Relation-Theoretic Coincidence and Common Fixed Point Results in Extended Rectangular b-Metric Spaces with Applications

Abstract

The objective of this paper is to obtain new relation-theoretic coincidence and common fixed point results for some mappings F and g via hybrid contractions and auxiliary functions in extended rectangular b-metric spaces, which improve the existing results and give some relevant results. Finally, some nontrivial examples and applications to justify the main results.

1. Introduction and Preliminaries

Throughout the article, we denote, by $\mathbb{R}$, the set of all real numbers; by ${\mathbb{R}}_{+}$, the set of all non-negative real numbers; and by $\mathbb{N}$, the set of all non-negative integers. At the beginning, we retrace several known metric-type spaces, which will be useful in the following.

In 1993, Czerwik [1] formally introduced and studied this interesting generalized metric space named b-metric space. Since then, many scholars have extended and developed fixed point theorems in b-metric spaces. Recent studies of fixed point theorems in b-metric spaces can be seen in [2,3,4].

Definition 1

([1]). Let $\Omega \ne \varnothing$ and $s⩾1$ be a given real number. If a function $d:\Omega \times \Omega \to {\mathbb{R}}_{+}$ satisfies the following conditions:

$\left(d_1\right)$$d(u,v)=0$ if and only if $u=v$;

$\left(d_2\right)$$d(u,v)=d(v,u)$, for all $u,v\in \Omega$;

$\left(d_3\right)$$d(u,v)⩽s\left[d\right(u,w)+d(w,v\left)\right]$, for all $u,v,w\in \Omega$,

then d is said to be a b-metric, and $(\Omega ,d)$ is said to be a b-metric space with the coefficient s.

In 2000, a generalized metric that replaces the triangular inequality with quadrilateral inequality was proposed by Branciari [5].

Definition 2

([5]). Let $\Omega \ne \varnothing$. For all $u,v\in \Omega$ and all distinct points $w,t\in \Omega \backslash \{u,v\}$, if a function $d_r:\Omega \times \Omega \to [0,\infty )$ satisfies the following conditions:

$\left(d1\right)$$d_r(u,v)=0\iff u=v$;

$\left(d2\right)$$d_r(u,v)=d_r(v,u)$; and

$\left(d3\right)$$d_r(u,v)⩽d_r(u,w)+d_r(w,t)+d_r(t,v)$,

then $d_r$ is said to be a rectangular metric and $(\Omega ,d_r)$ is said to be a rectangular metric space (Branciari distance space).

In 2015, rectangular b-metric was raised by George et al. [6], which is a development of b-metric and rectangular metric.

Definition 3

([6]). Let $\Omega \ne \varnothing$ and $s⩾1$ be a given real number. If, for all $u,v\in \Omega$ and for all distinct points $w,t\in \Omega \backslash \{u,v\}$, a function $d_{rb}:\Omega \times \Omega \to {\mathbb{R}}_{+}$ satisfies the following conditions:

$\left(d_{rb}1\right)$$d_{rb}(u,v)=0$ if and only if $u=v$;

$\left(d_{rb}2\right)$$d_{rb}(u,v)=d_{rb}(v,u)$; and

$\left(d_{rb}3\right)$$d_{rb}(u,v)⩽s[d_{rb}(u,w)+d_{rb}(w,t)+d_{rb}(t,v)]$,

then $d_{rb}$ is said to be a rectangular b-metric and $(\Omega ,d_{rb})$ is said to be a rectangular b-metric space with the coefficient s.

In 2017, a binary function proposed by Kamran et al. [7] was used to introduce a novel metric-type space.

Definition 4

([7]). Let $\Omega \ne \varnothing$ and $\theta :\Omega \times \Omega \to [1,\infty )$. A function $d_{\theta }:\Omega \times \Omega \to [0,\infty )$ is said to be an extended b-metric if it satisfies the following conditions:

$\left(d_{\theta }1\right)$$d_{\theta }(u,v)=0$ if and only if $u=v$;

$\left(d_{\theta }2\right)$$d_{\theta }(u,v)=d_{\theta }(v,u)$, for all $u,v\in \Omega$;

$\left(d_{\theta }3\right)$$d_{\theta }(u,v)⩽\theta (u,v)[d_{\theta }(u,w)+d_{\theta }(w,v)]$, for all $u,v,w\in \Omega$,

then $(\Omega ,d_{\theta })$ is said to be an extended b-metric space with θ.

In 2019, inspired by [5,7], Asim et al. [8] presented a more generalized metric space called extended rectangular b-metric space(also extended Branciari b-distance in [9]).

Definition 5

([8]). Let $\Omega \ne \varnothing$ and $\xi :\Omega \times \Omega \to [1,\infty )$. A function $d_{\xi }:\Omega \times \Omega \to [0,\infty )$ is said to be an extended rectangular b-metric, if for all $u,v\in \Omega$ and all distinct points $w,t\in \Omega \backslash \{u,v\}$, $d_{\xi }$ satisfies the following conditions:

$\left(d_{\xi }1\right)$$d_{\xi }(u,v)=0\iff u=v$;

$\left(d_{\xi }2\right)$$d_{\xi }(u,v)=d_{\xi }(v,u)$; and

$\left(d_{\xi }3\right)$$d_{\xi }(u,v)⩽\xi (u,v)[d_{\xi }(u,w)+d_{\xi }(w,t)+d_{\xi }(t,v)]$,

then $(\Omega ,d_{\xi })$ is said to be an extended rectangular b-metric space.

Remark 1.

The relationship between these types of metric spaces are shown in Figure 1.

Now, we review some topological properties of the extended rectangular b-metric space.

Definition 6

([8]). Let $(\Omega ,d_{\xi })$ be an extended rectangular b-metric space.

$\left(i\right)$ a sequence $\left\{u_n\right\}$ in Ω is said to be a Cauchy sequence if ${\displaystyle \underset{n,m\to \infty }{lim}}{d}_{\xi }(u_n,u_m)=0$;

$\left(ii\right)$ a sequence $\left\{u_n\right\}$ in Ω is said to be convergent to u if ${\displaystyle \underset{n\to \infty }{lim}}{d}_{\xi }(u_n,u)=0$; and

$\left(iii\right)$$(\Omega ,d_{\xi })$ is said to be complete if every Cauchy sequence in Ω convergent to some point in Ω.

Next, we introduce the simulation function was introduced by Khojasteh et al. [10]. It plays an important role in recent studies on the fixed point theory, which has inspired many scholars. Some results via simulation functions can be referred to [11,12,13,14].

Definition 7

([10]). A function $\eta :{\mathbb{R}}_{+}\times {\mathbb{R}}_{+}\to \mathbb{R}$ is said to be a simulation function, if it satisfies the following conditions:

$\left(\eta 1\right)$$\eta (0,0)=0$;

$\left(\eta 2\right)$$\eta (u,v)<v-u,foru,v>0$; and which

$\left(\eta 3\right)$ if $\left\{u_n\right\},\left\{v_n\right\}$ are sequences in $(0,\infty )$ such that ${\displaystyle \underset{n\to \infty }{lim}}{u}_n={\displaystyle \underset{n\to \infty }{lim}}{v}_n>0$, then

$${\displaystyle \underset{n\to \infty }{lim\; sup}}\eta (u_n,v_n)<0.$$

We denote the set of all simulation functions by $\mathcal{Z}$.

Definition 8

([10]). Let $(\Omega ,d)$ be a metric space, $F:\Omega \to \Omega$ be a mapping and $\eta \in \mathcal{Z}$. Then, T is called a $\mathcal{Z}$-contraction with respect to η if the following condition holds:

$$\eta \left(d\right(Fu,Fv),d(u,v\left)\right)⩾0,$$

where $u,v\in \Omega$, with $u\ne v$.

Theorem 1

([10]). Every $\mathcal{Z}$-contraction on a complete metric space has a unique fixed point.

Another new variant of Banach contraction principle with binary relation is proposed by Alam and Imdad [15] on complete metric spaces. In this case, the contraction condition is relatively weaker than the usual contraction, since it only needs to keep those elements that are related under the binary relation, not the whole space. With the introduction of binary relations, the study of fixed point theory is more colorful.

For instance, Al-Sulami et al. [15] raised $(\theta ,\Re )$ contraction by binary relation and applied it to nonlinear matrix equations, Alfaqih et al. [16] proposed ${(F,\Re )}_g$-contraction in the metric space with a binary relation and investigated the existence and uniqueness of a solution of integral equation of Volterra type, Zadal and Sarwar [17] obtained common fixed point for two mappings in the case of binary relation. Now, we recall some basic definitions of binary relations, which play an important role in our main results.

Definition 9

([18]). Let $\Omega \ne \varnothing$ and ℜ be a binary relation on Ω. For any $u\Re v$ or $(u,v)\in \Re$, where $u,v\in \Omega$, we say that "u is ℜ-related to v" or "u relates to v under ℜ".

Definition 10

([18]). Let $\Omega \ne \varnothing$, ℜ be a binary relation on Ω and $F:\Omega \to \Omega$ be a mapping.

$\left(i\right)$ A sequence $\left\{u_n\right\}$ is called an ℜ-preserving sequence if $u_n\Re u_{n+1}$, for all $n\in \mathbb{N}$.

$\left(ii\right)$ A binary relation ℜ on Ω is said to be F-closed if $Fu\Re Fv$, whenever $u\Re v$.

$\left(iii\right)$ A binary relation ℜ on Ω is said to be d-self-closed if for any sequence $\left\{u_n\right\}\subseteq \Omega$ such that $\left\{u_n\right\}$ is ℜ-preserving with $u_n\to u\in \Omega$, there exists a subsequence $\left\{u_{n_k}\right\}$ of $\left\{u_n\right\}$ such that $u_{n_k}\Re u$ or $u\Re u_{n_k}$, for all $k\in \mathbb{N}$.

$\left(iv\right)$ A binary relation ℜ on Ω is said to be transitive if $u\Re v$ and $v\Re w$ implies that $u\Re w$.

Definition 11

([18]). For $u,v\in \Omega$, a path of length $p\in \mathbb{N}$ in ℜ from u to v is a finite sequence $\{u_0,u_1,\cdots ,u_p\}$ such that $u_0=u,u_p=v$ and $u_i\Re u_{i+1}$ for all $i\in \{0,1,\cdots ,p-1\}$.

In addition, Alam and Imdad [19] utilized some relatively weaker notions to prove results on the existence and uniqueness of coincidence points involving a pair of mappings defined on a metric space endowed with an arbitrary binary relation. For completeness, we first review some of the relevant definitions that are known.

Definition 12

([19]). Let $(\Omega ,d)$ be a metric space, ℜ be a binary relation on Ω and $F,g:\Omega \to \Omega$ be two mappings.

$\left(i\right)$ The set Ω is ℜ-complete if every ℜ-preserving Cauchy sequence in Ω converges to a limit in Ω.

$\left(ii\right)$ A binary relation ℜ on Ω is said to be $(F,g)$-closed if $Fu\Re Fv$, whenever $gu\Re gv$.

$\left(iii\right)$ A binary relation ℜ on Ω is said to be $(g,d)$-self-closed if for any sequence $\left\{u_n\right\}\subseteq \Omega$ such that $\left\{u_n\right\}$ is ℜ-preserving with ${\displaystyle \underset{n\to \infty }{lim}}{u}_n=u$, there exists a subsequence $\left\{u_{n_k}\right\}$ of $u_n$ such that $g{u}_{n_k}\Re gu$ or $gu\Re g{u}_{n_k}$, for all $k\in \mathbb{N}$.

$\left(iv\right)$ F is ℜ-continuous at $u\in \Omega$ if, for any ℜ-preserving sequence, such that ${\displaystyle \underset{n\to \infty }{lim}}{u}_n=u$, we have ${\displaystyle \underset{n\to \infty }{lim}}F{u}_n=Fu$. Moreover, F is called ℜ-continuous if it is ℜ-continuous at each point of Ω.

$\left(v\right)$ F is $(g,\Re )$-continuous at x if for any sequence $\left\{u_n\right\}\subseteq \Omega$ such that $\left\{g{u}_n\right\}$ is ℜ-preserving with ${\displaystyle \underset{n\to \infty }{lim}}g{u}_n=gu$, we have ${\displaystyle \underset{n\to \infty }{lim}}F{u}_n=Fu$. Moreover, F is called $(g,\Re )$-continuous if it is $(g,\Re )$-continuous at each point of Ω.

$\left(vi\right)$$(F,g)$ is ℜ-compatible if for any sequence $\left\{u_n\right\}\subseteq \Omega$ such that $\left\{g{u}_n\right\}$ and $\left\{F{u}_n\right\}$ are ℜ-preserving and ${\displaystyle \underset{n\to \infty }{lim}}g{u}_n={\displaystyle \underset{n\to \infty }{lim}}F{u}_n=u\in \Omega$, we have ${\displaystyle \underset{n\to \infty }{lim}}d(Fg{u}_n,gF{u}_n)=0$.

$\left(vii\right)$ A subset $E\subseteq \Omega$ is said to be ℜ-connected, if for any $u,v\in E$, there exists a path in ℜ from u to v.

Definition 13

([19]). Let $(\Omega ,d)$ be a metric space and F and g are two self-mappings on Ω. Then,

$\left(i\right)$ a point $u\in \Omega$ is called a coincidence point of F and g if $gu=Fu$;

$\left(ii\right)$ if $u\in \Omega$ is a coincidence point of F and g, and there exists a point $\overline{u}$ such that $\overline{u}=gu=Fu$, then $\overline{u}$ is called a point of coincidence of F and g;

$\left(iii\right)$ if $u\in \Omega$ is a coincidence fixed point of F and g and $u=gu=Fu$, then u is called a common fixed point of F and g; and

$\left(iv\right)$ F and g are called weakly compatible if for all $u\in \Omega$ with $Fu=gu$ implies $F\left(gu\right)=g\left(Fu\right)$.

Theorem 2

([19]). Let $(\Omega ,d)$ be a metric space with a binary relation ℜ, and $▵$ be an ℜ-complete subspace of Ω. F and g are two self-mappings on Ω, which satisfy

$$d(Fu,Fv)⩽kd(gu,gv),forallgu\Re gv,$$

where $k\in (0,1)$. In addition, if F and g satisfy the following conditions:

$\left(i\right)$ there exists $v_0\in \Omega$ such that $g{v}_0\Re F{v}_0$;

$\left(ii\right)$ ℜ is $(F,g)$-closed;

$\left(iii\right)$$F(\Omega )\subseteq (▵\cap g(\Omega \left)\right)$; and

$\left(iv\right)$$\left(a\right)$$▵\subseteq g(\Omega )$ is ℜ-complete; and

        $\left(b\right)$ one of the conditions satisfies:

     $\left(1\right)$ F is $(g,\Re )$-continuous;

     $\left(2\right)$ F and g are continuous; and

     $\left(3\right)$ ${\Re |}_{\Omega }$ is d-self-closed,

or alternatively,

$\left(i{v}^{\prime }\right)$$\left(a^{\prime }\right)$ F and g are ℜ-compatible;

         $\left(b^{\prime }\right)$ g is ℜ-continuous; and

         $\left(c\right)$ one of the conditions satisfies:

     $\left(1^{\prime }\right)$ f is ℜ-continuous; and

     $\left(2^{\prime }\right)$ ℜ is $(g,d)$-self-closed,

then F and g have a coincidence point.

The following lemma plays a crucial role in proving the main results of this paper.

Lemma 1

([19]). Let Ω be a nonempty set and $g:\Omega \to \Omega$. Then, there exists a subset E of Ω such that $g\left(E\right)=g(\Omega )$ and $g:E\to E$ is one to one.

Through the above inspiration, we can understand that the extended rectangular b-metric spaces are a type of generalized metric spaces including metric spaces, rectangular metric spaces and b-metric spaces. As far as we know, in metric space, rectangular metric and b-metric space, there are also some contractions that have not been studied; thus, we intend to study the coincidence point and common fixed point results for some mappings F and g in the extended rectangular b-metric with a binary relation ℜ, which develops the results of [1,6,8,14,18,19,20,21,22,23].

2. Main Results

In this section, we introduce an auxiliary function before we begin our discussion of the main results. Let $\Psi$ be the set of all increasing functions $\psi :[0,\infty )\to [0,\infty )$ satisfying the following condition: ${\displaystyle \underset{n\to \infty }{lim}}{\psi }^n\left(t\right)=0,\mathrm{for}\mathrm{all}t>0.$

Remark 2.

If $\psi \in \Psi$, then $\psi \left(t\right)<t$, for all $t>0$.

Theorem 3.

Let $(\Omega ,d_{\xi })$ be an extended rectangular b-metric space with a binary relation ℜ such that ℜ is $(F,g)$-closed, and $▵$ be an ℜ-complete subspace of Ω. F and g are two self-mappings on Ω, which satisfy $F(\Omega )\subseteq (▵\cap g(\Omega \left)\right)$ and

$$\eta (d_{\xi }(Fu,Fv),\psi \left(M_{F,g}(u,v)\right))⩾0,forallgu\Re gv,$$

(1)

where $\eta \in \mathcal{Z}$, $\psi \in \Psi$ and

$$\begin{array}{cc}\hfill M_{F,g}(u,v)& =max\{d_{\xi }(gu,gv),d_{\xi }(gu,Fu),d_{\xi }(Fv,gv),\hfill \\ \hfill & \frac{d_{\xi }(gv,Fv)(1+d_{\xi }(gu,Fu))}{1+d_{\xi }(gu,gv)},\frac{d_{\xi }(gu,Fu)(1+d_{\xi }(gv,Fv))}{1+d_{\xi }(gu,gv)}\}.\hfill \end{array}$$

In addition, if F and g satisfy the following conditions:

$\left(i\right)$ there exists $v_0\in \Omega$ such that $g{v}_0\Re F{v}_0$ and $g{v}_0\Re F{v}_1$, where $v_1$ is such that $g{v}_1=F{v}_0$;

$\left(ii\right)$ for $v_0$ in $\left(i\right)$, we have ${\displaystyle \underset{n\to \infty }{lim\; sup}}\frac{{\psi }^{n+1}\left(t\right)}{{\psi }^n\left(t\right)}\xi (u_{n+1},u_p)<1$, where for all p,$n\in \mathbb{N}$, $u_n=F{v}_n=g{v}_{n+1}$ and $t\in (0,d_{\xi }(u_0,u_1)]$ with $u_0\ne u_1$;

$\left(iii\right)$$\left(a\right)$$▵\subseteq g(\Omega )$;

     $\left(b\right)$ F is $(g,\Re )$-continuous or F and g are continuous or ${\Re |}_{g(\Omega )}$ is $d_{\xi }$-self-closed and $d_{\xi }(gw,Fw)>0$, where $w\in \Omega$, such that

$${\displaystyle \underset{t\to d_{\xi }(gw,Fw)}{lim\; sup}}\psi \left(t\right)<\frac{d_{\xi }(gw,Fw)}{\xi (Fw,gw)}or{\displaystyle \underset{t\to d_{\xi }(gw,Fw)}{lim\; sup}}\psi \left(t\right)<\frac{d_{\xi }(gw,Fw)}{\xi (gw,Fw)};$$

or alternatively,

$\left(ii{i}^{\prime }\right)$ if $d_{\xi }$ is continuous, $(F,g)$ is ℜ-compatible, and g and F are ℜ-continuous,

then F and g have a coincidence point.

Proof. 

For $gu\Re gv$, by (1) and $\left(\eta 1\right)$, it is easy to show that

$$d_{\xi }(Fu,Fv)⩽\psi \left(M_{F,g}(u,v)\right),\mathrm{for}{M}_{F,g}(u,v)\ne 0.$$

(2)

Considering $F(\Omega )\subseteq (▵\cap g(\Omega \left)\right)$, we deduce that $F(\Omega )\subseteq g(\Omega )$. Now, we define two sequences $\left\{u_n\right\}$ and $\left\{v_n\right\}$ by $u_n=F{v}_n=g{v}_{n+1}$. By $g{v}_0\Re F{v}_0$ and ℜ is $(F,g)$-closed, it follows that

$$g{v}_0\Re F{v}_0\Rightarrow g{v}_0\Re g{v}_1\Rightarrow F{v}_0\Re F{v}_1\Rightarrow g{v}_1\Re g{v}_2.$$

(3)

Combining (3) with ℜ is $(F,g)$-closed, we have

$$g{v}_1\Re g{v}_2\Rightarrow F{v}_1\Re F{v}_2\Rightarrow g{v}_2\Re g{v}_3.$$

(4)

Repeating the above process, we can find

$$F{v}_n\Re F{v}_{n+1}$$

(5)

and

$$g{v}_n\Re g{v}_{n+1}.$$

(6)

By $g{v}_0\Re F{v}_1$ and ℜ is $(F,g)$-closed, we obtain

$$g{v}_0\Re F{v}_1\Rightarrow g{v}_0\Re g{v}_2\Rightarrow F{v}_0\Re F{v}_2\Rightarrow g{v}_1\Re g{v}_3.$$

(7)

Taking (7), $\left(i\right)$ and ℜ is $(F,g)$-closed in mind, we find

$$g{v}_1\Re g{v}_3\Rightarrow F{v}_1\Re F{v}_3\Rightarrow g{v}_2\Re g{v}_4.$$

(8)

Repeating the above process, it follows that

$$F{v}_n\Re F{v}_{n+2}$$

(9)

and

$$g{v}_n\Re g{v}_{n+2}.$$

(10)

If there exists $n_0\in \mathbb{N}$ such that $u_{n_0}=u_{n_0+1}$, that is, $g{v}_{n_0+1}=F{v}_{n_0+1}$, then $v_{n_0+1}$ is the coincidence point of F and g. The proof is complete.

Now, suppose that $u_n\ne u_{n+1}$, for all $n\in \mathbb{N}$. Let $u=v_n$, $v=v_{n+1}$ in (2), by (6), we have

$$\begin{array}{cc}\hfill d_{\xi }(u_n,u_{n+1})& =d_{\xi }(F{v}_n,F{v}_{n+1})\hfill \\ \hfill & ⩽\psi \left(M_{F,g}(v_n,v_{n+1})\right)\hfill \\ \hfill & =\psi (max\{d_{\xi }(g{v}_n,g{v}_{n+1}),d_{\xi }(g{v}_n,F{v}_n),d_{\xi }(F{v}_{n+1},g{v}_{n+1}),\hfill \\ \hfill & \frac{d_{\xi }(g{v}_{n+1},F{v}_{n+1})(1+d_{\xi }(g{v}_n,F{v}_n))}{1+d_{\xi }(g{v}_n,g{v}_{n+1})},\hfill \\ \hfill & \frac{d_{\xi }(g{v}_n,F{v}_n)(1+d_{\xi }(g{v}_{n+1},F{v}_{n+1}))}{1+d_{\xi }(g{v}_n,g{v}_{n+1})}\left\}\right)\hfill \\ \hfill & =\psi (max\{d_{\xi }(u_{n-1},u_n),d_{\xi }(u_{n-1},u_n),d_{\xi }(u_{n+1},u_n),\hfill \\ \hfill & \frac{d_{\xi }(u_n,u_{n+1})(1+d_{\xi }(u_{n-1},u_n))}{1+d_{\xi }(u_{n-1},u_n)},\hfill \\ \hfill & \frac{d_{\xi }(u_{n-1},u_n)(1+d_{\xi }(u_n,u_{n+1}))}{1+d_{\xi }(u_{n-1},u_n)}\left\}\right)\hfill \\ \hfill & =\psi (max\{d_{\xi }(u_{n-1},u_n),d_{\xi }(u_n,u_{n+1})).\hfill \end{array}$$

(11)

If

$$max\{d_{\xi }(u_{n-1},u_n),d_{\xi }(u_n,u_{n+1})\}=d_{\xi }(u_n,u_{n+1}),$$

by (11) and Remark 2, we gain

$$d_{\xi }(u_n,u_{n+1})⩽\psi \left(d_{\xi }(u_n,u_{n+1})\right)<d_{\xi }(u_n,u_{n+1}).$$

This is a contradiction. Thus,

$$max\{d_{\xi }(u_{n-1},u_n),d_{\xi }(u_n,u_{n+1})\}=d_{\xi }(u_{n-1},u_n).$$

In view of (11), we can deduce that

$$d_{\xi }(u_n,u_{n+1})⩽\psi \left(d_{\xi }(u_{n-1},u_n)\right).$$

(12)

By (12), we acquire

$$d_{\xi }(u_n,u_{n+1})⩽\psi \left(d_{\xi }(u_{n-1},u_n)\right)⩽\cdots ⩽{\psi }^n\left(d_{\xi }(u_0,u_1)\right).$$

(13)

Taking the limits on the both sides of (13), we have

$${\displaystyle \underset{n\to \infty }{lim}}{d}_{\xi }(u_n,u_{n+1})=0.$$

(14)

Now, we show that $u_n\ne u_m$, for all $n\ne m$ $\in \mathbb{N}$. If there exist $n_0$, $m_0\in \mathbb{N}$ such that $u_{n_0}=u_{m_0}$ with $n_0<m_0$, we have

$$\begin{array}{cc}\hfill d_{\xi }(u_{n_0},u_{n_0+1})& =d_{\xi }(u_{n_0},F{v}_{n_0+1})\hfill \\ \hfill & =d_{\xi }(u_{m_0},F{v}_{n_0+1})\hfill \\ \hfill & =d_{\xi }(F{v}_{m_0},F{v}_{n_0+1})\hfill \\ \hfill & ⩽\psi \left(M_{F,g}(v_{m_0},v_{n_0+1})\right)\hfill \\ \hfill & =\psi (max\{d_{\xi }(g{v}_{m_0},g{v}_{n_0+1}),d_{\xi }(g{v}_{m_0},F{v}_{m_0}),d_{\xi }(F{v}_{n_0+1},g{v}_{n_0+1}),\hfill \\ \hfill & \frac{d_{\xi }(g{v}_{n_0+1},F{v}_{n_0+1})(1+d_{\xi }(g{v}_{m_0},F{v}_{m_0}))}{1+d_{\xi }(g{v}_{m_0},g{v}_{n_0+1})},\hfill \\ \hfill & \frac{d_{\xi }(g{v}_{m_0},F{v}_{m_0})(1+d_{\xi }(g{v}_{n_0+1},F{v}_{n_0+1}))}{1+d_{\xi }(g{v}_{m_0},g{v}_{n_0+1})}\left\}\right)\hfill \\ \hfill & =\psi (max\{d_{\xi }(u_{m_0-1},u_{m_0}),d_{\xi }(u_{m_0-1},u_{m_0}),d_{\xi }(u_{n_0+1},u_{n_0}),\hfill \\ \hfill & \frac{d_{\xi }(u_{n_0},u_{n_0+1})(1+d_{\xi }(u_{m_0-1},u_{m_0}))}{1+d_{\xi }(u_{m_0-1},u_{m_0})},\hfill \\ \hfill & \frac{d_{\xi }(u_{m_0-1},u_{m_0})(1+d_{\xi }(u_{n_0},u_{n_0+1}))}{1+d_{\xi }(u_{m_0-1},u_{m_0})}\left\}\right)\hfill \\ \hfill & =\psi (d_{\xi }(u_{n_0},u_{n_0+1})\hfill \\ \hfill & <d_{\xi }(u_{n_0},u_{n_0+1}),\hfill \end{array}$$

which contradicts $d_{\xi }(u_{n_0},u_{n_0+1})>0$. Thus, $u_n\ne u_m$, for all $n,m\in \mathbb{N}$.

Letting $u=v_n$, $v=v_{n+2}$ in (2), by (10), we obtain

$$\begin{array}{cc}\hfill d_{\xi }(u_n,u_{n+2})& =d_{\xi }(F{v}_n,F{v}_{n+2})\hfill \\ \hfill & ⩽\psi \left(M_{F,g}(v_n,v_{n+2})\right)\hfill \\ \hfill & =\psi (max\{d_{\xi }(g{v}_n,g{v}_{n+2}),d_{\xi }(g{v}_n,F{v}_n),d_{\xi }(F{v}_{n+2},g{v}_{n+2}),\hfill \\ \hfill & \frac{d_{\xi }(g{v}_{n+2},F{v}_{n+2})(1+d_{\xi }(g{v}_n,F{v}_n))}{1+d_{\xi }(g{v}_n,g{v}_{n+2})},\hfill \\ \hfill & \frac{d_{\xi }(g{v}_n,F{v}_n)(1+d_{\xi }(g{v}_{n+2},F{v}_{n+2}))}{1+d_{\xi }(g{v}_n,g{v}_{n+2})}\left\}\right)\hfill \\ \hfill & =\psi (max\{d_{\xi }(u_{n-1},u_{n+1}),d_{\xi }(u_{n-1},u_n),d_{\xi }(u_{n+2},u_{n+1}),\hfill \\ \hfill & \frac{d_{\xi }(u_{n+2},u_{n+1})(1+d_{\xi }(u_{n-1},u_n))}{1+d_{\xi }(u_{n-1},u_{n+1})},\hfill \\ \hfill & \frac{d_{\xi }(u_{n-1},u_n)(1+d_{\xi }(u_{n+2},u_{n+1}))}{1+d_{\xi }(u_{n-1},u_{n+1})}\left\}\right)\hfill \\ \hfill & ⩽\psi (max\{d_{\xi }(u_{n-1},u_{n+1}),d_{\xi }(u_{n-1},u_n),\hfill \\ \hfill & \frac{d_{\xi }(u_{n-1},u_n)(1+d_{\xi }(u_{n-1},u_n))}{1+d_{\xi }(u_{n-1},u_{n+1})}\left\}\right)\hfill \\ \hfill & =\psi \left(A_n\right),\hfill \end{array}$$

(15)

where

$$A_n=max\{d_{\xi }(u_{n-1},u_{n+1}),d_{\xi }(u_{n-1},u_n),\frac{d_{\xi }(u_{n-1},u_n)(1+d_{\xi }(u_{n-1},u_n))}{1+d_{\xi }(u_{n-1},u_{n+1})}\}.$$

If $A_n=d_{\xi }(u_{n-1},u_{n+1}).$ By (15), we have

$$d_{\xi }(u_n,u_{n+2})⩽\psi \left(d_{\xi }(u_{n-1},u_{n+1})\right)⩽\cdots ⩽{\psi }^n\left(d_{\xi }(u_0,u_2)\right).$$

(16)

If $A_n=d_{\xi }(u_{n-1},u_n),$ from (13) and (15), we gain

$$d_{\xi }(u_n,u_{n+2})⩽\psi \left(d_{\xi }(u_{n-1},u_n)\right)⩽{\psi }^{n-1}\left(d_{\xi }(u_0,u_1)\right).$$

(17)

If $A_n=\frac{d_{\xi }(u_{n-1},u_n)(1+d_{\xi }(u_{n-1},u_n))}{1+d_{\xi }(u_{n-1},u_{n+1})},$ combining (13), (15) with Remark 2, we acquire

$$\begin{array}{cc}\hfill d_{\xi }(u_n,u_{n+2})& ⩽\psi (\frac{d_{\xi }(u_{n-1},u_n)(1+d_{\xi }(u_{n-1},u_n))}{1+d_{\xi }(u_{n-1},u_{n+1})})\hfill \\ \hfill & <\frac{d_{\xi }(u_{n-1},u_n)(1+d_{\xi }(u_{n-1},u_n))}{1+d_{\xi }(u_{n-1},u_{n+1})}\hfill \\ \hfill & <d_{\xi }(u_{n-1},u_n)(1+d_{\xi }(u_{n-1},u_n))\hfill \\ \hfill & ⩽{\psi }^{n-1}\left(d_{\xi }(u_0,u_1)\right)(1+{\psi }^{n-1}\left(d_{\xi }(u_0,u_1)\right)).\hfill \end{array}$$

(18)

Taking the limits on the both sides of (16), (17) and (18), by ${\displaystyle \underset{n\to \infty }{lim}}{\psi }^n\left(t\right)=0,\mathrm{for}\mathrm{all}t>0$, we find

$${\displaystyle \underset{n\to \infty }{lim}}{d}_{\xi }(u_n,u_{n+2})=0.$$

(19)

Now, we show that $\left\{u_n\right\}$ is a Cauchy sequence. The next discussion can be divided into the following cases.

Case I: when $m=n+2k+1$ with $k⩾1$. By $\left(d_{\xi }3\right)$ and (13), for all $n\in \mathbb{N}$, we have

$$\begin{array}{cc}\hfill d_{\xi }(u_n,u_{n+2k+1})& ⩽\xi (u_n,u_{n+2k+1})[(d_{\xi }(u_n,u_{n+1})+d_{\xi }(u_{n+1},u_{n+2})+d_{\xi }(u_{n+2},u_{n+2k+1})]\hfill \\ \hfill & =\xi (u_n,u_{n+2k+1})[d_{\xi }(u_n,u_{n+1})+d_{\xi }(u_{n+1},u_{n+2})]+\xi (u_n,u_{n+2k+1})d_{\xi }(u_{n+2},u_{n+2k+1})\hfill \\ \hfill & ⩽\xi (u_n,u_{n+2k+1})(d_n+d_{n+1})+\xi (u_n,u_{n+2k+1})\xi (u_{n+2},u_{n+2k+1})(d_{n+2}+d_{n+3})\hfill \\ \hfill & +\xi (u_n,u_{n+2k+1})\xi (u_{n+2},u_{n+2k+1})d_{\xi }(u_{n+2},u_{n+2k+1})\hfill \\ \hfill & ⩽\xi (u_n,u_{n+2k+1})(d_n+d_{n+1})+\xi (u_n,u_{n+2k+1})\xi (u_{n+2},u_{n+2k+1})(d_{n+2}+d_{n+3})+\cdots \hfill \\ \hfill & +\xi (u_n,u_{n+2k+1})\xi (u_{n+2},u_{n+2k+1})\cdots \xi (u_{n+2k-2},u_{n+2k+1})(d_{n+2k-2}+d_{n+2k-1})\hfill \\ \hfill & +\xi (u_n,u_{n+2k+1})\xi (u_{n+2},u_{n+2k+1})\cdots \xi (u_{n+2k-2},u_{n+2k+1})d_{\xi }(u_{n+2k},u_{n+2k+1})\hfill \\ \hfill & ⩽\xi (u_n,u_{n+2k+1})({\psi }^n\left(G_0\right)+{\psi }^{n+1}\left(G_0\right))+\xi (u_{n+2},u_{n+2m+1})({\psi }^{n+2}\left(G_0\right)+{\psi }^{n+3}\left(G_0\right))+\cdots \hfill \\ \hfill & +\xi (u_n,u_{n+2k+1})\xi (u_{n+2},u_{n+2k+1})\cdots \xi (u_{n+2k-2},u_{n+2k+1})({\psi }^{n+2k-2}\left(G_0\right)+{\psi }^{n+2k-1}\left(G_0\right))\hfill \\ \hfill & +\xi (u_n,u_{n+2k+1})\xi (u_{n+2},u_{n+2k+1})\cdots \xi (u_{n+2k-2},u_{n+2k+1}){\psi }^{n+2k}\left(G_0\right)\hfill \\ \hfill & ⩽\xi (u_0,u_{n+2k+1})\xi (u_1,u_{n+2k+1})\xi (u_2,u_{n+2k+1})\cdots \xi (u_n,u_{n+2k+1})[{\psi }^n\left(G_0\right)\hfill \\ \hfill & +\xi (u_{n+1},u_{n+2k+1}){\psi }^{n+1}\left(G_0\right)]+\xi (u_0,u_{n+2k+1})\xi (u_1,u_{n+2k+1})\xi (u_2,u_{n+2k+1})\cdots \hfill \\ \hfill & \times \xi (u_{n+2},u_{n+2k+1})[{\psi }^{n+2}\left(G_0\right)+\xi (u_{n+3},u_{n+2k+1}){\psi }^{n+3}\left(G_0\right)]+\cdots +\xi (u_0,u_{n+2k+1})\hfill \\ \hfill & \times \xi (u_1,u_{n+2k+1})\xi (u_2,u_{n+2k+1})\cdots \xi (u_{n+2k-2},u_{n+2k+1})[{\psi }^{n+2k-2}\left(G_0\right)\hfill \\ \hfill & +\xi (u_{n+2k-1},u_{n+2k+1}){\psi }^{n+2k-1}\left(G_0\right)]+\xi (u_0,u_{n+2k+1})\xi (u_1,u_{n+2k+1})\xi (u_2,u_{n+2k+1})\hfill \\ \hfill & \times \cdots \xi (u_{n+2k},u_{n+2k+1}){\psi }^{n+2k}\left(G_0\right)\hfill \\ \hfill & =\sum _{i=n}^{n+2k}{\psi }^i\left(G_0\right)\prod _{j=0}^i\xi (u_j,u_{n+2k+1}),\hfill \end{array}$$

(20)

where $d_n=d_{\xi }(u_n,u_{n+1})$ and ${\psi }^n\left(G_0\right)={\psi }^n\left(d_{\xi }(u_0,u_1)\right)$, for all $n\in \mathbb{N}$. Let

$$S_n=\sum _{i=0}^n{\psi }^i\left(G_0\right)\prod _{j=0}^i\xi (u_j,u_{n+2k+1}).$$

By (20), we obtain

$$d_{\xi }(u_n,u_{n+2k+1})⩽S_{n+2k}-S_{n-1}.$$

(21)

Suppose that $u_n={\psi }^n\left(G_0\right){\displaystyle \prod _{j=0}^n}\xi (u_j,u_{n+2k+1})$. We have

$$\frac{u_{n+1}}{u_n}=\frac{{\psi }^{n+1}\left(G_0\right){\displaystyle \prod _{j=0}^{n+1}}\xi (u_j,u_{n+2k+1})}{{\psi }^n\left(G_0\right){\displaystyle \prod _{j=0}^n}\xi (u_j,u_{n+2k+1})}=\frac{{\psi }^{n+1}\left(G_0\right)}{{\psi }^n\left(G_0\right)}\xi (u_{n+1},u_{n+2k+1}).$$

By $\left(ii\right)$ and Ratio test, we deduce that the series ${\displaystyle \sum _{i=0}^{\infty }}{\psi }^i\left(G_0\right){\displaystyle \prod _{j=0}^i}\xi (u_j,u_{n+2k+1})$ is convergent. Letting $n\to \infty$ in (21), we have

$$d_{\xi }(u_n,u_m)\to 0,n\to \infty .$$

Case II: when $m=n+2k$ with $k⩾1$. By $\left(d_{\xi }3\right)$ and (13), for all $n\in \mathbb{N}$, we obtain

$$\begin{array}{cc}\hfill d_{\xi }(u_n,u_{n+2k})& ⩽\xi (u_n,u_{n+2k})[(d_{\xi }(u_n,u_{n+2})+d_{\xi }(u_{n+2},u_{n+3})+d_{\xi }(u_{n+3},u_{n+2k})]\hfill \\ \hfill & =\xi (u_n,u_{n+2k})[d_{\xi }(u_n,u_{n+2})+d_{\xi }(u_{n+2},u_{n+3})]+\xi (u_n,u_{n+2k})d_{\xi }(u_{n+3},u_{n+2k})\hfill \\ \hfill & ⩽\xi (u_n,u_{n+2k})(d_{\xi }(u_n,u_{n+2})+d_{n+2})+\xi (u_n,u_{n+2k})\xi (u_{n+3},u_{n+2k})(d_{n+3}+d_{n+4})\hfill \\ \hfill & +\xi (u_n,u_{n+2k})\xi (u_{n+3},u_{n+2k})d_{\xi }(u_{n+5},u_{n+2k})\hfill \\ \hfill & ⩽\xi (u_n,u_{n+2k})(d_{\xi }(u_n,u_{n+2})+d_{n+2})+\xi (u_n,u_{n+2k})\xi (u_{n+3},u_{n+2k})(d_{n+3}+d_{n+4})+\cdots \hfill \\ \hfill & +\xi (u_n,u_{n+2k})\xi (u_{n+3},u_{n+2k})\cdots \xi (u_{n+2k-3},u_{n+2k})(d_{n+2k-3}+d_{n+2k-2}+d_{n+2k-1})\hfill \\ \hfill & ⩽\xi (u_n,u_{n+2k})(d_{\xi }(u_n,u_{n+2})+{\psi }^{n+2}\left(G_0\right))+\xi (x_n,u_{n+2k})\xi (u_{n+3},u_{n+2k})\hfill \\ & ({\psi }^{n+3}\left(G_0\right)+{\psi }^{n+4}\left(G_0\right))+\cdots +\xi (u_n,u_{n+2k})\hfill \\ \hfill & \xi (u_{n+3},u_{n+2k})\cdots \xi (u_{n+2k-3},u_{n+2k})({\psi }^{n+2k-3}\left(G_0\right)+{\psi }^{n+2k-2}\left(G_0\right)+{\psi }^{n+2k-1}\left(G_0\right))\hfill \\ \hfill & <\xi (u_n,u_{n+2k})d_{\xi }(u_n,u_{n+2})+\xi (u_0,u_{n+2k})\xi (u_1,u_{n+2k})\xi (u_2,u_{n+2k})\cdots \hfill \\ \hfill & \xi (u_{n+2},u_{n+2k}){\psi }^{n+2}\left(G_0\right)+\xi (u_0,u_{n+2k})\xi (u_1,u_{n+2k})\xi (x_2,u_{n+2k})\cdots \hfill \\ \hfill & \xi (u_{n+3},u_{n+2k})[{\psi }^{n+3}\left(G_0\right)+\xi (u_{n+4},u_{n+2k}){\psi }^{n+4}\left(G_0\right)]+\cdots +\hfill \\ \hfill & \xi (u_0,u_{n+2k})\xi (u_1,u_{n+2k})\xi (u_2,u_{n+2k})\cdots \xi (u_{n+2k-3},u_{n+2k})[{\psi }^{n+2k-3}\left(G_0\right)\hfill \\ \hfill & +\xi (u_{n+2k-2},u_{n+2k}){\psi }^{n+2k-2}\left(G_0\right)]+\xi (u_0,u_{n+2k})\xi (u_1,u_{n+2k})\hfill \\ \hfill & \xi (u_2,u_{n+2k})\cdots \xi (u_{n+2k-1},u_{n+2k}){\psi }^{n+2k-1}\left(G_0\right)\hfill \\ \hfill & =\xi (u_n,u_{n+2k})d_{\xi }(u_n,u_{n+2})+\sum _{i=n+2}^{n+2k-1}{\psi }^i\left(G_0\right)\prod _{j=0}^i\xi (u_j,u_{n+2k}),\hfill \end{array}$$

(22)

where ${\psi }^n\left(G_0\right)={\psi }^n\left(d_{\xi }(u_0,u_1)\right)$ and $d_n=d_{\xi }(u_n,u_{n+1})$. For all $n\in \mathbb{N}$, assume that

$$R_n=\sum _{i=0}^n{\psi }^i\left(G_0\right)\prod _{j=0}^i\xi (u_j,u_{n+2k}).$$

According to (22), we find

$$d_{\xi }(u_n,u_{n+2k})<\xi (u_n,u_{n+2k})d_{\xi }(u_n,u_{n+2})+R_{n+2k-1}-R_{n+1}.$$

(23)

Now, let $w_n={\psi }^n\left(G_0\right){\displaystyle \prod _{j=0}^n}\xi (u_j,u_{n+2k})$. It follows that

$$\frac{w_{n+1}}{w_n}=\frac{{\psi }^{n+1}\left(G_0\right){\displaystyle \prod _{j=0}^{n+1}}\xi (u_j,u_{n+2k})}{{\psi }^n\left(G_0\right){\displaystyle \prod _{j=0}^n}\xi (u_j,u_{n+2k})}=\frac{{\psi }^{n+1}\left(G_0\right)}{{\psi }^n\left(G_0\right)}\xi (u_{n+1},u_{n+2k}).$$

In a similar way as in thecase I, we deduce that the series ${\displaystyle \sum _{i=0}^{\infty }}{\psi }^i\left(G_0\right){\displaystyle \prod _{j=0}^i}\xi (u_j,u_{n+2k})$ is convergent. Taking the limits on the both sides of (23), by (19), we have

$$d_{\xi }(u_n,u_m)\to 0,n\to \infty .$$

In both Cases, ${\displaystyle \underset{n,m\to \infty }{lim}}{d}_{\xi }(u_n,u_m)=0.$ Thus, $\left\{u_n\right\}$ is a Cauchy sequence.

Now, we show that F and g have a coincidence point. We discuss the following cases:

Case I: $\left(iii\right)$ holds.

Since $(▵,d_{\xi })$ is ℜ-complete, $F(\Omega )\subseteq ▵$, $u_n=F{v}_n=g{v}_{n+1}$ and (6), there exists $u\in ▵$ such that

$${\displaystyle \underset{n\to \infty }{lim}}{d}_{\xi }(g{v}_n,u)=0.$$

(24)

Considering $▵\subseteq g(\Omega )$; thus, there exists $v\in \Omega$ such that $u=gv$. That is,

$${\displaystyle \underset{n\to \infty }{lim}}{d}_{\xi }(g{v}_n,gv)=0.$$

(25)

By $u_n=F{v}_n=g{v}_{n+1}$, we have

$${\displaystyle \underset{n\to \infty }{lim}}{d}_{\xi }(F{v}_n,gv)=0.$$

(26)

If there exists an infinite subsequence $\left\{u_{n_k}\right\}$ of $\left\{u_n\right\}$ such that $u_{n_k}=Fv$ or $u_{n_k}=gv$, then it will lead to a contradiction with $u_n\ne u_m$, for all $n\ne m$ $\in \mathbb{N}$. Thus, we assume that $u_n\ne Fv$ and $u_n\ne gv$ for all $n\in \mathbb{N}$.

If F is $(g,\Re )$-continuous. Thinking about (6) and (25), we obtain

$${\displaystyle \underset{n\to \infty }{lim}}{d}_{\xi }(F{v}_n,Fv)=0.$$

(27)

By $\left(d_{\xi }3\right)$, it follows that

$$\begin{array}{cc}\hfill d_{\xi }(Fv,gv)& ⩽\xi (Fv,gv)[d_{\xi }(Fv,F{v}_n)+d_{\xi }(F{v}_n,F{v}_{n+1})+d_{\xi }(F{v}_{n+1},gv)]\hfill \\ \hfill & =\xi (Fv,gv)[d_{\xi }(Fv,F{v}_n)+d_{\xi }(u_n,u_{n+1})+d_{\xi }(F{v}_{n+1},gv)]\hfill \end{array}$$

(28)

Taking the limits on the both sides of (28), keep (14), (26) and (27) in mind, we deduce that

$$d_{\xi }(Fv,gv)=0.$$

Thus, v is a coincidence point of F and g.

Assume that F and g are continuous. By Lemma 1, it is not difficult to find that there exists $E\subset \Omega$ such that $g\left(E\right)=g(\Omega )$ and $g:E\to E$ is one to one. Define a function $T:g\left(E\right)\to g\left(E\right)$ by $Fe=Tge$, where $e\in E$. Clearly, T is well-defined. Since F and g are continuous, we deduce that, T is continuous as well. Without loss of generality, we choose $\left\{v_n\right\}\subseteq E$ and $v\in E$. By (25), we obtain

$$\begin{array}{cc}\hfill {\displaystyle \underset{n\to \infty }{lim}}{d}_{\xi }(F{v}_n,Fv)& ={\displaystyle \underset{n\to \infty }{lim}}{d}_{\xi }(Tg{v}_n,Fv)\hfill \\ & ={\displaystyle \underset{n\to \infty }{lim}}{d}_{\xi }(Tg{v}_n,Tgv)\hfill \\ & =0,\hfill \end{array}$$

that is (27) holds. Taking the limits on the both sides of (28), keep (14), (26) and (27) in mind, we deduce that

$$d_{\xi }(Fv,gv)=0.$$

Then, v is a coincidence point of F and g.

If ${\Re |}_{g\Omega }$ is $d_{\xi }$-self closed, form (6) and (25), there exists a subsequence $\left\{g{v}_{n_k}\right\}$ of $\left\{g{v}_n\right\}$ satisfying

$$g{v}_{n_k}\Re gv\mathrm{or}gv\Re g{v}_{n_k}.$$

(29)

Assume that $g{v}_{n_k}\Re gv$. Let $u=v_{n_k}$ in (2), keep (29) in mind, we have

$$\begin{array}{cc}\hfill d_{\xi }(F{v}_{n_k},Fv)& ⩽\psi \left(M_{F,g}(v_{n_k},v)\right)\hfill \\ \hfill & =\psi (max\{d_{\xi }(g{v}_{n_k},gv),d_{\xi }(g{v}_{n_k},F{v}_{n_k}),d_{\xi }(Fv,gv),\hfill \\ \hfill & \frac{d_{\xi }(gv,Fv)(1+d_{\xi }(g{v}_{n_k},F{v}_{n_k}))}{1+d_{\xi }(g{v}_{n_k},gv)},\frac{d_{\xi }(g{v}_{n_k},F{v}_{n_k})(1+d_{\xi }(gv,Fv))}{1+d_{\xi }(g{v}_{n_k},gv)}\left\}\right)\hfill \\ \hfill & =\psi (max\{d_{\xi }(g{v}_{n_k},gx),d_{\xi }(u_{n_k-1},u_{n_k}),d_{\xi }(Fv,gv),\hfill \\ \hfill & \frac{d_{\xi }(gv,Fv)(1+d_{\xi }(u_{n_k-1},u_{n_k}))}{1+d_{\xi }(g{v}_{n_k},gv)},\frac{d_{\xi }(u_{n_k-1},u_{n_k})(1+d_{\xi }(gv,Fv))}{1+d_{\xi }(g{v}_{n_k},gv)}\left\}\right).\hfill \end{array}$$

(30)

Taking the super limits on the both sides of (30), we gain

$${\displaystyle \underset{k\to \infty }{lim\; sup}}{d}_{\xi }(F{v}_{n_k},Fv)⩽{\displaystyle \underset{t\to d_{\xi }(gv,Fv)}{lim\; sup}}\psi \left(t\right).$$

(31)

Taking the super limits on the both sides of (28), according to (14), (25) and (31), we have

$$d_{\xi }(Fv,gv)⩽\xi (Fv,gv){\displaystyle \underset{t\to d_{\xi }(gv,Fv)}{lim\; sup}}\psi \left(t\right).$$

This leads to a contradiction with

$${\displaystyle \underset{t\to d_{\xi }(gv,Fv)}{lim\; sup}}\psi \left(t\right)<\frac{d_{\xi }(gv,Fv)}{\xi (Fv,gv)}.$$

Thus, $d_{\xi }(gv,Fv)=0$.

If $gv\Re g{v}_{n_k}$, by the similar discussion and keep

$${\displaystyle \underset{t\to d_{\xi }(gw,Fw)}{lim\; sup}}\psi \left(t\right)<\frac{d_{\xi }(gw,Fw)}{\xi (gw,Fw)}$$

in mind, we can also find $d_{\xi }(gv,Fv)=0$.

Case II: $\left(ii{i}^{\prime }\right)$ holds.

By $F(\Omega )\subseteq (▵\cap g(\Omega \left)\right)$, $▵$ being an ℜ-complete and the construction of the sequence $\left\{u_n\right\}$, there exists $u\in (\Delta \cap g\Omega )$ such that

$${\displaystyle \underset{n\to \infty }{lim}}F{v}_n=u$$

(32)

and

$${\displaystyle \underset{n\to \infty }{lim}}g{v}_n=u.$$

(33)

If F and g are ℜ-continuous, we obtain

$${\displaystyle \underset{n\to \infty }{lim}}gF{v}_n=gu$$

(34)

and

$${\displaystyle \underset{n\to \infty }{lim}}Fg{v}_n=Fu.$$

(35)

Considering (32), (33) and $(F,g)$ is ℜ-compatible, we gain

$${\displaystyle \underset{n\to \infty }{lim}}{d}_{\xi }(Fg{v}_n,gF{v}_n)=0.$$

(36)

Clearly, by (34)–(36) and $d_{\xi }$ is continuous, we have

$$d_{\xi }(Fu,gu)=0.$$

The proof is complete. □

Example 1.

Let $\Omega =[0,1]$ with $u\Re v$ if and only if $u,v\in [\frac{1}{32},\frac{1}{16}]$ and $d_{\xi }(u,v)$ $=\frac{{(u-v)}^2}{2}$ with $\xi (u,v)=u+v+4$ for all $u,v\in \Omega$. Suppose that $\Delta =[0,\frac{5}{32}]$, clearly, $(\Omega ,d_{\xi })$ is an extended rectangular b-metric space and Δ is ℜ-complete. Indeed, $d_{\xi }$ is generated from standard metric, for every ℜ-preserving Cauchy sequence $\left\{u_n\right\}$ in Ω, we acquire sequence $\left\{u_n\right\}$ converges to a point in Ω. Define the mappings $F,g:\Omega \to \Omega$ by

$$\begin{array}{c}\hfill Fu=\left\{\begin{array}{ccc}& \frac{u}{2},\hfill & \hfill ifu\in [0,\frac{1}{4}],\\ & \frac{1}{8},\hfill & \hfill otherwise.\end{array}\right.\end{array}$$

and

$$\begin{array}{c}\hfill gu=\left\{\begin{array}{ccc}& \frac{u}{2},\hfill & \hfill ifu\in [0,\frac{1}{2}],\\ & \frac{1}{4},\hfill & \hfill otherwise.\end{array}\right.\end{array}$$

Clearly, $F(\Omega )\subseteq \Delta \subseteq g(\Omega )$, ℜ is $(F,g)$-closed. Indeed, for all $gu\Re gv$, we obtain $u,v\in [\frac{1}{16},\frac{1}{8}]$, then $Fu,Fv\in [\frac{1}{32},\frac{1}{16}]$, that is $Fu\Re Fv$. Suppose that a sequence $\left\{u_n\right\}\subseteq \Omega$ and a point $u\in \Omega$ such that ${\displaystyle \underset{n\to \infty }{lim}}{u}_n=u$. For mapping F, if $u_n\in [0,\frac{1}{4}]$, by the definitions of function $d_{\xi }$ and mapping F, we have $u\in [0,\frac{1}{4}]$, $F{u}_n=\frac{u_n}{2}$ and $Fu=\frac{u}{2}$. Then, ${\displaystyle \underset{n\to \infty }{lim}}F{u}_n=Fu$. If $u_n\in (\frac{1}{4},1]$, by the definitions of function $d_{\xi }$ and mapping F, we have $u\in [\frac{1}{4},1]$ and $F{u}_n=\frac{1}{8}$ and $Fu=\frac{1}{8}$. Then, ${\displaystyle \underset{n\to \infty }{lim}}F{u}_n=Fu$. Thus, mapping F is continuous. By similarly discuss, we can also find g is continuous. In addition, there exists $v_0=\frac{1}{32}$ such that $g{v}_0\Re F{v}_0$ and $g{v}_0\Re F{v}_1$, where $v_1$ is such that $g{v}_1=F{v}_0$. Take $\eta (u,v)=\frac{1}{2}v-u$ and

$$\begin{array}{c}\hfill \psi \left(t\right)=\left\{\begin{array}{ccc}& \frac{2}{9}t,\hfill & \hfill ift\in [0,1],\\ & \frac{260}{1161},\hfill & \hfill otherwise.\end{array}\right.\end{array}$$

For all $t\in (0,d_{\xi }(u_0,u_1)]$ and for all $p\in \mathbb{N}$, we have

$$\begin{array}{cc}\hfill {\displaystyle \underset{n\to \infty }{lim\; sup}}\frac{{\psi }^{n+1}\left(t\right)}{{\psi }^n\left(t\right)}\xi (u_{n+1},u_p)& ={\displaystyle \underset{n\to \infty }{lim\; sup}}\frac{2}{9}(4+u_{n+1}+u_p)\hfill \\ \hfill & ={\displaystyle \underset{n\to \infty }{lim\; sup}}\frac{2}{9}(4+\frac{v_0}{2}+\frac{v_0}{2})\hfill \\ \hfill & =\frac{2}{9}(4+\frac{1}{32})\hfill \\ \hfill & <1.\hfill \end{array}$$

Now, we show that F and g satisfy condition (1). Indeed, for all $gu\Re gv$,

$$\begin{array}{cc}\hfill \frac{1}{2}\psi \left(M_{F,g}(u,v)\right)-d_{\xi }(Fu,Fv)& ⩾\frac{1}{9}max\{d_{\xi }(gu,Fu),d_{\xi }(gv,Fv)\}-\frac{{(\frac{u}{2}-\frac{v}{2})}^2}{2}\hfill \\ \hfill & =\frac{1}{9}max\{\frac{9u^2}{8},\frac{9v^2}{8}\}-\frac{{(u-v)}^2}{8}\hfill \\ \hfill & ⩾(\frac{1\times 9}{9}-1)d_{\xi }(Fu,Fv)\hfill \\ \hfill & ⩾0.\hfill \end{array}$$

Thus, by Theorem 3, there exists $v=\frac{1}{32}$ such that $F\left(\frac{1}{32}\right)=g\left(\frac{1}{32}\right)$.

Example 2.

Let $\Omega =[0,3)$, $u\Re v$ if and only if $(u,v)\in [0,\frac{1}{8}]\times [0,\frac{1}{8}]$ and $d_{\xi }(u,v)$ $={(u-v)}^2$ with $\xi (u,v)=u+v+4$, for all $u,v\in \Omega$. Define the mappings $F,g:\Omega \to \Omega$ by

$$\begin{array}{c}\hfill Fu=\left\{\begin{array}{ccc}& \frac{u}{4},\hfill & \hfill ifu\in [0,\frac{1}{2}],\\ & 2,\hfill & \hfill ifu\in (\frac{1}{2},3).\end{array}\right.\end{array}$$

and

$$\begin{array}{c}\hfill gu=\left\{\begin{array}{ccc}& u,\hfill & \hfill ifu\in [0,\frac{1}{2}],\\ & 2,\hfill & \hfill ifu\in (\frac{1}{2},3).\end{array}\right.\end{array}$$

Clearly, $F(\Omega )\subseteq \Delta \subseteq g(\Omega )$, ℜ is $(F,g)$-closed and F is $(g,\Re )$-continuous. Indeed, for all $gu\Re gv$, we obtain $u,v\in [0,\frac{1}{8}]$, then $Fu,Fv\in [0,\frac{1}{32}]$, that is $Fu\Re Fv$. For any sequence $\left\{u_n\right\}\subseteq \Omega$ such that $\left\{g{u}_n\right\}$ is ℜ-preserving with ${\displaystyle \underset{n\to \infty }{lim}}g{u}_n=gu$, we have sequence $\left\{u_n\right\}\subseteq [0,\frac{1}{32}]$ and $u\in [0,\frac{1}{32}]$, so ${\displaystyle \underset{n\to \infty }{lim}}F{u}_n=Fu$. We can find that both F and g are not continuous at $u=\frac{1}{2}$, and Δ is ℜ-complete via $d_{\xi }$ is generated from standard metrics, where $\Delta =[0,\frac{1}{2}]\cup \left\{2\right\}$. In addition, there exists $v_0=\frac{1}{8}$ such that $g{v}_0\Re F{v}_0$ and $g{v}_0\Re F{v}_1$, where $v_1$ with $g{v}_1=F{v}_0$. Take $\eta (u,v)=\frac{1}{2}v-u$ and $\psi \left(t\right)=\frac{4}{17}t$, for all $t\in [0,\infty )$.

For every $t\in (0,d_{\xi }(u_0,u_1)]$, we have

$$\begin{array}{cc}\hfill {\displaystyle \underset{n\to \infty }{lim\; sup}}\frac{{\psi }^{n+1}\left(t\right)}{{\psi }^n\left(t\right)}\xi (u_{n+1},u_p)& ={\displaystyle \underset{n\to \infty }{lim\; sup}}\frac{4}{17}(4+u_{n+1}+u_p)\hfill \\ \hfill & ={\displaystyle \underset{n\to \infty }{lim\; sup}}\frac{4}{17}(4+\frac{v_0}{4^{n+1}}+\frac{v_0}{4^{p+1}})\hfill \\ \hfill & =\frac{4}{17}(4+\frac{1}{8})\hfill \\ \hfill & <1.\hfill \end{array}$$

Now, we show that condition (1) for F and g holds. Indeed, for all $gu\Re gv$,

$$\begin{array}{cc}\hfill \eta (d_{\xi }(Fu,Fv),M_{F,g}(u,v))& =\frac{1}{2}\psi \left(M_{F,g}(u,v)\right)-d_{\xi }(Fu,Fv)\hfill \\ \hfill & ⩾\frac{2}{17}{d}_{\xi }(gu,gv)-{(\frac{u}{4}-\frac{v}{4})}^2\hfill \\ \hfill & =(\frac{2}{17}-\frac{1}{16})d_{\xi }(gu,gv)\hfill \\ \hfill & ⩾0.\hfill \end{array}$$

So, by Theorem 3, there exists $v=0$ such that $F0=g0$. Further, we claim that the common fixed point theorems in [20,21] are not valid in proving the existence of common fixed points of F and g. Indeed, for $u=0,v=2$, $d_{\xi }(Fu,Fv)>k_1{d}_{\xi }(gu,gv)$ and $d_{\xi }(Fu,Fv)>k_2[d_{\xi }(gu,Fu)+d_{\xi }(gv,Fv)]$, where $k_1\in (0,1)$, $k_2\in (0,\frac{1}{2})$.

According to Examples 1 and 2, we find that the coincidence point of F and g is not unique. Thus, Theorem 3 shows only the existence of coincidence point of F and g. Now, we add some conditions to show that the point of coincidence of F and g is unique.

Theorem 4.

In addition the assumption in Theorem 3, we also suppose the following condition:

$\left(iv\right)$ If $gu\Re gv$ or $gv\Re gu$, for all u, v $\in C(F,g)$, where $C(F,g)=\{u\in \Omega :Fu=gu\}$,

then the point of coincidence of F and g is unique.

Proof. 

Assume that there exist $u,v\in C(F,g)$ with $d_{\xi }(Fu,Fv)>0$. If $gu\Re gv$, by (2), we have

$$\begin{array}{cc}\hfill d_{\xi }(Fu,Fv)& ⩽\psi \left(M_{F,g}(u,v)\right)\hfill \\ \hfill & =\psi (max\{d_{\xi }(gu,gv),d_{\xi }(gu,Fu),d_{\xi }(Fv,gv),\hfill \\ \hfill & \frac{d_{\xi }(gv,Fv)(1+d_{\xi }(gu,Fu))}{1+d_{\xi }(gu,gv)},\frac{d_{\xi }(gu,Fu)(1+d_{\xi }(gv,Fv))}{1+d_{\xi }(gu,gv)}\left\}\right)\hfill \\ \hfill & =\psi (max\{d_{\xi }(Fu,Fv),0,0,0,0\})\hfill \\ \hfill & =\psi \left(d_{\xi }(Fu,Fv)\right)\hfill \\ \hfill & <d_{\xi }(Fu,Fv),\hfill \end{array}$$

which leads to a contradiction with $d_{\xi }(Fu,Fv)>0$. Thus, $d_{\xi }(Fu,Fv)=0$. If $gv\Re gu$, by the similar discussion, we have $d_{\xi }(Fu,Fv)=0$. The proof is complete. □

Theorem 4 shows that the point of coincidence of F and g is unique. Now, we add a condition to show that F and g have a unique common fixed point.

Theorem 5.

Except for the assumption in Theorem 4, if $(F,g)$ is weakly compatible, then F and g have a unique common fixed point.

Proof. 

By Theorem 3, there exists $v\in \Omega$ such that $Fv=gv$. Assume that $u=Fv=gv$. Since $(F,g)$ is weakly compatible, we have $Fu=Fgv=gFv=gu$. By Theorem 4, we have $Fu=gu=Fv=gv=u$. Thus, u is the common point of F and g. Suppose that there exists s such that $s=Fs=gs$ and $s\ne u$. By $s\ne u$, we have $Fs\ne Fu$—a contradiction. Thus, $s=u$. The proof is complete. □

Remark 3.$\left(i\right)$ By the proof of Theorem 3, we only use the property $\left(\eta 1\right)$ of function η.

$\left(ii\right)$ In the proofs of Theorem 3, Theorem 4 and Theorem 5, we can find that we mainly use (2) instead of (1). Thus, if we replace (1) with

$$d_{\xi }(Fu,Fv)⩽\psi \left(M_{F,g}(u,v)\right),forallgu\Re gv,$$

in Theorem 3, these results still hold.

$\left(iii\right)$ We observe that

$$\begin{array}{cc}\hfill \frac{d_{\xi }(g{v}_n,g{v}_{n+1})d_{\xi }(g{v}_{n+1},F{v}_{n+1})}{1+d_{\xi }(g{v}_n,F{v}_n)}& ⩽d_{\xi }(u_n,u_{n+1});\hfill \\ \hfill \frac{d_{\xi }(g{v}_n,g{v}_{n+2})d_{\xi }(g{v}_{n+2},F{v}_{n+2})}{1+d_{\xi }(g{v}_n,F{v}_n)}& =\frac{d_{\xi }(u_{n-1},u_{n+1})d_{\xi }(u_{n+1},u_{n+2})}{1+d_{\xi }(u_{n-1},u_n)};\hfill \\ \hfill \frac{d_{\xi }(g{v}_{n_k},gv)d_{\xi }(gv,Fv)}{1+d_{\xi }(g{v}_{n_k},F{v}_{n_k})}& =\frac{d_{\xi }(u_{n_k-1},gv)d_{\xi }(gv,Fv)}{1+d_{\xi }(u_{n_k-1},u_{n_k})};\hfill \\ \hfill \frac{d_{\xi }(g{v}_n,F{v}_n)d_{\xi }(g{v}_{n+1},F{v}_{n+1})}{1+d_{\xi }(F{v}_n,F{v}_{n+1})}& ⩽d_{\xi }(u_{n-1},u_n);\hfill \\ \hfill \frac{d_{\xi }(g{v}_n,F{v}_n)d_{\xi }(g{v}_{n+2},F{v}_{n+2})}{1+d_{\xi }(F{v}_n,F{v}_{n+2})}& =\frac{d_{\xi }(u_{n-1},u_n)d_{\xi }(u_{n+1},u_{n+2})}{1+d_{\xi }(u_n,u_{n+2})};\hfill \\ \hfill \frac{d_{\xi }(g{v}_n,F{v}_n)d_{\xi }(gv,Fv)}{1+d_{\xi }(F{v}_n,Fv)}& =\frac{d_{\xi }(u_{n-1},u_n)d_{\xi }(gv,Fv)}{1+d_{\xi }(u_n,Fv)}.\hfill \end{array}$$

Thus, we add $\frac{d_{\xi }(gu,gv)d_{\xi }(gv,Fv)}{1+d_{\xi }(gu,Fu)}$ and $\frac{d_{\xi }(gu,Fu)d_{\xi }(gv,Fv)}{1+d_{\xi }(Fu,Fv)}$ to $M_{F,g}(u,v)$, the above results still hold.

3. Corollaries

Corollary 1.

Let $(\Omega ,d_{\xi })$ be an extended rectangular b-metric space with a binary relation ℜ. F is a self-mapping on Ω, which satisfies

$$\eta (d_{\xi }(Fu,Fv),\psi \left(M(u,v)\right))⩾0,forallu\Re v,$$

(37)

where $\eta \in \mathcal{Z}$, $\psi \in \Psi$ and

$$\begin{array}{cc}\hfill M(u,v)& =max\{d_{\xi }(u,v),d_{\xi }(u,Fu),d_{\xi }(Fv,v),\hfill \\ \hfill & \frac{d_{\xi }(v,Fv)(1+d_{\xi }(u,Fu))}{1+d_{\xi }(u,v)},\frac{d_{\xi }(u,Fu)(1+d_{\xi }(v,Fv))}{1+d_{\xi }(u,v)}\}.\hfill \end{array}$$

In addition, if F satisfies the following conditions:

$\left(i\right)$ There exists $v_0\in \Omega$ such that $v_0\Re F{v}_0$ and $v_0\Re F^2{v}_0$.

$\left(ii\right)$ ℜ is F-closed.

$\left(iii\right)$ For $v_0$ in $\left(i\right)$, we have ${\displaystyle \underset{n\to \infty }{lim\; sup}}\frac{{\psi }^{n+1}\left(t\right)}{{\psi }^n\left(t\right)}\xi (v_{n+1},v_p)<1$, where $p\in \mathbb{N}$, $v_{n+1}=F{v}_n$ and $t\in (0,d_{\xi }(v_0,v_1)]$ with $v_0\ne v_1$.

$\left(iv\right)$ There exists $▵\subseteq \Omega$ such that $F(\Omega )\subseteq ▵$ and $(▵,d_{\xi })$ is ℜ-complete.

$\left(v\right)$ One of the conditions holds:

        $\left(a\right)$ F is ℜ-continuous; or

        $\left(b\right)$ ${\Re |}_{\Omega }$ is $d_{\xi }$-self-closed and for all $w\in \Omega$ with $d_{\xi }(w,Fw)>0$ such that

$${\displaystyle \underset{t\to d_{\xi }(w,Fw)}{lim\; sup}}\psi \left(t\right)<\frac{d_{\xi }(w,Fw)}{\xi (Fw,w)}or{\displaystyle \underset{t\to d_{\xi }(w,Fw)}{lim\; sup}}\psi \left(t\right)<\frac{d_{\xi }(w,Fw)}{\xi (w,Fw)},$$

then F has a fixed point.

In addition, if

$\left(vi\right)$$u\Re v$ or $v\Re u$, for all u, v with $u=Fu$ and $v=Fv$,

then F has a unique fixed point.

Proof. 

Take $g=I$ (the identity map) in Theorem 5, it is clear that the result is true. □

Corollary 2.

Let $(\Omega ,d_{\xi })$ be an extended rectangular b-metric space with a binary relation ℜ and $▵$ be an ℜ-complete subspace of Ω. F and g are self-mappings on Ω, which satisfy $F(\Omega )\subseteq \left(g\right(\Omega )\cap ▵)$, and

$$d_{\xi }(Fu,Fv)⩽k{M}_{F,g}(u,v),forallgu\Re gv,$$

(38)

where $k\in (0,1)$ and

$$\begin{array}{cc}\hfill M_{F,g}(u,v)& =max\{d_{\xi }(gu,gv),d_{\xi }(gu,Fu),d_{\xi }(Fv,gv),\hfill \\ \hfill & \frac{d_{\xi }(gv,Fv)(1+d_{\xi }(gu,Fu))}{1+d_{\xi }(gu,gv)},\frac{d_{\xi }(gu,Fu)(1+d_{\xi }(gv,Fv))}{1+d_{\xi }(gu,gv)}\}.\hfill \end{array}$$

In addition, if F and g satisfy the following conditions:

$\left(i\right)$ there exists $v_0\in \Omega$ such that $g{v}_0\Re F{v}_0$ and $g{v}_0\Re F{v}_1$, where $v_1$ is such that $g{v}_1=F{v}_0$;

$\left(ii\right)$ ℜ is $(F,g)$-closed;

$\left(iii\right)$ for $v_0$ in $\left(i\right)$, we have ${\displaystyle \underset{n\to \infty }{lim\; sup}}\xi (u_{n+1},u_p)<\frac{1}{k}$, where $p\in \mathbb{N}$, $u_n=F{v}_n=g{v}_{n+1}$ and $t\in (0,d_{\xi }(u_0,u_1)]$ with $u_0\ne u_1$; and

$\left(iv\right)$$\left(a\right)$$▵\subseteq g(\Omega )$; and

     $\left(b\right)$ F is $(g,\Re )$-continuous or F and g are continuous or ${\Re |}_{g(\Omega )}$ is $d_{\xi }$-self-closed and $d_{\xi }(gw,Fw)>0$, where $w\in \Omega$, such that $\xi (Fw,w)<\frac{1}{k}$ or $\xi (w,Fw)<\frac{1}{k};$

or alternatively,

$\left(i{v}^{\prime }\right)$$d_{\xi }$ is continuous and $(F,g)$ is ℜ-compatible, and g and F are ℜ-continuous;

$\left(v\right)$ if $gu\Re gv$ or $gv\Re gu$, for all u, v with $gu=Fu$ and $gv=Fv$; and

$\left(vi\right)$$(F,g)$ is weakly compatible,

then F and g have a unique fixed point.

Proof. 

By Remark 3, if $\psi \left(u\right)=ku$, where $k\in (0,1)$, it is clear that the result is true. □

Remark 4.

Let $\Re ={\Omega }^2$ and $M_{F,g}(u,v)=d_{\xi }(gu,gv)$ in Corollary 2, we can find the results of Hassen et al. [20].

Corollary 3.

Let $(\Omega ,d_{\xi })$ be an extended rectangular b-metric space with a binary relation ℜ. F is a self-mappings on Ω, which satisfies

$$d_{\xi }(Fu,Fv)⩽kM(u,v),forallu\Re v,$$

where $k\in (0,1)$ and

$$\begin{array}{cc}\hfill M(u,v)& =max\{d_{\xi }(u,v),d_{\xi }(u,Fu),d_{\xi }(Fv,v),\hfill \\ \hfill & \frac{d_{\xi }(v,Fv)(1+d_{\xi }(u,Fu))}{1+d_{\xi }(u,v)},\frac{d_{\xi }(u,Fu)(1+d_{\xi }(v,Fv))}{1+d_{\xi }(u,v)}\}.\hfill \end{array}$$

In addition, if F satisfies the following conditions:

$\left(i\right)$ there exists $v_0\in \Omega$ such that $v_0\Re F{v}_0$ and $v_0\Re F^2{v}_0$;

$\left(ii\right)$ ℜ is F-closed;

$\left(iii\right)$ for $v_0$ in $\left(i\right)$, we have ${\displaystyle \underset{n\to \infty }{lim\; sup}}\xi (v_{n+1},v_p)<\frac{1}{k}$, where $p\in \mathbb{N}$, $v_{n+1}=F{v}_n$;

$\left(iv\right)$$\left(a\right)$ there exists $▵$ such that $F(\Omega )\subseteq ▵$ and $(▵,d_{\xi })$ is ℜ-complete; and

       $\left(b\right)$ F is ℜ-continuous or ${\Re |}_{\Omega }$ is $d_{\xi }$-self-closed and $d_{\xi }(w,Fw)>0$, where $w\in \Omega$, such that $\xi (Fw,w)<\frac{1}{k}$ or $\xi (w,Fw)<\frac{1}{k};$ and

$\left(v\right)$ if $u\Re v$ or $v\Re u$, for all u, v with $u=Fu$ and $v=Fv$,

then F has a unique fixed point.

Proof. 

By Corollary 2, let $g=I$, it is clear that the result is true. □

Example 3.

Let $\Omega =[1,4]$ with $\Re ={[1,2]}^2$, $d_{\xi }(u,v)=|u-v|$ with $\xi (u,v)=u+v+1$ for all $u,v\in \Omega$. Clearly, $(\Omega ,d_{\xi })$ be an ℜ-complete extended rectangular b-metric space. Consider that the mapping $F:\Omega \to \Omega$ is defined by

$$F\left(u\right)=\left\{\begin{array}{c}\frac{7}{4},ifu\in [1,2];\hfill \\ \frac{u}{10},otherwise.\hfill \end{array}\right.$$

Then, for all $u\Re v$, we obtain $u,v\in {[1,2]}^2$, then $Fu=Fv=\frac{7}{4}\in [1,2]$, that is $Fu\Re Fv$. Since ℜ is F-closed. for any sequence $\left\{u_n\right\}\subseteq \Omega$ such that $\left\{u_n\right\}$ is ℜ-preserving with ${\displaystyle \underset{n\to \infty }{lim}}{u}_n=u$, we obtain that $u_n\in [1,2]$, for all $n\in \mathbb{N}$ and $u\in [1,2]$, then $Fu=F{u}_n=\frac{7}{4}$, for all $n\in \mathbb{N}$, that is, ${\Re |}_{\Omega }$ is $d_{\xi }$-self-closed. Moreover, there exists $v_0=\frac{7}{4}$ such that $v_0\Re F{v}_0$ and $v_0\Re F^2{v}_0$. Clearly,

$$d_{\xi }(Fu,Fv)⩽\frac{1}{10}\left(M(u,v)\right),forallu\Re v,$$

and for all $w\in \Omega$ with $d_{\xi }(w,Fw)>0$ satisfies $\xi (Fw,w)<10$ and ${\displaystyle \underset{n\to \infty }{lim\; sup}}\xi (v_{n+1},v_p)<10.$ Thus, by Corollary 3, $\frac{7}{4}$ is the unique fixed point of F.

Corollary 4.

Let $(\Omega ,d_{\xi })$ be an extended rectangular b-metric space. F is a self-mapping on Ω, which satisfies

$$d_{\xi }(Fu,Fv)⩽kM(u,v),forallu,v\in \Omega ,$$

where $k\in [0,1)$ and

$$\begin{array}{cc}\hfill M(u,v)& =max\{d_{\xi }(u,v),d_{\xi }(u,Fu),d_{\xi }(Fv,v),\hfill \\ \hfill & \frac{d_{\xi }(v,Fv)(1+d_{\xi }(u,Fu))}{1+d_{\xi }(u,v)},\frac{d_{\xi }(u,Fu)(1+d_{\xi }(v,Fv))}{1+d_{\xi }(u,v)}\}.\hfill \end{array}$$

In addition, if F satisfies the following conditions:

$\left(i\right)$ there exists $v_0\in \Omega$ such that ${\displaystyle \underset{n\to \infty }{lim\; sup}}\xi (v_{n+1},v_p)<\frac{1}{k}$, where $p\in \mathbb{N}$ and $v_{n+1}=F{v}_n$;

$\left(ii\right)$ there exists $▵\subseteq \Omega$ such that $F(\Omega )\subseteq ▵$ and $(▵,d_{\xi })$ is complete; and

$\left(iii\right)$ one of the conditions holds:

        $\left(a\right)$ F is continuous; or

        $\left(b\right)$ for all $w\in \Omega$ with $d_{\xi }(w,Fw)>0$ such that $k<\frac{1}{\xi (Fw,w)}$ or $k<\frac{1}{\xi (w,Fw)},$

then F has a unique fixed point.

Proof. 

Let $\Re ={\Omega }^2$, by Corollary 3, the proof is complete. □

Corollary 5.

Let $(\Omega ,d_{\xi })$ be an extended rectangular b-metric space with a binary relation ℜ. Assume that F is a self-mapping on Ω, which satisfies

$$\begin{array}{cc}\hfill d_{\xi }(Fu,Fv)& ⩽a_1{d}_{\xi }(u,v)+a_2{d}_{\xi }(u,Fu)+a_3{d}_{\xi }(Fv,v)+\hfill \\ \hfill & a_4\frac{d_{\xi }(v,Fv)(1+d_{\xi }(u,Fu))}{1+d_{\xi }(u,v)}+a_5\frac{d_{\xi }(u,Fu)(1+d_{\xi }(v,Fv))}{1+d_{\xi }(u,v)},forallu\Re v,\hfill \end{array}$$

where ${\displaystyle \sum _{i=1}^5}{a}_i\in (0,1).$ In addition, if F satisfies the following conditions:

$\left(i\right)$ there exists $v_0\in \Omega$ such that $v_0\Re F{v}_0$ and $v_0\Re F^2{v}_0$;

$\left(ii\right)$ ℜ is F-closed;

$\left(iii\right)$ for $v_0$ in $\left(i\right)$, we have ${\displaystyle \underset{n\to \infty }{lim\; sup}}\xi (v_{n+1},v_p)<\frac{1}{{\displaystyle \sum _{i=1}^5}{a}_i}$, where $p\in \mathbb{N}$ and $v_{n+1}=F{v}_n$;

$\left(iv\right)$ there exists $▵\subseteq \Omega$ such that $F(\Omega )\subseteq ▵$ and $(▵,d_{\xi })$ is ℜ-complete;

$\left(v\right)$ one of the conditions holds:

        $\left(a\right)$ F is ℜ-continuous; or

        $\left(b\right)$ ${\Re |}_{\Omega }$ is $d_{\xi }$-self-closed and for all $w\in \Omega$ with $d_{\xi }(w,Fw)>0$ such that

$$\sum _{i=1}^5{a}_i<\frac{1}{\xi (Fw,w)}or\sum _{i=1}^5{a}_i<\frac{1}{\xi (w,Fw)};$$

and

$\left(vi\right)$ if $u\Re v$ or $v\Re u$, for all u, v with $Fu=u$ and $Fv=v$,

then F has a unique fixed point.

Proof. 

For all $u\Re v$,

$$\begin{array}{cc}\hfill d_{\xi }(Fu,Fv)& ⩽a_1{d}_{\xi }(u,v)+a_2{d}_{\xi }(u,Fu)+a_3{d}_{\xi }(Fv,v)+\hfill \\ \hfill & a_4\frac{d_{\xi }(v,Fv)(1+d_{\xi }(u,Fu))}{1+d_{\xi }(u,v)}+a_5\frac{d_{\xi }(u,Fu)(1+d_{\xi }(v,Fv))}{1+d_{\xi }(u,v)}\hfill \\ \hfill & ⩽\sum _{i=1}^5{a}_{i}max\{d_{\xi }(u,v),d_{\xi }(u,Fu),d_{\xi }(Fv,v),\hfill \\ \hfill & \frac{d_{\xi }(v,Fv)(1+d_{\xi }(u,Fu))}{1+d_{\xi }(u,v)},\frac{d_{\xi }(u,Fu)(1+d_{\xi }(v,Fv))}{1+d_{\xi }(u,v)}\}\hfill \\ \hfill & =kM(u,v),\hfill \end{array}$$

where $k={\displaystyle \sum _{i=1}^5}{a}_i$. By Corollary 4, the proof is complete. □

Remark 5.$\left(i\right)$ In Corollary 5, take $a_i=0,i=2,3,4,5$, our results generalized the results of Alam et al. [18] to extended rectangular b-metric spaces.

$\left(ii\right)$ In Corollary 5, if $a_i=0,i=2,3,5$, then we develop the result of Hossain et al. [23] into extended rectangular b-metric space.

Corollary 6.

Let $(\Omega ,d_{\xi })$ be an extended rectangular b-metric space. F is a self-mapping on Ω, which satisfies

$$\begin{array}{cc}\hfill d_{\xi }(Fu,Fv)& ⩽a_1{d}_{\xi }(u,v)+a_2{d}_{\xi }(u,Fu)+a_3{d}_{\xi }(Fv,v)+\hfill \\ \hfill & a_4\frac{d_{\xi }(v,Fv)(1+d_{\xi }(u,Fu))}{1+d_{\xi }(u,v)}+a_5\frac{d_{\xi }(u,Fu)(1+d_{\xi }(v,Fv))}{1+d_{\xi }(u,v)},forallu,v\in \Omega ,\hfill \end{array}$$

where ${\displaystyle \sum _{i=1}^5}{a}_i\in (0,1)$. In addition, if F satisfies the following conditions:

$\left(i\right)$ there exists $v_0\in \Omega$ such that ${\displaystyle \underset{n\to \infty }{lim\; sup}}\xi (v_{n+1},v_p)<\frac{1}{{\displaystyle \sum _{i=1}^5}{a}_i}$, where $p\in \mathbb{N}$ and $v_{n+1}=F{v}_n$;

$\left(ii\right)$ there exists a set $▵\subseteq \Omega$ such that $F(\Omega )\subseteq ▵$ and $(▵,d_{\xi })$ is complete; and

$\left(iii\right)$ one of the conditions holds:

        $\left(a\right)$ F is continuous; or

        $\left(b\right)$ for all $w\in \Omega$ with $d_{\xi }(w,Fw)>0$ such that ${\displaystyle \sum _{i=1}^5}{a}_i<\frac{1}{\xi (Fw,w)}$ or ${\displaystyle \sum _{i=1}^5}{a}_i<\frac{1}{\xi (w,Fw)},$

then F has a unique fixed point.

Proof. 

Let $\Re ={\Omega }^2$, by Corollary 5, the proof is complete. □

Remark 6.$\left(i\right)$ In Corollary 6, if $a_i=0,i=2,3,4,5$, we can obtain the Banach type fixed point theorem.

$\left(ii\right)$ In Corollary 6, if $a_i=0,i=1,4,5$, we can find the Kannan type fixed point theorem.

$\left(iii\right)$ In Corollary 6, if $a_i=0,i=2,3,5$, we can develop the result of Dass et al. [22] into extended rectangular b-metric space.

4. Applications

4.1. Application to Ordinary Differential Equations with Periodic Boundary Value

In this section, we apply our results to show the existence of solutions to the following ordinary differential equations with periodic boundary value.

$$\begin{array}{c}\hfill \left\{\begin{array}{ccc}& u^{\prime }\left(t\right)=f(t,u\left(t\right)),\hfill & \hfill t\in [0,T],\\ & u\left(0\right)=u\left(T\right),\hfill \end{array}\right.\end{array}$$

(39)

where $T\in (0,\infty )$ is a constant, $u\left(t\right):[0,T]\to \mathbb{R}$ and $f:[0,T]\times \mathbb{R}\to \mathbb{R}$ is continuous. It is clear that the solution of (39) is equivalent to the following integral equation

$$u\left(t\right)={\int }_0^{T}G(t,s)[f(s,u\left(s\right))+\lambda u\left(s\right)]ds,t\in [0,T],$$

(40)

where $\lambda >0$ and

$$\begin{array}{c}\hfill G(t,s)=\left\{\begin{array}{ccc}& \frac{e^{\lambda (T+s-t)}}{e^{\lambda T}-1},\hfill & \hfill 0⩽s<t⩽T,\\ & \frac{e^{\lambda (s-t)}}{e^{\lambda T}-1},\hfill & \hfill 0⩽t<s⩽T.\end{array}\right.\end{array}$$

Let $C\left(\right[0,T],\mathbb{R})$ be the set of all continuous real value functions defined on $[0,T]$. For all $u,v\in C\left(\right[0,T],\mathbb{R})$, we define two functions $\xi (u,v)$, $d_{\xi }(u,v)$ and a mapping F by $\xi (u,v)=\left|u\right|+\left|v\right|+4$,

$$d_{\xi }(u,v)=\underset{t\in [0,T]}{max}{|u\left(t\right)-v\left(t\right)|}^2,$$

and

$$F\left(u\left(t\right)\right)={\int }_0^{T}G(t,s)[f(s,u\left(s\right))+\lambda u\left(s\right)]ds,t\in [0,T].$$

(41)

Clearly, $(d_{\xi },C([0,T],\mathbb{R}))$ is a complete extended rectangular b-metric space and F is continuous.

Theorem 6.

If the following conditions hold,

$\left(i\right)$ there exist $\lambda ,\mu >0$ with $\mu <{\lambda }^2$ and $\psi \in \Psi$ such that

$$0⩽f(t,u)+\lambda u-\left[f\right(t,v)+\lambda v],forallu⩽v$$

and

$${|f(t,u)+\lambda u-[f(t,v)+\lambda v]|}^2⩽\mu \psi {(\underset{t\in [0,T]}{max}|u\left(t\right)-v\left(t\right)|}^2),forallu⩽v;$$

$\left(ii\right)$ (39) has a lower solution, that is, there exists $u_0\left(t\right)\in$ $C\left(\right[0,T],\mathbb{R})$ such that

$$\begin{array}{c}\hfill \left\{\begin{array}{ccc}& u_0^{\prime }\left(t\right)⩽f(t,u_0\left(t\right)),\hfill & \hfill t\in [0,T],\\ & u_0\left(0\right)⩽u_0\left(T\right);and\hfill \end{array}\right.\end{array}$$

$\left(iii\right)$ for $u_0$ in $\left(ii\right)$, we have ${\displaystyle \underset{n\to \infty }{lim\; sup}}\frac{{\psi }^{n+1}\left(t\right)}{{\psi }^n\left(t\right)}\xi (u_{n+1},u_p)<1$, where $p,n\in \mathbb{N}$, $u_{n+1}=F{u}_n$ and $t\in (0,d_{\xi }(u_0,u_1)]$ with $u_0\ne u_1$, then the ordinary differential equation with periodic boundary value (39) has a solution.

Proof. 

First, we define a binary relation ℜ by $u\Re v$ if and only if $u\left(t\right)⩽v\left(t\right)$, for all $t\in [0,T]$. Clearly, considering $\left(ii\right)$, we have $u_0\left(t\right)⩽F{u}_0\left(t\right)$. By $0⩽f(t,u)+\lambda u-\left[f\right(t,v)+\lambda v]$, for all $u⩽v$, $u_0\left(t\right)⩽F{u}_0\left(t\right)$ and the definition of F, we have $F\left(u_0\left(t\right)\right)⩽F^2\left(u_0\left(t\right)\right)$.

We can easily deduce that $u_0\left(t\right)⩽F{u}_0\left(t\right)$ and $u_0\left(t\right)⩽F^2\left(u_0\left(t\right)\right)$. By the definition of ℜ, there exists $u_0\left(t\right)$ such that $u_0\left(t\right)\Re F\left(u_0\left(t\right)\right)$ and $u_0\left(t\right)\Re F^2\left(u_0\left(t\right)\right)$. We can conclude that ℜ is F-closed via $0⩽f(t,u)+\lambda u-\left[f\right(t,v)+\lambda v]$, for all $u⩽v$, $u_0\left(t\right)⩽F{u}_0\left(t\right)$ and the definitions of F and ℜ. Now, we prove that F satisfies (37). Indeed, for all $u\Re v$, we have

$$\begin{array}{cc}\hfill {|Fu\left(t\right)-Fv\left(t\right)|}^2& ={|{\int }_0^{T}G(t,s)[f(s,u\left(s\right))+\lambda u\left(s\right)]ds-{\int }_0^{T}G(t,s)[f(s,v\left(s\right))+\lambda v\left(s\right)]ds|}^2\hfill \\ \hfill & ={|{\int }_0^{T}G(t,s)\{[f(s,u\left(s\right))+\lambda u\left(s\right)]-[f(s,v\left(s\right))+\lambda v\left(s\right)]\}ds|}^2\hfill \\ \hfill & ⩽\underset{t\in [0,T]}{max}{\left|\{[f(s,u\left(s\right))+\lambda u\left(s\right)]-[f(s,v\left(s\right))+\lambda v\left(s\right)]\}\right|}^2{|{\int }_0^{T}G(t,s)ds|}^2\hfill \\ \hfill & ⩽\mu \psi (\underset{t\in [0,T]}{max}{|u\left(t\right)-v\left(t\right)|}^2){|\underset{t\in [0,T]}{max}{\int }_0^{T}G(t,s)ds|}^2\hfill \\ \hfill & ⩽\mu \psi \left(M(u,v)\right){|\underset{t\in [0,T]}{max}{\int }_0^{T}G(t,s)ds|}^2\hfill \\ \hfill & =\mu \psi \left(M(u,v)\right)\times \frac{1}{{\lambda }^2}\hfill \\ \hfill & =\frac{\mu }{{\lambda }^2}\psi \left(M(u,v)\right),\hfill \end{array}$$

that is, $d_{\xi }(Fu,Fv)⩽\frac{\mu }{{\lambda }^2}\psi \left(M(u,v)\right)$. Therefore, all conditions of Corollary 1 are satisfied, and thus (39) has a solution. □

4.2. Application to Linear Matrix Equations

In this section, of the paper, Corollary 3 is used to prove the existence of solutions to a class of linear matrix equations. For convenience, we first give the following notations:

We denote $M^m$ is the set of all complex number matrices of order m, $H^m$ is the set of all Hermitian matrices of order m, $P^m$ and $H_{+}^m$ represent the set of all $m\times m$ positive matrices and $m\times m$ positive semi-definite matrices, respectively. Clearly, $P^m\subseteq H^m\subseteq M^m$, $H_{+}^m\subseteq H^m$. Here, $A_1\succ O$ (O represents null matrix of same order) and $A_1⪰O$ mean that $A_1\in P^m$ and $A_1\in H_{+}^m$, respectively; for $A_1-A_2⪰O$ and $A_1-A_2\succ O$, we will use $A_1⪰A_2$ and $A_1\succ A_2$, respectively.

In the section, we investigate the existence of the solution to the following linear matrix equations:

$$U=G+\sum _{i=1}^m{A}_i^{*}U{A}_i+\sum _{i=1}^m{B}_i^{*}U{B}_i,$$

(42)

where $G\in P^m$, $A_i$, $B_i$ are arbitrary $m\times m$ matrices for each i. We use the metric $d(A,B)={\parallel A-B\parallel }_{tr,X}={\parallel X^{\frac{1}{2}}(A-B)X^{\frac{1}{2}}\parallel }_{tr}$, which is induced by the norm ${\parallel A\parallel }_{tr}={\displaystyle \sum _{i=1}^n}{\sigma }_i\left(A\right)$, where $X\in P^m$, $A,B\in H^m$ and ${\sigma }_i\left(A\right)$, $i=1,2,3,\cdots ,n,$ are the singular values of $A\in M^m.$ Clearly, the set $H^m$ equipped with the metric d is a complete metric space, then $(H^m,d)$ is a complete extended rectangular b-metric space with respect to $\xi =3$.

Define ℜ and mapping $F:H^m\to H^m$ by $A\Re B$ iff $B-A\in H^m$ and

$$F\left(U\right)=G+\sum _{i=1}^m{A}_i^{*}U{A}_i+\sum _{i=1}^m{B}_i^{*}U{B}_i,\mathrm{for}\mathrm{all}A,B\in H^m.$$

Note that the solutions of the matrix Equation (42) are the fixed point of the mapping F, furthermore, the mapping F is continuous in $H^m$, ℜ is F-closed and there exists $U_0$ such that $U_0\Re F\left(U_0\right)$ and $U_0\Re F^2\left(U_0\right)$.

To establish the existence result, we introduce the following Lemmas.

Lemma 2

([24]). If $A,B\in H_{+}^m$, then $0⩽tr\left(AB\right)⩽\parallel A\parallel tr\left(B\right).$

Lemma 3

([24]). If $A\in H^m$ such that $A\prec I_n$, then $\parallel A\parallel <1$.

Theorem 7.

If $X\in P^m$, ${\displaystyle \sum _{i=1}^m}{A}_i^{*}X{A}_i\prec \frac{1}{7}X$ and ${\displaystyle \sum _{i=1}^m}{B}_i^{*}X{B}_i\prec \frac{1}{7}X$, then the mapping F has a fixed point in $H^m$.

Proof. 

Suppose that $U,V\in H^m$ and $U\Re V$. Consider

$$\begin{array}{cc}\hfill {\parallel F\left(U\right)-F\left(V\right)\parallel }_{tr,X}& =tr\left(X^{\frac{1}{2}}(F\left(U\right)-F\left(V\right))X^{\frac{1}{2}}\right)\hfill \\ \hfill & =tr\left(\sum _{i=1}^m\left\{X^{\frac{1}{2}}(A_i^{*}(U-V)A_i+B_i^{*}(U-V)B_i{X}^{\frac{1}{2}})\right\}\right)\hfill \\ \hfill & =tr(\sum _{i=1}^m{X}^{\frac{1}{2}}{A}_i^{*}(U-V)A_i{X}^{\frac{1}{2}}+\sum _{i=1}^m{X}^{\frac{1}{2}}{B}_i^{*}(U-V)B_i{X}^{\frac{1}{2}})\hfill \\ \hfill & =\sum _{i=1}^{m}tr(X^{\frac{1}{2}}{A}_i^{*}(U-V)A_i{X}^{\frac{1}{2}}+X^{\frac{1}{2}}{B}_i^{*}(U-V)B_i{X}^{\frac{1}{2}})\hfill \\ \hfill & =\sum _{i=1}^{m}tr\left(X^{\frac{1}{2}}{A}_i^{*}(U-V)A_i{X}^{\frac{1}{2}}\right)+\sum _{i=1}^{m}tr\left(X^{\frac{1}{2}}{B}_i^{*}(U-V)B_i{X}^{\frac{1}{2}}\right)\hfill \\ \hfill & =\sum _{i=1}^{m}tr\left(A_i^{*}X{A}_i(U-V)\right)+\sum _{i=1}^{m}tr\left(B_i^{*}X{B}_i(U-V)\right)\hfill \\ \hfill & =\sum _{i=1}^{m}tr\left(A_i^{*}X{A}_i{X}^{-\frac{1}{2}}{X}^{\frac{1}{2}}(U-V)X^{-\frac{1}{2}}{X}^{\frac{1}{2}}\right)+\sum _{i=1}^{m}tr\left(B_i^{*}X{B}_i{X}^{-\frac{1}{2}}{X}^{\frac{1}{2}}(U-V)X^{-\frac{1}{2}}{X}^{\frac{1}{2}}\right)\hfill \\ \hfill & =\sum _{i=1}^{m}tr\left(X^{-\frac{1}{2}}{A}_i^{*}X{A}_i{X}^{-\frac{1}{2}}{X}^{\frac{1}{2}}(U-V)X^{\frac{1}{2}}\right)+\sum _{i=1}^{m}tr\left(X^{-\frac{1}{2}}{B}_i^{*}X{B}_i{X}^{-\frac{1}{2}}{X}^{\frac{1}{2}}(U-V)X^{\frac{1}{2}}\right)\hfill \\ \hfill & =tr\left(\sum _{i=1}^m{X}^{-\frac{1}{2}}{A}_i^{*}X{A}_i{X}^{-\frac{1}{2}}{X}^{\frac{1}{2}}(U-V)X^{\frac{1}{2}}\right)+tr\left(\sum _{i=1}^m{X}^{-\frac{1}{2}}{B}_i^{*}X{B}_i{X}^{-\frac{1}{2}}{X}^{\frac{1}{2}}(U-V)X^{\frac{1}{2}}\right)\hfill \\ \hfill & ⩽\parallel \sum _{i=1}^m{X}^{-\frac{1}{2}}{A}_i^{*}X{A}_i{X}^{-\frac{1}{2}}{\parallel \parallel (U-V)\parallel }_{tr,X}+\parallel \sum _{i=1}^m{X}^{-\frac{1}{2}}{B}_i^{*}X{B}_i{X}^{-\frac{1}{2}}{\parallel \parallel (U-V)\parallel }_{tr,X}\hfill \\ \hfill & =(\parallel \sum _{i=1}^m{X}^{-\frac{1}{2}}{A}_i^{*}X{A}_i{X}^{-\frac{1}{2}}\parallel +\parallel \sum _{i=1}^m{X}^{-\frac{1}{2}}{B}_i^{*}X{B}_i{X}^{-\frac{1}{2}}{\parallel )\parallel (U-V)\parallel }_{tr,X}\hfill \\ \hfill & ={k\parallel (U-V)\parallel }_{tr,X}\hfill \\ \hfill & ⩽kM(U,V),\hfill \end{array}$$

where $k=\parallel {\displaystyle \sum _{i=1}^m}{X}^{-\frac{1}{2}}{A}_i^{*}X{A}_i{X}^{-\frac{1}{2}}\parallel +\parallel {\displaystyle \sum _{i=1}^m}{X}^{-\frac{1}{2}}{B}_i^{*}X{B}_i{X}^{-\frac{1}{2}}\parallel$ and

$$\begin{array}{cc}\hfill M(U,V)& =max\{\parallel (U-V){\parallel }_{tr,X},\parallel (U-F\left(U\right)){\parallel }_{tr,X},\parallel (V-F\left(V\right)){\parallel }_{tr,X},\hfill \\ \hfill & \frac{{\parallel (V-F\left(V\right))\parallel }_{tr,X}{(1+\parallel (U--F\left(U\right))\parallel }_{tr,X})}{1+{\parallel (U-V)\parallel }_{tr,X}},\frac{{\parallel (U-F\left(U\right))\parallel }_{tr,X}{(1+\parallel (V-F\left(V\right))\parallel }_{tr,X})}{1+{\parallel (U-V)\parallel }_{tr,X}}\}.\hfill \end{array}$$

By Lemma 3, we have $k<\frac{2}{7}$. Mapping F and ℜ satisfy the conditions of the Corollary 3; therefore, F has a fixed point, and linear matrix Equation (42) has a solution. □

5. Conclusions

In this paper, some new relation-theoretic coincidence and common fixed point results of for some mappings of F and g are obtained by using hybrid contractions and auxiliary functions in extended rectangular b-metric space. We improve and expand some recent results. Furthermore, we use instances and applications to justify the results. Finally, regarding the main results of this paper, we draw some corollaries. Due to the importance of the fixed point theory, we consider possible future research directions.

These are potential works in the future:

(i)

replace or weak some conditions in our main theorems;

(ii)

extend our results to another metric spaces; and

(iii)

use our contraction to study the problem of fixed-circle and fixed-disc [25,26,27,28,29] in different generalized metric spaces.