(ψ,ϕ)-Contractions under a Class of Transitive Binary Relations

Abstract

The present article is devoted to prove some fixed point results for $(\psi ,\varphi )$-contractions in the framework of metric space equipped with a locally finitely $\mathcal{G}$-transitive relation. The results proved in this article improve and weaken some existing fixed point results available in the literature. Finally, an example is provided for attesting to the credibility of my results.

1. Introduction

The classical Banach contraction principle (abbreviated as: BCP) established by Banach [1] was widely accepted due to its simplicity as well as applicability. Ran and Reurings [2] and Nieto and Rodríguez-López [3] extended and improved BCP in the framework of partially ordered metric spaces. Later, Alam and Imdad [4] investigated a novel generalization of BCP utilizing an arbitrary relation instead of partial order.

On the other hand, several diversified extensions of BCP are obtained by employing more general contractivity conditions. A self-map $\mathcal{G}$ on a metric space $(\mathbb{W},\varrho )$ is referred to as a linear contraction if there exists $k\in [0,1)$ satisfying

$$\varrho (\mathcal{G}r,\mathcal{G}t)\le k\varrho (r,t)\mathrm{for} \mathrm{all} r, t \in \mathbb{W}.$$

(1)

In inequality (1), the constant k plays a crucial role. Several researchers replaced the constant k by a test function (say, $\varphi$) depending upon the contractivity conditions. Any map $\varphi :[0,\infty )\to [0,\infty )$ satisfying $\varphi \left(s\right)<s$ for each $s>0$ is referred as a control function. Any self-map $\mathcal{G}$ on a metric space $(\mathbb{W},\varrho )$ is known as $\varphi -\mathrm{contraction}$ if it satisfies

$$\varrho (\mathcal{G}r,\mathcal{G}t)\le \varphi \left(\varrho \right(r,t\left)\right)\mathrm{for} \mathrm{all} r, t \in \mathbb{W}.$$

Indeed, the notion of $\varphi$-contraction was introduced by Browder [5] in 1968 and was later generalized by Boyd and Wong [6] and Mukherjea [7] and Jotic [8].

Following Khan et al. [9], a map $\psi :[0,\infty )\to [0,\infty )$ is referred to as an altering distance map if it satisfies the following axioms:

(i)

$\psi \left(s\right)=0$ iff $s=0$,

(ii)

$\psi$ remains increasing and continuous.

Dutta and Choudhury [10] obtained a novel generalization of BCP employing a pair of auxiliary functions.

Theorem 1

([10]). Assume that $(\mathbb{W},\varrho )$ remains a complete metric space and $\mathcal{G}$ remains a map from $\mathbb{W}$ to itself. If ϕ and ψ remain two altering distance functions satisfying

$$\psi \left(\varrho \right(\mathcal{G}r,\mathcal{G}t\left)\right)\le \psi \left(\varrho \right(r,t\left)\right)-\varphi \left(\varrho \right(r,t\left)\right)$$

for all $r,t\in \mathbb{W}$, then $\mathcal{G}$ admits a unique fixed point.

Later on, various authors improved Theorem 1, e.g., Doric [11], Popescu [12], Luong and Thuan [13], Alam et al. [14] and similar others. Shahzad et al. [15] slightly modified the $(\psi ,\varphi )$-contractions by introducing the following one:

$$\psi \left(\varrho \right(\mathcal{G}r,\mathcal{G}t\left)\right)\le \varphi \left(\varrho \right(r,t\left)\right).$$

For a recent development of metric fixed point theory, refer to [16,17,18] and the reference therein. The intent of the present article is to establish fixed point results for $(\psi ,\varphi )$-contractions in the sense of Shahzad et al. [15] involving a locally finitely $\mathcal{G}$-transitive relation. An example is also presented, which attests to the credibility of my results.

2. Preliminaries

This section involves some relevant notions and auxiliary results needed in main results. $\mathbb{N}$ and ${\mathbb{N}}_0$ denote, respectively, the set of natural numbers and that of whole numbers. Any subset $\mathfrak{R}$ of ${\mathbb{W}}^2$ is referred to as a binary relation (or simply, a relation) on $\mathbb{W}$. Throughout this section, we assume that $\mathbb{W}$ remains a set, $\mathfrak{R}$ remains a relation on $\mathbb{W}$ and $\mathcal{G}:\mathbb{W}\to \mathbb{W}$ remains a map.

Definition 1

([4]). Two elements $r,t\in \mathbb{W}$ are termed as $\mathfrak{R}$-comparative (often denoted by $[r,t]\in \mathfrak{R}$) if either $(r,t)\in \mathfrak{R}$ or $(t,r)\in \mathfrak{R}$.

Definition 2

([19]). $\mathfrak{R}$ is termed as a complete relation if $[r,t]\in \mathfrak{R}$, for every $r,t\in \mathbb{W}$.

Definition 3

([19]). ${\mathfrak{R}}^{-1}:=\{(r,t)\in {\mathbb{W}}^2:(t,r)\in \mathfrak{R}\}$ is termed as inverse (transpose) of $\mathfrak{R}$.

Definition 4

([19]). The relation ${\mathfrak{R}}^s:=\mathfrak{R}\cup {\mathfrak{R}}^{-1}$ is termed as symmetric closure of $\mathfrak{R}$.

Proposition 1

([4]). $(r,t)\in {\mathfrak{R}}^s⟺[r,t]\in \mathfrak{R}.$

Definition 5

([19]). Given $\mathbb{Z}\subseteq \mathbb{W}$, the restriction of $\mathfrak{R}$ on $\mathbb{Z}$ is defined as

$${\mathfrak{R}|}_{\mathbb{Z}}:=\mathfrak{R}\cap {\mathbb{Z}}^2.$$

Clearly, ${\mathfrak{R}|}_{\mathbb{Z}}$ is a relation on $\mathbb{Z}$.

Definition 6

([4]). $\mathfrak{R}$ is known as $\mathcal{G}$-closed if for every $r,t\in \mathbb{W}$ with $(r,t)\in \mathfrak{R}$, we have

$$(\mathcal{G}r,\mathcal{G}t)\in \mathfrak{R}.$$

Example 1

([20]). Consider the set $\mathbb{W}=(0,1]$ equipped with a relation $\mathfrak{R}=\{(r,t):\frac{1}{4}\le r\le t\le \frac{1}{3}\mathrm{or}\frac{1}{2}\le r\le t\le 1\}$. Moreover, assume that $\mathcal{G}:\mathbb{W}\to \mathbb{W}$ is a map defined by

$$\mathcal{G}\left(r\right)=\left\{\begin{array}{c}\frac{1}{4}\mathrm{if}0\le r<\frac{1}{2}\hfill \\ 1\mathrm{if}\frac{1}{2}\le r\le 1.\hfill \end{array}\right.$$

Then, it remains to easily check that $\mathcal{R}$ is $\mathcal{G}$-closed.

Proposition 2

([21]). For each $n\in {\mathbb{N}}_0$, $\mathfrak{R}$ is ${\mathcal{G}}^n$-closed whenever $\mathfrak{R}$ is $\mathcal{G}$-closed.

Definition 7

([4]). Any sequence $\left\{r_n\right\}\subset \mathbb{W}$ satisfying $(r_n,r_{n+1})\in \mathfrak{R}$, $\forall n\in {\mathbb{N}}_0$ is known as $\mathfrak{R}$-preserving.

Definition 8

([22]). $(\mathbb{W},\varrho )$ is known as $\mathfrak{R}$-complete if each $\mathfrak{R}$-preserving Cauchy sequence in $\mathbb{W}$ converges.

It is clear that any complete metric space remains an $\mathfrak{R}$-complete whatever $\mathfrak{R}$. Moreover, in case $\mathfrak{R}={\mathbb{W}}^2$, these two concepts coincide.

Example 2

([20]). Consider $\mathbb{W}=(0,1]$ equipped with standard metric ϱ. On $\mathbb{W}$, endow a relation $\mathfrak{R}=\{(r,t):\frac{1}{4}\le r\le t\le \frac{1}{3}\mathrm{or}\frac{1}{2}\le r\le t\le 1\}$. Then, $(\mathbb{W},\varrho )$ remains $\mathfrak{R}$-complete, however, it is not complete.

Definition 9

([22]). $\mathcal{G}$ is known as $\mathfrak{R}$-continuous at $r\in \mathbb{W}$ if for every $\mathfrak{R}$-preserving sequence $\left\{r_n\right\}$ of elements of $\mathbb{W}$ satisfying $r_n\stackrel{\varrho }{⟶}r$, we have

$$\mathcal{G}\left(r_n\right)\stackrel{\varrho }{⟶}\mathcal{G}\left(r\right).$$

Moreover, if $\mathcal{G}$ remains $\mathfrak{R}$-continuous at each point of $\mathbb{W}$, then it is known as $\mathfrak{R}$-continuous.

It is clear that any continuous map remains an $\mathfrak{R}$-continuous whatever $\mathfrak{R}$. Moreover, in case $\mathfrak{R}={\mathbb{W}}^2$, these two concepts coincide.

Example 3

([20]). Consider $\mathbb{W}=(0,1]$ equipped with standard metric ϱ. On $\mathbb{W}$, endow a relation $\mathfrak{R}=\{(r,t):\frac{1}{4}\le r\le t\le \frac{1}{3}\mathrm{or}\frac{1}{2}\le r\le t\le 1\}$. If $\mathcal{G}$ remains a map from $\mathbb{W}$ to $\mathbb{W}$ given by

$$\mathcal{G}\left(r\right)=\left\{\begin{array}{c}\frac{1}{4}\mathrm{if}0\le r<\frac{1}{2}\hfill \\ 1\mathrm{if}\frac{1}{2}\le r\le 1.\hfill \end{array}\right..$$

Then, $\mathcal{G}$ remains $\mathfrak{R}$-continuous, however, it is not continuous.

Definition 10

([4]). $\mathfrak{R}$ is known as a ϱ-self-closed relation if every $\mathfrak{R}$-preserving sequence $\left\{r_n\right\}$ (of elements of $\mathbb{W}$) satisfying $r_n\stackrel{\varrho }{⟶}r$ (for some $r\in \mathbb{W}$) admits a subsequence $\left\{r_{n_{\alpha }}\right\}$ satisfying $[r_{n_{\alpha }},r]\in \mathfrak{R}$, for all $\alpha \in {\mathbb{N}}_0.$

Definition 11

([23]). Any subset $\mathbb{Z}$ of $\mathbb{W}$ is termed as $\mathfrak{R}$-directed if for every pair $r,t\in \mathbb{Z}$, $\mathrm{\exists }$ satisfying $(r,s)\in \mathfrak{R}$ and $(t,s)\in \mathfrak{R}$.

Definition 12

([24]). A path of length k in a relation $\mathfrak{R}$ from r to t (whereas $r,t\in \mathbb{W}$) remains a finite sequence $\{s_0,s_1,\dots ,s_k\}$⊂$\mathbb{W}$ satisfying

(i)

$s_0=r$ and $s_k=t$,

(ii)

$(s_i,s_{i+1})$∈$\mathfrak{R},0\le i\le k-1$.

Definition 13

([21]). A subset $\mathbb{Z}$ of $\mathbb{W}$ is termed as an $\mathfrak{R}$-connected set if each pair of elements of $\mathbb{Z}$ admits a path between them.

Definition 14

([19]). $\mathfrak{R}$ is termed as transitive if $(r,s)\in \mathfrak{R}$ and $(s,t)\in \mathfrak{R}$ implies $(r,t)\in \mathfrak{R}$.

Definition 15

([21,25]). $\mathfrak{R}$ is termed as $\mathcal{G}$-transitive if for every triplets $r,s,t\in \mathbb{W}$ satisfying $(\mathcal{G}r,\mathcal{G}s),(\mathcal{G}s,\mathcal{G}t)\in \mathfrak{R}$, we have

$$(\mathcal{G}r,\mathcal{G}t)\in \mathfrak{R}.$$

Definition 16

([26]). Given $m\in {\mathbb{N}}_0$, $m\ge 2$, a relation $\mathfrak{R}$ is termed as m-transitive if for any $r_0,r_1,\dots ,r_m\in \mathbb{W}$ satisfying $(r_{j-1},r_j)\in \mathfrak{R}$, for $1\le j\le m,$ we have

$$(r_0,r_m)\in \mathfrak{R}.$$

Thus, a 2-transitive relation means the transitive relation. $\mathfrak{R}$ is known as a finitely transitive relation if it remains m-transitive for some $m\ge 2$ (cf. [27]).

Definition 17

([27]). $\mathfrak{R}$ is termed as locally finitely transitive if for every countably infinite subset $\mathbb{Z}$ of $\mathbb{W}$, $\exists m=m\left(\mathbb{Z}\right)\ge 2$, such that ${\mathfrak{R}|}_{\mathbb{Z}}$ remains m-transitive.

To make the two independent concepts ($\mathcal{G}$-transitivity and locally finitely transitivity) compatible, Alam et al. [28] initiated yet a new concept of transitivity as follows:

Definition 18

([28]). $\mathfrak{R}$ is termed as locally finitely $\mathcal{G}$-transitive if for every every countably infinite subset of $\mathbb{Z}$ of $\mathcal{G}\left(\mathbb{W}\right)$, $\exists m=m\left(\mathbb{Z}\right)\ge 2$, such that ${\mathfrak{R}|}_{\mathbb{Z}}$ remains m-transitive.

In lieu of above the definitions, it is clear that the class of locally $\mathcal{G}$-transitive binary relation includes the classes of other types of transitive binary relations.

The statement of the relation-theoretic analogue of BCP established by Alam and Imdad [4] is given as:

Theorem 2

([4,20,22]). Assume that $(\mathbb{W},\varrho )$ remains a metric space, $\mathfrak{R}$ remains a relation on $\mathbb{W}$ and $\mathcal{G}:\mathbb{W}\to \mathbb{W}$ remains a map. Moreover,

(i)

$(\mathbb{W},\varrho )$remains$\mathfrak{R}$-complete;

(ii)

$\mathfrak{R}$remains$\mathcal{G}$-closed;

(iii)

either$\mathcal{G}$remains$\mathfrak{R}$-continuous or$\mathfrak{R}$remains ϱ-self-closed;

(iv)

$\exists r_0\in \mathbb{W}$such that$(r_0,\mathcal{G}{r}_0)\in \mathfrak{R}$;

(v)

$\exists k\in [0,1)$such that

$$\varrho (\mathcal{G}r,\mathcal{G}t)\le k\varrho (r,t),\forall r,t\in \mathbb{W}\mathrm{with}(r,t)\in \mathfrak{R}.$$

Then,$\mathcal{G}$admits a fixed point. Moreover, if$\mathcal{G}\left(\mathbb{W}\right)$remains${\mathfrak{R}}^s$-connected, then$\mathcal{G}$admits a unique fixed point.

Finally, the following two known results are stated.

Lemma 1

([29]). If the sequence $\left\{r_n\right\}$, in a metric space $(\mathbb{W},\varrho )$, is not a Cauchy, then we are able to find an $ϵ>0$ and two subsequences $\left\{r_{n_{\alpha }}\right\}$ and $\left\{r_{p_{\alpha }}\right\}$ of $\left\{r_n\right\}$ satisfying

(i)

$\alpha \le p_{\alpha }<n_{\alpha }$, $\mathrm{\forall }\alpha \in \mathbb{N}$;

(ii)

$\varrho (r_{p_{\alpha }},r_{n_{\alpha }})\ge ϵ$;

(iii)

$\varrho (r_{p_{\alpha }},r_{s_{\alpha }})<ϵ,\forall$$s_{\alpha }\in \{p_{\alpha }+1,p_{\alpha }+2,\dots ,n_{\alpha }-2,n_{\alpha }-1\}$.

Moreover, if $\underset{n\to \infty }{lim}\varrho (r_n,r_{n+1})=0$, then

$$\underset{\alpha \to \infty }{lim}\varrho (r_{p_{\alpha }},r_{n_{\alpha }+s})=ϵ,\forall s\in {\mathbb{N}}_0.$$

Lemma 2

([27]). Assume that $\mathbb{W}$ is a set equipped with a relation $\mathfrak{R}$ and $\left\{z_n\right\}\subset \mathbb{W}$ remains $\mathfrak{R}$-preserving sequence. Furthermore, suppose that $\mathfrak{R}$ remains m-transitive on $\mathbb{Z}=\{z_n:n\in {\mathbb{N}}_0\}$, then

$$(z_n,z_{n+1+s(m-1)})\in \mathfrak{R},\forall n,s\in {\mathbb{N}}_0.$$

3. Main Results

In what follows, $\mathrm{\Sigma }$ denotes the class of all pair of auxiliary functions $(\psi ,\varphi )$, wherein $\psi ,\varphi :[0,\infty )\to [0,\infty )$ enjoy the following properties:

$\left({\mathrm{\Sigma }}_1\right)$

If the sequence $\left\{t_n\right\}\subset (0,\infty )$ verifies $\psi \left(t_{n+1}\right)\le \varphi \left(t_n\right)$, $\forall n\in \mathbb{N}$, then $t_n\to 0$.

$\left({\mathrm{\Sigma }}_2\right)$

Whenever two convergent sequences $\left\{s_n\right\},\left\{t_n\right\}\subset [0,\infty )$ have a common limit L such that $\psi \left(t_n\right)\le \varphi \left(s_n\right)$, $\forall n\in \mathbb{N}$ and $L<s_n$, then $L=0$.

The above family of pair of functions is suggested by Shahzad et al. [15].

Theorem 3.

Assume that $(\mathbb{W},\varrho )$ remains a metric space, $\mathfrak{R}$ remains a relation on $\mathbb{W}$ and $\mathcal{G}:\mathbb{W}\to \mathbb{W}$ remains a map. Moreover,

(a) 

$(\mathbb{W},\varrho )$ remains $\mathfrak{R}$-complete;

(b) 

$\mathfrak{R}$ remains $\mathcal{G}$-closed as well as locally finitely $\mathcal{G}$-transitive;

(c) 

$\mathcal{G}$ remains $\mathfrak{R}$-continuous;

(d) 

$\exists r_0\in \mathbb{W}$ such that $(r_0,\mathcal{G}{r}_0)\in \mathfrak{R}$;

(e) 

$\exists (\psi ,\varphi )\in \mathrm{\Sigma }$ such that

$$\psi \left(\varrho \right(\mathcal{G}r,\mathcal{G}t)\le \varphi (\varrho (r,t)),\forall r,t\in \mathbb{W} with (r,t)\in \mathfrak{R},$$

then $\mathcal{G}$ admits a fixed point.

Proof. 

By assumption $\left(d\right)$, choose $r_0\in \mathbb{W}(\mathcal{G},\mathfrak{R})$, then we have $(r_0,\mathcal{G}{r}_0)\in \mathfrak{R}$. Construct the sequence $\left\{r_n\right\}\subset \mathbb{W}$ such that

$$r_n={\mathcal{G}}^n\left(r_0\right)=\mathcal{G}\left(r_{n-1}\right),\forall n\in \mathbb{N}.$$

(2)

As $(r_0,\mathcal{G}{r}_0)\in \mathfrak{R}$, assumption $\left(b\right)$ and Proposition 2, we have

$$({\mathcal{G}}^n{r}_0,{\mathcal{G}}^{n+1}{r}_0)\in \mathfrak{R},$$

which, in lieu of (2), becomes

$$(r_n,r_{n+1})\in \mathfrak{R},\forall n\in {\mathbb{N}}_0.$$

(3)

This means that $\left\{r_n\right\}$ remains $\mathfrak{R}$-preserving.

If there exists $n_0\in {\mathbb{N}}_0$ such that $\varrho (r_{n_0},r_{n_0+1})=0$, then by (2) $r_{n_0}$ remains a fixed point of $\mathcal{G}$. Otherwise, in case ${\varrho }_n:=\varrho (r_n,r_{n+1})>0$ for all $n\in {\mathbb{N}}_0$, we use assumption $\left(e\right)$ to obtain

$$\begin{array}{c}\hfill \psi \left(\varrho (r_{n+1},r_{n+2})\right)=\psi \left(\varrho (\mathcal{G}{r}_n,\mathcal{G}{r}_{n+1})\right)\le \varphi \left(\varrho (r_n,r_{n+1})\right)\end{array}$$

so that

$$\psi \left({\varrho }_{n+1}\right)\le \varphi \left({\varrho }_n\right),n\in \mathbb{N}.$$

Making use of the axiom $\left({\mathrm{\Sigma }}_1\right)$, we obtain

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}{\varrho }_n=\underset{n\to \infty }{lim}\varrho (r_n,r_{n+1})=0.\end{array}$$

(4)

Suppose that $\left\{r_n\right\}$ is not a Cauchy sequence. Consequently, Lemma 1 guarantees the existence of $ϵ>0$ and two subsequences $\left\{r_{n_{\alpha }}\right\}$ and $\left\{r_{p_{\alpha }}\right\}$ of $\left\{r_n\right\}$ satisfying $\alpha \le p_{\alpha }<n_{\alpha }$, $\varrho (r_{p_{\alpha }},r_{n_{\alpha }})\ge ϵ$ and $\varrho (r_{p_{\alpha }},r_{s_{\alpha }})<ϵ$, where $s_{\alpha }\in \{p_{\alpha }+1,p_{\alpha }+2,\dots ,n_{\alpha }-2,n_{\alpha }-1\}$. Moreover, due to the availability of (4), we have

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}\varrho (r_{p_{\alpha }},r_{n_{\alpha }+s})=ϵ,\forall s\in {\mathbb{N}}_0.\end{array}$$

(5)

Since $\left\{r_n\right\}\subset \mathcal{G}\left(\mathbb{W}\right)$, the range $\mathbb{Z}=\{r_n:n\in {\mathbb{N}}_0\}$ remains a denumerable subset of $\mathcal{G}\left(\mathbb{W}\right)$. Therefore, using locally finitely $\mathcal{G}$-transitivity of $\mathfrak{R}$, we are able to find a natural number $m=m\left(\mathbb{Z}\right)\ge 2$, such that ${\mathfrak{R}|}_{\mathbb{Z}}$ remains m-transitive.

Since $p_{\alpha }<n_{\alpha }$ and $m-1>0$, therefore applying division algorithm, one obtains

$$\begin{array}{l}{n}_{\alpha }-p_{\alpha }=(m-1)({\mu }_{\alpha }-1)+(m-{\eta }_{\alpha })\\ {\mu }_{\alpha }-1\ge 0,0\le m-{\eta }_{\alpha }<m-1\\ \iff \left\{\begin{array}{c}{n}_{\alpha }+{\eta }_{\alpha }=p_{\alpha }+1+(m-1){\mu }_{\alpha }\hfill \\ {\mu }_{\alpha }\ge 1,1<{\eta }_{\alpha }\le m.\hfill \end{array}\right.\end{array}$$

It can be noticed that ${\mu }_{\alpha }$ and ${\eta }_{\alpha }$ remain suitable numbers, so that the value of ${\eta }_{\alpha }\in (1,m]$ may be considered finitely. Consequently, we are able to choose subsequences $\left\{r_{n_{\alpha }}\right\}$ and $\left\{r_{p_{\alpha }}\right\}$ of $\left\{r_n\right\}$(verifying (5)) in such a way that ${\eta }_{\alpha }$ becomes a constant (say, $\eta$). We have

$$m_{\alpha }^{\prime }=n_{\alpha }+\eta =p_{\alpha }+1+(m-1){\mu }_{\alpha },$$

(6)

whereas $\eta$$(1<\eta \le m)$ remains constant. Making use of (5) and (6), one obtains

$$\underset{\alpha \to \infty }{lim}\varrho (r_{p_{\alpha }},r_{m_{\alpha }^{\prime }})=\underset{\alpha \to \infty }{lim}\varrho (r_{p_{\alpha }},r_{n_{\alpha }+\eta })=ϵ.$$

(7)

Using triangular inequality, we have

$$\varrho (r_{p_{\alpha }+1},r_{m_{\alpha }^{\prime }+1})\le \varrho (r_{p_{\alpha }+1},r_{p_{\alpha }})+\varrho (r_{p_{\alpha }},r_{m_{\alpha }^{\prime }})+\varrho (r_{m_{\alpha }^{\prime }},r_{m_{\alpha }^{\prime }+1})$$

(8)

and

$$\varrho (r_{p_{\alpha }},r_{m_{\alpha }^{\prime }})\le \varrho (r_{p_{\alpha }},r_{p_{\alpha }+1})+\varrho (r_{p_{\alpha }+1},r_{m_{\alpha }^{\prime }+1})+\varrho (r_{m_{\alpha }^{\prime }+1},r_{m_{\alpha }^{\prime }})$$

or

$$\begin{array}{ccc}\hfill \varrho (r_{p_{\alpha }},r_{m_{\alpha }^{\prime }})-\varrho (r_{p_{\alpha }},r_{p_{\alpha }+1})-\varrho (r_{m_{\alpha }^{\prime }+1},r_{m_{\alpha }^{\prime }})& \le & \varrho (r_{p_{\alpha }+1},r_{m_{\alpha }^{\prime }+1}).\hfill \end{array}$$

(9)

Letting $\alpha \to \infty$ in (8) and (9) and using (4) and (7), we obtain

$$\underset{\alpha \to \infty }{lim}\varrho (r_{p_{\alpha }+1},r_{m_{\alpha }^{\prime }+1})=ϵ.$$

(10)

Due to the availability of (6) and Lemma 2, we obtain $\varrho (r_{p_{\alpha }},r_{m_{\alpha }^{\prime }})\in \mathfrak{R}$. Furthermore, by assumption $\left(e\right)$, we obtain

$$\begin{array}{c}\hfill \psi \left(\varrho (r_{p_{\alpha }+1},r_{m_{\alpha }^{\prime }+1})\right)=\psi \left(\varrho (\mathcal{G}{r}_{p_{\alpha }},\mathcal{G}{r}_{m_{\alpha }^{\prime }})\right)\le \varphi \left(\varrho (r_{p_{\alpha }},r_{m_{\alpha }^{\prime }})\right).\end{array}$$

Applying condition $\left({\mathrm{\Sigma }}_2\right)$ to $\{s_{\alpha }=\varrho (r_{p_{\alpha }},r_{m_{\alpha }^{\prime }})\}$, $\{t_{\alpha }=\varrho (r_{p_{\alpha }+1},r_{m_{\alpha }^{\prime }+1})$ and $L=ϵ$, we find that $ϵ=0$, which remains a contradiction. Hence, $\left\{r_n\right\}$ is Cauchy. By $\mathfrak{R}$-completeness of $\mathbb{W}$, $\exists r\in \mathbb{W}$ such that $r_n\stackrel{\varrho }{\to }r$. The $\mathfrak{R}$-continuity of $\mathcal{G}$ gives rise to $\mathcal{G}\left(r_n\right)\stackrel{\varrho }{\to }\mathcal{G}\left(r\right)$, which in view of (2), reduces to $r_{n+1}\stackrel{\varrho }{\to }r.$ Finally, by uniqueness of limit, we obtain $\mathcal{G}\left(r\right)=r$. □

Theorem 4.

Theorem 3 remains valid if assumption $\left(c\right)$ is replaced by the following condition:

$\left(c^{\prime }\right)$

$\mathfrak{R}$ is ϱ-self-closed, and the pair $(\psi ,\varphi )$ verifies the following property:

If $\left\{s_n\right\},\left\{t_n\right\}\subset [0,\infty )$ verifies $s_n\to 0$ and $\psi \left(t_n\right)\le \varphi \left(s_n\right)$, $\forall n\in \mathbb{N}$, then $t_n\to 0$.

Proof. 

Similar to previous result, it can be shown that

$$r_n\stackrel{\varrho }{\to }r.$$

Using the assumption $\left(c^{\prime }\right)$, we show that r remains a fixed point of $\mathcal{G}$. Since $\left\{r_n\right\}$ remains an $\mathfrak{R}$-preserving sequence satisfying $r_n\stackrel{\varrho }{\to }r$, therefore by $\varrho$-self-closedness of $\mathfrak{R}$, there exists a subsequence of $\left\{r_{n_k}\right\}$ of $\left\{r_n\right\}$ satisfying $[r_{n_k},r]\in \mathfrak{R}$ for all $k\in {\mathbb{N}}_0$. Therefore, using assumption $\left(e\right)$, one obtains

$$\begin{array}{c}\hfill \psi \left(\varrho (r_{n_k+1},\mathcal{G}r)\right)=\psi \left(\varrho (\mathcal{G}{r}_{n_k},\mathcal{G}r)\right)\le \varphi \left(\varrho (r_{n_k},r)\right).\end{array}$$

Due to the fact that $r_{n_k}\stackrel{\varrho }{\to }x$, and the continuity of $\varrho$, we have $\varrho (r_{n_k},x)\to 0$ as $k\to \infty$. Therefore, making use of the property of the pair $(\psi ,\varphi )$, we have

$$\varrho (r_{n_k+1},\mathcal{G}r)\to 0\mathrm{as}k\to \infty$$

so that

$$r_{n_k+1}\stackrel{\varrho }{\to }\mathcal{G}\left(r\right).$$

By uniqueness of limit, one obtains $\mathcal{G}\left(r\right)=r$. □

Finally, the following uniqueness result is presented.

Theorem 5.

Under the hypotheses of Theorem 3 (or Theorem 4), if $\mathcal{G}\left(\mathbb{W}\right)$ remains ${\mathfrak{R}}^s$-connected, then $\mathcal{G}$ admits a unique fixed point.

Proof. 

In view of Theorem 3 (or Theorem 4), if r and t remain two fixed points of $\mathcal{G}$, then

$${\mathcal{G}}^n\left(r\right)=r\text{ and }{\mathcal{G}}^n\left(t\right)=t,\forall n\in {\mathbb{N}}_0.$$

Clearly $r,t\in \mathcal{G}\left(\mathbb{W}\right)$. By the ${\mathfrak{R}}^s$-connectedness of $\mathcal{G}\left(\mathbb{W}\right)$, we can find a path $\{s_0,s_1,s_2,\dots ,s_k\}$ in ${\mathfrak{R}}^s$ from r to t so that

$$\begin{array}{c}\hfill s_0=r,s_k=t\mathrm{and}[s_i,s_{i+1}]\in \mathfrak{R},\forall i=0,1,\dots ,k-1.\end{array}$$

(11)

As $\mathfrak{R}$ is $\mathcal{G}$-closed, we have

$$\begin{array}{c}\hfill [{\mathcal{G}}^n{s}_i,{\mathcal{G}}^n{s}_{i+1}]\in \mathfrak{R},\forall n\in {\mathbb{N}}_0\mathrm{and}\forall i=0,1,\dots ,k-1.\end{array}$$

(12)

Denote

$${\delta }_n^i:=\varrho ({\mathcal{G}}^n{s}_i,{\mathcal{G}}^n{s}_{i+1}),\forall n\in {\mathbb{N}}_0\mathrm{and}\forall i=0,1,\dots ,k-1.$$

We show that

$$\begin{array}{ccc}\hfill \underset{n\to \infty }{lim}{\delta }_n^i& =& 0.\hfill \end{array}$$

(13)

Assume that

$${\delta }_{n_0}^i=\varrho ({\mathcal{G}}^{n_0}{s}_i,{\mathcal{G}}^{n_0}{s}_{i+1})=0\text{ for some }{n}_0\in {\mathbb{N}}_0,$$

which gives rise ${\mathcal{G}}^{n_0}\left(s_i\right)={\mathcal{G}}^{n_0}\left(s_{i+1}\right)$. Now, applying (2), one obtains ${\mathcal{G}}^{n_0+1}\left(s_i\right)={\mathcal{G}}^{n_0+1}\left(s_{i+1}\right)$. Hence, ${\delta }_{n_0+1}^i=0$. Therefore, by mathematical induction, one obtains ${\delta }_n^i=0$, ∀$n\ge n_0$, thereby implying t $\underset{n\to \infty }{lim}{\delta }_n^i=0$.

As either case, one can assume that ${\delta }_n^i>0$∀$n\in {\mathbb{N}}_0$. Making use of (12) together with condition $\left(e\right)$, one obtains

$$\begin{array}{ccc}\hfill \psi \left({\delta }_{n+1}^i\right)& =& \psi \left(\varrho ({\mathcal{G}}^{n+1}{s}_i,{\mathcal{G}}^{n+1}{s}_{i+1})\right)\hfill \\ \hfill & =& \psi \left(\varrho (\mathcal{G}\left({\mathcal{G}}^n{s}_i\right),\mathcal{G}\left({\mathcal{G}}^n{s}_{i+1}\right))\right)\hfill \\ \hfill & \le & \varphi \left(\varrho ({\mathcal{G}}^n{s}_i,{\mathcal{G}}^n{s}_{i+1})\right)\hfill \\ \hfill & =& \varphi \left({\delta }_n^i\right)\hfill \end{array}$$

so that

$$\begin{array}{ccc}\hfill \psi \left({\delta }_{n+1}^i\right)& \le & \varphi \left({\delta }_n^i\right).\hfill \end{array}$$

Applying the property $\left({\mathrm{\Sigma }}_1\right)$, the above inequality yields that

$$\underset{n\to \infty }{lim}{\delta }_n^i=0.$$

Hence, in both cases, (13) is proved. Consequently, one obtains

$$\begin{array}{ccc}\hfill \varrho (r,t)& =& \varrho ({\mathcal{G}}^n{s}_0,{\mathcal{G}}^n{s}_k)\hfill \\ \hfill & \le & {\delta }_n^0+{\delta }_n^1+\cdots +{\delta }_n^{k-1}\hfill \\ \hfill & \to & 0asn\to \infty \hfill \end{array}$$

so that $r=t$. □

Corollary 1.

Theorem 5 is valid if “${\mathfrak{R}}^s$-connectedness of $\mathcal{G}\left(\mathbb{W}\right)$” is replaced by one of the following:

(i) $\mathcal{G}\left(\mathbb{W}\right)$ remains ${\mathfrak{R}}^s$-directed;

(ii) ${\mathfrak{R}|}_{\mathcal{G}\left(\mathbb{W}\right)}$ remains complete.

Proof. 

Assume that condition (i) holds. Take $r,t\in \mathcal{G}\left(\mathbb{W}\right)$. Then, by assumption (i), one can find $z\in \mathbb{W}$ satisfying $[x,z]\in \mathfrak{R}$ and $[y,z]\in \mathfrak{R}$. This implies that $\{x,z,y\}$ remains a path of length 2 in ${\mathfrak{R}}^s$ from r to t. Therefore, $\mathcal{G}\left(\mathbb{W}\right)$ remains ${\mathfrak{R}}^s$-connected and hence by Theorem 5, the conclusion holds.

If the assumption (ii) holds, then for each $r,t\in \mathcal{G}\left(\mathbb{W}\right)$, we have $[r,t]\in \mathfrak{R}$. This implies that $\{r,t\}$ remains a path of length 1 in ${\mathfrak{R}}^s$ from r to t. Consequently, $\mathcal{G}\left(\mathbb{W}\right)$ remains ${\mathfrak{R}}^s$-connected and hence by Theorem 5, the conclusion holds. □

For $\mathfrak{R}={\mathbb{W}}^2$, Theorem 5 reduces to:

Corollary 2.

Assume that $(\mathbb{W},\varrho )$ remains a complete metric space and $\mathcal{G}$ remains a map from $\mathbb{W}$ to itself. If there exists $(\psi ,\varphi )\in \mathrm{\Sigma }$ satisfying

$$\psi \left(\varrho \right(\mathcal{G}r,\mathcal{G}t\left)\right)\le \varphi \left(\varrho \right(r,t\left)\right)$$

for all $r,t\in \mathbb{W}$, then $\mathcal{G}$ admits a unique fixed point.

4. An Illustrative Example

Now, I give an example in support of Theorem 3.

Example 4.

Consider $\mathbb{W}=[0,1]\cup \mathbb{N}$ with a metric ϱ and a relation $\mathfrak{R}$ defined by

$$\varrho (r,t)=\left\{\begin{array}{cc}|r-t|,\hfill & ifr,t\in [0,1]andr\ne t;\hfill \\ r+t,\hfill & if(r,t)\notin [0,1]\times [0,1]andr\ne t;\hfill \\ 0,\hfill & ifr=t.\hfill \end{array}\right.$$

$$\mathfrak{R}=\{(r,t)\in {\mathbb{W}}^2:r>t\},$$

then $(\mathbb{W},\varrho )$ is a $\mathfrak{R}$-complete metric space.

Define $\psi ,\varphi :[0,\infty )\to [0,\infty )$ as follows:

$$\psi \left(s\right)=\left\{\begin{array}{cc}ln\left(\frac{1}{12}+\frac{5}{12}s\right),\hfill & if0\le s\le 1\hfill \\ ln\left(\frac{1}{12}+\frac{4}{12}s\right),\hfill & ifs>1\hfill \end{array}\right.$$

and

$$\varphi \left(s\right)=\left\{\begin{array}{cc}ln\left(\frac{1}{12}+\frac{3}{12}s\right),\hfill & if0\le s\le 1\hfill \\ ln\left(\frac{1}{12}+\frac{2}{12}s\right),\hfill & ifs>1.\hfill \end{array}\right.$$

Clearly, $(\psi ,\varphi )\in \mathrm{\Sigma }$. Assume that $\mathcal{G}:\mathbb{W}\to \mathbb{W}$ is a map defined by

$$\mathcal{G}\left(r\right)=\left\{\begin{array}{cc}r/5,\hfill & ifr\in [0,1),\hfill \\ 3/125,\hfill & ifr\in \mathbb{N}.\hfill \end{array}\right.$$

Take $r,t\in \mathbb{W}$ with $(r,t)\in \mathfrak{R}$, then $r>t$. Then, we have the following cases:

Case-1: When $r\in [0,1]$, then we have

$$\begin{array}{ccc}\hfill \psi \left(\varrho \right(\mathcal{G}r,\mathcal{G}t\left)\right)& =& ln\left(\frac{1}{12}+\frac{5}{12}\varrho (\mathcal{G}r,\mathcal{G}t)\right)\hfill \\ \hfill & =& ln\left(\frac{1}{12}+\frac{5}{12}|\mathcal{G}r-\mathcal{G}t|\right)\hfill \\ \hfill & =& ln\left(\frac{1}{12}+\frac{1}{12}|r-t|\right)\hfill \\ \hfill & \le & \varphi \left(\varrho \right(r,t\left)\right).\hfill \end{array}$$

Case-2: When $r\in \mathbb{N}-\left\{0\right\}$. If $t\in [0,1)$, then we have

$$\begin{array}{ccc}\hfill \psi \left(\varrho \right(\mathcal{G}r,\mathcal{G}t\left)\right)& =& ln\left(\frac{1}{12}+\frac{5}{12}\varrho (\mathcal{G}r,\mathcal{G}t)\right)\hfill \\ \hfill & =& ln\left(\frac{1}{12}+\frac{5}{12}|\mathcal{G}r-\mathcal{G}t|\right)\hfill \\ \hfill & \le & ln\left(\frac{1}{12}+\frac{5}{12}\left(\frac{3}{125}+\frac{t}{5}\right)\right)\hfill \\ \hfill & \le & ln\left(\frac{1}{12}+\frac{1}{100}+\frac{t}{12}\right)\hfill \\ \hfill & \le & \varphi \left(\varrho (r,t)\right)\left((as\frac{1}{100}+\frac{t}{12}\le \frac{1}{12}(r+t))\right).\hfill \end{array}$$

Otherwise, if $t\in \mathbb{N}$, then we have

$$\begin{array}{ccc}\hfill \psi \left(\varrho \right(\mathcal{G}r,\mathcal{G}t\left)\right)& =& ln\left(\frac{1}{12}+\frac{5}{12}\varrho (\mathcal{G}r,\mathcal{G}t)\right)\hfill \\ \hfill & =& ln\left(\frac{1}{12}\right)\hfill \\ \hfill & \le & \varphi \left(\varrho \right(r,t\left)\right).\hfill \end{array}$$

Therefore, $\mathcal{G}$ is $\mathfrak{R}$-continuous and satisfies assumption $\left(e\right)$ of Theorem 3. Notice that here $\mathfrak{R}$ remains locally finitely $\mathcal{G}$-transitive. Moreover, the relation $\mathfrak{R}$ remains $\mathcal{G}$-closed. The rest of the conditions of Theorems 3 and 5 are also satisfied. Consequently, $\mathcal{G}$ admits a unique fixed point (namely: $r=0$).

5. Conclusions

In this manuscript, the fixed point results in the framework of natural structure, namely, metric space (abbreviated as: MS) endowed with a locally finitely $\mathcal{G}$-transitive relation employing a pair of auxiliary functions, were proved. For future works, the analogues of these results can be proved in generalized metrical structure (such as, semi MS, quasi MS, pseudo MS, multiplicative MS, dislocated space, D-MS, 2-MS, S-MS, G-MS, b-MS, partial MS, cone MS, complex-valued MS, fuzzy MS, $JS$-MS, modular space and rectangular MS) endowed with locally finitely $\mathcal{G}$-transitive relations.