Banach Fixed Point Theorem in Extended b_v(s)-Metric Spaces ^†

Abstract

We define the class of extended $b_v\left(s\right)$-metric spaces by replacing the real number $s\ge 1$ with a strictly increasing continuous function $\varphi$ in the definition of a $b_v\left(s\right)$-metric space. Also, we presented an example for this newly introduced space and exhibited that in a particular situation, the class of extended $b_v\left(s\right)$-metric spaces reduces to the class of $b_v\left(s\right)$-metric spaces. Afterwards, we establish a fixed point theorem which ensured the existence of a fixed point for the self-map satisfying the Banach contractive condition in the context of this newly defined space. Moreover, we compared the proved result with the existing fixed point theorems in the literature.

1. Introduction and Preliminaries

As the solution of certain problems of nonlinear analysis relies heavily on fixed-point theory, one of the key findings of metric fixed-point theory, the Banach contraction principle, is the subject of intense investigation. For the last three decades, the generalization of the metric space is one of the key directions in which the contraction principle is continuously flourishing. In 1989, Bakhtin [1] pioneered the idea of b-metric space while moving in the same direction that was conventionally defined in 1993 by Czerwik [2]. On the other side, Branciari [3] generalized the metric space by replacing the triangle inequality with the quadrilateral inequality, and later on, it was referred to as rectangular metric space by the researchers. In 2015, George et al. [4] defined the class of rectangular b-metric spaces as an extension of the class of b-metric spaces. By the ideas of Bakhtin [1] and George et al. [4], a notion of $b_v\left(s\right)$- metric space is announced by Mitrović and Radenović [5] in the recent past, which in general is an extension of v-generalized metric spaces, b-metric spaces and rectangular b-metric spaces.

Very recently, Mustafa et al. [6] presented a new notion, namely extended rectangular b-metric space, as an extension of rectangular b-metric spaces, which was inspired by the idea of Parvaneh and Ghoncheh [7] for defining the concept of extended b-metric space. In a similar manner, we here introduce the concept of extended $b_v\left(s\right)$-metric space and ensure the existence of a fixed point for the self-map that satisfies the Banach contractive condition in this space, i.e., we present the analogue of the Banach contraction principle in this newly defined space.

Here, we give a few definitions and findings that will be important in the discussion that follows.

Definition 1 

([4]). Let Ω be a nonempty set and $s\ge 1$ be a real number. A mapping $d_{rb}:\Omega \times \Omega \to [0,\infty )$ is said to be a rectangular b-metric on Ω if the following axioms hold for all $x,y\in \Omega$:

1. 

$d_{rb}(x,y)=0$ if and only if $x=y$,

2. 

$d_{rb}(x,y)=d_{rb}(y,x)$,

3. 

$d_{rb}(x,y)\le s[d_{rb}(x,{\omega }_1)+d_{rb}({\omega }_1,{\omega }_2)+d_{rb}({\omega }_2,y)]$ for all distinct points ${\omega }_1,{\omega }_2\in \Omega -\{x,y\}$.

• The pair $(\Omega ,d_{rb})$ is called a rectangular b-metric space.

In 2019, Mustafa et al. [6] follow the idea of Parveneh and Ghoncheh [7] and define the notion of extended rectangular b-metric space by utilizing the following class of strictly increasing continuous functions:

$$\begin{array}{ccc}\hfill \Psi & =& \{\varphi :[0,\infty )\to [0,\infty ),\varphi \mathrm{is}\mathrm{a}\mathrm{strictly}\mathrm{increasing}\mathrm{continuous}\mathrm{function}\hfill \\ & & \mathrm{with}t\le \varphi \left(t\right)\mathrm{and}\varphi \left(0\right)=0\}\hfill \end{array}$$

Definition 2 

([6]). Let Ω be a nonempty set and $\varphi \in \Psi$. A mapping $d_{rb}^e:\Omega \times \Omega \to [0,\infty )$ is said to be an extended rectangular b-metric on Ω if the following axioms hold for all $x,y\in \Omega$:

1. 

$d_{rb}^e(x,y)=0$ if and only if $x=y$,

2. 

$d_{rb}^e(x,y)=d_{rb}^e(y,x)$,

3. 

$d_{rb}^e(x,y)\le \varphi [d_{rb}^e(x,{\omega }_1)+d_{rb}^e({\omega }_1,{\omega }_2)+d_{rb}^e({\omega }_2,y)]$ for all distinct points ${\omega }_1,{\omega }_2\in \Omega -\{x,y\}$.

• The pair $(\Omega ,d_{rb}^e)$ is called an extended rectangular b-metric space.

The concept of $b_v\left(s\right)$-metric space given by Mitrović and Radenović [5] (also called v-generalized b-metric space in Kadyan et al. [8]) is defined as under:

Definition 3 

([5]). Let Ω be a nonempty set and $s\ge 1$ be a real number. A mapping $d_{v_{g}b}:\Omega \times \Omega \to [0,\infty )$ is said to be a $b_v\left(s\right)$-metric on Ω if the following axioms hold for all $x,y\in \Omega$:

1. 

$d_{v_{g}b}(x,y)=0$ if and only if $x=y$,

2. 

$d_{v_{g}b}(x,y)=d_{v_{g}b}(y,x)$,

3. 

$d_{v_{g}b}(x,y)\le s[d_{v_{g}b}(x,{\omega }_1)+d_{v_{g}b}({\omega }_1,{\omega }_2)+....+d_{v_{g}b}({\omega }_v,y)]$ for all distinct points ${\omega }_1,{\omega }_2,$$...,{\omega }_v\in \Omega -\{x,y\}$, where v is a natural number.

• The pair $(\Omega ,d_{v_{g}b})$ is called a $b_v\left(s\right)$-metric space.

2. Main Result

We start by introducing the concept of extended $b_v\left(s\right)$-metric space.

Definition 4. 

Let Ω be a nonempty set and $\varphi \in \Psi$. A mapping $d_{v_{g}b}^e:\Omega \times \Omega \to [0,\infty )$ is said to be an extended $b_v\left(s\right)$-metric on Ω if the following axioms hold for all $x,y\in \Omega$:

1. 

$d_{v_{g}b}^e(x,y)=0$ if and only if $x=y$,

2. 

$d_{v_{g}b}^e(x,y)=d_{v_{g}b}^e(y,x)$,

3. 

$d_{v_{g}b}^e(x,y)\le \varphi [d_{v_{g}b}^e(x,{\omega }_1)+d_{v_{g}b}^e({\omega }_1,{\omega }_2)+....+d_{v_{g}b}^e({\omega }_v,y)]$ for all distinct points ${\omega }_1,{\omega }_2,$$...,{\omega }_v\in \Omega -\{x,y\}$, where v is a natural number.

• The pair $(\Omega ,d_{v_{g}b}^e)$ is said to be an extended $b_v\left(s\right)$-metric space.

Remark 1. 

In respect of this newly introduced space, it is important to note that

(1)

If $v=1$, then it reduces to an extended b-metric space.

(2)

If $v=2$, then it reduces to an extended rectangular b-metric space.

(3)

If we define $\varphi \left(t\right)=st$, where $s\ge 1$, then it reduces to a $b_v\left(s\right)$-metric space.

(4)

If we take $\varphi \left(t\right)=t$, then it reduces to a v-generalized metric space.

In light of the above remark, it is clear to mention that the class of extended $b_v\left(s\right)$-metric spaces is larger than the class of extended rectangular b-metric spaces and the class of $b_v\left(s\right)$-metric spaces.

Now, we construct an example for the extended $b_v\left(s\right)$-metric space.

Example 1. 

Let $(\Omega ,d_{vg})$ be a v-generalized metric space. Define $\rho (x,y)=e^{d_{vg}(x,y)}-e^{-d_{vg}(x,y)}$. We will show that $(\Omega ,\rho )$ is an extended $b_v\left(s\right)$-metric space with $\varphi \left(x\right)=e^x-e^{-x}\forall x\ge 0$.

It is easy to verify conditions 1 and 2 of Definition 4. Now, we verify condition 3 as follows:

$$\begin{array}{ccc}\hfill \rho (x,y)& =& e^{d_{vg}(x,y)}-e^{-d_{vg}(x,y)}\hfill \\ & \le & e^{d_{vg}(x,{\omega }_1)+d_{vg}({\omega }_1,{\omega }_2)+\cdots +d_{vg}({\omega }_v,y)}-e^{-(d_{vg}(x,{\omega }_1)+d_{vg}({\omega }_1,{\omega }_2)+\cdots +d_{vg}({\omega }_v,y))}\hfill \\ & \le & e^{\left[e^{d_{vg}(x,{\omega }_1)}-e^{-d_{vg}(x,{\omega }_1)}+e^{d_{vg}({\omega }_1,{\omega }_2)}-e^{-d_{vg}({\omega }_1,{\omega }_2)}+\cdots +e^{d_{vg}({\omega }_v,y)}-e^{-d_{vg}({\omega }_v,y)}\right]}\hfill \\ & & -e^{-\left[e^{d_{vg}(x,{\omega }_1)}-e^{-d_{vg}(x,{\omega }_1)}+e^{d_{vg}({\omega }_1,{\omega }_2)}-e^{-d_{vg}({\omega }_1,{\omega }_2)}+\cdots +e^{d_{vg}({\omega }_v,y)}-e^{-d_{vg}({\omega }_v,y)}\right]}\hfill \\ & \le & e^{[\rho (x,{\omega }_1)+\rho ({\omega }_1,{\omega }_2)+\cdots +\rho ({\omega }_v,y)]}-e^{-[\rho (x,{\omega }_1)+\rho ({\omega }_1,{\omega }_2)+\cdots +\rho ({\omega }_v,y)]}\hfill \\ & =& \varphi [\rho (x,{\omega }_1)+\rho ({\omega }_1,{\omega }_2)+\cdots +\rho ({\omega }_v,y)].\hfill \end{array}$$

Thus, all the conditions of Definition 4 are satisfied, and hence, $(\Omega ,\rho )$ is an extended $b_v\left(s\right)$-metric space.

The convergence of a sequence and Cauchy sequence in an extended $b_v\left(s\right)$-metric space is defined in the usual sense as mentioned under:

Definition 5. 

Let $(\Omega ,d_{v_{g}b}^e)$ be an extended $b_v\left(s\right)$-metric space. A sequence ${\zeta }_n\in \Omega$ is said to be

(1) 

A convergent sequence (converges to a point $\zeta \in \Omega$) if for each $ϵ>0,\exists$ a positive integer N such that $d_{v_{g}b}^e({\zeta }_n,\zeta )<ϵ$ for all $n\ge N$. Symbolically, it may be written as ${\zeta }_n\to \zeta$ whenever $n\to \infty$.

(2) 

A Cauchy sequence if for each $ϵ>0,\exists$ a positive integer N such that $d_{v_{g}b}^e({\zeta }_n,{\zeta }_{n+p})<ϵ$ for all $n\ge N$ and $p>0$. It is also denoted as ${lim}_{n\to \infty }{d}_{v_{g}b}^e({\zeta }_n,{\zeta }_{n+p})=0$.

The extended $b_v\left(s\right)$-metric space $(\Omega ,d_{v_{g}b}^e)$ is said to be complete if every Cauchy sequence in $\Omega$ converges in $\Omega$ itself. Now, we establish the following lemmas which are required to prove our forthcoming result.

Lemma 1. 

Let $(\Omega ,d_{v_{g}b}^e)$ be an extended $b_v\left(s\right)$-metric space and $\left\{{\zeta }_n\right\}$ be a sequence of distinct points in Ω. Furthermore, if $d_{v_{g}b}^e({\zeta }_m,{\zeta }_n)\le \eta d_{v_{g}b}^e({\zeta }_{m-1},{\zeta }_{n-1})\forall m,n$, where $\eta \in [0,1)$. Then, the sequence $\left\{{\zeta }_n\right\}$ is Cauchy.

Proof. 

As $d_{v_{g}b}^e({\zeta }_m,{\zeta }_n)\le \eta d_{v_{g}b}^e({\zeta }_{m-1},{\zeta }_{n-1})$ for all $m,n\in \mathbb{N}$, it follows that

$$\begin{array}{ccc}\hfill d_{v_{g}b}^e({\zeta }_{m+k},{\zeta }_{n+k})& \le & \eta d_{v_{g}b}^e({\zeta }_{m+k-1},{\zeta }_{n+k-1})\hfill \\ & \le & {\eta }^2{d}_{v_{g}b}^e({\zeta }_{m+k-2},{\zeta }_{n+k-2})\hfill \\ & ⋮& \hfill \\ & \le & {\eta }^k{d}_{v_{g}b}^e({\zeta }_m,{\zeta }_n)\mathrm{for}\mathrm{all}m,n.\hfill \end{array}$$

(1)

Furthermore, it is obvious to say that

$$d_{v_{g}b}^e({\zeta }_n,{\zeta }_{n+1})\le {\eta }^n{d}_{v_{g}b}^e({\zeta }_0,{\zeta }_1)\mathrm{for}\mathrm{all}n.$$

(2)

Using condition (3) of Definition 4, we have

$$\begin{array}{ccc}\hfill d_{v_{g}b}^e({\zeta }_n,{\zeta }_{n+p})& \le & \varphi [d_{v_{g}b}^e({\zeta }_n,{\zeta }_{n+1})+d_{v_{g}b}^e({\zeta }_{n+1},{\zeta }_{n+2})+\cdots \hfill \\ & & +d_{v_{g}b}^e({\zeta }_{n+v-1},{\zeta }_{n+v})+d_{v_{g}b}^e({\zeta }_{n+v},{\zeta }_{n+p})]\hfill \\ & \le & \varphi [{\eta }^n{d}_{v_{g}b}^e({\zeta }_0,{\zeta }_1)+{\eta }^{n+1}{d}_{v_{g}b}^e({\zeta }_0,{\zeta }_1)+\cdots \hfill \\ & & +{\eta }^{n+v-1}{d}_{v_{g}b}^e({\zeta }_0,{\zeta }_1)+{\eta }^n{d}_{v_{g}b}^e({\zeta }_v,{\zeta }_p)]\hfill \\ & =& \varphi \left[\frac{{\eta }^n(1-{\eta }^v)}{1-\eta }{d}_{v_{g}b}^e({\zeta }_0,{\zeta }_1)+{\eta }^n{d}_{v_{g}b}^e({\zeta }_v,{\zeta }_p)\right].\hfill \end{array}$$

(3)

Since $\eta \in [0,1)$, then we concluded that ${lim}_{n\to \infty }{d}_{v_{g}b}^e({\zeta }_n,{\zeta }_{n+p})\le \varphi \left(0\right)=0$. Thus, the sequence $\left\{{\zeta }_n\right\}$ is Cauchy. □

In an extended $b_v\left(s\right)$-metric space, the sequence may converge to more than one point (see Example 2 of [8]). The following lemma ensures that the Cauchy sequence in an extended $b_v\left(s\right)$-metric space converges to at most one point in the given space.

Lemma 2. 

Let $(\Omega ,d_{v_{g}b}^e)$ be an extended $b_v\left(s\right)$-metric space and $\left\{{\zeta }_n\right\}$ be a Cauchy sequence of distinct points in Ω. Then, $\left\{{\zeta }_n\right\}$ can converge to at most one point.

Proof. 

Suppose that the sequence ${\zeta }_n$ converges to two distinct points of $\Omega$, say $\zeta$ and $\zeta *$. Then, there exists a real number $r>0$ such that for all $n\ge r$, the terms of the sequence $\left\{{\zeta }_n\right\}$ are distinct from $\zeta$ and $\zeta *$. Let $ϵ>0$ be given; then, there exist positive integers $l_1,l_2,l_3$ (as the sequence $\left\{{\zeta }_n\right\}$ is Cauchy and converges to $\zeta$ and $\zeta *$) such that

$$d_{v_{g}b}^e({\zeta }_n,{\zeta }_m)<\frac{ϵ}{(v+1)}\forall n,m\ge l_1,$$

$$d_{v_{g}b}^e({\zeta }_n,\zeta )<\frac{ϵ}{(v+1)}\forall n\ge l_2,$$

and

$$d_{v_{g}b}^e({\zeta }_n,\zeta *)<\frac{ϵ}{(v+1)}\forall n\ge l_3.$$

Define $l=max\{r,l_1,l_2,l_3\}$. Then,

$$\begin{array}{ccc}\hfill d_{v_{g}b}^e(\zeta ,{\zeta }^{*})& \le & \varphi [d_{v_{g}b}^e(\zeta ,{\zeta }_{n+v-2})+d_{v_{g}b}^e({\zeta }_{n+v-2},{\zeta }_{n+v-3})+\cdots \hfill \\ & & +d_{v_{g}b}^e({\zeta }_{n+1},{\zeta }_n)+d_{v_{g}b}^e({\zeta }_n,{\zeta }_l)+d_{v_{g}b}^e({\zeta }_l,{\zeta }^{*})]\hfill \\ & <& \varphi \left[\frac{ϵ}{(v+1)}+\frac{ϵ}{(v+1)}+\cdots +\frac{ϵ}{(v+1)}\right]=\varphi \left(ϵ\right),\mathrm{for}n\ge l.\hfill \end{array}$$

That implies ${\varphi }^{-1}\left(d_{v_{g}b}^e(\zeta ,\zeta *)\right)<ϵ$ and hence $\zeta ={\zeta }^{*}$ as $ϵ>0$ is arbitrary and $\varphi \left(0\right)=0$. So, the sequence $\left\{{\zeta }_n\right\}$ converges to a unique point in $\Omega$. □

Now, we prove the Banach fixed point theorem in extended $b_v\left(s\right)$-metric space.

Theorem 1. 

Let $(\Omega ,d_{v_{g}b}^e)$ be an extended $b_v\left(s\right)$-metric space and $T:\Omega \to \Omega$ be a mapping satisfying

$$d_{v_{g}b}^e(T\alpha ,T\beta )\le \eta d_{v_{g}b}^e(\alpha ,\beta )\mathrm{for}\mathrm{all}\alpha ,\beta \in \Omega ,$$

(4)

where $\eta \in [0,1)$. Then, T has a unique fixed point provided the space Ω is complete.

Proof. 

Choose an arbitrary element ${\zeta }_0\in \Omega$ and construct a sequence $\left\{{\zeta }_n\right\}$ as ${\zeta }_{n+1}=T{\zeta }_n$ for each $n\in \mathbb{N}\cup \left\{0\right\}$. Suppose ${\zeta }_n\ne {\zeta }_{n+1}$ for all $n\ge 0$; otherwise, ${\zeta }_n$ is a fixed point of T for some n and the result holds. Firstly, we prove that the terms of the sequence $\left\{{\zeta }_n\right\}$ are distinct. On the contrary, if ${\zeta }_n={\zeta }_m$ for some $n>m$, then ${\zeta }_n={\zeta }_{m+k}$ for some $k\ge 1$ that implies ${\zeta }_{m+1}={\zeta }_{m+k+1}$. Thus, by using inequality (4), we obtain that

$$\begin{array}{ccc}\hfill d_{v_{g}b}^e({\zeta }_{m+1},{\zeta }_m)& =& d_{v_{g}b}^e({\zeta }_{m+k+1},{\zeta }_{m+k})\hfill \\ & =& d_{v_{g}b}^e(T{\zeta }_{m+k},T{\zeta }_{m+k-1})\hfill \\ & \le & \eta d_{v_{g}b}^e({\zeta }_{m+k},{\zeta }_{m+k-1})\hfill \\ & \le & {\eta }^2{d}_{v_{g}b}^e({\zeta }_{m+k-1},{\zeta }_{m+k-2})\hfill \\ & ⋮& \hfill \\ & \le & {\eta }^k{d}_{v_{g}b}^e({\zeta }_{m+1},{\zeta }_m)\hfill \\ & <& d_{v_{g}b}^e({\zeta }_{m+1},{\zeta }_m).\hfill \end{array}$$

(5)

Therefore, our supposition that ${\zeta }_n={\zeta }_m$ for some $n>m$ is not true, and hence, the terms of the sequence $\left\{{\zeta }_n\right\}$ are distinct. Moreover, inequality (4) gives that $d_{v_{g}b}^e({\zeta }_m,{\zeta }_n)\le \eta d_{v_{g}b}^e({\zeta }_{m-1},{\zeta }_{n-1})$ for all $m,n$. Due to Lemma 1, it follows that the sequence $\left\{{\zeta }_n\right\}$ is Cauchy in Ω, and it converges to some point, say $\zeta \in \Omega$, as the space $(\Omega ,d_{v_{g}b}^e)$ is complete.

Now, we claim that the point $\zeta \in \Omega$ is a fixed point of map T. As the sequence $\left\{{\zeta }_n\right\}$ converges to ζ and ${\zeta }_n=T^n{\zeta }_0$, then $T^n{\zeta }_0\to \zeta$. Again, due to inequality(4), we have

$$\begin{array}{ccc}\hfill d_{v_{g}b}^e(T^{n+1}{\zeta }_0,T\zeta )& \le & \eta d_{v_{g}b}^e(T^n{\zeta }_0,\zeta )\hfill \\ & =& \eta d_{v_{g}b}^e({\zeta }_n,\zeta ).\hfill \end{array}$$

Taking $n\to \infty$, we obtain that $d_{v_{g}b}^e(T^{n+1}{\zeta }_0,T\zeta )\to 0$ and consequently ${\zeta }_{n+1}\to T\zeta$. Therefore, on account of Lemma 2, it follows that $T\zeta =\zeta$. Thus, ζ is a fixed point of the map T. If ${\zeta }_1,{\zeta }_2$ are two fixed points of T in the space Ω, then in lieu of inequality (4), we obtain

$$d_{v_{g}b}^e({\zeta }_1,{\zeta }_2)\le \eta d_{v_{g}b}^e({\zeta }_1,{\zeta }_2)<d_{v_{g}b}^e({\zeta }_1,{\zeta }_2).$$

This is not true, and thus, the fixed point of T is unique. □

Remark 2. 

If we take $\varphi \left(t\right)=st$, where $s\ge 1$, then we obtain the Banach contraction principle in $b_v\left(s\right)$-metric space: that means we obtain Theorem 2.1 of [5] and Theorem 9 of [9] as a special case of Theorem 1. If we take $v=2$, then we obtain the Banach fixed point theorem for extended rectangular b-metric space, and hence, Theorem 2.1 of [10] is a particular case of Theorem 1.

3. Discussion

The space we introduced here is generalizing not only the metric space but also the various generalized versions of metric space existing in the literature, for example, b-metric space, rectangular metric space, rectangular b-metric space, etc. Therefore, the fixed point result presented here is important in the sense that it extends the existing Banach contraction principle in extended rectangular b-metric spaces and $b_v\left(s\right)$-metric spaces.