Fixed Point Theorems via Binary Relations and Generalized Altering Distance Functions with Applications to Boundary Value Problems

Abstract

This manuscript seeks to study couple fixed-point findings over a relational extended contraction through a locally, finitely transitive binary relation and a pair of generalized altering distance functions. Various prior discoveries are expanded, refined, modified, compiled, and enhanced by our findings. We craft some instances for demonstrating the practical relevance of our findings. We embark on our research in an attempt to recognize a unique solution to a boundary value problem connected with a second-order ordinary differential equation.

1. Introduction

Metric fixed point theory is reliable, as its concepts may be adapted to numerous different fields. The classical BCP appears as a considerable and primordial achievement of metric fixed point theory. Fortunately, BCP promises the availability of a unique fixed point for a contraction inequality defined in a complete MS. This outcome further incorporates an iteration technique to estimate the unique fixed point. BCP has continued to generalize in an extensive variety of aspects throughout the past century.

Alam and Imdad [1] initiated an inventive and instinctual rendition of BCP, whereby MS comprises a BR and the given mapping preserves BR. Numerous discoveries have been made along these lines. We quote a handful of them through [2,3,4,5,6,7,8,9,10,11,12] and references therein. These relational contractions remain weaker than Banach contractions, especially as they are applied to elements that are connected via BR. These findings deduce the classical fixed point results under universal BR. These outcomes serve a purpose for detecting unique solutions to certain sorts of matrix equations, BVP, and nonlinear integral equations.

The proposal of an extended contraction map was recently laid out by Pant [13], who offered the following fixed point result.

Theorem 1 ([13]).

Let ϝ be a self-map on a complete MS $(\mathbf{H},e)$. If $\kappa \in [0,1)$ is a constant that verifies the following:

(1)$\begin{array}{c}\hfill e(\mathsf{ϝ}z,\mathsf{ϝ}w))\le \kappa ·e(z,w),\forall z,w\in \mathbf{H},with[z\ne \mathsf{ϝ}\left(z\right)orw\ne \mathsf{ϝ}\left(w\right)],\end{array}$

Then, ϝ admits a fixed point.

Moreover, the corresponding unique fixed point result is also established, as indicated below.

Theorem 2 ([13]).

Alongside the presumptions of Theorem 1, ϝ admits a unique fixed point iff the condition (1) is satisfied for every pair $z\ne w$ in $\mathbf{H}$.

The class of extended contractions of Pant is so large that it also covers some classes of nonexpansive maps. The domain and fixed point set of maps verifying Theorem 1 admit interesting algebraic, geometric, and dynamical features. Indeed, extended contraction maps either incorporate a unique fixed point or all points eventually become fixed points; in the latter scenario, it takes only a limited number of iterations to attain the fixed point from any initial point. As an application, a strategy for calculating the cardinality of the fixed point sets of such mappings was also offered by Pant [13].

The extended versions of Boyd–Wong and Matkowski contractions were investigated by Pant [14], who utilized them to determine the unique solution of a second-order BVP. In the same continuation, Pant [15] developed the above idea for $(ϵ-\delta )$–nonexpansive maps, working with this to present certain fixed point theorems. In the setup of relational MS, the extended versions of almost nonlinear contractions were described by Alshaban et al. [16], Boyd–Wong contractions were investigated by Filali and Khan [17], and almost Matkowski contractions were presented by Filali et al. [18].

A function $\alpha :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ is outlined as an altering distance function (cf. [19]) if the following are true:

(i). ${\alpha }^{-1}\left(0\right)=0$;

(ii). $\alpha$ remains increasing and continuous.

In conjunction with the aforementioned concept, Khan et al. [19] explored the following extension of BCP.

Theorem 3 ([19]).

Let $(\mathbf{H},e)$ be a complete MS and ϝ: $\mathbf{H}\to \mathbf{H}$ a map. If ∃ a constant $0\le \kappa <1$ and an altering distance function α that enjoy the following:

(2)$\alpha \left(e\right(\mathsf{ϝ}z,\mathsf{ϝ}w\left)\right)\le \kappa ·\alpha \left(e\right(z,w\left)\right),\forall z,w\in \mathbf{H}.$

Then, ϝ possesses a unique fixed point.

In particular, Theorem 3 turns into the usual BCP for $\alpha =I$ (identity map). Yan et al. [20] generalized the concept of altering distance functions in 2012 by inserting another function $\beta$ associated with a weaker class of altering distance functions for the $c·\alpha$ occurring in the RHS of (2). Being more precise, they proposed the following contraction criterion:

(3)$\alpha \left(e\right(\mathsf{ϝ}\mathrm{z},\mathsf{ϝ}\mathrm{w}\left)\right)\le \beta \left(e\right(\mathrm{z},\mathrm{w}\left)\right).$

Notably, both auxiliary functions fulfil the linked requirement $\beta \left(t\right)<\alpha \left(t\right)$, $\forall t>0$. The findings of Yan et al. [20] were enhanced by Alsulami et al. [21] and Su [22] by influencing the characteristics of the auxiliary functions utilized in the contraction inequality (3). As a follow-up, Sawangsup and Sintunavarat [23] demonstrated, in relational MS, a specific pair of generalized altering distance functions and employed them to analyze a nonlinear matrix equation.

The primary emphasis in this study is to improve the results of Pant [13] by extending the contraction condition and the (ambient) MS, which are worth noting in the following respects:

The setup of ordinary MS” utilized in Pant’s Theorem [13] is enlarged to relational MS”. To ascertain the existence of fixed points for nonlinear contractions, transitivity of the underlying BR is additionally required. But as the transitivity requirement is very restrictive, we utilize an optimal condition of transitivity, namely, locally finitely ϝ-transitive BR.

The extended linear contraction condition (1) is generalized to an extended nonlinear contraction associated with a pair of generalized altering distance functions. Furthermore, our contraction condition is preserved with the underlying BR. The geometry of the involved BR reveals that the contraction condition initiated here must be satisfied only for relationally comparable elements rather than for all elements.

Employing the above-mentioned ambient space and contraction condition, we present results on the validity and uniqueness of fixed points. We provide a few exemplary instances to clarify the key findings. To assist with our insights, we address a result dealing with the occurrence of a unique solution of a certain two-point BVP affiliated with an ODE of order two.

2. Preliminaries

A BR $\mathbf{S}$ on a set $\mathbf{H}$ is defined to be a subset of ${\mathbf{H}}^2$. In upcoming notions, $\mathbf{H}$ is ambient set, e is metric on $\mathbf{H}$, $\mathbf{S}$ is BR on $\mathbf{H}$ and ϝ: $\mathbf{H}\to \mathbf{H}$ is a map. We say the following:

Definition 1 ([1]).

$z,w\in \mathbf{H}$ remain $\mathbf{S}$-comparative, represented by $[z,w]\in \mathbf{S}$, provided

$(z,w)\in \mathbf{S}or(w,z)\in \mathbf{S}.$

Definition 2 ([24]).

${\mathbf{S}}^{-1}:=\{(z,w)\in {\mathbf{H}}^2:(w,z)\in \mathbf{S}\}$ serves as inverse of $\mathbf{S}$.

Definition 3 ([24]).

${\mathbf{S}}^s:=\mathbf{S}\cup {\mathbf{S}}^{-1}$ serves as symmetric closure of $\mathbf{S}$.

Proposition 1 ([1]).

$(z,w)\in {\mathbf{S}}^s⟺[z,w]\in \mathbf{S}.$

Definition 4 ([24]).

A BR on $\mathbf{K}\subseteq \mathbf{H}$ described by

${\mathbf{S}|}_{\mathbf{K}}:=\mathbf{S}\cap {\mathbf{K}}^2$

is restriction of $\mathbf{S}$ to $\mathbf{K}$.

Definition 5 ([1]).

$\mathbf{S}$ is named as ϝ-closed if the following is true:

$(\mathsf{ϝ}z,\mathsf{ϝ}w)\in \mathbf{S},\forall z,w\in \mathbf{H};(z,w)\in \mathbf{S}.$

Definition 6 ([1]).

A sequence $\left\{z_n\right\}\subset \mathbf{H}$ is $\mathbf{S}$-preserving if $(z_n,z_{n+1})\in \mathbf{S}$∀$n\in {\mathbb{N}}_0$.

Definition 7 ([2]).

$(\mathbf{H},e)$ retains $\mathbf{S}$-complete MS provided every $\mathbf{S}$-preserving Cauchy sequence converges.

Definition 8 ([2]).

ϝ remains $\mathbf{S}$-continuous provided for any $z\in \mathbf{H}$ and any $\mathbf{S}$-preserving sequence $\left\{z_n\right\}\subset \mathbf{H}$ verifying $z_n\stackrel{e}{⟶}z$,

$\mathsf{ϝ}\left(z_n\right)\stackrel{e}{⟶}\mathsf{ϝ}\left(z\right).$

Definition 9 ([1]).

$\mathbf{S}$ serves as e-self-closed provided for every $\mathbf{S}$-preserving convergent sequence, the terms of some subsequence of it retain $\mathbf{S}$-comparative with the limit of the sequence.

Definition 10 ([25]).

Given $z,w\in \mathbf{H}$, a finite set $\{r_0,r_1,\dots ,r_l\}$⊂$\mathbf{H}$ retains a path (with length $l\in \mathbb{N}$) in $\mathbf{S}$ from $z$ to $w$ provided the following are true:

(i). $r_0=z$ and $r_l=w$;

(ii). $(r_{𝚤},r_{𝚤+1})$∈$\mathbf{S},0\le 𝚤\le l-1$.

Definition 11 ([4]).

A set $\mathbf{K}\subseteq \mathbf{H}$ retains an $\mathbf{S}$-connected set if any two elements of $\mathbf{K}$ joins a path.

Definition 12 ([26]).

Given $\aleph \in \mathbb{N}\setminus \left\{1\right\}$, $\mathbf{S}$ is ℵ-transitive if for any $z_0,z_1,\dots ,z_n\in \mathbf{H}$ verifying $(z_{i-1},z_i)\in \mathbf{S}$, for $1\le i\le \aleph ,$ we have the following:

$(z_0,z_n)\in \mathbf{S}.$

Definition 13 ([27,28]).

$\mathbf{S}$ remains a finitely transitive BR provided $\exists \aleph \in \mathbb{N}\setminus \left\{1\right\}$ for which $\mathbf{S}$ is ℵ-transitive.

Definition 14 ([3]).

$\mathbf{S}$ remains locally finitely ϝ-transitive provided for every countable subset $\mathbf{K}\subseteq \mathsf{ϝ}\left(\mathbf{H}\right)$, ${\mathbf{S}|}_{\mathbf{K}}$ retains finitely transitive.

Proposition 2 ([4]).

If $\mathbf{S}$ remains ϝ-closed, then for each $n\in {\mathbb{N}}_0$, $\mathbf{S}$ must be ${\mathsf{ϝ}}^n$-closed.

Lemma 1 ([29]).

Let $\left\{z_n\right\}$ be a non-Cauchy sequence in a MS $(\mathbf{H},e)$. Then, ∃ $\epsilon >0$ and a couple of subsequences $\left\{z_{m_k}\right\}$ and $\left\{z_{n_k}\right\}$ of $\left\{z_m\right\}$ verifying the following:

$k\le n_k<m_k$∀$k\in \mathbb{N}$,

$e(z_{n_k},z_{m_k})\ge \epsilon$,

$e(z_{n_k},z_{s_k})<\epsilon$, ∀$s_k\in \{n_k+1,n_k+2,\dots ,m_k-2,m_k-1\}$.

Additionally, if $\underset{n\to \infty }{lim}e(z_n,z_{n+1})=0$, then the following is true:

$\underset{k\to \infty }{lim}e(z_{n_k},z_{n_k+\sigma })={\epsilon }^{+},\forall \sigma \in {\mathbb{N}}_0.$

Lemma 2 ([27]).

Let $\mathbf{S}$ be BR on a set $\mathbf{H}$, which is ℵ-transitive on $\mathbf{K}=\{z_n:n\in {\mathbb{N}}_0\}$, whereas $\left\{z_n\right\}\subset \mathbf{H}$ is $\mathbf{S}$-preserving sequence. Then, we have the following:

$(z_n,z_{n+1+\lambda (\aleph -1)})\in \mathbf{S},\forall n,\lambda \in {\mathbb{N}}_0.$

$\Sigma$ will denote the collection of the pair $(\alpha ,\beta )$ of functions $\alpha ,\beta :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ verifying the following axioms:(Σ₁)

$\beta \left(\mathsf{t}\right)<\alpha \left(\mathsf{t}\right),\forall \mathsf{t}\in {\mathbb{R}}^{+}/\left\{0\right\}$;

(Σ₂)

$\alpha$ is increasing, lower semicontinuous function and ${\alpha }^{-1}\left(0\right)=0$;

(Σ₃)

$\beta$ is a right upper semicontinuous function and $\beta \left(0\right)=0$.

Some typical examples of the functions of family $\Sigma$ are given by the following:

$\alpha \left(\mathsf{t}\right)=\left\{\begin{array}{cc}ln(\mathsf{t}+1)\hfill & \mathrm{if} \mathsf{t}\le 1\hfill \\ 3\mathsf{t}/4\hfill & \mathrm{if} \mathsf{t}>1\hfill \end{array}\right.$ and $\beta \left(\mathsf{t}\right)=2\mathsf{t}/3;$

$\alpha \left(\mathsf{t}\right)={\mathsf{t}}^2$ and $\beta \left(\mathsf{t}\right)=\left\{\begin{array}{cc}{\mathsf{t}}^3\hfill & \mathrm{if} \mathsf{t}\le 1/2\hfill \\ \mathsf{t}-3/8\hfill & \mathrm{if} \mathsf{t}>1/2;\hfill \end{array}\right.$

$\alpha \left(\mathsf{t}\right)=\mathsf{t}{e}^{\mathsf{t}}$ and $\beta \left(\mathsf{t}\right)=\left\{\begin{array}{cc}{\mathsf{t}}^2\hfill & \mathrm{if} \mathsf{t}\le 1\hfill \\ e^{\mathsf{t}}-1\hfill & \mathrm{if} \mathsf{t}>1.\hfill \end{array}\right.$

Lemma 3.

Let $(\alpha ,\beta )\in \Sigma$. If $\left\{a_n\right\}\subset {\mathbb{R}}^{+}\setminus \left\{0\right\}$ is a sequence such that the following is true:

(4) $\alpha \left(a_{n+1}\right)<\beta \left(a_n\right),\forall n\in \mathbb{N},$

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

Proof. 

By axiom (${\Sigma }_1$), we conclude the following:

$\alpha \left(a_{n+1}\right)\le \beta \left(a_n\right)<\alpha \left(a_n\right).$

Using monotone-property of $\alpha$, we attain the following:

$a_{n+1}<a_n,\forall n\in \mathbb{N}.$

It yields that $\left\{a_n\right\}\subset {\mathbb{R}}^{+}/\left\{0\right\}$ is decreasing, which is already bounded below. Consequently, ∃$\lambda \ge 0$ that verifies $a_n\stackrel{\mathbb{R}}{⟶}{\lambda }^{+}$ as $n\to \infty$.

Allowing the lower limit in (4) and by lower semicontinuity of $\alpha$ and right upper semicontinuity of $\beta$, we conclude the following:

$\begin{array}{ccc}\hfill \alpha \left(\lambda \right)& \le & \underset{n\to \infty }{lim\; inf}\alpha \left(a_n\right)\hfill \\ & \le & \underset{n\to \infty }{lim\; inf}\beta \left(a_n\right)\hfill \\ & \le & \underset{n\to \infty }{lim\; sup}\beta \left(a_n\right)\hfill \\ & \le & \beta \left(\lambda \right),\hfill \end{array}$

which by axiom (${\Sigma }_1$) gives rise to $\lambda =0$. □

Because of the the symmetry of the metric e, we drew a couple of conclusions.

Proposition 3.

In the course of that $(\alpha ,\beta )\in \Sigma$, the subsequent conditions are essentially the same: (i)

$\alpha \left(e\right(\mathsf{ϝ}z,\mathsf{ϝ}w\left)\right)\le \beta \left(e\right(z,w\left)\right),\forall (z,w)\in \mathbf{S},verifying\left[\mathsf{ϝ}\right(z)\ne zor\mathsf{ϝ}(w)\ne w],$

(ii)

$\alpha \left(e\right(\mathsf{ϝ}z,\mathsf{ϝ}w\left)\right)\le \beta \left(e\right(z,w\left)\right),\forall [z,w]\in \mathbf{S},verifying\left[\mathsf{ϝ}\right(z)\ne zor\mathsf{ϝ}(w)\ne w].$

Proposition 4.

In the course of that $(\alpha ,\beta )\in \Sigma$, the subsequent conditions are essentially the same: (i)

$\alpha \left(e\right(\mathsf{ϝ}z,\mathsf{ϝ}w\left)\right)\le \beta \left(e\right(z,w\left)\right),\forall (z,w)\in \mathbf{S},$

(ii)

$\alpha \left(e\right(\mathsf{ϝ}z,\mathsf{ϝ}w\left)\right)\le \beta \left(e\right(z,w\left)\right),\forall [z,w]\in \mathbf{S}.$

3. Main Results

Throughout the section, we assume that $(\mathbf{H},e)$ is a MS with a BR $\mathbf{S}$ and $\mathsf{ϝ}:\mathbf{H}\to \mathbf{H}$ is a map. Consider the following conditions:

(a). $(\mathbf{H},e)$ is $\mathbf{S}$-complete;

(b). $\mathbf{S}$ retains locally finitely $\mathsf{ϝ}$-transitive and $\mathsf{ϝ}$-closed;

(c). $\exists {\mathrm{z}}_0\in \mathbf{H}$ with $({\mathrm{z}}_0,\mathsf{ϝ}{\mathrm{z}}_0)\in \mathbf{S}$;

(d). $\mathsf{ϝ}$ retains $\mathbf{S}$-continuous, or $\mathbf{S}$ remains e-self-closed.

We now intend to demonstrate the fixed point findings for a relational extended contraction through generalized altering distance functions.

Theorem 4.

Under the arguments $\left(a\right)$–$\left(d\right)$, assume that ∃$(\alpha ,\beta )\in \Sigma$ that verify

(5)$\alpha \left(e\right(\mathsf{ϝ}z,\mathsf{ϝ}w\left)\right)\le \beta \left(e\right(z,w\left)\right),\forall (z,w)\in \mathbf{S},verifying\left[\mathsf{ϝ}\right(z)\ne zor\mathsf{ϝ}(w)\ne w].$

Then, ϝ possesses a fixed point.

Proof. 

To wind up the conclusion, we will carry out a few steps.

Step–1. Considering ${\mathrm{z}}_0\in \mathbf{H}$, we formulate the sequence $\left\{{\mathrm{z}}_n\right\}\subset \mathbf{H}$ that fulfills the following:

(6)${\mathrm{z}}_n:=\mathsf{ϝ}\left({\mathrm{z}}_{n-1}\right)={\mathsf{ϝ}}^n\left({\mathrm{z}}_0\right),\forall n\in \mathbb{N}.$

Step–2. We argue that $\left\{{\mathrm{z}}_n\right\}$ serves as a $\mathbf{S}$-preserving. By assertion $\left(c\right)$, $\mathsf{ϝ}$-closedness of $\mathbf{S}$ and Proposition 2, we attain the following:

$({\mathsf{ϝ}}^n{\mathrm{z}}_0,{\mathsf{ϝ}}^{n+1}{\mathrm{z}}_0)\in \mathbf{S},$

Which, owing to (6), deduces the following:

(7)$({\mathrm{z}}_n,{\mathrm{z}}_{n+1})\in \mathbf{S},\forall n\in {\mathbb{N}}_0.$

Step–3. Write $e_n:=e({\mathrm{z}}_n,{\mathrm{z}}_{n+1})$. If $e_{m_0}=e({\mathrm{z}}_{m_0},{\mathrm{z}}_{m_0+1})=0$ for some $m_0\in {\mathbb{N}}_0$, then using (6), we conclude $\mathsf{ϝ}\left({\mathrm{z}}_{m_0}\right)={\mathrm{z}}_{m_0}$. Thus, ${\mathrm{z}}_{m_0}$ retains a fixed point of $\mathsf{ϝ}$ and so our task is completed. If $e_n>0,\forall n\in {\mathbb{N}}_0$, then we go ahead with Step–4.

Step–4. We argue that $\left\{{\mathrm{z}}_n\right\}$ retains semi-Cauchy, i.e., $\underset{n\to \infty }{lim}\omega ({\mathrm{z}}_n,{\mathrm{z}}_{n+1})=0$. As $e_n>0$, we attain ${\mathrm{z}}_n\ne \mathsf{ϝ}\left({\mathrm{z}}_n\right),\forall n\in {\mathbb{N}}_0$. Employing hypothesis (5)–(7), we arrive at the following:

$\alpha ({\mathrm{z}}_{n+1},{\mathrm{z}}_{n+2})=\alpha \left(e(\mathsf{ϝ}{\mathrm{z}}_n,\mathsf{ϝ}{\mathrm{z}}_{n+1})\right)\le \beta (e({\mathrm{z}}_n,{\mathrm{z}}_{n+1})$

so that

(8)$\alpha \left(e_{n+1}\right)\le \beta \left(e_n\right),\forall n\in {\mathbb{N}}_0.$

From (8) and Lemma 3, we attain the following:

(9)$\underset{n\to \infty }{lim}e({\mathrm{z}}_{n+1},{\mathrm{z}}_n)=0.$

Step–5. We argue that $\left\{{\mathrm{z}}_n\right\}$ is Cauchy. In the event $\left\{{\mathrm{z}}_n\right\}$ fails to be Cauchy, by Lemma 1, we find $\epsilon >0$ and two subsequences of $\left\{{\mathrm{z}}_n\right\}$, say $\left\{{\mathrm{z}}_{m_k}\right\}$ and $\left\{{\mathrm{z}}_{n_k}\right\}$ that verify $k\le n_k<m_k$, $e({\mathrm{z}}_{n_k},{\mathrm{z}}_{m_k})\ge \epsilon$ and $e({\mathrm{z}}_{n_k},{\mathrm{z}}_{s_k})<\epsilon$ wherein $s_k\in \{n_k+1,n_k+2,\dots ,m_k-2,m_k-1\}$. Also, due to (9), we obtain the following:

(10)$\begin{array}{c}\hfill \underset{n\to \infty }{lim}e({\mathrm{z}}_{n_k},{\mathrm{z}}_{m_k+s})={\epsilon }^{+}\forall s\in {\mathbb{N}}_0.\end{array}$

 As $\left\{{\mathrm{z}}_n\right\}\subset \mathsf{ϝ}\left(\mathbf{H}\right)$, the range $\mathbf{K}=\{{\mathrm{z}}_n:n\in {\mathbb{N}}_0\}$ is a enumerable subset of $\mathsf{ϝ}\left(\mathbf{H}\right)$. Hence, using locally finitely $\mathsf{ϝ}$-transitivity of $\mathbf{S}$, we may detect an integer $\aleph =\aleph \left(\mathbf{K}\right)\ge 2$ such that ${\mathbf{S}|}_{\mathbf{K}}$ retains ℵ-transitive.

 Obviously, $n_k<m_k$ and $\aleph -1>0$. By division algorithm, we attain the following:

$$\begin{gathered} m_k-n_k=(\aleph -1)(a_k-1)+(\aleph -b_k) \\ {a}_k-1\ge 0,0\le \aleph -b_k<\aleph -1 \end{gathered}$$

$\iff \left\{\begin{array}{cc}{m}_k+b_k=n_k+1+(\aleph -1)a_k\hfill & \\ a_k\ge 1,1<b_k\le \aleph .\hfill \end{array}\right.$

Here, the numbers $a_k$ and $b_k$ satisfies $1<b_k\le \aleph$. By not affecting generality, it is feasible to pick out the subsequences $\left\{{\mathrm{z}}_{m_k}\right\}$ and $\left\{{\mathrm{z}}_{n_k}\right\}$ of $\left\{{\mathrm{z}}_n\right\}$ (satisfying (10)) in which $b_k$ retains a constant $\eta$. Therefore, we conclude the following:

(11)$p_k=m_k+\eta =n_k+1+(\aleph -1)a_k.$

From (10) and (11), we obtain the following:

(12)$\underset{k\to \infty }{lim}e({\mathrm{z}}_{n_k},{\mathrm{z}}_{p_k})=\underset{k\to \infty }{lim}e({\mathrm{z}}_{n_k},{\mathrm{z}}_{m_k+\eta })={\epsilon }^{+}.$

Employing triangle inequality, we find the following:

(13)$e({\mathrm{z}}_{n_k+1},{\mathrm{z}}_{p_k+1})\le e({\mathrm{z}}_{n_k+1},{\mathrm{z}}_{n_k})+e({\mathrm{z}}_{n_k},{\mathrm{z}}_{p_k})+e({\mathrm{z}}_{p_k},{\mathrm{z}}_{p_k+1})$

and

$e({\mathrm{z}}_{n_k},{\mathrm{z}}_{p_k})\le e({\mathrm{z}}_{n_k},{\mathrm{z}}_{n_k+1})+e({\mathrm{z}}_{n_k+1},{\mathrm{z}}_{p_k+1})+e({\mathrm{z}}_{p_k+1},{\mathrm{z}}_{p_k})$

or

(14)$\begin{array}{ccc}\hfill e({\mathrm{z}}_{n_k},{\mathrm{z}}_{p_k})-e({\mathrm{z}}_{n_k},{\mathrm{z}}_{n_k+1})-e({\mathrm{z}}_{p_k+1},{\mathrm{z}}_{p_k})& \le & e({\mathrm{z}}_{n_k+1},{\mathrm{z}}_{p_k+1}).\hfill \end{array}$

Taking $k\to \infty$ in (13) and (14) and employing (9) and (12), we attain the following:

(15)$\underset{k\to \infty }{lim}e({\mathrm{z}}_{n_k+1},{\mathrm{z}}_{p_k+1})={\epsilon }^{+}.$

Implementing (11) and Lemma 2, we attain the following: $e({\mathrm{z}}_{n_k},{\mathrm{z}}_{p_k})\in \mathbf{S}$. Now, due to hypothesis (5), we obtain the following:

$\begin{array}{c}\hfill \alpha \left(e({\mathrm{z}}_{n_k+1},{\mathrm{z}}_{p_k+1})\right)=\alpha \left(e(\mathsf{ϝ}{\mathrm{z}}_{n_k},\mathsf{ϝ}{\mathrm{z}}_{p_k})\right)\le \beta \left(e({\mathrm{z}}_{n_k},{\mathrm{z}}_{p_k})\right).\end{array}$

Proceeding the lower limit in (16) and by (12) and (15), lower semicontinuity of $\alpha$ and right upper semicontinuity of $\beta$, we conclude

$\alpha \left(\epsilon \right)\le \beta \left(\epsilon \right).$

Employing axiom (${\Sigma }_1$) above inequality determines that $\epsilon =0$; which retains a contradiction. This concludes that $\left\{{\mathrm{z}}_n\right\}$ is Cauchy. Also, as $\left\{{\mathrm{z}}_n\right\}$ is a $\mathbf{S}$-preserving, by hypothesis $\left(a\right)$, ∃$\overline{\mathrm{z}}\in \mathbf{H}$ verifying ${\mathrm{z}}_n\stackrel{e}{⟶}\overline{\mathrm{z}}$.

Step–6. Through the hypothesis $\left(d\right)$, we establish that the desired fixed point of $\mathsf{ϝ}$ is $\overline{\mathrm{z}}$. If $\mathsf{ϝ}$ retains $\mathbf{S}$-continuous, then

$\overline{\mathrm{z}}=\underset{n\to \infty }{lim}{\mathrm{z}}_{n+1}=\underset{n\to \infty }{lim}\mathsf{ϝ}\left({\mathrm{z}}_n\right)=\mathsf{ϝ}\left(\overline{\mathrm{z}}\right)$

pointing to that $\overline{\mathrm{z}}$ serves as a fixed point and our task is finished.

 If $\mathbf{S}$ is e-self-closed, then $\left\{{\mathrm{z}}_n\right\}$ determines a subsequence $\left\{{\mathrm{z}}_{m_k}\right\}$ for which

$[{\mathrm{z}}_{m_k},\overline{\mathrm{z}}]\in \mathbf{S},\forall k\in \mathbb{N}.$

Employing hypothesis (5) and Proposition 3, we get

(16)$\alpha \left(e({\mathrm{z}}_{m_k+1},\mathsf{ϝ}\overline{\mathrm{z}})\right)=\alpha \left(e(\mathsf{ϝ}{\mathrm{z}}_{m_k},\mathsf{ϝ}\overline{\mathrm{z}})\right)\le \beta \left(e({\mathrm{z}}_{m_k},\overline{\mathrm{z}})\right).$

 Letting lower limit in (16) and due to lower semicontinuity of $\alpha$ and right upper semicontinuity of $\beta$, we obtain the following:

$\alpha \left(e(\overline{\mathrm{z}},\mathsf{ϝ}\overline{\mathrm{z}})\right)\le \underset{k\to \infty }{lim\; inf}\alpha \left(e({\mathrm{z}}_{m_k+1},\mathsf{ϝ}\overline{\mathrm{z}})\right)\le \underset{k\to \infty }{lim\; sup}\beta \left(e({\mathrm{z}}_{m_k},\overline{\mathrm{z}})\right)\le \beta \left(0\right)$

so that

$\alpha \left(e(\overline{\mathrm{z}},\mathsf{ϝ}\overline{\mathrm{z}})\right)=\beta \left(0\right)$

which using axiom (${\Sigma }_3$) yields that

$\alpha \left(e(\overline{\mathrm{z}},\mathsf{ϝ}\overline{\mathrm{z}})\right)=0.$

Using axiom (${\Sigma }_2$), above equation determines $e(\overline{\mathrm{z}},\mathsf{ϝ}\overline{\mathrm{z}})=0$ so that $\mathsf{ϝ}\left(\overline{\mathrm{z}}\right)=\overline{\mathrm{z}}$. Hence, $\overline{\mathrm{z}}$ retains a fixed point as desired. □

The following outcome is an analogue of the main finding of Alam et al. [3].

Corollary 1.

Under the arguments $\left(a\right)$–$\left(d\right)$, if ∃ right upper semicontinuous function $\phi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ verifying $\phi \left(0\right)=0$ and $\phi \left(t\right)<t,\forall t>0$ such that

$e(\mathsf{ϝ}z,\mathsf{ϝ}w))\le \phi (e(z,w)),\forall (z,w)\in \mathbf{S},with[z\ne \mathsf{ϝ}\left(z\right)orw\ne \mathsf{ϝ}\left(w\right)],$

then ϝ possesses a fixed point.

Proof. 

Putting $\alpha =I$, the identity on ${\mathbb{R}}^{+}$ and $\beta =\phi$ in Theorem 4, we get the conclusion. □

Corollary 2([14]).

Let $(\mathbf{H},e)$ be a complete MS a and $\mathsf{ϝ}:\mathbf{H}\to \mathbf{H}$ be a map. If ∃ right upper semicontinuous function $\phi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ verifying $\phi \left(0\right)=0$ and $\phi \left(t\right)<t,\forall t>0$ such that

$e(\mathsf{ϝ}z,\mathsf{ϝ}w))\le \phi (e(z,w)),\forall (z,w)\in \mathbf{S},with[z\ne \mathsf{ϝ}\left(\mathrm{z}\right)orw\ne \mathsf{ϝ}\left(w\right)],$

then ϝ possesses a fixed point.

Proof. 

The result holds by setting $\mathbf{S}:={\mathbf{H}}^2$ in Corollary 1. □

Theorem 5.

Under the arguments $\left(a\right)$–$\left(d\right)$ if (i)

∃$(\alpha ,\beta )\in \Sigma$ with

$\alpha \left(e\right(\mathsf{ϝ}z,\mathsf{ϝ}w\left)\right)\le \beta \left(e\right(z,w\left)\right),\forall (z,w)\in \mathbf{S},$

and (ii)

$\mathsf{ϝ}\left(\mathbf{H}\right)$ remains ${\mathbf{S}}^s$-connected,

then ϝ enjoys a unique fixed point.

Proof. 

Overwhelmingly evident that the assertion (5), of Theorem 4 holds if $\left(i\right)$ does. As the conclusion of Theorem 4, if $\mathrm{z}$ and $\mathrm{w}$ are two fixed points of $\mathsf{ϝ}$, then

${\mathsf{ϝ}}^n\left(\mathrm{z}\right)=\mathrm{z}and{\mathsf{ϝ}}^n\left(\mathrm{w}\right)=\mathrm{w},\forall n\in {\mathbb{N}}_0.$

Obviously, $\mathrm{z},\mathrm{w}\in \mathsf{ϝ}\left(\mathbf{H}\right)$. Employing hypothesis $\left(ii\right)$, there is a path $\{{\mathsf{r}}_0,{\mathsf{r}}_1,{\mathsf{r}}_2,\dots ,{\mathsf{r}}_l\}$ between $\mathrm{z}$ to $\mathrm{w}$ verifying the following:

(17)$\begin{array}{c}\hfill {\mathsf{r}}_0=\mathrm{z},{\mathsf{r}}_l=\mathrm{w}\mathrm{and}[{\mathsf{r}}_{𝚤},{\mathsf{r}}_{𝚤+1}]\in \mathbf{S},\forall 𝚤\in \{0,1,\dots ,l-1\}.\end{array}$

By $\mathsf{ϝ}$-closedness of $\mathbf{S}$, we attain the following:

(18)$\begin{array}{c}\hfill [{\mathsf{ϝ}}^n{\mathsf{r}}_{𝚤},{\mathsf{ϝ}}^n{\mathsf{r}}_{𝚤+1}]\in \mathbf{S},\forall n\in {\mathbb{N}}_0\mathrm{and}\forall 𝚤\in \{0,1,\dots ,l-1\}.\end{array}$

Denote

${\varrho }_n^{𝚤}:=e({\mathsf{ϝ}}^n{\mathsf{r}}_{𝚤},{\mathsf{ϝ}}^n{\mathsf{r}}_{𝚤+1})\forall n\in {\mathbb{N}}_{0}and\forall 𝚤\in \{0,1,\dots ,l-1\}.$

We will deduce that

(19)$\begin{array}{ccc}\hfill \underset{n\to \infty }{lim}{\varrho }_n^{𝚤}& =& 0.\hfill \end{array}$

Let us fix ı. Consider the case

$\mathrm{for} \mathrm{some}{m}_0\in {\mathbb{N}}_0,{\varrho }_{m_0}^{𝚤}=e({\mathsf{ϝ}}^{m_0}{\mathsf{r}}_{𝚤},{\mathsf{ϝ}}^{m_0}{\mathsf{r}}_{𝚤+1})=0,$

which gives rise to ${\mathsf{ϝ}}^{m_0}\left({\mathsf{r}}_{𝚤}\right)={\mathsf{ϝ}}^{m_0}\left({\mathsf{r}}_{𝚤+1}\right)$. Now, using (6), we get ${\mathsf{ϝ}}^{m_0+1}\left({\mathsf{r}}_{𝚤}\right)={\mathsf{ϝ}}^{m_0+1}\left({\mathsf{r}}_{𝚤+1}\right)$ so that ${\varrho }_{m_0+1}^{𝚤}=0$, which inductively determines ${\varrho }_n^{𝚤}=0$∀$n\ge m_0$. Thus, we obtain $\underset{n\to \infty }{lim}{\varrho }_n^{𝚤}=0$.

Alternatively, we have ${\varrho }_n^{𝚤}>0$∀$n\in {\mathbb{N}}_0$. Through (18), hypothesis $\left(i\right)$ and Proposition 4, we conclude the following:

$\begin{array}{ccc}\hfill \alpha \left({\varrho }_{n+1}^{𝚤}\right)& =& \alpha \left(e({\mathsf{ϝ}}^{n+1}{\mathsf{r}}_{𝚤},{\mathsf{ϝ}}^{n+1}{\mathsf{r}}_{𝚤+1})\right)\hfill \\ & =& \alpha \left(e(\mathsf{ϝ}\left({\mathsf{ϝ}}^n{\mathsf{r}}_{𝚤}\right),\mathsf{ϝ}\left({\mathsf{ϝ}}^n{\mathsf{r}}_{𝚤+1}\right))\right)\hfill \\ & \le & \beta \left(e({\mathsf{ϝ}}^n{\mathsf{r}}_{𝚤},{\mathsf{ϝ}}^n{\mathsf{r}}_{𝚤+1})\right)\hfill \\ & =& \beta \left({\varrho }_n^{𝚤}\right)\hfill \end{array}$

implying thereby

(20)$\alpha \left({\varrho }_{n+1}^{𝚤}\right)\le \beta \left({\varrho }_n^{𝚤}\right).$

By (20) and Lemma 3, we find

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

Hence, (19) is verified for each ı$(0\le 𝚤\le l-1)$. Through triangular-inequality, we arrive at the following:

$\begin{array}{ccc}\hfill e(\mathrm{z},\mathrm{w})& =& e({\mathsf{ϝ}}^n{\mathsf{r}}_0,{\mathsf{ϝ}}^n{\mathsf{r}}_l)\hfill \\ & \le & {\varrho }_n^0+{\varrho }_n^1+\cdots +{\varrho }_l^{n-1}\hfill \\ & \to & 0asn\to \infty \hfill \end{array}$

so that $\mathrm{z}=\mathrm{w}$. Hence, $\mathsf{ϝ}$ enjoys a unique fixed point. □

4. Illustrative Examples

We will offer the subsequent instances to throw spotlight on our findings.

Example 1.

Take $\mathbf{H}=[-2,-1]\cup [1,2]$ with the Euclidean metric e. Define a BR $\mathbf{S}$ on $\mathbf{H}$ by

$\mathbf{S}=\{(z,w)\in {\mathbf{H}}^2:z<w\}.$

Clearly, $\mathbf{S}$ is a e-self-closed binary BR and $(\mathbf{H},e)$ forms a $\mathbf{S}$-complete MS.

The map $\mathsf{ϝ}:\mathbf{H}\to \mathbf{H}$ is outlined as follows:

$\mathsf{ϝ}\left(z\right)=\left\{\begin{array}{cc}-1\hfill & if-2\le z\le -1,\hfill \\ 1\hfill & if1\le z\le 2.\hfill \end{array}\right.$

Naturally, $\mathbf{S}$ is locally finitely ϝ-transitive and ϝ-closed BR. Also $z_0=-2\in \mathbf{H}$ (and hence $\mathsf{ϝ}\left(z_0\right)=-1$) satisfies $(z_0,\mathsf{ϝ}{z}_0)\in \mathbf{S}$.

Define the pair $(\alpha ,\beta )\in \Sigma$ of auxiliary functions by

$\alpha \left(t\right)=\left\{\begin{array}{cc}0,\hfill & ift=0\hfill \\ lnt,\hfill & ift>0\hfill \end{array}\right.$

and

$\beta \left(t\right)=\left\{\begin{array}{cc}t/2,\hfill & ift\le 2\hfill \\ 2,\hfill & ift>2.\hfill \end{array}\right.$

The contraction prerequisite of Theorem 4 can then be readily confirmed. Thus far, each prerequisite in Theorems 4 has been established. In turns, ϝ has a fixed point.

Noting that $\mathsf{ϝ}\left(\mathbf{H}\right)$ is not ${\mathbf{S}}^s$-connected as there is no path between $-1$ and 1. Consequently, Theorem 5 cannot be applied to this example. Here 1 and $-1$ are two fixed point of ϝ.

Example 2.

Take $\mathbf{H}={\mathbb{R}}^{+}$ with Euclidean metric e. Construct a BR $\mathbf{S}$ on $\mathbf{H}$ by the following:

$\mathbf{S}:=\{(z,w)\in {\mathbf{H}}^2:z^2+2z=w^2+2w\}.$

Clearly, the MS $(\mathbf{H},e)$ forms a $\mathbf{S}$-complete.

Define a map $\mathsf{ϝ}:\mathbf{H}\to \mathbf{H}$ by the following:

$\mathsf{ϝ}\left(z\right)=ln(z^2+2z+1),\forall z\in \mathbf{H}.$

Then $\mathbf{S}$ retains locally finitely ϝ-transitive and ϝ-closed BR while ϝ remains $\mathbf{S}$-continuous. Also $z_0=0\in \mathbf{H}$ satisfies $(0,\mathsf{ϝ}0)\in \mathbf{S}$.

Now, we prove that $\mathbf{S}$ is ϝ-closed. Take $(z,w)\in \mathbf{S}$. Next, we possess the following:

$\mathsf{ϝ}\left(z\right)=ln(z^2+2z+1)=ln(w^2+2w+1)=\mathsf{ϝ}\left(w\right)$

yielding thereby

${\left(\mathsf{ϝ}z\right)}^2+2\mathsf{ϝ}z={\left(\mathsf{ϝ}w\right)}^2+2\mathsf{ϝ}w.$

It implies that $(\mathsf{ϝ}z,\mathsf{ϝ}w)\in \mathbf{S}$ and hence $\mathbf{S}$ is ϝ-closed.

Define the pair $(\alpha ,\beta )\in \Sigma$ of auxiliary functions by the following:

(21) $\alpha \left(t\right)=\left\{\begin{array}{cc}ln(t+1),& if t\le 1,\\ 3t/4,& if t>1\end{array}\right.$

and $\beta \left(t\right)=2t/3$.

Finally, for each pair $z,w\in \mathbf{H}$ with $(z,w)\in \mathbf{S}$, the following condition is also satisfied.

$\alpha \left(e\right(\mathsf{ϝ}z,\mathsf{ϝ}w\left)\right)\le \beta \left(e\right(z,w\left)\right).$

Thus far, each prerequisite in Theorems 4 has been established. In turns, ϝ has a fixed point, e.g., $\overline{z}=0$ is one of the fixed points of ϝ.

Example 3.

Take $\mathbf{H}=[2,4]$ with Euclidean metric e. Construct a BR $\mathbf{S}$ on $\mathbf{H}$ by the following:

$\mathbf{S}=\left\{\right(2,2),(2,3),(3,2),(3,3),(2,4),(3,4\left)\right\}.$

Clearly, $(\mathbf{H},e)$ forms a $\mathbf{S}$-complete MS.

The map $\mathsf{ϝ}:\mathbf{H}\to \mathbf{H}$ is outlined as follows:

$\mathsf{ϝ}\left(z\right)=\left\{\begin{array}{c}2\mathrm{if}2\le z\le 3\hfill \\ 3\mathrm{if}3<z\le 4.\hfill \end{array}\right.$

Then $\mathbf{S}$ retains locally finitely ϝ-transitive and ϝ-closed BR , while $\mathsf{ϝ}\left(\mathbf{H}\right)$ is ${\mathbf{S}}^s$-connected. Let $\left\{z_n\right\}\subset \mathbf{H}$ be a $\mathbf{S}$-preserving sequence along-with $z_n\stackrel{e}{⟶}\widehat{z}$ and $(z_n,z_{n+1})\in \mathbf{S},\forall n\in \mathbb{N}$. But $(z_n,z_{n+1})\notin \{(2,4),(3,4)\}$, so $(z_n,z_{n+1})\in \{(2,2),(2,3),(3,2),(3,3)\},\forall n\in \mathbb{N}$ and hence $\left\{z_n\right\}\subset \{2,3\}$. Next $\{2,3\}$ being closed provides $[z_n,\widehat{z}]\in \mathbf{S}$. Thus, $\mathbf{S}$ is e-self-closed.

Define the pair $(\alpha ,\beta )\in \Sigma$ of auxiliary functions by the following:

$\alpha \left(t\right)=t^2$

and

$\beta \left(t\right)={\displaystyle \frac{t^2}{t^2+1}}.$

The contraction prerequisite of Theorem 5 can then be readily confirmed. Additionally, remaining presumptions of Theorem 5 are met. Consequently, ϝ enjoys a unique fixed point, which is $\overline{z}=2$.

5. An Application

Consider the following two-point BVP connected to the second order ODE:

(22)$\left\{\begin{array}{c}{\mathrm{z}}^{″}+\mathcal{M}(r,\mathrm{z})=0,r\in [0,1]\\ \mathrm{z}\left(0\right)=\mathrm{z}\left(1\right)=0.\end{array}\right.$

Theorem 6.

In conjunction to BVP (22), if $\mathcal{M}:[0,1]\times \mathbb{R}\to {\mathbb{R}}^{+}$ is monotonically increasing in second variable and continuous. Also, $\exists 0\le \hslash \le 8$ such that

(23)$\mathcal{M}(r,q)-\mathcal{M}(r,p)\le \hslash \sqrt{ln\left[{(q-p)}^2+1\right]},\forall p,q\in \mathbb{R};p\le q.$

Then the BVP (22) owns a unique nonnegative solution.

Proof. 

Noting that $\mathrm{z}\in {\mathcal{C}}^2[0,1]$ admits a solution of (22) iff $\mathrm{z}\in \mathcal{C}[0,1]$ solves the following equation:

$\mathrm{z}\left(r\right)={\int }_0^1\mathbf{G}(r,\mathrm{s})\mathcal{M}\left(\mathrm{s},\mathrm{z}\left(\mathrm{s}\right)\right)d\mathrm{s},\mathrm{for}\mathrm{each}r\in [0,1],$

whereas Green function $\mathbf{G}(r,\mathrm{s})$ is given by the following:

$\mathbf{G}(r,\mathrm{s})=\left\{\begin{array}{cc}r(1-\mathrm{s}),& 0\le r\le \mathrm{s}\le 1\\ \mathrm{s}(1-r),& 0\le \mathrm{s}\le r\le 1.\end{array}\right.$

Consider the following cone:

$\mathbf{H}=\left\{\mathrm{z}\in \mathcal{C}[0,1]:\mathrm{z}\left(r\right)\ge 0,\mathrm{for}\mathrm{each}r\in [0,1]\right\}.$

Construct a BR $\mathsf{ϝ}$ on $\mathbf{H}$ by the following:

$\mathbf{S}=\left\{(\mathrm{z},\mathrm{w})\in {\mathbf{H}}^2:\mathrm{z}\left(r\right)\le \mathrm{w}\left(r\right),\mathrm{for}\mathrm{each}r\in [0,1]\right\}.$

On $\mathbf{H}$, consider a metric e defined by the following:

$e(\mathrm{z},\mathrm{w})=sup\left\{\left|\mathrm{z}\right(r)-\mathrm{w}(r\left)\right|:r\in [0,1]\right\}.$

Define an operator $\mathsf{ϝ}:\mathbf{H}\to \mathbf{H}$ by the following:

$\left(\mathsf{ϝ}\mathrm{z}\right)\left(r\right)={\int }_0^1\mathbf{G}(r,\mathrm{s})\mathcal{M}(\mathrm{s},\mathrm{z}\left(\mathrm{s}\right))d\mathrm{s},\forall \mathrm{z}\in \mathbf{H}.$

We will confirm all the hypotheses of Theorem 5.

(a) Clearly, the MS$(\mathbf{H},e)$ is $\mathbf{S}$-complete.

(b) $\mathbf{S}$ being a partial ordering relation is locally finitely $\mathsf{ϝ}$-transitive. Let $\mathrm{z},\mathrm{w}\in \mathbf{H}$ such that $(\mathrm{z},\mathrm{w})\in \mathbf{S}$. Then, for each $r\in [0,1]$, we attain $\mathrm{z}\left(r\right)\le \mathrm{w}\left(r\right)$. Employing monotonicity of $\mathcal{M}$ in second variable, for each $r\in [0,1]$, we obtain the following:

$\left(\mathsf{ϝ}\mathrm{z}\right)\left(r\right)={\int }_0^1\mathbf{G}(r,\mathrm{s})\mathcal{M}(\mathrm{s},\mathrm{z}\left(\mathrm{s}\right))d\mathrm{s}\le {\int }_0^1\mathbf{G}(r,\mathrm{s})\mathcal{M}(\mathrm{s},\mathrm{w}\left(\mathrm{s}\right))d\mathrm{s}=\left(\mathsf{ϝ}\mathrm{w}\right)\left(r\right)$

so that $(\mathsf{ϝ}\mathrm{z},\mathsf{ϝ}\mathrm{w})\in \mathbf{S}$. Hence, $\mathbf{S}$ is $\mathsf{ϝ}$-closed.

(c) As $\mathcal{M}$ and $\mathbf{G}$ both are nonnegative functions, therefore zero operator $\mathsf{0}\in \mathbf{H}$ verifies for all $r\in [0,1]$ that

$\mathsf{0}\left(r\right)=0\le {\int }_0^1\mathbf{G}(r,\mathrm{s})\mathcal{M}(\mathrm{s},0)d\mathrm{s}=\left(\mathsf{ϝ}\mathsf{0}\right)\left(r\right)$

thereby yielding the following:

$(\mathsf{0},\mathsf{ϝ}\mathsf{0})\in \mathbf{S}.$

(d) Let $\left\{{\mathrm{z}}_n\right\}\subset \mathbf{H}$ be a $\mathbf{S}$-preserving sequence that converges to $\tilde{\mathrm{z}}\in \mathbf{H}$. For every $r\in [0,1]$, therefore ${\mathrm{z}}_n\left(r\right)\le \tilde{\mathrm{z}}\left(r\right),\forall n\in \mathbb{N}$. This implies that $({\mathrm{z}}_n,\tilde{\mathrm{z}})\in \mathbf{S},\forall n\in \mathbb{N}$. Thus, $\mathbf{S}$ retains e-self-closed.

(i) Let $(\mathrm{z},\mathrm{w})\in \mathbf{S}$. For every $r\in [0,1]$, therefore we attain $\mathrm{z}\left(r\right)\le \mathrm{w}\left(r\right)$. By (23), we find the following:

(24)$\begin{array}{cc}\hfill e(\mathsf{ϝ}\mathrm{w},\mathsf{ϝ}\mathrm{z})& =\underset{r\in [0,1]}{sup}\left|\left(\mathsf{ϝ}\mathrm{w}\right)\left(r\right)-\left(\mathsf{ϝ}\mathrm{z}\right)\left(r\right)\right|\hfill \\ \hfill & =\underset{r\in [0,1]}{sup}\left(\left(\mathsf{ϝ}\mathrm{w}\right)\left(r\right)-\left(\mathsf{ϝ}\mathrm{z}\right)\left(r\right)\right)\hfill \\ \hfill & =\underset{r\in [0,1]}{sup}{\int }_0^1\mathbf{G}(r,\mathrm{s})(\mathcal{M}(\mathrm{s},\mathrm{w}\left(\mathrm{s}\right))-\mathcal{M}(\mathrm{s},\mathrm{z}\left(\mathrm{s}\right)))d\mathrm{s}\hfill \\ \hfill & \le \underset{r\in [0,1]}{sup}{\int }_0^1\mathbf{G}(r,\mathrm{s})\hslash \sqrt{ln\left[{(\mathrm{w}-\mathrm{z})}^2+1\right]}\hfill \\ \hfill & \le \underset{r\in [0,1]}{sup}{\int }_0^1\mathbf{G}(r,\mathrm{s})\hslash \sqrt{\left[ln\left|\right|\mathrm{w}-{\mathrm{z}\left|\right|}^2+1\right]}d\mathrm{s}\hfill \\ \hfill & =\hslash \sqrt{ln\left[\left|\right|\mathrm{w}-{\mathrm{z}\left|\right|}^2+1\right]}\underset{r\in [0,1]}{sup}{\int }_0^1\mathbf{G}(r,\mathrm{s})d\mathrm{s}.\hfill \end{array}$

It can be easily verified that

${\int }_0^1\mathbf{G}(r,\mathrm{s})d\mathrm{s}={\displaystyle \frac{-r^2}{2}}+{\displaystyle \frac{r}{2}}$

implying thereby

$\underset{r\in [0,1]}{sup}{\int }_0^1\mathbf{G}(r,\mathrm{s})d\mathrm{s}={\displaystyle \frac{1}{8}}.$

From the last equation, the inequality (24) becomes

$\begin{array}{cc}\hfill e(\mathsf{ϝ}\mathrm{w},\mathsf{ϝ}\mathrm{z})& \le {\displaystyle \frac{\hslash }{8}}\sqrt{ln\left[\left|\right|\mathrm{w}-{\mathrm{z}\left|\right|}^2+1\right]}\hfill \\ \hfill & \le \sqrt{ln\left[\left|\right|\mathrm{w}-{\mathrm{z}\left|\right|}^2+1\right]}(\mathrm{by} \mathrm{the} \mathrm{hypothesis} 0\le \hslash \le 8)\hfill \\ \hfill & =\sqrt{ln\left[e{(\mathrm{z},\mathrm{w})}^2+1\right]}\hfill \end{array}$

so that

$e{(\mathsf{ϝ}\mathrm{w},\mathsf{ϝ}\mathrm{z})}^2\le ln\left[e{(\mathrm{z},\mathrm{w})}^2+1\right].$

Define $\alpha \left(\mathrm{t}\right)={\mathrm{t}}^2$ and $\beta \left(\mathrm{t}\right)=ln({\mathrm{t}}^2+1)$. Then, $(\alpha ,\beta )\in \Sigma$. Hence, the foregoing inequality reduces to

$\alpha \left(e\right(\mathsf{ϝ}\mathrm{z},\mathsf{ϝ}\mathrm{w}\left)\right)\le \beta \left(e\right(\mathrm{z},\mathrm{w}\left)\right).$

(ii) Let $\mathrm{z},\mathrm{w}\in \mathbf{H}$; so $\mathsf{ϝ}\left(\mathrm{z}\right),\mathsf{ϝ}\left(\mathrm{w}\right)\in \mathsf{ϝ}\left(\mathbf{H}\right)$. Write $\mathsf{u}:=max\{\mathsf{ϝ}\mathrm{z},\mathsf{ϝ}\mathrm{w}\}$, then we attain $(\mathsf{ϝ}\mathrm{z},\mathsf{u})\in \mathbf{S}$ and $(\mathsf{ϝ}\mathrm{w},\mathsf{u})\in \mathbf{S}$. It follows that $\mathsf{ϝ}\left(\mathbf{H}\right)$ is a ${\mathbf{S}}^{\mathrm{s}}$-connected set.

Thus, all assertions of Theorem 5 are met. Consequently, $\mathsf{ϝ}$ admits a unique fixed point, say $\overline{\mathrm{z}}$. Further, as $\overline{\mathrm{z}}\in \mathbf{H}$, $\overline{\mathrm{z}}$ retains the unique (nonnegative) solution of (22) as desired. □

6. Conclusions

We explored certain outcomes in an MS through a locally finitely $\mathsf{ϝ}$-transitive BR$\mathbf{S}$ under relational extended contraction via a pair of generalized altering distance functions. In the hypotheses of our existing finding, the underlying BR is required to be locally finitely $\mathsf{ϝ}$-transitive and $\mathsf{ϝ}$-closed. Meanwhile, to obtain the uniqueness result, connectedness property besides stronger condition of contractivity is required. To bring up the significance of the arguments and the breadth of our conclusions, we also included a few concrete instances and an application to second-order BVP. The outcomes of this research included an optimal contraction condition which solely applies to the pairs of comparative elements, not all elements.

In the following, we shall discuss some special features of our work, which demonstrates the utility of our outcomes over corresponding existing ones.

6.1. Comparison with Recent Results

Very recently, some relation-theoretic variants of Theorem 1 are found by Alshaban et al. [16], Filali and Khan [17], and Filali et al. [18] employing the following contraction condition involving a single auxiliary function $\varphi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$:

(25)$e(\mathsf{ϝ}\mathrm{z},\mathsf{ϝ}\mathrm{w})\le \varphi \left(e\right(\mathrm{z},\mathrm{w}\left)\right),\forall (\mathrm{z},\mathrm{w})\in \mathbf{S},\mathrm{verifying} \left[\mathsf{ϝ}\right(\mathrm{z})\ne \mathrm{z}or\mathsf{ϝ}(\mathrm{w})\ne \mathrm{w}].$

Indeed, the earlier authors [16,17,18] assumed that the involved auxiliary functions are that utilized in Bianchini-Grandolfi type, Boyd-Wong type, and Matkowski type (compatible with Berinde’s almost contraction). In present work, we employed the following contraction condition involving two auxiliary function $\alpha ,\beta :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$:

(26)$\alpha \left(e\right(\mathsf{ϝ}\mathrm{z},\mathsf{ϝ}\mathrm{w}\left)\right)\le \beta \left(e\right(\mathrm{z},\mathrm{w}\left)\right),\forall (\mathrm{z},\mathrm{w})\in \mathbf{S},\mathrm{verifying} \left[\mathsf{ϝ}\right(\mathrm{z})\ne \mathrm{z}or\mathsf{ϝ}(\mathrm{w})\ne \mathrm{w}].$

Observe that for $\alpha =I$ (the identity map) and $\beta =\varphi$, the contraction condition (26) reduces to the condition (25) in the sense of [17]. Thus far, our outcomes generalize the outcomes of Filali and Khan [17], which extends the corresponding outcomes of Alshaban et al. [16]. However, our findings are totally different from that of Filali et al. [18].

6.2. Complicated BVP

Noting that the BVP appeared in our work is independent from Lipschitz constant and involved a Logarithm condition of the form (23). Consequently, such problems cannot be solved by BVP and similar other ordinary fixed point outcomes.

6.3. Weakening Transitivity Condition

In our outcomes, an finite transitive (such as: 2-transitive, or 3-transitive,….) BR has been considered. Indeed, the usual transitivity coincides with 2-transitivity. Additionally, such finite transitivity of BR is being assured also in a local way on the range of set, which means that transitivity condition must be hold only on denumerable sets of $\mathsf{ϝ}\left(\mathbf{H}\right)$.

6.4. Employing Relational Connectedness

In order to uniqueness theorem in relational MS, several noted authors utilized the concepts of complete BR and directed sets. Recall that a BR $\mathbf{S}$ is termed as complete if $[\mathrm{z},\mathrm{w}]\in \mathbf{S}$, for all $\mathrm{z},\mathrm{w}\in \mathbf{H}$. Also, $\mathbf{H}$ is named as ${\mathbf{S}}^{\mathrm{s}}$-directed when for each pair $\mathrm{z},\mathrm{w}\in \mathbf{H}$, there exists $\mathsf{r}\in \mathbf{H}$ verifying $[\mathrm{z},\mathsf{r}]\in \mathbf{S}$ and $[\mathrm{w},\mathsf{r}]\in \mathbf{S}$. In these two concepts, each pair of elements joins a path of lengths 1 and 2, respectively. Apart from this, we employed more general concept (${\mathbf{S}}^{\mathrm{s}}$-connectedness of $\mathsf{ϝ}\left(\mathbf{H}\right)$), whereas each pair of elements joins a path of any length.

6.5. Possible Future Directions

Taking into account the vital role of relation-theoretic fixed point paradigm, we incorporate into consideration the following potential future research projects:

Improving the characteristics of generalized altering distance functions;

Generalizing the outcomes of Pant [15] in the setup of relational MS;

To enhance our outcomes in various metrical structures, viz, cone MS, semi-MS, quasi MS, fuzzy MS etc.;

Extending our outcomes for a couple of self-mappings by investigating the common fixed point outcomes;

On implementing our findings to first-order periodic BVP, nonlinear matrix equations, and integral equations.

Author Contributions

Methodology: M.F.A. and F.A.K.; conceptualization: D.F. and S.B.A.; investigation: A.A. and M.Z.A.; formal analysis: M.F.A. and S.B.A.; writing—original draft preparation: F.A.K. and M.Z.A.; writing—review and editing: D.F. and A.A.; project administration: A.A. and M.Z.A.; funding acquisition: D.F.; supervision: F.A.K. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

This article comprises the data drawn up in the present research. With a suitable request, supplementary details can be obtained precisely from the corresponding authors.

Conflicts of Interest

The authors affirm that they have no competing interests.

Notations and Abbreviations

This manuscript incorporates the following symbols and acronyms:

$\mathbb{N}$	Set of positive integers;
${\mathbb{N}}_0:=$	$\mathbb{N}\cup \left\{0\right\}$;
$\mathbb{R}$	Set of real numbers;
${\mathbb{R}}^{+}$	$[0,\infty )$;
BCP	Banach contraction principle;
RHS	right hand side;
MS	metric space;
ODE	ordinary differential equation;
BVP	boundary value problems;
$\mathcal{C}\left(J\right)$	Collection of real-valued continuous functions defined on the interval J;
${\mathcal{C}}^n\left(J\right)$	Collection of real-valued $n^{\mathrm{th}}$ times differentiable continuous functions defined on the interval J.

Footnotes

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