On a Common Jungck Type Fixed Point Result in Extended Rectangular b-Metric Spaces

Abstract

In this paper, we present a Jungck type common fixed point result in extended rectangular b-metric spaces. We also give some examples and a known common fixed point theorem in extended b-metric spaces.

1. Introduction

The notion of b-metric spaces was first introduced by Bakhtin [1] and Czerwik [2]. This metric type space has been generalized in several directions. Among of them, we may cite, extended b-metric spaces [3], controlled metric spaces [4] and double controlled metric spaces [5]. Within another vision, Branciari [6] initiated rectangular metric spaces. In same direction, Asim et al. [7] included a control function to initiate the concept of extended rectangular b-metric spaces, as a generalization of rectangular b-metric spaces [8].

Definition 1

([7]). Let X be a nonempty set and $e:X\times X\to [1,\infty )$ be a function. If $d_e:X\times X\to [0,\infty )$ is such that

• (ERbM1) $d_e(\omega ,\mathsf{\Omega })=0$ iff $\omega =\mathsf{\Omega }$;
• (ERbM2) $d_e(\omega ,\mathsf{\Omega })=d_e(\mathsf{\Omega },\omega )$;
• (ERbM3) $d_e(\omega ,\mathsf{\Omega })\le e(\omega ,\mathsf{\Omega })[d_e(\omega ,\zeta )+d_e(\zeta ,\sigma )+d_e(\sigma ,\mathsf{\Omega })];$

for all $\omega ,\mathsf{\Omega }\in X$ and all distinct elements $\zeta ,\sigma \in X\backslash \{\omega ,\mathsf{\Omega }\},$ then $d_e$ is an extended rectangular b-metric on X with mapping e.

Definition 2

([7]). Let $(X,d_e)$ be an extended rectangular b-metric space, $\left\{{\mathsf{\Omega }}_n\right\}$ be a sequence in X and $\mathsf{\Omega }\in X$.

(a) 

$\left\{{\mathsf{\Omega }}_n\right\}$ converges to Ω, if for each $\tau >0$ there is $n_0\in \mathbb{N}$ so that $d_e({\mathsf{\Omega }}_n,\mathsf{\Omega })<\tau$ for any $n>n_0$. We write it as $\underset{n\to \infty }{lim}{\mathsf{\Omega }}_n=\mathsf{\Omega }$ or ${\mathsf{\Omega }}_n\to \mathsf{\Omega }$ as $n\to \infty .$

(b) 

$\left\{{\mathsf{\Omega }}_n\right\}$ is Cauchy if for each $\tau >0$ there is $n_0\in \mathbb{N}$ so that $d_e({\mathsf{\Omega }}_n,{\mathsf{\Omega }}_{n+p})<\tau$ for any $n>n_0$ and $p>0$.

(c) 

$(X,d)$ is complete if each Cauchy sequence is convergent.

Note that the topology of rectangular metric spaces need not be Hausdorff. For more examples, see the papers of Sarma et al. [9] and Samet [10]. The topological structure of rectangular metric spaces is not compatible with the topology of classic metric spaces, see Example 7 in the paper of Suzuki [11]. Going in same direction, extended rectangular b-metric spaces can not be Hausdorff. The following example (a variant of Example 1.7 of George et al. [8]) explains this fact.

Example 1.

Let $X={\mathsf{\Gamma }}_1\cup {\mathsf{\Gamma }}_2$, where ${\mathsf{\Gamma }}_1=\{\frac{1}{n},n\in \mathbb{N}\}$ and ${\mathsf{\Gamma }}_2$ is the set of all positive integers. Define $d_e:X\times X\to [0,\infty )$ so that $d_e$ is symmetric and for all $\mathsf{\Omega },\omega \in X$,

$$d_e(\mathsf{\Omega },\omega )=\left\{\begin{array}{c}0,if\mathsf{\Omega }=\omega ,\hfill \\ 8,if\mathsf{\Omega },\omega \in {\mathsf{\Gamma }}_1,\hfill \\ \frac{2}{n},if\mathsf{\Omega }\in {\mathsf{\Gamma }}_{1}and\omega \in \{2,3\},\hfill \\ 4otherwise.\hfill \end{array}\right.$$

Here, $(X,d_e)$ is an extended rectangular b-metric space with $e(\mathsf{\Omega },\omega )=2$. Note that there exist no ${\tau }_1,{\tau }_2>0$ such that $B_{{\tau }_1}\left(2\right)\cap B_{{\tau }_2}\left(3\right)=\varnothing$ (where $B_x\left(\tau \right)$ denotes the ball of center x and radius τ). That is, $(X,d_e)$ is not Hausdorff.

The main result of Jungck [12] is following.

Theorem 1

([12]). If f and H are commuting self-maps on a complete metric space $(X,d)$ such that $f\left(X\right)\subseteq H\left(X\right)$, H is continuous and

$$d(f\mathsf{\Omega },f\omega )\le \delta d(H\mathsf{\Omega },H\omega ),$$

(1)

for all $\mathsf{\Omega },\omega \in X$, where $0<\delta <1$, then there is a unique common fixed point of f and H.

Our goal is to get the analogue of Theorem 1 in the setting of extended rectangular b-metric spaces. Some examples are also provided.

2. Main Results

Definition 3.

Let X be a nonempty set and $f,H$ be two commuting self-mappings of X so that $f\left(X\right)\subseteq H\left(X\right)$. Then $(f,H)$ is called a Jungck pair of mappings on X.

Example 2.

Let $X=\mathbb{R}\times \mathbb{R}$. Define $f,H:X\to X$ by $f(\omega ,\mathsf{\Omega })=(2\omega ,(\mathsf{\Omega }/2)+3)$ and $H(\omega ,\mathsf{\Omega })=(3\omega ,(\mathsf{\Omega }/3)+4)$. Then $f\left(H\right(\omega ,\mathsf{\Omega }\left)\right)=(6\omega ,(\mathsf{\Omega }/6)+5)=H\left(f\right(\omega ,\mathsf{\Omega }\left)\right)$, so that $(f,H)$ is a Jungck pair of mappings on X.

Lemma 1.

Let X be a nonempty set and $(f,H)$ be a Jungck pair of mappings on X. Given ${\mathsf{\Omega }}_0\in X$. Then there is a sequence $\left\{{\mathsf{\Omega }}_n\right\}$ in X so that $H{\mathsf{\Omega }}_{n+1}=f{\mathsf{\Omega }}_n,n\ge 0.$

Proof. 

For such ${\mathsf{\Omega }}_0\in X$, $f{\mathsf{\Omega }}_0$ and $H{\mathsf{\Omega }}_0$ are well defined. Since $f{\mathsf{\Omega }}_0\in H\left(X\right)$, there is ${\mathsf{\Omega }}_1\in X$ so that $H{\mathsf{\Omega }}_1=f{\mathsf{\Omega }}_0$. Going in same direction, we arrive to $H{\mathsf{\Omega }}_{n+1}=f{\mathsf{\Omega }}_n$. □

Definition 4.

Let $(f,H)$ be a Jungck pair of mappings on a nonempty set X. Given $e:X\times X\to [1,\infty )$. Let $\left\{{\mathsf{\Omega }}_n\right\}$ be a sequence such that $H{\mathsf{\Omega }}_{n+1}=f{\mathsf{\Omega }}_n,$ for each $n\ge 0$. Then $\left\{{\mathsf{\Omega }}_n\right\}$ is called a $(f,H)$ Jungck sequence in X. We say that $\left\{{\mathsf{\Omega }}_n\right\}$ is e-bounded if $\underset{n,m\to \infty }{lim\; sup}e(H{\mathsf{\Omega }}_n,H{\mathsf{\Omega }}_m)<\infty$.

Remark 1.

1. 

If $H=id,\left(id\right(\omega )=\omega ,\omega \in X)$ then a $(f,id)$ Jungck sequence is a Picard sequence.

2. 

Note that each sequence in a rectangular b-metric space with coefficient $s\ge 1$ (see [8]) is e-bounded ($e({\mathsf{\Omega }}_m,{\mathsf{\Omega }}_n)=s,$ for all $m,n\in \mathbb{N}$).

Theorem 2.

Let $(f,H)$ be a Jungck pair of mappings on a complete extended rectangular b-metric space $(X,d_e)$ so that

$$d_e(f\mathsf{\Omega },f\omega )\le \rho d_e(H\mathsf{\Omega },H\omega ),$$

(2)

for all $\mathsf{\Omega },\omega \in X,$ where $0<\rho <1.$ If H is continuous and there is an e-bounded $(f,H)$ Jungck sequence, then there is a unique common fixed point of f and H.

Proof. 

Let $\left\{{\mathsf{\Omega }}_n\right\}$ be an e-bounded $(f,H)$ Jungck sequence. Then for ${\mathsf{\Omega }}_0\in X$, $f{\mathsf{\Omega }}_{n+1}=H{\mathsf{\Omega }}_n$, for each $n\ge 0$. We show that $\left\{f{\mathsf{\Omega }}_n\right\}$ is Cauchy. From (2), we have

$$\begin{array}{ccc}\hfill d_e(H{\mathsf{\Omega }}_{m+k},H{\mathsf{\Omega }}_{n+k})& =& d_e(f{\mathsf{\Omega }}_{m+k-1},f{\mathsf{\Omega }}_{n+k-1})\hfill \\ & \le & \rho d_e(H{\mathsf{\Omega }}_{m+k-1},H{\mathsf{\Omega }}_{n+k-1}).\hfill \end{array}$$

So,

$$d_e(H{\mathsf{\Omega }}_{m+k},H{\mathsf{\Omega }}_{n+k})\le {\rho }^k{d}_e(H{\mathsf{\Omega }}_m,H{\mathsf{\Omega }}_n),$$

(3)

for each $k\in \mathbb{N}$.

Case 1:

If $H{\mathsf{\Omega }}_n=H{\mathsf{\Omega }}_{n+1}$ for some n, define $\theta :=f{\mathsf{\Omega }}_n=H{\mathsf{\Omega }}_n.$ We claim that $f\theta =H\theta =\theta$ and $\theta$ is unique. First,

$$f\theta =fH{\mathsf{\Omega }}_n=Hf{\mathsf{\Omega }}_n=H\theta .$$

Let $d_e(\theta ,f\theta )>0.$ Here,

$$\begin{array}{ccc}\hfill d_e\left(\theta ,f\theta \right)& =& d_e\left(f{\mathsf{\Omega }}_n,f\theta \right)\hfill \\ & \le & \rho d_e\left(H{\mathsf{\Omega }}_n,H\theta \right)\hfill \\ & =& \rho d_e\left(\theta ,H\theta \right)\hfill \\ & =& \rho d_e\left(\theta ,f\theta \right)\hfill \\ & <& d_e\left(\theta ,f\theta \right),\hfill \end{array}$$

which is a contradiction. Recall that (2) yields that $f{\mathsf{\Omega }}_n=H{\mathsf{\Omega }}_n=\theta$ is the unique common fixed point of f and H.

Case 2:

If $H{\mathsf{\Omega }}_n\ne H{\mathsf{\Omega }}_{n+1}$ for all $n\ge 0,$ then $H{\mathsf{\Omega }}_n\ne H{\mathsf{\Omega }}_{n+k}$ for all $n\ge 0$ and $k\ge 1.$ Namely, if $H{\mathsf{\Omega }}_n=H{\mathsf{\Omega }}_{n+k}$ for some $n\ge 0$ and $k\ge 1$, we have that

$$\begin{array}{ccc}\hfill d_e(H{\mathsf{\Omega }}_{n+1},H{\mathsf{\Omega }}_{n+k+1})& =& d_e(f{\mathsf{\Omega }}_n,f{\mathsf{\Omega }}_{n+k})\hfill \\ & \le & \rho d_e(H{\mathsf{\Omega }}_n,H{\mathsf{\Omega }}_{n+k})\hfill \\ & =& 0.\hfill \end{array}$$

So, $H{\mathsf{\Omega }}_{n+1}=H{\mathsf{\Omega }}_{n+k+1}$. Then (3) implies that

$$d_e(H{\mathsf{\Omega }}_{n+1},H{\mathsf{\Omega }}_n)=d_e(H{\mathsf{\Omega }}_{n+k+1},H{\mathsf{\Omega }}_{n+k})\le {\rho }^k{d}_e(H{\mathsf{\Omega }}_{n+1},H{\mathsf{\Omega }}_n)<d_e(H{\mathsf{\Omega }}_{n+1},H{\mathsf{\Omega }}_n).$$

It is a contradiction. Thus we assume that $H{\mathsf{\Omega }}_n\ne H{\mathsf{\Omega }}_m$ for all integers $n\ne m$. Note that $H{\mathsf{\Omega }}_{m+k}\ne H{\mathsf{\Omega }}_{n+k}$ for any $k\in \mathbb{N}$. Also, $H{\mathsf{\Omega }}_{n+k},H{\mathsf{\Omega }}_{m+k}\in X\backslash \{H{\mathsf{\Omega }}_n,H{\mathsf{\Omega }}_m\}.$ Since $(X,d_e)$ is an extended rectangular b-metric space, by (ERbM3), we get

$$\begin{array}{ccc}\hfill d_e(H{\mathsf{\Omega }}_m,H{\mathsf{\Omega }}_n)& \le & e(H{\mathsf{\Omega }}_m,H{\mathsf{\Omega }}_n)[d_e(H{\mathsf{\Omega }}_m,H{\mathsf{\Omega }}_{m+n_0})+d_e(H{\mathsf{\Omega }}_{m+n_0},H{\mathsf{\Omega }}_{n+n_0})\hfill \\ & +& d_e(H{\mathsf{\Omega }}_{n+n_0},H{\mathsf{\Omega }}_n)],\hfill \end{array}$$

where $n_0\in \mathbb{N}$ so that $lim\underset{n,m\to \infty }{sup}e(H{\mathsf{\Omega }}_m,H{\mathsf{\Omega }}_n)<\frac{1}{{\rho }^{n_0}}.$ Then

$$\begin{array}{ccc}\hfill d_e(H{\mathsf{\Omega }}_m,H{\mathsf{\Omega }}_n)& \le & e(H{\mathsf{\Omega }}_m,H{\mathsf{\Omega }}_n)[{\rho }^m{d}_e(H{\mathsf{\Omega }}_0,H{\mathsf{\Omega }}_{n_0})+{\rho }^{n_0}{d}_e(H{\mathsf{\Omega }}_m,H{\mathsf{\Omega }}_n)\hfill \\ & +& {\rho }^n{d}_e(H{\mathsf{\Omega }}_0,H{\mathsf{\Omega }}_{n_0})].\hfill \end{array}$$

So,

$$(1-e(H{\mathsf{\Omega }}_m,H{\mathsf{\Omega }}_n){\rho }^{n_0})d_e(H{\mathsf{\Omega }}_m,H{\mathsf{\Omega }}_n)\le e(H{\mathsf{\Omega }}_m,H{\mathsf{\Omega }}_n)({\rho }^m+{\rho }^n)d_e(H{\mathsf{\Omega }}_0,H{\mathsf{\Omega }}_{n_0}).$$

From this, we obtain

$$d_e(H{\mathsf{\Omega }}_m,H{\mathsf{\Omega }}_n)\le \frac{e(H{\mathsf{\Omega }}_m,H{\mathsf{\Omega }}_n)({\rho }^m+{\rho }^n)}{1-e(H{\mathsf{\Omega }}_m,H{\mathsf{\Omega }}_n){\rho }^{n_0}}{d}_e(H{\mathsf{\Omega }}_0,H{\mathsf{\Omega }}_{n_0}).$$

(4)

Thus $\left\{H{\mathsf{\Omega }}_n\right\}$ is Cauchy in $H\left(X\right)$, which is complete, so there is $u\in X$ so that

$$\underset{n\to \infty }{lim}H{\mathsf{\Omega }}_n=\underset{n\to \infty }{lim}f{\mathsf{\Omega }}_{n-1}=u.$$

(5)

The continuity of H together with (2) implies that f is itself continuous. The commutativity of f and H leads to

$$Hu=H\left(\underset{n\to \infty }{lim}f{\mathsf{\Omega }}_n\right)=\underset{n\to \infty }{lim}Hf{\mathsf{\Omega }}_n=\underset{n\to \infty }{lim}fH{\mathsf{\Omega }}_n=f\left(\underset{n\to \infty }{lim}H{\mathsf{\Omega }}_n\right)=fu.$$

(6)

Let $v=Hu=fu$. Then

$$fv=fHu=Hfu=Hv.$$

(7)

If $fu\ne fv$, by (2) we find that

$$\begin{array}{ccc}\hfill d_e(fu,fv)& \le & \rho d_e(Hu,Hv)\hfill \\ & =& \rho d_e(fu,fv)\hfill \\ & <& d_e(fu,fv).\hfill \end{array}$$

It is a contradiction, hence $fu=fv$. Thus,

$$fv=Hv=v.$$

Condition (2) yields that v is the unique common fixed point. □

Example 3.

If we take in Example 3.1. of [7], $H=id$ and f as

$$f1=f2=f3=f4=2andf5=1,$$

then all the other conditions of Theorem 2 are satisfied, and so f and H have a unique fixed point, which is, $\theta =2$. Here, the space $(X,d_e)$ is extended rectangular b-metric space, but it is not extended b-metric space. Hence Theorem 2 generalizes, compliments and improves several known results in existing literature.

A variant of Banach theorem in extended rectangular b-metric spaces is given as follows.

Theorem 3.

Let $(X,d_e)$ be a complete extended rectangular b-metric space and $f:X\to X$ be so that

$$d_e(f\mathsf{\Omega },f\omega )\le \rho d_e(\mathsf{\Omega },\omega )$$

(8)

for all $\mathsf{\Omega },\omega \in X$, where $\rho \in [0,1).$ If there is an e-bounded Picard sequence in X, then f has a unique fixed point.

Remark 2.

Theorem 3.1 in [7] is a consequence of Theorem 3. Indeed, instead of condition $\underset{n,m\to \infty }{lim}{d}_e({\mathsf{\Omega }}_n,{\mathsf{\Omega }}_m)<\frac{1}{\rho }$ of Theorem 3.1 in [7], we used a weaker condition, that is, $lim\underset{n,m\to \infty }{sup}{d}_e({\mathsf{\Omega }}_n,{\mathsf{\Omega }}_m)<\infty$.

3. A Jungck Theorem in Extended b-Metric Spaces

Let $(X,d_e)$ be an extended b-metric space (see Definition 3 in [3]) and $\left\{{\mathsf{\Omega }}_n\right\}$ be a $(f,H)$ e-bounded Jungck sequence in X. Then

$$\begin{array}{ccc}\hfill d_e(H{\mathsf{\Omega }}_m,H{\mathsf{\Omega }}_n)& \le & e(H{\mathsf{\Omega }}_m,H{\mathsf{\Omega }}_n)[d_e(H{\mathsf{\Omega }}_m,H{\mathsf{\Omega }}_{m+n_0})+d_e(H{\mathsf{\Omega }}_{m+n_0},H{\mathsf{\Omega }}_n)]\hfill \\ & \le & e(H{\mathsf{\Omega }}_m,H{\mathsf{\Omega }}_n)[d_e(H{\mathsf{\Omega }}_m,H{\mathsf{\Omega }}_{m+n_0})+\hfill \\ & & e(H{\mathsf{\Omega }}_{m+n_0},H{\mathsf{\Omega }}_n)[d_e(H{\mathsf{\Omega }}_{m+n_0},H{\mathsf{\Omega }}_{n+n_0})+d_e(H{\mathsf{\Omega }}_{n+n_0},H{\mathsf{\Omega }}_n)]]\hfill \\ & \le & e(H{\mathsf{\Omega }}_m,H{\mathsf{\Omega }}_n)e(H{\mathsf{\Omega }}_{m+n_0},H{\mathsf{\Omega }}_n)[d_e(H{\mathsf{\Omega }}_m,H{\mathsf{\Omega }}_{m+n_0})+\hfill \\ & & d_e(H{\mathsf{\Omega }}_{m+n_0},H{\mathsf{\Omega }}_{n+n_0})+d_e(H{\mathsf{\Omega }}_{n+n_0},H{\mathsf{\Omega }}_n)].\hfill \end{array}$$

Since $\left\{{\mathsf{\Omega }}_n\right\}$ is a $(f,H)$ e-bounded Jungck sequence, we find that

$$lim\underset{m,n\to \infty }{sup}e(H{\mathsf{\Omega }}_m,H{\mathsf{\Omega }}_n)e(H{\mathsf{\Omega }}_{m+n_0},H{\mathsf{\Omega }}_n)<\infty .$$

By Theorem 2, we obtain the following.

Theorem 4.

Let $(f,H)$ be a Jungck pair of mappings on a complete extended b-metric space $(X,d_e)$ so that

$$d_e(f\mathsf{\Omega },f\omega )\le \rho d_e(H\mathsf{\Omega },H\omega ),$$

(9)

for all $\mathsf{\Omega },\omega \in X,$ where $0<\rho <1.$ If H is continuous and there is an e-bounded $(f,H)$ Jungck sequence, then f and H have a unique common fixed point.

Remark 3.

By Theorem 4, we obtain the Banach contraction principle in extended b-metric spaces. It improves Theorem 2.1 in [13], Theorem 2 in [3] and Theorem 2.1 in [14]. Also Theorem 3 generalizes an open problem raised by George et al. [8].

Example 4.

Let $X=[0,\infty ),e:X\times X\to [1,\infty )$. Consider $d_e:X\times X\to [0,\infty )$ as

$$d_e(\mathsf{\Omega },\omega )={(\mathsf{\Omega }-\omega )}^2,$$

where $e(\mathsf{\Omega },\omega )=\mathsf{\Omega }+\omega +2$. Then $(X,d_e)$ is an extended b-metric space. Define $f\mathsf{\Omega }=\frac{3\mathsf{\Omega }}{4}.$ Then (8) holds for $\rho =\frac{9}{16}.$ Let ${\mathsf{\Omega }}_0\in X$ and ${\mathsf{\Omega }}_n=f^n{\mathsf{\Omega }}_0,n\in \mathbb{N}.$ Then $\underset{m,n\to \infty }{lim}e({\mathsf{\Omega }}_m,{\mathsf{\Omega }}_n)=2$. So, $\underset{m,n\to \infty }{lim}e({\mathsf{\Omega }}_m,{\mathsf{\Omega }}_n)>\frac{16}{9}$ and Theorem 3.1 in [7] is not applicable. Applying Theorem 3, we conclude that f has a unique fixed point.