Common Fixed Point Results on a Double-Controlled Metric Space for Generalized Rational-Type Contractions with Application

Abstract

In this manuscript, we prove several common fixed point theorems for generalized rational-type contraction mappings under several conditions in the context of double-controlled metric spaces. Further, we utilize a double-controlled metric space equipped with a graph to prove rational-type common fixed point theorems. Furthermore, we establish non-trivial examples to show the validity of the main results. These results improve and generalize already known results. At the end, we solve the Fredholm-type integral equation by utilizing the main results.

1. Introduction

The concept of fixed points has been utilized on a global scale in many disciplines of research and engineering [1,2]. Fixed point results were used to show that there are solutions to ordinary boundary value problems and fractional boundary value problems with integral-type boundary conditions, as described by Karapinar et al. [3]. In 1906, Frechet [4] presented the notion of metric spaces (MSs). MS techniques have been used for decades in a variety of applications, including image classification, protein classification, and Internet search engines. In order to extend metric techniques like Banach’s theorem to non-Hausdorff topologies, Matthews [5] introduced a symmetric generalized metric for such topologies. Mustafa and Sims [6] established the notion of generalized MS and worked on completeness and compactness. Azam et al. [7] proved a common fixed point theorem by utilizing contractive mappings for a pair of mappings in the framework of complex-valued MSs. Mitrovic and Radenovic [8] established the notion of  $b_v\left(s\right)$-metric space as a generalization of MS, rectangular MS, b-MS, rectangular b-MS, v-generalized MS Banach, and proved fixed point results for Reich contractions in  $bv\left(s\right)$-MS.

In 1993, Czerwik [9] established the concept of b-MS as a generalization of MS. He used a constant  $s\ge 1$  on the right side of the triangle inequality; if we consider  $s=1$  then b-MS becomes an MS. In 2017, Kamran et al. [10] presented the notion of an extended b-MS as a generalization of b-MS. They replaced a constant  $s\ge 1$  with a function  $\sigma :\Xi \times \Xi \to \left[1,\infty \right)$  and proved a Banach version of contraction mapping in the framework of extended b-MS. Recently, Mlaiki et al. [11] extended the notion of extended b-MS by utilizing the function  $\sigma :\Xi \times \Xi \to \left[1,\infty \right)$  with both terms separately on the right side of the triangular inequality and introduced controlled MS (CMS). Mustafa et al. [12] used the idea of extended b-MS and presented the notion of extended rectangular b-MS. Lateef [13] proved the fixed point theorem for Kannan-type contraction mapping in the context of CMSs. Abuloha et al. [14] derived several fixed point results for CMSs by utilizing the class of functions denoted by  ${\psi }_{\gamma }.$  For more related results in the setting of controlled-type MSs, see [15,16].

Abdeljawad et al. [17] presented double-controlled MS (DCMS) as a generalization of controlled MS by utilizing two functions  $\sigma ,\gamma :\Xi \times \Xi \to \left[1,\infty \right),$  with both terms separately on the right side of triangular inequality as follows:

$$d\left(\varpi ,\omega \right)\le \alpha \left(\varpi ,\varkappa \right)d\left(\varpi ,\varkappa \right)+\gamma \left(\varkappa ,\omega \right)d\left(\varkappa ,\omega \right)\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}\varpi ,\varkappa ,\omega \in \Xi .$$

Lateef [18] proved Fisher type fixed point results in controlled metric spaces. Farhan et al. [19] proved numerous fixed point results for  $(\alpha ,F)$-contraction and Reich-type contraction in the setting of DCMSs and partially ordered DCMSs. The authors in [20,21,22,23,24] generalized the notion of DCMSs by utilizing intuitionistic fuzzy sets and neutrosophic sets, and proved fixed point theorems with several applications. Latif [25] introduced the concept of neutrosophic delta–beta-connected topological spaces. Touqeer and Rasool [26] utilized a neutrosophic approach for decision making. Bousselsal Mostefaoui [27] proved some common fixed point results in partial metric spaces for generalized rational-type contraction mappings. Rao et al. [28] proved the existence and uniqueness of Suzuki-type results in  $S_b$-metric spaces with application to integral equations. Chandok and Kim [29] derived a fixed point theorem in ordered metric spaces for generalized contraction mappings that satisfy rational-type expressions.

In this paper, we prove some common fixed point theorems for generalized rational-type contraction mappings and rational-type contractions equipped with graphs in the setting of DCMSs. Further, we solve the integral equation by using the main results. Our results generalize several existing results in [11,18,20].

2. Preliminaries

This section contains some basic definitions.

Definition 1 ([9]).

Suppose  $\Xi \ne \varnothing$  and  $s\ge 1$  is any real number. The pair  $\left(\Xi ,∆,s\right)$  is called b-MS if a function  $∆:\Xi \times \Xi \to \left[0,\infty \right),$  verifying the following axioms:

• (bMS1)  $∆\left(\varpi ,\varkappa \right)\ge 0$  and  $∆\left(\varpi ,\varkappa \right)=0$  if and only if  $\varpi =\varkappa ,$ 
• (bMS2)  $∆\left(\varpi ,\varkappa \right)=∆\left(\varkappa ,\varpi \right),$ 
• (bMS3)  $∆\left(\varpi ,\omega \right)\le s\left[∆\left(\varpi ,\varkappa \right)+∆\left(\varkappa ,\omega \right)\right],$ 

for all  $\varpi ,\varkappa ,\omega \in \Xi .$ 

Kamran et al. [10] established the following notion of extended b-MS in 2017.

Definition 2 ([10]).

Assume that  $\Xi \ne \varnothing$  and  $\sigma :\Xi \times \Xi \to \left[1,\infty \right).$  The triplet  $\left(\Xi ,∆,\sigma \right)$  is called extended b-MS if a function  $∆:\Xi \times \Xi \to \left[0,\infty \right),$  verifying the following axioms:

• (EbMS1)  $∆\left(\varpi ,\varkappa \right)\ge 0$  and  $∆\left(\varpi ,\varkappa \right)=0$  if and only if  $\varpi =\varkappa ,$ 
• (EbMS2)  $∆\left(\varpi ,\varkappa \right)=∆\left(\varkappa ,\varpi \right),$ 
• (EbMS3)  $∆\left(\varpi ,\omega \right)\le \sigma \left(\varpi ,\omega \right)\left[∆\left(\varpi ,\varkappa \right)+∆\left(\varkappa ,\omega \right)\right],$ 

for all  $\varpi ,\varkappa ,\omega \in \Xi .$ 

Mlaiki et al. [11] presented the following notion of CMS in 2018.

Definition 3 ([11]).

Assume that  $\Xi \ne \varnothing$  and  $\sigma :\Xi \times \Xi \to \left[1,\infty \right).$  The pair  $\left(\Xi ,{∆}_{\sigma }\right)$  is called a CMS if a function  $∆:\Xi \times \Xi \to [0,\infty )$  , verifying the following axioms:

• (CMS1)  ${∆}_{\sigma }\left(\varpi ,\varkappa \right)\ge 0$  and  ${∆}_{\sigma }\left(\varpi ,\varkappa \right)=0$  if and only if  $\varpi =\varkappa ,$ 
• (CMS2)  ${∆}_{\sigma }\left(\varpi ,\varkappa \right)={∆}_{\sigma }\left(\varkappa ,\varpi \right),$ 
• (CMS3)  ${∆}_{\sigma }\left(\varpi ,\omega \right)\le \sigma \left(\varpi ,\varkappa \right){∆}_{\sigma }\left(\varpi ,\varkappa \right)+\sigma \left(\varkappa ,\omega \right){∆}_{\sigma }\left(\varkappa ,\omega \right),$ 

for all  $\varpi ,\varkappa ,\omega \in \Xi .$ 

Definition 4 ([16]).

Assume that  $\Xi \ne \varnothing$  and  $\sigma ,\gamma :\Xi \times \Xi \to \left[1,\infty \right).$  The quadruple  $\left(\Xi ,\Delta ,\sigma ,\gamma \right)$  is called double CMS (DCMS) if a function  $∆:\Xi \times \Xi \to \left[0,\infty \right),$  verifying the following axioms:

• (DCMS1)  $∆\left(\varpi ,\varkappa \right)\ge 0$  and  $∆\left(\varpi ,\varkappa \right)=0$  if and only if  $\varpi =\varkappa ,$ 
• (DCMS2)  $∆\left(\varpi ,\varkappa \right)=∆\left(\varkappa ,\varpi \right),$ 
• (DCMS3)  $∆\left(\varpi ,\omega \right)\le \sigma \left(\varpi ,\varkappa \right)∆\left(\varpi ,\varkappa \right)+\gamma \left(\varkappa ,\omega \right)∆\left(\varkappa ,\omega \right),$ 

for all  $\varpi ,\varkappa ,\omega \in \Xi .$ 

Definition 5.

Let  $\left(\Xi ,\Delta ,\sigma ,\gamma \right)$  be a DCMS and  ${\left\{{\varpi }_n\right\}}_{n\in \mathbb{N}}$  be a sequence in  $\Xi .$  Then a sequence  $\left\{{\varpi }_n\right\}$  is said to be convergent to  $\varpi \in \Xi$  if, for each  $\epsilon >0,$  there exists  $N\in \mathbb{N}$  with  $N=N\left(\epsilon \right)$  such that  $∆\left({\varpi }_n,\varpi \right)<\epsilon$  for all  $n\ge N.$  Further, we can write

$$\underset{n\to \infty }{\mathit{lim}}{\varpi }_n=\varpi .$$

a sequence  $\left\{{\varpi }_n\right\}$  is called Cauchy if, for each  $\epsilon >0,$  there exists  $N\in \mathbb{N}$  with  $N=N\left(\epsilon \right)$  such that  $∆\left({\varpi }_m,{\varpi }_n\right)<\epsilon$  for all  $m,n\ge N.$ 

The DCMS  $\left(\Xi ,\Delta ,\sigma ,\gamma \right)$  is said to be complete if every Cauchy sequence is convergent in  $\Xi$  .

3. Main Result

In this section, we will prove common fixed point results in DCMS.

Theorem 1.

Suppose  $\left(\Xi ,\Delta ,\sigma ,\gamma \right)$  is a complete DCMS,  $T_1,T_2:\Xi \to \Xi$  and there exists  $k_1,k_2:\Xi \times \Xi \to [0,1)$  such that

(I)

 $k_1\left(T_2{T}_1\varpi ,\varkappa \right)\le k_1(\varpi ,\varkappa )$  and  $k_1\left(\varpi ,T_1{T}_2\varkappa \right)\le k_1\left(\varpi ,\varkappa \right),$ 

(II)

 $k_2\left(T_2{T}_1\varpi ,\varkappa \right)\le k_2(\varpi ,\varkappa )$  and  $k_2\left(\varpi ,T_1{T}_2\varkappa \right)\le k_2\left(\varpi ,\varkappa \right),$ 

(III)

 $k_1\left(\varpi ,\varkappa \right)+k_2\left(\varpi ,\varkappa \right)<1,$ 

(IV)

$$∆\left(T_1\varpi ,T_2\varkappa \right)\le k_1\left(\varpi ,\varkappa \right)∆\left(\varpi ,\varkappa \right)+k_2\left(\varpi ,\varkappa \right)\frac{∆\left(\varpi ,T_1\varpi \right)∆\left(\varpi ,T_2\varpi \right)}{1+∆\left(\varpi ,\varkappa \right)},forall\varpi ,\varkappa \in \Xi ,$$

(1)

for  ${\varpi }_0\in \Xi ,$  a sequence  ${\left\{{\varpi }_j\right\}}_{j\ge 0}$  is defined as  ${\varpi }_{2j+1}=T_1{\varpi }_{2j}$  and  ${\varpi }_{2j+2}=T_2{\varpi }_{2j+1}$  for every  $j\ge 0.$  Suppose that

$$\underset{m\ge 1}{\mathit{sup}}\underset{i\to \infty }{\mathit{lim}}\frac{\sigma \left({\varpi }_{i+1},{\varpi }_{i+2}\right)\gamma \left({\varpi }_{i+1},{\varpi }_m\right)}{\sigma \left({\varpi }_i,{\varpi }_{i+1}\right)}<\frac{1}{k},$$

(2)

where  $\frac{k_1({\varpi }_0,{\varpi }_1)}{1-k_2({\varpi }_0,{\varpi }_1)}=k.$  Moreover, assume that  $\underset{j\to +\infty }{\mathit{lim}}\sigma \left({\varpi }_j,\varpi \right),\underset{j\to +\infty }{\mathit{lim}}\sigma \left(\varpi ,{\varpi }_j\right),$   $\underset{j\to +\infty }{\mathit{lim}}\gamma \left({\varpi }_j,\varpi \right)$  and  $\underset{j\to +\infty }{\mathit{lim}}\gamma \left(\varpi ,{\varpi }_j\right)$  exist and are finite, then there exists a unique fixed point  ${\varpi }^{*}\in \Xi$  such that  $T_1{\varpi }^{*}=T_2{\varpi }^{*}={\varpi }^{*}.$ 

Proof. 

Let  ${\varpi }_0\in \Xi .$  We construct  $\left\{{\varpi }_j\right\}$  in  $\Xi$  by  ${\varpi }_{2j+1}=T_1{\varpi }_{2j}$  and  ${\varpi }_{2j+2}=T_2{\varpi }_{2j+1}$  for each  $j\ge 0.$  From Assumption (1), we obtain

$$∆\left({\varpi }_{2j+1},{\varpi }_{2j+2}\right)=∆\left(T_1{\varpi }_{2j},T_2{\varpi }_{2j+1}\right)$$

$$\begin{array}{l}\le k_1\left({\varpi }_{2j},{\varpi }_{2j+1}\right)∆\left({\varpi }_{2j},{\varpi }_{2j+1}\right)+k_2\left({\varpi }_{2j},{\varpi }_{2j+1}\right)\frac{∆\left({\varpi }_{2j},T_1{\varpi }_{2j}\right)∆\left({\varpi }_{2j+1},T_2{\varpi }_{2j+1}\right)}{1+∆\left({\varpi }_{2j},{\varpi }_{2j+1}\right)}\\ \le k_1\left(T_1{T}_2{\varpi }_{2j-2},{\varpi }_{2j+1}\right)∆\left({\varpi }_{2j},{\varpi }_{2j+1}\right)\\ +k_2\left(T_1{T}_2{\varpi }_{2j-2},{\varpi }_{2j+1}\right)\frac{∆\left({\varpi }_{2j},T_1{\varpi }_{2j}\right)∆\left({\varpi }_{2j+1},T_2{\varpi }_{2j+1}\right)}{1+∆\left({\varpi }_{2j},{\varpi }_{2j+1}\right)}\\ \le k_1\left({\varpi }_{2j-2},{\varpi }_{2j+1}\right)∆\left({\varpi }_{2j},{\varpi }_{2j+1}\right)+k_2\left({\varpi }_{2j-2},{\varpi }_{2j+1}\right)∆\left({\varpi }_{2j+1},{\varpi }_{2j+2}\right),\\ =k_1\left(T_2{T}_1{\varpi }_{2j-4},{\varpi }_{2j+1}\right)∆\left({\varpi }_{2j},{\varpi }_{2j+1}\right)+k_2\left(T_2{T}_1{\varpi }_{2j-4},{\varpi }_{2j+1}\right)∆\left({\varpi }_{2j+1},{\varpi }_{2j+2}\right)\\ \le k_1\left({\varpi }_{2j-4},{\varpi }_{2j+1}\right)∆\left({\varpi }_{2j},{\varpi }_{2j+1}\right)+k_2\left({\varpi }_{2j-4},{\varpi }_{2j+1}\right)∆\left({\varpi }_{2j+1},{\varpi }_{2j+2}\right)\\ \le \dots \le k_1\left({\varpi }_0,{\varpi }_{2j+1}\right)∆\left({\varpi }_{2j},{\varpi }_{2j+1}\right)+k_2\left({\varpi }_0,{\varpi }_{2j+1}\right)∆\left({\varpi }_{2j+1},{\varpi }_{2j+2}\right),\\ =k_1\left({\varpi }_0,T_1{T}_2{\varpi }_{2j-1}\right)∆\left({\varpi }_{2j},{\varpi }_{2j+1}\right)+k_2\left({\varpi }_0,T_1{T}_2{\varpi }_{2j-1}\right)∆\left({\varpi }_{2j+1},{\varpi }_{2j+2}\right)\\ \le k_1\left({\varpi }_0,{\varpi }_{2j-1}\right)∆\left({\varpi }_{2j},{\varpi }_{2j+1}\right)+k_2\left({\varpi }_0,{\varpi }_{2j-1}\right)∆\left({\varpi }_{2j+1},{\varpi }_{2j+2}\right)\\ \le \dots \le k_1\left({\varpi }_0,{\varpi }_1\right)∆\left({\varpi }_{2j},{\varpi }_{2j+1}\right)+k_2\left({\varpi }_0,{\varpi }_1\right)∆\left({\varpi }_{2j+1},{\varpi }_{2j+2}\right).\end{array}$$

This implies that

$$∆\left({\varpi }_{2j+1},{\varpi }_{2j+2}\right)\le \left(\frac{k_1\left({\varpi }_0,{\varpi }_1\right)}{1-k_2\left({\varpi }_0,{\varpi }_1\right)}\right)∆\left({\varpi }_{2j},{\varpi }_{2j+1}\right).$$

Similarly,

$$∆\left({\varpi }_{2j+2},{\varpi }_{2j+3}\right)=∆\left(T_2{\varpi }_{2j+1},T_1{\varpi }_{2j+2}\right)=∆\left(T_1{\varpi }_{2j+2},T_2{\varpi }_{2j+1}\right)$$

$$\begin{array}{l}\le k_1\left({\varpi }_{2j+2},{\varpi }_{2j+1}\right)∆\left({\varpi }_{2j+2},{\varpi }_{2j+1}\right)\\ +k_2\left({\varpi }_{2j+2},{\varpi }_{2j+1}\right)\frac{∆\left({\varpi }_{2j+2},T_1{\varpi }_{2j+2}\right)∆\left({\varpi }_{2j+1},T_1{\varpi }_{2j+1}\right)}{1+∆\left({\varpi }_{2j+2},{\varpi }_{2j+1}\right)},\\ =k_1\left({\varpi }_{2j+2},T_1{T}_2{\varpi }_{2j-1}\right)∆\left({\varpi }_{2j+2},{\varpi }_{2j+1}\right)\\ +k_2\left({\varpi }_{2j+2},T_1{T}_2{\varpi }_{2j-1}\right)\frac{∆\left({\varpi }_{2j+2},{\varpi }_{2j+3}\right)∆\left({\varpi }_{2j+1},{\varpi }_{2j+2}\right)}{1+∆\left({\varpi }_{2j+2},{\varpi }_{2j+1}\right)}\\ \le k_1\left({\varpi }_{2j+2},{\varpi }_{2j-1}\right)∆\left({\varpi }_{2j+2},{\varpi }_{2j+1}\right)+k_2\left({\varpi }_{2j+2},{\varpi }_{2j-1}\right)∆\left({\varpi }_{2j+2},{\varpi }_{2j+3}\right),\\ =k_1\left({{\varpi }_{2j+2},T}_1{T}_2{\varpi }_{2j-3}\right)∆\left({\varpi }_{2j+2},{\varpi }_{2j+1}\right)+k_2\left({{\varpi }_{2j+2},T}_1{T}_2{\varpi }_{2j-3}\right)∆\left({\varpi }_{2j+2},{\varpi }_{2j+3}\right)\\ \le k_1\left({\varpi }_{2j+2},{\varpi }_{2j-3}\right)∆\left({\varpi }_{2j+2},{\varpi }_{2j+1}\right)+k_2\left({\varpi }_{2j+2},{\varpi }_{2j-3}\right)∆\left({\varpi }_{2j+2},{\varpi }_{2j+3}\right)\\ \le \dots \le k_1\left({{\varpi }_{2j+2},\varpi }_0\right)∆\left({\varpi }_{2j+2},{\varpi }_{2j+1}\right)+k_2\left({{\varpi }_{2j+2},\varpi }_0\right)∆\left({\varpi }_{2j+2},{\varpi }_{2j+3}\right),\\ =k_1\left(T_2{T}_1{\varpi }_{2j},{\varpi }_0\right)∆\left({\varpi }_{2j+2},{\varpi }_{2j+1}\right)+k_2\left(T_2{T}_1{\varpi }_{2j},{\varpi }_0\right)∆\left({\varpi }_{2j+2},{\varpi }_{2j+3}\right)\\ \le k_1\left({\varpi }_{2j},{\varpi }_0\right)∆\left({\varpi }_{2j+2},{\varpi }_{2j+1}\right)+k_2\left({\varpi }_{2j},{\varpi }_0\right)∆\left({\varpi }_{2j+2},{\varpi }_{2j+3}\right)\\ \le \dots \le k_1\left({\varpi }_1,{\varpi }_0\right)∆\left({\varpi }_{2j+2},{\varpi }_{2j+1}\right)+k_2\left({\varpi }_1,{\varpi }_0\right)∆\left({\varpi }_{2j+2},{\varpi }_{2j+3}\right)\\ =k_1\left({\varpi }_0,{\varpi }_1\right)∆\left({\varpi }_{2j+2},{\varpi }_{2j+1}\right)+k_2\left({\varpi }_0,{\varpi }_1\right)∆\left({\varpi }_{2j+2},{\varpi }_{2j+3}\right).\end{array}$$

This implies that

$$∆\left({\varpi }_{2j+2},{\varpi }_{2j+3}\right)\le \left(\frac{k_1\left({\varpi }_0,{\varpi }_1\right)}{1-k_2\left({\varpi }_0,{\varpi }_1\right)}\right)∆\left({\varpi }_{2j+1},{\varpi }_{2j+2}\right)=k∆\left({\varpi }_{2j+1},{\varpi }_{2j+2}\right).$$

By pursuing in this direction, we obtain

$$∆\left({\varpi }_j,{\varpi }_{j+1}\right)\le k∆\left({\varpi }_{j-1},{\varpi }_j\right)=k^2∆\left({\varpi }_{2j-2},{\varpi }_{j-1}\right)\le \dots \le k^j∆\left({\varpi }_0,{\varpi }_1\right).$$

Thus,

$$∆\left({\varpi }_j,{\varpi }_{j+1}\right)\le k^j∆\left({\varpi }_0,{\varpi }_1\right).$$

(3)

Now for  $j<m,$  for all  $j,m\in \mathbb{N},$  we deduce that

$$\begin{array}{l}∆\left({\varpi }_j,{\varpi }_m\right)\le \sigma \left({\varpi }_j,{\varpi }_{j+1}\right)∆\left({\varpi }_j,{\varpi }_{j+1}\right)+\gamma \left({\varpi }_{j+1},{\varpi }_m\right)∆\left({\varpi }_{j+1},{\varpi }_m\right)\\ \le \sigma \left({\varpi }_j,{\varpi }_{j+1}\right)∆\left({\varpi }_j,{\varpi }_{j+1}\right)+\gamma \left({\varpi }_{j+1},{\varpi }_m\right)\sigma \left({\varpi }_{j+1},{\varpi }_{j+2}\right)∆\left({\varpi }_{j+1},{\varpi }_{j+2}\right)\\ +\gamma ({\varpi }_{j+1},{\varpi }_m)\gamma \left({\varpi }_{j+2},{\varpi }_m\right)∆\left({\varpi }_{j+2},{\varpi }_m\right)\\ \le \sigma \left({\varpi }_j,{\varpi }_{j+1}\right)∆\left({\varpi }_j,{\varpi }_{j+1}\right)+\gamma \left({\varpi }_{j+1},{\varpi }_m\right)\sigma \left({\varpi }_{j+1},{\varpi }_{j+2}\right)∆\left({\varpi }_{j+1},{\varpi }_{j+2}\right)\\ +\gamma \left({\varpi }_{j+1},{\varpi }_m\right)\gamma \left({\varpi }_{j+2},{\varpi }_m\right)\sigma \left({\varpi }_{j+2},{\varpi }_{j+3}\right)∆\left({\varpi }_{j+2},{\varpi }_{j+3}\right)\\ +\gamma \left({\varpi }_{j+1},{\varpi }_m\right)\gamma \left({\varpi }_{j+2},{\varpi }_m\right)\gamma \left({\varpi }_{j+3},{\varpi }_m\right)∆\left({\varpi }_{j+3},{\varpi }_m\right)\\ \le \dots \le \sigma \left({\varpi }_j,{\varpi }_{j+1}\right)∆\left({\varpi }_j,{\varpi }_{j+1}\right)+{\displaystyle \sum _{i=j+1}^{m-2}}\left({\displaystyle \prod _{t=j+1}^i}\gamma \left({\varpi }_t,{\varpi }_m\right)\right)\sigma \left({\varpi }_j,{\varpi }_{j+1}\right)∆\left({\varpi }_j,{\varpi }_{j+1}\right)\\ +{\displaystyle \prod _{i=j+1}^{m-1}}\gamma \left({\varpi }_i,{\varpi }_m\right)∆\left({\varpi }_{m1},{\varpi }_m\right).\end{array}$$

This further implies that

$$\begin{array}{l}∆\left({\varpi }_j,{\varpi }_m\right)\le \sigma \left({\varpi }_j,{\varpi }_{j+1}\right)∆\left({\varpi }_j,{\varpi }_{j+1}\right)+{\displaystyle \sum _{i=j+1}^{m-2}}\left({\displaystyle \prod _{t=j+1}^i}\gamma \left({\varpi }_t,{\varpi }_m\right)\right)\sigma \left({\varpi }_j,{\varpi }_{j+1}\right)∆\left({\varpi }_j,{\varpi }_{j+1}\right)\\ +\left({\displaystyle \prod _{i=j+1}^{m-1}}\gamma \left({\varpi }_i,{\varpi }_m\right)\right)∆\left({\varpi }_{m-1},{\varpi }_m\right).\\ \le \sigma \left({\varpi }_j,{\varpi }_{j+1}\right)k^j∆\left({\varpi }_0,{\varpi }_1\right)+{\displaystyle \sum _{i=j+1}^{m-2}}\left({\displaystyle \prod _{t=j+1}^i}\gamma \left({\varpi }_t,{\varpi }_m\right)\right)\sigma \left({\varpi }_i,{\varpi }_{i+1}\right)k^i∆\left({\varpi }_0,{\varpi }_1\right)\\ +\left({\displaystyle \prod _{i=j+1}^{m-1}}\gamma \left({\varpi }_i,{\varpi }_m\right)\right)k^{m-1}∆\left({\varpi }_0,{\varpi }_1\right).\\ =\sigma \left({\varpi }_j,{\varpi }_{j+1}\right)k^j∆\left({\varpi }_0,{\varpi }_1\right)+{\displaystyle \sum _{i=j+1}^{m-1}}\left({\displaystyle \prod _{t=j+1}^i}\gamma \left({\varpi }_t,{\varpi }_m\right)\right)\sigma \left({\varpi }_i,{\varpi }_{i+1}\right)k^i∆\left({\varpi }_0,{\varpi }_1\right).\end{array}$$

Thus

$$∆\left({\varpi }_j,{\varpi }_m\right)\le \sigma \left({\varpi }_j,{\varpi }_{j+1}\right)k^j∆\left({\varpi }_0,{\varpi }_1\right)+\sum _{i=j+1}^{m-1}\left(\prod _{t=j+1}^i\gamma \left({\varpi }_t,{\varpi }_m\right)\right)\sigma \left({\varpi }_i,{\varpi }_{i+1}\right)k^i∆\left({\varpi }_0,{\varpi }_1\right).$$

(4)

Let

$${\psi }_{∆}=\sum _{i=j+1}^{m-1}\left(\prod _{t=j+1}^i\gamma \left({\varpi }_t,{\varpi }_m\right)\right)\sigma \left({\varpi }_i,{\varpi }_{i+1}\right)k^i∆\left({\varpi }_0,{\varpi }_1\right).$$

From (4), we obtain

$$∆\left({\varpi }_j,{\varpi }_m\right)\le ∆\left({\varpi }_0,{\varpi }_1\right)[\sigma \left({\varpi }_j,{\varpi }_{j+1}\right)k^j+{(\psi }_{m-1}-{\psi }_j)].$$

(5)

Since  $\sigma \left(\varpi ,\varkappa \right)\ge 1,$  and by employing the ratio test  $\underset{j\to \infty }{\lim }{\psi }_j$  exists. Thus  $\left\{{\psi }_j\right\}$  is a Cauchy sequence.

Therefore, taking  $j,m\to +\infty$  in inequality (5), we obtain

$$\underset{j,m\to \infty }{\lim }∆\left({\varpi }_j,{\varpi }_m\right)=0.$$

(6)

Hence,  $\left\{{\varpi }_j\right\}$  is a Cauchy sequence in DCMS. As  $\left(\Xi ,\mathsf{\Delta },\sigma ,\gamma \right)$  is complete, so there exists  ${\varpi }^{*}\in \Xi$  such that

$$\underset{j\to +\infty }{\lim }∆\left({\varpi }_j,{\varpi }^{*}\right)=0.$$

(7)

Hence,  ${\varpi }_j\to {\varpi }^{*}$  as  $j\to +\infty .$  From conditions (III) and (IV), we deduce that

$$\begin{array}{l}∆\left({\varpi }^{*},T_1{\varpi }^{*}\right)\le \sigma \left({\varpi }^{*},{\varpi }_{2j+2}\right)∆\left({\varpi }^{*},{\varpi }_{2j+2}\right)+\gamma \left({\varpi }_{2j+2},T_1{\varpi }^{*}\right)∆\left({\varpi }_{2j+2},T_1{\varpi }^{*}\right)\\ =\sigma \left({\varpi }^{*},{\varpi }_{2j+2}\right)∆\left({\varpi }^{*},{\varpi }_{2j+2}\right)+\gamma \left({\varpi }_{2j+2},T_1{\varpi }^{*}\right)∆\left(T_2{\varpi }_{2j+1},T_1{\varpi }^{*}\right)\\ =\sigma \left({\varpi }^{*},{\varpi }_{2j+2}\right)∆\left({\varpi }^{*},{\varpi }_{2j+2}\right)+\gamma \left({\varpi }_{2j+2},T_1{\varpi }^{*}\right)∆\left(T_1{\varpi }^{*},T_2{\varpi }_{2j+1}\right)\\ =\sigma \left({\varpi }^{*},{\varpi }_{2j+2}\right)∆\left({\varpi }^{*},{\varpi }_{2j+2}\right)\\ +\gamma \left({\varpi }_{2j+2},T_1{\varpi }^{*}\right)\left[\begin{array}{c}{k}_1\left({\varpi }^{*},{\varpi }_{2j+1}\right)∆\left({\varpi }^{*},{\varpi }_{2j+1}\right)\\ +k_2\left({\varpi }^{*},{\varpi }_{2j+1}\right)\frac{∆\left({\varpi }^{*},T_1{\varpi }_{2j+2}\right)∆\left({\varpi }_{2j+1},{\varpi }_{2j+2}\right)}{1+∆\left({\varpi }^{*},{\varpi }_{2j+1}\right)}\end{array}\right]\\ =\sigma \left({\varpi }^{*},{\varpi }_{2j+2}\right)∆\left({\varpi }^{*},{\varpi }_{2j+2}\right)\\ +\gamma \left({\varpi }_{2j+2},T_1{\varpi }^{*}\right)\left[\begin{array}{c}{k}_1\left({\varpi }^{*},{\varpi }_{2j+1}\right)∆\left({\varpi }^{*},{\varpi }_{2j+1}\right)\\ +k_2\left({\varpi }^{*},{\varpi }_{2j+1}\right)\frac{∆\left({\varpi }^{*},T_1{\varpi }^{*}\right)∆\left({\varpi }_{2j+1},{\varpi }_{2j+2}\right)}{1+∆\left({\varpi }^{*},{\varpi }_{2j+1}\right)}\end{array}\right].\end{array}$$

Letting  $j\to +\infty$  and applying Equation (7), which contradicts  $∆\left({\varpi }^{*},T_1{\varpi }^{*}\right)>0.$  That is,  $∆\left({\varpi }^{*},T_1{\varpi }^{*}\right)=0.$  We have  ${\varpi }^{*}=T_1{\varpi }^{*}.$  On the same lines, we can examine that  ${\varpi }^{*}=T_2{\varpi }^{*}.$  Hence,  $T_1$  and  $T_2$  has a common fixed point  ${\varpi }^{*}.$ 

Now, we examine the uniqueness of the fixed point  ${\varpi }^{*}.$  Let another fixed point  ${\varpi }^{’}\in \Xi$  , which is different from  ${\varpi }^{*}$  such that  ${\varpi }^{’}=T_1{\varpi }^{’}=T_2{\varpi }^{’}.$  We have

$$\begin{gathered} ∆\left({\varpi }^{*},{\varpi }^{’}\right)=∆\left(T_1{\varpi }^{*},T_2{\varpi }^{’}\right)\le k_1\left({\varpi }^{*},{\varpi }^{’}\right)∆\left({\varpi }^{*},{\varpi }^{’}\right)+k_2\left({\varpi }^{*},{\varpi }^{’}\right)\frac{∆\left({\varpi }^{*},{T_1\varpi }^{’}\right)∆\left({\varpi }^{’},{T_2\varpi }^{’}\right)}{1+∆\left({\varpi }^{*},{\varpi }^{’}\right)} \\ =k_1\left({\varpi }^{*},{\varpi }^{’}\right)∆\left({\varpi }^{*},{\varpi }^{’}\right). \end{gathered}$$

Since  $k_1\left({\varpi }^{\mathrm{*}},{\varpi }^{\mathrm{’}}\right)\in \left[0,1\right),$  we have  $∆\left({\varpi }^{\mathrm{*}},{\varpi }^{\mathrm{’}}\right)=0$  . Thus, we obtain  ${\varpi }^{\mathrm{*}}={\varpi }^{\mathrm{’}},$  which shows that  ${\varpi }^{\mathrm{*}}$  is unique. □

Example 1.

Let  $\Xi =\left[0,1\right].$  Now we define  $∆:\Xi \times \Xi \to [0,\infty )$  by

$$∆\left(\varpi ,\varkappa \right)={\left(\varpi +\varkappa \right)}^2,$$

where  $\sigma \left(\varpi ,\varkappa \right)=\varpi +\varkappa +2$  and  $\gamma \left(\varpi ,\varkappa \right)={\varpi }^2+{\varkappa }^2+1$  for all  $\varpi ,\varkappa \in \Xi .$  Now, we define  $T_1,T_2:\Xi \to \Xi$  by  $T_1\varpi =\frac{\varpi }{3},$  and  $T_2\varpi =\frac{\varpi }{4},$  for  $\varpi \in \mathbb{R}.$  Choose  $k_1,k_2:\Xi \times \Xi \to [0,1)$  by

$$k_1\left(\varpi ,\varkappa \right)=\frac{16+\varpi +\varkappa }{144}and{k}_2\left(\varpi ,\varkappa \right)=\frac{15+\varpi +\varkappa }{144}.$$

Then, evidently,

$$k_1\left(\varpi ,\varkappa \right)+k_2\left(\varpi ,\varkappa \right)<1.$$

Now,

$$k_1\left(T_2{T}_1\varpi ,\varkappa \right)=\frac{1}{9}+\frac{\varpi }{1726}+\frac{\varkappa }{144}\le \frac{16+\varpi +\varkappa }{144}=k_1\left(\varpi ,\varkappa \right),$$

and

$$k_1\left(\varpi ,T_2{T}_1\varkappa \right)=\frac{1}{9}+\frac{\varpi }{144}+\frac{\varkappa }{1726}\le \frac{16+\varpi +\varkappa }{144}=k_1\left(\varpi ,\varkappa \right),$$

and

$$\begin{gathered} k_2\left(T_2{T}_1\varpi ,\varkappa \right)=\frac{5}{46}+\frac{\varpi }{1726}+\frac{\varkappa }{144}\le \frac{15+\varpi +\varkappa }{144}=k_1\left(\varpi ,\varkappa \right), \\ {k}_2\left(\varpi ,T_2{T}_1\varkappa \right)=\frac{5}{46}+\frac{\varpi }{144}+\frac{\varkappa }{1726}\le \frac{15+\varpi +\varkappa }{144}=k_1\left(\varpi ,T_2{T}_1\varkappa \right). \end{gathered}$$

Take  ${\varpi }_0=0,$  so (2) is satisfied. Let  $\varpi ,\varkappa \in \Xi$  . Then,

$$\begin{gathered} ∆\left(T_1\varpi ,T_2\varkappa \right)=\frac{{\left(4\varpi +3\varkappa \right)}^2}{144}\le \frac{{\left(4\varpi +4\varkappa \right)}^2}{144} \\ \le \frac{16+\varpi +\varkappa }{144}{\left(\varpi +\varkappa \right)}^2+\frac{15+\varpi +\varkappa }{144}\frac{{\left(\frac{5\varpi }{4}\right)}^2{\left(\frac{6\varkappa }{4}\right)}^2}{1+{\left(\varpi +\varkappa \right)}^2}, \\ {k}_1\left(\varpi ,\varkappa \right)∆\left(\varpi ,\varkappa \right)+k_2\left(\varpi ,\varkappa \right)\frac{∆\left(\varpi ,T_1\varpi \right)∆(\varkappa ,T_2\varkappa )}{1+∆(\varpi ,\varkappa )}. \end{gathered}$$

Hence, Theorem 1 is fulfilled and  ${\varpi }^{*}=0\in \Xi$  is a fixed point, that is,  $T_1{\varpi }^{*}=T_2{\varpi }^{*}={\varpi }^{*}.$ 

By setting  $T_1=T_2=T$  in Theorem 1, we have the following corollary.

Corollary 1.

Suppose  $\left(\Xi ,\Delta ,\sigma ,\gamma \right)$  is a complete DCMS,  $T:\Xi \to \Xi$  and there exists  $k_1,k_2:\Xi \times \Xi \to [0,1)$  such that

(I)

 $k_1\left(T\varpi ,\varkappa \right)\le k_1(\varpi ,\varkappa )$  and  $k_1\left(\varpi ,T\varkappa \right)\le k_1\left(\varpi ,\varkappa \right),$ 

(II)

 $k_2\left(T\varpi ,\varkappa \right)\le k_2(\varpi ,\varkappa )$  and  $k_2\left(\varpi ,T\varkappa \right)\le k_2\left(\varpi ,\varkappa \right),$ 

(III)

 $k_1\left(\varpi ,\varkappa \right)+k_2\left(\varpi ,\varkappa \right)<1,$ 

(IV)

 $∆\left(T\varpi ,T\varkappa \right)\le k_1\left(\varpi ,\varkappa \right)∆\left(\varpi ,\varkappa \right)+k_2\left(\varpi ,\varkappa \right)\frac{∆\left(\varpi ,T\varpi \right)∆\left(\varpi ,T\varpi \right)}{1+∆\left(\varpi ,\varkappa \right)},forall\varpi ,\varkappa \in \Xi ,$ 

for  ${\varpi }_0\in \Xi ,$  a sequence  ${\left\{{\varpi }_j\right\}}_{j\ge 0}$  is defined as  ${\varpi }_{j+1}=T{\varpi }_j$  for each  $j\ge 0.$  Assume that

$$\underset{m\ge 1}{\mathit{sup}}\underset{i\to \infty }{\mathit{lim}}\frac{\sigma \left({\varpi }_{i+1},{\varpi }_{i+2}\right)\gamma \left({\varpi }_{i+1},{\varpi }_m\right)}{\sigma \left({\varpi }_i,{\varpi }_{i+1}\right)}<\frac{1}{k},$$

where  $\frac{k_1({\varpi }_0,{\varpi }_1)}{1-k_2({\varpi }_0,{\varpi }_1)}=k.$  Moreover, assume that  $\underset{j\to +\infty }{\mathit{lim}}\sigma \left({\varpi }_j,\varpi \right),\underset{j\to +\infty }{\mathit{lim}}\sigma \left(\varpi ,{\varpi }_j\right),$   $\underset{j\to +\infty }{\mathit{lim}}\gamma \left({\varpi }_j,\varpi \right)$  and  $\underset{j\to +\infty }{\mathit{lim}}\gamma \left(\varpi ,{\varpi }_j\right)$  exist and are finite, then there exists a unique point  ${\varpi }^{*}\in \Xi$  such that  $T{\varpi }^{*}={\varpi }^{*}.$ 

4. Deduced Results

Theorem 2.

Suppose  $\left(\Xi ,\Delta ,\sigma ,\gamma \right)$  is a complete DCMS,  $T_1,T_2:\Xi \to \Xi$  and there exists  ${\tau }_1,{\tau }_2:\Xi \to [0,1)$  such that

(I)

 ${\tau }_1\left(T_1\varpi ,\varkappa \right)\le {\tau }_1(\varpi ,\varkappa )$  and  ${\tau }_2\left(T_1\varpi \right)\le {\tau }_1\left(\varpi \right),$ 

(II)

 ${\tau }_1\left(T_2\varpi \right)\le {\tau }_1\left(\varpi \right)$  and  ${\tau }_2\left(T_2\varpi \right)\le {\tau }_2\left(\varpi \right),$ 

(III)

 $\left(k_1+k_2\right)\varpi <1,$ 

(IV)

 $∆\left(T_1\varpi ,T_2\varkappa \right)\le {\tau }_1\left(\varpi ,\varkappa \right)∆\left(\varpi ,\varkappa \right)+{\tau }_2\left(\varpi ,\varkappa \right)\frac{∆\left(\varpi ,T_1\varpi \right)∆\left(\varkappa ,T_2\varkappa \right)}{1+∆\left(\varpi ,\varkappa \right)},forall\varpi ,\varkappa \in \Xi ,$ 

for  ${\varpi }_0\in \Xi ,$  a sequence  ${\left\{{\varpi }_j\right\}}_{j\ge 0}$  is defined as  ${\varpi }_{2j+1}=T_1{\varpi }_{2j}$  and  ${\varpi }_{2j+2}=T_2{\varpi }_{2j+1}$  for every  $j\ge 0.$  Let

$$\underset{m\ge 1}{\mathit{sup}}\underset{i\to \infty }{\mathit{lim}}\frac{\sigma \left({\varpi }_{i+1},{\varpi }_{i+2}\right)\gamma \left({\varpi }_{i+1},{\varpi }_m\right)}{\sigma \left({\varpi }_i,{\varpi }_{i+1}\right)}<\frac{1}{\tau },$$

where  $\frac{{\tau }_1({\varpi }_0,)}{1-{\tau }_2\left({\varpi }_0\right)}=\tau .$  Furthermore, let  $\underset{j\to +\infty }{\mathit{lim}}\sigma \left({\varpi }_j,\varpi \right),\underset{j\to +\infty }{\mathit{lim}}\sigma \left(\varpi ,{\varpi }_j\right),$   $\underset{j\to +\infty }{\mathit{lim}}\gamma \left({\varpi }_j,\varpi \right)$  and  $\underset{j\to +\infty }{\mathit{lim}}\gamma \left(\varpi ,{\varpi }_j\right)$  exist and are finite, then there exists a unique point  ${\varpi }^{*}\in \Xi$  such that  $T_1{\varpi }^{*}=T_2{\varpi }^{*}={\varpi }^{*}.$ 

Proof. 

Define  $k_1,k_2:\Xi \times \Xi \to [0,1)$  by  ${\tau }_1\left(\varpi ,\varkappa \right)\le {\tau }_1\left(\varpi \right)$  and  $k_2\left(\varpi ,\varkappa \right)\le {\tau }_2\left(\varpi \right),\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}\varpi ,\varkappa \in \Xi .$ 

(i)

 $k_1\left(T_2{T}_1\varpi ,\varkappa \right)={\tau }_1\left(T_2{T}_1\varpi \right)\le {\tau }_1\left(T_1\varpi \right)\le {\tau }_1\left(\varpi \right)=k_1(\varpi ,\varkappa )$  and  $k_1\left(\varpi ,T_1{T}_2\varkappa \right)={\tau }_1\left(\varpi \right)={\tau }_1\left(\varpi ,\varkappa \right),$ 

(ii)

 $k_1\left(T_2{T}_1\varpi ,\varkappa \right)={\tau }_2\left(T_2{T}_1\varpi \right)\le {\tau }_2\left(T_1\varpi \right)\le {\tau }_2\left(\varpi \right)=k_1(\varpi ,\varkappa )$  and  $k_2\left(\varpi ,T_1{T}_2\varkappa \right)={\tau }_2\left(\varpi \right)={\tau }_{12}\left(\varpi ,\varkappa \right),$ 

(iii)

 $k_1\left(\varpi ,\varkappa \right)+k_2\left(\varpi ,\varkappa \right)={\tau }_1\left(\varpi \right)+{\tau }_2\left(\varpi \right)<1,$ 

(iv)

 $∆\left(T_1\varpi ,T_2\varkappa \right)\le {\tau }_1\left(\varpi \right)∆\left(\varpi ,\varkappa \right)+{\tau }_2\left(\varpi \right)\frac{∆\left(\varpi ,T_1\varpi \right)∆\left(\varkappa ,T_2\varkappa \right)}{1+∆\left(\varpi ,\varkappa \right)}$ 

$$=k_1\left(\varpi ,\varkappa \right)∆\left(\varpi ,\varkappa \right)+k_2\left(\varpi ,\varkappa \right)\frac{∆\left(\varpi ,T_1\varpi \right)∆\left(\varkappa ,T_2\varkappa \right)}{1+∆\left(\varpi ,\varkappa \right)}.$$

By Theorem 2,  $T_1\mathrm{a}\mathrm{n}\mathrm{d}{T}_2$  have a unique common fixed point. □

Corollary 2.

Suppose  $\left(\Xi ,\Delta ,\sigma ,\gamma \right)$  be a complete DCMS,  $T:\Xi \to \Xi$  and there exists  ${\tau }_1,{\tau }_2:\Xi \to [0,1)$  such that

(I)

 ${\tau }_1\left(T\varpi ,\varkappa \right)\le {\tau }_1(\varpi ,\varkappa )$  and  ${\tau }_2\left(T\varpi \right)\le {\tau }_1\left(\varpi \right),$ 

(II)

 $\left({\tau }_1+{\tau }_2\right)\varpi <1,$ 

(III)

 $∆\left(T\varpi ,T\varkappa \right)\le {\tau }_1\left(\varpi ,\varkappa \right)∆\left(\varpi ,\varkappa \right)+{\tau }_2\left(\varpi ,\varkappa \right)\frac{∆\left(\varpi ,T\varpi \right)∆\left(\varkappa ,T\varkappa \right)}{1+∆\left(\varpi ,\varkappa \right)},forall\varpi ,\varkappa \in \Xi ,\left(2\right)$ 

for  ${\varpi }_0\in \Xi ,$  we set  $\frac{{\tau }_1({\varpi }_0,)}{1-{\tau }_2\left({\varpi }_0\right)}=\tau .$  Assume that

$$\underset{m\ge 1}{\mathit{sup}}\underset{i\to \infty }{\mathit{lim}}\frac{\sigma \left({\varpi }_{i+1},{\varpi }_{i+2}\right)\gamma \left({\varpi }_{i+1},{\varpi }_m\right)}{\sigma \left({\varpi }_i,{\varpi }_{i+1}\right)}<\frac{1}{\tau },$$

where  ${\varpi }_{j+1}=T{\varpi }_j$  for each  $j\ge 0.$  Moreover, assume that  $\underset{j\to +\infty }{\mathit{lim}}\sigma \left({\varpi }_j,\varpi \right),\underset{j\to +\infty }{\mathit{lim}}\sigma \left(\varpi ,{\varpi }_j\right),$   $\underset{j\to +\infty }{\mathit{lim}}\gamma \left({\varpi }_j,\varpi \right)$  and  $\underset{j\to +\infty }{\mathit{lim}}\gamma \left(\varpi ,{\varpi }_j\right)$  exist and are finite, then there exists a unique point  ${\varpi }^{*}\in \Xi$  such that  $T{\varpi }^{*}={\varpi }^{*}.$ 

Proof. 

Immediate by considering  $T_1=T_2=T$  in the Theorem 2. □

Theorem 3.

Suppose  $\left(\Xi ,\Delta ,\sigma ,\gamma \right)$  is a complete DCMS,  $T:\Xi \to \Xi$  and there exists  ${\tau }_1,{\tau }_2:\Xi \to [0,1)$  such that

(I)

 ${\tau }_1\left(T^j\varpi ,\varkappa \right)\le {\tau }_1(\varpi ,\varkappa )$  and  ${\tau }_2\left(T^j\varpi \right)\le {\tau }_1\left(\varpi \right),$ 

(II)

 $\left({\tau }_1+{\tau }_2\right)\varpi <1,$ 

(III)

 $∆\left(T^j\varpi ,T^j\varkappa \right)\le {\tau }_1\left(\varpi ,\varkappa \right)∆\left(\varpi ,\varkappa \right)+{\tau }_2\left(\varpi ,\varkappa \right)\frac{∆\left(\varpi ,T^j\varpi \right)∆\left(\varkappa ,T^j\varkappa \right)}{1+∆\left(\varpi ,\varkappa \right)},forall\varpi ,\varkappa \in \Xi ,\left(3\right)$ 

for  ${\varpi }_0\in \Xi ,$  we set  $\frac{{\tau }_1({\varpi }_0,)}{1-{\tau }_2\left({\varpi }_0\right)}=\tau .$  Assume that

$$\underset{m\ge 1}{\mathit{sup}}\underset{i\to \infty }{\mathit{lim}}\frac{\sigma \left({\varpi }_{i+1},{\varpi }_{i+2}\right)\gamma \left({\varpi }_{i+1},{\varpi }_m\right)}{\sigma \left({\varpi }_i,{\varpi }_{i+1}\right)}<\frac{1}{\tau },$$

where  ${\varpi }_{j+1}=T{\varpi }_j$  for all  $j\ge 0.$  Furthermore, let  $\underset{j\to +\infty }{\mathit{lim}}\sigma \left({\varpi }_j,\varpi \right),\underset{j\to +\infty }{\mathit{lim}}\sigma \left(\varpi ,{\varpi }_j\right),$   $\underset{j\to +\infty }{\mathit{lim}}\gamma \left({\varpi }_j,\varpi \right)$  and  $\underset{j\to +\infty }{\mathit{lim}}\gamma \left(\varpi ,{\varpi }_j\right)$  exist and are finite, then there exists a unique point  ${\varpi }^{*}\in \Xi$  such that  $T{\varpi }^{*}={\varpi }^{*}.$ 

Proof. 

Using Theorem 1, we obtain  $T^j{\varpi }^{*}={\varpi }^{*}.$  Now, as

$${T^j(T\varpi }^{*})={T(T^j\varpi }^{*})=T{\varpi }^{*}.$$

So,  $T^j$  has a fixed point  $T{\varpi }^{*}.$  Hence,  $T{\varpi }^{*}={\varpi }^{*}.$  Meanwhile  $T^j$  has a unique fixed point, so  $T$  has a fixed point  ${\varpi }^{*}.$  □

Corollary 3.

Suppose  $\left(\Xi ,\Delta ,\sigma ,\gamma \right)$  is a complete DCMS,  $T_1,T_2:\Xi \to \Xi$  and there exists  ${\gamma }_1,{\gamma }_2\in [0,1)$  with  ${\gamma }_1+{\gamma }_2<1$  such that

$$∆\left(T_1\varpi ,T_2\varkappa \right)\le {\gamma }_1∆\left(\varpi ,\varkappa \right)+{\gamma }_2\frac{∆\left(\varpi ,T_1\varpi \right)∆\left(\varkappa ,T_2\varkappa \right)}{1+∆\left(\varpi ,\varkappa \right)},forall\varpi ,\varkappa \in \Xi ,$$

for  ${\varpi }_0\in \Xi ,$  a sequence  ${\left\{{\varpi }_j\right\}}_{j\ge 0}$  is generated as  ${\varpi }_{2j+1}=T_1{\varpi }_{2j}$  and  ${\varpi }_{2j+2}=T_2{\varpi }_{2j+1}$  for every  $j\ge 0.$  Let

$$\underset{m\ge 1}{\mathit{sup}}\underset{i\to \infty }{\mathit{lim}}\frac{\sigma \left({\varpi }_{i+1},{\varpi }_{i+2}\right)\gamma \left({\varpi }_{i+1},{\varpi }_m\right)}{\sigma \left({\varpi }_i,{\varpi }_{i+1}\right)}<\frac{1}{\Omega },$$

where  $\frac{{\gamma }_1}{1-{\gamma }_2}=\Omega .$  Moreover, assume that  $\underset{j\to +\infty }{\mathit{lim}}\sigma \left({\varpi }_j,\varpi \right),\underset{j\to +\infty }{\mathit{lim}}\sigma \left(\varpi ,{\varpi }_j\right),$   $\underset{j\to +\infty }{\mathit{lim}}\gamma \left({\varpi }_j,\varpi \right)$  and  $\underset{j\to +\infty }{\mathit{lim}}\gamma \left(\varpi ,{\varpi }_j\right)$  exist and are finite, then there exists a unique point  ${\varpi }^{*}\in \Xi$  such that  $T_1{\varpi }^{*}=T_2{\varpi }^{*}={\varpi }^{*}.$ 

Proof. 

Immediate by letting  ${\gamma }_1\left(.\right)={\gamma }_1$  and  ${\gamma }_2\left(.\right)={\gamma }_2$  in Theorem 2. □

Corollary 4.

Suppose  $\left(\Xi ,\Delta ,\sigma ,\gamma \right)$  is a complete DCMS,  $T:\Xi \to \Xi$  and there exist  ${\gamma }_1,{\gamma }_2\in [0,1)$  with  ${\gamma }_1+{\gamma }_2<1$  such that

$$∆\left(T\varpi ,T\varkappa \right)\le {\gamma }_1∆\left(\varpi ,\varkappa \right)+{\gamma }_2\frac{∆\left(\varpi ,T\varpi \right)∆\left(\varkappa ,T\varkappa \right)}{1+∆\left(\varpi ,\varkappa \right)},forall\varpi ,\varkappa \in \Xi ,$$

for  ${\varpi }_0\in \Xi ,$  we set  $\frac{{\gamma }_1}{1-{\gamma }_2}=\Omega .$  Suppose that

$$\underset{m\ge 1}{\mathit{sup}}\underset{i\to \infty }{\mathit{lim}}\frac{\sigma \left({\varpi }_{i+1},{\varpi }_{i+2}\right)\gamma \left({\varpi }_{i+1},{\varpi }_m\right)}{\sigma \left({\varpi }_i,{\varpi }_{i+1}\right)}<\frac{1}{\Omega },$$

where  ${\varpi }_{j+1}=T{\varpi }_j$  for all  $j\ge 0.$  Moreover, assume that  $\underset{j\to +\infty }{\mathit{lim}}\sigma \left({\varpi }_j,\varpi \right),\underset{j\to +\infty }{\mathit{lim}}\sigma \left(\varpi ,{\varpi }_j\right),$   $\underset{j\to +\infty }{\mathit{lim}}\gamma \left({\varpi }_j,\varpi \right)$  and  $\underset{j\to +\infty }{\mathit{lim}}\gamma \left(\varpi ,{\varpi }_j\right)$  exist and are finite, then there exists a unique point  ${\varpi }^{*}\in \Xi$  such that  $T{\varpi }^{*}={\varpi }^{*}.$ 

Corollary 5.

Suppose  $\left(\Xi ,\Delta ,\sigma ,\gamma \right)$  is a complete DCMS,  $T:\Xi \to \Xi$  and there exists  ${\gamma }_1,{\gamma }_2\in [0,1)$  with  ${\gamma }_1+{\gamma }_2<1$  such that

$$∆\left(T^j\varpi ,T^j\varkappa \right)\le {\gamma }_1∆\left(\varpi ,\varkappa \right)+{\gamma }_2\frac{∆\left(\varpi ,T^j\varpi \right)∆\left(\varkappa ,T^j\varkappa \right)}{1+∆\left(\varpi ,\varkappa \right)},forall\varpi ,\varkappa \in \Xi ,$$

for  ${\varpi }_0\in \Xi ,$  we set  $\frac{{\gamma }_1}{1-{\gamma }_2}=\Omega .$  Assume that

$$\underset{m\ge 1}{\mathit{sup}}\underset{i\to \infty }{\mathit{lim}}\frac{\sigma \left({\varpi }_{i+1},{\varpi }_{i+2}\right)\gamma \left({\varpi }_{i+1},{\varpi }_m\right)}{\sigma \left({\varpi }_i,{\varpi }_{i+1}\right)}<\frac{1}{\Omega },$$

where  ${\varpi }_{j+1}=T{\varpi }_j$  for all  $j\ge 0$  Moreover, assume that  $\underset{j\to +\infty }{\mathit{lim}}\sigma \left({\varpi }_j,\varpi \right),\underset{j\to +\infty }{\mathit{lim}}\sigma \left(\varpi ,{\varpi }_j\right),$   $\underset{j\to +\infty }{\mathit{lim}}\gamma \left({\varpi }_j,\varpi \right)$  and  $\underset{j\to +\infty }{\mathit{lim}}\gamma \left(\varpi ,{\varpi }_j\right)$  exist and are finite, then there exists a unique point  ${\varpi }^{*}\in \Xi$  such that  $T{\varpi }^{*}={\varpi }^{*}.$ 

Corollary 6.

Suppose  $\left(\Xi ,\Delta ,\sigma ,\gamma \right)$  is a complete DCMS,  $T:\Xi \to \Xi$  and there exist  ${\gamma }_1,{\gamma }_2\in [0,1)$  with  ${\gamma }_1+{\gamma }_2<1$  such that

$$∆\left(T^j\varpi ,T^j\varkappa \right)\le {\gamma }_1∆\left(\varpi ,\varkappa \right),forall\varpi ,\varkappa \in \Xi ,$$

for  ${\varpi }_0\in \Xi ,$  we set  $\frac{{\gamma }_1}{1-{\gamma }_1}=\Omega .$  Suppose that

$$\underset{m\ge 1}{\mathit{sup}}\underset{i\to \infty }{\mathit{lim}}\frac{\sigma \left({\varpi }_{i+1},{\varpi }_{i+2}\right)\gamma \left({\varpi }_{i+1},{\varpi }_m\right)}{\sigma \left({\varpi }_i,{\varpi }_{i+1}\right)}<\frac{1}{\Omega },$$

where  ${\varpi }_{j+1}=T{\varpi }_j$  for all  $j\ge 0$  Moreover, assume that  $\underset{j\to +\infty }{\mathit{lim}}\sigma \left({\varpi }_j,\varpi \right),\underset{j\to +\infty }{\mathit{lim}}\sigma \left(\varpi ,{\varpi }_j\right),$   $\underset{j\to +\infty }{\mathit{lim}}\gamma \left({\varpi }_j,\varpi \right)$  and  $\underset{j\to +\infty }{\mathit{lim}}\gamma \left(\varpi ,{\varpi }_j\right)$  exist and are finite, then there exists a unique point  ${\varpi }^{*}\in \Xi$  such that  $T{\varpi }^{*}={\varpi }^{*}.$ 

5. Application in Graphs

Suppose $\left(\Xi ,\mathsf{\Delta },\sigma ,\gamma \right)$  is a complete DCMS and a directed graph  $G$ . Assume  $G^{-1}$  is a graph that we obtain from  $G$  by altering the direction of  $E\left(G\right).$  Thus,

$$E\left(G^{-1}\right)=\left\{\left(\varpi ,\varkappa \right)\in \Xi \times \Xi :\left(\varkappa ,\varpi \right)\in E\left(G\right)\right\}.$$

Definition 6.

An arbitrary point  $\varpi \in \Xi$  is a common fixed point of  $\left(T_1,T_2\right),$  if  $T_1\left(\varpi \right)=T_2\left(\varpi \right)=\varpi .$  By  $CFix\left(T_1,T_2\right),$  we represent the set of all common fixed point of  $\left(T_1,T_2\right),$  i.e.,

$$CFix\left(T_1,T_2\right)=\left\{\varpi \in \Xi :T_1\left(\varpi \right)=T_2\left(\varpi \right)=\varpi \right\}.$$

Definition 7.

Suppose that  $T_1,T_2:\Xi \to \Xi$  are two mappings on complete DCMS  $\left(\Xi ,\Delta ,\sigma ,\gamma \right)$  equipped with a directed graph  $G$  . For any  $\varpi \in \Xi ,$   $\left(T_1,T_2\right)$  is called a  $G-$Orbital cyclic pair, if

$$\begin{gathered} \left(\varpi ,T_1\varpi \right)\in E\left(G\right)\Rightarrow \left(T_1\varpi ,T_2\left(T_1\varpi \right)\right)\in E\left(G\right), \\ \left(\varpi ,T_2\varpi \right)\in E\left(G\right)\Rightarrow \left(T_2\varpi ,T_1\left(T_1\varpi \right)\right)\in E\left(G\right). \end{gathered}$$

Consider the below sets

$$\begin{gathered} {\Xi }^{T_1}=\left\{\varpi \in \Xi :\left(\varpi ,T_1\varpi \right)\in E\left(G\right)\right\}, \\ {\Xi }^{T_2}=\left\{\varpi \in \Xi :\left(\varpi ,T_2\varpi \right)\in E\left(G\right)\right\}. \end{gathered}$$

Remark 1.

If the pair  $\left(T_1,T_2\right)$  is a  $G-$Orbital cyclic pair, then  ${\Xi }^{T_1}\ne \varphi ifandonlyif{\Xi }^{T_2}\ne \varphi .$ 

Proof. 

Let  ${\varpi }_0\in {\Xi }^{T_1}.$  Then,  $\left({\varpi }_0,T_1{\varpi }_0\right)\in E\left(G\right)\Rightarrow \left(T_1{\varpi }_0,T_2\left(T_1{\varpi }_0\right)\right)\in E\left(G\right).$  If  ${\varpi }_1=T_1{\varpi }_0,$  then we obtain  $\left({\varpi }_1,T_2{\varpi }_1\right)\in E\left(G\right),$  thus  ${\Xi }^{T_2}\ne \varphi .$  □

Theorem 4.

Let  $\left(\Xi ,\Delta ,\sigma ,\gamma \right)$  be a complete DCMS equipped with a directed graph  $G$  and  $T_1,T_2:\Xi \to \Xi$  is a  $G-$Orbital cyclic pair. Suppose there exist  ${\tau }_1\in [0,1)$  such that

(i)

 ${\Xi }^{T_1}\ne \varphi ,$ 

(ii)

for all  $\varpi \in {\Xi }^{T_1}$  and  $\varkappa \in {\Xi }^{T_2},$ 

$$∆\left(T_1\varpi ,T_2\varkappa \right)\le {\tau }_1\mathit{max}\left\{∆\left(\varpi ,\varkappa \right),∆\left(\varpi ,T_1\varpi \right),∆\left(\varkappa ,T_2\varkappa \right)\right\};$$

(8)

(iii)

for all  ${\left\{{\varpi }_n\right\}}_{j\in \mathbb{N}}\subset \Xi ,$  one has  $\left({\varpi }_j,{\varpi }_{j+1}\right)\in E\left(G\right),$ 

$$\underset{m\le 1}{\mathit{sup}}\underset{i\to \infty }{\mathit{lim}}\frac{\sigma \left({\varpi }_{i+1},{\varpi }_{i+2}\right)\gamma \left({\varpi }_{i+1},{\varpi }_m\right)}{\sigma \left({\varpi }_i,{\varpi }_{i+1}\right)}<\frac{1}{\tau },$$

(9)

where  $\tau =\frac{{\tau }_1}{1-{\tau }_1}$ 

(iv)

 $T_1$  and  $T_2$  are continuous, or for all  ${\left({\varpi }_j\right)}_{j\in \mathbb{N}}\subset \Xi ,$  with  ${\varpi }_j\to \varpi$  as  $j\to +\infty ,$  and  $\left({\varpi }_j,{\varpi }_{j+1}\right)\in E\left(G\right)$  for  $j\in \mathbb{N},$  we have  $\varpi \in {\Xi }^{T_1}\cap {\Xi }^{T_2}.$  In these conditions,  $CFix\left(T_1,T_2\right)\ne \varphi ;$ 

(v)

For all  $\varpi \in \Xi ,$  we have  $\underset{j\to +\infty }{\mathit{lim}}\sigma \left({\varpi }_j,\varpi \right),\underset{j\to +\infty }{\mathit{lim}}\sigma \left(\varpi ,{\varpi }_j\right),$   $\underset{j\to +\infty }{\mathit{lim}}\gamma \left({\varpi }_j,\varpi \right)$  and  $\underset{j\to +\infty }{\mathit{lim}}\gamma \left(\varpi ,{\varpi }_j\right)$  exists and finite;

(vi)

If  $\left({\varpi }^{*},{\varpi }^{’}\right)\in CFix\left(T_1,T_2\right)$  implies  ${\varpi }^{*}\in {\Xi }^{T_1}$  and  ${\varpi }^{’}\in {\Xi }^{T_2},$  then the pair  $\left(T_1,T_2\right)$  has a unique common fixed point.

Proof. 

Let  ${\varpi }_0\in {\Xi }^{T_1}.$  Thus  ${(\varpi }_0,T_1{\varpi }_0)\in E(G)$  . As the pair  $\left(T_1,T_2\right)$  is a  $G-$  Orbital cyclic, we obtain  $\left({T_1\varpi }_0,T_2{T}_1{\varpi }_0\right)\in E\left(G\right).$  Construct  ${\varpi }_1$  by  ${\varpi }_1=T_1{\varpi }_0,$  we have  $\left({\varpi }_1,T_2{\varpi }_1\right)\in E\left(G\right)$  and from here  $\left({T_2\varpi }_1,T_1{T}_2{\varpi }_1\right)\in E\left(G\right).$  Denoting by  ${\varpi }_2=T_2{\varpi }_1,$  we have  $\left({\varpi }_2,T_1{\varpi }_2\right)\in E\left(G\right).$  In the same manner, we obtain a sequence  ${\left({\varpi }_j\right)}_{j\in \mathbb{N}}$  with  ${\varpi }_{2j}=T_2{\varpi }_{2j-1}$  and  ${\varpi }_{2j+1}=T_1{\varpi }_{2j},$  such that  $\left({\varpi }_{2j},{\varpi }_{2j+1}\right)\in E\left(G\right).$  we assume that  ${\varpi }_j\ne {\varpi }_{j+1}.$  If, there exist $j_0\in \mathbb{N}$  such that  ${\varpi }_{j0}={\varpi }_{j0+1},$  then from the fact that  $∆\subset E\left(G\right),\left({\varpi }_{j0},{\varpi }_{j0+1}\right)\in E\left(G\right)$  and  ${\varpi }^{*}={\varpi }_{j0}$  is a fixed point of  $T_1.$  Now, for  ${\varpi }^{*}\in \mathrm{C}\mathrm{F}\mathrm{i}\mathrm{x}\left(T_1,T_2\right)$  , we deliberate the two cases for  $j_0.$  If  $j_0=2j,$  then  ${\varpi }_{2j}={\varpi }_{2j+1}=T_1{\varpi }_{2j}$  and thus,  ${\varpi }_{2j}$  is a fixed point of  $T_1.$  Assume that  ${\varpi }_{2j}={\varpi }_{2j+1}=T_1{\varpi }_{2j}$  but  $∆\left(T_1{\varpi }_{2j},T_2{\varpi }_{2j+1}\right)>0,$  and let  $\varpi ={\varpi }_{2j}\in {\Xi }^{T_1}$  and  $\varkappa ={\varpi }_{2j+1}\in {\Xi }^{T_2}.$  So

$$\begin{array}{l}0<∆\left({\varpi }_{2j+1},{\varpi }_{2j+2}\right)=∆\left(T_1{\varpi }_{2j},T_2{\varpi }_{2j+1}\right)\\ \le {\tau }_1\max \left\{∆\left({\varpi }_{2j},{\varpi }_{2j+1}\right),∆\left({\varpi }_{2j},T_1{\varpi }_{2j}\right),∆\left({\varpi }_{2j+1},T_2{\varpi }_{2j+1}\right)\right\};\\ ={\tau }_1\max \left\{∆\left({\varpi }_{2j},{\varpi }_{2j+1}\right),∆\left({\varpi }_{2j},{\varpi }_{2j+1}\right),∆\left({\varpi }_{2j+1},{\varpi }_{2j+2}\right)\right\};\\ ={\tau }_1∆\left({\varpi }_{2j+1},{\varpi }_{2j+2}\right).\end{array}$$

This is a contradiction of the fact  ${\tau }_1<1.$  Therefore,  ${\varpi }_{2j}$  is a fixed point of  $T_2$ . In the same manner, if  $j_0$  is an odd number, then there exists ${\varpi }^{*}\in \Xi$  such that  $T_1{\varpi }^{*}=T_2{\varpi }^{*}={\varpi }^{*}.$  So, we assume that  ${\varpi }_j\ne {\varpi }_{j+1}$  for each  $j\in \mathbb{N}.$  Now, we examine that  ${\left({\varpi }_j\right)}_{j\in \mathbb{N}}$  is a Cauchy sequence. We discuss the below two cases:

Case 1.

$$\begin{array}{l}0<∆\left({\varpi }_{2j+1},{\varpi }_{2j+2}\right)=∆\left(T_1{\varpi }_{2j},T_2{\varpi }_{2j+1}\right)\\ \le {\tau }_1\mathit{max}\left\{∆\left({\varpi }_{2j},{\varpi }_{2j+1}\right),∆\left({\varpi }_{2j},T_1{\varpi }_{2j}\right),∆\left({\varpi }_{2j+1},T_2{\varpi }_{2j+1}\right)\right\}\\ ={\tau }_1\mathit{max}\left\{∆\left({\varpi }_{2j},{\varpi }_{2j+1}\right),∆\left({\varpi }_{2j},{\varpi }_{2j+1}\right),∆\left({\varpi }_{2j+1},{\varpi }_{2j+2}\right)\right\};\\ ={\tau }_1\mathit{max}\left\{∆\left({\varpi }_{2j},{\varpi }_{2j+1}\right),∆\left({\varpi }_{2j+1},{\varpi }_{2j+2}\right)\right\};\\ \le {\tau }_1\left[∆\left({\varpi }_{2j},{\varpi }_{2j+1}\right)+∆\left({\varpi }_{2j+1},{\varpi }_{2j+2}\right)\right],\end{array}$$

that is

$$\left(1-{\tau }_1\right)∆\left({\varpi }_{2j+1},{\varpi }_{2j+2}\right)\le {\tau }_1∆\left({\varpi }_{2j},{\varpi }_{2j+1}\right),$$

which implies

$$∆\left({\varpi }_{2j+1},{\varpi }_{2j+2}\right)\le \frac{{\tau }_1}{1-{\tau }_1}∆\left({\varpi }_{2j},{\varpi }_{2j+1}\right)$$

(10)

Case 2.

 $\varpi ={\varpi }_{2j}\in {\Xi }^{T_1}$  and  $\varkappa ={\varpi }_{2j-1}\in {\Xi }^{T_2}.$ 

$$\begin{array}{l}0<∆\left({\varpi }_{2j+1},{\varpi }_{2j}\right)=∆\left(T_1{\varpi }_{2j},T_2{\varpi }_{2j-1}\right)\\ \le {\tau }_1\mathit{max}\left\{∆\left({\varpi }_{2j},{\varpi }_{2j-1}\right),∆\left({\varpi }_{2j},T_1{\varpi }_{2j}\right),∆\left({\varpi }_{2j-1},T_2{\varpi }_{2j-1}\right)\right\}\\ ={\tau }_1\mathit{max}\left\{∆\left({\varpi }_{2j},{\varpi }_{2j-1}\right),∆\left({\varpi }_{2j},{\varpi }_{2j+1}\right),∆\left({\varpi }_{2j-1},{\varpi }_{2j}\right)\right\};\\ ={\tau }_1\mathit{max}\left\{∆\left({\varpi }_{2j-1},{\varpi }_{2j}\right),∆\left({\varpi }_{2j},{\varpi }_{2j+1}\right)\right\};\\ \le {\tau }_1\left[∆\left({\varpi }_{2j-1},{\varpi }_{2j}\right)+∆\left({\varpi }_{2j},{\varpi }_{2j+1}\right)\right],\end{array}$$

that is

$$\left(1-{\tau }_1\right)∆\left({\varpi }_{2j+1},{\varpi }_{2j}\right)\le {\tau }_1∆\left({\varpi }_{2j},{\varpi }_{2j-1}\right),$$

which implies

$$∆\left({\varpi }_{2j},{\varpi }_{2j+1}\right)\le \frac{{\tau }_1}{1-{\tau }_1}∆\left({\varpi }_{2j-1},{\varpi }_{2j}\right).$$

(11)

Since  $\tau =\frac{{\tau }_1}{1-{\tau }_1},$  we have

$$∆\left({\varpi }_j,{\varpi }_{j+1}\right)\le \tau ∆\left({\varpi }_{j-1},{\varpi }_j\right).$$

(12)

Thus, we have

$$∆\left({\varpi }_j,{\varpi }_{j+1}\right)\le \tau ∆\left({\varpi }_{j-1},{\varpi }_j\right)\le {\tau }^2∆\left({\varpi }_{j-2},{\varpi }_{j-1}\right)\le \dots \le {\tau }^j∆\left({\varpi }_0,{\varpi }_1\right).$$

For each  $j,m\in \mathbb{N}\left(j<m\right),$  we have

$$\begin{array}{l}∆\left({\varpi }_j,{\varpi }_m\right)\le \sigma \left({\varpi }_j,{\varpi }_{j+1}\right)∆\left({\varpi }_j,{\varpi }_{j+1}\right)+\gamma \left({\varpi }_{j+1},{\varpi }_m\right)∆\left({\varpi }_{j+1},{\varpi }_m\right)\\ \le \sigma \left({\varpi }_j,{\varpi }_{j+1}\right)∆\left({\varpi }_j,{\varpi }_{j+1}\right)+\gamma \left({\varpi }_{j+1},{\varpi }_m\right)\sigma \left({\varpi }_{j+1},{\varpi }_{j+2}\right)∆\left({\varpi }_{j+1},{\varpi }_{j+2}\right)\\ +\gamma \left({\varpi }_{j+1},{\varpi }_m\right)\gamma \left({\varpi }_{j+2},{\varpi }_m\right)∆\left({\varpi }_{j+2},{\varpi }_m\right)\\ \le \sigma \left({\varpi }_j,{\varpi }_{j+1}\right)∆\left({\varpi }_j,{\varpi }_{j+1}\right)+\gamma \left({\varpi }_{j+1},{\varpi }_m\right)\sigma \left({\varpi }_{j+1},{\varpi }_{j+2}\right)∆\left({\varpi }_{j+1},{\varpi }_{j+2}\right)\\ +\gamma \left({\varpi }_{j+1},{\varpi }_m\right)\gamma \left({\varpi }_{j+2},{\varpi }_m\right)\sigma \left({\varpi }_{j+2},{\varpi }_{j+3}\right)∆\left({\varpi }_{j+2},{\varpi }_{j+3}\right)\\ +\gamma \left({\varpi }_{j+1},{\varpi }_m\right)\gamma \left({\varpi }_{j+2},{\varpi }_m\right)\gamma \left({\varpi }_{j+3},{\varpi }_m\right)∆\left({\varpi }_{j+3},{\varpi }_m\right)\\ \le \dots \le \sigma \left({\varpi }_j,{\varpi }_{j+1}\right)∆\left({\varpi }_j,{\varpi }_{j+1}\right)+{\displaystyle \sum _{i=j+1}^{m-2}}\left({\displaystyle \prod _{t=j+1}^i}\gamma \left({\varpi }_t,{\varpi }_m\right)\right)\sigma \left({\varpi }_j,{\varpi }_{j+1}\right)∆\left({\varpi }_j,{\varpi }_{j+1}\right)\\ +{\displaystyle \prod _{i=j+1}^{m-1}}\gamma \left({\varpi }_i,{\varpi }_m\right)∆\left({\varpi }_{m1},{\varpi }_m\right).\end{array}$$

This further implies that

$$\begin{array}{l}∆\left({\varpi }_j,{\varpi }_m\right)\le \sigma \left({\varpi }_j,{\varpi }_{j+1}\right)∆\left({\varpi }_j,{\varpi }_{j+1}\right)+{\displaystyle \sum _{i=j+1}^{m-2}}\left({\displaystyle \prod _{t=j+1}^i}\gamma \left({\varpi }_t,{\varpi }_m\right)\right)\sigma \left({\varpi }_j,{\varpi }_{j+1}\right)∆\left({\varpi }_j,{\varpi }_{j+1}\right)\\ +\left({\displaystyle \prod _{i=j+1}^{m-1}}\gamma \left({\varpi }_i,{\varpi }_m\right)\right)∆\left({\varpi }_{m-1},{\varpi }_m\right)\\ \le \sigma \left({\varpi }_j,{\varpi }_{j+1}\right){\tau }^j∆\left({\varpi }_0,{\varpi }_1\right)+{\displaystyle \sum _{i=j+1}^{m-2}}\left({\displaystyle \prod _{t=j+1}^i}\gamma \left({\varpi }_t,{\varpi }_m\right)\right)\sigma \left({\varpi }_i,{\varpi }_{i+1}\right){\tau }^i∆\left({\varpi }_0,{\varpi }_1\right)\\ +\left({\displaystyle \prod _{i=j+1}^{m-1}}\gamma \left({\varpi }_i,{\varpi }_m\right)\right){\tau }^{m-1}∆\left({\varpi }_0,{\varpi }_1\right)\\ =\sigma \left({\varpi }_j,{\varpi }_{j+1}\right){\tau }^j∆\left({\varpi }_0,{\varpi }_1\right)+{\displaystyle \sum _{i=j+1}^{m-1}}\left({\displaystyle \prod _{t=j+1}^i}\gamma \left({\varpi }_t,{\varpi }_m\right)\right)\sigma \left({\varpi }_i,{\varpi }_{i+1}\right){\tau }^i∆\left({\varpi }_0,{\varpi }_1\right).\end{array}$$

$$∆\left({\varpi }_j,{\varpi }_m\right)\le \sigma \left({\varpi }_j,{\varpi }_{j+1}\right)k^j∆\left({\varpi }_0,{\varpi }_1\right)+\sum _{i=j+1}^{m-1}\left(\prod _{t=j+1}^i\gamma \left({\varpi }_t,{\varpi }_m\right)\right)\sigma \left({\varpi }_i,{\varpi }_{i+1}\right)k^i∆\left({\varpi }_0,{\varpi }_1\right).$$

(13)

Let

$${\psi }_v=\sum _{i=j+1}^v\left(\prod _{t=j+1}^i\gamma \left({\varpi }_t,{\varpi }_m\right)\right)\sigma \left({\varpi }_i,{\varpi }_{i+1}\right)k^i∆\left({\varpi }_0,{\varpi }_1\right).$$

Then, from (13), we obtain

$$∆\left({\varpi }_j,{\varpi }_m\right)\le ∆\left({\varpi }_0,{\varpi }_1\right)[{\tau }^j\sigma \left({\varpi }_j,{\varpi }_{j+1}\right)+{(\psi }_{m-1}-{\psi }_j)].$$

(14)

Since  $\sigma \left(\varpi ,\varkappa \right)\ge 1,$  and by employing the ratio test, then  $\underset{j\to \infty }{\mathit{lim}}{\psi }_j$  exists. Clearly, letting  $j,m\to +\infty$  in (14), we obtain

$$\underset{j,m\to \infty }{\mathit{lim}}∆\left({\varpi }_j,{\varpi }_m\right)=0.$$

(15)

Hence,  $\left\{{\varpi }_j\right\}$  is a Cauchy sequence in  $\left(\Xi ,∆,\sigma ,\gamma \right).$  So, there exist  $s{\varpi }^{*}\in \Xi ,$  we have

$$\underset{j\to +\infty }{\mathit{lim}}∆\left({\varpi }_j,{\varpi }^{*}\right)=0.$$

(16)

That is  ${\varpi }_j\to {\varpi }^{*}$  as  $j\to +\infty .$  It is obvious that

$$\underset{j\to +\infty }{\mathit{lim}}{\varpi }_{2j}=\underset{j\to +\infty }{\mathit{lim}}{\varpi }_{2j+1}={\varpi }^{*}.$$

(17)

As  $T_1$  and  $T_2$  are continuous, so we have

$$\begin{gathered} {\varpi }^{*}=\underset{j\to +\infty }{\mathit{lim}}{\varpi }_{2j+1}=\underset{j\to +\infty }{\mathit{lim}}{T}_1\left({\varpi }_{2j}\right)=T_1\left({\varpi }^{*}\right), \\ {\varpi }^{*}=\underset{j\to +\infty }{\mathit{lim}}{\varpi }_{2j+2}=\underset{j\to +\infty }{\mathit{lim}}{T}_2\left({\varpi }_{2j+1}\right)=T_2\left({\varpi }^{*}\right), \end{gathered}$$

Now letting  $\varpi ={\varpi }^{*}\in {\Xi }^{T_1}$  and  $\varkappa ={\varpi }_{2j+2}\in {\Xi }^{T_2},$  we have

$$\begin{array}{l}0<∆\left(T_1{\varpi }^{*},{\varpi }_{2j+2}\right)=∆\left(T_1{\varpi }^{*},{T_2\varpi }_{2j+1}\right)\\ \le {\tau }_1\mathit{max}\left\{∆\left({\varpi }^{*},{\varpi }_{2j+1}\right),∆\left({\varpi }^{*},T_1{\varpi }^{*}\right),∆\left({\varpi }_{2j+1},T_2{\varpi }_{2j+1}\right)\right\},\\ ={\tau }_1\mathit{max}\left\{∆\left({\varpi }^{*},{\varpi }_{2j+1}\right),∆\left({\varpi }^{*},T_1{\varpi }^{*}\right),∆\left({\varpi }_{2j+1},{\varpi }_{2j+2}\right)\right\}.\end{array}$$

Taking  $j\to +\infty$  and by (17), it is immediate that  $∆\left({\varpi }^{*},T_1{\varpi }^{*}\right)=0.$  This yields that  ${\varpi }^{*}=T_1{\varpi }^{*}.$  Similarly, suppose that  $\varpi ={\varpi }_{2j+1}\in {\Xi }^{T_1}$  and  $\varkappa ={\varpi }^{*}\in {\Xi }^{T_2},$  we have

$$\begin{array}{l}0<∆\left({\varpi }_{2j+2},T_2{\varpi }^{*}\right)=∆\left({T_1\varpi }_{2j},T_2{\varpi }^{*}\right)\\ \le {\tau }_1\mathit{max}\left\{∆\left({\varpi }_{2j},{\varpi }^{*}\right),∆\left({\varpi }_{2j},T_1{\varpi }_{2j}\right),∆\left({\varpi }^{*},T_2{\varpi }^{*}\right)\right\},\\ ={\tau }_1\mathit{max}\left\{∆\left({\varpi }_{2j},{\varpi }^{*}\right),∆\left({\varpi }_{2j},{\varpi }_{2j+1}\right),∆\left({\varpi }^{*},T_2{\varpi }^{*}\right)\right\}.\end{array}$$

Taking  $j\to +\infty$  and by applying (17), we obtain  $∆\left({\varpi }^{*},T_2{\varpi }^{*}\right)=0.$  This yields that  ${\varpi }^{*}=T_{11iavethatconcludethatasesfor2}{\varpi }^{*}.$  □

Corollary 7.

Let  $\left(\Xi ,\Delta ,\sigma ,\gamma \right)$  be a complete DCMS equipped with a directed graph  $G$  and  $T:\Xi \to \Xi$  is a  $G-$  Orbital cyclic. Assume that there exist  $s{\tau }_1\in [0,1)$  such that

(i)

 ${\Xi }^T\ne \varphi ,$ 

(ii)

for all  $\varpi ,\varkappa \in {\Xi }^T,$ 

$$∆\left(T\varpi ,T\varkappa \right)\le {\tau }_1\mathit{max}\left\{∆\left(\varpi ,\varkappa \right),∆\left(\varpi ,T\varpi \right),∆\left(\varkappa ,T\varkappa \right)\right\},$$

(iii)

for all  ${\left\{{\varpi }_n\right\}}_{j\in \mathbb{N}}\subset \Xi ,$  one has  $\left({\varpi }_j,{\varpi }_{j+1}\right)\in E\left(G\right),$ 

$$\underset{m\le 1}{\mathit{sup}}\underset{i\to \infty }{\mathit{lim}}\frac{\sigma \left({\varpi }_{i+1},{\varpi }_{i+2}\right)\gamma \left({\varpi }_{i+1},{\varpi }_m\right)}{\sigma \left({\varpi }_i,{\varpi }_{i+1}\right)}<\frac{1}{\tau },$$

(18)

where  $\tau =\frac{{\tau }_1}{1-{\tau }_1},$ 

(iv)

 $T$  is continuous, or for all  ${\left({\varpi }_j\right)}_{j\in \mathbb{N}}\subset \Xi ,$  with  ${\varpi }_j\to \varpi$  as  $j\to +\infty ,$  and  $\left({\varpi }_j,{\varpi }_{j+1}\right)\in E\left(G\right)$  for  $j\in \mathbb{N},$  we obtain  $\varpi \in {\Xi }^T,$ 

(v)

for all  $\varpi \in \Xi ,$  we assume  $\underset{j\to +\infty }{\mathit{lim}}\sigma \left({\varpi }_j,\varpi \right),\underset{j\to +\infty }{\mathit{lim}}\sigma \left(\varpi ,{\varpi }_j\right),$   $\underset{j\to +\infty }{\mathit{lim}}\gamma \left({\varpi }_j,\varpi \right)$  and  $\underset{j\to +\infty }{\mathit{lim}}\gamma \left(\varpi ,{\varpi }_j\right)$  exists and are finite,

then  $T$  has a unique fixed point.

Example 2.

Suppose  $\Xi =\left\{0,1,2,3,4\right\}.$  Define  $∆:\Xi \times \Xi \to [0,+\infty )$  by

$$∆\left(\varpi ,\varkappa \right)={\left|\varpi -\varkappa \right|}^2$$

and  $\sigma :\Xi \times \Xi \to [1,+\infty )$  by

$$\sigma \left(\varpi ,\varkappa \right)=1+\varpi +\varkappa$$

and

$$\gamma \left(\varpi ,\varkappa \right)=2+{\varpi }^2+{\varkappa }^2$$

for all  $\varpi ,\varkappa \in \Xi .$  Then,  $\left(\Xi ,\Delta ,\sigma ,\gamma \right)$  is a complete DCMS. Now define  $T:\Xi \to \Xi$  by  $T\varpi =0for\varpi \in \left\{0,1\right\},$  and  $T\varpi =1for\varpi \in \left\{2,3\right\}.$ 

Moreover, let a directed graph by  $\mathrm{G}=\left\{\left(0,1\right),\left(\mathrm{0,2}\right),\left(2,3\right),\left(\mathrm{0,0}\right),\left(\mathrm{1,1}\right),\left(\mathrm{2,2}\right)\left(\mathrm{3,3}\right)\right\}.$  Then, Corollary 3 is fulfilled with  ${\mathsf{\tau }}_1=\frac{1}{3}$  and  $\mathrm{T}$  has a unique fixed point  ${\mathsf{\varpi }}^{\mathrm{*}}=0.$ 

6. Application to Integral Equations

In this part, we examine the solution of the following Fredholm equation:

$$\varpi \left(t\right)={\int }_0^{1}K\left(t,s,\varpi \left(t\right)\right)ds,$$

(19)

for all  $t\in \left[0,1\right],$  where  $K\left(t,s,\varpi \left(t\right)\right)$  is a continuous function from  $\left[0,1\right]\times [0,1]$  into  $\mathbb{R}$ . Suppose  $\Xi =C\left(\left[0,1\right],\mathbb{R}\right).$  Define  $∆:\Xi \times \Xi \to [1,+\infty )$  by

$$∆\left(\varpi ,\varkappa \right)=\underset{t\in [0,1]}{\sup }\left(\frac{\left|\varpi \left(t\right)\right|+\left|\varkappa \left(t\right)\right|}{2}\right).$$

Then,  $\left(\Xi ,\mathsf{\Delta },\sigma ,\gamma \right)$  is a complete DCMS with  $\sigma \left(\varpi ,\varkappa \right)=1+\varpi +\varkappa ,$  and  $\gamma \left(\varpi ,\varkappa \right)=2+{\varpi }^2+{\varkappa }^2.$ 

Theorem 5.

Assume that

(a)

 $\left|K\left(t,s,\varpi \left(t\right)\right)\right|+\left|K\left(t,s,\varkappa \left(t\right)\right)\right|\le {\tau }_1\left(\underset{t\in \left[0,1\right]}{\mathit{sup}}\left|\varpi \left(t\right)\right|+\left|\varkappa \left(t\right)\right|\right)\left(\left|\varpi \left(t\right)\right|+\left|\varkappa \left(t\right)\right|\right)$

for some  ${\tau }_1\to \Xi \to \left[0,1\right);$

(b)

 $K\left(t,s,{\int }_0^{1}K\left(t,s,\varpi \left(t\right)\right)ds,\right)<K\left(t,s,\varpi \left(t\right)\right)$

for all  $t,s\in \left[0,1\right].$  Then, there is a unique solution to the integral Equation (19).

Proof. 

Define the mapping  $T:\Xi \to \Xi$  by

$$T\varpi \left(t\right)={\int }_0^{1}K\left(t,s,\varpi \left(t\right)\right)ds.$$

Then

$$∆\left(T\varpi ,T\varkappa \right)=\underset{t\in [0,1]}{\sup }\left(\frac{\left|\varpi \left(t\right)\right|+\left|\varkappa \left(t\right)\right|}{2}\right).$$

Now

$$\begin{array}{l}∆\left(T\varpi \left(t\right),T\varkappa \left(t\right)\right)=\frac{\left|T\varpi \left(t\right)\right|+\left|T\varkappa \left(t\right)\right|}{2}\\ =\frac{\left|{\int }_0^{1}K\left(t,s,\varpi \left(t\right)\right)ds\right|+\left|{\int }_0^{1}K\left(t,s,\varkappa \left(t\right)\right)ds\right|}{2}\\ \le \frac{{\int }_0^1\left|K\left(t,s,\varpi \left(t\right)\right)ds\right|+{\int }_0^1\left|K\left(t,s,\varkappa \left(t\right)\right)ds\right|}{2}\\ =\frac{{\int }_0^1\left(\left|K\left(t,s,\varpi \left(t\right)\right)\right|+\left|K\left(t,s,\varkappa \left(t\right)\right)\right|\right)ds}{2}\\ =\frac{{\int }_0^1{\tau }_1\left(\underset{t\in \left[0,1\right]}{\sup }\left|\varpi \left(t\right)\right|+\left|\varkappa \left(t\right)\right|\right)\left(\left|\varpi \left(t\right)\right|+\left|\varkappa \left(t\right)\right|\right)}{2}\\ \le {\tau }_1\left(\underset{t\in \left[0,1\right]}{\sup }\left|\varpi \left(t\right)\right|+\left|\varkappa \left(t\right)\right|\right)∆\left(\varpi \left(t\right),\varkappa \left(t\right)\right).\end{array}$$

Also, we observe that

$$\sigma \left(\varpi ,\varkappa \right)=\frac{1}{{\tau }_1\left(\underset{t\in \left[0,1\right]}{\sup }\left|\varpi \left(t\right)\right|+\left|\varkappa \left(t\right)\right|\right)}.$$

Hence, all of the requirements for Corollary 6 have been met. As a result, Equation (19) has a unique solution. □

7. Conclusions

In this manuscript, we proved two common fixed point results for generalized rational-type contractions under some conditions in the context of DCMS. Further, we extended and proved rational-type contractions equipped with graphs in DCMS. Several non-trivial examples and an application are presented to show the validity of the main results. The given results are improved and generalized to the existing ones in [11,18,20]. These results can be generalized by utilizing the notions in [19,21,22,23,24,25,26].