Results for Nonlinear-Prešić Contractions in Relational Metric Spaces

Abstract

This article aims to adopt some notions for mapping $\mathfrak{f}:{\mathfrak{X}}^k\to \mathfrak{X}$, (where integer k is positive) and to prove the nonlinear-Prešić-type results on metric spaces employing a $\mathfrak{f}$-reflexive and locally finitely $\mathfrak{f}$-transitive binary relation (not necessarily partial order). The outcomes proven herewith are extended and generalized to several fixed point findings of literature. Lastly, examples are provided to support the applicability of these outcomes.

1. Introduction

The celebrated Banach contraction principle (BCP) was established in 1922 by Banach [1]. Several notable extensions and generalizations of this fundamental result can be found in the literature, often in an abstract setting. In this regard, we suggest readers consult the recent works of Cvetković [2,3,4] and the references therein. Naturally, an element $\omega \in \mathfrak{X}$ that ensures $\mathfrak{f}\left(\omega \right)=\omega$ is referred to as a fixed point of $\mathfrak{f}$ given a mapping of $\mathfrak{f}:\mathfrak{X}\to \mathfrak{X}$. In 1965, Prešić [5,6] initiated the notion of the fixed point for the map $\mathfrak{f}:{\mathfrak{X}}^k\to \mathfrak{X}$ as follows:

Definition 1

([5,6]). Consider a nonempty set $\mathfrak{X}$ and a mapping $\mathfrak{f}:{\mathfrak{X}}^k\to \mathfrak{X}$. A point $\omega \in \mathfrak{X}$ is a fixed point of $\mathfrak{f}$ if

$$\mathfrak{f}\underset{ktimes}{\left(\underbrace{\omega ,\omega ,\cdots ,\omega }\right)}=\omega .$$

Prešić [5,6] was the author of one such notable result, which is follows:

Theorem 1

(Prešić [5]). Consider a complete metric space $(\mathfrak{X},d)$ along with Prešić map $\mathfrak{f}:{\mathfrak{X}}^k\to \mathfrak{X}$ enjoying Prešić contraction: $d(\mathfrak{f}({\omega }_1,{\omega }_2,\cdots ,{\omega }_k),\mathfrak{f}({\omega }_2,{\omega }_3,\cdots ,{\omega }_{k+1}))\le {\sum }_{i=1}^k{\alpha }_{i}d({\omega }_i,{\omega }_{i+1})$ $\forall {\omega }_1,{\omega }_2,\cdots ,{\omega }_k,{\omega }_{k+1}\in \mathfrak{X}$ with ${\sum }_{i=1}^k{\alpha }_i<1,$ where ${\alpha }_i^{\prime }s$ are non negative constants. Then, $Fix\left(\mathfrak{f}\right)$ is singleton, where $Fix\left(\mathfrak{f}\right)=\left\{\omega \right\}$$(i.e.,\mathfrak{f}(\omega ,\omega ,\cdots ,\omega )=\omega )$. Moreover, if for ${\omega }_1,{\omega }_2,\cdots ,{\omega }_k\in \mathfrak{X}$, ${\omega }_{n+k}=\mathfrak{f}\underset{ktimes}{\left(\underbrace{{\omega }_n,{\omega }_{n+1},\cdots ,{\omega }_{n+k-1}}\right)}$ for $n\in \mathbb{N}$, then the sequence $\left\{{\omega }_n\right\}$ is convergent and ${lim}_{n\to \infty }{\omega }_n=\mathfrak{f}({lim}_{n\to \infty }{\omega }_n,{lim}_{n\to \infty }{\omega }_n,\cdots ,{lim}_{n\to \infty }{\omega }_n)$.

Notation $Fix\left(\mathfrak{f}\right)$ is defined latter. For $k=1$, the above finding deduces the BCP as follows:

Theorem 2

(Banach [1]). If $(\mathfrak{X},d)$ is a complete metric space along with a contraction map $\mathfrak{f}:\mathfrak{X}\to \mathfrak{X}$, then $Fix\left(\mathfrak{f}\right)$ is singleton. Moreover, for any ${\omega }_0\in \mathfrak{X}$, ${\omega }_{n+1}=\mathfrak{f}\left({\omega }_n\right)$ (for $n\in \mathbb{N}$) tends to a point in $Fix\left(\mathfrak{f}\right)$.

Opoitsev [7,8] introduced a weaker definition of a fixed point in 1975, particularly for $k=2$. This notion is based on the suppositions that $\mathfrak{f}(\omega ,\chi )=\omega$ and $\mathfrak{f}(\chi ,\omega )=\chi$, rather than $\mathfrak{f}(\omega ,\omega )=\omega$. Therefore, for $\chi =\omega$, this goes down to Definition 1. Based on this idea, Opoitsev and Khurodze [9] proved specific outcomes for nonlinear functions on ordered Banach spaces. Despite using the term “coupled fixed point”, this notion was erroneously revisited in 1987 by Guo and Lakshmikantham [10] regarding mixed monotone operators given on a real Banach space bestowed with the partial ordering by a cone.

Definition 2

([7,10]). Consider a nonempty set $\mathfrak{X}$ and a map $\mathfrak{f}:{\mathfrak{X}}^2\to \mathfrak{X}$. If $\mathfrak{f}(\omega ,\chi )=\omega \mathrm{and}\mathfrak{f}(\chi ,\omega )=\chi ,$ then element $(\omega ,\chi )\in {\mathfrak{X}}^2$ is a coupled fixed point of $\mathfrak{f}$

Several authors, including Berinde and Borcut [11] and Karapinar and Luong [12], introduced the notions of tripled and quadrupled fixed points in an effort to expand the idea of coupled fixed point from ${\mathfrak{X}}^2$ to ${\mathfrak{X}}^3$ and ${\mathfrak{X}}^4$, respectively. This concept is extended for the mapping $\mathfrak{f}:{\mathfrak{X}}^n\to \mathfrak{X}$ by various authors as multidimensional fixed point (see Roldán et al. [13,14,15], Akhadkulov et al. [16], Karapınar et al. [17], Rad et al. [18], and Alam et al. [19]).

In 1986, Turinici [20] initiated the idea of ordered-theoretic fixed point. Later, Ran and Reurings [21], and Nieto and Rodríguez-Loṕez [22] extended BCP to ordered metric spaces via the class of linear contractions. Thereafter, Alam and Imdad [23] extended BCP under amorphous binary relation. On the other hand, BCP was enlarged to a class of nonlinear contractions by Browder [24], Boyd and Wong [25], Mukherjea [26], and Jotić [27]. Subsequently, Agarwal et al. [28] and O’Regan and Petruşel [29] enhanced certain order-theoretic fixed-point outcomes under various kinds of control functions. Further, Alam et al. [30] and Alam and Imdad [31] demonstrated the outcomes regarding the appropriate classes of nonlinear contractions in additionally using “locally finitely $\mathfrak{f}$-transitive” and “locally $\mathfrak{f}$-transitive” relations respectively, which are relatively weaker forms of transitive relations. For further details on transitivity, the reader can consult [32,33,34,35,36,37,38]. In 2014, Shukla and Radenović [39] proved an ordered-theoretic version of Boyd–Wong-type Prešić results.

The aim of this paper is to extend the order-theoretic Prešić-type results of Shukla and Radenović [39] to a set of nonlinear contraction-type mappings using $\mathfrak{f}$-reflexivity and locally finitely $\mathfrak{f}$-transitivity of ℜ (relation version of Prešić-type results), considering several results from the existing literature, namely: results contained in Prešić [5,6], Shukla and Radenović [39], and Alam et al. [30], as well as covering several other fixed point results.

2. Preliminaries

In the subsequent discussion, we consider the following:

Given a nonempty set $\mathfrak{X}$, the binary relation (relation) ℜ on $\mathfrak{X}$ stands for arbitrary (amorphous) subset of ${\mathfrak{X}}^2$, and d indicates a metric on $\mathfrak{X}$ and Prešić mapping as $\mathfrak{f}:{\mathfrak{X}}^k\to \mathfrak{X}$ (keeping in mind that ${\mathfrak{X}}^k$ represents the product of $\mathfrak{X}$, k times, with k being any positive integer). Additionally, we choose ${\mathbb{N}}_0$ and $\mathbb{N}$ as the sets of whole numbers and natural numbers, correspondingly. ${\Re |}_U:=\Re \cap U^2$ is the restriction of ℜ on U, for a subset $U\subseteq \mathfrak{X}$. We say that:

Definition 3

([23]). ω and $\chi \in \mathfrak{X}$ are “ℜ-comparative" if either $(\omega ,\chi )\in \Re$ or $(\chi ,\omega )\in \Re$. This is symbolized by $[x,y]\in \Re$.

Definition 4

([40]). ${\Re }^{-1}:=\{(\omega ,\chi )\in {\mathfrak{X}}^2:(\chi ,\omega )\in \Re \}$ is the inverse of ℜ.

Definition 5

([40]). “${\Re }^s:=\Re \cup {\Re }^{-1}$" is the symmetric closure of ℜ.

Proposition 1

([23]). $(\omega ,\chi )\in {\Re }^s⟺[x,y]\in \Re .$

Definition 6

([23]). $\left\{{\omega }_n\right\}\subset \mathfrak{X}$ is “ℜ-preserving" if $({\omega }_n,{\omega }_{n+1})\in \Re \forall n\in {\mathbb{N}}_0.$

Definition 7

([41]). $(\mathfrak{X},d)$ is “ℜ-complete". If every Cauchy sequence in $\mathfrak{X}$ that is ℜ-preserving, this will be convergent in $\mathfrak{X}$.

Definition 8.

ℜ is “$\mathfrak{f}$-closed" if for any $\omega ,\chi \in \mathfrak{X}$,

$$(\omega ,\chi )\in \Re \Rightarrow \left(\mathfrak{f}\right(\omega ,\omega ,\cdots ,\omega ),\mathfrak{f}(\chi ,\chi ,\cdots ,\chi \left)\right)\in \Re .$$

Proposition 2

([41]). $\mathfrak{f}$-closedness of ℜ implies that $\mathfrak{f}$-closedness of ${\Re }^s$.

Definition 9

([31]). ℜ is “locally $\mathfrak{f}$-transitive" if for every countable infinite subset U of $\mathfrak{f}\left({\mathfrak{X}}^k\right)$, ${\Re |}_U$ remains transitive.

Definition 10

([35]). Given $N\in {\mathbb{N}}_0,N\ge 2$, ℜ is “N-transitive” if ${\omega }_0,{\omega }_1,...,{\omega }_N\in \mathfrak{X}$, $i\in \{0,1,2,\cdots ,N\}$ with $({\omega }_{i-1},{\omega }_i)\in \Re \Rightarrow ({\omega }_0,{\omega }_N)\in \Re .$

Definition 11

([38]). ℜ is “locally finitely transitive", if for every countable-infinite subset $U\subset \mathfrak{X}$, ∃$N=N\left(U\right)\ge 2$, such that ${\Re |}_U$ is N-transitive.

Definition 12

([30]). ℜ “locally finitely $\mathfrak{f}$-transitive," if for every countable-infinite subset U of $\mathfrak{f}\left({\mathfrak{X}}^k\right)$, ∃$N=N\left(U\right))\ge 2$, where ${\Re |}_U$ remains N-transitive.

Definition 13.

$\mathfrak{f}$ is ℜ-continuous at $({\omega }^1,{\omega }^2,\cdots ,{\omega }^k)\in {\mathfrak{X}}^k$ if for any ℜ-preserving sequences $\left\{{\omega }_n^i\right\}$ ($1\le i\le k$) with ${\omega }_n^i\stackrel{d}{\to }{w}^i$, we have $\mathfrak{f}({\omega }_n^1,{\omega }_n^2,\cdots ,{\omega }_n^k)\stackrel{d}{⟶}\mathfrak{f}({\omega }^1,{\omega }^2,\cdots ,{\omega }^k)$. Additionally, if $\mathfrak{f}$ is ℜ-continuous at each point of ${\mathfrak{X}}^k$, then $\mathfrak{f}$ is ℜ-continuous.

Definition 14.

ℜ is “$\mathfrak{f}$-reflexive" if ${\Re |}_{\mathfrak{f}\left({\mathfrak{X}}^k\right)}$ is reflexive.

Remark 1.

Upon setting $k=1$, Definitions 8, 9, 12, and 13, are reduced to Definition 2.12 of [23], Definition 11 of [30], and Definition 2.16 of [31], Definition 3.20 [41], respectively.

Definition 15

([23]). ℜ is “d-self-closed" if, for any sequence, $\left\{{\omega }_n\right\}$ provides ℜ-preserving, such that ${\omega }_n\stackrel{d}{⟶}\omega$, there exists sub-sequence $\left\{{\omega }_{n_k}\right\}\mathrm{of}\left\{{\omega }_n\right\}\mathrm{with}[{\omega }_{n_k},\omega ]\in \Re \forall k\in {\mathbb{N}}_0$.

Definition 16

([42]). $E\subseteq \mathfrak{X}$ is “ℜ-directed” if for each $\omega ,\chi \in E$, ∃$\varsigma \in \mathfrak{X}$, such that $(\omega ,\varsigma )\in \Re$ and $(\chi ,\varsigma )\in \Re$.

Definition 17

([43]). For each $\omega ,\chi \in \mathfrak{X}$, “l” is the length of the path ($l\in \mathbb{N}$) in ℜ from ω to χ, which is a set of finite points of $\{{\varsigma }_0,{\varsigma }_1,{\varsigma }_2,...,{\varsigma }_l\}\subset \mathfrak{X}$, verifying:

(i) ${\varsigma }_0=\omega \mathrm{and}{\varsigma }_l=\chi$,

(ii) $({\varsigma }_i,{\varsigma }_{i+1})\in \Re$, $i\in \{0,1,2,\cdots ,l-1\}$.

Definition 18

([41]). $U\subseteq \mathfrak{X}$ is “ℜ-connected”, if for each $\omega ,\chi \in U$, there exists a path in ℜ from ω to χ.

The following notations are utilized in the present discussion:

(i) 

$Fix\left(\mathfrak{f}\right):=\{\omega \in \mathfrak{X}:\omega =\mathfrak{f}\underset{ktimes}{\left(\underbrace{\omega ,\omega ,\cdots ,\omega }\right)}\}$ is the set of all fixed points of $\mathfrak{f}$,

(ii) 

${\mathfrak{X}}^k(\mathfrak{f},\Re ):=\{\omega \in \mathfrak{X}:(\omega ,\mathfrak{f}\underset{ktimes}{\left(\underbrace{\omega ,\omega ,\cdots ,\omega }\right)})\in \Re \}$.

Notice that for $k=1,$ the above notions lead to the usual notions of the fixed point and $\mathfrak{X}(\mathfrak{f},\Re )$ of the self-map.

Inspired by the notion introduced by Shukla and Radenović [39], we adopted the following:

We denote by $\mathsf{\Psi }$: set all such maps $\psi :{[0,\infty )}^k\to [0,\infty )$ (where integer k is positive) verifying the following conditions:

(Ψ₁)

: ${\displaystyle \underset{\sigma \to \varrho }{lim\; sup}\psi \underset{ktimes}{\underbrace{(\sigma ,\sigma ,\cdots ,\sigma )}}<\varrho }$ for each $\varrho >0$,

(Ψ₂)

: $\psi \underset{ktimes}{\underbrace{(\varrho ,\varrho ,\cdots ,\varrho )}}<\varrho$ for each $\varrho >0$,

(Ψ₃)

: $\psi \underset{ktimes}{\underbrace{(\varrho ,0,\cdots ,0)}}+\psi \underset{ktimes}{\underbrace{(0,\varrho ,\cdots ,0)}}+\cdots +\psi \underset{ktimes}{\underbrace{(0,0,\cdots ,\varrho )}}\le \psi \underset{ktimes}{\underbrace{(\varrho ,\varrho ,\cdots ,\varrho )}}$ for each $\varrho \in [0,\infty )$.

The aforementioned class reduces to the following family of control functions shown in Alam et al. [30] for $k=1$. “$\mathsf{\Phi }=\{\psi :[0,\infty )\to [0,\infty ):\psi \left(\varrho \right)<\varrho \mathrm{for}\mathrm{each}\varrho >0\mathrm{and}{lim\; sup}_{\sigma \to \varrho }\psi \left(\sigma \right)<\varrho \mathrm{for}\mathrm{each}\varrho >0\}$ [30]”.

Proposition 3.

If $\psi \in \mathsf{\Psi }$, then the following Prešić contractions are equivalent:

(I) 

$d(\mathfrak{f}({\omega }_1,{\omega }_2,\cdots ,{\omega }_k),\mathfrak{f}({\omega }_2,{\omega }_3,\cdots ,{\omega }_{k+1}))\le \psi (d({\omega }_1,{\omega }_2),d({\omega }_2,{\omega }_3),\cdots ,d({\omega }_k,{\omega }_{k+1}))\forall {\omega }_1,{\omega }_2,\cdots ,{\omega }_k,{\omega }_{k+1}\in \mathfrak{X}$ with $({\omega }_i,{\omega }_{i+1})\in \Re$ for $1\le i\le k+1$.

(II) 

$d(\mathfrak{f}({\omega }_1,{\omega }_2,\cdots ,{\omega }_k),\mathfrak{f}({\omega }_2,{\omega }_3,\cdots ,{\omega }_{k+1}))\le \psi (d({\omega }_1,{\omega }_2),d({\omega }_2,{\omega }_3),\cdots ,d({\omega }_k,{\omega }_{k+1}))\forall {\omega }_1,{\omega }_2,\cdots ,{\omega }_k,{\omega }_{k+1}\in \mathfrak{X}$ with $[{\omega }_i,{\omega }_{i+1}]\in \Re$ for $1\le i\le k+1$.

Proof. 

It is simple to compute $\left(II\right)$⟹$\left(I\right)$. To assert that $\left(I\right)$⟹$\left(II\right)$, choose ${\omega }_1,{\omega }_2,\cdots ,{\omega }_k,{\omega }_{k+1}\in \mathfrak{X}$, such that $[{\omega }_i,{\omega }_{i+1}]\in \Re$ for $1\le i\le k+1$. If $({\omega }_i,{\omega }_{i+1})\in \Re$ for $1\le i\le k+1$ from $\left(I\right)$, $\left(II\right)$ follows immediately. Otherwise, if $({\omega }_{i+1},{\omega }_i)\in \Re$ for $1\le i\le k+1$, then by $\left(I\right)$ and the symmetry of property of d, we can obtain the conclusion. □

Now, we prove an auxiliary result w.r.t the class $\mathsf{\Psi }$, as follows:

Proposition 4.

Suppose $\psi \in \mathsf{\Psi }.$ If $\left\{{\varrho }_n\right\}\subset (0,\infty )$ is a sequence such that ${\varrho }_{n+1}\le \psi ({\varrho }_n,{\varrho }_n,\cdots ,{\varrho }_n)\forall n\in {\mathbb{N}}_0$, then ${\displaystyle \underset{n\to \infty }{lim}{\varrho }_n=0.}$

Proof. 

Given

$$\begin{array}{c}\hfill {\varrho }_{n+1}\le \varphi ({\varrho }_n,{\varrho }_n,\cdots ,{\varrho }_n)\forall n\in {\mathbb{N}}_0.\end{array}$$

(1)

As ${\varrho }_n>0$ and owing to the use of $\left({\mathsf{\Psi }}_1\right)$, we have ${\varrho }_{n+1}\le \psi ({\varrho }_n,{\varrho }_n,\cdots ,{\varrho }_n)<{\varrho }_n$$\forall n\in {\mathbb{N}}_0,$ so that $\left\{{\varrho }_n\right\}$ is a decreasing sequence in $(0,\infty )$, it is also bounded below by 0 (as ${\varrho }_n>0$). Therefore, there exists $\varrho \ge 0$ such that

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}{\varrho }_n=\varrho .\end{array}$$

(2)

We claim that

$$\underset{n\to \infty }{lim}{\varrho }_n=0.$$

On the contrary, assume that $\varrho >0$. Tending the limit superior in (1) and (2) (as $n\to \infty$) besides utilizing $\left({\mathsf{\Psi }}_1\right)$, we have

$$\varrho =\underset{n\to \infty }{lim}{\varrho }_{n+1}\le \underset{n\to \infty }{lim\; sup}\psi ({\varrho }_n,{\varrho }_n,\cdots ,{\varrho }_n)=\underset{{\varrho }_n\to \varrho }{lim\; sup}\psi ({\varrho }_n,{\varrho }_n,\cdots ,{\varrho }_n)<\varrho ,$$

which meets the contradiction. Hence, $\varrho =0.$ □

Proposition 5

([44]). Let $\psi \in \mathsf{\Phi }.$ If $\left\{{\varrho }_n\right\}\subset (0,\infty )$ is a sequence such that ${\varrho }_{n+1}\le \psi \left({\varrho }_n\right)\forall n\in {\mathbb{N}}_0$, then ${\displaystyle \underset{n\to \infty }{lim}{\varrho }_n=0.}$

Alam et al. [30] suggest that a relation-theoretic approach to nonlinear BCP is as under:

Theorem 3

([30]). Suppose that $(\mathfrak{X},d)$ is a metric space, ℜ is a relation on $\mathfrak{X}$, while $\mathfrak{f}:\mathfrak{X}\to \mathfrak{X}$ remains a mapping. Also, suppose the following hypotheses are accepted:

(a) 

$(\mathfrak{X},d)$ is ℜ-complete,

(b) 

ℜ is $\mathfrak{f}$-closed and locally finitely $\mathfrak{f}$-transitive,

(c) 

Either $\mathfrak{f}$ is ℜ-continuous or ℜ is d-self-closed,

(d) 

$\mathfrak{X}(\mathfrak{f},\Re )\ne \varnothing$,

(e) 

There exists $\psi \in \mathsf{\Phi }$, such that $d(\mathfrak{f}x,\mathfrak{f}y)\le \psi \left(d\right(\omega ,\chi \left)\right)\forall \omega ,\chi \in \mathfrak{X}$ with $(\omega ,\chi )\in \Re$.

Then, $Fix\left(\mathfrak{f}\right)\ne \varnothing$. Additionally, if $\mathfrak{f}\left(\mathfrak{X}\right)$ is ${\Re }^s$-connected, then $Fix\left(\mathfrak{f}\right)$ is singleton.

Last, but not least, we include the next two established lemmas, which we employ to prove our novel developments:

Lemma 1

([32]). If $\left\{{\omega }_n\right\}$ in $(\mathfrak{X},d)$ is not a Cauchy sequence, then there exist $ϵ>0$ and two sub-sequences $\left\{{\omega }_{n_p}\right\}$ and $\left\{{\omega }_{m_p}\right\}$ of $\left\{{\omega }_n\right\}$, such that

(i) 

$p\le m_p<n_k\forall p\in \mathbb{N}$,

(ii) 

$d({\omega }_{m_p},{\omega }_{n_p})\ge ϵ,$

(iii) 

$d({\omega }_{m_p},{\omega }_{{\varrho }_p})<ϵ\forall {\varrho }_p\in \{m_p+1,m_p+2,...,n_p-2,n_p-1\}.$

Additionally, if $\left\{{\omega }_n\right\}$ also verifies $\underset{n\to \infty }{lim}d({\omega }_n,{\omega }_{n+1})=0$, then

$$\underset{p\to \infty }{lim}d({\omega }_{m_p},{\omega }_{n_p+t})=ϵ\forall t\in {\mathbb{N}}_0.$$

Lemma 2

([38]). If ℜ is a “N-transitive” on $U:=\{{\varsigma }_n:n\in {\mathbb{N}}_0\}$ for some $N\in \mathbb{N}$ with $N\ge 2$ (provided $\left\{{\varsigma }_n\right\}\subset \mathfrak{X}$ with $({\varsigma }_n,{\varsigma }_{n+1})\in \Re$$\forall n\in {\mathbb{N}}_0$), then

$$({\varsigma }_n,{\varsigma }_{n+1+r(N-1)})\in \Re \forall n,r\in {\mathbb{N}}_0.$$

3. Main Results

Firstly, we determine that the nonlinear Prešić mapping possesses a fixed point under the class $\mathsf{\Psi }$ (already discussed):

Theorem 4.

Suppose that $(\mathfrak{X},d)$ is a metric space, ℜ is a relation on $\mathfrak{X}$ while $\mathfrak{f}:{\mathfrak{X}}^k\to \mathfrak{X}$ remains a Prešić mapping. Also, suppose the following hypotheses are accepted:

(a) 

$(\mathfrak{X},d)$ is ℜ-complete,

(b) 

${\mathfrak{X}}^k(\mathfrak{f},\Re )$ is nonempty,

(c) 

ℜ is $\mathfrak{f}$-closed, $\mathfrak{f}$-reflexive, and locally finitely $\mathfrak{f}$-transitive,

(d) 

either $\mathfrak{f}$ is ℜ-continuous or ℜ is d-self-closed,

(e) 

there exists $\psi \in \mathsf{\Psi }$, such that nonlinear Prešić contraction holds: $\mathrm{i}.\mathrm{e}.,$

$d(\mathfrak{f}({\omega }_1,{\omega }_2,\cdots ,{\omega }_k),\mathfrak{f}({\omega }_2,{\omega }_3,\cdots ,{\omega }_{k+1}))\le \psi (d({\omega }_1,{\omega }_2),d({\omega }_2,{\omega }_3),\cdots ,d({\omega }_k,{\omega }_{k+1}))\forall {\omega }_1,{\omega }_2,\cdots ,{\omega }_k,{\omega }_{k+1}\in \mathfrak{X}$ with $({\omega }_i,{\omega }_{i+1})\in \Re$ for $1\le i\le k+1$.

Then, $Fix\left(\mathfrak{f}\right)\ne \varnothing$.

Proof. 

We prove the result in four steps:

Step-1. We construct a ℜ-preserving sequence $\left\{{\omega }_n\right\}$ with the initial point ${\omega }_0\in \mathfrak{X}$ as follows:

As ${\mathfrak{X}}^k(\mathfrak{f},\Re )\ne \varnothing$. Fix ${\omega }_0\in {\mathfrak{X}}^k(\mathfrak{f},\Re )$ $\mathrm{i}.\mathrm{e}.,$ $({\omega }_0,\mathfrak{f}({\omega }_0,{\omega }_0,\cdots ,{\omega }_0))\in \Re$ as $\mathfrak{f}\left({\mathfrak{X}}^k\right)\subset \mathfrak{X}$ so ∃${\omega }_1\in \mathfrak{X}$, such that ${\omega }_1=\mathfrak{f}({\omega }_0,{\omega }_0,\cdots ,{\omega }_0)$. Therefore, $({\omega }_0,{\omega }_1)\in \Re$. As ℜ is $\mathfrak{f}$-closed, we obtain $(\mathfrak{f}({\omega }_0,{\omega }_0,\cdots ,{\omega }_0),\mathfrak{f}({\omega }_1,{\omega }_1,\cdots ,{\omega }_1))\in \Re .$ Again, as $\mathfrak{f}\left({\mathfrak{X}}^k\right)\subset \mathfrak{X}$, ∃${\omega }_2\in \mathfrak{X}$, such that ${\omega }_2=\mathfrak{f}({\omega }_1,{\omega }_1,\cdots ,{\omega }_1).$ Therefore, $({\omega }_1,{\omega }_2)\in \Re$. Inductively, we obtain $\left\{{\omega }_n\right\}$ in $\mathfrak{X}$, such that

$$\begin{array}{c}\hfill {\omega }_{n+1}=\mathfrak{f}({\omega }_n,{\omega }_n,\cdots ,{\omega }_n)\forall n\in {\mathbb{N}}_0.\end{array}$$

(3)

with

$$\begin{array}{c}\hfill ({\omega }_n,{\omega }_{n+1})\in \Re \forall n\in {\mathbb{N}}_0.\end{array}$$

(4)

Therefore, $\left\{{\omega }_n\right\}$ is ℜ-preserving.

Step-2. We show that ${lim}_{n\to \infty }d({\omega }_{n+1},{\omega }_n)=0$. Set ${\delta }_n:=d({\omega }_n,{\omega }_{n+1})$, $n\in {\mathbb{N}}_0$. We claim that

$$\begin{array}{c}\hfill {\displaystyle \underset{n\to \infty }{lim}{\delta }_n=\underset{n\to \infty }{lim}d({\omega }_{n+1},{\omega }_n)=0.}\end{array}$$

(5)

Now, if ${\delta }_{n_0}=d({\omega }_{n_0},{\omega }_{n_0+1})=0$ for some $n_0\in {\mathbb{N}}_0$, then in light of (3), we have $\mathfrak{f}({\omega }_{n_0},{\omega }_{n_0},\cdots ,{\omega }_{n_0})={\omega }_{n_0}$ so that ${\omega }_{n_0}\in Fix\left(\mathfrak{f}\right)$; hence, we are finished.

Otherwise, if ${\delta }_n>0$$\forall n\in {\mathbb{N}}_0,$ and employing triangular inequality, we obtain for all $n\in {\mathbb{N}}_0$,

$$\begin{array}{ccc}\hfill {\delta }_{n+1}=d({\omega }_{n+1},{\omega }_{n+2})& =& d(\mathfrak{f}({\omega }_n,{\omega }_n,\cdots ,{\omega }_n),\mathfrak{f}({\omega }_{n+1},{\omega }_{n+1},\cdots ,{\omega }_{n+1}))\hfill \\ & \le & d(\mathfrak{f}({\omega }_n,{\omega }_n,\cdots ,{\omega }_n),\mathfrak{f}({\omega }_n,\cdots ,{\omega }_n,{\omega }_{n+1}))\hfill \\ & & +d(\mathfrak{f}({\omega }_n,\cdots ,{\omega }_n,{\omega }_{n+1}),\mathfrak{f}({\omega }_n,\cdots ,{\omega }_{n+1},{\omega }_{n+1}))\hfill \\ & & +\cdots +d(\mathfrak{f}({\omega }_n,{\omega }_{n+1},\cdots ,{\omega }_{n+1}),\mathfrak{f}({\omega }_{n+1},{\omega }_{n+1},\cdots ,{\omega }_{n+1})).\hfill \end{array}$$

Now, when applying Prešić contraction $\left(e\right)$ to (4), the $\mathfrak{f}$-reflexivity of ℜ and using $\left({\mathsf{\Psi }}_3\right)$, then the above inequality deduces for all $n\in {\mathbb{N}}_0$ that

$$\begin{array}{ccc}\hfill {\delta }_{n+1}=d({\omega }_{n+1},{\omega }_{n+2})& \le & \psi (0,\cdots ,0,d({\omega }_n,{\omega }_{n+1}))+\psi (0,\cdots ,0,d({\omega }_n,{\omega }_{n+1}),0)\hfill \\ & & +\cdots +\psi (d({\omega }_n,{\omega }_{n+1}),0,\cdots ,0).\hfill \end{array}$$

so that

$$\begin{array}{ccc}\hfill {\delta }_{n+1}=d({\omega }_{n+1},{\omega }_{n+2})& =& \psi (0,\cdots ,0,{\delta }_n)+\psi (0,\cdots ,0,{\delta }_n,0)+\cdots ++\psi ({\delta }_n,0,\cdots ,0)\hfill \\ & \le & \psi ({\delta }_n,{\delta }_n,\cdots ,{\delta }_n).\hfill \end{array}$$

Thus, owing to use of Proposition 4, ${\displaystyle \underset{n\to \infty }{lim}{\delta }_n}$ exists and ${\displaystyle \underset{n\to \infty }{lim}{\delta }_n=0.}$

Step-3. We demonstrate that the sequence $\left\{{\omega }_n\right\}$ is Cauchy. Assuming $\left\{{\omega }_n\right\}$ is not Cauchy, then in light of Lemma 1, there exist $ϵ>0$, sub-sequences $\left\{{\omega }_{n_p}\right\}$ and $\left\{{\omega }_{m_p}\right\}$ of $\left\{{\omega }_n\right\}$, such that $p\le m_p<n_p$, $d({\omega }_{m_p},{\omega }_{n_p})\ge ϵ$ and $d({\omega }_{m_p},{\omega }_{{\varrho }_p})<ϵ$ where ${\varrho }_p\in \{m_p+1,m_p+2,...,n_p-2,n_p-1\}.$ Again, in light of Lemma 1 and (5), we infer

$$\begin{array}{c}\hfill \underset{p\to \infty }{lim}d({\omega }_{m_p},{\omega }_{n_{p+\varrho }})=ϵ\forall \varrho \in {\mathbb{N}}_0.\end{array}$$

(6)

In view of (1), $\left\{{\omega }_n\right\}\subset \mathfrak{f}\left({\mathfrak{X}}^k\right)$; hence, the range $U:=\{{\omega }_n:n\in {\mathbb{N}}_0\}$ ($\mathrm{i}.\mathrm{e}.,$$\left\{{\omega }_n\right\}$) is a countable subset of $\mathfrak{f}\left({\mathfrak{X}}^k\right)$). Since ℜ is locally finitely $\mathfrak{f}$-transitive, ∃$N\in \mathbb{N}$ such that $N=N\left(E\right)\ge 2$, ${\Re |}_U$ remains N-transitive.

As $m_p<n_p$ and $N-1>0$, we have then by the Division Rule—(“Given integers a and b, with $b>0$, the Division Rule states that there exist unique integers q and r satisfying $a=qb+r$ $0\le r<b$. The integers q and r are called the quotient and remainder, respectively (cf. [45])”).

$$n_p-m_p=(N-1)({\mu }_p-1)+(N-{\eta }_p)$$

$${\mu }_P-1\ge 0,0\le N-{\eta }_p<N-1.$$

$$⟺\left\{\begin{array}{c}{n}_p+{\eta }_P=m_p+1+(N-1){\mu }_p\hfill \\ {\mu }_p\ge 1,1<{\eta }_p\le N.\hfill \end{array}\right.$$

Here, ${\eta }_p$ can assume a finite integral number in $(1,N]$ since ${\mu }_p$ and ${\eta }_p$ are appropriate natural numbers. Thus, without a loss of generality, we can select two sub-sequences $\left\{{\omega }_{n_p}\right\}$ and $\left\{{\omega }_{m_p}\right\}$ of $\left\{{\omega }_n\right\}$ (verifying (6)), such that a constant ${\eta }_p$, denoted as $\eta$, that is independent of p. Write

$$\begin{array}{c}\hfill m_p^{\prime }:=n_p+\eta =m_p+1+(N-1){\mu }_p,\end{array}$$

(7)

where the constant is $\eta (1<\eta \le N)$. From (6) and (7), we conclude

$$\begin{array}{c}\hfill \underset{p\to \infty }{lim}d({\omega }_{m_p},{\omega }_{m_p^{\prime }})=\underset{p\to \infty }{lim}d({\omega }_{m_p},{\omega }_{n_p+\eta })=ϵ.\end{array}$$

(8)

Using triangular inequality, we obtain

$$\begin{array}{ccc}\hfill d({\omega }_{m_p},{\omega }_{m_p^{\prime }})& \le & d({\omega }_{m_p},{\omega }_{m_p+1})+d({\omega }_{m_{p+1}},{\omega }_{m_p^{\prime }+1})+d({\omega }_{m_p^{\prime }+1},{\omega }_{m_p^{\prime }}).\hfill \end{array}$$

(9)

and

$$\begin{array}{ccc}\hfill d({\omega }_{m_p+1},{\omega }_{m_p^{\prime }+1})& \le & d({\omega }_{m_p+1},{\omega }_{m_p})+d({\omega }_{m_p},{\omega }_{m_p^{\prime }})+d({\omega }_{m_p^{\prime }},{\omega }_{m_p^{\prime }+1}).\hfill \end{array}$$

(10)

Therefore, using (9) and (10), we have

$$\begin{array}{c}d({\omega }_{m_p},{\omega }_{m_p^{\prime }})-d({\omega }_{m_p},{\omega }_{m_p+1})-d({\omega }_{m_p^{\prime }+1},{\omega }_{m_p^{\prime }})\le d({\omega }_{m_{p+1}},{\omega }_{m_p^{\prime }+1})\hfill \\ \hfill \le d({\omega }_{m_p+1},{\omega }_{m_p})+d({\omega }_{m_p},{\omega }_{m_p^{\prime }})+d({\omega }_{m_p^{\prime }},{\omega }_{m_p^{\prime }+1}).\end{array}$$

Placing $p\to \infty$ in the above inequality and using (5) and (8), gives rise to

$$\begin{array}{c}\hfill \underset{p\to \infty }{lim}d({\omega }_{m_p+1},{\omega }_{m_p^{\prime }+1})=ϵ.\end{array}$$

(11)

Utilizing the triangular inequality, we get

$$\begin{array}{ccc}\hfill d({\omega }_{m_p+1},{\omega }_{m_p^{\prime }+1})& =& d(\mathfrak{f}({\omega }_{m_p},{\omega }_{m_p},\cdots ,{\omega }_{m_p}),\mathfrak{f}({\omega }_{m_p^{\prime }},{\omega }_{m_p^{\prime }},\cdots ,{\omega }_{m_p^{\prime }}))\hfill \\ & \le & d(\mathfrak{f}({\omega }_{m_p},{\omega }_{m_p},\cdots ,{\omega }_{m_p}),\mathfrak{f}({\omega }_{m_p},\cdots ,{\omega }_{m_p},{\omega }_{m_p^{\prime }}))\hfill \\ & & +d(\mathfrak{f}({\omega }_{m_p},\cdots ,{\omega }_{m_p},{\omega }_{m_p^{\prime }}),\mathfrak{f}({\omega }_{m_p},\cdots ,{\omega }_{m_p},{\omega }_{m_p^{\prime }},{\omega }_{m_p^{\prime }}))\hfill \\ & & +\cdots +d(\mathfrak{f}({\omega }_n,{\omega }_{m_p^{\prime }},\cdots ,{\omega }_{m_p^{\prime }}),\mathfrak{f}({\omega }_{m_p^{\prime }},{\omega }_{m_p^{\prime }},\cdots ,{\omega }_{m_p^{\prime }})).\hfill \end{array}$$

Denote ${\lambda }_p:=d({\omega }_{m_p},{\omega }_{m_p^{\prime }}).$ Lemma 2 and (7) lead us to a conclusion that $({\omega }_{m_p},{\omega }_{m_p^{\prime }})\in \Re$, again employing Prešić contraction $\left(e\right)$ to (4), $\mathfrak{f}$-reflexivity of ℜ and using $\left({\mathsf{\Psi }}_3\right)$, then, the above inequality deduces for all $p\in {\mathbb{N}}_0$ that

$$\begin{array}{ccc}\hfill d({\omega }_{m_p+1},{\omega }_{m_p^{\prime }+1})& \le & \psi (0,\cdots ,0,{\lambda }_p)+\psi (0,\cdots ,0,{\lambda }_p,0)+\cdots +\psi ({\lambda }_p,0,\cdots ,0)\hfill \\ & \le & \psi ({\lambda }_p,{\lambda }_p,\cdots ,{\lambda }_p)\hfill \end{array}$$

so that

$$\begin{array}{c}\hfill d({\omega }_{m_p+1},{\omega }_{m_p^{\prime }+1})\le \psi ({\lambda }_p,{\lambda }_p,\cdots ,{\lambda }_p).\end{array}$$

(12)

Utilizing ${\lambda }_p\to ϵ$ (w.r.t. real line) as $p⟶\infty$ (in view of (8)) and employing $\left({\mathsf{\Psi }}_1\right)$, we have

$$\begin{array}{c}\hfill \underset{p\to \infty }{lim\; sup}\psi ({\lambda }_p,{\lambda }_p,\cdots ,{\lambda }_p)=\underset{\rho \to ϵ}{lim\; sup}\psi (\rho ,\rho ,\cdots ,\rho )<ϵ.\end{array}$$

(13)

Applying the superior limit as $p\to \infty$ in (12), as well as utilizing (11) and (13), we obtain

$$ϵ=\underset{p\to \infty }{lim\; sup}d({\omega }_{m_p+1},{\omega }_{m_p^{\prime }+1})\le \underset{p\to \infty }{lim\; sup}\psi ({\lambda }_p,{\lambda }_p,\cdots ,{\lambda }_p)<ϵ,$$

which meets the contradiction. Therefore, $\left\{{\omega }_n\right\}$ is a Cauchy sequence that enjoys (4). Considering the ℜ-completeness of $(\mathfrak{X},d)$, ∃$\omega \in \mathfrak{X}$ with ${\omega }_n\stackrel{d}{⟶}\omega$.

Step-4. We demonstrate that a fixed point of $\mathfrak{f}$ is $\omega$. To validate this, consider ℜ-continuity of $\mathfrak{f}$. As $\left\{{\omega }_n\right\}$ is ℜ-preserving with ${\omega }_n\stackrel{d}{\to }w$, the ℜ-continuity of $\mathfrak{f}$ yields that ${\omega }_{n+1}=\mathfrak{f}({\omega }_n,{\omega }_n,\cdots ,{\omega }_n)\stackrel{d}{⟶}\mathfrak{f}(\omega ,\omega ,\cdots ,\omega )$, as ${\omega }_n\to w$ (as $n\to \infty$) uniquely in ℜ-complete space $(\mathfrak{X},d)$, we obtain $\mathfrak{f}(\omega ,\omega ,\cdots ,\omega )=\omega$, $\mathrm{i}.\mathrm{e}.,Fix\left(\mathfrak{f}\right)\ne \varnothing$.

Alternately, consider d-self-closedness of ℜ. As $\left\{{\omega }_n\right\}$ is ℜ-preserving such that ${\omega }_n\stackrel{d}{⟶}\omega$, which implies that there exists $\left\{{\omega }_{n_p}\right\}$ of $\left\{{\omega }_n\right\}$ with $[{\omega }_{n_p},\omega ]\in \Re (\forall p\in {\mathbb{N}}_0)$. On using Prešić contraction $\left(e\right)$, Proposition 3 and $[{\omega }_{n_p},\omega ]\in \Re$ with ${\omega }_{n_p}\stackrel{d}{⟶}\omega$, we get

$$\begin{array}{ccc}\hfill d({\omega }_{n_p+1},\mathfrak{f}(\omega ,\omega ,\cdots ,\omega ))& =& d(\mathfrak{f}({\omega }_{n_p},{\omega }_{n_p},\cdots ,{\omega }_{n_p}),\mathfrak{f}(\omega ,\omega ,\cdots ,\omega ))\hfill \\ & \le & \psi (d({\omega }_{n_p},\omega ),d({\omega }_{n_p},\omega ),\cdots ,d({\omega }_{n_p},\omega ))\forall p\in {\mathbb{N}}_0.\hfill \end{array}$$

We assert that

$$\begin{array}{ccc}\hfill d({\omega }_{n_p+1},\mathfrak{f}(\omega ,\omega ,\cdots ,\omega ))& \le & d({\omega }_{n_p},\omega )\forall p\in \mathbb{N}.\hfill \end{array}$$

Consider the partition of $\mathbb{N}$ as ${\mathbb{N}}^0\cup {\mathbb{N}}^{+}=\mathbb{N}$ and ${\mathbb{N}}^0\cap {\mathbb{N}}^{+}=\varnothing$ (mutually disjoint) in order to demonstrate the assertion.

(i) 

$d({\omega }_{n_p},\omega )=0\forall p\in {\mathbb{N}}^0,$

(ii) 

$d({\omega }_{n_p},\omega )>0\forall p\in {\mathbb{N}}^{+}$.

For case $\left(i\right)$, we have $d(\mathfrak{f}({\omega }_{n_p},{\omega }_{n_p},\cdots ,{\omega }_{n_p}),\mathfrak{f}(\omega ,\omega ,\cdots ,\omega ))=0\le d({\omega }_{n_p},\omega )$$\forall p\in {\mathbb{N}}^0.$ In case $\left(ii\right)$ using Prešić contraction $\left(e\right)$ to $[{\omega }_{n_p},\omega ]\in \Re$ and in the light of $\left({\mathsf{\Psi }}_2\right)$, we obtain $d({\omega }_{n_p+1},\mathfrak{f}(\omega ,\omega ,\cdots ,\omega ))\le \psi (d({\omega }_{n_p},\omega ),d({\omega }_{n_p},\omega ),\cdots ,d({\omega }_{n_p},\omega ))<d({\omega }_{n_p},\omega )$ for all $p\in {\mathbb{N}}^{+}.$ Thus, in all cases, we obtain $d({\omega }_{n_p+1},\mathfrak{f}(\omega ,\omega ,\cdots ,\omega ))\le d({\omega }_{n_p},\omega )\forall p\in \mathbb{N}$. Now, using ${\omega }_{n_p}\stackrel{d}{⟶}\omega$ as $p\to \infty$ gives rise ${\omega }_{n_p+1}\stackrel{d}{⟶}\mathfrak{f}(\omega ,\omega ,\cdots ,\omega )$. Owing to the unique property of the convergence sequence, we have $\mathfrak{f}(\omega ,\omega ,\cdots ,\omega )=\omega$ so that $Fix\left(\mathfrak{f}\right)\ne \varnothing$. This concludes the proof. □

Corollary 1.

If in the hypotheses of Theorem 4, locally finitely $\mathfrak{f}$-transitivity of ℜ is replaced by locally finitely transitivity or lacally $\mathfrak{f}$-transtivity of ℜ, then $Fix\left(\mathfrak{f}\right)\ne \varnothing$.

Proof. 

As every locally finitely transitive relation is locally finitely $\mathfrak{f}$-transitive. Also, every lacally $\mathfrak{f}$-transtive relation is locally finitely $\mathfrak{f}$-transitive. Thus, in both cases. ℜ is locally finitely $\mathfrak{f}$-transitive. Hence, the conclusion follows from Theorem 4. □

We offer a uniqueness result that is consistent with Theorem 4.

Theorem 5.

Taking into account every hypothesis from Theorem 4, provided that the following hypothesis is accepted:

(u) 

: $\mathfrak{f}\left({\mathfrak{X}}^k\right)$ is ${\Re }^s$-connected,

then $Fix\left(\mathfrak{f}\right)$ is singleton.

Proof. 

In light of Theorem 4, $Fix\left(\mathfrak{f}\right)\ne \varnothing$. Choose $\omega ,\chi \in Fix\left(\mathfrak{f}\right)$, then

$$\begin{array}{c}\hfill \mathfrak{f}(\omega ,\omega ,\cdots ,\omega )=\omega \mathrm{and}\mathfrak{f}(\chi ,\chi ,\cdots ,\chi )=\chi .\end{array}$$

(14)

From the assumption $\left(u\right)$, ∃ a path of some finite length l in ${\Re }^s$ from $\omega$ to $\chi$ (say: $\{{\varsigma }_0,{\varsigma }_1,{\varsigma }_2,...,{\varsigma }_l\}$), so that

$$\begin{array}{c}\hfill {\varsigma }_0=\omega ,{\varsigma }_l=\chi \mathrm{and}[{\varsigma }_i,{\varsigma }_{i+1}]\in \Re ,\mathrm{for}\mathrm{each}i\in \{0,1,2,\cdots ,l-1\}.\end{array}$$

(15)

As ℜ is $\mathfrak{f}$-closed and using Propositions 1 and 2, for each $i\in \{0,1,2,\cdots ,l-1\}$, we have

$$\begin{array}{c}\hfill [\mathfrak{f}({\varsigma }_i,{\varsigma }_i,\cdots ,{\varsigma }_i),\mathfrak{f}({\varsigma }_{i+1},{\varsigma }_{i+1},\cdots ,{\varsigma }_{i+1})]\in \Re .\end{array}$$

(16)

Setting ${\chi }^i=\mathfrak{f}({\varsigma }_i,{\varsigma }_i,\cdots ,{\varsigma }_i)$ and ${\chi }_{n+1}^i=\mathfrak{f}({\chi }_n^i,{\chi }_n^i,\cdots ,{\chi }_n^i),$ $\mathrm{for}\mathrm{each}i\in \{0,1,2,\cdots ,l-1\}$ and for each n $\in {\mathbb{N}}_0$. We claim that

$$\begin{array}{c}\hfill [{\chi }_n^i,{\chi }_n^{i+1}]\in \Re (i\in \{0,1,2,\cdots ,l-1\}\mathrm{and}\forall n\in {\mathbb{N}}_0).\end{array}$$

(17)

For $n=0$, ${\chi }_0^i={\chi }^i$ for each i, $(0\le i\le l-1)$ and using (16), so (17) is true for $n=0$. Suppose (17) is valid for $n=r$ $\mathrm{i}.\mathrm{e}.,$ $[{\chi }_r^i,{\chi }_r^{i+1}]\in \Re (i\in \{0,1,2,\cdots ,l-1\}$). Again, utilizing $\mathfrak{f}$-closedness of ℜ, we obtain $[{\chi }_{r+1}^i,{\chi }_{r+1}^{i+1}]\in \Re$. Hence, the claim is valid for $n=r+1.$ Thus, inductively (17) is valid for all $n\in {\mathbb{N}}_0.$

Set ${\varrho }_n^i:=d({\chi }_n^i,{\chi }_n^{i+1})$. We assert that

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}{\varrho }_n^i=0,\mathrm{for}\mathrm{each}i\in \{0,1,2,\cdots ,l-1\}.\end{array}$$

(18)

Fix $i\in \{0,1,2,\cdots ,l-1\}$, we discuss two cases. Firstly, choose ${\varrho }_{n_0}^i=d({\chi }_{n_0}^i,{\chi }_{n_0}^{i+1})=0$ for some $n_0\in {\mathbb{N}}_0$, $\mathrm{i}.\mathrm{e}.,$ $\mathfrak{f}({\chi }_{n_0}^i,{\chi }_{n_0}^i,\cdots ,{\chi }_{n_0}^i)=\mathfrak{f}({\chi }_{n_0}^{i+1},{\chi }_{n_0}^{i+1},\cdots ,{\chi }_{n_0}^{i+1})$, which gives rise to $\mathfrak{f}({\chi }_{n_0+1}^i,{\chi }_{n_0+1}^i,\cdots ,{\chi }_{n_0+1}^i)=\mathfrak{f}({\chi }_{n_0+1}^{i+1},{\chi }_{n_0+1}^{i+1},\cdots ,{\chi }_{n_0+1}^{i+1})$. Consequently, we obtain ${\varrho }_{n_0+1}^i=d({\chi }_{n_0+1}^i,{\chi }_{n_0+1}^{i+1})=0$. Hence, from the induction on n, we obtain ${\varrho }_n^i=0\forall n\ge n_0,$ so that $\underset{n\to \infty }{lim}{\varrho }_n^i=0$. Secondly, assume that ${\varrho }_n^i>0\forall n\in {\mathbb{N}}_0$. Then, employing Prešić contraction $\left(e\right)$ to (17) and triangle inequality, we obtain

$$\begin{array}{c}\hfill {\varrho }_{n+1}^i=d({\chi }_{n+1}^i,{\chi }_{n+1}^{i+1})=d(\mathfrak{f}({\chi }_n^i,{\chi }_n^i,\cdots ,{\chi }_n^i),\mathfrak{f}({\chi }_n^{i+1},{\chi }_n^{i+1},\cdots ,{\chi }_n^{i+1}))\le \psi ({\varrho }_n^i,{\varrho }_n^i,\cdots ,{\varrho }_n^i)\end{array}$$

so that

$$\begin{array}{c}\hfill {\varrho }_{n+1}^i\le \psi ({\varrho }_n^i,{\varrho }_n^i,\cdots ,{\varrho }_n^i).\end{array}$$

(19)

Tending $n\to \infty$ in (19) and using Proposition 4, we have

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

Thus, in each case, (18) is proven.

In light of (14), (15), (17), (18) and triangular property of d, we have

$$d(\omega ,\chi )=d(\mathfrak{f}({\chi }_n^0,{\chi }_n^0,\cdots ,{\chi }_n^0),\mathfrak{f}({\chi }_n^l,{\chi }_n^l,\cdots ,{\chi }_n^l))\le {\varrho }_n^0+{\varrho }_n^1+\cdots +{\varrho }_n^{l-1}\to 0asn\to \infty$$

so that $\omega =\chi$. Thus, $Fix\left(\mathfrak{f}\right)$ is singleton. □

It is required emphasized that ℜ is defined as complete ($\mathrm{i}.\mathrm{e}.,$ “ℜ is complete”), if $(\omega ,\chi )\in \Re$ or $(\chi ,\omega )\in \Re$ for all $\omega ,\chi \in \mathfrak{X}$.

Corollary 2.

Taking into account every hypothesis from Theorem 5, if we replace condition $\left(u\right)$ by one of the following conditions:

(u′) 

${\Re |}_{\mathfrak{f}\left({\mathfrak{X}}^k\right)}$ is complete,

(u″) 

$\mathfrak{f}\left({\mathfrak{X}}^k\right)$ is ${\Re }^s$-directed.

Then, $Fix\left(\mathfrak{f}\right)$ is singleton.

We omit the proof of the above result as it is similar to that of Corollary 3.4 evidence given in [31].

We immediately finish a few special demonstrations, which are identified as fixed-point results in the literature.

(1)

For $k=1,$ and taking $\Re ={\mathfrak{X}}^2$, Theorem 5 reduces to a nonlinear fixed point Theorem.

(2)

Upon setting the $\Re ={\mathfrak{X}}^2$ (universal relation) and $\psi ({\varrho }_1,{\varrho }_2,\cdots ,{\varrho }_k)={\sum }_{i=1}^k{\alpha }_i{\varrho }_i$ with ${\sum }_{i=1}^k{\alpha }_i<1,$ in Theorem 5, we conclude Theorem 1 (due to Prešić [5]).

(3)

For $k=1,$ and locally finitely $\mathfrak{f}$-transitive relation (without $\mathfrak{f}$-reflexivity of ℜ), in Theorem 5, we obtained Theorem 3 (due to Alam et al. [30]). Henceforth, the results are contained in [32,35].

(4)

If we replace the condition $\left({\mathsf{\Psi }}_1\right)$ of $\mathsf{\Psi }$ by ${\displaystyle \left({\mathsf{\Psi }}_1^{\prime }\right):=\{\underset{n\to \infty }{lim\; sup}\psi ({\varrho }_n,{\varrho }_n,\cdots ,{\varrho }_n)\le \psi (t,t,\cdots ,t),\mathrm{whenever}{\varrho }_n\in {\mathbb{R}}_{+}\forall n\in \mathbb{N}\mathrm{and}{\varrho }_n\downarrow t\ge 0\}}$, denote the class of such functions satisfying ${\mathsf{\Psi }}_2,{\mathsf{\Psi }}_3,\mathrm{and}{\mathsf{\Psi }}_1^{\prime }$ by $\mathsf{\Theta }$. However, if we replace the class $\mathsf{\Psi }$ by class $\mathsf{\Theta },$ then the earlier utilized relation (such as locally finitely $\mathfrak{f}$-transitive relation) does not hold. In this case, we employ class $\mathsf{\Theta }$ along with locally $\mathfrak{f}$-transitive relation in Theorem 5, and we deduce a sharpened version (in the context that ℜ is not a partial order) of Corollary 6 due to Shukla and Radenović [39].

4. Illustrative Examples

Ultimately, we implement a few examples to show how Theorems 4 and 5 are more useful than the comparable previously established results.

Example 1.

Consider that $\mathfrak{X}=[0,\infty )$ enjoys the standard metric $d.$ Let $\Re :=\{(0,0),(1,2),(2,3),(3,0),(1,0),(2,0)\}\cup \{(\omega ,\chi )\in {\mathfrak{X}}^2:\omega -\chi \ge 0\forall \omega ,\chi \in [0,1)\}$ be a relation. For $k=2$, $\mathrm{i}.\mathrm{e}.,$ on ${\mathfrak{X}}^2$, define a Prešić mapping $\mathfrak{f}:{\mathfrak{X}}^2\to \mathfrak{X}$, by $\mathfrak{f}(\omega ,\chi )={\displaystyle \frac{\omega +\chi }{1+\omega +\chi }}\forall \omega ,\chi \in \mathfrak{X}$. Notice that, ℜ is neither reflexive nor transitive, but ℜ is $\mathfrak{f}$-reflexive and locally finitely $\mathfrak{f}$-transitive. Also, ℜ is $\mathfrak{f}$-closed and $(\mathfrak{X},d)$ is ℜ-complete. Consider an auxiliary map $\psi :[0,\infty )\times [0,\infty )\to [0,\infty )$ by $\psi ({\varrho }_1,{\varrho }_2)={\displaystyle \frac{{\varrho }_1+{\varrho }_2}{1+{\varrho }_1+{\varrho }_2}}\forall {\varrho }_1,{\varrho }_2\in [0,\infty ).$ Clearly, $\psi \in \mathsf{\Psi }\subset \mathsf{\Theta }.$ For all $(\omega ,\chi )\in \Re$, nonlinear Prešić condition $\left(e\right)$ can be easily verified. For any sequence $\left\{{\omega }_n\right\}$ in $\mathfrak{X}$ enjoying

$$\begin{array}{c}\hfill ({\omega }_n,{\omega }_{n+1})\in \Re ,\forall n\in \mathbb{N}\mathrm{with}{\omega }_n\stackrel{d}{⟶}\omega ,\end{array}$$

we have $\omega \in [0,1)$. Hence, we can choose sub-sequence $\left\{{\omega }_{n_k}\right\}$ of the sequence $\left\{{\omega }_n\right\}$, such that $[{\omega }_{n_k},x]\in \Re$$\forall k\in \mathbb{N}$. Therefore, ℜ is d-self-closed. Theorem 4’s assumptions are all met. Note that $Fix\left(\mathfrak{f}\right)\ne \varnothing$. Clearly, ${\Re |}_{\mathfrak{f}\left(\mathfrak{X}\right)}$ is complete. Hence Corollary 2 implies a guarantee of the uniqueness, so that $Fix\left(\mathfrak{f}\right)$ is singleton (namely $Fix\left(\mathfrak{f}\right)=\left\{0\right\}$). But, Corollary 6 of Shukla and Radenović [39] cannot be applied to the present situation because ℜ is not transitive, indicating that ℜ is not a partial order. This supports the importance of our findings.

Example 2.

Let’s assume that $\mathfrak{X}=[0,1]$ is equipped with the standard metric d. For $k=2$, define a Prešić map $\mathfrak{f}:{\mathfrak{X}}^2\to \mathfrak{X}$ by

$$\mathfrak{f}(\omega ,\chi )=\left\{\begin{array}{cc}{\displaystyle \frac{\omega }{\omega +1}},\hfill & (\omega ,\chi )\in \{(\omega ,\chi )\in {\mathfrak{X}}^2:\forall \omega ,\chi \in \mathfrak{X}\}\setminus \{(\omega ,\omega )\in {\mathfrak{X}}^2:\forall \omega \in [0,1]\},\hfill \\ {\displaystyle \frac{\chi }{\chi +1}},\hfill & (\omega ,\chi )\in \{(\omega ,\omega )\in {\mathfrak{X}}^2:\forall \omega \in [0,1]\}.\hfill \end{array}\right.$$

Let $\Re :=\{(\omega ,\chi )\in {\mathfrak{X}}^2:\omega -\chi >0\}\cup \{(\omega ,\omega )\in {\mathfrak{X}}^2:\forall \omega \in [0,1)\}\subset {\mathfrak{X}}^2$, then ℜ is not reflexive, but it is $\mathfrak{f}$-reflexive and transitive. Since transitivity implies locally finitely $\mathfrak{f}$-transitivity. Clearly, $(\mathfrak{X},d)$ is ℜ-complete, $0\in {\mathfrak{X}}^2(\mathfrak{f},\Re )$ and ℜ is $\mathfrak{f}$-closed. Now, define a control map $\psi :[0,\infty )\times [0,\infty )\to [0,\infty )$ by $\psi ({\varrho }_1,{\varrho }_2)={\displaystyle \frac{{\varrho }_1}{1+|\frac{{\varrho }_1}{2}-{\varrho }_2|}}\forall {\varrho }_1,{\varrho }_2\in [0,\infty ).$ It can be easily seen that ψ is a member of $\mathsf{\Psi }.$ Now, for all $(\omega ,\chi )\mathrm{and}(\chi ,\varsigma )\in \Re$, we obtain

$$\begin{array}{ccc}\hfill d(\mathfrak{f}(\omega ,\chi ),\mathfrak{f}(\chi ,\varsigma ))=|{\displaystyle \frac{\omega }{1+\omega }}-{\displaystyle \frac{\chi }{1+\chi }}|& =& \left|{\displaystyle \frac{(\omega -\chi )}{1+\omega +\chi +\omega \chi }}\right|\hfill \\ & \le & {\displaystyle \frac{(\omega -\chi )}{1+(\omega -\chi )+(\chi -\varsigma )}}\hfill \\ & \le & {\displaystyle \frac{(\omega -\chi )}{1+\frac{(\omega -\chi )}{2}+(\chi -\varsigma )}}\hfill \\ & \le & {\displaystyle \frac{(\omega -\chi )}{1+|\frac{(\omega -\chi )}{2}-(\chi -\varsigma )|}}\hfill \\ & =& {\displaystyle \frac{d(\omega ,\chi )}{1+|\frac{d(\omega ,\chi )}{2}-d(\chi ,\varsigma )|}}\hfill \end{array}$$

so that

$$d\left(\mathfrak{f}\right(\omega ,\chi ),\mathfrak{f}(\chi ,\varsigma \left)\right)\le \psi \left(d\right(\omega ,\chi ),d(\chi ,\varsigma \left)\right).$$

Hence, $\mathfrak{f}$ satisfies the hypothesis $\left(e\right)$ (w. r. t. ψ) of Theorem 4. Moreover, the remaining conditions of Theorem 5 can easily be verified, leading one to conclude that $Fix\left(\mathfrak{f}\right)=\left\{0\right\}$.

Notice that for sufficiently small positive ϵ and taking $\omega =ϵ$ and $y=z=0$, for the linear Prešić contraction ($k=2$) of Theorem 1, we have

$${\displaystyle \frac{ϵ}{1+ϵ}}=d(\mathfrak{f}(ϵ,0),\mathfrak{f}(0,0))\le {\alpha }_{1}d(ϵ,0)+{\alpha }_{2}d(0,0)={\alpha }_1ϵ,$$

which implies that

$${\displaystyle \frac{1}{1+ϵ}}\le {\alpha }_1.$$

As ϵ is any positive number, when ϵ tends to 0, it yields ${\alpha }_1\ge 1$, which contradicts the fact that ${\alpha }_1+{\alpha }_2<1$. Therefore, $\mathfrak{f}$ is not linear Prešić contraction (for $k=2$). Hence, Example 2 cannot be covered by Theorem 1 (due to Prešić [5]). This shows the genuineness of our newly obtained findings.

5. Conclusions

In the presented work, we show the existence as well as uniqueness of fixed point for nonlinear contraction-type mappings in the metric space, wherein transitive binary relations are provided in a weakened form within the metric space namely the locally finitely $\mathfrak{f}$-transitive relation. We also show the utility of our findings by adopting some examples over earlier known corresponding findings in the existing literature. Further, it has already been pointed out that our findings, via specific binary relation and family of control functions are a genuine extension of several noted findings contained in Prešić [5], Shukla and Radenović [39], and Alam et al. [30]. For further development of our established findings, our results should be generalized via ambient relational metric spaces using employing different classes of control functions, using Branciari distance spaces, b-metric spaces, partial metric spaces, and various other generalized metric spaces $\mathrm{e}\mathrm{t}\mathrm{c}$.