The Existence and Uniquenes Solution of Nonlinear Integral Equations via Common Fixed Point Theorems

Abstract

In this paper, we prove some common fixed-point theorems on complex partial metric space. The presented results generalize and expand some of the well-known results in the literature. We also explore some of the applications of our key results.

1. Introduction

Azam et al. [1] introduced the concept of complex-valued metric spaces and studied some fixed point theorems for mappings satisfying a rational inequality.

Two years later, in [2], Rao et al. discussed for the first time the idea of complex-valued b-metric spaces.

In 2017, Dhivya and Marudai [3] introduced the concept of complex partial metric space and suggested a plan to expand the results, as well as proving common fixed-point theorems under the rational expression contraction condition. This idea has been followed by Gunaseelan [4], who introduced the concept of complex partial b-metric spaces and discussed some results of fixed-point theory for self-mappings in these new spaces.

In [5], Prakasam and Gunaseelan proved the existence and uniqueness of a common fixed-point (with an illustrative example) theorem using CLR and E.A. properties in complex partial b-metric spaces. Their proved results generalize and extend some of the well-known results in the literature.

In [6], Gunaseelan et al. proved a fixed-point theorem in complex partial b-metric spaces under a contraction mapping. They also gave some applications of their main results.

In this paper, we prove some common fixed-point theorems on complex partial metric space.

2. Preliminaries

Let $\mathfrak{C}$ be the set of complex numbers and ${\tau }_1,{\tau }_2,{\tau }_3\in \mathfrak{C}$. Define a partial order ⪯ on $\mathfrak{C}$ as follows:

${\tau }_1⪯{\tau }_2$ if and only if $\mathcal{R}\left({\tau }_1\right)\le \mathcal{R}\left({\tau }_2\right)$, $\mathcal{I}\left({\tau }_1\right)\le \mathcal{I}\left({\tau }_2\right)$.

Consequently, one can infer that ${\tau }_1⪯{\tau }_2$ if one of the following conditions is satisfied:

(i)

$\mathcal{R}\left({\tau }_1\right)=\mathcal{R}\left({\tau }_2\right)$, $\mathcal{I}\left({\tau }_1\right)<\mathcal{I}\left({\tau }_2\right)$,

(ii)

$\mathcal{R}\left({\tau }_1\right)<\mathcal{R}\left({\tau }_2\right)$, $\mathcal{I}\left({\tau }_1\right)=\mathcal{I}\left({\tau }_2\right)$,

(iii)

$\mathcal{R}\left({\tau }_1\right)<\mathcal{R}\left({\tau }_2\right)$, $\mathcal{I}\left({\tau }_1\right)<\mathcal{I}\left({\tau }_2\right)$,

(iv)

$\mathcal{R}\left({\tau }_1\right)=\mathcal{R}\left({\tau }_2\right)$, $\mathcal{I}\left({\tau }_1\right)=\mathcal{I}\left({\tau }_2\right)$.

In particular, we write ${\tau }_1⋨{\tau }_2$ if ${\tau }_1\ne {\tau }_2$, and one of $\left(i\right),\left(ii\right)$ and $\left(iii\right)$ is satisfied and we write ${\tau }_1\prec {\tau }_2$ if only $\left(iii\right)$ is satisfied. Notice that

(a)

If $0⪯{\tau }_1⋨{\tau }_2$, then $|{\tau }_1|<|{\tau }_2|,$

(b)

If ${\tau }_1⪯{\tau }_2$ and ${\tau }_2\prec {\tau }_3$, then ${\tau }_1\prec {\tau }_3$,

(c)

If $\eta ,\gamma \in \mathbb{R}$ and $\eta \le \gamma$, then $\eta {\tau }_1⪯\gamma {\tau }_1$ for all $0⪯{\tau }_1\in \mathfrak{C}$.

Here ${\mathfrak{C}}_{+}(=\{(\aleph ,\mathfrak{y})|\aleph ,\mathfrak{y}\in {\mathbb{R}}_{+}\})$ and ${\mathbb{R}}_{+}(=\{\aleph \in \mathbb{R}|\aleph \ge 0\})$ denote the set of non-negative complex numbers and the set of non negative real numbers, respectively.

Now, let us recall some basic concepts and notations that will be used below.

Definition 1 

( [3]).  A complex partial metric on a non-void set G is a function ${\varrho }_{cb}:G\times G\to {\mathbb{C}}^{+}$ such that for all $\theta ,\omega ,\vartheta \in G$:

(i) 

$0⪯{\varrho }_{cb}(\theta ,\theta )⪯{\varrho }_{cb}(\theta ,\omega )\left(smallself-distances\right)$

(ii) 

${\varrho }_{cb}(\theta ,\omega )={\varrho }_{cb}(\omega ,\theta )\left(symmetry\right)$

(iii) 

${\varrho }_{cb}(\theta ,\theta )={\varrho }_{cb}(\theta ,\omega )={\varrho }_{cb}(\omega ,\omega )$ if and only if $\theta =\omega \left(equality\right)$

(iv) 

${\varrho }_{cb}(\theta ,\omega )⪯{\varrho }_{cb}(\theta ,\vartheta )+{\varrho }_{cb}(\vartheta ,\omega )-{\varrho }_{cb}(\vartheta ,\vartheta )\left(triangularity\right)$.

A complex partial metric space is a pair $(G,{\varrho }_{cb})$ such that G is a non-void set and ${\varrho }_{cb}$ is the complex partial metric on G.

Definition 2

( [3]).  Let $(G,{\wp }_{cb})$ be a complex partial metric space. Let $\{{\theta }_n\}$ be any sequence in G. Then

(i) 

The sequence $\{{\theta }_n\}$ is said to converge to θ, if ${lim}_{n\to \infty }{\wp }_{cb}({\theta }_n,\theta )={\wp }_{cb}(\theta ,\theta )$.

(ii) 

The sequence $\{{\theta }_n\}$ is said to be a Cauchy sequence in $(G,{\wp }_{cb})$ if

${lim}_{n,m\to \infty }{\wp }_{cb}({\theta }_n,{\theta }_m)$ exists and is finite.

(iii) 

$(G,{\wp }_{cb})$ is said to be a complete complex partial metric space if for every Cauchy sequence $\{{\theta }_n\}$ in G there exists $\theta \in G$ such that

${lim}_{n,m\to \infty }{\wp }_{cb}({\theta }_n,{\theta }_m)={lim}_{n\to \infty }{\wp }_{cb}({\theta }_n,\theta )={\wp }_{cb}(\theta ,\theta )$.

(iv) 

A mapping $\mathsf{\Pi }:G\to G$ is said to be continuous at ${\theta }_0\in G$ if for every $ϵ>0$, there exists $\delta >0$ such that $\mathsf{\Pi }\left(B_{{\wp }_{cb}}({\theta }_0,\delta )\right)\subset B_{{\wp }_{cb}}\left(\mathsf{\Pi }({\theta }_0,ϵ)\right)$.

Definition 3

( [3]).  Let Π and Ψ be self-mappings of non-void set G. A point $\aleph \in G$ is called a common fixed point of Π and Ψ if $\aleph =\mathsf{\Pi }\aleph =\mathsf{\Psi }\aleph$.

Theorem 1

( [3]).  Let $(G,⪯)$ be a partially ordered set and suppose that there exists a complex partial metric ${\varrho }_{cb}$ in G such that $(G,{\varrho }_{cb})$ is a complete complex partial metric space. Let $\mathsf{\Pi },\mathsf{\Psi }:G\to G$ be a pair of weakly increasing mappings, and suppose that for every comparable $\aleph ,\mathfrak{y}\in G$ we have either

$$\begin{array}{c}\hfill {\varrho }_{cb}(\mathsf{\Pi }\aleph ,\mathsf{\Psi }\mathfrak{y})⪯a\frac{{\varrho }_{cb}(\aleph ,\mathsf{\Pi }\aleph ){\varrho }_{cb}(\mathfrak{y},\mathsf{\Psi }\mathfrak{y})}{{\varrho }_{cb}(\aleph ,\mathfrak{y})}+b{\varrho }_{cb}(\aleph ,\mathfrak{y})\end{array}$$

for ${\varrho }_{cb}(\aleph ,\mathfrak{y})\ne 0$ with $a\ge 0,b\ge 0$, $a+b<1$, or

$$\begin{array}{c}\hfill {\varrho }_{cb}(\mathsf{\Pi }\aleph ,\mathsf{\Psi }\mathfrak{y})=0\mathrm{if}{\varrho }_{cb}(\aleph ,\mathfrak{y})=0.\end{array}$$

If Π or Ψ is continuous, then Π and Ψ have a common fixed point $\propto \in G$ and ${\varrho }_{cb}(\propto ,\propto )=0$.

Inspired by Theorem 1, here we prove some common fixed-point theorems on complex partial metric space with an application. For complex partial metric space, we will use the CPMS notation.

3. Main Results

Theorem 2.

Let $(G,{\wp }_{cb})$ be a complete CPMS and $\mathsf{\Pi },\mathsf{\Psi }:G\to G$ be two continuous mappings such that

$$\begin{array}{cc}\hfill {\wp }_{cb}(\mathsf{\Pi }\theta ,\mathsf{\Psi }\omega )& ⪯⋏max\{{\wp }_{cb}(\theta ,\omega ),{\wp }_{cb}(\theta ,\mathsf{\Pi }\theta ),{\wp }_{cb}(\omega ,\mathsf{\Psi }\omega ),\hfill \\ \hfill & {\displaystyle \frac{1}{2}}({\wp }_{cb}(\theta ,\mathsf{\Psi }\omega )+{\wp }_{cb}(\omega ,\mathsf{\Pi }\theta ))\},\hfill \end{array}$$

(1)

for all $\theta ,\omega \in G$, where $0\le ⋏<1$ and ${\wp }_{cb}(\mathsf{\Pi }\theta ,\mathsf{\Psi }\omega )\ne 0$. Then, the pair $(\mathsf{\Pi },\mathsf{\Psi })$ has a unique common fixed point and ${\wp }_{cb}({\theta }^{*},{\theta }^{*})=0$.

Proof. 

Let ${\theta }_0$ be arbitrary point in G and define a sequence $\{{\theta }_n\}$ as follows:

$$\begin{array}{c}\hfill {\theta }_{2n+1}=\mathsf{\Pi }{\theta }_{2n}and{\theta }_{2n+2}=\mathsf{\Psi }{\theta }_{2n+1},n=0,1,2,\dots \end{array}$$

(2)

Then by (1) and (2), we obtain

$$\begin{array}{cc}\hfill {\wp }_{cb}({\theta }_{2n+1},{\theta }_{2n+2})& ={\wp }_{cb}(\mathsf{\Pi }{\theta }_{2n},\mathsf{\Psi }{\theta }_{2n+1})\hfill \\ \hfill & ⪯⋏max\{{\wp }_{cb}({\theta }_{2n},{\theta }_{2n+1}),{\wp }_{cb}({\theta }_{2n},\mathsf{\Pi }{\theta }_{2n}),{\wp }_{cb}({\theta }_{2n+1},\mathsf{\Psi }{\theta }_{2n+1}),\hfill \\ \hfill & {\displaystyle \frac{1}{2}}({\wp }_{cb}({\theta }_{2n},\mathsf{\Psi }{\theta }_{2n+1})+{\wp }_{cb}({\theta }_{2n+1},\mathsf{\Pi }{\theta }_{2n}))\}\hfill \\ \hfill & ⪯⋏max\{{\wp }_{cb}({\theta }_{2n},{\theta }_{2n+1}),{\wp }_{cb}({\theta }_{2n},{\theta }_{2n+1}),{\wp }_{cb}({\theta }_{2n+1},{\theta }_{2n+2}),\hfill \\ \hfill & {\displaystyle \frac{1}{2}}({\wp }_{cb}({\theta }_{2n},{\theta }_{2n+2})+{\wp }_{cb}({\theta }_{2n+1},{\theta }_{2n+1}))\}\hfill \\ \hfill & ⪯⋏max\{{\wp }_{cb}({\theta }_{2n},{\theta }_{2n+1}),{\wp }_{cb}({\theta }_{2n+1},{\theta }_{2n+2}),\hfill \\ \hfill & {\displaystyle \frac{1}{2}}({\wp }_{cb}({\theta }_{2n},{\theta }_{2n+1})+{\wp }_{cb}({\theta }_{2n+1},{\theta }_{2n+2})-{\wp }_{cb}({\theta }_{2n+1},{\theta }_{2n+1})\hfill \\ \hfill & +{\wp }_{cb}({\theta }_{2n+1},{\theta }_{2n+1}))\}\hfill \\ \hfill & =⋏max\{{\wp }_{cb}({\theta }_{2n},{\theta }_{2n+1}),{\wp }_{cb}({\theta }_{2n+1},{\theta }_{2n+2}),\hfill \\ \hfill & {\displaystyle \frac{1}{2}}({\wp }_{cb}({\theta }_{2n},{\theta }_{2n+1})+{\wp }_{cb}({\theta }_{2n+1},{\theta }_{2n+2}))\}\hfill \end{array}$$

Case I:

If $max\{{\wp }_{cb}({\theta }_{2n},{\theta }_{2n+1}),{\wp }_{cb}({\theta }_{2n+1},{\theta }_{2n+2}),{\displaystyle \frac{1}{2}}({\wp }_{cb}({\theta }_{2n},{\theta }_{2n+1})+{\wp }_{cb}({\theta }_{2n+1},{\theta }_{2n+2}))\}$$={\wp }_{cb}({\theta }_{2n+1},{\theta }_{2n+2})$, then we have

$$\begin{array}{c}\hfill {\wp }_{cb}({\theta }_{2n+1},{\theta }_{2n+2})⪯⋏{\wp }_{cb}({\theta }_{2n+1},{\theta }_{2n+2}).\end{array}$$

This implies $⋏\ge 1$, which is a contradiction.

Case II:

If $max\{{\wp }_{cb}({\theta }_{2n},{\theta }_{2n+1}),{\wp }_{cb}({\theta }_{2n+1},{\theta }_{2n+2}),{\displaystyle \frac{1}{2}}({\wp }_{cb}({\theta }_{2n},{\theta }_{2n+1})+{\wp }_{cb}({\theta }_{2n+1},{\theta }_{2n+2}))\}$$={\wp }_{cb}({\theta }_{2n},{\theta }_{2n+1})$, then we have

$$\begin{array}{c}\hfill {\wp }_{cb}({\theta }_{2n+1},{\theta }_{2n+2})⪯⋏{\wp }_{cb}({\theta }_{2n},{\theta }_{2n+1}).\end{array}$$

(3)

From the next step, we have

$$\begin{array}{cc}\hfill {\wp }_{cb}({\theta }_{2n+2},{\theta }_{2n+3})& ⪯⋏max\{{\wp }_{cb}({\theta }_{2n+1},{\theta }_{2n+2}),{\wp }_{cb}({\theta }_{2n+2},{\theta }_{2n+3}),\hfill \\ \hfill & {\displaystyle \frac{1}{2}}({\wp }_{cb}({\theta }_{2n+1},{\theta }_{2n+2})+{\wp }_{cb}({\theta }_{2n+2},{\theta }_{2n+3}))\}.\hfill \end{array}$$

The following three cases arise.

Case IIa:

$$\begin{array}{c}\hfill {\wp }_{cb}({\theta }_{2n+2},{\theta }_{2n+3})⪯⋏{\wp }_{cb}({\theta }_{2n+2},{\theta }_{2n+3}),\end{array}$$

which implies $⋏\ge 1$, which is a contradiction.

Case IIb:

$$\begin{array}{c}\hfill {\wp }_{cb}({\theta }_{2n+2},{\theta }_{2n+3})⪯⋏{\displaystyle \frac{1}{2}}({\wp }_{cb}({\theta }_{2n+1},{\theta }_{2n+2})+{\wp }_{cb}({\theta }_{2n+2},{\theta }_{2n+3})).\end{array}$$

This implies that

$$\begin{array}{c}\hfill {\wp }_{cb}({\theta }_{2n+2},{\theta }_{2n+3})⪯{\displaystyle \frac{⋏}{(2-⋏)}}{\wp }_{cb}({\theta }_{2n+1},{\theta }_{2n+2}).\end{array}$$

(4)

Since $a:={\displaystyle \frac{⋏}{2-⋏}}<1$, we get ${\wp }_{cb}({\theta }_{n+1},{\theta }_{n+2})⪯a{\wp }_{cb}({\theta }_n,{\theta }_{n+1})$. Therefore ${\{{\theta }_n\}}_{n\in \mathbb{N}}$ is a Cauchy sequence in G.

Case IIc:

$$\begin{array}{c}\hfill {\wp }_{cb}({\theta }_{2n+2},{\theta }_{2n+3})⪯⋏{\wp }_{cb}({\theta }_{2n+1},{\theta }_{2n+2}).\end{array}$$

(5)

From (3) and (5), $\forall n=0,1,2,\dots$, we get

$$\begin{array}{c}\hfill {\wp }_{cb}({\theta }_{n+1},{\theta }_{n+2})⪯⋏{\wp }_{cb}({\theta }_n,{\theta }_{n+1})⪯\dots ⪯{⋏}^{n+1}{\wp }_{cb}({\theta }_0,{\theta }_1).\end{array}$$

For $m,n\in \mathbb{N}$, with $m>n$, we have

$$\begin{array}{cc}\hfill {\wp }_{cb}({\theta }_n,{\theta }_m)& ⪯{\wp }_{cb}({\theta }_n,{\theta }_{n+1})+{\wp }_{cb}({\theta }_{n+1},{\theta }_m)-{\wp }_{cb}({\theta }_{n+1},{\theta }_{n+1})\hfill \\ \hfill & ⪯{\wp }_{cb}({\theta }_n,{\theta }_{n+1})+{\wp }_{cb}({\theta }_{n+1},{\theta }_m)\hfill \\ \hfill & ⪯{\wp }_{cb}({\theta }_n,{\theta }_{n+1})+{\wp }_{cb}({\theta }_{n+1},{\theta }_{n+2})+{\wp }_{cb}({\theta }_{n+2},{\theta }_m)\hfill \\ \hfill & -{\wp }_{cb}({\theta }_{n+2},{\theta }_{n+2})\hfill \\ \hfill & ⪯{\wp }_{cb}({\theta }_n,{\theta }_{n+1})+{\wp }_{cb}({\theta }_{n+1},{\theta }_{n+2})+{\wp }_{cb}({\theta }_{n+2},{\theta }_m)\hfill \\ \hfill & ⪯{\wp }_{cb}({\theta }_n,{\theta }_{n+1})+{\wp }_{cb}({\theta }_{n+1},{\theta }_{n+2})+{\wp }_{cb}({\theta }_{n+2},{\theta }_{n+3})\hfill \\ \hfill & +\dots +{\wp }_{cb}({\theta }_{m-2},{\theta }_{m-1})+{\wp }_{cb}({\theta }_{m-1},{\theta }_m).\hfill \end{array}$$

Moreover, by using (5), we get

$$\begin{array}{cc}\hfill {\wp }_{cb}({\theta }_n,{\theta }_m)& ⪯{⋏}^n{\wp }_{cb}({\theta }_0,{\theta }_1)+{⋏}^{n+1}{\wp }_{cb}({\theta }_0,{\theta }_1)+{⋏}^{n+2}{\wp }_{cb}({\theta }_0,{\theta }_1)\hfill \\ \hfill & +\dots +{⋏}^{m-2}{\wp }_{cb}({\theta }_0,{\theta }_1)+{⋏}^{m-1}{\wp }_{cb}({\theta }_0,{\theta }_1)\hfill \\ \hfill & =\sum _{i=1}^{m-n}{⋏}^{i+n-1}{\wp }_{cb}({\theta }_0,{\theta }_1).\hfill \end{array}$$

Therefore

$$\begin{array}{cc}\hfill |{\wp }_{cb}({\theta }_n,{\theta }_m)|& \le \sum _{i=1}^{m-n}{⋏}^{i+n-1}|{\wp }_{cb}({\theta }_0,{\theta }_1)|=\sum _{t=n}^{m-1}{⋏}^t\left|{\wp }_{cb}({\theta }_0,{\theta }_1)\right|\hfill \\ \hfill & \le \sum _{i=n}^{\infty }\left|{\wp }_{cb}({\theta }_0,{\theta }_1)\right|\hfill \\ \hfill & ={\displaystyle \frac{{⋏}^n}{1-⋏}}\left|{\wp }_{cb}({\theta }_0,{\theta }_1)\right|.\hfill \end{array}$$

Then, we have

$$\begin{array}{cc}\hfill |{\wp }_{cb}({\theta }_n,{\theta }_m)|& \le {\displaystyle \frac{{⋏}^n}{1-⋏}}\left|{\wp }_{cb}({\theta }_0,{\theta }_1)\right|\to 0asn\to \infty .\hfill \end{array}$$

Hence, $\{{\theta }_n\}$ is a Cauchy sequence in G.

Case III:

If $max\{{\wp }_{cb}({\theta }_{2n},{\theta }_{2n+1}),{\wp }_{cb}({\theta }_{2n+1},{\theta }_{2n+2}),{\displaystyle \frac{1}{2}}({\wp }_{cb}({\theta }_{2n},{\theta }_{2n+1})+{\wp }_{cb}({\theta }_{2n+1},{\theta }_{2n+2}))\}$$={\displaystyle \frac{1}{2}}({\wp }_{cb}({\theta }_{2n},{\theta }_{2n+1})+{\wp }_{cb}({\theta }_{2n+1},{\theta }_{2n+2}))$.

Then, we have

$$\begin{array}{c}\hfill {\wp }_{cb}({\theta }_{2n+1},{\theta }_{2n+2})⪯{\displaystyle \frac{⋏}{2}}({\wp }_{cb}({\theta }_{2n},{\theta }_{2n+1})+{\wp }_{cb}({\theta }_{2n+1},{\theta }_{2n+2}))\end{array}$$

Hence,

$$\begin{array}{c}\hfill {\wp }_{cb}({\theta }_{2n+1},{\theta }_{2n+2})⪯{\displaystyle \frac{⋏}{2-⋏}}{\wp }_{cb}({\theta }_{2n},{\theta }_{2n+1}).\end{array}$$

(6)

For the next step, we have

$$\begin{array}{cc}\hfill {\wp }_{cb}({\theta }_{2n+2},{\theta }_{2n+3})& ⪯⋏max\{{\wp }_{cb}({\theta }_{2n+1},{\theta }_{2n+2}),{\wp }_{cb}({\theta }_{2n+2},{\theta }_{2n+3}),\hfill \\ \hfill & {\displaystyle \frac{1}{2}}({\wp }_{cb}({\theta }_{2n+1},{\theta }_{2n+2})+{\wp }_{cb}({\theta }_{2n+2},{\theta }_{2n+3}))\}.\hfill \end{array}$$

Then, we have the following three cases:

Case IIIa:

$$\begin{array}{c}\hfill {\wp }_{cb}({\theta }_{2n+2},{\theta }_{2n+3})⪯⋏{\wp }_{cb}({\theta }_{2n+2},{\theta }_{2n+3}),\end{array}$$

which implies $⋏\ge 1$, which is a contradiction.

Case IIIb:

$$\begin{array}{c}\hfill {\wp }_{cb}({\theta }_{2n+2},{\theta }_{2n+3})⪯⋏{\wp }_{cb}({\theta }_{2n+1},{\theta }_{2n+2}).\end{array}$$

(7)

Then by (6) and (7), we get ${\wp }_{cb}({\theta }_{n+1},{\theta }_{n+2})⪯\gamma {\wp }_{cb}({\theta }_n,{\theta }_{n+1})$, where $\gamma =max\left\{⋏,{\displaystyle \frac{⋏}{2-⋏}}\right\}<1$. Hence ${\{{\theta }_n\}}_{n\in \mathbb{N}}$ is a Cauchy sequence in G.

Case IIIc:

$$\begin{array}{c}\hfill {\wp }_{cb}({\theta }_{2n+2},{\theta }_{2n+3})⪯{\displaystyle \frac{1}{2}}({\wp }_{cb}({\theta }_{2n+1},{\theta }_{2n+2})+{\wp }_{cb}({\theta }_{2n+2},{\theta }_{2n+3})).\end{array}$$

Hence, we obtain

$$\begin{array}{c}\hfill {\wp }_{cb}({\theta }_{2n+2},{\theta }_{2n+3})⪯{\displaystyle \frac{⋏}{(2-⋏)}}{\wp }_{cb}({\theta }_{2n+1},{\theta }_{2n+2}).\end{array}$$

(8)

Using (6) and (8) yields

$$\begin{array}{c}\hfill {\wp }_{cb}({\theta }_{n+1},{\theta }_{n+2})⪯\wr {\wp }_{cb}({\theta }_n,{\theta }_{n+1}),\end{array}$$

(9)

where $0\le \wr ={\displaystyle \frac{⋏}{2-⋏}}<1$.

Then, $\forall n=0,1,2,\dots ,$ and we get

$$\begin{array}{c}\hfill {\wp }_{cb}({\theta }_{n+1},{\theta }_{n+2})⪯\wr {\wp }_{cb}({\theta }_n,{\theta }_{n+1})⪯\dots ⪯{\wr }^{n+1}{\wp }_{cb}({\theta }_0,{\theta }_1).\end{array}$$

For $m,n\in \mathbb{N}$, with $m>n$, we have

$$\begin{array}{cc}\hfill {\wp }_{cb}({\theta }_n,{\theta }_m)& ⪯{\wp }_{cb}({\theta }_n,{\theta }_{n+1})+{\wp }_{cb}({\theta }_{n+1},{\theta }_m)-{\wp }_{cb}({\theta }_{n+1},{\theta }_{n+1})\hfill \\ \hfill & ⪯{\wp }_{cb}({\theta }_n,{\theta }_{n+1})+{\wp }_{cb}({\theta }_{n+1},{\theta }_m)\hfill \\ \hfill & ⪯{\wp }_{cb}({\theta }_n,{\theta }_{n+1})+{\wp }_{cb}({\theta }_{n+1},{\theta }_{n+2})+{\wp }_{cb}({\theta }_{n+2},{\theta }_m)\hfill \\ \hfill & -{\wp }_{cb}({\theta }_{n+2},{\theta }_{n+2})\hfill \\ \hfill & ⪯{\wp }_{cb}({\theta }_n,{\theta }_{n+1})+{\wp }_{cb}({\theta }_{n+1},{\theta }_{n+2})+{\wp }_{cb}({\theta }_{n+2},{\theta }_m)\hfill \\ \hfill & ⪯{\wp }_{cb}({\theta }_n,{\theta }_{n+1})+{\wp }_{cb}({\theta }_{n+1},{\theta }_{n+2})+{\wp }_{cb}({\theta }_{n+2},{\theta }_{n+3})\hfill \\ \hfill & +\dots +{\wp }_{cb}({\theta }_{m-2},{\theta }_{m-1}))+{\wp }_{cb}({\theta }_{m-1},{\theta }_m).\hfill \end{array}$$

Using (9), we get

$$\begin{array}{cc}\hfill {\wp }_{cb}({\theta }_n,{\theta }_m)& ⪯{\wr }^n{\wp }_{cb}({\theta }_0,{\theta }_1)+{\wr }^{n+1}{\wp }_{cb}({\theta }_0,{\theta }_1)+{\wr }^{n+2}{\wp }_{cb}({\theta }_0,{\theta }_1)\hfill \\ \hfill & +\dots +{\wr }^{m-2}{\wp }_{cb}({\theta }_0,{\theta }_1)+{\wr }^{m-1}{\wp }_{cb}({\theta }_0,{\theta }_1)\hfill \\ \hfill & =\sum _{i=1}^{m-n}{\wr }^{i+n-1}{\wp }_{cb}({\theta }_0,{\theta }_1).\hfill \end{array}$$

Therefore,

$$\begin{array}{cc}\hfill |{\wp }_{cb}({\theta }_n,{\theta }_m)|& \le \sum _{i=1}^{m-n}{\wr }^{i+n-1}|{\wp }_{cb}({\theta }_0,{\theta }_1)|=\sum _{t=n}^{m-1}{\wr }^t\left|{\wp }_{cb}({\theta }_0,{\theta }_1)\right|\hfill \\ \hfill & \le \sum _{i=n}^{\infty }{\wr }^t\left|{\wp }_{cb}({\theta }_0,{\theta }_1)\right|\hfill \\ \hfill & ={\displaystyle \frac{{\wr }^n}{1-\wr }}\left|{\wp }_{cb}({\theta }_0,{\theta }_1)\right|.\hfill \end{array}$$

Hence, we have

$$\begin{array}{cc}\hfill |{\wp }_{cb}({\theta }_n,{\theta }_m)|& \le {\displaystyle \frac{{\wr }^n}{1-\wr }}\left|{\wp }_{cb}({\theta }_0,{\theta }_1)\right|\to 0asn\to \infty .\hfill \end{array}$$

Hence, $\{{\theta }_n\}$ is a Cauchy sequence in G. In all cases above discussed, we get the sequence ${\{{\theta }_n\}}_{n\in \mathbb{N}}$, which is a Cauchy sequence. Since G is complete, there exists ${\theta }^{*}\in G$ such that ${\theta }_n\to {\theta }^{*}$ as $n\to \infty$ and

$$\begin{array}{c}\hfill {\wp }_{cb}({\theta }^{*},{\theta }^{*})=\underset{n\to \infty }{lim}{\wp }_{cb}({\theta }^{*},{\theta }_n)=\underset{n\to \infty }{lim}{\wp }_{cb}({\theta }_n,{\theta }_n)=0\end{array}$$

By the continuity of $\mathsf{\Pi }$, it follows that ${\theta }_{2n+1}=\mathsf{\Pi }{\theta }_{2n}\to \mathsf{\Pi }{\theta }^{*}$ as $n\to \infty$.

$$\begin{array}{c}\hfill i.e.,{\wp }_{cb}(\mathsf{\Pi }{\theta }^{*},\mathsf{\Pi }{\theta }^{*})=\underset{n\to \infty }{lim}{\wp }_{cb}(\mathsf{\Pi }{\theta }^{*},\mathsf{\Pi }{\theta }_{2n})=\underset{n\to \infty }{lim}{\wp }_{cb}(\mathsf{\Pi }{\theta }_{2n},\mathsf{\Pi }{\theta }_{2n}).\end{array}$$

However,

$$\begin{array}{c}\hfill {\wp }_{cb}(\mathsf{\Pi }{\theta }^{*},\mathsf{\Pi }{\theta }^{*})=\underset{n\to \infty }{lim}{\wp }_{cb}(\mathsf{\Pi }{\theta }_{2n},\mathsf{\Pi }{\theta }_{2n})=\underset{n\to \infty }{lim}{\wp }_{cb}({\theta }_{2n+1},{\theta }_{2n+1})=0.\end{array}$$

Next, we have to prove that ${\theta }^{*}$ is a fixed point of $\mathsf{\Pi }$.

$$\begin{array}{c}\hfill {\wp }_{cb}(\mathsf{\Pi }{\theta }^{*},{\theta }^{*})⪯{\wp }_{cb}(\mathsf{\Pi }{\theta }^{*},\mathsf{\Pi }{\theta }_{2n})+{\wp }_{cb}(\mathsf{\Pi }{\theta }_{2n},{\theta }^{*})-{\wp }_{cb}(\mathsf{\Pi }{\theta }_{2n},\mathsf{\Pi }{\theta }_{2n}).\end{array}$$

As $n\to \infty$, we obtain $|{\wp }_{cb}(\mathsf{\Pi }{\theta }^{*},{\theta }^{*})|\le 0$. Thus, ${\wp }_{cp}(\mathsf{\Pi }{\theta }^{*},{\theta }^{*})=0$. Hence ${\wp }_{cb}({\theta }^{*},{\theta }^{*})={\wp }_{cb}({\theta }^{*},\mathsf{\Pi }{\theta }^{*})={\wp }_{cb}(\mathsf{\Pi }{\theta }^{*},\mathsf{\Pi }{\theta }^{*})=0$ and $\mathsf{\Pi }{\theta }^{*}={\theta }^{*}$. In the same way, we have ${\theta }^{*}\in G$ such that ${\theta }_n\to {\theta }^{*}$ as $n\to \infty$ and

$$\begin{array}{c}\hfill {\wp }_{cb}({\theta }^{*},{\theta }^{*})=\underset{n\to \infty }{lim}{\wp }_{cb}({\theta }^{*},{\theta }_n)=\underset{n\to \infty }{lim}{\wp }_{cb}({\theta }_n,{\theta }_n)=0\end{array}$$

By the continuity of $\mathsf{\Pi }$, it follows ${\theta }_{2n+2}=\mathsf{\Psi }{\theta }_{2n+1}\to \mathsf{\Psi }{\theta }^{*}$ as $n\to \infty$.

$$\begin{array}{c}\hfill i.e.,{\wp }_{cb}(\mathsf{\Psi }{\theta }^{*},\mathsf{\Psi }{\theta }^{*})=\underset{n\to \infty }{lim}{\wp }_{cb}(\mathsf{\Psi }{\theta }^{*},\mathsf{\Psi }{\theta }_{2n+1})=\underset{n\to \infty }{lim}{\wp }_{cb}(\mathsf{\Psi }{\theta }_{2n+1},\mathsf{\Psi }{\theta }_{2n+1}).\end{array}$$

However,

$$\begin{array}{c}\hfill {\wp }_{cb}(\mathsf{\Psi }{\theta }^{*},\mathsf{\Psi }{\theta }^{*})=\underset{n\to \infty }{lim}{\wp }_{cb}(\mathsf{\Psi }{\theta }_{2n+1},\mathsf{\Psi }{\theta }_{2n+1})=\underset{n\to \infty }{lim}{\wp }_{cb}({\theta }_{2n+2},{\theta }_{2n+2})=0.\end{array}$$

Next we have to prove that ${\theta }^{*}$ is a fixed point of $\mathsf{\Psi }$.

$$\begin{array}{c}\hfill {\wp }_{cb}(\mathsf{\Psi }{\theta }^{*},{\theta }^{*})⪯{\wp }_{cb}(\mathsf{\Psi }{\theta }^{*},\mathsf{\Psi }{\theta }_{2n+1})+{\wp }_{cb}(\mathsf{\Psi }{\theta }_{2n+1},{\theta }^{*})-{\wp }_{cb}(\mathsf{\Psi }{\theta }_{2n+1},\mathsf{\Pi }{\theta }_{2n+1}).\end{array}$$

As $n\to \infty$, we obtain $|{\wp }_{cb}(\mathsf{\Psi }{\theta }^{*},{\theta }^{*})|\le 0$. Thus, ${\wp }_{cp}(\mathsf{\Psi }{\theta }^{*},{\theta }^{*})=0$. Hence, ${\wp }_{cb}({\theta }^{*},{\theta }^{*})={\wp }_{cb}({\theta }^{*},\mathsf{\Psi }{\theta }^{*})={\wp }_{cb}(\mathsf{\Psi }{\theta }^{*},\mathsf{\Psi }{\theta }^{*})=0$ and $\mathsf{\Psi }{\theta }^{*}={\theta }^{*}$. Therefore, ${\theta }^{*}$ is a common fixed point of the pair $(\mathsf{\Pi },\mathsf{\Psi })$.

To prove uniqueness, let us consider ${\omega }^{*}\in G$ is another common fixed point for the pair $(\mathsf{\Pi },\mathsf{\Psi })$. Then

$$\begin{array}{cc}\hfill {\wp }_{cb}({\theta }^{*},{\omega }^{*})& ={\wp }_{cb}(\mathsf{\Pi }{\theta }^{*},\mathsf{\Psi }{\omega }^{*})\hfill \\ \hfill & ⪯⋏max\{{\wp }_{cb}({\theta }^{*},{\omega }^{*}),{\wp }_{cb}({\theta }^{*},\mathsf{\Pi }{\theta }^{*}),{\wp }_{cb}({\omega }^{*},\mathsf{\Psi }{\omega }^{*}),\hfill \\ \hfill & {\displaystyle \frac{1}{2}}({\wp }_{cb}({\theta }^{*},\mathsf{\Psi }{\omega }^{*})+{\wp }_{cb}({\omega }^{*},\mathsf{\Pi }{\theta }^{*}))\}\hfill \\ \hfill & ⪯⋏max\{{\wp }_{cb}({\theta }^{*},{\omega }^{*}),{\wp }_{cb}({\theta }^{*},{\theta }^{*}),{\wp }_{cb}({\omega }^{*},{\omega }^{*}),\hfill \\ \hfill & {\displaystyle \frac{1}{2}}({\wp }_{cb}({\theta }^{*},{\omega }^{*})+{\wp }_{cb}({\omega }^{*},{\theta }^{*}))\}\hfill \\ \hfill & ⪯⋏{\wp }_{cb}({\theta }^{*},{\omega }^{*}).\hfill \end{array}$$

This implies that ${\theta }^{*}={\omega }^{*}$. □

In the absence of the continuity condition for the mappings $\mathsf{\Pi }$ and $\mathsf{\Psi }$, we get the the following theorem.

Theorem 3.

Let $(G,{\wp }_{cb})$ be a complete CPMS and $\mathsf{\Pi },\mathsf{\Psi }:G\to G$ be two mappings such that

$$\begin{array}{cc}\hfill {\wp }_{cb}(\mathsf{\Pi }\theta ,\mathsf{\Psi }\omega )& ⪯⋏max\{{\wp }_{cb}(\theta ,\omega ),{\wp }_{cb}(\theta ,\mathsf{\Pi }\theta ),{\wp }_{cb}(\omega ,\mathsf{\Psi }\omega ),\hfill \\ \hfill & {\displaystyle \frac{1}{2}}({\wp }_{cb}(\theta ,\mathsf{\Psi }\omega )+{\wp }_{cb}(\omega ,\mathsf{\Pi }\theta ))\},\hfill \end{array}$$

(10)

for all $\theta ,\omega \in G$, where $0\le ⋏<1$ and ${\wp }_{cb}(\mathsf{\Pi }\theta ,\mathsf{\Psi }\omega )\ne 0$. Then, the pair $(\mathsf{\Pi },\mathsf{\Psi })$ has a unique common fixed point and ${\wp }_{cb}({\theta }^{*},{\theta }^{*})=0$.

Proof. 

Following from Theorem 2, we get that the sequence $\{{\theta }_n\}$ is a Cauchy sequence. Since G is complete, there exists ${\theta }^{*}\in G$ such that ${\theta }_n\to {\theta }^{*}$ as $n\to \infty$.

Since $\mathsf{\Pi }$ and $\mathsf{\Psi }$ are not continuous, we have ${\wp }_{cb}({\theta }^{*},\mathsf{\Pi }{\theta }^{*})=\vartheta >0$.

Then, we estimate

$$\begin{array}{cc}\hfill \vartheta & ={\wp }_{cb}({\theta }^{*},\mathsf{\Pi }{\theta }^{*})\hfill \\ \hfill & ⪯{\wp }_{cb}({\theta }^{*},{\theta }_{2i+2})+{\wp }_{cb}({\theta }_{2i+2},\mathsf{\Pi }{\theta }^{*})-{\wp }_{cb}({\theta }_{2i+2},{\theta }_{2i+2})\hfill \\ \hfill & ⪯{\wp }_{cb}({\theta }^{*},{\theta }_{2i+2})+{\wp }_{cb}({\theta }_{2i+2},\mathsf{\Pi }{\theta }^{*})\hfill \\ \hfill & ⪯{\wp }_{cb}({\theta }^{*},{\theta }_{2i+2})+{\wp }_{cb}(\mathsf{\Psi }{\theta }_{2i+1},\mathsf{\Pi }{\theta }^{*})\hfill \\ \hfill & ⪯{\wp }_{cb}({\theta }^{*},{\theta }_{2i+2})+⋏max\{{\wp }_{cb}({\theta }_{2i+1},{\theta }^{*}),{\wp }_{cb}({\theta }_{2i+1},\mathsf{\Psi }{\theta }_{2i+1}),{\wp }_{cb}({\theta }^{*},\mathsf{\Pi }{\theta }^{*}),\hfill \\ \hfill & {\displaystyle \frac{1}{2}}({\wp }_{cb}({\theta }_{2i+1},\mathsf{\Pi }{\theta }^{*})+{\wp }_{cb}({\theta }^{*},\mathsf{\Psi }{\theta }_{2i+1}))\}\hfill \\ \hfill & ⪯{\wp }_{cb}({\theta }^{*},{\theta }_{2i+2})+⋏max\{{\wp }_{cb}({\theta }_{2i+1},{\theta }^{*}),{\wp }_{cb}({\theta }_{2i+1},{\theta }_{2i+2}),{\wp }_{cb}({\theta }^{*},\mathsf{\Pi }{\theta }^{*}),\hfill \\ \hfill & {\displaystyle \frac{1}{2}}({\wp }_{cb}({\theta }_{2i+1},\mathsf{\Pi }{\theta }^{*})+{\wp }_{cb}({\theta }^{*},{\theta }_{2i+2}))\}\hfill \\ \hfill & ⪯{\wp }_{cb}({\theta }^{*},{\theta }_{2i+2})+⋏{\wp }_{cb}({\theta }^{*},\mathsf{\Pi }{\theta }^{*})\hfill \\ \hfill & ⪯s{\wp }_{cb}({\theta }^{*},{\theta }_{2i+2})+⋏\vartheta .\hfill \end{array}$$

This yields

$$\begin{array}{c}\hfill \left|\vartheta \right|\le |{\wp }_{cb}({\theta }^{*},{\theta }_{2i+2})|+⋏|\vartheta |.\end{array}$$

By definition,

$$\underset{i}{lim}{\wp }_{cb}({\theta }^{*},{\theta }_{2i+2})={\wp }_{cb}({\theta }^{*},{\theta }^{*})$$

From the Cauchy property of $\left({\theta }_n\right)$, the above limit is zero, and then,

$$\begin{array}{c}\hfill \left|\vartheta \right|\le ⋏\left|\vartheta \right|.\end{array}$$

Hence, $⋏\ge 1$, which is a contradiction. Then ${\theta }^{*}=\mathsf{\Pi }{\theta }^{*}$. In the same way, we obtain ${\theta }^{*}=\mathsf{\Psi }{\theta }^{*}$. Hence ${\theta }^{*}$ is a common fixed point for the pair $(\mathsf{\Pi },\mathsf{\Psi })$ and ${\wp }_{cb}({\theta }^{*},{\theta }^{*})={\wp }_{cb}({\theta }^{*},\mathsf{\Psi }{\theta }^{*})={\wp }_{cb}(\mathsf{\Psi }{\theta }^{*},\mathsf{\Psi }{\theta }^{*})=0$. Uniqueness of the common fixed point ${\theta }^{*}$ follows from Theorem 2. □

For $\mathsf{\Pi }=\mathsf{\Psi }$, we get the following fixed points results on CPMS.

Theorem 4.

Let $(G,{\wp }_{cb})$ be a complete CPMS and $\mathsf{\Pi }:G\to G$ be a continuous mapping such that

$$\begin{array}{cc}\hfill {\wp }_{cb}(\mathsf{\Pi }\theta ,\mathsf{\Pi }\omega )& ⪯⋏max\{{\wp }_{cb}(\theta ,\omega ),{\wp }_{cb}(\theta ,\mathsf{\Pi }\theta ),{\wp }_{cb}(\omega ,\mathsf{\Pi }\omega ),\hfill \\ \hfill & {\displaystyle \frac{1}{2}}({\wp }_{cb}(\theta ,\mathsf{\Pi }\omega )+{\wp }_{cb}(\omega ,\mathsf{\Pi }\theta ))\},\hfill \end{array}$$

(11)

for all $\theta ,\omega \in G$, where $0\le ⋏<1$ and ${\wp }_{cb}(\mathsf{\Pi }\theta ,\mathsf{\Pi }\omega )\ne 0$. Then the pair Π has a unique fixed point and ${\wp }_{cb}({\theta }^{*},{\theta }^{*})=0$.

Remark 1.

Similarly, we get a fixed point result in the absence of continuity condition for the mapping Π.

Corollary 1.

Let $(G,{\wp }_{cb})$ be a complete CPMS and $\mathsf{\Psi }:G\to G$ be a continuous mapping such that

$$\begin{array}{cc}\hfill {\wp }_{cb}({\mathsf{\Psi }}^n\theta ,{\mathsf{\Psi }}^n\omega )& ⪯⋏max\{{\wp }_{cb}(\theta ,\omega ),{\wp }_{cb}(\theta ,{\mathsf{\Psi }}^n\theta ),{\wp }_{cb}(\omega ,{\mathsf{\Psi }}^n\omega ),\hfill \\ \hfill & {\displaystyle \frac{1}{2}}({\wp }_{cb}(\theta ,{\mathsf{\Psi }}^n\omega )+{\wp }_{cb}(\omega ,{\mathsf{\Psi }}^n\theta ))\},\hfill \end{array}$$

for all $\theta ,\omega \in G$, where $0\le ⋏<1$, ${\wp }_{cb}({\mathsf{\Psi }}^n\theta ,{\mathsf{\Psi }}^n\omega )\ne 0$ and $n\in \mathbb{N}$. Then, Ψ has a unique fixed point and ${\wp }_{cb}({\theta }^{*},{\theta }^{*})=0$.

Proof. 

By Theorem 2, we get ${\theta }^{*}\in G$ such that ${\mathsf{\Psi }}^n{\theta }^{*}={\theta }^{*}$ and ${\wp }_{cb}({\theta }^{*},{\theta }^{*})=0$. Then, we get

$$\begin{array}{cc}\hfill {\wp }_{cb}(\mathsf{\Psi }{\theta }^{*},{\theta }^{*})& ={\wp }_{cb}(\mathsf{\Psi }{\mathsf{\Psi }}^n{\theta }^{*},{\mathsf{\Psi }}^n{\theta }^{*})={\wp }_{cb}({\mathsf{\Psi }}^n\mathsf{\Psi }{\theta }^{*},{\mathsf{\Psi }}^n{\theta }^{*})\hfill \\ \hfill & ⪯⋏max\{{\wp }_{cb}(\mathsf{\Psi }{\theta }^{*},{\theta }^{*}),{\wp }_{cb}(\mathsf{\Psi }{\theta }^{*},{\mathsf{\Psi }}^n\mathsf{\Psi }{\theta }^{*}),{\wp }_{cb}({\theta }^{*},{\mathsf{\Psi }}^n{\theta }^{*}),\hfill \\ \hfill & {\displaystyle \frac{1}{2}}({\wp }_{cb}(\mathsf{\Psi }{\theta }^{*},{\mathsf{\Psi }}^n{\theta }^{*})+{\wp }_{cb}({\theta }^{*},{\mathsf{\Psi }}^n\mathsf{\Psi }{\theta }^{*}))\}\hfill \\ \hfill & ⪯⋏max\{{\wp }_{cb}(\mathsf{\Psi }{\theta }^{*},{\theta }^{*}),{\wp }_{cb}(\mathsf{\Psi }{\theta }^{*},\mathsf{\Psi }{\theta }^{*}),{\wp }_{cb}({\theta }^{*},{\theta }^{*}),\hfill \\ \hfill & {\displaystyle \frac{1}{2}}({\wp }_{cb}(\mathsf{\Psi }{\theta }^{*},{\theta }^{*})+{\wp }_{cb}({\theta }^{*},\mathsf{\Psi }{\theta }^{*}))\}\hfill \\ \hfill & =⋏{\wp }_{cb}(\mathsf{\Psi }{\theta }^{*},{\theta }^{*}).\hfill \end{array}$$

Hence ${\mathsf{\Psi }}^n{\theta }^{*}=\mathsf{\Psi }{\theta }^{*}={\theta }^{*}$. Then $\mathsf{\Psi }$ has a unique fixed point. □

Remark 2.

From the above Corollary 1, similarly, we get a fixed-point result in the absence of continuity condition for the mapping Ψ.

Next, we present a new generalization of a common fixed point theorem on CPMS.

Theorem 5.

Let $(G,{\wp }_{cb})$ be a complete CPMS and $\mathsf{\Pi },\mathsf{\Psi }:G\to G$ be two continuous mappings such that

$$\begin{array}{cc}\hfill {\wp }_{cb}(\mathsf{\Pi }\theta ,\mathsf{\Psi }\omega )& ⪯⋏max\left\{{\wp }_{cb}(\theta ,\omega ),{\displaystyle \frac{{\wp }_{cb}(\theta ,\mathsf{\Pi }\theta ){\wp }_{cb}(\omega ,\mathsf{\Psi }\omega )}{1+{\wp }_{cb}(\theta ,\omega )}},{\displaystyle \frac{{\wp }_{cb}(\theta ,\mathsf{\Pi }\theta ){\wp }_{cb}(\mathsf{\Pi }\theta ,\mathsf{\Psi }\omega )}{1+{\wp }_{cb}(\theta ,\omega )}}\right\},\hfill \end{array}$$

(12)

for all $\theta ,\omega \in G$, where $0\le ⋏<1$ and ${\wp }_{cb}(\mathsf{\Pi }\theta ,\mathsf{\Psi }\omega )\ne 0$. Then, the pair $(\mathsf{\Pi },\mathsf{\Psi })$ has a unique common fixed point and ${\wp }_{cb}({\theta }^{*},{\theta }^{*})=0$.

Proof. 

Let ${\theta }_0$ be arbitrary point in G and define a sequence $\{{\theta }_n\}$ as follows:

$$\begin{array}{c}\hfill {\theta }_{2n+1}=\mathsf{\Pi }{\theta }_{2n}and{\theta }_{2n+2}=\mathsf{\Psi }{\theta }_{2n+1},n=0,1,2,\dots \end{array}$$

(13)

Then, by (12) and (13), we obtain

$$\begin{array}{cc}\hfill {\wp }_{cb}({\theta }_{2n+1},{\theta }_{2n+2})& ={\wp }_{cb}(\mathsf{\Pi }{\theta }_{2n},\mathsf{\Psi }{\theta }_{2n+1})\hfill \\ \hfill & ⪯⋏max\{{\wp }_{cb}({\theta }_{2n},{\theta }_{2n+1}),{\displaystyle \frac{{\wp }_{cb}({\theta }_{2n},{\theta }_{2n+1}){\wp }_{cb}(\mathsf{\Psi }{\theta }_{2n+1},\mathsf{\Pi }{\theta }_{2n})}{1+{\wp }_{cb}({\theta }_{2n},{\theta }_{2n+1})}},\hfill \\ \hfill & {\displaystyle \frac{{\wp }_{cb}({\theta }_{2n},\mathsf{\Pi }{\theta }_{2n},){\wp }_{cb}(\mathsf{\Pi }{\theta }_{2n},\mathsf{\Psi }{\theta }_{2n+1})}{1+{\wp }_{cb}({\theta }_{2n},{\theta }_{2n+1})}}\}\hfill \\ \hfill & ⪯⋏max\{{\wp }_{cb}({\theta }_{2n},{\theta }_{2n+1}),{\displaystyle \frac{{\wp }_{cb}({\theta }_{2n},{\theta }_{2n+1}){\wp }_{cb}({\theta }_{2n+1},{\theta }_{2n+2})}{1+{\wp }_{cb}({\theta }_{2n},{\theta }_{2n+1})}},\hfill \\ \hfill & {\displaystyle \frac{{\wp }_{cb}({\theta }_{2n},{\theta }_{2n+1}){\wp }_{cb}({\theta }_{2n+1},{\theta }_{2n+2})}{1+{\wp }_{cb}({\theta }_{2n},{\theta }_{2n+1})}}\}\hfill \\ \hfill & ⪯⋏max\{{\wp }_{cb}({\theta }_{2n},{\theta }_{2n+1}),{\wp }_{cb}({\theta }_{2n+1},{\theta }_{2n+2})\}.\hfill \end{array}$$

If $max\{{\wp }_{cb}({\theta }_{2n},{\theta }_{2n+1}),{\wp }_{cb}({\theta }_{2n+1},{\theta }_{2n+2})\}={\wp }_{cb}({\theta }_{2n+1},{\theta }_{2n+2})$, then

$$\begin{array}{c}\hfill {\wp }_{cb}({\theta }_{2n+1},{\theta }_{2n+2})⪯⋏{\wp }_{cb}({\theta }_{2n+1},{\theta }_{2n+2}).\end{array}$$

This shows that $⋏\ge 1$, which is a contradiction. Therefore

$$\begin{array}{c}\hfill {\wp }_{cb}({\theta }_{2n+1},{\theta }_{2n+2})⪯⋏{\wp }_{cb}({\theta }_{2n},{\theta }_{2n+1}).\end{array}$$

(14)

Similarly, we obtain

$$\begin{array}{c}\hfill {\wp }_{cb}({\theta }_{2n+2},{\theta }_{2n+3})⪯⋏{\wp }_{cb}({\theta }_{2n+1},{\theta }_{2n+2}).\end{array}$$

(15)

From (14) and (15), $\forall n=0,1,2,\dots$, we get

$$\begin{array}{c}\hfill {\wp }_{cb}({\theta }_{n+1},{\theta }_{n+2})⪯⋏{\wp }_{cb}({\theta }_n,{\theta }_{n+1})⪯\dots ⪯{⋏}^{n+1}{\wp }_{cb}({\theta }_0,{\theta }_1).\end{array}$$

(16)

For $m,n\in \mathbb{N}$, with $m>n$, we have

$$\begin{array}{cc}\hfill {\wp }_{cb}({\theta }_n,{\theta }_m)& ⪯{\wp }_{cb}({\theta }_n,{\theta }_{n+1})+{\wp }_{cb}({\theta }_{n+1},{\theta }_m)-{\wp }_{cb}({\theta }_{n+1},{\theta }_{n+1})\hfill \\ \hfill & ⪯{\wp }_{cb}({\theta }_n,{\theta }_{n+1})+{\wp }_{cb}({\theta }_{n+1},{\theta }_m)\hfill \\ \hfill & ⪯{\wp }_{cb}({\theta }_n,{\theta }_{n+1})+{\wp }_{cb}({\theta }_{n+1},{\theta }_{n+2})+{\wp }_{cb}({\theta }_{n+2},{\theta }_m)\hfill \\ \hfill & -{\wp }_{cb}({\theta }_{n+2},{\theta }_{n+2})\hfill \\ \hfill & ⪯{\wp }_{cb}({\theta }_n,{\theta }_{n+1})+{\wp }_{cb}({\theta }_{n+1},{\theta }_{n+2})+{\wp }_{cb}({\theta }_{n+2},{\theta }_m)\hfill \\ \hfill & ⪯{\wp }_{cb}({\theta }_n,{\theta }_{n+1})+{\wp }_{cb}({\theta }_{n+1},{\theta }_{n+2})+{\wp }_{cb}({\theta }_{n+2},{\theta }_{n+3})\hfill \\ \hfill & +\dots +{\wp }_{cb}({\theta }_{m-2},{\theta }_{m-1})+s^{m-n}{\wp }_{cb}({\theta }_{m-1},{\theta }_m).\hfill \end{array}$$

By using (16), we get

$$\begin{array}{cc}\hfill {\wp }_{cb}({\theta }_n,{\theta }_m)& ⪯{⋏}^n{\wp }_{cb}({\theta }_0,{\theta }_1)+{⋏}^{n+1}{\wp }_{cb}({\theta }_0,{\theta }_1)+{⋏}^{n+2}{\wp }_{cb}({\theta }_0,{\theta }_1)\hfill \\ \hfill & +\dots +{⋏}^{m-2}{\wp }_{cb}({\theta }_0,{\theta }_1)+{⋏}^{m-1}{\wp }_{cb}({\theta }_0,{\theta }_1)\hfill \\ \hfill & =\sum _{i=1}^{m-n}{⋏}^{i+n-1}{\wp }_{cb}({\theta }_0,{\theta }_1).\hfill \end{array}$$

Therefore,

$$\begin{array}{cc}\hfill |{\wp }_{cb}({\theta }_n,{\theta }_m)|& \le \sum _{i=1}^{m-n}{⋏}^{i+n-1}|{\wp }_{cb}({\theta }_0,{\theta }_1)|=\sum _{i=1}^{m-n}{⋏}^t\left|{\wp }_{cb}({\theta }_0,{\theta }_1)\right|\hfill \\ \hfill & \le \sum _{i=n}^{\infty }{⋏}^t\left|{\wp }_{cb}({\theta }_0,{\theta }_1)\right|\hfill \\ \hfill & ={\displaystyle \frac{{⋏}^n}{1-⋏}}\left|{\wp }_{cb}({\theta }_0,{\theta }_1)\right|.\hfill \end{array}$$

Hence, we have

$$\begin{array}{cc}\hfill |{\wp }_{cb}({\theta }_n,{\theta }_m)|& \le {\displaystyle \frac{{⋏}^n}{1-⋏}}\left|{\wp }_{cb}({\theta }_0,{\theta }_1)\right|\to 0asn\to \infty .\hfill \end{array}$$

Hence, $\{{\theta }_n\}$ is a Cauchy sequence in G. Since G is complete, there exists ${\theta }^{*}\in G$ such that ${\theta }_n\to {\theta }^{*}$ as $n\to \infty$ and

$$\begin{array}{c}\hfill {\wp }_{cb}({\theta }^{*},{\theta }^{*})=\underset{n\to \infty }{lim}{\wp }_{cb}({\theta }^{*},{\theta }_n)=\underset{n\to \infty }{lim}{\wp }_{cb}({\theta }_n,{\theta }_n)=0.\end{array}$$

Since $\mathsf{\Psi }$ is continuous, it yields

$$\begin{array}{c}\hfill {\theta }^{*}=\underset{n\to \infty }{lim}{\theta }_{2n+2}=\underset{n\to \infty }{lim}\mathsf{\Psi }{\theta }_{2n+1}=\mathsf{\Psi }\underset{n\to \infty }{lim}{\theta }_{2n+1}=\mathsf{\Psi }{\theta }^{*}.\end{array}$$

Similarly, by the continuity of $\mathsf{\Pi }$, we get ${\theta }^{*}=\mathsf{\Pi }{\theta }^{*}$. Then the pair $(\mathsf{\Pi },\mathsf{\Psi })$ has a common fixed point. To prove uniqueness, let us consider that ${\omega }^{*}\in G$ is another common fixed point for the pair $(\mathsf{\Pi },\mathsf{\Psi })$. Then

$$\begin{array}{cc}\hfill {\wp }_{cb}({\theta }^{*},{\omega }^{*})& ={\wp }_{cb}(\mathsf{\Pi }{\theta }^{*},\mathsf{\Psi }{\omega }^{*})\hfill \\ \hfill & ⪯⋏max\{{\wp }_{cb}({\theta }^{*},{\omega }^{*}),{\displaystyle \frac{{\wp }_{cb}({\theta }^{*},\mathsf{\Pi }{\theta }^{*}){\wp }_{cb}({\omega }^{*},\mathsf{\Psi }{\omega }^{*})}{1+{\wp }_{cb}({\theta }^{*},{\omega }^{*})}},\hfill \\ \hfill & {\displaystyle \frac{{\wp }_{cb}({\theta }^{*},\mathsf{\Pi }{\theta }^{*}){\wp }_{cb}(\mathsf{\Psi }{\omega }^{*},\mathsf{\Pi }{\theta }^{*})}{1+{\wp }_{cb}({\theta }^{*},{\omega }^{*})}}\}\hfill \\ \hfill & ⪯⋏{\wp }_{cb}({\theta }^{*},{\omega }^{*})\hfill \end{array}$$

This implies that ${\theta }^{*}={\omega }^{*}$. □

In the absence of the continuity condition for the mapping $\mathsf{\Pi }$ and $\mathsf{\Psi }$ in the Theorem 5, we obtain the following result.

Theorem 6.

Let $(G,{\wp }_{cb})$ be a complete CPMS and $\mathsf{\Pi },\mathsf{\Psi }:G\to G$ be two mappings such that

$$\begin{array}{cc}\hfill {\wp }_{cb}(\mathsf{\Pi }\theta ,\mathsf{\Psi }\omega )& ⪯⋏max\left\{{\wp }_{cb}(\theta ,\omega ),{\displaystyle \frac{{\wp }_{cb}(\theta ,\mathsf{\Pi }\theta ){\wp }_{cb}(\omega ,\mathsf{\Psi }\omega )}{1+{\wp }_{cb}(\theta ,\omega )}},{\displaystyle \frac{{\wp }_{cb}(\theta ,\mathsf{\Pi }\theta ){\wp }_{cb}(\mathsf{\Pi }\theta ,\mathsf{\Psi }\omega )}{1+{\wp }_{cb}(\theta ,\omega )}}\right\},\hfill \end{array}$$

(17)

for all $\theta ,\omega \in G$, where $0\le ⋏<1$ and ${\wp }_{cb}(\mathsf{\Pi }\theta ,\mathsf{\Psi }\omega )\ne 0$. Then the pair $(\mathsf{\Pi },\mathsf{\Psi })$ has a unique common fixed point and ${\wp }_{cb}({\theta }^{*},{\theta }^{*})=0$.

Proof. 

Following from Theorem 5, we get that the sequence $\{{\theta }_n\}$ is a Cauchy sequence. Since G is complete, then there exists ${\theta }^{*}\in G$ such that ${\theta }_n\to {\theta }^{*}$ as $n\to \infty$ and

$$\begin{array}{c}\hfill {\wp }_{cb}({\theta }^{*},{\theta }^{*})=\underset{n\to \infty }{lim}{\wp }_{cb}({\theta }^{*},{\theta }_n)=\underset{n\to \infty }{lim}{\wp }_{cb}({\theta }_n,{\theta }_n)=0.\end{array}$$

Since $\mathsf{\Pi }$ and $\mathsf{\Psi }$ are not continuous, we have ${\wp }_{cb}({\theta }^{*},\mathsf{\Pi }{\theta }^{*})=\vartheta >0$.

Then, we estimate

$$\begin{array}{cc}\hfill \vartheta & ={\wp }_{cb}({\theta }^{*},\mathsf{\Pi }{\theta }^{*})\hfill \\ \hfill & ⪯{\wp }_{cb}({\theta }^{*},{\theta }_{2i+2})+{\wp }_{cb}({\theta }_{2i+2},\mathsf{\Pi }{\theta }^{*})-{\wp }_{cb}({\theta }_{2i+2},{\theta }_{2i+2})\hfill \\ \hfill & ⪯{\wp }_{cb}({\theta }^{*},{\theta }_{2i+2})+{\wp }_{cb}(\mathsf{\Pi }{\theta }^{*},{\theta }_{2i+2})\hfill \\ \hfill & ⪯{\wp }_{cb}({\theta }^{*},{\theta }_{2i+2})+{\wp }_{cb}(\mathsf{\Pi }{\theta }^{*},\mathsf{\Psi }{\theta }_{2i+1})\hfill \\ \hfill & ⪯{\wp }_{cb}({\theta }^{*},{\theta }_{2i+2})+⋏max\{{\wp }_{cb}({\theta }^{*},{\theta }_{2i+1}),{\displaystyle \frac{{\wp }_{cb}({\theta }^{*},\mathsf{\Pi }{\theta }^{*}){\wp }_{cb}({\theta }_{2i+1},\mathsf{\Psi }{\theta }_{2i+1})}{1+{\wp }_{cb}({\theta }^{*},{\theta }_{2i+1})}},\hfill \\ \hfill & {\displaystyle \frac{{\wp }_{cb}({\theta }^{*},\mathsf{\Pi }{\theta }_{*}){\wp }_{cb}(\mathsf{\Pi }{\theta }^{*},\mathsf{\Psi }{\theta }_{2i+1})}{1+{\wp }_{cb}({\theta }^{*},{\theta }_{2i+1})}}\}\hfill \\ \hfill & ⪯{\wp }_{cb}({\theta }^{*},{\theta }_{2i+2})+⋏max\{{\wp }_{cb}({\theta }^{*},{\theta }_{2i+1}),{\displaystyle \frac{{\wp }_{cb}({\theta }^{*},\mathsf{\Pi }{\theta }^{*}){\wp }_{cb}({\theta }_{2i+1},{\theta }_{2i+2})}{1+{\wp }_{cb}({\theta }^{*},{\theta }_{2i+1})}}\},\hfill \\ \hfill & {\displaystyle \frac{{\wp }_{cb}({\theta }^{*},\mathsf{\Pi }{\theta }^{*}){\wp }_{cb}(\mathsf{\Pi }{\theta }^{*},{\theta }_{2i+2})}{1+{\wp }_{cb}({\theta }^{*},{\theta }_{2i+1})}}\}\hfill \\ \hfill & ⪯{\wp }_{cb}({\theta }^{*},{\theta }_{2i+2})+⋏{\wp }_{cb}{({\theta }^{*},\mathsf{\Pi }{\theta }^{*})}^2\hfill \\ \hfill & ⪯{\wp }_{cb}({\theta }^{*},{\theta }_{2i+2})+⋏{\vartheta }^2.\hfill \end{array}$$

This yields

$$\begin{array}{c}\hfill \left|\vartheta \right|\le |{\wp }_{cb}({\theta }^{*},{\theta }_{2i+2}){|+⋏|\vartheta |}^2.\end{array}$$

Hence, $⋏\ge 1$, which is a contradiction. Then ${\theta }^{*}=\mathsf{\Pi }{\theta }^{*}$. In the same way, we obtain ${\theta }^{*}=\mathsf{\Psi }{\theta }^{*}$. Hence, ${\theta }^{*}$ is a common fixed point for the pair $(\mathsf{\Pi },\mathsf{\Psi })$. For uniqueness of the common fixed point, ${\theta }^{*}$ follows from Theorem 5. □

For $\mathsf{\Pi }=\mathsf{\Psi }$, we get the following fixed-points results on CPMS.

Theorem 7.

Let $(G,{\wp }_{cb})$ be a complete CPMS and $\mathsf{\Pi }:G\to G$ be a continuous mapping such that

$$\begin{array}{cc}\hfill {\wp }_{cb}(\mathsf{\Pi }\theta ,\mathsf{\Pi }\omega )& ⪯⋏max\left\{{\wp }_{cb}(\theta ,\omega ),{\displaystyle \frac{{\wp }_{cb}(\theta ,\mathsf{\Pi }\theta ){\wp }_{cb}(\omega ,\mathsf{\Pi }\omega )}{1+{\wp }_{cb}(\theta ,\omega )}},{\displaystyle \frac{{\wp }_{cb}(\theta ,\mathsf{\Pi }\theta ){\wp }_{cb}(\mathsf{\Pi }\theta ,\mathsf{\Pi }\omega )}{1+{\wp }_{cb}(\theta ,\omega )}}\right\},\hfill \end{array}$$

for all $\theta ,\omega \in G$, where $0\le ⋏<1$ and ${\wp }_{cb}(\mathsf{\Pi }\theta ,\mathsf{\Pi }\omega )\ne 0$. Then, Π has a unique fixed point and ${\wp }_{cb}({\theta }^{*},{\theta }^{*})=0$.

Remark 3.

Similarly, in the absence of continuity condition, we can get a fixed point result on Π.

Corollary 2.

Let $(G,{\wp }_{cb})$ be a complete CPMS and $\mathsf{\Pi }:G\to G$ be a continuous mapping such that

$$\begin{array}{cc}\hfill {\wp }_{cb}({\mathsf{\Pi }}^n\theta ,{\mathsf{\Pi }}^n\omega )& ⪯⋏max\{{\wp }_{cb}(\theta ,\omega ),{\displaystyle \frac{{\wp }_{cb}(\theta ,{\mathsf{\Pi }}^n\theta ){\wp }_{cb}(\omega ,{\mathsf{\Pi }}^n\omega )}{1+{\wp }_{cb}(\theta ,\omega )}},\hfill \\ \hfill & {\displaystyle \frac{{\wp }_{cb}(\theta ,{\mathsf{\Pi }}^n\theta ){\wp }_{cb}({\mathsf{\Pi }}^n\theta ,\mathsf{\Pi }\omega )}{1+{\wp }_{cb}(\theta ,\omega )}}\},\hfill \end{array}$$

for all $\theta ,\omega \in G$, where $0\le ⋏<1$ and ${\wp }_{cb}({\mathsf{\Pi }}^n\theta ,{\mathsf{\Pi }}^n\omega )\ne 0$. Then Π has a unique fixed point and ${\wp }_{cb}({\theta }^{*},{\theta }^{*})=0$.

Proof. 

By Theorem 5, we get ${\theta }^{*}\in G$ such that ${\mathsf{\Pi }}^n{\theta }^{*}={\theta }^{*}$ and ${\wp }_{cb}({\theta }^{*},{\theta }^{*})=0$. Then we get

$$\begin{array}{cc}\hfill {\wp }_{cb}(\mathsf{\Pi }{\theta }^{*},{\theta }^{*})& ={\wp }_{cb}(\mathsf{\Pi }{\mathsf{\Pi }}^n{\theta }^{*},{\mathsf{\Pi }}^n{\theta }^{*})={\wp }_{cb}({\mathsf{\Pi }}^n\mathsf{\Pi }{\theta }^{*},{\mathsf{\Pi }}^n{\theta }^{*})\hfill \\ \hfill & ⪯⋏max\{{\wp }_{cb}(\mathsf{\Pi }{\theta }^{*},{\theta }^{*}),{\displaystyle \frac{{\wp }_{cb}(\mathsf{\Pi }{\theta }^{*},{\mathsf{\Pi }}^n\mathsf{\Pi }{\theta }^{*}){\wp }_{cb}({\theta }^{*},{\mathsf{\Pi }}^n{\theta }^{*})}{1+{\wp }_{cb}(\mathsf{\Pi }{\theta }^{*},{\theta }^{*})}},\hfill \\ \hfill & {\displaystyle \frac{{\wp }_{cb}(\mathsf{\Pi }{\theta }^{*},{\mathsf{\Pi }}^n\mathsf{\Pi }{\theta }^{*}){\wp }_{cb}({\mathsf{\Pi }}^n\mathsf{\Pi }{\theta }^{*},{\mathsf{\Pi }}^n{\theta }^{*})}{1+{\wp }_{cb}(\mathsf{\Pi }{\theta }^{*},{\theta }^{*})}}\}\hfill \\ \hfill & ⪯⋏max\{{\wp }_{cb}(\mathsf{\Pi }{\theta }^{*},{\theta }^{*}),{\displaystyle \frac{{\wp }_{cb}(\mathsf{\Pi }{\theta }^{*},\mathsf{\Pi }{\mathsf{\Pi }}^n{\theta }^{*}){\wp }_{cb}({\theta }^{*},{\mathsf{\Pi }}^n{\theta }^{*})}{1+{\wp }_{cb}(\mathsf{\Pi }{\theta }^{*},{\theta }^{*})}},\hfill \\ \hfill & {\displaystyle \frac{{\wp }_{cb}(\mathsf{\Pi }{\theta }^{*},\mathsf{\Pi }{\mathsf{\Pi }}^n{\theta }^{*}){\wp }_{cb}(\mathsf{\Pi }{\mathsf{\Pi }}^n{\theta }^{*},{\mathsf{\Pi }}^n{\theta }^{*})}{1+{\wp }_{cb}(\mathsf{\Pi }{\theta }^{*},{\theta }^{*})}}\}\hfill \\ \hfill & =⋏{\wp }_{cb}(\mathsf{\Pi }{\theta }^{*},{\theta }^{*}).\hfill \end{array}$$

Hence ${\mathsf{\Pi }}^n{\theta }^{*}=\mathsf{\Pi }{\theta }^{*}={\theta }^{*}$. Then, $\mathsf{\Pi }$ has a unique fixed point. □

Remark 4.

From the above corollary 2, similarly, we get a fixed point result in the absence of continuity condition for the mapping Π.

Example 1.

Let $G=\{1,2,3,4\}$ be endowed with the order $\theta ⪯\omega$ if and only if $\theta \le \omega$. Then, ⪯ is a partial order in G. Define the complex partial metric space ${\wp }_{cb}:G\times G\to {\mathbb{C}}^{+}$ as follows:

$(\theta ,\omega )$	${\wp }_{cb}(\theta ,\omega )$
(1,1), (2,2)	0
(1,2),(2,1),(1,3),(3,1),(2,3),(3,2),(3,3)	$e^{ix}$
(1,4),(4,1),(2,4),(4,2),(3,4),(4,3),(4,4)	$3e^{ix}$

Obviously, $(G,{\wp }_{cb})$ is a complete CPMS for $x\in [0,\frac{\pi }{2}]$. Define $\mathsf{\Pi },\mathsf{\Psi }:G\to G$ by $\mathsf{\Pi }\theta =1$,

$$\mathsf{\Psi }\left(\theta \right)=\left\{\begin{array}{cc}1\hfill & \mathrm{if}\theta \in \{1,2,3\}\hfill \\ 2\hfill & \mathrm{if}\theta =4.\hfill \end{array}\right.$$

Clearly Π and Ψ are continuous functions. Now, for $⋏=\frac{1}{3}$, we consider the following cases:

(A) 

If $\theta =1$ and $\omega \in G-\{4\}$, then $\mathsf{\Pi }\left(\theta \right)=\mathsf{\Psi }\left(\omega \right)=1$ and the conditions of Theorem 2 are satisfied.

(B) 

If $\theta =1$, $\omega =4$, then $\mathsf{\Pi }\theta =1$, $\mathsf{\Psi }\omega =2$,

$$\begin{array}{cc}\hfill {\wp }_{cb}(\mathsf{\Pi }\theta ,\mathsf{\Psi }\omega )=e^{ix}& ⪯3⋏e^{ix}\hfill \\ \hfill & =⋏max\{3e^{ix},0,3e^{ix},\frac{1}{2}(e^{ix}+3e^{ix})\}\hfill \\ \hfill & =⋏max\{{\wp }_{cb}(\theta ,\omega ),{\wp }_{cb}(\theta ,\mathsf{\Pi }\theta ),{\wp }_{cb}(\omega ,\mathsf{\Psi }\omega ),\hfill \\ \hfill & {\displaystyle \frac{1}{2}}({\wp }_{cb}(\theta ,\mathsf{\Psi }\omega )+{\wp }_{cb}(\omega ,\mathsf{\Pi }\theta ))\},\hfill \end{array}$$

(C) 

If $\theta =2$, $\omega =4$, then $\mathsf{\Pi }\theta =1$, $\mathsf{\Psi }\omega =2$,

$$\begin{array}{cc}\hfill {\wp }_{cb}(\mathsf{\Pi }\theta ,\mathsf{\Psi }\omega )=e^{ix}& ⪯3⋏e^{ix}\hfill \\ \hfill & =⋏max\{3e^{ix},e^{ix},3e^{ix},\frac{1}{2}(0+3e^{ix})\}\hfill \\ \hfill & =⋏max\{{\wp }_{cb}(\theta ,\omega ),{\wp }_{cb}(\theta ,\mathsf{\Pi }\theta ),{\wp }_{cb}(\omega ,\mathsf{\Psi }\omega ),\hfill \\ \hfill & {\displaystyle \frac{1}{2}}({\wp }_{cb}(\theta ,\mathsf{\Psi }\omega )+{\wp }_{cb}(\omega ,\mathsf{\Pi }\theta ))\},\hfill \end{array}$$

(D) 

If $\theta =3$, $\omega =4$, then $\mathsf{\Pi }\theta =1$, $\mathsf{\Psi }\omega =2$,

$$\begin{array}{cc}\hfill {\wp }_{cb}(\mathsf{\Pi }\theta ,\mathsf{\Psi }\omega )=e^{ix}& ⪯3⋏e^{ix}\hfill \\ \hfill & =⋏max\{3e^{ix},e^{ix},3e^{ix},\frac{1}{2}(e^{ix}+3e^{ix})\}\hfill \\ \hfill & =⋏max\{{\wp }_{cb}(\theta ,\omega ),{\wp }_{cb}(\theta ,\mathsf{\Pi }\theta ),{\wp }_{cb}(\omega ,\mathsf{\Psi }\omega ),\hfill \\ \hfill & {\displaystyle \frac{1}{2}}({\wp }_{cb}(\theta ,\mathsf{\Psi }\omega )+{\wp }_{cb}(\omega ,\mathsf{\Pi }\theta ))\},\hfill \end{array}$$

(E) 

If $\theta =4$, $\omega =4$, then $\mathsf{\Pi }\theta =2$, $\mathsf{\Psi }\omega =2$,

$$\begin{array}{cc}\hfill {\wp }_{cb}(\mathsf{\Pi }\theta ,\mathsf{\Psi }\omega )=e^{ix}& ⪯3⋏e^{ix}\hfill \\ \hfill & =⋏max\{3e^{ix},3e^{ix},3e^{ix},\frac{1}{2}(3e^{ix}+3e^{ix})\}\hfill \\ \hfill & =⋏max\{{\wp }_{cb}(\theta ,\omega ),{\wp }_{cb}(\theta ,\mathsf{\Pi }\theta ),{\wp }_{cb}(\omega ,\mathsf{\Psi }\omega ),\hfill \\ \hfill & {\displaystyle \frac{1}{2}}({\wp }_{cb}(\theta ,\mathsf{\Psi }\omega )+{\wp }_{cb}(\omega ,\mathsf{\Pi }\theta ))\},\hfill \end{array}$$

Moreover, for $⋏=\frac{1}{3}$, with $⋏<1$, the conditions of Theorem 2 are satisfied. Therefore, 1 is the unique common fixed point of Π and Ψ.

4. Application

Consider the following systems of nonlinear integral equations:

$$\begin{array}{c}\hfill w\left(s\right)=\mathfrak{F}\left(s\right)+{\int }_a^b{T}_1(s,p,w\left(p\right))dp,\end{array}$$

(18)

and

$$\begin{array}{c}\hfill z\left(s\right)=\mathfrak{F}\left(s\right)+{\int }_a^b{T}_2(s,p,z\left(p\right))dp,\end{array}$$

(19)

where

(i)

$\mathfrak{F}:[a,b]\to {\mathbb{R}}^n$ is a continuous mapping and $\mathfrak{F}\left(s\right)$ is a given function in $(C\left([a,b]\right),{\mathbb{R}}^n)$,

(ii)

$w\left(s\right)$ and $z\left(s\right)$ are unknown variables for each $s\in J=[a,b]$, $b>a\ge 0$,

(iii)

$T_1(s,p)$ and $T_2(s,p)$ are deterministic kernels defined for $s,p\in J=[a,b]$.

In this section, we present an existence theorem for a common solution to (18) and (19) that belongs to $G=(C\left(J\right),{\mathbb{R}}^n)$ (the set of continuous functions defined on J) by using the obtained result in Theorem 2. We consider the continuous mappings $\mathsf{\Pi },\mathsf{\Psi }:G\to G$ given by

$$\begin{array}{c}\hfill \mathsf{\Pi }w\left(s\right)=\mathfrak{F}\left(s\right)+{\int }_a^b{T}_1(s,p,w\left(p\right))dp,w\in G,s\in J,\end{array}$$

and

$$\begin{array}{c}\hfill \mathsf{\Psi }z\left(s\right)=\mathfrak{F}\left(s\right)+{\int }_a^b{T}_2(s,p,z\left(p\right))dp,z\in G,s\in J,\end{array}$$

Then, the existence of a common solution to the nonlinear integral Equations (18) and (19) is equivalent to the existence of a common fixed point of $\mathsf{\Pi }$ and $\mathsf{\Psi }$. It is well known that G, endowed with the metric ${\wp }_{cb}$, defined by

$$\begin{array}{c}\hfill {\wp }_{cb}(w,z)=\underset{s\in J}{sup}|w\left(s\right)-z\left(s\right)|+2,\end{array}$$

for all $w,z\in G$, is a complete CPMS. G can also be equipped with the partial order ⪯ given by

$$\begin{array}{c}\hfill w,z\in G,w⪯z\mathrm{if}\mathrm{and}\mathrm{only}w\left(s\right)\ge z\left(s\right),\mathrm{for}\mathrm{all}s\in J.\end{array}$$

Further, let us consider a system of nonlinear integral equation as (18) and (19) under the following condition hold:

(A)

$T_1,T_2:J\times J\times {\mathbb{R}}^n\to {\mathbb{R}}^n$ are continuous functions satisfying

$$\begin{array}{c}\hfill |T_1(s,p,w\left(p\right))-T_2(s,p,z\left(p\right))|⪯\frac{S(w,z)}{(b-a)e^t}-\frac{2}{b-a},\forall t>0,\end{array}$$

where

$$\begin{array}{cc}\hfill S(w,z)& =max\{{\wp }_{cb}(w,z),{\wp }_{cb}(w,\mathsf{\Pi }w),{\wp }_{cb}(z,\mathsf{\Psi }z),\hfill \\ \hfill & \frac{1}{2}({\wp }_{cb}(w,\mathsf{\Psi }z)+{\wp }_{cb}(z,\mathsf{\Pi }w))\}.\hfill \end{array}$$

Theorem 8.

Let $(C\left(J\right),{\mathbb{R}}^n,{\wp }_{cb})$ be a complete CPMS; then, the system (18) and (19) under condition (A) have a unique common solution.

Proof. 

For $w,z\in (C\left(J\right),{\mathbb{R}}^n)$ and $s\in J$, we define the continuous mappings $\mathsf{\Pi },\mathsf{\Psi }:G\to G$ by

$$\begin{array}{c}\hfill \mathsf{\Pi }w\left(s\right)=\mathfrak{F}\left(s\right)+{\int }_a^b{T}_1(s,p,w\left(p\right))dp,\end{array}$$

and

$$\begin{array}{c}\hfill \mathsf{\Psi }z\left(s\right)=\mathfrak{F}\left(s\right)+{\int }_a^b{T}_2(s,p,z\left(p\right))dp.\end{array}$$

Then, we have

$$\begin{array}{cc}\hfill {\wp }_{cb}(\mathsf{\Pi }w\left(s\right),\mathsf{\Psi }z\left(s\right))& =\underset{s\in J}{sup}|\mathsf{\Pi }w\left(s\right)-\mathsf{\Psi }z\left(s\right)|+2\hfill \\ \hfill & ⪯{\int }_a^b|T_1(s,p,w\left(p\right))-T_2(s,p,z\left(p\right))|dp+2\hfill \\ \hfill & ⪯{\int }_a^b\left(\frac{S(w,z)}{(b-a)e^t}-\frac{2}{b-a}\right)dp+2\hfill \\ \hfill & =\frac{S(w,z)}{e^t}\hfill \\ \hfill & =⋏S(w,z)\hfill \\ \hfill & =⋏max\{{\wp }_{cb}(w,z),{\wp }_{cb}(w,\mathsf{\Pi }w),{\wp }_{cb}(z,\mathsf{\Psi }z),\hfill \\ \hfill & \frac{1}{2}({\wp }_{cb}(w,\mathsf{\Psi }z)+{\wp }_{cb}(z,\mathsf{\Pi }w))\}.\hfill \end{array}$$

Hence, all the conditions of Theorem 2 are satisfied for $0<⋏=\frac{1}{e^t}<1$ with $t>0$. Therefore the system of nonlinear integral Equations (18) and (19) have a unique common solution. □

5. Conclusions

In this paper, we proved some common fixed-point theorems on complex partial metric space. An illustrative example and application on complex partial metric space is given.