Double-Composed Metric Spaces

Abstract

The double-controlled metric-type space $(X,\mathcal{D})$ is a metric space in which the triangle inequality has the form $\mathcal{D}(\eta ,\mu )\le {\zeta }_1(\eta ,\theta )\mathcal{D}(\eta ,\theta )+{\zeta }_2(\theta ,\mu )\mathcal{D}(\theta ,\mu )$ for all $\eta ,\theta ,\mu \in X$. The maps ${\zeta }_1,{\zeta }_2:X\times X\to [1,\infty )$ are called control functions. In this paper, we introduce a novel generalization of a metric space called a double-composed metric space, where the triangle inequality has the form $\mathcal{D}(\eta ,\mu )\le \alpha \left(\mathcal{D}(\eta ,\theta )\right)+\beta \left(\mathcal{D}(\theta ,\mu )\right)$ for all $\eta ,\theta ,\mu \in X$. In our new space, the control functions $\alpha ,\beta :[0,\infty )\to [0,\infty )$ are composed of the metric $\mathcal{D}$ in the triangle inequality, where the control functions ${\zeta }_1,{\zeta }_2:X\times X\to [1,\infty )$ in a double-controlled metric-type space are multiplied with the metric $\mathcal{D}$. We establish some fixed-point theorems along with the examples and applications.

1. Introduction

Over the last few decades, numerous generalizations of the usual metric space have been constructed in the field of fixed-point theory. As a result of the discovery of these generalized metric spaces, researchers have proven fixed-point theorems similar to the Banach fixed-point theorem, the Kannan fixed-point theorem, and several [1,2,3,4,5,6,7,8,9]. By employing a constant in the right hand side of the triangle inequality, Czerwik [10] and Bakhtin [11] proposed the concept of the fascinating generalized metric space known as $b-$metric space. The topology of b-metric spaces is quite different than the topology of usual metric space. In 2017, the definition of $b-$metric space was generalized to so called extended $b-$metric space by Kamran et al. [12], and the related fixed-point theorems were also established. In 2018, Mlaiki et al. [13] further generalized the extended $b-$metric spaces to controlled metric spaces by employing a binary control function on the right side of triangle inequality. They also established the corresponding Banach fixed-point result in the same space. In 2019, Lattef [14] established a Kannan-type fixed-point result in controlled metric spaces. In 2020, Ahmad et al. [15] established a fixed-point result for Reich-type contractions in controlled metric spaces. As a further generalization of controlled metric spaces, Abdeljawad et al. [16] introduced double-controlled metric-type spaces (DCMTS for short) by employing two binary control functions on the right side of triangle inequality. Additionally, they established the corresponding Banach-type and Kannan-type fixed-point result in the same space. In 2020, Mlaiki [17] introduced double-controlled metric-like spaces as a further generalization of double-controlled metric-type spaces.

The goal of this paper is to develop a novel generalization of a metric space called double-composed metric space (DCMS for short). In a double-controlled metric-type space, the triangle has the form $\mathcal{D}(\eta ,\mu )\le {\psi }_1(\eta ,\theta )\mathcal{D}(\eta ,\theta )+{\psi }_2(\theta ,\mu )\mathcal{D}(\theta ,\mu )$ for all $\eta ,\theta ,\mu \in X$, where the binary control functions ${\zeta }_1,{\zeta }_2:X\times X\to [1,\infty )$ are multiplied with the metric d. In this paper, we replace the triangle inequality in the usual metric space by $\mathcal{D}(\eta ,\mu )\le \alpha \left(\mathcal{D}(\eta ,\theta )\right)+\beta \left(\mathcal{D}(\theta ,\mu )\right)$ for all $\eta ,\theta ,\mu \in X$, where the control functions $\alpha ,\beta :[0,\infty )\to [0,\infty )$ are composed of the metric d in the triangle inequality. The composition and multiplication of two functions being two independent operations, we note that DCMS and DCMTC are two independent generalizations of a metric space.

2. Preliminaries

This work on DCMS has been motivated by DCMTS, and due to the apparent similarities in the triangle inequalities in the two spaces, it is worth to mention here the definition of DCMTS and the related fixed-point results.

We begin with a definition of the extended $b-$metric spaces introduced by Kamran et al. [12].

Definition 1. 

Let $\mathcal{W}$ be a non-empty set and ${\zeta }_1:\mathcal{W}\times \mathcal{W}\to [1,\infty )$. A function $\mathcal{D}:\mathcal{W}\times \mathcal{W}\to [0,\infty )$ is called an extended $b-$metric type if it satisfies:

• $\mathcal{D}(\eta ,\theta )=0$ if and only if $\eta =\theta$ for all $\eta ,\theta \in \mathcal{W}$;
• $\mathcal{D}(\eta ,\theta )=\mathcal{D}(\theta ,\eta )$ for all $\eta ,\theta \in \mathcal{W}$;
• $\mathcal{D}(\eta ,\mu )\le {\zeta }_1(\eta ,\theta )[\mathcal{D}(\eta ,\theta )+\mathcal{D}(\theta ,\mu )]$ for all $\eta ,\theta ,\mu \in \mathcal{W}$.

The pair $\left(\mathcal{W},\mathcal{D}\right)$ is called extended $b-$metric space.

Mlaiki et al. [13] proposed the following new generalization of extended $b-$metric spaces called controlled metric-type spaces.

Definition 2. 

Let $\mathcal{W}$ be a non-empty set and ${\zeta }_1:\mathcal{W}\times \mathcal{W}\to [1,\infty )$. A function $\mathcal{D}:\mathcal{W}\times \mathcal{W}\to [0,\infty )$ is called a controlled metric type if it satisfies:

• $\mathcal{D}(\eta ,\theta )=0$ if and only if $\eta =\theta$ for all $\eta ,\theta \in \mathcal{W}$;
• $\mathcal{D}(\eta ,\theta )=\mathcal{D}(\theta ,\eta )$ for all $\eta ,\theta \in \mathcal{W}$;
• $\mathcal{D}(\eta ,\mu )\le {\zeta }_1(\eta ,\theta )\mathcal{D}(\eta ,\theta )+{\zeta }_1(\theta ,\mu )\mathcal{D}(\theta ,\mu )$ for all $\eta ,\theta ,\mu \in \mathcal{W}$.

The pair $\left(\mathcal{W},\mathcal{D}\right)$ is called a controlled metric-type space.

In [16], Abdeljawad et al. proposed the following generalization of a controlled metric-type space and named it a double-controlled metric-type space [DCMTS].

Definition 3 

(DCMTS). Let $\mathcal{W}$ be a non-empty set and ${\zeta }_1,{\zeta }_2:\mathcal{W}\times \mathcal{W}\to [1,\infty )$. A function $\mathcal{D}:\mathcal{W}\times \mathcal{W}\to [0,\infty )$ is called a double-controlled metric type if it satisfies:

• $\mathcal{D}(\eta ,\theta )=0$ if and only if $\eta =\theta$ for all $\eta ,\theta \in \mathcal{W}$
• $\mathcal{D}(\eta ,\theta )=\mathcal{D}(\theta ,\eta )$ for all $\eta ,\theta \in \mathcal{W}$
• $\mathcal{D}(\eta ,\mu )\le {\zeta }_1(\eta ,\theta )\mathcal{D}(\eta ,\theta )+{\zeta }_2(\theta ,\mu )\mathcal{D}(\theta ,\mu )$ for all $\eta ,\theta ,\mu \in \mathcal{W}$.

The pair $\left(\mathcal{W},\mathcal{D}\right)$ is called a double-controlled metric-type space.

Example 1 

([16]). Let $\mathcal{W}=[0,\infty )$. Define $\mathcal{D}:\mathcal{W}\times \mathcal{W}\to [0,\infty )$ by

$$\mathcal{D}(\eta ,\theta )=\left\{\begin{array}{cc}0,\hfill & ⟺\eta =\theta \hfill \\ \frac{1}{\eta },\hfill & \mathit{if}\eta \ge 1\mathit{and}\theta \in [0,1)\hfill \\ \frac{1}{\theta },\hfill & \mathit{if}\theta \ge 1\mathit{and}\eta \in [0,1)\hfill \\ 1,\hfill & \mathit{if}\mathrm{not}.\hfill \end{array}\right.$$

Consider ${\zeta }_1,{\zeta }_2:{\mathcal{W}}^2\to [1,\infty )$ as

$${\zeta }_1(\eta ,\theta )=\left\{\begin{array}{cc}\eta ,\hfill & \mathit{if}\eta ,\theta \ge 1,\hfill \\ 1,\hfill & \mathit{if}\mathrm{not}\hfill \end{array}\mathit{and}{\zeta }_2(\eta ,\theta )=\left\{\begin{array}{cc}1,\hfill & \mathit{if}\eta ,\theta <1\hfill \\ max\{\eta ,\theta \},\hfill & \mathit{if}\mathrm{not}\hfill \end{array}\right.\right.$$

The pair $(\mathcal{W},\mathcal{D})$ is a double-controlled metric-type space.

Abdeljawad et al. [16] proved the following fixed-point results in a double-controlled type metric.

Theorem 1. 

Let $(\mathcal{W},\mathcal{D})$ be a complete DCMTS with ${\zeta }_1,{\zeta }_2:\mathcal{W}\times \mathcal{W}\to [1,\infty )$. Let $F:\mathcal{W}\to \mathcal{W}$ be a mapping such that

$$\mathcal{D}(F\eta ,F\theta )\le \kappa \mathcal{D}(\eta ,\theta ).$$

for all $\eta ,\theta \in \mathcal{W}$, where $\kappa \in (0,1)$. For $x_0\in \mathcal{W}$, take $x_n=F^n{x}_0$. Assume that

$$\underset{m\ge 1}{sup}\underset{i\to \infty }{lim}\frac{{\zeta }_1\left(x_{i+1},x_{i+2}\right)}{{\zeta }_1\left(x_i,x_{i+1}\right)}{\zeta }_2\left(x_{i+1},x_m\right)<\frac{1}{\kappa }.$$

Further, suppose that, for every $\eta \in \mathcal{W}$, we have ${lim}_{n\to \infty }{\zeta }_1(\eta ,x_n)$ and ${lim}_{n\to \infty }{\zeta }_2\left(x_n,\eta \right)$ exist and are finite. Then, F has a unique fixed point.

Theorem 2. 

Let $(\mathcal{W},\mathcal{D})$ be a complete DCMTS with ${\zeta }_1,{\zeta }_2:\mathcal{W}\times \mathcal{W}\to [1,\infty )$. Let $F:\mathcal{W}\to \mathcal{W}$ be a mapping such that

$$\mathcal{D}(F\eta ,F\theta )\le \kappa \left[\mathcal{D}\right(\eta ,F\eta )+\mathcal{D}(\theta ,F\theta \left)\right].$$

for all $\eta ,\theta \in \mathcal{W}$, where $\kappa \in (0,\frac{1}{2})$. For $x\in \mathcal{W}$, take $x=F^n{x}_0$. Assume that

$$\underset{m\ge 1}{sup}\underset{i\to \infty }{lim}\frac{{\zeta }_1\left(x_{i+1},x_{i+2}\right)}{{\zeta }_1\left(x_i,x_{i+1}\right)}{\zeta }_2\left(x_{i+1},x_m\right)<\frac{1-k}{k}.$$

Further, suppose that, for every $\eta \in \mathcal{W}$, ${lim}_{n\to \infty }{\zeta }_1(\eta ,x_n)$ exists and is finite, and ${lim}_{n\to \infty }{\zeta }_2\left(x_n,\eta \right)<\frac{1}{\kappa }$. Furthermore, if $\mathcal{D}(x,x)=0$ for every fixed point x of F, then the fixed point of F is necessarily unique.

Now, we introduce a new space, namely, double-composed metric space (DCMS).

Definition 4 

(DCMS). Let $\mathcal{W}$ be a non-empty set and $\alpha ,\beta :[0,\infty )\to [0,\infty )$ be two non-constant functions. A function $\mathcal{D}:\mathcal{W}\times \mathcal{W}\to [0,\infty )$ is called a double-composed metric if it satisfies:

• $\mathcal{D}(\eta ,\theta )=0$ if and only if $\eta =\theta$ for all $\eta ,\theta \in \mathcal{W}$
• $\mathcal{D}(\eta ,\theta )=\mathcal{D}(\theta ,\eta )$ for all $\eta ,\theta \in \mathcal{W}$
• $\mathcal{D}(\eta ,\mu )\le \alpha \left(\mathcal{D}(\eta ,\theta )\right)+\beta \left(\mathcal{D}(\theta ,\mu )\right)$ for all $\eta ,\theta ,\mu \in \mathcal{W}$.

The pair $\left(\mathcal{W},\mathcal{D}\right)$ is a called double-composed metric space.

Remark 1. 

We note that DCMS and DCMTS are two independent generalizations of the metrics space, for the former involves the multiplication of control functions, whereas the later one involves the composition of functions in the triangle inequality.

Remark 2. 

Every metric space is a DCMS with the control functions $\alpha \left(\eta \right)=\beta \left(\eta \right)=\eta$, but the converse need not be true, as shown by the following example.

Example 2. 

Let $\mathcal{W}=\mathbb{R}$. Define functions $\alpha ,\beta :[0,\infty )\to [0,\infty )$ by $\alpha \left(\eta \right)=\beta \left(\eta \right)=e^{\eta }$. Define a function $\mathcal{D}:\mathcal{W}\times \mathcal{W}\to [0,\infty )$ by

$$\mathcal{D}(\eta ,\theta )={(\eta -\theta )}^2$$

Then, $(\mathcal{W},\mathcal{D})$ is a DCMS with control functions α and β; however, it is not a metric space in the usual sense.

Proof. 

For $\eta =1,\theta =0$ and $\mu =-2$, we observe that the usual triangle inequality $\mathcal{D}(\eta ,\mu )\le \mathcal{D}(\eta ,\theta )+\mathcal{D}(\theta ,\mu )$ is not satisfied.

Now, we prove the triangle inequality for DCMS. Consider the function $f\left(\eta \right)=2{\eta }^2-e^{{\eta }^2}$ on $\mathbb{R}$ with global maximum given by $log\left(4\right)-2$ at $\eta =\pm \sqrt{log\left(2\right)}$. Therefore, for all $\kappa ,v\in \mathbb{R}$, we have

$$\begin{array}{cc}\hfill (2{\kappa }^2-e^{{\kappa }^2})+(2v^2-e^{v^2})& \le f\left(\kappa \right)+f\left(v\right)\hfill \\ \hfill & \le log\left(4\right)-2+log\left(4\right)-2\hfill \\ \hfill & =2log\left(4\right)-4\hfill \end{array}$$

(1)

We shall use the following classical inequality:

$${(\kappa +v)}^2\le 2({\kappa }^2+v^2)$$

Using inequality (1), the above inequality may be manipulated as follows:

$$\begin{array}{cc}\hfill {(\kappa +v)}^2& \le 2({\kappa }^2+v^2)\hfill \\ \hfill {(\kappa +v)}^2-(e^{{\kappa }^2}+e^{v^2})& \le 2{\kappa }^2-e^{{\kappa }^2}+2v^2-e^{v^2}\hfill \\ & \le 2log\left(4\right)-4\hfill \\ & \le 0\hfill \end{array}$$

that is,

$$\begin{array}{c}\hfill {(\kappa +v)}^2\le (e^{{\kappa }^2}+e^{v^2})\end{array}$$

(2)

Now, for all $\eta ,\theta ,\mu \in \mathcal{W}$, letting $\kappa =\eta -\theta$ and $v=\theta -\mu$ in the inequality (2), we obtain

$$\begin{array}{cc}\hfill {(\eta -\theta +\theta -\mu )}^2& \le e^{{(\eta -\theta )}^2}+e^{{(\theta -\mu )}^2}\hfill \\ \hfill \mathcal{D}(\eta ,\mu )& \le \alpha \left({(\eta -\theta )}^2\right)+\beta \left({(\theta -\mu )}^2\right)\hfill \\ & \le \alpha \left(\mathcal{D}\right(\eta -\theta \left)\right)+\beta \left(\mathcal{D}\right(\theta -\mu \left)\right)\hfill \end{array}$$

This proves the triangle inequality. □

Example 3. 

Let $\mathcal{W}=\{1,2,3\}$. Define two non-constant functions $\alpha ,\beta :[0,\infty )\to [0,\infty )$ by $\alpha \left(\eta \right)=\eta +700$ and $\beta \left(\eta \right)=\eta$. Define a function $\mathcal{D}:\mathcal{W}\times \mathcal{W}\to [0,\infty )$ by

$$\mathcal{D}(1,1)=\mathcal{D}(2,2)=\mathcal{D}(3,3)=0$$

$$\mathcal{D}(1,2)=\mathcal{D}(2,1)=70$$

$$\mathcal{D}(1,3)=\mathcal{D}(3,1)=1200$$

$$\mathcal{D}(2,3)=\mathcal{D}(3,2)=700$$

Then, $(\mathcal{W},\mathcal{D})$ is a DCMS with control functions α and β; however, it is not a metric space in the usual sense.

Proof. 

First note that $\mathcal{D}(1,3)\le \mathcal{D}(1,2)+\mathcal{D}(2,3)$ is not satisfied, so that $(\mathcal{W},\mathcal{D})$ is not a metric space. Now, we prove that $(\mathcal{W},\mathcal{D})$ is a DCMS. The first two conditions of Definition 4 are trivially satisfied by $\mathcal{D}$. We establish the triangle inequality.

$$\alpha \left(\mathcal{D}\right(1,3\left)\right)+\beta \left(\mathcal{D}\right(3,2\left)\right)=\alpha \left(1200\right)+\beta \left(700\right)=(1200+700)+700\ge 70=\mathcal{D}(1,2)$$

$$\alpha \left(\mathcal{D}\right(1,2\left)\right)+\beta \left(\mathcal{D}\right(2,3\left)\right)=\alpha \left(70\right)+\beta \left(700\right)=(70+700)+700\ge 1200=\mathcal{D}(1,3)$$

$$\alpha \left(\mathcal{D}\right(2,1\left)\right)+\beta \left(\mathcal{D}\right(1,3\left)\right)=\alpha \left(70\right)+\beta \left(1200\right)=(70+700)+1200\ge 700=\mathcal{D}(2,3)$$

Therefore, $\mathcal{D}(\eta ,\mu )\le \alpha \left(\mathcal{D}(\eta ,\theta )\right)+\beta \left(\mathcal{D}(\theta ,\mu )\right)$ for all $\eta ,\theta ,\mu \in \mathcal{W}$. □

Example 4. 

Let $\mathcal{W}=l_n\left(\mathbb{R}\right)$ with $0<p<1$ where $l_p\left(\mathbb{R}\right):=\left\{\left\{x_n\right\}\subset \mathbb{R}:{\sum }_{n=1}^{\infty }{\left|x_n\right|}^p<\infty \right\}$. Define $\mathcal{D}:\mathcal{W}\times \mathcal{W}\to [0,\infty )$ as

$$\mathcal{D}(\eta ,\theta )={\left(\sum _{n=1}^{\infty }{\left|{\eta }_n-{\theta }_n\right|}^p\right)}^{1/n}$$

where $\eta =\left\{{\eta }_n\right\},\theta =\left\{{\theta }_n\right\}$. Then $(\mathcal{W},\mathcal{D})$ is a DCMS with control functions $\alpha \left(\eta \right)=\beta \left(\eta \right)=2^{1/p}\eta .$

Example 5. 

Let $\mathcal{W}=L_p[0,1]$ be the space of all real functions $\eta \left(t\right),t\in [0,1]$ such that ${\int }_0^1\left|\eta \left(t\right)\right|dt<\infty$ with $0<p<1$. Define $\mathcal{D}:\mathcal{W}\times \mathcal{W}\to [0,\infty )$ as

$$\mathcal{D}(\eta \left(t\right),\theta \left(t\right))={\left({\int }_0^1{|\eta \left(t\right)-\theta \left(t\right)|}^{p}dt\right)}^{1/p},$$

where $\eta \left(t\right),\theta \left(t\right)\in \mathcal{W}$. Then, $(\mathcal{W},\mathcal{D})$ is a DCMS with control functions $\alpha \left(\eta \right)=\beta \left(\eta \right)=2^{1/p}\eta .$

Now, we define the topology of double-composed metric spaces.

Definition 5. 

Let $\left(\mathcal{W},\mathcal{D}\right)$ be a double-composed metric space. Let $x_0\in \mathcal{W}$ and $ϵ>0$; the open ball centered at $x_0$ is defined as $B(x_0,x)=\{x\in \mathcal{W}:\mathcal{D}(x_0,x)<ϵ\}.$

Remark 3. 

It is easy to see that the double-composed metric $\mathcal{D}$ generates the $T_0$ topology on $\mathcal{W}$, where the base of the topology is given by $\{B(x_0,ϵ):x_0\in \mathcal{X},ϵ>0\}.$

It is fascinating to investigate other topological concepts such as compactness, connectedness, and so on in the framework of a double-composed metric space.

Definition 6. 

Let $\left(\mathcal{W},\mathcal{D}\right)$ be a double-composed metric space. For each sequence $\left\{x_n\right\}\in \mathcal{W}$, we say

1.

That $\left\{x_n\right\}$ a Cauchy sequence if ${\displaystyle \underset{n,m\to \infty }{lim}d\left(x_n,x_m\right)}$ exists and is finite;

2.

That $\left\{x_n\right\}$ converges to x if ${\displaystyle \underset{n\to \infty }{lim}d\left(x_n,x\right)=0}$;

3.

That $\left(\mathcal{W},\mathcal{D}\right)$ is complete if every Cauchy sequence in $\mathcal{W}$ is convergent to some point in $\mathcal{W}$.

A convergent sequence in a double-composed metric space may not have a unique limit. For the uniqueness of the limit, we have the following result.

Proposition 1. 

When $\left(\mathcal{W},\mathcal{D}\right)$ be a double-composed metric space with two non-constant and continuous control functions $\alpha ,\beta :[0,\infty )\to [0,\infty )$ satisfying $\alpha \left(0\right)+\beta \left(0\right)=0$, the every convergent sequence has a unique limit.

Proof. 

Suppose that the sequence $\left\{x_n\right\}\in \mathcal{W}$ converges to s and t in $\mathcal{W}.$ By the definition of convergence, we have ${\displaystyle \underset{n\to \infty }{lim}\mathcal{D}\left(x_n,s\right)=0}$ and ${\displaystyle \underset{n\to \infty }{lim}\mathcal{D}\left(x_n,t\right)=0}$. By the triangle inequality in DCMS, we obtain

$$\mathcal{D}(s,t)\le \alpha \left(\mathcal{D}(s,x_n)\right)+\beta \left(\mathcal{D}(x_n,t)\right)$$

Since $\alpha$ and $\beta$ are continuous, taking limit in the above inequality, we obtain

$$\begin{array}{cc}\hfill \mathcal{D}(s,t)& {\displaystyle \le \alpha \left(\underset{n\to \infty }{lim}\mathcal{D}(s,x_n)\right)+\beta \left(\underset{n\to \infty }{lim}\mathcal{D}(x_n,t)\right)}\hfill \\ & =\alpha \left(0\right)+\beta \left(0\right)\hfill \\ & =0\hfill \end{array}$$

This implies that $\mathcal{D}(s,t)=0$, which further implies that $s=t$. □

3. Main Results

The following result is analogous to Banach fixed-point theorem.

Theorem 3. 

Let $(\mathcal{W},\mathcal{D})$ be a complete double-composed metric space with non-constant control functions $\alpha ,\beta :[0,\infty )\to [0,\infty )$. Let $F:\mathcal{W}\to \mathcal{W}$ be a mapping satisfying

$$\mathcal{D}(F\eta ,F\theta )\le k\mathcal{D}(\eta ,\theta )$$

(3)

for all $\eta ,\theta \in \mathcal{W}$ and $k\in (0,1).$ For $x_0\in \mathcal{W}$, define a sequence $\left\{x_n\right\}$ by $x_n=F^n{x}_0$. Suppose that the following conditions are satisfied:

1.

α and β are continuous and non-decreasing functions with $\alpha \left(0\right)+\beta \left(0\right)=0$, and β is sub-additive.

2.

${\displaystyle \underset{m,n\to \infty }{lim}\left[\sum _{i=m}^{n-2}{\beta }^{i-m}\alpha \left(k^i\mathcal{D}(x_0,x_1)\right)+{\beta }^{n-m-1}\left(k^{n-1}\mathcal{D}(x_0,x_1))\right)\right]=0}$ where ${\beta }^{i-m}$$\left(\alpha \left(k^i\mathcal{D}(x_0,x_1))\right)\right)$ and ${\beta }^{n-m-1}\left(k^{n-1}\mathcal{D}(x_0,x_1)\right)$ denote the composite functions.

Then, F has a unique fixed point.

Proof. 

Let $x_0\in \mathcal{W}$. Define a sequence $\left\{x_n\right\}$ in $\mathcal{W}$ with $x_n=F^n{x}_0$ so that $x_{n+1}=F{x}_n$ for all $n\in \mathbb{N}$. We have

$$\begin{array}{cc}\hfill \mathcal{D}\left(x_n,x_{n+1}\right)& =\mathcal{D}\left(F{x}_{n-1},F{x}_n\right)\hfill \\ \hfill & \le k\mathcal{D}\left(x_{n-1},x_n\right)\hfill \\ \hfill & =k\mathcal{D}\left(F{x}_{n-2},F{x}_{n-1}\right)\hfill \\ \hfill & \le k^2\mathcal{D}\left(x_{n-2},x_{n-1}\right)\hfill \\ \hfill ⋮\\ \hfill & \le k^n\mathcal{D}\left(x_0,x_1\right).\hfill \end{array}$$

(4)

$$\mathcal{D}\left(x_n,x_{n+1}\right)\le k^n\mathcal{D}\left(x_0,x_1\right)$$

(5)

For $n\ge m$, by using the triangle inequality repeatedly, we obtain

$$\begin{array}{cc}\hfill \mathcal{D}(x_m,x_n)& \le \alpha \left(\mathcal{D}(x_m,x_{m+1})\right)+\beta \left(\mathcal{D}(x_{m+1},x_n)\right)\hfill \\ \hfill & \le \alpha \left(\mathcal{D}(x_m,x_{m+1})\right)+\beta [\alpha \left(\mathcal{D}(x_{m+1},x_{m+2})\right)+\beta \left(\mathcal{D}(x_{m+2},x_n)\right)]\hfill \\ \hfill & =\alpha \left(\mathcal{D}(x_m,x_{m+1})\right)+\beta \alpha \left(\mathcal{D}(x_{m+1},x_{m+2})\right)+{\beta }^2\left[\mathcal{D}(x_{m+2},x_n)\right]\hfill \\ \hfill & \le \alpha \left(\mathcal{D}(x_m,x_{m+1})\right)+\beta \alpha \left(\mathcal{D}(x_{m+1},x_{m+2})\right)+\hfill \\ \hfill & {\beta }^2[\alpha \left(\mathcal{D}(x_{m+2},x_{m+3})\right)+\beta \left(\mathcal{D}(x_{m+3},x_n)\right)]\hfill \\ \hfill & =\alpha \left(\mathcal{D}(x_m,x_{m+1})\right)+\beta \alpha \left(\mathcal{D}(x_{m+1},x_{m+2})\right)\hfill \\ \hfill & +{\beta }^2\alpha \left(\mathcal{D}(x_{m+2},x_{m+3})\right)+{\beta }^3\left[\mathcal{D}(x_{m+3},x_n)\right]\hfill \\ \hfill & \le \alpha \left(\mathcal{D}(x_m,x_{m+1})\right)+\beta \alpha \left(\mathcal{D}(x_{m+1},x_{m+2})\right)+{\beta }^2\alpha \left(\mathcal{D}(x_{m+2},x_{m+3})\right)\hfill \\ \hfill & +{\beta }^3[\alpha \left(\mathcal{D}(x_{m+3},x_{m+4})\right)+\beta \left(\mathcal{D}(x_{m+4},x_n)\right)]\hfill \\ \hfill & =\alpha \left(\mathcal{D}(x_m,x_{m+1})\right)+\beta \alpha \left(\mathcal{D}(x_{m+1},x_{m+2})\right)+{\beta }^2\alpha \left(\mathcal{D}(x_{m+2},x_{m+3})\right)\hfill \\ \hfill & +{\beta }^3\alpha \left(\mathcal{D}(x_{m+3},x_{m+4})\right)+{\beta }^4\left[\mathcal{D}(x_{m+4},x_n)\right]\hfill \\ \hfill ⋮\\ \hfill & \le \alpha \left(\mathcal{D}(x_m,x_{m+1})\right)+\beta \alpha \left(\mathcal{D}(x_{m+1},x_{m+2})\right)+{\beta }^2\alpha \left(\mathcal{D}(x_{m+2},x_{m+3})\right)+\hfill \\ & {\beta }^3\alpha \left(\mathcal{D}(x_{m+3},x_{m+4})\right)+{\beta }^4\left[\mathcal{D}(x_{m+4},x_n)\right]+...+{\beta }^{n-2}[\alpha \left(\mathcal{D}(x_{n-2},x_{n-1})\right)\hfill \\ \hfill & +\beta \left(\mathcal{D}(x_{n-1},x_n)\right)]\hfill \\ \hfill & =\alpha \left(\mathcal{D}(x_m,x_{m+1})\right)+\beta \alpha \left(\mathcal{D}(x_{m+1},x_{m+2})\right)+{\beta }^2\alpha \left(\mathcal{D}(x_{m+2},x_{m+3})\right)+\hfill \\ \hfill & {\beta }^3\alpha \left(\mathcal{D}(x_{m+3},x_{m+4})\right)+{\beta }^4\left[\mathcal{D}(x_{m+4},x_n)\right]+...+{\beta }^{n-m-2}\alpha \left(\mathcal{D}(x_{n-2},x_{n-1})\right)\hfill \\ \hfill & +{\beta }^{n-m-1}\left(\mathcal{D}(x_{n-1},x_n)\right)\hfill \\ \hfill & =\sum _{i=m}^{n-2}{\beta }^{i-m}\alpha \left(\mathcal{D}(x_i,x_{i+1})\right)+{\beta }^{n-m-1}\left(\mathcal{D}(x_{n-1},x_n)\right).\hfill \end{array}$$

(6)

Note the we have used sub-additivity of $\beta$ repeatedly to establish the inequality (6). Since $\alpha$ and $\beta$ are non-decreasing functions, the compositions ${\beta }^{i-m}\alpha \left(\mathcal{D}(x_i,x_{i+1})\right)$ and ${\beta }^{n-m-1}\left(\mathcal{D}(x_{n-1},x_n)\right)$ are also non-decreasing. Using inequality (5) in (6), we obtain

$$\mathcal{D}(x_m,x_n)\le \sum _{i=m}^{n-2}{\beta }^{i-m}\alpha \left(k^i\mathcal{D}(x_0,x_1)\right)+{\beta }^{n-m-1}\left(k^{n-1}\mathcal{D}(x_0,x_1))\right)$$

(7)

Letting $m,n$ tend to infinity in (7), by condition (1) of Theorem 3, we obtain

$$\mathcal{D}(x_m,x_n)=0$$

(8)

Therefore, the sequence $\left\{x_n\right\}$ is cauchy in $\mathcal{W}$. Since $\mathcal{W}$ is a complete double-composed metric space, the sequence $\left\{x_n\right\}$ converges to a point $s\in \mathcal{W}$, that is,

$${\displaystyle \underset{n\to \infty }{lim}\mathcal{D}(x_n,s)=0}$$

(9)

Next, we prove that s is a fixed point of F, that is, $Fs=s$. By the triangle inequality, we have

$$\begin{array}{cc}\hfill \mathcal{D}(s,Fs)& \le \alpha \left(\mathcal{D}(s,x_n)\right)+\beta \left(\mathcal{D}(x_n,Fs)\right)\hfill \\ \hfill & =\alpha \left(\mathcal{D}(s,x_n)\right)+\beta \left(\mathcal{D}(F{x}_{n-1},Fs)\right)\hfill \\ \hfill & \le \alpha \left(\mathcal{D}(s,x_n)\right)+\beta \left(k\mathcal{D}(x_{n-1},s)\right)\hfill \end{array}$$

(10)

Since $\alpha$ and $\beta$ are continuous, using the condition (2) from Theorem 3 and inequality (9) in (10), we obtain

$$\begin{array}{cc}\hfill \mathcal{D}(s,Fs)& {\displaystyle \le \underset{n\to \infty }{lim}\left[\alpha \left(\mathcal{D}(s,x_n)\right)+\beta \left(k\mathcal{D}(x_{n-1},s)\right)\right]}\hfill \\ \hfill & {\displaystyle =\alpha \left(\underset{n\to \infty }{lim}\mathcal{D}(s,x_n)\right)+\beta \left(\underset{n\to \infty }{lim}k\mathcal{D}(x_{n-1},s)\right)}\hfill \\ \hfill & =\alpha \left(0\right)+\beta \left(0\right)\hfill \\ \hfill & =0\hfill \end{array}$$

(11)

Finally, we prove that F has a unique fixed point. Suppose there are two fixed points $s,t\in \mathcal{W},s\ne t$. We have $\mathcal{D}(s,t)=\mathcal{D}(Fs,Ft)\le k\mathcal{D}(s,t)$, which implies $\mathcal{D}(s,t)=0$ since $k\in (0,1).$ This further implies that $s=t$. □

Corollary 1. 

Letting $\alpha \left(\eta \right)=\beta \left(\eta \right)=\eta$ in Theorem 3, we obtain the classical Banach fixed-point theorem.

Proof. 

With $\alpha \left(\eta \right)=\beta \left(\eta \right)=\eta$, we observe that all the conditions of Theorem 3 are satisfied. In particular, we have

$$\begin{array}{cc}& {\displaystyle \underset{m,n\to \infty }{lim}\left[\sum _{i=m}^{n-2}{\beta }^{i-m}\alpha \left(k^i\mathcal{D}(x_0,x_1)\right)+{\beta }^{n-m-1}\left(k^{n-1}\mathcal{D}(x_0,x_1))\right)\right]}\hfill \\ & {\displaystyle =\mathcal{D}(x_0,x_1)\underset{m,n\to \infty }{lim}\left[\sum _{i=m}^{n-2}{k}^i+k^{n-1}\right]}\hfill \\ & {\displaystyle =\mathcal{D}(x_0,x_1)\underset{m,n\to \infty }{lim}\left[\frac{k^n-k^{m+1}}{(k-1)k}+k^{n-1}\right]}\hfill \\ & {\displaystyle =\mathcal{D}(x_0,x_1)\underset{m,n\to \infty }{lim}\left[\frac{k^n-k^m}{(k-1)}\right]}\hfill \\ & =0\hfill \end{array}$$

□

The following result is similar to a Kannan-type fixed-point theorem.

Theorem 4. 

Let $(\mathcal{W},\mathcal{D})$ be a complete double-composed metric space with non-constant control functions $\alpha ,\beta :[0,\infty )\to [0,\infty )$. Let $F:\mathcal{W}\to \mathcal{W}$ be a mapping satisfying

$$\mathcal{D}(F\eta ,F\theta )\le k\left[\mathcal{D}\right(\eta ,T\eta )+\mathcal{D}(\theta ,T\theta \left)\right]$$

(12)

for all $\eta ,\theta \in \mathcal{W}$ and $k\in (0,\frac{1}{2}).$ For $x_0\in \mathcal{W}$, define a sequence $\left\{x_n\right\}$ by $x_n=F^n{x}_0$. Suppose that the following conditions are satisfied:

1.

${\displaystyle \underset{m,n\to \infty }{lim}\left[\sum _{i=m}^{n-2}{\beta }^{i-m}\alpha \left(r^i\mathcal{D}(x_0,x_1)\right)+{\beta }^{n-m-1}\left(r^{n-1}\mathcal{D}(x_0,x_1))\right)\right]=0}$ where $r=\frac{k}{1-k}$, and ${\beta }^{i-m}\left(\alpha \left(r^i\mathcal{D}(x_0,x_1))\right)\right)$ and ${\beta }^{n-m-1}\left(r^{n-1}\mathcal{D}(x_0,x_1)\right)$ denote the composite functions.

2.

α and β are continuous and non-decreasing functions with $\alpha \left(0\right)=0$, and β is sub-additive with $\beta \left(k\eta \right)<x$ for every $\eta \in \mathcal{W}$ and $k\in (0,1).$

Then, F has a unique fixed point.

Proof. 

Let $x_0\in \mathcal{W}$ and define a sequence $\left\{x_n\right\}$ in $\mathcal{W}$ inductively by taking $x_n=F{x}_{n-1}$, $n\ge 1$. Set ${\mathcal{D}}_n=\mathcal{D}(x_n,x_{n+1})$; then, we have

$$\begin{array}{cc}\hfill {\mathcal{D}}_n=\mathcal{D}\left(x_n,x_{n+1}\right)& =\mathcal{D}\left(F{x}_{n-1},F{x}_n\right)\hfill \\ \hfill & \le k\left[\mathcal{D}(x_{n-1},F{x}_{n-1})+\mathcal{D}(x_n,F{x}_n)\right]\hfill \\ \hfill & =k\left[\mathcal{D}(x_{n-1},x_n)+\mathcal{D}(x_n,x_{n+1})\right]\hfill \\ \hfill & \le k\left[{\mathcal{D}}_{n-1}+{\mathcal{D}}_n)\right]\hfill \end{array}$$

(13)

which implies,

$${\mathcal{D}}_n\le r{\mathcal{D}}_{n-1}$$

(14)

where $r=\frac{k}{1-k}<1$ as $k\in (0,\frac{1}{2})$.

Thus, we have

$$\mathcal{D}\left(x_n,x_{n+1}\right)={\mathcal{D}}_n\le r{\mathcal{D}}_{n-1}\le r^2{\mathcal{D}}_{n-2}\le ...\le r^n\mathcal{D}\left(x_0,x_1\right)$$

(15)

Similar to inequality (6) in Theorem 3, for all $n\ge m$, we have

$$\mathcal{D}(x_m,x_n)\le \sum _{i=m}^{n-2}{\beta }^i\alpha \left(\mathcal{D}(x_i,x_{i+1})\right)+{\beta }^{n-1}\left(\mathcal{D}(x_{n-1},x_n)\right)$$

(16)

Using inequality (15) in (16), we obtain

$$\mathcal{D}(x_m,x_n)\le \sum _{i=m}^{n-2}{\beta }^{i-m}\alpha \left(r^i\mathcal{D}(x_0,x_1)\right)+{\beta }^{n-m-1}\left(r^{n-1}\mathcal{D}(x_0,x_1))\right)$$

(17)

Letting $m,n$ tend to infinity in (17), and by condition (1) of Theorem 4, we obtain

$$\mathcal{D}(x_m,x_n)=0$$

(18)

Therefore, the sequence $\left\{x_n\right\}$ is Cauchy in $\mathcal{W}$. Since $\mathcal{W}$ is a complete double-composed metric space, the sequence $\left\{x_n\right\}$ converges to a point $s\in \mathcal{W}$, that is,

$${\displaystyle \underset{n\to \infty }{lim}\mathcal{D}(x_n,s)=0}$$

(19)

Next, we prove that s is a fixed point of F. Assume that $Fs\ne s$. Using inequality (12), we have

$$\begin{array}{cc}\hfill 0<\mathcal{D}(s,Fs)& \le \alpha \left(\mathcal{D}(s,x_n)\right)+\beta \left(\mathcal{D}(x_n,Fs)\right)\hfill \\ \hfill & =\alpha \left(\mathcal{D}(s,x_n)\right)+\beta \left(\mathcal{D}(F{x}_{n-1},Fs)\right)\hfill \\ \hfill & \le \alpha \left(\mathcal{D}(s,x_n)\right)+\beta \left(k\mathcal{D}(x_{n-1},F{x}_{n-1})+k\mathcal{D}(s,Fs)\right)\hfill \\ \hfill & =\alpha \left(\mathcal{D}(s,x_n)\right)+\beta \left(k\mathcal{D}(x_{n-1},x_n)+k\mathcal{D}(s,Fs)\right).\hfill \end{array}$$

(20)

Since $\alpha$ and $\beta$ are continuous, taking limit both sides in inequality (20) and using (19), we obtain

$$0<\mathcal{D}(s,Fs)\le \alpha \left(0\right)+\beta \left(k\mathcal{D}(s,Fs)\right)$$

(21)

By condition (2) of Theorem 4, we have

$$\alpha \left(0\right)=0,\beta \left(k\mathcal{D}(s,Fs)\right)<\mathcal{D}(s,Fs)$$

(22)

Using inequality (22) in (22), we have

$$0<\mathcal{D}(s,Fs)<\mathcal{D}(s,Fs),$$

which is a contradiction. Hence, $Fs=s.$ Finally, we prove the uniqueness of the fixed point. Suppose that F has two distinct fixed points x and $\delta$, that $Fx=x$ and $F\delta =\delta$. We have,

$$\mathcal{D}(x,\delta )=\mathcal{D}(Fx,F\delta )\le k\left[\mathcal{D}(x,Fx)+\mathcal{D}(\delta ,F\delta )\right]=k\left[\mathcal{D}(x,x)+\mathcal{D}(\delta ,\delta )\right]=0.$$

This implies $\mathcal{D}(x,\delta )=0$, which further implies that $x=\delta$. □

4. Applications

The following example demonstrates the application of Theorem 3.

Example 6. 

Let $\mathcal{W}=[-\frac{\sqrt{2-\sqrt{2}}}{2},\frac{\sqrt{2-\sqrt{2}}}{2}]$. Define functions $\alpha ,\beta :[0,\infty )\to [0,\infty )$ by

$$\alpha \left(\eta \right)=\beta \left(\eta \right)=\left\{\begin{array}{cc}sin\left(4{sin}^{-1}\left(x\right)\right)\mathit{if}\hfill & 0\le \eta \le \frac{\sqrt{2-\sqrt{2}}}{2}\hfill \\ 1\mathit{if}\hfill & \frac{\sqrt{2-\sqrt{2}}}{2}<\eta <\infty \hfill \end{array}\right..$$

Define a function $\mathcal{D}:\mathcal{W}\times \mathcal{W}\to [0,\infty )$ by

$$\mathcal{D}(\eta ,\theta )={(\eta -\theta )}^2$$

Similar to the proof of Example 2, it is not difficult to see that $(\mathcal{W},\mathcal{D})$ is a complete DCMS with control functions $\alpha \left(\eta \right)$ and $\beta \left(\eta \right)$. Define a self-map $F:\mathcal{W}\to \mathcal{W}$ by $F\eta =\frac{\eta }{2}.$ Then, F has a unique fixed point.

Proof. 

First note that

$$\mathcal{D}(F\eta ,F\theta )={\left(\frac{\eta }{2}-\frac{\theta }{2}\right)}^2\le \frac{1}{4}{(\eta -\theta )}^4=k\mathcal{D}(\eta ,\theta ),$$

so that F satisfies the contraction condition with $k=\frac{1}{4}$. We also have $\alpha ,\beta$ continuous and non-decreasing with $\alpha \left(0\right)+\beta \left(0\right)=0.$ Next, we claim that $\beta$ is the sub-additive function. There is a classical result from the real analysis stating that if a function f on $[0,\infty )$ is concave, and $f\left(0\right)\ge 0$, then f is sub-additive. It can be easily verified that $\beta$ is indeed a concave function on its domain and therefore is sub-additive as well.

Now we have,

$$\beta \left(\eta \right)=sin\left(4{sin}^{-1}\left(\eta \right)\right)$$

$${\beta }^2\left(\eta \right)=\beta \left(\beta \left(\eta \right)\right)=sin\left(4{sin}^{-1}(sin\left(4{sin}^{-1}\left(\eta \right)\right))\right)=sin\left(4^2{sin}^{-1}\left(\eta \right)\right)$$

$${\beta }^3\left(\eta \right)=\beta \left({\beta }^2\left(\eta \right)\right)=\beta (sin\left(4^2{sin}^{-1}\left(\eta \right)\right))=sin\left(4{sin}^{-1}(sin\left(4^2{sin}^{-1}\left(\eta \right)\right))\right)=sin\left(4^3{sin}^{-1}\left(\eta \right)\right)$$

In general, we observe that for all natural numbers i and m, we have

$${\beta }^i\left(\eta \right)=sin\left(4^i{sin}^{-1}\left(\eta \right)\right)$$

(23)

Since $\alpha =\beta$, we further compute

$${\beta }^{i-m}\alpha \left(\eta \right)={\beta }^{i-m}\left(\beta \left(\eta \right)\right)=sin\left(4^{i-m+1}{sin}^{-1}\left(\eta \right)\right)$$

(24)

Fix $x_0=0\in X$, then $x_1=F{x}_0=\frac{0}{2}=0$ so that $d(x_0,x_1)={(0-0)}^2=0.$

Using inequality (23) and (24), we obtain

$$\begin{array}{cc}& {\displaystyle \underset{m,n\to \infty }{lim}\left[\sum _{i=m}^{n-2}{\beta }^{i-m}\alpha \left(k^i\mathcal{D}(x_0,x_1)\right)+{\beta }^{n-m-1}\left(k^{n-1}\mathcal{D}(x_0,x_1))\right)\right]}\hfill \\ & {\displaystyle =\underset{m,n\to \infty }{lim}\left[\sum _{i=m}^{n-2}sin\left(4^{i-m+1}{sin}^{-1}\left(k^{i}d(x_0,x_1)\right)\right)+sin\left(4^{n-m-1}{sin}^{-1}\left(k^{n-1}d(x_0,x_1)\right)\right)\right]}\hfill \\ & {\displaystyle =\underset{m,n\to \infty }{lim}\left[\sum _{i=m}^{n-2}sin\left(4^{i-m+1}{sin}^{-1}\left(0\right)\right)+sin\left(4^{n-m-1}{sin}^{-1}\left(0\right)\right)\right]}\hfill \\ & =0\hfill \end{array}$$

There, F satisfies all of the conditions of Theorem 3, implying that F has a unique fixed point given by $\eta =0$.

We note that $(\mathcal{W},\mathcal{D})$ is not the usual metric space; hence, the classical Banach contraction principle is not applicable. □

5. Conclusions

In this article, we developed a novel generalization of a metric space, called a double-composed metric space. We have provided some examples of our new space. We have also established Banach- and Kannan-type fixed-point theorems in double-composed metric spaces. We have illustrated the application of Banach-type fixed-point theorem with an example. Our new generalization would further enrich the study of fixed-point theory.

We propose some open questions for future work.

i.

Establish the new fixed-point results in a double-composed metric space for the Chatterjee contraction, the Reich contraction, the Hardy–Rogers contraction, and other rational-type contraction mappings.

ii.

Establish some non-trivial applications of Theorems 3 and 4. In particular, provide a non-trivial application of Theorem 3 to the theory of non-linear integral equations.

iii.

Finally, we propose an important direction for further research in the framework of a double-composed metric space. When there is no unique fixed point, one method for generalizing fixed-point results is to analyze the geometric properties of the set of fixed points. The fixed-circle problem (see [18]) and the fixed-figure problem (see [19,20]) have been introduced accordingly.