions are satisfied:

(a) 

$\ell \left({\mathcal{F}}_1\xi \right)\le \ell \left(\xi \right)$ and $\ell \left({\mathcal{F}}_2\xi \right)\le \ell \left(\xi \right)$

$\mathfrak{m}\left({\mathcal{F}}_1\xi \right)\le \mathfrak{m}\left(\xi \right)$ and $\mathfrak{m}\left({\mathcal{F}}_2\xi \right)\le \mathfrak{m}\left(\xi \right),$

$\mathfrak{r}\left({\mathcal{F}}_1\xi \right)\le \mathfrak{r}\left(\xi \right)$ and $\mathfrak{r}\left({\mathcal{F}}_2\xi \right)\le \mathfrak{r}\left(\xi \right);$

(b) 

$\ell \left(\xi \right)+2\mathfrak{m}\left(\xi \right)+\mathfrak{r}\left(\xi \right)<1;$

(c) 

$$\begin{array}{ccc}\hfill d_{\mathfrak{E}}\left({\mathcal{F}}_1\xi ,{\mathcal{F}}_2\varsigma \right)& ⪯& \ell \left(\xi \right)d_{\mathfrak{E}}\left(\xi ,\varsigma \right)+\mathfrak{m}\left(\xi \right)\left(d_{\mathfrak{E}}\left(\xi ,{\mathcal{F}}_1\xi \right)+d_{\mathfrak{E}}\left(\varsigma ,{\mathcal{F}}_2\varsigma \right)\right)\hfill \\ & & +\mathfrak{r}\left(\xi \right)\frac{d_{\mathfrak{E}}\left(\xi ,{\mathcal{F}}_1\xi \right)d_{\mathfrak{E}}\left(\varsigma ,{\mathcal{F}}_2\varsigma \right)}{1+d_{\mathfrak{E}}\left(\xi ,\varsigma \right)}\hfill \end{array}$$

for all $\xi ,\varsigma \in$$X;$

(d) 

for each ${\xi }_0\in X,$$\lambda =\frac{\ell \left({\xi }_0\right)+\mathfrak{m}\left({\xi }_0\right)}{1-\mathfrak{m}\left({\xi }_0\right)-\mathfrak{r}\left({\xi }_0\right)}<1$ and ${lim}_{n,k\to \infty }\phi \left({\xi }_n,{\xi }_k\right)\lambda <1$ hold, then ${\mathcal{F}}_1$ and ${\mathcal{F}}_2$ have a unique common fixed point.

Proof. 

Take $\hslash :X\to [0,1)$ as $\hslash \left(\xi \right)=0$ in Theorem 1. □

Corollary 3.

Let $\mathcal{F}:\left(X,d_{\mathfrak{E}},\phi \right)\to \left(X,d_{\mathfrak{E}},\phi \right)$. If there exist the mappings $\ell ,\mathfrak{m},\mathfrak{r}:X\to [0,1)$ such that the following conditions are satisfied:

(a) 

$\ell \left(\mathcal{F}\xi \right)\le \ell \left(\xi \right)$,

$\mathfrak{m}\left({\mathcal{F}}_1\xi \right)\le \mathfrak{m}\left(\xi \right)$,

$\mathfrak{r}\left({\mathcal{F}}_1\xi \right)\le \mathfrak{r}\left(\xi \right)$;

(b) 

$\ell \left(\xi \right)+2\mathfrak{m}\left(\xi \right)+\mathfrak{r}\left(\xi \right)<1;$

(c) 

$$\begin{array}{ccc}\hfill d_{\mathfrak{E}}\left(\mathcal{F}\xi ,\mathcal{F}\varsigma \right)& ⪯& \ell \left(\xi \right)d_{\mathfrak{E}}\left(\xi ,\varsigma \right)+\mathfrak{m}\left(\xi \right)\left(d_{\mathfrak{E}}\left(\xi ,\mathcal{F}\xi \right)+d_{\mathfrak{E}}\left(\varsigma ,{\mathcal{F}}_2\varsigma \right)\right)\hfill \\ & & +\mathfrak{r}\left(\xi \right)\frac{d_{\mathfrak{E}}\left(\xi ,\mathcal{F}\xi \right)d_{\mathfrak{E}}\left(\varsigma ,\mathcal{F}\varsigma \right)}{1+d_{\mathfrak{E}}\left(\xi ,\varsigma \right)}\hfill \end{array}$$

for all $\xi ,\varsigma \in$$X;$

(d) 

for each ${\xi }_0\in X,$$\lambda =\frac{\ell \left({\xi }_0\right)+\mathfrak{m}\left({\xi }_0\right)}{1-\mathfrak{m}\left({\xi }_0\right)-\mathfrak{r}\left({\xi }_0\right)}<1$ and ${lim}_{n,k\to \infty }\phi \left({\xi }_n,{\xi }_k\right)\lambda <1$ hold, then $\mathcal{F}$ has a unique fixed point.

Proof. 

Take ${\mathcal{F}}_1={\mathcal{F}}_2=\mathcal{F}$ in Corollary 2. □

Corollary 4.

Let ${\mathcal{F}}_1,{\mathcal{F}}_2:\left({X,d}_{\mathfrak{E}},\phi \right)\to \left({X,d}_{\mathfrak{E}},\phi \right)$. If there exist the mappings $\ell ,\mathfrak{m},\hslash :X\to [0,1)$ such that the following conditions are satisfied:

(a) 

$\ell \left({\mathcal{F}}_1\xi \right)\le \ell \left(\xi \right)$ and $\ell \left({\mathcal{F}}_2\xi \right)\le \ell \left(\xi \right)$

$\mathfrak{m}\left({\mathcal{F}}_1\xi \right)\le \mathfrak{m}\left(\xi \right)$ and $\mathfrak{m}\left({\mathcal{F}}_2\xi \right)\le \mathfrak{m}\left(\xi \right),$

$\hslash \left({\mathcal{F}}_1\xi \right)\le \hslash \left(\xi \right)$ and $\hslash \left({\mathcal{F}}_2\xi \right)\le \hslash \left(\xi \right);$

(b) 

$\ell \left(\xi \right)+2\mathfrak{m}\left(\xi \right)+\hslash \left(\xi \right)<1;$

(c) 

$$\begin{array}{ccc}\hfill d_{\mathfrak{E}}\left({\mathcal{F}}_1\xi ,{\mathcal{F}}_2\varsigma \right)& ⪯& \ell \left(\xi \right)d_{\mathfrak{E}}\left(\xi ,\varsigma \right)+\mathfrak{m}\left(\xi \right)\left(d_{\mathfrak{E}}\left(\xi ,{\mathcal{F}}_1\xi \right)+d_{\mathfrak{E}}\left(\varsigma ,{\mathcal{F}}_2\varsigma \right)\right)\hfill \\ & & +\hslash \left(\xi \right)\frac{d_{\mathfrak{E}}\left(\xi ,{\mathcal{F}}_2\varsigma \right)d_{\mathfrak{E}}\left(\varsigma ,{\mathcal{F}}_1\xi \right)}{1+d_{\mathfrak{E}}\left(\xi ,\varsigma \right)}\hfill \end{array}$$

for all $\xi ,\varsigma \in$$X;$

(d) 

for each ${\xi }_0\in X,$$\lambda =\frac{\ell \left({\xi }_0\right)+\mathfrak{m}\left({\xi }_0\right)}{1-\mathfrak{m}\left({\xi }_0\right)}<1$ and ${lim}_{n,k\to \infty }\phi \left({\xi }_n,{\xi }_k\right)\lambda <1$ hold, then ${\mathcal{F}}_1$ and ${\mathcal{F}}_2$ have a unique common fixed point.

Proof. 

Take $\mathfrak{r}:X\to [0,1)$ as $\mathfrak{r}\left(\xi \right)=0$ in Theorem 1. □

Corollary 5.

Let $\mathcal{F}:\left(X,d_{\mathfrak{E}},\phi \right)\to \left(X,d_{\mathfrak{E}},\phi \right).$ If there exist the mappings $\ell ,\mathfrak{m},\hslash :X\to [0,1)$ such that the following conditions are satisfied:

(a) 

$\ell \left(\mathcal{F}\xi \right)\le \ell \left(\xi \right)$,

$\mathfrak{m}\left(\mathcal{F}\xi \right)\le \mathfrak{m}\left(\xi \right)$

$\hslash \left(\mathcal{F}\xi \right)\le \hslash \left(\xi \right)$;

(b) 

$\ell \left(\xi \right)+2\mathfrak{m}\left(\xi \right)+\hslash \left(\xi \right)<1;$

(c) 

$$\begin{array}{ccc}\hfill d_{\mathfrak{E}}\left(\mathcal{F}\xi ,\mathcal{F}\varsigma \right)& ⪯& \ell \left(\xi \right)d_{\mathfrak{E}}\left(\xi ,\varsigma \right)+\mathfrak{m}\left(\xi \right)\left(d_{\mathfrak{E}}\left(\xi ,\mathcal{F}\xi \right)+d_{\mathfrak{E}}\left(\varsigma ,\mathcal{F}\varsigma \right)\right)\hfill \\ & & +\hslash \left(\xi \right)\frac{d_{\mathfrak{E}}\left(\xi ,\mathcal{F}\varsigma \right)d_{\mathfrak{E}}\left(\varsigma ,\mathcal{F}\xi \right)}{1+d_{\mathfrak{E}}\left(\xi ,\varsigma \right)}\hfill \end{array}$$

for all $\xi ,\varsigma \in$$X;$

(d) 

for each ${\xi }_0\in X,$$\lambda =\frac{\ell \left({\xi }_0\right)+\mathfrak{m}\left({\xi }_0\right)}{1-\mathfrak{m}\left({\xi }_0\right)}<1$ and ${lim}_{n,k\to \infty }\phi \left({\xi }_n,{\xi }_k\right)\lambda <1$ hold, then $\mathcal{F}$ has a unique fixed point.

Proof. 

Take ${\mathcal{F}}_1={\mathcal{F}}_2=\mathcal{F}$ in above Corollary. □

Corollary 6.

Let ${\mathcal{F}}_1,{\mathcal{F}}_2:\left(X,d_{\mathfrak{E}},\phi \right)\to \left(X,d_{\mathfrak{E}},\phi \right)$. If there exist the mappings $\ell ,\mathfrak{r},\hslash :X\to [0,1)$ such that the following conditions are satisfied:

(a) 

$\ell \left({\mathcal{F}}_1\xi \right)\le \ell \left(\xi \right)$ and $\ell \left({\mathcal{F}}_2\xi \right)\le \ell \left(\xi \right)$

$\mathfrak{r}\left({\mathcal{F}}_1\xi \right)\le \mathfrak{r}\left(\xi \right)$ and $\mathfrak{r}\left({\mathcal{F}}_2\xi \right)\le \mathfrak{r}\left(\xi \right),$

$\hslash \left({\mathcal{F}}_1\xi \right)\le \hslash \left(\xi \right)$ and $\hslash \left({\mathcal{F}}_2\xi \right)\le \hslash \left(\xi \right);$

(b) 

$\ell \left(\xi \right)+2\mathfrak{m}\left(\xi \right)+\mathfrak{r}\left(\xi \right)+\hslash \left(\xi \right)<1;$

(c) 

$$\begin{array}{ccc}\hfill d_{\mathfrak{E}}\left({\mathcal{F}}_1\xi ,{\mathcal{F}}_2\varsigma \right)& ⪯& \ell \left(\xi \right)d_{\mathfrak{E}}\left(\xi ,\varsigma \right)\hfill \\ & & +\mathfrak{r}\left(\xi \right)\frac{d_{\mathfrak{E}}\left(\xi ,{\mathcal{F}}_1\xi \right)d_{\mathfrak{E}}\left(\varsigma ,{\mathcal{F}}_2\varsigma \right)}{1+d_{\mathfrak{E}}\left(\xi ,\varsigma \right)}\hfill \\ & & +\hslash \left(\xi \right)\frac{d_{\mathfrak{E}}\left(\xi ,{\mathcal{F}}_2\varsigma \right)d_{\mathfrak{E}}\left(\varsigma ,{\mathcal{F}}_1\xi \right)}{1+d_{\mathfrak{E}}\left(\xi ,\varsigma \right)}\hfill \end{array}$$

for all $\xi ,\varsigma \in$$X;$

(d) 

for each ${\xi }_0\in X,$$\lambda =\frac{\ell \left({\xi }_0\right)}{1--\mathfrak{r}\left({\xi }_0\right)}<1$ and ${lim}_{n,k\to \infty }\phi \left({\xi }_n,{\xi }_k\right)\lambda <1$ hold, then ${\mathcal{F}}_1$ and ${\mathcal{F}}_2$ have a unique common fixed point.

Proof. 

Take $\mathfrak{m}:X\to [0,1)$ as $\mathfrak{m}\left(\xi \right)=0$ in Theorem 1. □

Corollary 7.

Let $\mathcal{F}:\left(X,d_{\mathfrak{E}},\phi \right)\to \left(X,d_{\mathfrak{E}},\phi \right)$. If there exist mappings $\ell ,\mathfrak{r},\hslash :X\to [0,1)$ such that the following conditions are satisfied:

(a) 

$\ell \left(\mathcal{F}\xi \right)\le \ell \left(\xi \right)$,

$\mathfrak{r}\left(\mathcal{F}\xi \right)\le \mathfrak{r}\left(\xi \right),$

$\hslash \left(\mathcal{F}\xi \right)\le \hslash \left(\xi \right)$;

(b) 

$\ell \left(\xi \right)+\mathfrak{r}\left(\xi \right)+\hslash \left(\xi \right)<1;$

(c) 

$$\begin{array}{ccc}\hfill d_{\mathfrak{E}}\left(\mathcal{F}\xi ,\mathcal{F}\varsigma \right)& ⪯& \ell \left(\xi \right)d_{\mathfrak{E}}\left(\xi ,\varsigma \right)\hfill \\ & & +\mathfrak{r}\left(\xi \right)\frac{d_{\mathfrak{E}}\left(\xi ,\mathcal{F}\xi \right)d_{\mathfrak{E}}\left(\varsigma ,\mathcal{F}\varsigma \right)}{1+d_{\mathfrak{E}}\left(\xi ,\varsigma \right)}\hfill \\ & & +\hslash \left(\xi \right)\frac{d_{\mathfrak{E}}\left(\xi ,\mathcal{F}\varsigma \right)d_{\mathfrak{E}}\left(\varsigma ,\mathcal{F}\xi \right)}{1+d_{\mathfrak{E}}\left(\xi ,\varsigma \right)}\hfill \end{array}$$

for all $\xi ,\varsigma \in$$X;$

(d) 

for each ${\xi }_0\in X,$$\lambda =\frac{\ell \left({\xi }_0\right)}{1-\mathfrak{r}\left({\xi }_0\right)}<1$ and ${lim}_{n,k\to \infty }\phi \left({\xi }_n,{\xi }_k\right)\lambda <1$ hold, then $\mathcal{F}$ has a unique fixed point.

Proof. 

Take ${\mathcal{F}}_1={\mathcal{F}}_2=\mathcal{F}$ in above Corollary. □

Corollary 8.

Let ${\mathcal{F}}_1,{\mathcal{F}}_2:\left(X,d_{\mathfrak{E}},\phi \right)\to \left(X,d_{\mathfrak{E}},\phi \right)$. If there exist mappings $\ell ,\mathfrak{m}:X\to [0,1)$ such that the following conditions are satisfied:

(a) 

$\ell \left({\mathcal{F}}_1\xi \right)\le \ell \left(\xi \right)$ and $\ell \left({\mathcal{F}}_2\xi \right)\le \ell \left(\xi \right)$

$\mathfrak{m}\left({\mathcal{F}}_1\xi \right)\le \mathfrak{m}\left(\xi \right)$ and $\mathfrak{m}\left({\mathcal{F}}_2\xi \right)\le \mathfrak{m}\left(\xi \right);$

(b) 

$\ell \left(\xi \right)+2\mathfrak{m}\left(\xi \right)<1;$

(c) 

$$d_{\mathfrak{E}}\left({\mathcal{F}}_1\xi ,{\mathcal{F}}_2\varsigma \right)⪯\ell \left(\xi \right)d_{\mathfrak{E}}\left(\xi ,\varsigma \right)+\mathfrak{m}\left(\xi \right)\left(d_{\mathfrak{E}}\left(\xi ,{\mathcal{F}}_1\xi \right)+d_{\mathfrak{E}}\left(\varsigma ,{\mathcal{F}}_2\varsigma \right)\right)$$

for all $\xi ,\varsigma \in$$X;$

(d) 

for each ${\xi }_0\in X,$$\lambda =\frac{\ell \left({\xi }_0\right)+\mathfrak{m}\left({\xi }_0\right)}{1-\mathfrak{m}\left({\xi }_0\right)}<1$ and ${lim}_{n,k\to \infty }\phi \left({\xi }_n,{\xi }_k\right)\lambda <1$ hold, then ${\mathcal{F}}_1$ and ${\mathcal{F}}_2$ have a unique common fixed point.

Proof. 

Take $\mathfrak{r}\left(\xi \right)=0$ and $\hslash \left(\xi \right)=0$ in Theorem 1. □

Corollary 9 

(Carmel Pushpa Raj et al. [10]). Let ${\mathcal{F}}_1,{\mathcal{F}}_2:\left(X,d_{\mathfrak{E}},\phi \right)\to \left(X,d_{\mathfrak{E}},\phi \right)$. Assume that there exist some constants $\ell ,\mathfrak{m}\in [0,1)$ such that $\ell +2\mathfrak{m}<1$ and

$$d_{\mathfrak{E}}\left({\mathcal{F}}_1\xi ,{\mathcal{F}}_2\varsigma \right)⪯\ell d_{\mathfrak{E}}\left(\xi ,\varsigma \right)+\mathfrak{m}\left(d_{\mathfrak{E}}\left(\xi ,{\mathcal{F}}_1\xi \right)+d_{\mathfrak{E}}\left(\varsigma ,{\mathcal{F}}_2\varsigma \right)\right)$$

for all $\xi ,\varsigma \in$$X.$ Moreover, if for each ${\xi }_0\in X,$${lim}_{n,k\to \infty }\phi \left({\xi }_n,{\xi }_k\right)\lambda <1$ holds with $\lambda =\frac{\ell +\mathfrak{m}}{1-\mathfrak{m}}<1,$ then ${\mathcal{F}}_1$ and ${\mathcal{F}}_2$ have a unique common fixed point.

Proof. 

Take $\ell (·)=\ell$ and $\mathfrak{m}(·)=\mathfrak{m}$ in Corollary 8. □

Corollary 10.

Let $\mathcal{F}:\left(X,d_{\mathfrak{E}},\phi \right)\to \left(X,d_{\mathfrak{E}},\phi \right)$. If there exist the mappings $\ell ,\mathfrak{m}:X\to [0,1)$ such that the following conditions are satisfied:

(a) 

$\ell \left(\mathcal{F}\xi \right)\le \ell \left(\xi \right)$,

$\mathfrak{m}\left(\mathcal{F}\xi \right)\le \mathfrak{m}\left(\xi \right)$;

(b) 

$\ell \left(\xi \right)+2\mathfrak{m}\left(\xi \right)<1;$

(c) 

$$d_{\mathfrak{E}}\left(\mathcal{F}\xi ,\mathcal{F}\varsigma \right)⪯\ell \left(\xi \right)d_{\mathfrak{E}}\left(\xi ,\varsigma \right)+\mathfrak{m}\left(\xi \right)\left(d_{\mathfrak{E}}\left(\xi ,\mathcal{F}\xi \right)+d_{\mathfrak{E}}\left(\varsigma ,\mathcal{F}\varsigma \right)\right)$$

for all $\xi ,\varsigma \in$$X;$

(d) 

for each ${\xi }_0\in X,$$\lambda =\frac{\ell \left({\xi }_0\right)+\mathfrak{m}\left({\xi }_0\right)}{1-\mathfrak{m}\left({\xi }_0\right)}<1$ and ${lim}_{n,k\to \infty }\phi \left({\xi }_n,{\xi }_k\right)\lambda <1$ hold, then $\mathcal{F}$ has a unique fixed point.

Proof. 

Take ${\mathcal{F}}_1={\mathcal{F}}_2=\mathcal{F}$ in Corollary 8. □

Corollary 11.

Let ${\mathcal{F}}_1,{\mathcal{F}}_2:\left(X,d_{\mathfrak{E}},\phi \right)\to \left(X,d_{\mathfrak{E}},\phi \right)$. If there exist the constants $\ell ,\mathfrak{m},\mathfrak{r},\hslash \in [0,1)$ with $\ell +2\mathfrak{m}+\mathfrak{r}+\hslash <1$ such that

$$\begin{array}{ccc}\hfill d_{\mathfrak{E}}\left({\mathcal{F}}_1\xi ,{\mathcal{F}}_2\varsigma \right)& ⪯& \ell d_{\mathfrak{E}}\left(\xi ,\varsigma \right)+\mathfrak{m}\left(d_{\mathfrak{E}}\left(\xi ,{\mathcal{F}}_1\xi \right)+d_{\mathfrak{E}}\left(\varsigma ,{\mathcal{F}}_2\varsigma \right)\right)\hfill \\ & & +\mathfrak{r}\frac{d_{\mathfrak{E}}\left(\xi ,{\mathcal{F}}_1\xi \right)d_{\mathfrak{E}}\left(\varsigma ,{\mathcal{F}}_2\varsigma \right)}{1+d_{\mathfrak{E}}\left(\xi ,\varsigma \right)}\hfill \\ & & +\hslash \frac{d_{\mathfrak{E}}\left(\xi ,{\mathcal{F}}_2\varsigma \right)d_{\mathfrak{E}}\left(\varsigma ,{\mathcal{F}}_1\xi \right)}{1+d_{\mathfrak{E}}\left(\xi ,\varsigma \right)}\hfill \end{array}$$

for all $\xi ,\varsigma \in$$X.$ Also, for ${\xi }_0\in X$ and $\lambda =\frac{\ell +\mathfrak{m}}{1-\mathfrak{m}-\mathfrak{r}}<1,$${lim}_{n,k\to \infty }\phi \left({\xi }_n,{\xi }_k\right)\lambda <1$ hold, then ${\mathcal{F}}_1$ and ${\mathcal{F}}_2$ have a unique common fixed point.

Proof. 

Take $\ell \left(\xi \right)=\ell ,$$\mathfrak{m}\left(\xi \right)=\mathfrak{m},\mathfrak{r}\left(\xi \right)=\mathfrak{r}$ and $\hslash \left(\xi \right)=\hslash$ in Theorem 1. □

Corollary 12.

Let $\mathcal{F}:\left(X,d_{\mathfrak{E}},\phi \right)\to \left(X,d_{\mathfrak{E}},\phi \right)$, if there exist the constants $\ell ,\mathfrak{m},\mathfrak{r},\hslash \in [0,1)$ with $\ell +2\mathfrak{m}+\mathfrak{r}+\hslash <1$ such that

$$\begin{array}{ccc}\hfill d_{\mathfrak{E}}\left(\mathcal{F}\xi ,\mathcal{F}\varsigma \right)& ⪯& \ell d_{\mathfrak{E}}\left(\xi ,\varsigma \right)+\mathfrak{m}\left(d_{\mathfrak{E}}\left(\xi ,\mathcal{F}\xi \right)+d_{\mathfrak{E}}\left(\varsigma ,\mathcal{F}\varsigma \right)\right)\hfill \\ & & +\mathfrak{r}\frac{d_{\mathfrak{E}}\left(\xi ,\mathcal{F}\xi \right)d_{\mathfrak{E}}\left(\varsigma ,\mathcal{F}\varsigma \right)}{1+d_{\mathfrak{E}}\left(\xi ,\varsigma \right)}\hfill \\ & & +\hslash \frac{d_{\mathfrak{E}}\left(\xi ,\mathcal{F}\varsigma \right)d_{\mathfrak{E}}\left(\varsigma ,\mathcal{F}\xi \right)}{1+d_{\mathfrak{E}}\left(\xi ,\varsigma \right)}\hfill \end{array}$$

for all $\xi ,\varsigma \in$$X.$ Also, for ${\xi }_0\in X$ and $\lambda =\frac{\ell +\mathfrak{m}}{1-\mathfrak{m}-\mathfrak{r}}<1,$${lim}_{n,k\to \infty }\phi \left({\xi }_n,{\xi }_k\right)\lambda <1$ hold, then $\mathcal{F}$ has a unique fixed point.

Corollary 13.

Let ${\mathcal{F}}_1,{\mathcal{F}}_2:\left(X,d_{\mathfrak{E}},\phi \right)\to \left(X,d_{\mathfrak{E}},\phi \right)$, if there exist $\ell ,\mathfrak{r},\hslash \in [0,1)$ with $\ell +\mathfrak{r}+\hslash <1$ such that

$$\begin{array}{ccc}\hfill d_{\mathfrak{E}}\left({\mathcal{F}}_1\xi ,{\mathcal{F}}_2\varsigma \right)& ⪯& \ell d_{\mathfrak{E}}\left(\xi ,\varsigma \right)\hfill \\ & & +\mathfrak{r}\frac{d_{\mathfrak{E}}\left(\xi ,{\mathcal{F}}_1\xi \right)d_{\mathfrak{E}}\left(\varsigma ,{\mathcal{F}}_2\varsigma \right)}{1+d_{\mathfrak{E}}\left(\xi ,\varsigma \right)}\hfill \\ & & +\hslash \frac{d_{\mathfrak{E}}\left(\xi ,{\mathcal{F}}_2\varsigma \right)d_{\mathfrak{E}}\left(\varsigma ,{\mathcal{F}}_1\xi \right)}{1+d_{\mathfrak{E}}\left(\xi ,\varsigma \right)}\hfill \end{array}$$

for all $\xi ,\varsigma \in$$X.$ Also, for ${\xi }_0\in X$ and $\lambda =\frac{\ell }{1-\mathfrak{r}}<1,$${lim}_{n,k\to \infty }\phi \left({\xi }_n,{\xi }_k\right)\lambda <1$ holds, then ${\mathcal{F}}_1$ and ${\mathcal{F}}_2$ have unique common fixed point.

Proof. 

Take $\mathfrak{m}=0$ in Corollary 11. □

Corollary 14.

Let $\mathcal{F}:\left(X,d_{\mathfrak{E}},\phi \right)\to \left(X,d_{\mathfrak{E}},\phi \right)$, if there exist the constants $\ell ,\mathfrak{r},\hslash \in [0,1)$ with $\ell +\mathfrak{r}+\hslash <1$ such that

$$\begin{array}{ccc}\hfill d_{\mathfrak{E}}\left(\mathcal{F}\xi ,\mathcal{F}\varsigma \right)& ⪯& \ell d_{\mathfrak{E}}\left(\xi ,\varsigma \right)\hfill \\ & & +\mathfrak{r}\frac{d_{\mathfrak{E}}\left(\xi ,\mathcal{F}\xi \right)d_{\mathfrak{E}}\left(\varsigma ,\mathcal{F}\varsigma \right)}{1+d_{\mathfrak{E}}\left(\xi ,\varsigma \right)}\hfill \\ & & +\hslash \frac{d_{\mathfrak{E}}\left(\xi ,\mathcal{F}\varsigma \right)d_{\mathfrak{E}}\left(\varsigma ,\mathcal{F}\xi \right)}{1+d_{\mathfrak{E}}\left(\xi ,\varsigma \right)}\hfill \end{array}$$

for all $\xi ,\varsigma \in$$X.$ Also, for ${\xi }_0\in X$ and $\lambda =\frac{\ell }{1-\mathfrak{r}}<1,$${lim}_{n,k\to \infty }\phi \left({\xi }_n,{\xi }_k\right)\lambda <1$ holds, then $\mathcal{F}$ has a unique fixed point.

Corollary 15.

Setting ${\mathcal{F}}_1={\mathcal{F}}_2=\mathcal{F}$ in above Corollary.

Example 6.

Let $X=[0,+\infty )$ and $\phi :X\times X\to [1,\infty )$ be a function defined by

$$\phi (\xi ,\varsigma )=1+\xi +\varsigma$$

and $d_{\mathfrak{E}}:X\times X\to \mathbb{C}$ by

$$d_{\mathfrak{E}}(\xi ,\varsigma )=\left|\xi -\varsigma \right|+i\left|\xi -\varsigma \right|$$

for all $\xi ,\varsigma \in$$X.$ Then, $(X,d_{\mathfrak{E}})$ is complete CVEbMS. Define a self mapping $\mathcal{F}:X$$\to X$ by

$$\mathcal{F}\xi =\frac{1}{3}(2-\xi ).$$

Then all the assumptions of Corollary 14 are satisfied for any $\mathfrak{r},\hslash$ and $\ell =\frac{1}{2}.$ Thus $\frac{1}{2}$ is the unique fixed point of mapping $\mathcal{F}.$

Corollary 16.

Let $\mathcal{F}:\left(X,d_{\mathfrak{E}},\phi \right)\to \left(X,d_{\mathfrak{E}},\phi \right)$. If there exist the mappings $\ell ,\mathfrak{m},\mathfrak{r},\hslash :X\to [0,1)$ and some natural number $n\in \mathbb{N}$ such that the following conditions are satisfied:

(a) 

$\ell \left({\mathcal{F}}^n\xi \right)\le \ell \left(\xi \right),$

$\mathfrak{m}\left({\mathcal{F}}^n\xi \right)\le \mathfrak{m}\left(\xi \right),$

$\mathfrak{r}\left({\mathcal{F}}^n\xi \right)\le \mathfrak{r}\left(\xi \right),$

$\hslash \left({\mathcal{F}}^n\xi \right)\le \hslash \left(\xi \right),$

where ${\mathcal{F}}^n$ is the n-th iterate of $\mathcal{F}$;

(b) 

$\ell \left(\xi \right)+2\mathfrak{m}\left(\xi \right)+\mathfrak{r}\left(\xi \right)+\hslash \left(\xi \right)<1;$

(c) 

$$\begin{array}{ccc}\hfill d_{\mathfrak{E}}\left({\mathcal{F}}^n\xi ,{\mathcal{F}}^n\varsigma \right)& ⪯& \ell \left(\xi \right)d_{\mathfrak{E}}\left(\xi ,\varsigma \right)+\mathfrak{m}\left(\xi \right)\left(d_{\mathfrak{E}}\left(\xi ,{\mathcal{F}}^n\xi \right)+d_{\mathfrak{E}}\left(\varsigma ,{\mathcal{F}}^n\varsigma \right)\right)\hfill \\ & & +\mathfrak{r}\left(\xi \right)\frac{d_{\mathfrak{E}}\left(\xi ,{\mathcal{F}}^n\xi \right)d_{\mathfrak{E}}\left(\varsigma ,{\mathcal{F}}^n\varsigma \right)}{1+d_{\mathfrak{E}}\left(\xi ,\varsigma \right)}\hfill \\ & & +\hslash \left(\xi \right)\frac{d_{\mathfrak{E}}\left(\xi ,{\mathcal{F}}^n\varsigma \right)d_{\mathfrak{E}}\left(\varsigma ,{\mathcal{F}}^n\xi \right)}{1+d_{\mathfrak{E}}\left(\xi ,\varsigma \right)}\hfill \end{array}$$

for all $\xi ,\varsigma \in$$X;$

(d) 

for each ${\xi }_0\in X,$$\lambda =\frac{\ell \left({\xi }_0\right)+\mathfrak{m}\left({\xi }_0\right)}{1-\mathfrak{m}\left({\xi }_0\right)-\mathfrak{r}\left({\xi }_0\right)}<1$ and ${lim}_{n,k\to \infty }\phi \left({\xi }_n,{\xi }_k\right)\lambda <1$ hold, whenever the sequence {${\xi }_n$} is defined by

$${\xi }_{n+1}=\mathcal{F}{\xi }_n,$$

then $\mathcal{F}$ has unique fixed point.

Proof. 

From the Corollary 1, we have $\xi \in X$ such that ${\mathcal{F}}^n\xi =\xi .$ Now from

$$\begin{array}{ccc}\hfill d_{\mathfrak{E}}\left(\mathcal{F}\xi ,\xi \right)& =& d_{\mathfrak{E}}\left({\mathcal{FF}}^n\xi ,{\mathcal{F}}^n\xi \right)\hfill \\ & =& d_{\mathfrak{E}}\left({\mathcal{F}}^n\mathcal{F}\xi ,{\mathcal{F}}^n\xi \right)⪯\ell \left(\mathcal{F}\xi \right)d_{\mathfrak{E}}\left(\mathcal{F}\xi ,\xi \right)\hfill \\ & & +\mathfrak{m}\left(\xi \right)\left(d_{\mathfrak{E}}\left(\mathcal{F}\xi ,{\mathcal{F}}^n\mathcal{F}\xi \right)+d_{\mathfrak{E}}\left(\xi ,{\mathcal{F}}^n\xi \right)\right)\hfill \\ & & +\mathfrak{r}\left(\mathcal{F}\xi \right)\frac{d_{\mathfrak{E}}\left(\mathcal{F}\xi ,{\mathcal{F}}^n\mathcal{F}\xi \right)d_{\mathfrak{E}}\left(\xi ,{\mathcal{F}}^n\xi \right)}{1+d_{\mathfrak{E}}\left(\mathcal{F}\xi ,\xi \right)}\hfill \\ & & +\hslash \left(\mathcal{F}\xi \right)\frac{d_{\mathfrak{E}}\left(\mathcal{F}\xi ,\mathcal{F}\xi \right)d_{\mathfrak{E}}\left(\xi ,{\mathcal{F}}^n\mathcal{F}\xi \right)}{1+d_{\mathfrak{E}}\left(\mathcal{F}\xi ,\xi \right)}\hfill \\ & =& \ell \left(\mathcal{F}\xi \right)d_{\mathfrak{E}}\left(\mathcal{F}\xi ,\xi \right)+\mathfrak{m}\left(\xi \right)\left(d_{\mathfrak{E}}\left(\mathcal{F}\xi ,\mathcal{F}\xi \right)+d_{\mathfrak{E}}\left(\xi ,\xi \right)\right)\hfill \\ & & +\mathfrak{r}\left(\mathcal{F}\xi \right)\frac{d_{\mathfrak{E}}\left(\mathcal{F}\xi ,\mathcal{F}\xi \right)d_{\mathfrak{E}}\left(\xi ,\xi \right)}{1+d_{\mathfrak{E}}\left(\mathcal{F}\xi ,\xi \right)}\hfill \\ & & +\hslash \left(\mathcal{F}\xi \right)\frac{d_{\mathfrak{E}}\left(\mathcal{F}\xi ,\mathcal{F}\xi \right)d_{\mathfrak{E}}\left(\xi ,\mathcal{F}\xi \right)}{1+d_{\mathfrak{E}}\left(\mathcal{F}\xi ,\xi \right)}\hfill \\ & =& \ell \left(\mathcal{F}\xi \right)d_{\mathfrak{E}}\left(\mathcal{F}\xi ,\xi \right)\hfill \end{array}$$

which implies that

$$\left|d_{\mathfrak{E}}\left(\mathcal{F}\xi ,\xi \right)\right|\le \ell \left(\mathcal{F}\xi \right)\left|d_{\mathfrak{E}}\left(\mathcal{F}\xi ,\xi \right)\right|$$

which is possible only whenever $\left|d_{\mathfrak{E}}\left(\mathcal{F}\xi ,\xi \right)\right|=0.$ Thus, $\mathcal{F}\xi =\xi .$ □

Remark 1.

Defining the mappings $\ell ,\mathfrak{m},\mathfrak{r},\hslash :X\to [0,1)$ equal to 0 appropriately in Theorem 1 and Corollaries 1–16, one can obtain a number of common fixed point results, which are generally contemporary results in the background of complex valued extended b-metric space.

4. Results in Complex Valued b-Metric Spaces

If we take $\phi :X\times X\to [1,\infty )$ as $\phi (\xi ,\varsigma )=s\ge 1$ in Definition 4, then CVEbMS is reduced to CVbMS. Throughout this section, we consider $\left(X,d_{cb},s\right)$ as complete CVbMS with $s\ge 1.$

Corollary 17.

Let ${\mathcal{F}}_1,{\mathcal{F}}_2:\left(X,d_{cb},s\right)\to \left(X,d_{cb},s\right)$. If there exist the mappings $\ell ,\mathfrak{m},\mathfrak{r},\hslash :X\to [0,1)$ such that the following conditions are satisfied:

(a) 

$\ell \left({\mathcal{F}}_1\xi \right)\le \ell \left(\xi \right)$ and $\ell \left({\mathcal{F}}_2\xi \right)\le \ell \left(\xi \right)$

$\mathfrak{m}\left({\mathcal{F}}_1\xi \right)\le \mathfrak{m}\left(\xi \right)$ and $\mathfrak{m}\left({\mathcal{F}}_2\xi \right)\le \mathfrak{m}\left(\xi \right),$

$\mathfrak{r}\left({\mathcal{F}}_1\xi \right)\le \mathfrak{r}\left(\xi \right)$ and $\mathfrak{r}\left({\mathcal{F}}_2\xi \right)\le \mathfrak{r}\left(\xi \right),$

$\hslash \left({\mathcal{F}}_1\xi \right)\le \hslash \left(\xi \right)$ and $\hslash \left({\mathcal{F}}_2\xi \right)\le \hslash \left(\xi \right);$

(b) 

$\ell \left(\xi \right)+2\mathfrak{m}\left(\xi \right)+\mathfrak{r}\left(\xi \right)+\hslash \left(\xi \right)<1;$

(c) 

$$\begin{array}{ccc}\hfill d_{cb}\left({\mathcal{F}}_1\xi ,{\mathcal{F}}_2\varsigma \right)& ⪯& \ell \left(\xi \right)d_{cb}\left(\xi ,\varsigma \right)+\mathfrak{m}\left(\xi \right)\left(d_{cb}\left(\xi ,{\mathcal{F}}_1\xi \right)+d_{cb}\left(\varsigma ,{\mathcal{F}}_2\varsigma \right)\right)\hfill \\ & & +\mathfrak{r}\left(\xi \right)\frac{d_{cb}\left(\xi ,{\mathcal{F}}_1\xi \right)d_{cb}\left(\varsigma ,{\mathcal{F}}_2\varsigma \right)}{1+d_{cb}\left(\xi ,\varsigma \right)}\hfill \\ & & +\hslash \left(\xi \right)\frac{d_{cb}\left(\xi ,{\mathcal{F}}_2\varsigma \right)d_{cb}\left(\varsigma ,{\mathcal{F}}_1\xi \right)}{1+d_{cb}\left(\xi ,\varsigma \right)}\hfill \end{array}$$

for all $\xi ,\varsigma \in$$X,$ then ${\mathcal{F}}_1$ and ${\mathcal{F}}_2$ have a unique common fixed point.

Proof. 

Take $\phi :X\times X\to [1,\infty )$ as $\phi (\xi ,\varsigma )=s\ge 1$ in Theorem 1. □

Corollary 18 

(Kumar et al. [7]). Let ${\mathcal{F}}_1,{\mathcal{F}}_2:\left(X,d_{cb},s\right)\to \left(X,d_{cb},s\right)$. If there exist some constants $\ell ,\mathfrak{r},\hslash \in [0,1)$ with $\ell +\mathfrak{r}+\hslash <1$ such that

$$\begin{array}{ccc}\hfill d_{cb}\left({\mathcal{F}}_1\xi ,{\mathcal{F}}_2\varsigma \right)& ⪯& \ell d_{cb}\left(\xi ,\varsigma \right)\hfill \\ & & +\mathfrak{r}\frac{d_{cb}\left(\xi ,{\mathcal{F}}_1\xi \right)d_{cb}\left(\varsigma ,{\mathcal{F}}_2\varsigma \right)}{1+d_{cb}\left(\xi ,\varsigma \right)}\hfill \\ & & +\hslash \frac{d_{cb}\left(\xi ,{\mathcal{F}}_2\varsigma \right)d_{cb}\left(\varsigma ,{\mathcal{F}}_1\xi \right)}{1+d_{cb}\left(\xi ,\varsigma \right)}\hfill \end{array}$$

for all $\xi ,\varsigma \in$$X,$ then ${\mathcal{F}}_1$ and ${\mathcal{F}}_2$ have unique common fixed point.

Proof. 

Take $\ell ,\mathfrak{m},\mathfrak{r},\hslash :X\to [0,1)$ as $\ell (·)=\ell ,$$\mathfrak{r}(·)=\mathfrak{r},\hslash (·)=\hslash$ and $\mathfrak{m}(·)=0$ in Corollary 17. □

Corollary 19 

(Mukheimer [6]). Let ${\mathcal{F}}_1,{\mathcal{F}}_2:\left(X,d_{cb},s\right)\to \left(X,d_{cb},s\right)$. If there exist some constants $\ell ,\mathfrak{r}\in [0,1)$ with $\ell +\mathfrak{r}<1$ such that

$$\begin{array}{ccc}\hfill d_{cb}\left({\mathcal{F}}_1\xi ,{\mathcal{F}}_2\varsigma \right)& ⪯& \ell d_{cb}\left(\xi ,\varsigma \right)\hfill \\ & & +\mathfrak{r}\frac{d_{cb}\left(\xi ,{\mathcal{F}}_1\xi \right)d_{cb}\left(\varsigma ,{\mathcal{F}}_2\varsigma \right)}{1+d_{cb}\left(\xi ,\varsigma \right)}\hfill \end{array}$$

for all $\xi ,\varsigma \in$$X,$ then ${\mathcal{F}}_1$ and ${\mathcal{F}}_2$ have unique common fixed point.

Proof. 

Take $\hslash =0$ in Corollary 18. □

5. Results in Complex Valued Metric Spaces

If we consider $\phi :X\times X\to [1,\infty )$ as $\phi (\xi ,\varsigma )=1$ in Definition 4, then CVEbMS is reduced to CVMS. Throughout this section, we consider $\left(X,d_{cv}\right)$ as complete CVMS.

Corollary 20.

Let ${\mathcal{F}}_1,{\mathcal{F}}_2:\left(X,d_{cv}\right)\to \left(X,d_{cv}\right)$. If there exist the mappings $\ell ,\mathfrak{m},\mathfrak{r},\hslash :X\to [0,1)$ such that the following conditions are satisfied:

(a) 

$\ell \left({\mathcal{F}}_1\xi \right)\le \ell \left(\xi \right)$ and $\ell \left({\mathcal{F}}_2\xi \right)\le \ell \left(\xi \right)$

$\mathfrak{m}\left({\mathcal{F}}_1\xi \right)\le \mathfrak{m}\left(\xi \right)$ and $\mathfrak{m}\left({\mathcal{F}}_2\xi \right)\le \mathfrak{m}\left(\xi \right),$

$\mathfrak{r}\left({\mathcal{F}}_1\xi \right)\le \mathfrak{r}\left(\xi \right)$ and $\mathfrak{r}\left({\mathcal{F}}_2\xi \right)\le \mathfrak{r}\left(\xi \right),$

$\hslash \left({\mathcal{F}}_1\xi \right)\le \hslash \left(\xi \right)$ and $\hslash \left({\mathcal{F}}_2\xi \right)\le \hslash \left(\xi \right);$

(b) 

$\ell \left(\xi \right)+2\mathfrak{m}\left(\xi \right)+\mathfrak{r}\left(\xi \right)+\hslash \left(\xi \right)<1;$

(c) 

$$\begin{array}{ccc}\hfill d_{cv}\left({\mathcal{F}}_1\xi ,{\mathcal{F}}_2\varsigma \right)& ⪯& \ell \left(\xi \right)d_{cv}\left(\xi ,\varsigma \right)+\mathfrak{m}\left(\xi \right)\left(d_{cv}\left(\xi ,{\mathcal{F}}_1\xi \right)+d_{cv}\left(\varsigma ,{\mathcal{F}}_2\varsigma \right)\right)\hfill \\ & & +\mathfrak{r}\left(\xi \right)\frac{d_{cv}\left(\xi ,{\mathcal{F}}_1\xi \right)d_{cv}\left(\varsigma ,{\mathcal{F}}_2\varsigma \right)}{1+d_{cv}\left(\xi ,\varsigma \right)}\hfill \\ & & +\hslash \left(\xi \right)\frac{d_{cv}\left(\xi ,{\mathcal{F}}_2\varsigma \right)d_{cv}\left(\varsigma ,{\mathcal{F}}_1\xi \right)}{1+d_{cv}\left(\xi ,\varsigma \right)}\hfill \end{array}$$

for all $\xi ,\varsigma \in$$X,$ then ${\mathcal{F}}_1$ and ${\mathcal{F}}_2$ have a unique common fixed point.

Proof. 

Take $\phi :X\times X\to [1,\infty )$ as $\phi (\xi ,\varsigma )=1$ in Theorem 1. □

Corollary 21 

(Sitthikul et al. [4]). Let ${\mathcal{F}}_1,{\mathcal{F}}_2:\left(X,d_{cv}\right)\to \left(X,d_{cv}\right)$. If there exist the mappings $\ell ,\mathfrak{r},\hslash :X\to [0,1)$ such that the following conditions are satisfied:

(a) 

$\ell \left({\mathcal{F}}_1\xi \right)\le \ell \left(\xi \right)$ and $\ell \left({\mathcal{F}}_2\xi \right)\le \ell \left(\xi \right),$

$\mathfrak{r}\left({\mathcal{F}}_1\xi \right)\le \mathfrak{r}\left(\xi \right)$ and $\mathfrak{r}\left({\mathcal{F}}_2\xi \right)\le \mathfrak{r}\left(\xi \right),$

$\hslash \left({\mathcal{F}}_1\xi \right)\le \hslash \left(\xi \right)$ and $\hslash \left({\mathcal{F}}_2\xi \right)\le \hslash \left(\xi \right);$

(b) 

$\ell \left(\xi \right)+\mathfrak{r}\left(\xi \right)+\hslash \left(\xi \right)<1;$

(c) 

$$\begin{array}{ccc}\hfill d_{cv}\left({\mathcal{F}}_1\xi ,{\mathcal{F}}_2\varsigma \right)& ⪯& \ell \left(\xi \right)d_{cv}\left(\xi ,\varsigma \right)\hfill \\ & & +\mathfrak{r}\left(\xi \right)\frac{d_{cv}\left(\xi ,{\mathcal{F}}_1\xi \right)d_{cv}\left(\varsigma ,{\mathcal{F}}_2\varsigma \right)}{1+d_{cv}\left(\xi ,\varsigma \right)}\hfill \\ & & +\hslash \left(\xi \right)\frac{d_{cv}\left(\xi ,{\mathcal{F}}_2\varsigma \right)d_{cv}\left(\varsigma ,{\mathcal{F}}_1\xi \right)}{1+d_{cv}\left(\xi ,\varsigma \right)}\hfill \end{array}$$

for all $\xi ,\varsigma \in$$X,$ then ${\mathcal{F}}_1$ and ${\mathcal{F}}_2$ have a unique common fixed point.

Proof. 

Take $\mathfrak{m}:X\to [0,1)$ as $\mathfrak{m}\left(\xi \right)=0$ in Corollary 20. □

Corollary 22 

(Rouzkard et al. [2]). Let ${\mathcal{F}}_1,{\mathcal{F}}_2:\left(X,d_{cv}\right)\to \left(X,d_{cv}\right)$. If there exist some constants $\ell ,\mathfrak{r},\hslash \in [0,1)$ such that $\ell +\mathfrak{r}+\hslash <1$ and

$$\begin{array}{ccc}\hfill d_{cv}\left({\mathcal{F}}_1\xi ,{\mathcal{F}}_2\varsigma \right)& ⪯& \ell d_{cv}\left(\xi ,\varsigma \right)\hfill \\ & & +\mathfrak{r}\frac{d_{cv}\left(\xi ,{\mathcal{F}}_1\xi \right)d_{cv}\left(\varsigma ,{\mathcal{F}}_2\varsigma \right)}{1+d_{cv}\left(\xi ,\varsigma \right)}\hfill \\ & & +\hslash \frac{d_{cv}\left(\xi ,{\mathcal{F}}_2\varsigma \right)d_{cv}\left(\varsigma ,{\mathcal{F}}_1\xi \right)}{1+d_{cv}\left(\xi ,\varsigma \right)}\hfill \end{array}$$

for all $\xi ,\varsigma \in$$X,$ then ${\mathcal{F}}_1$ and ${\mathcal{F}}_2$ have a unique common fixed point.

Proof. 

Take $\ell ,\mathfrak{r},\hslash :X\to [0,1)$ as $\ell (·)=\ell ,$$\mathfrak{r}(·)=\mathfrak{r},\hslash (·)=\hslash$ in Corollary 21. □

Corollary 23.

Let ${\mathcal{F}}_1,{\mathcal{F}}_2:\left(X,d_{cv}\right)\to \left(X,d_{cv}\right)$. If there exist the mappings $\ell ,\mathfrak{r},\hslash :X\to [0,1)$ such that the following conditions are satisfied:

(a) 

$\ell \left({\mathcal{F}}_1\xi \right)\le \ell \left(\xi \right)$ and $\ell \left({\mathcal{F}}_2\xi \right)\le \ell \left(\xi \right),$

$\mathfrak{r}\left({\mathcal{F}}_1\xi \right)\le \mathfrak{r}\left(\xi \right)$ and $\mathfrak{r}\left({\mathcal{F}}_2\xi \right)\le \mathfrak{r}\left(\xi \right);$

(b) 

$\ell \left(\xi \right)+\mathfrak{r}\left(\xi \right)<1;$

(c) 

$$\begin{array}{ccc}\hfill d_{cv}\left({\mathcal{F}}_1\xi ,{\mathcal{F}}_2\varsigma \right)& ⪯& \ell \left(\xi \right)d_{cv}\left(\xi ,\varsigma \right)\hfill \\ & & +\mathfrak{r}\left(\xi \right)\frac{d_{cv}\left(\xi ,{\mathcal{F}}_1\xi \right)d_{cv}\left(\varsigma ,{\mathcal{F}}_2\varsigma \right)}{1+d_{cv}\left(\xi ,\varsigma \right)}\hfill \end{array}$$

for all $\xi ,\varsigma \in$$X,$ then ${\mathcal{F}}_1$ and ${\mathcal{F}}_2$ have a unique common fixed point.

Proof. 

Take $\hslash :X\to [0,1)$ as $\hslash \left(\xi \right)=0$ in Corollary 21. □

Corollary 24 

(Sintunavarat et al. [3]). Let $\mathcal{F}:\left(X,d_{cv}\right)\to \left(X,d_{cv}\right)$. If there exist the mappings $\ell ,\mathfrak{r},\hslash :X\to [0,1)$ such that the following conditions are satisfied:

(a) 

$\ell \left(\mathcal{F}\xi \right)\le \ell \left(\xi \right)$,

$\mathfrak{r}\left(\mathcal{F}\xi \right)\le \mathfrak{r}\left(\xi \right)$;

(b) 

$\ell \left(\xi \right)+\mathfrak{r}\left(\xi \right)<1;$

(c) 

$$\begin{array}{ccc}\hfill d_{cv}\left(\mathcal{F}\xi ,\mathcal{F}\varsigma \right)& ⪯& \ell \left(\xi \right)d_{cv}\left(\xi ,\varsigma \right)\hfill \\ & & +\mathfrak{r}\left(\xi \right)\frac{d_{cv}\left(\xi ,\mathcal{F}\xi \right)d_{cv}\left(\varsigma ,\mathcal{F}\varsigma \right)}{1+d_{cv}\left(\xi ,\varsigma \right)}\hfill \end{array}$$

for all $\xi ,\varsigma \in$$X,$ then $\mathcal{F}$ has a unique fixed point.

Proof. 

Taking ${\mathcal{F}}_1={\mathcal{F}}_2=\mathcal{F}$ in above Corollary 20. □

Corollary 25 

(Azam et al. [1]). Let ${\mathcal{F}}_1,{\mathcal{F}}_2:\left(X,d_{cv}\right)\to \left(X,d_{cv}\right).$ If there exist some non-negative constants $\ell ,\mathfrak{r}\in [0,1)$ with $\ell +\mathfrak{r}<1$ such that

$$d_{cv}\left({\mathcal{F}}_1\xi ,{\mathcal{F}}_2\varsigma \right)⪯\ell d_{cv}\left(\xi ,\varsigma \right)+\mathfrak{r}\frac{d_{cv}\left(\xi ,{\mathcal{F}}_1\xi \right)d_{cv}\left(\varsigma ,{\mathcal{F}}_2\varsigma \right)}{1+d_{cv}\left(\xi ,\varsigma \right)},$$

for all $\xi ,\varsigma \in$ X, then ${\mathcal{F}}_1$ and ${\mathcal{F}}_2$ have unique common fixed point.

Proof. 

Define $\ell ,\mathfrak{r}:X\to [0,1)$ by $\ell \left(\xi \right)=\ell$ and $\mathfrak{r}\left(\xi \right)=\mathfrak{r}$ for all $\xi \in$X in Corollary 23. □

6. Applications

Fixed point theory is a very important tool to solve differential and integral equations used to obtain solutions of different mathematical models, dynamical systems, models of economy, game theory, physics, computer science, engineering, neural networks and many others. In this section, let us give an application of our fixed point theorem to Urysohn integral equations.

Theorem 2.

Let $X=C([a,b],{\mathbb{R}}^n)$, $a>0$ and $d_{\mathfrak{E}}:X\times X\to \mathbb{C}$ be defined as

$$d_{\mathfrak{E}}(\xi ,\varsigma )=\underset{t\in \left[a,b\right]}{max}∥\xi \left(t\right)-\varsigma \left(t\right)∥\sqrt{1+a^2}{e}^{i{tan}^{-1}a}$$

and $\phi :X\times X\to [1,\infty )$ be defined by $\phi (\xi ,\varsigma )=1+\xi +\varsigma .$ Then, ($X,d_{\mathfrak{E}},\phi$) is complete CVEbMS. Take the Urysohn integral equations

$$\xi \left(t\right)={\int }_a^b{K}_1(t,s,\xi \left(s\right))d_{\mathfrak{E}}s+\alpha \left(t\right),$$

(6)

$$\xi \left(t\right)={\int }_a^b{K}_2(t,s,\xi \left(s\right))d_{\mathfrak{E}}s+\beta \left(t\right),$$

(7)

for all $t\in [a,b]\subset \mathbb{R}$, $\xi ,\alpha ,\beta \in X$.

Suppose that $K_1,K_2:[a,b]\times [a,b]\times {\mathbb{R}}^n\to {\mathbb{R}}^n$ are such that ${\Im }_{\xi },{\Re }_{\xi }\in X$ for each $\xi \in X,$ where,

$${\Im }_{\xi }\left(t\right)={\int }_a^b{K}_1(t,s,\xi \left(s\right))d_{\mathfrak{E}}s,{\Re }_{\xi }\left(t\right)={\int }_a^b{K}_2(t,s,\xi \left(s\right))d_{\mathfrak{E}}s.$$

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

If there exists $\ell ,\hslash :C([a,b],{\mathbb{R}}^n)\to [0,1)$ such that for every $\xi ,\varsigma \in X,$ these conditions are satisfied:

(a) 

$\ell \left({\Im }_{\xi }+\alpha \right)\le \ell \left(\xi \right)$ and $\ell \left({\Re }_{\xi }+\beta \right)\le \ell \left(\xi \right)$,

$\mathfrak{m}\left({\Im }_{\xi }+\alpha \right)\le \mathfrak{m}\left(\xi \right)$ and $\mathfrak{m}\left({\Re }_{\xi }+\beta \right)\le \mathfrak{m}\left(\xi \right),$

$\mathfrak{r}\left({\Im }_{\xi }+\alpha \right)\le \mathfrak{r}\left(\xi \right)$ and $\mathfrak{r}\left({\Re }_{\xi }+\beta \right)\le \mathfrak{r}\left(\xi \right),$

$\hslash \left({\Im }_{\xi }+\alpha \right)\le \hslash \left(\xi \right)$ and $\hslash \left({\Re }_{\xi }+\beta \right)\le \hslash \left(\xi \right);$

(b) 

$\ell \left(\xi \right)+2\mathfrak{m}\left(\xi \right)+\mathfrak{r}\left(\xi \right)+\hslash \left(\xi \right)<1;$

(c) 

$$\begin{array}{ccc}& & ∥{\Im }_{\xi }\left(t\right)-{\Re }_{\varsigma }\left(t\right)+\alpha \left(t\right)-\beta \left(t\right)∥\sqrt{1+a^2}{e}^{i{tan}^{-1}a}\hfill \\ & \precsim & \ell \left(\xi \right)A\left(\xi ,\varsigma \right)\left(t\right)+\mathfrak{m}\left(\xi \right)B\left(\xi ,\varsigma \right)\left(t\right)+\mathfrak{r}\left(\xi \right)C\left(\xi ,\varsigma \right)\left(t\right)+\hslash \left(\xi \right)D\left(\xi ,\varsigma \right)\left(t\right)\hfill \end{array}$$

where

$$\begin{array}{ccc}\hfill A\left(\xi ,\varsigma \right)\left(t\right)& =& ∥\xi \left(t\right)-\varsigma \left(t\right)∥\sqrt{1+a^2}{e}^{i{tan}^{-1}a},\hfill \\ \hfill B\left(\xi ,\varsigma \right)\left(t\right)& =& \left(∥{\Im }_{\xi }\left(t\right)+\alpha \left(t\right)-\xi \left(t\right)∥+∥{\Re }_{\varsigma }\left(t\right)+\beta \left(t\right)-\varsigma \left(t\right)∥\right)\sqrt{1+a^2}{e}^{i{tan}^{-1}a},\hfill \\ \hfill C\left(\xi ,\varsigma \right)\left(t\right)& =& \frac{∥{\Im }_{\xi }\left(t\right)+\alpha \left(t\right)-\xi \left(t\right)∥∥{\Re }_{\varsigma }\left(t\right)+\beta \left(t\right)-\varsigma \left(t\right)∥}{1+\underset{t\in \left[a,b\right]}{max}A\left(\xi ,\varsigma \right)\left(t\right)}\sqrt{1+a^2}{e}^{i{tan}^{-1}a},\hfill \\ \hfill D\left(\xi ,\varsigma \right)\left(t\right)& =& \frac{∥{\Re }_{\varsigma }\left(t\right)+\beta \left(t\right)-\xi \left(t\right)∥∥{\Im }_{\xi }\left(t\right)+\alpha \left(t\right)-\varsigma \left(t\right)∥}{1+\underset{t\in \left[a,b\right]}{max}A\left(\xi ,\varsigma \right)\left(t\right)}\sqrt{1+a^2}{e}^{i{tan}^{-1}a},\hfill \end{array}$$

then the system of integral Equations (6) and (7) have a unique common solution.

Proof. 

Define ${\mathcal{F}}_1,{\mathcal{F}}_2:X\to X$ by

$${\mathcal{F}}_1\xi ={\Im }_{\xi }+\alpha ,{\mathcal{F}}_2\xi ={\Re }_{\xi }+\beta .$$

Then,

$$d_{\mathfrak{E}}\left({\mathcal{F}}_1\xi ,{\mathcal{F}}_2\varsigma \right)=\underset{t\in \left[a,b\right]}{max}∥{\Im }_{\xi }\left(t\right)-{\Re }_{\varsigma }\left(t\right)+\alpha \left(t\right)-\beta \left(t\right)∥\sqrt{1+a^2}{e}^{i{tan}^{-1}a},$$

$$d_{\mathfrak{E}}\left(\xi ,\varsigma \right)=\underset{t\in \left[a,b\right]}{max}A\left(\xi ,\varsigma \right)\left(t\right),$$

$$d_{\mathfrak{E}}(\xi ,{\mathcal{F}}_1\xi )+d_{\mathfrak{E}}(\varsigma ,{\mathcal{F}}_2\varsigma )=\underset{t\in \left[a,b\right]}{max}B\left(\xi ,\varsigma \right)\left(t\right)$$

$$\frac{d_{\mathfrak{E}}(\xi ,{\mathcal{F}}_1\xi )d_{\mathfrak{E}}(\varsigma ,{\mathcal{F}}_2\varsigma )}{1+d_{\mathfrak{E}}(\xi ,\varsigma )}=\underset{t\in \left[a,b\right]}{max}C\left(\xi ,\varsigma \right)\left(t\right)$$

$$\frac{d_{\mathfrak{E}}(\xi ,{\mathcal{F}}_1\varsigma )d_{\mathfrak{E}}(\varsigma ,{\mathcal{F}}_2\xi )}{1+d_{\mathfrak{E}}(\xi ,\varsigma )}=\underset{t\in \left[a,b\right]}{max}D\left(\xi ,\varsigma \right)\left(t\right).$$

It is very simple to check that □

(a)

$\ell \left({\mathcal{F}}_1\xi \right)\le \ell \left(\xi \right)$ and $\ell \left({\mathcal{F}}_2\xi \right)\le \ell \left(\xi \right),$

$\mathfrak{m}\left({\mathcal{F}}_1\xi \right)\le \mathfrak{m}\left(\xi \right)$ and $\mathfrak{m}\left({\mathcal{F}}_2\xi \right)\le \mathfrak{m}\left(\xi \right),$

$\mathfrak{r}\left({\mathcal{F}}_1\xi \right)\le \mathfrak{r}\left(\xi \right)$ and $\mathfrak{r}\left({\mathcal{F}}_2\xi \right)\le \mathfrak{r}\left(\xi \right),$

$\hslash \left({\mathcal{F}}_1\xi \right)\le \hslash \left(\xi \right),$ and $\hslash \left({\mathcal{F}}_2\xi \right)\le \hslash \left(\xi \right);$

(b)

$\ell \left(\xi \right)+2\mathfrak{m}\left(\xi \right)+\mathfrak{r}\left(\xi \right)+\hslash \left(\xi \right)<1;$

(c)

$$\begin{array}{ccc}\hfill d_{\mathfrak{E}}\left({\mathcal{F}}_1\xi ,{\mathcal{F}}_2\varsigma \right)& \precsim & \ell \left(\xi \right)d_{\mathfrak{E}}(\xi ,\varsigma )+\mathfrak{m}\left(\xi \right)\left(d_{\mathfrak{E}}(\xi ,{\mathcal{F}}_1\xi )+d_{\mathfrak{E}}(\varsigma ,{\mathcal{F}}_2\varsigma )\right)\hfill \\ & & +\mathfrak{r}\left(\xi \right)\frac{d_{\mathfrak{E}}(\xi ,{\mathcal{F}}_1\xi )d_{\mathfrak{E}}(\varsigma ,{\mathcal{F}}_2\varsigma )}{1+d_{\mathfrak{E}}(\xi ,\varsigma )}\hfill \\ & & +\hslash \left(\xi \right)\frac{d_{\mathfrak{E}}(\xi ,{\mathcal{F}}_1\varsigma )d_{\mathfrak{E}}(\varsigma ,{\mathcal{F}}_2\xi )}{1+d_{\mathfrak{E}}(\xi ,\varsigma )}\hfill \end{array}$$

for every $\xi ,\varsigma \in X$. Thus, all the assumptions of Theorem 1 are satisfied. So by Theorem 1, the Urysohn integral Equations (6) and (7) have a unique common solution.

7. Conclusions

Complex valued metric spaces and their different generalizations allow us to consider the distances between complex numbers. In this research article, we instigated a novel notion named CVEbMS, which is a combination of complex numbers and extended b-metric space. We proposed some common fixed points of single-valued mappings for generalized contractions containing certain control functions. We derived the leading results of results of Azam et al. [1], Rouzkard et al. [2], Sintunavarat et al. [3], Sitthikul et al. [4], Kumar et al. [7] and Carmel Pushpa Raj et al. [10] as consequences of our main theorem. To confirm the applicability of the obtained theorems, we examined the solution of the Urysohn integral equation as an application. Some non-trivial examples are also explored to show the originality of our main results.

The given results in this research work can be augmented to some multi-valued mappings and fuzzy mappings in the framework of CVEbMS. Additionally, common fixed point results for self and non-self mappings can be proved in this context. As utilizations of these outcomes in the background of CVEbMS, some differential and integral inclusions can be explored.