A Generalization of b-Metric Space and Some Fixed Point Theorems

Abstract

In this paper, inspired by the concept of b-metric space, we introduce the concept of extended b-metric space. We also establish some fixed point theorems for self-mappings defined on such spaces. Our results extend/generalize many pre-existing results in literature.

1. Introduction

The idea of b-metric was initiated from the works of Bourbaki [1] and Bakhtin [2]. Czerwik [3] gave an axiom which was weaker than the triangular inequality and formally defined a b-metric space with a view of generalizing the Banach contraction mapping theorem. Later on, Fagin et al. [4] discussed some kind of relaxation in triangular inequality and called this new distance measure as non-linear elastic mathing (NEM). Similar type of relaxed triangle inequality was also used for trade measure [5] and to measure ice floes [6]. All these applications intrigued and pushed us to introduce the concept of extended b-metric space. So that the results obtained for such rich spaces become more viable in different directions of applications.

Definition 1.

Let X be a non empty set and $s\ge 1$ be a given real number. A function $d:X\times X\to [0,\infty )$ is called b-metric (Bakhtin [2], Czrerwik [3]) if it satisfies the following properties for each $x,y,z\in X$.

(b1): 

$d(x,y)=0\iff x=y;$

(b2): 

$d(x,y)=d(y,x);$

(b3): 

$d(x,z)\le s\left[d\right(x,y)+d(y,z\left)\right]$.

The pair $(X,d)$ is called a b-metric space.

Example 1.

1. Let $X:=l_p\left(\mathbb{R}\right)$ with $0<p<1$ where $l_p\left(\mathbb{R}\right):=\{\left\{x_n\right\}\subset \mathbb{R}:{\sum }_{n=1}^{\infty }|x_n{|}^p<\infty \}$. Define $d:X\times X\to {\mathbb{R}}^{+}$ as:

$$d(x,y)={(\sum _{n=1}^{\infty }{|x_n-y_n|}^p)}^{1/p}$$

where $x=\left\{x_n\right\},y=\left\{y_n\right\}$. Then d is a b-metric space [7,8,9] with coefficient $s=2^{1/p}$.

2. Let $X:=L_p[0,1]$ be the space of all real functions $x\left(t\right),t\in [0,1]$ such that ${\int }_0^1{\left|x\left(t\right)\right|}^p<\infty$ with $0<p<1$. Define $d:X\times X\to {\mathbb{R}}^{+}$ as:

$$d(x,y)={\left({\int }_0^1{|x\left(t\right)-y\left(t\right)|}^{p}dt\right)}^{1/p}$$

Then d is b-metric space [7,8,9] with coefficient $s=2^{1/p}.$

The above examples show that the class of b-metric spaces is larger than the class of metric spaces. When $s=1$, the concept of b-metric space coincides with the concept of metric space. For some details on subject see [7,8,9,10,11,12].

Definition 2.

Let $(X,d)$ be a b-metric space. A sequence $\left\{x_n\right\}$ in X is said to be:

(I) 

Cauchy [12] if and only if $d(x_n,x_m)\to 0$ as $n,m\to \infty ;$

(II) 

Convergent [12] if and only if there exist $x\in X$ such that $d(x_n,x)\to 0$ as $n\to \infty$ and we write ${\lim }_{n\to \infty }{x}_n=x;$

(III) 

The b-metric space $(X,d)$ is complete [12] if every Cauchy sequence is convergent.

In the following we recollect the extension of Banach contraction principle in case of b-metric spaces.

Theorem 1.

Let $(X,d)$ be a complete b-metric space with constant $s\ge 1$, such that b-metric is a continuous functional. Let $T:X\to X$ be a contraction having contraction constant $k\in [0,1)$ such that $ks<1.$ Then T has a unique fixed point [13].

2. Results

In this section, we introduce a new type of generalized metric space, which we call as an extended b-metric space. We also establish some fixed point theorems arising from this metric space.

Definition 3.

Let X be a non empty set and $\theta :X\times X\to [1,\infty ).$ A function $d_{\theta }:X\times X\to [0,\infty )$ is called an extended b-metric if for all $x,y,z\in X$ it satisfies:

• $\left(d_{\theta }1\right)$ $d_{\theta }(x,y)=0$ iff $x=y;$
• $\left(d_{\theta }2\right)$ $d_{\theta }(x,y)=d_{\theta }(y,x);$
• $\left(d_{\theta }3\right)$ $d_{\theta }(x,z)\le \theta (x,z)[d_{\theta }(x,y)+d_{\theta }(y,z)].$

The pair $(X,d_{\theta })$ is called an extended b-metric space.

Remark 1.

If $\theta (x,y)=s$ for $s\ge 1$ then we obtain the definition of a b-metric space.

Example 2.

Let $X=\{1,2,3\}.$ Define $\theta :X\times X\to {\mathbb{R}}^{+}$ and $d_{\theta }:X\times X\to {\mathbb{R}}^{+}$ as:

$$\theta (x,y)=1+x+y$$

$$d_{\theta }(1,1)=d_{\theta }(2,2)=d_{\theta }(3,3)=0$$

$$d_{\theta }(1,2)=d_{\theta }(2,1)=80,d_{\theta }(1,3)=d_{\theta }(3,1)=1000,d_{\theta }(2,3)=d_{\theta }(3,2)=600$$

Proof. 

$\left(d_{\theta }1\right)$ and $\left(d_{\theta }2\right)$ trivially hold. For $\left(d_{\theta }3\right)$ we have:

$$d_{\theta }(1,2)=80,\theta (1,2)\left[d_{\theta }(1,3)+d_{\theta }(3,2)\right]=4(1000+600)=6400$$

$$d_{\theta }(1,3)=1000,\theta (1,3)\left[d_{\theta }(1,2)+d_{\theta }(2,3)\right]=5(80+600)=3400$$

Similar calculations hold for $d_{\theta }(2,3)$. Hence for all $x,y,z\in X$

$$d_{\theta }(x,z)\le \theta (x,z)[d_{\theta }(x,y)+d_{\theta }(y,z)]$$

Hence $(X,d_{\theta })$ is an extended b-metric space. ☐

Example 3.

Let $X=C\left(\right[a,b],\mathbb{R})$ be the space of all continuous real valued functions define on $[a,b]$. Note that X is complete extended b-metric space by considering $d_{\theta }(x,y)={\sup }_{t\in [a,b]}{|x\left(t\right)-y\left(t\right)|}^2$, with $\theta (x,y)=\left|x\right(t\left)\right|+\left|y\right(t\left)\right|+2$, where $\theta :X\times X\to [1,\infty )$.

The concepts of convergence, Cauchy sequence and completeness can easily be extended to the case of an extended b-metric space.

Definition 4.

Let $(X,d_{\theta })$ be an extended b-metric space.

(i) 

A sequence $\left\{x_n\right\}$ in X is said to converge to $x\in X$, if for every $ϵ>0$ there exists $N=N\left(ϵ\right)\in \mathbb{N}$ such that $d_{\theta }(x_n,x)<ϵ,$ for all $n\ge N.$ In this case, we write ${\lim }_{n\to \infty }{x}_n=x.$

(ii) 

A sequence $\left\{x_n\right\}$ in X is said to be Cauchy, if for every $ϵ>0$ there exists $N=N\left(ϵ\right)\in \mathbb{N}$ such that $d_{\theta }(x_m,x_n)<ϵ,$ for all $m,n\ge N.$

Definition 5.

An extended b-metric space $(X,d_{\theta })$ is complete if every Cauchy sequence in X is convergent.

Note that, in general a b-metric is not a continuous functional and thus so is an extended b-metric.

Example 4.

Let $X=\mathbb{N}\bigcup \infty$ and let $d:X\times X\to \mathbb{R}$ be defined by [14]:

$$d(x,y)=\left\{\begin{array}{c}0ifm=n\hfill \\ |\frac{1}{m}-\frac{1}{n}|ifm,nareevenormn=\infty \hfill \\ 5ifm,nareoddandm\ne n\hfill \\ 2otherwise\hfill \end{array}\right.$$

Then $(X,d)$ is a b-metric with $s=3$ but it is not continuous.

Lemma 1.

Let $(X,d_{\theta })$ be an extended b-metric space. If $d_{\theta }$ is continuous, then every convergent sequence has a unique limit.

Our first theorem is an analogue of Banach contraction principle in the setting of extended b-metric space. Throughout this section, for the mapping $T:X\to X$ and $x_0\in X$, $\mathcal{O}\left(x_0\right)=\{x_0,T^2{x}_0,T^3{x}_0,\cdots \}$ represents the orbit of $x_0$.

Theorem 2.

Let $(X,d_{\theta })$ be a complete extended b-metric space such that $d_{\theta }$ is a continuous functional. Let $T:X\to X$ satisfy:

$$d_{\theta }(Tx,Ty)\le k{d}_{\theta }(x,y)forallx,y\in X$$

(1)

where $k\in [0,1)$ be such that for each $x_0\in X$, ${\lim }_{n,m\to \infty }\theta (x_n,x_m)<\frac{1}{k}$, here $x_n=T^n{x}_0$, $n=1,2,\cdots$. Then T has precisely one fixed point ξ. Moreover for each $y\in X$, $T^{n}y\to \xi$.

Proof. 

We choose any $x_0\in X$ be arbitrary, define the iterative sequence $\left\{x_n\right\}$ by:

$$x_0,T{x}_0=x_1,x_2=T{x}_1=T\left(T{x}_0\right)=T^2\left(x_0\right)\dots ,x_n=T^n{x}_0\dots .$$

Then by successively applying inequality (1) we obtain:

$$\begin{array}{ccc}\hfill d_{\theta }(x_n,x_{n+1})& \le & k^n{d}_{\theta }(x_0,x_1)\hfill \end{array}$$

(2)

By triangular inequality and (2), for $m>n$ we have:

$$\begin{array}{ccc}\hfill d_{\theta }(x_n,x_m)& \le & \theta (x_n,x_m)k^n{d}_{\theta }(x_0,x_1)+\theta (x_n,x_m)\theta (x_{n+1},x_m)k^{n+1}{d}_{\theta }(x_0,x_1)+\cdots +\hfill \\ & & \theta (x_n,x_m)\theta (x_{n+1},x_m)\theta (x_{n+2},x_m)...\theta (x_{m-2},x_m)\theta (x_{m-1},x_m)k^{m-1}{d}_{\theta }(x_0,x_1)\hfill \\ & \le & d_{\theta }(x_0,x_1)[\theta (x_1,x_m)\theta (x_2,x_m)\cdots \theta (x_{n-1},x_m)\theta (x_n,x_m)k^n+\hfill \\ & & \theta (x_1,x_m)\theta (x_2,x_m)\cdots \theta (x_n,x_m)\theta (x_{n+1},x_m)k^{n+1}+\cdots +\hfill \\ & & \theta (x_1,x_m)\theta (x_2,x_m)\cdots \theta (x_n,x_m)\theta (x_{n+1},x_m)...\theta (x_{m-2},x_m)\theta (x_{m-1},x_m)k^{m-1}]\hfill \end{array}$$

Since, ${\lim }_{n,m\to \infty }\theta (x_{n+1},x_m)k<1$ so that the series ${\sum }_{n=1}^{\infty }{k}^n{\prod }_{i=1}^n\theta (x_i,x_m)$ converges by ratio test for each $m\in \mathbb{N}$. Let:

$$S=\sum _{n=1}^{\infty }{k}^n\prod _{i=1}^n\theta (x_i,x_m),S_n=\sum _{j=1}^n{k}^j\prod _{i=1}^j\theta (x_i,x_m)$$

Thus for $m>n$ above inequality implies:

$$d_{\theta }(x_n,x_m)\le d_{\theta }(x_0,x_1)\left[S_{m-1}-S_n\right]$$

Letting $n\to \infty$ we conclude that $\left\{x_n\right\}$ is a Cauchy sequence. Since X is complete let $x_n\to \xi \in X$:

$$\begin{array}{ccc}\hfill d_{\theta }(T\xi ,\xi )& \le & \theta (T\xi ,\xi )[d_{\theta }(T\xi ,x_n)+d_{\theta }(x_n,\xi )]\hfill \\ & \le & \theta (T\xi ,\xi )[k{d}_{\theta }(\xi ,x_{n-1})+d_{\theta }(x_n,\xi )]\hfill \\ \hfill d_{\theta }(T\xi ,\xi )& \le & 0asn\to \infty \hfill \\ \hfill d_{\theta }(T\xi ,\xi )& =& 0\hfill \end{array}$$

Hence $\xi$ is a fixed point of T. Moreover uniqueness can easily be invoked by using inequality (1), since $k<1$. ☐

In the following we include another variant which is analogue to fixed point theorem by Hicks and Rhoades [15]. We need the following definition.

Definition 6.

Let $T:X\to X$ and for some $x_0\in X$, $\mathcal{O}\left(x_0\right)=\{x_0,f{x}_0,f^2{x}_0,\cdots \}$ be the orbit of $x_0$. A function G from X into the set of real numbers is said to be T-orbitally lower semi-continuous at $t\in X$ if $\left\{x_n\right\}\subset \mathcal{O}\left(x_0\right)$ and $x_n\to t$ implies $G\left(t\right)\le {\lim }_{n\to \infty }\inf G\left(x_n\right)$.

Theorem 3.

Let $(X,d_{\theta })$ be a complete extended b-metric space such that $d_{\theta }$ is a continuous functional. Let $T:X\to X$ and there exists $x_0\in X$ such that:

$$d_{\theta }(Ty,T^{2}y)\le k{d}_{\theta }(y,Ty)foreachy\in \mathcal{O}\left(x_0\right)$$

(3)

where $k\in [0,1)$ be such that for $x_0\in X$, ${\lim }_{n,m\to \infty }\theta (x_n,x_m)<\frac{1}{k}$, here $x_n=T^n{x}_0$, $n=1,2,\cdots$. Then $T^n{x}_0\to \xi \in X(asn\to \infty )$. Furthermore ξ is a fixed point of T if and only if $G\left(x\right)=d(x,Tx)$ is T-orbitally lower semi continuous at ξ.

Proof. 

For $x_0\in X$ we define the iterative sequence $\left\{x_n\right\}$ by:

$$x_0,T{x}_0=x_1,x_2=T{x}_1=T\left(T{x}_0\right)=T^2\left(x_0\right)\dots ,x_n=T^n{x}_0\dots .$$

Now for $y=T{x}_0$ by successively applying inequality (3) we obtain:

$$d_{\theta }(T^n{x}_0,T^{n+1}{x}_0)=d_{\theta }(x_n,x_{n+1})\le k^n{d}_{\theta }(x_0,x_1)$$

(4)

Following the same procedure as in the proof of Theorem 2 we conclude that $\left\{x_n\right\}$ is a Cauchy sequence. Since X is complete then $x_n=T^n{x}_0\to \xi \in X$. Assume that G is orbitally lower semi continuous at $\xi \in X$, then:

$$\begin{array}{ccc}\hfill d_{\theta }(\xi ,T\xi )& \le & \underset{n\to \infty }{\text{lim inf }}{d}_{\theta }(T^n{x}_0,T^{n+1}{x}_0)\hfill \end{array}$$

(5)

$$\begin{array}{cc}\le & \underset{n\to \infty }{\text{lim inf }}{k}^n{d}_{\theta }(x_0,x_1)=0\hfill \end{array}$$

(6)

Conversely, let $\xi =T\xi$ and $x_n\in O\left(x\right)$ with $x_n\to \xi$. Then:

$$G\left(\xi \right)=d(\xi ,T\xi )=0\le \underset{n\to \infty }{\text{lim inf }}G\left(x_n\right)=d(T^n{x}_0,T^{n+1}{x}_0)$$

(7)

 ☐

Remark 2.

When $\theta (x,y)=1$ a constant function then Theorem 3 reduces to main result of Hicks and Rhoades ([15] (Theorem 1)). Hence Theorem 3 extends/generalizes ([15] (Theorem 1) ).

Example 5.

Let $X=[0,\infty ).$ Define $d_{\theta }(x,y):X\times X\to {\mathbb{R}}^{+}$ and $\theta :X\times X\to [1,\infty )$ as:

$$d_{\theta }(x,y)={(x-y)}^2,\theta (x,y)=x+y+2$$

Then $d_{\theta }$ is a complete extended b-metric on X. Define $T:X\to X$ by $Tx=\frac{x}{2}.$ We have:

$$d_{\theta }(Tx,Ty)={\left(\frac{x}{2}-\frac{y}{2}\right)}^2\le \frac{1}{3}{(x-y)}^2=k{d}_{\theta }(x,y)$$

Note that for each $x\in X$, $T^{n}x=\frac{x}{2^n}.$ Thus we obtain:

$$\underset{m,n\to \infty }{\lim }\theta (T^{m}x,T^{n}x)=\underset{m,n\to \infty }{\lim }\left(\frac{x}{2^m}+\frac{x}{2^n}+2\right)<3$$

Therefore, all conditions of Theorem 3 are satisfied hence T has a unique fixed point.

Example 6.

Let $X=[0,\frac{1}{4}].$ Define $d_{\theta }(x,y):X\times X\to {\mathbb{R}}^{+}$ and $\theta :X\times X\to [1,\infty )$ as:

$$d_{\theta }(x,y)={(x-y)}^2,\theta (x,y)=x+y+2$$

Then $d_{\theta }$ is a complete extended b-metric on X. Define $T:X\to X$ by $Tx=x^2.$ We have:

$$d_{\theta }(Tx,Ty)\le \frac{1}{4}{d}_{\theta }(x,y)$$

Note that for each $x\in X$, $T^{n}x=x^{2n}.$ Thus we obtain:

$$\underset{m,n\to \infty }{\lim }\theta (T^{m}x,T^{n}x)<4$$

Therefore, all conditions of Theorem 3 are satisfied hence T has a unique fixed point.

3. Application

In this section, we give existence theorem for Fredholm integral equation. Let $X=C\left(\right[a,b],\mathbb{R})$ be the space of all continuous real valued functions define on $[a,b]$. Note that X is complete extended b-metric space by considering $d_{\theta }(x,y)={\sup }_{t\in [a,b]}{|x\left(t\right)-y\left(t\right)|}^2$, with $\theta (x,y)=\left|x\right(t\left)\right|+\left|y\right(t\left)\right|+2$, where $\theta :X\times X\to [1,\infty )$. Consider the Fredholm integral equation as:

$$x\left(t\right)={\int }_a^{b}M(t,s,x\left(s\right))ds+g\left(t\right),t,s\in [a,b]$$

(8)

where $g:[a,b]\to \mathbb{R}$ and $M:[a,b]\times [a,b]\times \mathbb{R}\to \mathbb{R}$ are continuous functions. Let $T:X\to X$ the operator given by:

$$Tx\left(t\right)={\int }_a^{b}M(t,s,x\left(s\right))ds+g\left(t\right)\mathrm{for}t,s\in [a,b]$$

where, the function $g:[a,b]\to \mathbb{R}$ and $M:[a,b]\times [a,b]\times \mathbb{R}\to \mathbb{R}$ are continuous. Further, assume that the following condition hold:

$$|M(t,s,x\left(s\right))-M(t,s,Tx\left(s\right))|\le \frac{1}{2}|x\left(s\right)-Tx\left(s\right)|\mathrm{for}\mathrm{each}t,s\in [a,b]\mathrm{and}x\in X$$

Then the integral Equation (8) has a solution.

We have to show that the operator T satisfies all the conditions of Theorem 3. For any $x\in X$ we have:

$$\begin{array}{ccc}\hfill {|Tx\left(t\right)-T\left(Tx\left(t\right)\right)|}^2& \le & {\left({\int }_a^b|M(t,s,x\left(s\right))-M(t,s,Tx\left(s\right))|ds\right)}^2\hfill \\ & \le & \frac{1}{4}{d}_{\theta }(x,Tx)\hfill \end{array}$$

All conditions of Theorem 3 follows by the hypothesis. Therefore, the operator T has a fixed point, that is, the Fredholm integral Equation (8) has a solution.