Relational Contractions of Matkowski–Berinde–Pant Type and an Application to Certain Fractional Differential Equations

Abstract

This paper concludes a few fixed-point outcomes involving almost Matkowski contraction-inequality of Pant type in a relational metric space. The findings established here enhance, expand, consolidate and develop several noted outcomes. In order to argue for our investigations, we construct some illustrative examples. We exploit our outcomes to analyze the availability of a (unique) positive solution to certain singular fractional differential equations.

1. Introduction

Fractional-order derivatives offer several advantages over traditional integer-order derivatives, particularly in modeling and control systems. The concept of FDEs remains an expansion of the differential equations involving fractional-order derivatives. FDEs are characterized recently due to their impressive development and accuracy to the realm of fractional calculus. For a deep description of FDEs, we refer the works contained [1,2,3,4,5]. Zhou et al. [6] and Zhai and Hao [7] discussed the solvability of FDEs using fixed-point theorems in partially ordered MS. On the other hand, Liang and Zhang [8] subsequently investigated the unique positive solution for a three-point BVP of FDE. The singular three-point BVP associated to FDEs were proved by Cabrera et al. [9] using order-theoretic fixed point theorems. Karapınar et al. [10] employed fixed-point theorems for large contractions to discuss the solvability to nonlinear fractional differential equations. Very recently, Abdou [11] solved certain nonlinear FDEs using fixed-point theorems in orthogonal MS.

The classical fractional BVP with $\mathsf{v}$ as a dependent variable and $\theta$ as an independent variable is described as

$$-{\mathbf{D}}^{\mathsf{p}}\mathsf{v}\left(\theta \right)=\hslash \left(\theta ,\mathsf{v}\left(\theta \right),{\mathbf{D}}^{{\alpha }_1}\mathsf{v}\left(\theta \right),{\mathbf{D}}^{{\alpha }_2}\mathsf{v}\left(\theta \right),\dots ,{\mathbf{D}}^{{\alpha }_{r-1}}\mathsf{v}\left(\theta \right)\right)$$

$$\left\{\begin{array}{c}{\mathbf{D}}^{{\alpha }_i}\mathsf{v}\left(0\right)=0,1\le i\le r-1,\hfill \\ {\mathbf{D}}^{{\alpha }_{r-1}+1}\mathsf{v}\left(0\right)=0,\hfill \\ {\mathbf{D}}^{{\alpha }_{r-1}}\mathsf{v}\left(1\right)={\displaystyle \sum _{j=1}^{m-2}}{e}_j{\mathbf{D}}^{{\alpha }_{r-1}}\mathsf{v}\left({\delta }_j\right),\hfill \end{array}\right.$$

(1)

The following definitions are used above:

• $r=3,4,5,\dots$ verifying $r-1<\mathsf{p}\le r$;
• $0<{\alpha }_1<{\alpha }_2<\dots <{\alpha }_{r-2}<{\alpha }_{r-1}$ and $r-3<{\alpha }_{r-1}<\mathsf{p}-2$;
• ${\mathbf{D}}^{\mathsf{p}}$ refers the standard Riemann–Liouville derivative;
• $\hslash \in \mathcal{C}\left([0,1]\times {\mathbb{R}}^r;{\mathbb{R}}^{+}\right)$;
• $e_j\in \mathbb{R}$ and $0<{\delta }_1<{\delta }_2<\dots <{\delta }_{m-1}<1$ with $0<{\displaystyle \sum _{j=1}^{m-2}}{e}_j{\delta }_j^{\mathsf{p}-{\alpha }_{r-1}-1}<1$.

The BCP serves as the cornerstone of metrical fixed-point theory. In accordance to this fundamental outcome, there is a contraction map on CMS. This finding also supplies a technique to predicate this (unique) fixed point. The vast majority of existing research contains a lot of generalizations of the BCP. $\mathsf{\Phi }$-contraction is a straightforward expanded contraction that was derived from conventional contraction by supplementing by the Lipschitz constant with a proper auxiliary function $\mathsf{\Phi }:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$. Browder [12] established a first fixed-point finding under $\mathsf{\Phi }$-contractions. Subsequently, Matkowski [13] expanded the Browder fixed-point finding incorporating the concept of comparison functions.

Quite recently, Pant [14] expanded BCP by investigating the following non-unique fixed point finding.

Theorem 1.

Let $\mathcal{P}$ be a self-map CMS $(\mathbf{V},\varpi )$. If $\exists \beta \in [0,1)$ with

$$\begin{array}{c}\hfill \varpi (\mathcal{P}\mathsf{z},\mathcal{P}\mathsf{w})\le \beta ·\varpi (\mathsf{z},\mathsf{w}),\forall \mathsf{z},\mathsf{w}\in \mathbf{V}with[\mathsf{z}\ne \mathcal{P}(\mathsf{z})or\mathsf{w}\ne \mathcal{P}(\mathsf{w}\left)\right],\end{array}$$

then, $\mathcal{P}$ owns a fixed point.

A generalization of Theorem 1 for $\mathsf{\Phi }$-contraction was subsequently proven by Pant [15].

In 2015, Alam and Imdad [16] established one more interesting and core variant of BCP with endowing an arbitrary BR on underlying MS wherein the contraction map preserves the given BR. During the foregoing decades, various researchers have sharpened and improved the relation-theoretic contraction principle, e.g., [17,18,19,20]. In the same continuation, a few authors investigated such types of outcomes in solving some typical fractional differential equations (cf. [21,22]).

The idea of “almost contraction” was invented by Berinde [23], in 2004, as follows:

Definition 1

([23]). A self-map $\mathcal{P}$ on an MS $(\mathbf{V},\varpi )$ is referred as an almost contraction if ∃$\beta \in (0,1)$ and $𝓁\in {\mathbb{R}}^{+}$ with

$$\varpi (\mathcal{P}\mathsf{z},\mathcal{P}\mathsf{w})\le \beta ·\varpi (\mathsf{z},\mathsf{w})+𝓁·\varpi (\mathsf{w},\mathcal{P}\mathsf{z}),\forall \mathsf{z},\mathsf{w}\in \mathbf{V}.$$

The above condition, by symmetry of $\varpi$, is equivalent to

$$\varpi (\mathcal{P}\mathsf{z},\mathcal{P}\mathsf{w})\le \beta ·\varpi (\mathsf{z},\mathsf{w})+𝓁·\varpi (\mathsf{z},\mathcal{P}\mathsf{w}),\forall \mathsf{z},\mathsf{w}\in \mathbf{V}.$$

Theorem 2

([23]). Every almost contraction on a CMS enjoys a fixed point.

The following subclass of almost contraction was established by Babu et al. [24] to investigate a uniqueness theorem associated with Theorem 2.

Definition 2

([24]). A self-map $\mathcal{P}$ on an MS $(\mathbf{V},\varpi )$ is referred to as a strict almost contraction if ∃$\beta \in (0,1)$ and $𝓁\in {\mathbb{R}}^{+}$ with

$$\varpi (\mathcal{P}\mathsf{z},\mathcal{P}\mathsf{w})\le \beta ·\varpi (\mathsf{z},\mathsf{w})+𝓁·min\left\{\varpi \right(\mathsf{z},\mathcal{P}\mathsf{z}),\varpi (\mathsf{w},\mathcal{P}\mathsf{w}),\varpi (\mathsf{z},\mathcal{P}\mathsf{w}),\varpi (\mathsf{w},\mathcal{P}\mathsf{z}\left)\right\},\forall \mathsf{z},\mathsf{w}\in \mathbf{V}.$$

Theorem 3

([24]). Every strict almost contraction on a CMS enjoys a unique fixed point.

Berinde and Păcurar [25] proved continuity of almost contractions on a fixed-point set. Furthermore, Berinde [26] investigated some fixed-point findings for almost Matkowski contractions. Turinici [27] presented the nonlinear formulation of almost contraction maps and employed the same to enhance Theorem 2 (see also Alfuraidan et al. [28]). Recently, Khan [29], Filali et al. [30] and Alshaban et al. [31] investigated some fixed-point findings under almost contractions in the context of relational MS.

In the following lines, we summarize two certain families of control functions utilizing in the concept of $\mathsf{\Phi }$-contractions.

Definition 3

([32]). A monotonic increasing function $\mathsf{\Phi }:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ is termed as comparison function if

$$\underset{\mathsf{n}\to \infty }{lim}{\mathsf{\Phi }}^{\mathsf{n}}\left(\mathsf{p}\right)=0,\forall \mathsf{p}\in {\mathbb{R}}^{+}\backslash \left\{0\right\}.$$

Definition 4

([32]). A monotonic increasing function $\mathsf{\Phi }:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ is termed as comparison function if

$$\sum _{\mathsf{n}=1}^{\infty }{\mathsf{\Phi }}^{\mathsf{n}}\left(\mathsf{p}\right)<\infty ,\forall \mathsf{p}\in {\mathbb{R}}^{+}\backslash \left\{0\right\}.$$

Obviously, each (c)-comparison function is a comparison function.

Remark 1.

Every comparison function Φ satisfies the following properties:

(i) 

$\mathsf{\Phi }\left(\mathsf{p}\right)<\mathsf{p},\forall \mathsf{p}\in {\mathbb{R}}^{+}\backslash \left\{0\right\}$;

(ii) 

$\underset{\mathsf{t}\to 0^{+}}{lim}\mathsf{\Phi }\left(\mathsf{t}\right)=\mathsf{\Phi }\left(0\right)=0$.

In the continuation, $\mathsf{\Gamma }$ will denote the collection of comparison functions and $\mathsf{\Omega }$ will denote the collection of functions $\varrho :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ verifying $\underset{\mathsf{t}\to 0^{+}}{lim}\varrho \left(\mathsf{t}\right)=\varrho \left(0\right)=0$. The class $\mathsf{\Omega }$ presented was suggested by Turinici [27] and improved by Alfuraidan et al. [28].

In the present article, we expand the recent fixed-point findings of Alshaban et al. [31] from (c)-comparison functions to comparison functions. Indeed, the resultant contraction-inequality subsumes the earlier contraction conditions: Matkowski contraction, almost contraction, relational contraction and Pant contraction. In the process, we prove the assessments on fixed-points in a relational MS. Nonlinear contractions usually require a transitivity condition on underlying BR in order to ensure the existence of a fixed-point. Due to the restrictive nature of a transitivity requirement, we adopt an optimum condition of transitivity (locally $\mathcal{P}$-transitive). For illustration of our outcomes, we constructed two instances. We deduce a number of classical fixed-point assessments, especially owing to Matkowski [13], Pant [15], Arif et al. [19], Babu et al. [24], Berinde [26], Turinici [27], Khan [29], Filali et al. [30] and similar others. To depict our findings, we evaluate a (unique) positive solution of a BVP concerning a singular FDE.

2. Preliminaries

On a set $\mathbf{V}$, by a BR $\mathbf{S}$, we mean any subset of ${\mathbf{V}}^2$. In keeping with the aforementioned definitions, $\mathbf{V}$ is a set, $\mathcal{P}:\mathbf{V}\to \mathbf{V}$ is a map, $\mathbf{S}$ is a BR on $\mathbf{V}$, and $\varpi$ is a metric on $\mathbf{V}$. We say that

Definition 5

([16]). Two elements $\mathsf{z},\mathsf{w}\in \mathbf{V}$ are $\mathbf{S}$-comparative and denoted by $[\mathsf{z},\mathsf{w}]\in \mathbf{S}$, if $(\mathsf{z},\mathsf{w})\in \mathbf{S}$ or $(\mathsf{w},\mathsf{z})\in \mathbf{S}$.

Definition 6

([33]). The BR ${\mathbf{S}}^{-1}:=\{(\mathsf{z},\mathsf{w})\in {\mathbf{V}}^2:(\mathsf{w},\mathsf{z})\in \mathbf{S}\}$ is the inverse of $\mathbf{S}$.

Definition 7

([33]). The BR ${\mathbf{S}}^s:=\mathbf{S}\cup {\mathbf{S}}^{-1}$ is the symmetric closure of $\mathbf{S}$.

Proposition 1

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

Proof. 

The observation is straightforward as

$$\begin{array}{ccc}\hfill (\mathsf{z},\mathsf{w})\in {\mathbf{S}}^s& \iff & (\mathsf{z},\mathsf{w})\in \mathbf{S}\cup {\mathbf{S}}^{-1}\hfill \\ & \iff & (\mathsf{z},\mathsf{w})\in \mathbf{S}\mathrm{or}(\mathsf{z},\mathsf{w})\in {\mathbf{S}}^{-1}\hfill \\ & \iff & (\mathsf{z},\mathsf{w})\in \mathbf{S}\mathrm{or}(\mathsf{w},\mathsf{z})\in \mathbf{S}\hfill \\ & \iff & [\mathsf{z},\mathsf{w}]\in \mathbf{S}.\hfill \end{array}$$

□

Definition 8

([16]). A sequence $\left\{{\mathsf{z}}_{\mathsf{n}}\right\}\subset \mathbf{V}$ satisfying $({\mathsf{z}}_{\mathsf{n}},{\mathsf{z}}_{\mathsf{n}+1})\in \mathbf{S}$, ∀$\mathsf{n}\in \mathbb{N}$ is $\mathbf{S}$-preserving.

Definition 9

([16]). $\mathbf{S}$ is ϖ-self-closed if for every convergent and $\mathbf{S}$-preserving sequence of $\mathbf{V}$, ∃ subsequence where the terms of this subsequence are $\mathbf{S}$-comparative with the limit.

Definition 10

([16]). $\mathbf{S}$ is $\mathcal{P}$-closed if $(\mathcal{P}\mathsf{z},\mathcal{P}\mathsf{w})\in \mathbf{S},$ for every $(\mathsf{z},\mathsf{w})\in \mathbf{S}$.

Proposition 2

([18]). $\mathbf{S}$ is ${\mathcal{P}}^{\mathsf{n}}$-closed if it is $\mathcal{P}$-closed.

Definition 11

([34]). A subset $\mathbf{W}\subseteq \mathbf{V}$$\mathbf{S}$-directed if every pair $\mathsf{u},\mathsf{w}\in \mathbf{W}$ admits an element $\mathsf{v}\in \mathbf{V}$ with $(\mathsf{u},\mathsf{v})\in \mathbf{S}$ and $(\mathsf{w},\mathsf{v})\in \mathbf{S}$.

Definition 12

([17]). $(\mathbf{V},\varpi )$ is $\mathbf{S}$-complete if each Cauchy and $\mathbf{S}$-preserving sequence in $\mathbf{V}$ converges.

Definition 13

([17]). $\mathcal{P}$ is $\mathbf{S}$-continuous if for each $\mathsf{z}\in \mathbf{V}$ and for any $\mathbf{S}$-preserving sequence $\left\{{\mathsf{z}}_{\mathsf{n}}\right\}\subset \mathbf{V}$, we have

$${\mathsf{z}}_{\mathsf{n}}\stackrel{\varpi }{⟶}\mathsf{z}⟹\mathcal{P}\left({\mathsf{z}}_{\mathsf{n}}\right)\stackrel{\varpi }{⟶}\mathcal{P}\left(\mathsf{z}\right).$$

Definition 14

([33]). A BR, on a subset $\mathbf{W}\subseteq \mathbf{V}$, defined by

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

is the restriction of $\mathbf{S}$ on $\mathbf{W}$.

Definition 15

([18]). $\mathbf{S}$ is locally $\mathcal{P}$-transitive if for each $\mathbf{S}$-preserving sequence $\left\{{\mathsf{w}}_n\right\}\subset \mathcal{P}\left(\mathbf{V}\right)$ (with range-set $\mathbf{W}=\{{\mathsf{w}}_{\mathsf{n}}:\mathsf{n}\in \mathbb{N}\})$, ${\mathbf{S}|}_{\mathbf{W}}$ is transitive.

In response to the symmetric axiom of $\varpi$, we constitute the forthcoming claims.

Proposition 3.

If $\mathsf{\Phi }\in \mathsf{\Gamma }$ and $\varrho \in \mathsf{\Omega }$, then two contraction-inequalities mentioned below are equivalent:

(i) 

$\varpi (\mathcal{P}\mathsf{z},\mathcal{P}\mathsf{w})\le \mathsf{\Phi }\left(\varpi \right(\mathsf{z},\mathsf{w}\left)\right)+min\left\{\varrho \right(\varpi (\mathsf{w},\mathcal{P}\mathsf{z})),\varrho (\varpi (\mathsf{z},\mathcal{P}\mathsf{w})\left)\right\},$

$\forall (\mathsf{z},\mathsf{w})\in \mathbf{S}with[\mathsf{z}\ne \mathcal{P}(\mathsf{z})or\mathsf{w}\ne \mathcal{P}(\mathsf{w}\left)\right]$;

(ii) 

$\varpi (\mathcal{P}\mathsf{z},\mathcal{P}\mathsf{w})\le \mathsf{\Phi }\left(\varpi \right(\mathsf{z},\mathsf{w}\left)\right)+min\left\{\varrho \right(\varpi (\mathsf{w},\mathcal{P}\mathsf{z})),\varrho (\varpi (\mathsf{z},\mathcal{P}\mathsf{w})\left)\right\},$

$\forall [\mathsf{z},\mathsf{w}]\in \mathbf{S}with[\mathsf{z}\ne \mathcal{P}(\mathsf{z})or\mathsf{w}\ne \mathcal{P}(\mathsf{w}\left)\right].$

Proposition 4.

If $\mathsf{\Phi }\in \mathsf{\Gamma }$ and $\varrho \in \mathsf{\Omega }$, then two contraction-inequalities mentioned below are equivalent:

(i) 

$\varpi (\mathcal{P}\mathsf{z},\mathcal{P}\mathsf{w})\le \mathsf{\Phi }\left(\varpi \right(\mathsf{z},\mathsf{w}\left)\right)+min\left\{\varrho \right(\varpi (\mathsf{w},\mathcal{P}\mathsf{z})),\varrho (\varpi (\mathsf{z},\mathcal{P}\mathsf{w})),\varrho (\varpi (\mathsf{z},\mathcal{P}\mathsf{z})),\varrho (\varpi (\mathsf{w},\mathcal{P}\mathsf{w})\left)\right\},$

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

(ii) 

$\varpi (\mathcal{P}\mathsf{z},\mathcal{P}\mathsf{w})\le \mathsf{\Phi }\left(\varpi \right(\mathsf{z},\mathsf{w}\left)\right)+min\left\{\varrho \right(\varpi (\mathsf{w},\mathcal{P}\mathsf{z})),\varrho (\varpi (\mathsf{z},\mathcal{P}\mathsf{w})),\varrho (\varpi (\mathsf{z},\mathcal{P}\mathsf{z})),\varrho (\varpi (\mathsf{w},\mathcal{P}\mathsf{w})\left)\right\},$

$\forall [\mathsf{z},\mathsf{w}]\in \mathbf{S}.$

3. Main Results

Hereby, we disclose the fixed-point findings in the structure of relational MS.

Theorem 4.

Assuming $(\mathbf{V},\varpi )$ is a MS comprising a BR $\mathbf{S}$ and $\mathcal{P}:\mathbf{V}\to \mathbf{V}$ is a map. Moreover, the following hold:

(a) 

$(\mathbf{V},\varpi )$ is $\mathbf{S}$-complete MS;

(b) 

$\exists z_0\in \mathbf{V}$ with $({\mathsf{z}}_0,\mathcal{P}{\mathsf{z}}_0)\in \mathbf{S}$;

(c) 

$\mathbf{S}$ is $\mathcal{P}$-closed and locally $\mathcal{P}$-transitive;

(d) 

$\mathcal{P}$ is $\mathbf{S}$-continuous or $\mathbf{S}$ is ϖ-self-closed;

(e) 

∃$\mathsf{\Phi }\in \mathsf{\Gamma }$ and $\varrho \in \mathsf{\Omega }$ satisfy

$$\begin{array}{c}\hfill \varpi (\mathcal{P}\mathsf{z},\mathcal{P}\mathsf{w})\le \mathsf{\Phi }\left(\varpi \right(\mathsf{z},\mathsf{w}\left)\right)+min\left\{\varrho \right(\varpi (\mathsf{w},\mathcal{P}\mathsf{z})),\varrho (\varpi (\mathsf{z},\mathcal{P}\mathsf{w})\left)\right\},\\ \hfill \forall (\mathsf{z},\mathsf{w})\in \mathbf{S}with[\mathsf{z}\ne \mathcal{P}(\mathsf{z})or\mathsf{w}\ne \mathcal{P}(\mathsf{w}\left)\right].\end{array}$$

Then, $\mathcal{P}$ owns a fixed point.

Proof. 

The task will be finished in the following stages:

Step–1. Define the following sequence $\left\{{\mathsf{z}}_{\mathsf{n}}\right\}\subset \mathbf{V}$:

$${\mathsf{z}}_{\mathsf{n}}={\mathcal{P}}^{\mathsf{n}}\left({\mathsf{z}}_0\right)=\mathcal{P}\left({\mathsf{z}}_{\mathsf{n}-1}\right),\forall \mathsf{n}\in \mathbb{N}.$$

(2)

Step–2. We will show that the sequence $\left\{{\mathsf{z}}_{\mathsf{n}}\right\}$ is $\mathbf{S}$-preserving. Utilizing $\left(b\right)$, $\mathcal{P}$-closedness of $\mathbf{S}$ and Proposition 2, we conclude

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

which on utilizing (2) becomes

$$({\mathsf{z}}_{\mathsf{n}},{\mathsf{z}}_{\mathsf{n}+1})\in \mathbf{S},\forall \mathsf{n}\in \mathbb{N}.$$

(3)

Step–3. Define ${\varpi }_{\mathsf{n}}:=\varpi ({\mathsf{z}}_{\mathsf{n}},{\mathsf{z}}_{\mathsf{n}+1})$. If for some $\mathsf{p}\in {\mathbb{N}}_0$ with ${\varpi }_{\mathsf{p}}=0$, then from (2), we conclude that ${\mathsf{w}}_{\mathsf{p}}={\mathsf{w}}_{{\mathsf{n}}_0+1}=\mathcal{P}\left({\mathsf{w}}_{\mathsf{p}}\right)$; so ${\mathsf{w}}_{\mathsf{p}}\in \mathrm{Fix}\left(\mathcal{P}\right)$ and so, we are finished. Unless we have ${\varpi }_{\mathsf{n}}>0$, ∀$\mathsf{n}\in {\mathbb{N}}_0$, so that we move to Step–4.

Step–4. We will show that the sequence $\left\{{\mathsf{z}}_{\mathsf{n}}\right\}$ is Cauchy. For each $\mathsf{n}\in {\mathbb{N}}_0$, we conclude that ${\mathsf{z}}_{\mathsf{n}-1}\ne {\mathsf{z}}_{\mathsf{n}}$. By $\left(e\right)$ and (2), we find

$$\begin{array}{ccc}\hfill \varpi ({\mathsf{z}}_{\mathsf{n}},{\mathsf{z}}_{\mathsf{n}+1})& \le & \mathsf{\Phi }\left(\varpi ({\mathsf{z}}_{\mathsf{n}-1},{\mathsf{z}}_{\mathsf{n}})\right)+\varrho \left(\varpi ({\mathsf{z}}_{\mathsf{n}},\mathcal{P}{\mathsf{z}}_{\mathsf{n}-1})\right)=\mathsf{\Phi }\left(\varpi ({\mathsf{z}}_{\mathsf{n}-1},{\mathsf{z}}_{\mathsf{n}})\right)+\varrho \left(0\right),\hfill \end{array}$$

i.e.,

$${\varpi }_{\mathsf{n}}\le \mathsf{\Phi }\left({\varpi }_{\mathsf{n}-1}\right),\forall \mathsf{n}\in \mathbb{N}.$$

Using monotonicity of $\mathsf{\Phi }$, last relation reduces to

$${\varpi }_{\mathsf{n}}\le {\mathsf{\Phi }}^{\mathsf{n}}\left({\varpi }_0\right),\forall \mathsf{n}\in \mathbb{N}.$$

(4)

Applying $\mathsf{n}\to \infty$ in (4) and using axiom (ii) of $\mathsf{\Phi }$, we conclude

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}{\varpi }_{\mathsf{n}}=0.\end{array}$$

(5)

Choose $\epsilon >0.$ Then by (5), we can determine $\mathsf{n}\in {\mathbb{N}}_0$, verifying

$$\begin{array}{c}\hfill {\varpi }_{\mathsf{n}}<\epsilon -\mathsf{\Phi }\left(\epsilon \right).\end{array}$$

(6)

Next, we will show that $\left\{{\mathsf{z}}_{\mathsf{n}}\right\}$ is Cauchy. Due to the monotonic property of $\mathsf{\Phi }$, (4) and (6), we attain

$$\begin{array}{ccc}\hfill \varpi ({\mathsf{z}}_{\mathsf{n}},{\mathsf{z}}_{\mathsf{n}+2})& \le & \varpi ({\mathsf{z}}_{\mathsf{n}},{\mathsf{z}}_{\mathsf{n}+1})+\varpi ({\mathsf{z}}_{\mathsf{n}+1},{\mathsf{z}}_{\mathsf{n}+2})={\varpi }_{\mathsf{n}}+{\varpi }_{\mathsf{n}+1}\hfill \\ & \le & {\varpi }_{\mathsf{n}}+\mathsf{\Phi }\left({\varpi }_{\mathsf{n}}\right)<\epsilon -\mathsf{\Phi }\left(\epsilon \right)+\mathsf{\Phi }(\epsilon -\mathsf{\Phi }\left(\epsilon \right))\hfill \\ & \le & \epsilon -\mathsf{\Phi }\left(\epsilon \right)+\mathsf{\Phi }\left(\epsilon \right)=\epsilon \hfill \end{array}$$

so that

$$\varpi ({\mathsf{z}}_{\mathsf{n}},{\mathsf{z}}_{\mathsf{n}+2})<\epsilon .$$

(7)

In lieu of (2), $\left\{{\mathsf{z}}_{\mathsf{n}}\right\}\subset \mathcal{P}\left(\mathbf{V}\right)$. Now, (3) and the locally $\mathcal{P}$-transitivity of $\mathbf{S}$ yield that $({\mathsf{z}}_{\mathsf{n}},{\mathsf{z}}_{\mathsf{n}+2})\in \mathbf{S}$. Hence, applying assumption $\left(e\right)$, we conclude that

$$\varpi ({\mathsf{z}}_{\mathsf{n}},{\mathsf{z}}_{\mathsf{n}+1})=\varpi (\mathcal{P}{\mathsf{z}}_{\mathsf{n}},\mathcal{P}{\mathsf{z}}_{\mathsf{n}+2})\le \mathsf{\Phi }\left(\varpi ({\mathsf{z}}_{\mathsf{n}},{\mathsf{z}}_{\mathsf{n}+2})\right),$$

which, making use of (7) and by monotonic property of $\mathsf{\Phi }$, reduces to

$$\varpi ({\mathsf{z}}_{\mathsf{n}},{\mathsf{z}}_{\mathsf{n}+1})\le \mathsf{\Phi }\left(\epsilon \right).$$

(8)

Using triangular inequality, (6) and (8), we conclude

$$\begin{array}{ccc}\hfill \varpi ({\mathsf{z}}_{\mathsf{n}},{\mathsf{z}}_{\mathsf{n}+3})& \le & {\varpi }_{\mathsf{n}}+\varpi ({\mathsf{z}}_{\mathsf{n}+1},{\mathsf{z}}_{\mathsf{n}+3})\hfill \\ & <& \epsilon -\mathsf{\Phi }\left(\epsilon \right)+\mathsf{\Phi }\left(\epsilon \right)=\epsilon .\hfill \end{array}$$

Using induction, we find

$$\varpi ({\mathsf{z}}_{\mathsf{n}},{\mathsf{z}}_{\mathsf{n}+\mathsf{p}})<\epsilon ,\forall \mathsf{p}\in \mathbb{N}.$$

Thus, $\left\{{\mathsf{z}}_{\mathsf{n}}\right\}$ is Cauchy and $\mathbf{S}$-preserving. By condition $\left(a\right)$, ∃${\mathsf{z}}^{*}\in \mathbf{V}$ with ${\mathsf{z}}_{\mathsf{n}}\stackrel{\varpi }{⟶}{\mathsf{z}}^{*}$.

Step–5. We will confirm that ${\mathsf{z}}^{*}$ is a fixed-point of $\mathcal{P}$. By $\left(d\right)$, if $\mathcal{P}$ remains $\mathbf{S}$-continuous, then the $\mathbf{S}$-preserving property of the sequence $\left\{{\mathsf{z}}_{\mathsf{n}}\right\}$ and the fact ${\mathsf{z}}_{\mathsf{n}}\stackrel{\varpi }{⟶}{\mathsf{z}}^{*}$ yield that

$${\mathsf{z}}_{\mathsf{n}+1}=\mathcal{P}\left({\mathsf{z}}_{\mathsf{n}}\right)\stackrel{\varpi }{⟶}\mathcal{P}\left({\mathsf{z}}^{*}\right)$$

implying thereby, $\mathcal{P}\left({\mathsf{z}}^{*}\right)={\mathsf{z}}^{*}$.

If $\mathbf{S}$ remains $\varpi$-self-closed, then $\left\{{\mathsf{z}}_{\mathsf{n}}\right\}$ admits a subsequence $\left\{{\mathsf{z}}_{{\mathsf{n}}_k}\right\}$ ensuring $[{\mathsf{z}}_{{\mathsf{n}}_k},{\mathsf{z}}^{*}]\in \mathbf{S},\forall k\in \mathbb{N}$. Define ${\sigma }_{\mathsf{n}}:=\varpi ({\mathsf{z}}^{*},{\mathsf{z}}_{\mathsf{n}})$. If $\mathcal{P}\left({\mathsf{z}}^{*}\right)={\mathsf{z}}^{*}$, then we are finished. If $\mathcal{P}\left({\mathsf{z}}^{*}\right)\ne {\mathsf{z}}^{*}$, then by condition $\left(e\right)$, Proposition 3 and $[{\mathsf{z}}_{{\mathsf{n}}_k},{\mathsf{z}}^{*}]\in \mathbf{S}$, we obtain

$$\begin{array}{ccc}\hfill \varpi ({\mathsf{z}}_{{\mathsf{n}}_k+1},\mathcal{P}{\mathsf{z}}^{*})& =& \varpi (\mathcal{P}{\mathsf{z}}_{{\mathsf{n}}_k},\mathcal{P}{\mathsf{z}}^{*})\hfill \\ & \le & \mathsf{\Phi }\left(\varpi ({\mathsf{z}}_{{\mathsf{n}}_k},{\mathsf{z}}^{*})\right)+min\left\{\varrho \left(\varpi ({\mathsf{z}}^{*},\mathcal{P}{\mathsf{z}}_{{\mathsf{n}}_k})\right),\varpi ({\mathsf{z}}_{{\mathsf{n}}_k},\mathcal{P}{\mathsf{z}}^{*})\right\}\hfill \\ & =& \mathsf{\Phi }\left({\sigma }_{{\mathsf{n}}_k}\right)+min\left\{\varrho \left({\sigma }_{{\mathsf{n}}_k+1}\right),\varpi ({\mathsf{z}}_{{\mathsf{n}}_k},\mathcal{P}{\mathsf{z}}^{*})\right\}.\hfill \end{array}$$

(9)

Now, ${\mathsf{z}}_{{\mathsf{n}}_k}\stackrel{\varpi }{⟶}{\mathsf{z}}^{*}$ implies that ${\sigma }_{{\mathsf{n}}_k}\stackrel{{\mathbb{R}}^{+}}{⟶}{0}^{+}$, whenever $k\to \infty$. Letting $k\to \infty$ in (9) and using Remark 1 and the property of $\mathsf{\Omega }$, we find

$$\begin{array}{ccc}\hfill \underset{k\to \infty }{lim}\varpi ({\mathsf{z}}_{{\mathsf{n}}_k+1},\mathcal{P}{\mathsf{z}}^{*})& \le & \underset{k\to \infty }{lim}\mathsf{\Phi }\left({\sigma }_{{\mathsf{n}}_k}\right)+min\left\{\underset{k\to \infty }{lim}\varrho \left({\sigma }_{{\mathsf{n}}_k+1}\right),\underset{k\to \infty }{lim}\varpi ({\mathsf{z}}_{{\mathsf{n}}_k},\mathcal{P}{\mathsf{z}}^{*})\right\}\hfill \\ & =& \underset{t\to 0^{+}}{lim}\mathsf{\Phi }\left(t\right)+min\left\{\underset{t\to 0^{+}}{lim}\varrho \left(t\right),\underset{k\to \infty }{lim}\varpi ({\mathsf{z}}_{{\mathsf{n}}_k},\mathcal{P}{\mathsf{z}}^{*})\right\}\hfill \\ & =& 0\hfill \end{array}$$

or, ${\mathsf{z}}_{{\mathsf{n}}_k+1}\stackrel{\varpi }{⟶}\mathcal{P}\left({\mathsf{z}}^{*}\right)$ implying $\mathcal{P}\left({\mathsf{z}}^{*}\right)={\mathsf{z}}^{*}$. Thus, ${\mathsf{z}}^{*}$ is a fixed point of $\mathcal{P}$. □

Theorem 5.

Along with the conditions $\left(a\right)$–$\left(d\right)$ of Theorem 4, if

(f) 

∃$\mathsf{\Phi }\in \mathsf{\Gamma }$ and $\varrho \in \mathsf{\Omega }$ with

$$\begin{array}{ccc}\hfill \varpi (\mathcal{P}\mathsf{z},\mathcal{P}\mathsf{w})& \le & \mathsf{\Phi }\left(\varpi \right(\mathsf{z},\mathsf{w}\left)\right)+min\left\{\varrho \right(\varpi (\mathsf{w},\mathcal{P}\mathsf{z})),\varrho (\varpi (\mathsf{z},\mathcal{P}\mathsf{w})),\varrho (\varpi (\mathsf{z},\mathcal{P}\mathsf{z})),\varrho (\varpi (\mathsf{w},\mathcal{P}\mathsf{w})\left)\right\},\hfill \\ & & \forall (\mathsf{z},\mathsf{w})\in \mathbf{S}\hfill \end{array}$$

and

(g) 

$\mathcal{P}\left(\mathbf{V}\right)$ is ${\mathbf{S}}^s$-directed,

then $\mathcal{P}$ enjoys a unique fixed point.

Proof. 

If $\left(f\right)$ is valid, then $\left(e\right)$ of Theorem 4 is valid. Employing Theorem 4, select two fixed points $\mathsf{z},\mathsf{w}$ of $\mathcal{P}$, i.e.,

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

(10)

As $\mathsf{z},\mathsf{w}\in \mathcal{P}\left(\mathbf{V}\right)$, by condition $\left(g\right)$, $\exists \mathsf{v}\in \mathbf{V}$ with $[\mathsf{z},\mathsf{v}]\in \mathbf{S}$ and $[\mathsf{w},\mathsf{v}]\in \mathbf{S}$. The $\mathcal{P}$-closedness of $\mathbf{S}$ along with Proposition 2 yields that

$$[{\mathcal{P}}^{\mathsf{n}}\mathsf{z},{\mathcal{P}}^{\mathsf{n}}\mathsf{v}]\in \mathbf{S}\mathrm{and}[{\mathcal{P}}^{\mathsf{n}}\mathsf{w},{\mathcal{P}}^{\mathsf{n}}\mathsf{v}]\in \mathbf{S},\forall \mathsf{n}\in \mathbb{N}.$$

(11)

Define ${\phi }_{\mathsf{n}}:=\varpi ({\mathcal{P}}^{\mathsf{n}}\mathsf{z},{\mathcal{P}}^{\mathsf{n}}\mathsf{v})$. We will reveal that

$${\displaystyle \underset{\mathsf{n}\to \infty }{lim}{\phi }_{\mathsf{n}}=\underset{\mathsf{n}\to \infty }{lim}\varpi ({\mathcal{P}}^{\mathsf{n}}\mathsf{z},{\mathcal{P}}^{\mathsf{n}}\mathsf{v})=0.}$$

(12)

Using (10), (11), condition $\left(f\right)$ and Proposition 4, we attain

$$\begin{array}{ccc}\hfill \varpi ({\mathcal{P}}^{\mathsf{n}+1}\mathsf{z},{\mathcal{P}}^{\mathsf{n}+1}\mathsf{v})& \le & \mathsf{\Phi }\left(\varpi ({\mathcal{P}}^{\mathsf{n}}\mathsf{z},{\mathcal{P}}^{\mathsf{n}}\mathsf{v})\right)+min\{\varrho \left(\varpi ({\mathcal{P}}^{\mathsf{n}}\mathsf{v},{\mathcal{P}}^{\mathsf{n}+1}\mathsf{z})\right),\varrho \left(\varpi ({\mathcal{P}}^{\mathsf{n}}\mathsf{z},{\mathcal{P}}^{\mathsf{n}+1}\mathsf{v})\right),\hfill \\ & & \varrho \left(\varpi ({\mathcal{P}}^{\mathsf{n}}\mathsf{z},{\mathcal{P}}^{\mathsf{n}+1}\mathsf{z})\right),\varrho \left(\varpi ({\mathcal{P}}^{\mathsf{n}}\mathsf{v},{\mathcal{P}}^{\mathsf{n}+1}\mathsf{v})\right)\},\hfill \\ & =& \mathsf{\Phi }\left(\varpi ({\mathcal{P}}^{\mathsf{n}}\mathsf{z},{\mathcal{P}}^{\mathsf{n}}\mathsf{v})\right),as\varpi ({\mathcal{P}}^{\mathsf{n}}\mathsf{z},{\mathcal{P}}^{\mathsf{n}+1}\mathsf{z})=\varpi (\mathsf{z},\mathsf{z})=0\hfill \end{array}$$

i.e.,

$${\phi }_{\mathsf{n}+1}\le \mathsf{\Phi }\left({\phi }_{\mathsf{n}}\right).$$

(13)

If there is some ${\mathsf{n}}_0\in \mathbb{N}$ for which ${\phi }_{{\mathsf{n}}_0}=0$, then we conclude that ${\mathcal{P}}^{{\mathsf{n}}_0}\left(\mathsf{z}\right)={\mathcal{P}}^{{\mathsf{n}}_0}\left(\mathsf{v}\right)$. This implies that ${\mathcal{P}}^{{\mathsf{n}}_0+1}\left(\mathsf{z}\right)={\mathcal{P}}^{{\mathsf{n}}_0+1}\left(\mathsf{v}\right)$. Thus, we find ${\phi }_{{\mathsf{n}}_0+1}=0$. Using induction, we obtain ${\phi }_{\mathsf{n}}=0,\forall \mathsf{n}\ge {\mathsf{n}}_0,$ so that $\underset{\mathsf{n}\to \infty }{lim}{\phi }_{\mathsf{n}}=0$. If ${\phi }_{\mathsf{n}}>0,\forall \mathsf{n}\in \mathbb{N}$, then by monotonic property $\mathsf{\Phi }$, (13) gives rise

$$\begin{array}{c}\hfill {\phi }_{\mathsf{n}+1}\le \mathsf{\Phi }\left({\phi }_{\mathsf{n}}\right)\le {\mathsf{\Phi }}^2\left({\phi }_{\mathsf{n}-1}\right)\le \cdots \le {\mathsf{\Phi }}^{\mathsf{n}}\left({\phi }_1\right)\end{array}$$

so that

$${\phi }_{\mathsf{n}+1}\le {\mathsf{\Phi }}^{\mathsf{n}}\left({\phi }_1\right).$$

Letting $\mathsf{n}\to \infty$ in last relation and by a characteristic of $\mathsf{\Phi }$, we attain

$${\displaystyle \underset{\mathsf{n}\to \infty }{lim}{\phi }_{\mathsf{n}+1}\le \underset{\mathsf{n}\to \infty }{lim}{\mathsf{\Phi }}^{\mathsf{n}}\left({\phi }_1\right)=0.}$$

Thus, (12) is proved. Likewise, we can find that

$${\displaystyle \underset{\mathsf{n}\to \infty }{lim}\varpi ({\mathcal{P}}^{\mathsf{n}}\mathsf{w},{\mathcal{P}}^{\mathsf{n}}\mathsf{v})=0.}$$

(14)

By (12) and (14), we obtain

$$\varpi (\mathsf{z},\mathsf{w})=\varpi ({\mathcal{P}}^{\mathsf{n}}\mathsf{z},{\mathcal{P}}^{\mathsf{n}}\mathsf{w})\le \varpi ({\mathcal{P}}^{\mathsf{n}}\mathsf{z},{\mathcal{P}}^{\mathsf{n}}\mathsf{v})+\varpi ({\mathcal{P}}^{\mathsf{n}}\mathsf{w},{\mathcal{P}}^{\mathsf{n}}\mathsf{v})\to 0\mathrm{as}\mathsf{n}\to \infty$$

so $\mathsf{z}=\mathsf{w}$. The conclusion has thus been arrived. □

4. Consequences

In the following portion, we will implement our outcomes to figure out various known fixed-point findings.

Particularly, for $\mathsf{\Phi }\left(\mathsf{p}\right)=\beta ·\mathsf{p}$ (where $\beta \in (0,1)$) and $\varrho \left(\mathsf{p}\right)=𝓁·\mathsf{p}$ (where $𝓁\in {\mathbb{R}}^{+}$), Theorem 4 deduces the following outcome. However, in this case, the the condition of locally $\mathcal{P}$-transitivity can be relaxed.

Corollary 1

(Khan [29]). Assuming $(\mathbf{V},\varpi )$ is an MS comprising a BR $\mathbf{S}$ and $\mathcal{P}:\mathbf{V}\to \mathbf{V}$ is a map. Also,

(a) 

$(\mathbf{V},\varpi )$ is $\mathbf{S}$-complete;

(b) 

$\exists {\mathsf{z}}_0\in \mathbf{V}$ with $({\mathsf{z}}_0,\mathcal{P}{\mathsf{z}}_0)\in \mathbf{S}$;

(c) 

$\mathbf{S}$ is $\mathcal{P}$-closed;

(d) 

$\mathcal{P}$ is $\mathbf{S}$-continuous or $\mathbf{S}$ is ϖ-self-closed;

(e) 

∃$\beta \in (0,1)$ and $𝓁\in {\mathbb{R}}^{+}$ with

$$\begin{array}{c}\hfill \varpi (\mathcal{P}\mathsf{z},\mathcal{P}\mathsf{w})\le \beta ·\varpi (\mathsf{z},\mathsf{w})+𝓁·\varpi (\mathsf{w},\mathcal{P}\mathsf{z}),\forall (\mathsf{z},\mathsf{w})\in \mathbf{S}.\end{array}$$

Then, $\mathcal{P}$ owns a fixed point.

Under the restriction $\varrho \left(\mathsf{p}\right)=𝓁·\mathsf{p}$ (where $𝓁\in {\mathbb{R}}^{+}$), Theorem 4 reduces to the following finding.

Corollary 2

(Filali et al. [30]). Assuming $(\mathbf{V},\varpi )$ is an MS comprising a BR $\mathbf{S}$ and $\mathcal{P}:\mathbf{V}\to \mathbf{V}$ is a map. Also,

(a) 

$(\mathbf{V},\varpi )$ is $\mathbf{S}$-complete;

(b) 

$\exists {\mathsf{z}}_0\in \mathbf{V}$ with $({\mathsf{z}}_0,\mathcal{P}{\mathsf{z}}_0)\in \mathbf{S}$;

(c) 

$\mathbf{S}$ is locally $\mathcal{P}$-transitive and $\mathcal{P}$-closed;

(d) 

$\mathcal{P}$ is $\mathbf{S}$-continuous or $\mathbf{S}$ is ϖ-self-closed;

(e) 

∃$\mathsf{\Phi }\in \mathsf{\Gamma }$ and $𝓁\in {\mathbb{R}}^{+}$, verifying

$$\begin{array}{c}\hfill \varpi (\mathcal{P}\mathsf{z},\mathcal{P}\mathsf{w})\le \mathsf{\Phi }\left(\varpi \right(\mathsf{z},\mathsf{w}\left)\right)+𝓁·min\left\{\varpi \right(\mathsf{w},\mathcal{P}\mathsf{z}),\varpi (\mathsf{z},\mathcal{P}\mathsf{w}\left)\right\},\forall (\mathsf{z},\mathsf{w})\in \mathbf{S}.\end{array}$$

Then, $\mathcal{P}$ owns a fixed point.

If we take $\varrho \left(\mathsf{p}\right)=0$ for all $𝓁\in {\mathbb{R}}^{+}$ in Theorem 4, then we find the following result.

Corollary 3

(Arif et al. [19]). Assuming $(\mathbf{V},\varpi )$ is an MS comprised with a BR $\mathbf{S}$ and $\mathcal{P}:\mathbf{V}\to \mathbf{V}$ is a map. Also,

(a) 

$(\mathbf{V},\varpi )$ is $\mathbf{S}$-complete;

(b) 

$\exists {\mathsf{z}}_0\in \mathbf{V}$ with $({\mathsf{z}}_0,\mathcal{P}{\mathsf{z}}_0)\in \mathbf{S}$;

(c) 

$\mathbf{S}$ is locally $\mathcal{P}$-transitive and $\mathcal{P}$-closed;

(d) 

$\mathcal{P}$ is $\mathbf{S}$-continuous or $\mathbf{S}$ is ϖ-self-closed;

(e) 

∃$\mathsf{\Phi }\in \mathsf{\Gamma }$ with

$$\begin{array}{c}\hfill \varpi (\mathcal{P}\mathsf{z},\mathcal{P}\mathsf{w})\le \mathsf{\Phi }\left(\varpi \right(\mathsf{z},\mathsf{w}\left)\right),\forall (\mathsf{z},\mathsf{w})\in \mathbf{S}.\end{array}$$

Then, $\mathcal{P}$ owns a fixed point.

Under universal relation $\mathbf{S}={\mathbf{V}}^2$, Theorem 4 deduces the following outcomes.

Corollary 4

(Turinici [27]). Assuming $(\mathbf{V},\varpi )$ is a CMS and $\mathcal{P}:\mathbf{V}\to \mathbf{V}$ is a map. If ∃$\beta \in [0,1)$ and $\varrho \in \mathsf{\Omega }$, verifying

$$\begin{array}{c}\hfill \varpi (\mathcal{P}\mathsf{z},\mathcal{P}\mathsf{w})\le \beta ·\varpi (\mathsf{z},\mathsf{w}))+\varrho (\varpi (\mathsf{w},\mathcal{P}\mathsf{z})),\forall \mathsf{z},\mathsf{w}\in \mathbf{V},\end{array}$$

then, $\mathcal{P}$ owns a fixed point.

Corollary 5

(Berinde [26]). Assuming $(\mathbf{V},\varpi )$ is a CMS and $\mathcal{P}:\mathbf{V}\to \mathbf{V}$ is a map. If ∃$\mathsf{\Phi }\in \mathsf{\Gamma }$ and $𝓁\in {\mathbb{R}}^{+}$, verifying

$$\begin{array}{c}\hfill \varpi (\mathcal{P}\mathsf{z},\mathcal{P}\mathsf{w})\le \mathsf{\Phi }\left(\varpi \right(\mathsf{z},\mathsf{w}\left)\right)+𝓁·\varpi (\mathsf{w},\mathcal{P}\mathsf{z}),\forall \mathsf{z},\mathsf{w}\in \mathbf{V},\end{array}$$

then, $\mathcal{P}$ owns a fixed point.

Corollary 6

(Pant [15]). Assuming $(\mathbf{V},\varpi )$ is a CMS and $\mathcal{P}:\mathbf{V}\to \mathbf{V}$ is a map. If ∃$\mathsf{\Phi }\in \mathsf{\Gamma }$, verifying

$$\begin{array}{c}\hfill \varpi (\mathcal{P}\mathsf{z},\mathcal{P}\mathsf{w})\le \mathsf{\Phi }\left(\varpi \right(\mathsf{z},\mathsf{w}\left)\right),\forall \mathsf{z},\mathsf{w}\in \mathbf{V}with[\mathsf{z}\ne \mathcal{P}(\mathsf{z})or\mathsf{w}\ne \mathcal{P}(\mathsf{w}\left)\right],\end{array}$$

then, $\mathcal{P}$ owns a fixed point.

Under universal relation $\mathbf{S}={\mathbf{V}}^2$, Theorem 5 deduces the following outcomes.

Corollary 7

(Babu et al. [24]). Assume that $(\mathbf{V},\varpi )$ is a CMS comprising a BR $\mathbf{S}$ and $\mathcal{P}:\mathbf{V}\to \mathbf{V}$ is a map. If ∃$\beta \in [0,1)$ and $𝓁\in {\mathbb{R}}^{+}$, verifying

$$\begin{array}{c}\hfill \varpi (\mathcal{P}\mathsf{z},\mathcal{P}\mathsf{w})\le \beta ·\varpi (\mathsf{z},\mathsf{w})+𝓁·min\left\{\varpi \right(\mathsf{w},\mathcal{P}\mathsf{z}),\varpi (\mathsf{z},\mathcal{P}\mathsf{w}),\varpi (\mathsf{z},\mathcal{P}\mathsf{z}),\varpi (\mathsf{w},\mathcal{P}\mathsf{w}\left)\right\},\forall \mathsf{z},\mathsf{w}\in \mathbf{V},\end{array}$$

then, $\mathcal{P}$ owns a unique fixed point.

Corollary 8

(Matkowski [13]). Assume that $(\mathbf{V},\varpi )$ is a CMS and $\mathcal{P}:\mathbf{V}\to \mathbf{V}$ is a map. If ∃$\mathsf{\Phi }\in \mathsf{\Gamma }$, verifying

$$\begin{array}{c}\hfill \varpi (\mathcal{P}\mathsf{z},\mathcal{P}\mathsf{w})\le \mathsf{\Phi }\left(\varpi \right(\mathsf{z},\mathsf{w}\left)\right),\forall \mathsf{z},\mathsf{w}\in \mathbf{V},\end{array}$$

then, $\mathcal{P}$ owns a unique fixed point.

5. Illustrative Examples

A number of examples concerning the Theorems 4 and 5 are offered in this part.

Example 1.

Consider $\mathbf{V}={\mathbb{R}}^{+}$ under Euclidean metric ϖ and a BR $\mathbf{S}:=\{(\mathsf{z},\mathsf{w})\in {\mathbf{V}}^2:\mathsf{z}-\mathsf{w}>0\}$. Define the map $\mathcal{P}:\mathbf{V}\to \mathbf{V}$ by $\mathcal{P}\left(\mathsf{z}\right)={\displaystyle \frac{\mathsf{z}}{\mathsf{z}+1}}$. Clearly, the BR $\mathbf{S}$ is locally $\mathcal{P}$-transitive, the MS $(\mathbf{V},\varpi )$ is $\mathbf{S}$-complete and $\mathcal{P}$ is $\mathbf{S}$-continuous.

Let $(\mathsf{z},\mathsf{w})\in \mathbf{S}$; then we attain $\mathsf{z}-\mathsf{w}>0$ and so,

$$\mathcal{P}\left(\mathsf{z}\right)-\mathcal{P}\left(\mathsf{w}\right)={\displaystyle \frac{\mathsf{z}-\mathsf{w}}{(\mathsf{z}+1)(\mathsf{w}+1)}}>0,$$

which concludes that $(\mathcal{P}\mathsf{z},\mathcal{P}\mathsf{w})\in \mathbf{S}$ so that $\mathbf{S}$ is $\mathcal{P}$-closed.

Define $\mathsf{\Phi }\in \mathsf{\Gamma }$ and $\varrho \in \mathsf{\Omega }$ by $\mathsf{\Phi }\left(t\right)={\displaystyle \frac{t}{t+1}}$ and $\varrho \left(t\right)=ln(1+t)$. Now, for all $(\mathsf{z},\mathsf{w})\in \mathbf{S}$, we have

$$\begin{array}{ccc}\hfill \varpi (\mathcal{P}\mathsf{z},\mathcal{P}\mathsf{w})& =& |{\displaystyle \frac{\mathsf{z}}{\mathsf{z}+1}}-{\displaystyle \frac{\mathsf{w}}{\mathsf{w}+1}}|=\left|{\displaystyle \frac{\mathsf{z}-\mathsf{w}}{1+\mathsf{z}+\mathsf{w}+\mathsf{z}\mathsf{w}}}\right|\hfill \\ & \le & {\displaystyle \frac{\mathsf{z}-\mathsf{w}}{1+(\mathsf{z}-\mathsf{w})}}={\displaystyle \frac{\varpi (\mathsf{z},\mathsf{w})}{1+\varpi (\mathsf{z},\mathsf{w})}}\hfill \\ & \le & \mathsf{\Phi }\left(\varpi \right(\mathsf{z},\mathsf{w}\left)\right)+min\left\{\varrho \right(\varpi (\mathsf{w},\mathcal{P}\mathsf{z})),\varrho (\varpi (\mathsf{z},\mathcal{P}\mathsf{w})),\varrho (\varpi (\mathsf{z},\mathcal{P}\mathsf{z})),\varrho (\varpi (\mathsf{w},\mathcal{P}\mathsf{w})\left)\right\}.\hfill \end{array}$$

It demonstrates that the argument $\left(f\right)$ of Theorem 5 is confirmed. Also, ${\mathsf{z}}_0=1$ satisfies the condition $\left(b\right)$. Finally, $\mathcal{P}\left(\mathbf{V}\right)$ is ${\mathbf{S}}^s$-directed since for every pair $\mathsf{z},\mathsf{w}\in \mathcal{P}\left(\mathbf{V}\right)$, the element $\mathsf{u}:=(\mathsf{z}+\mathsf{w})/2$ satisfies $[\mathsf{z},\mathsf{u}]\in \mathbf{S}$ and $[\mathsf{w},\mathsf{u}]\in \mathbf{S}$. Therefore, all the assumptions of Theorem 5 hold and hence $\mathcal{P}$ owns a unique fixed point, $\overline{\mathsf{z}}=0$.

In above example $\mathsf{\Phi }$ is not a (c)-comparison function. Therefore this example cannot be covered by corresponding theorems of Alshaban et al. [31]. This reveals that our results are more advantageous compared to the findings of Alshaban et al. [31].

Example 2.

Consider $\mathbf{V}=[0,1]$ under Euclidean metric ϖ and BR $\mathbf{S}=\mathbb{R}\times \mathbb{Q}$. Clearly, $(\mathbf{V},\varpi )$ is $\mathbf{S}$-complete MS. Let $\mathcal{P}$ be the identity map on $\mathbf{V}$. Then, $\mathbf{S}$ is $\mathcal{P}$-closed and $\mathcal{P}$ is $\mathbf{S}$-continuous.

Fix $\beta \in [0,1)$ and define $\mathsf{\Phi }\in \mathsf{\Gamma }$ and $\varrho \in \mathsf{\Omega }$ with $\mathsf{\Phi }\left(t\right)=\beta t$ and $\varrho \left(t\right)=t-\beta t$. For every $(\mathsf{z},\mathsf{w})\in \mathbf{S}$, the contraction-inequality of Theorem 4 is verified. In the same way, all the assertions of Theorem 4 hold; henceforth $\mathcal{P}$ owns a fixed point. In this example, $Fix\left(\mathcal{P}\right)=[0,1]$ and hence Theorem 5 cannot be applied.

6. Applications to Fractional Differential Equations

Consider the singular fractional BVP mentioned below

$$\left\{\begin{array}{c}{\mathbf{D}}_{0^{+}}^{\mathsf{p}}\mathsf{v}\left(\theta \right)+\hslash (\theta ,\mathsf{v}\left(\theta \right))=0,\forall \theta \in (0,1),\hfill \\ \mathsf{v}\left(0\right)={\mathsf{v}}^{\prime }\left(0\right)={\mathsf{v}}^{″}\left(0\right)=0,{\mathsf{v}}^{″}\left(1\right)=q{\mathsf{v}}^{″}\left(\delta \right),\hfill \end{array}\right.$$

(15)

in conjunction with the following presumptions:

• $3<\mathsf{p}\le 4$;
• $0<\delta <1$;
• $0<q{\delta }^{\mathsf{p}-3}<1$;
• $\hslash :[0,1]\times {\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ is continuous;
• ℏ retains singular at $\theta =0$, indicating that $\underset{\theta \to 0+}{lim}\hslash (\theta ,·)=\infty$.

Certainly, the BVP (15) is transformed into an integral equation given below:

$$\mathsf{v}\left(\theta \right)={\int }_0^1\mathbf{G}(\theta ,\tau )\hslash (\tau ,\mathsf{v}\left(\tau \right))d\tau +{\displaystyle \frac{q{\theta }^{\mathsf{p}-1}}{(\mathsf{p}-1)(\mathsf{p}-2)(1-q{\delta }^{\mathsf{p}-3})}}{\int }_0^1\mathbf{H}(\delta ,\tau )\hslash (\tau ,\mathsf{v}\left(\tau \right))d\tau$$

(16)

whereas Green function is

$$\mathbf{G}(\theta ,\tau )=\left\{\begin{array}{cc}{\displaystyle \frac{{\theta }^{\mathsf{p}-1}{(1-\tau )}^{\mathsf{p}-3}-{(\theta -\tau )}^{\mathsf{p}-1}}{\mathsf{\Gamma }\left(\mathsf{p}\right)}},\hfill & 0\le \tau \le \theta \le 1,\hfill \\ {\displaystyle \frac{{\theta }^{\mathsf{p}-1}{(1-\tau )}^{\mathsf{p}-3}}{\mathsf{\Gamma }\left(\mathsf{p}\right)}},\hfill & 0\le \theta \le \tau \le 1\hfill \end{array}\right.$$

and its second derivative $\mathbf{H}(\theta ,\tau ):={\displaystyle \frac{{\partial }^2\mathbf{G}(\theta ,\tau )}{\partial {\theta }^2}}$ becomes

$$\mathbf{H}(\theta ,\tau )=\left\{\begin{array}{cc}{\displaystyle \frac{(\mathsf{p}-1)(\mathsf{p}-2)}{\mathsf{\Gamma }\left(\mathsf{p}\right)}}\left[{\theta }^{\mathsf{p}-3}{(1-\tau )}^{\mathsf{p}-3}-{(\theta -\tau )}^{\mathsf{p}-3}\right],\hfill & 0\le \tau \le \theta \le 1,\hfill \\ {\displaystyle \frac{(\mathsf{p}-1)(\mathsf{p}-2)}{\mathsf{\Gamma }\left(\mathsf{p}\right)}}{\theta }^{\mathsf{p}-3}{(1-\tau )}^{\mathsf{p}-3},\hfill & 0\le \theta \le \tau \le 1.\hfill \end{array}\right.$$

$\mathsf{\Gamma }(·)$ and $\beta (·,·)$ denote the gamma and beta functions, respectively. Inspired by [8,9], we will compute a (unique) positive solution of (15).

Proposition 5

([9]). If $\mathbf{G}$ and $\mathbf{H}$ are described as above, then the following hold:

• $\mathbf{G}(\theta ,1)=0$;
• $\mathbf{G}(\theta ,\tau )\ge 0$ and $\mathbf{H}(\theta ,\tau )\ge 0$;
• $\mathbf{G}$ and $\mathbf{H}$ are continuous;
• $\underset{0\le \theta \le 1}{sup}{\int }_0^1\mathbf{G}(\theta ,\tau )d\tau ={\displaystyle \frac{2}{(\mathsf{p}-2)\mathsf{\Gamma }(\mathsf{p}+1)}}$;
• ${\int }_0^1\mathbf{H}(\delta ,\tau )d\tau ={\displaystyle \frac{{\delta }^{\mathsf{p}-3}(\mathsf{p}-1)(1-\delta )}{\mathsf{\Gamma }\left(\mathsf{p}\right)}}$.

Lemma 1.

If $\rho \in (0,1)$, then

$$\underset{0\le \theta \le 1}{sup}{\int }_0^1\mathbf{G}(\theta ,\tau ){\tau }^{-\rho }d\tau ={\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{p}\right)}}(\beta (1-\rho ,\mathsf{p}-2)-\beta (1-\rho ,\mathsf{p})).$$

Proof. 

Observe that

$$\begin{array}{cc}\hfill {\int }_0^1\mathbf{G}(\theta ,\tau ){\tau }^{-\rho }d\tau & ={\int }_0^{\theta }\mathbf{G}(\theta ,\tau ){\tau }^{-\rho }d\tau +{\int }_{\theta }^1\mathbf{G}(\theta ,\tau ){\tau }^{-\rho }d\tau \hfill \\ & ={\int }_0^{\theta }{\displaystyle \frac{{\theta }^{\mathsf{p}-1}{(1-\tau )}^{\mathsf{p}-3}-{(\theta -\tau )}^{\mathsf{p}-1}}{\mathsf{\Gamma }\left(\mathsf{p}\right)}}{\tau }^{-\rho }d\tau +{\int }_{\theta }^1{\displaystyle \frac{{\theta }^{\mathsf{p}-1}{(1-\tau )}^{\mathsf{p}-3}}{\mathsf{\Gamma }\left(\mathsf{p}\right)}}{\tau }^{-\rho }d\tau \hfill \\ & ={\int }_0^1{\displaystyle \frac{{\theta }^{\mathsf{p}-1}{(1-\tau )}^{\mathsf{p}-3}}{\mathsf{\Gamma }\left(\mathsf{p}\right)}}{\tau }^{-\rho }d\tau -{\int }_0^{\theta }{\displaystyle \frac{{(\theta -\tau )}^{\mathsf{p}-1}}{\mathsf{\Gamma }\left(\mathsf{p}\right)}}{\tau }^{-\mathsf{p}}d\tau \hfill \\ \hfill & ={\displaystyle \frac{{\theta }^{\mathsf{p}-1}}{\mathsf{\Gamma }\left(\mathsf{p}\right)}}{\int }_0^1{(1-\tau )}^{\mathsf{p}-3}{\tau }^{-\rho }d\tau -{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{p}\right)}}{\int }_0^{\theta }{(\theta -\tau )}^{\mathsf{p}-1}{\tau }^{-\rho }d\tau \hfill \\ \hfill & ={\displaystyle \frac{{\theta }^{\mathsf{p}-1}}{\mathsf{\Gamma }\left(\mathsf{p}\right)}}\beta (1-\rho ,\mathsf{p}-2)-{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{p}\right)}}\aleph ,\hfill \end{array}$$

(17)

where

$$\aleph ={\int }_0^{\theta }{(\theta -\tau )}^{\mathsf{p}-1}{\tau }^{-\rho }d\tau ={\int }_0^{\theta }{\left(1-{\displaystyle \frac{\tau }{\theta }}\right)}^{\mathsf{p}-1}{\theta }^{\mathsf{p}-1}{\tau }^{-\rho }d\tau ={\theta }^{\theta -\rho }{\int }_0^{\theta }{\left(1-{\displaystyle \frac{\tau }{\theta }}\right)}^{\mathsf{p}-1}{\left({\displaystyle \frac{\tau }{\theta }}\right)}^{-\rho }\theta d\tau .$$

Using the transformation $v=\tau /\theta$ (hence $\theta dv=d\tau$), the above integral gives rise

$$\begin{array}{c}\hfill \aleph ={\theta }^{\theta -\rho }{\int }_0^{\theta }{(1-v)}^{\mathsf{p}-1}{v}^{-\rho }dv={\theta }^{1-\rho }\beta (1-\rho ,\mathsf{p}).\end{array}$$

(18)

From (17) and (18), we conclude

$${\int }_0^1\mathbf{G}(\theta ,\tau ){\tau }^{-\rho }d\tau ={\displaystyle \frac{{\theta }^{\mathsf{p}-1}}{\mathsf{\Gamma }\left(\mathsf{p}\right)}}\beta (1-\rho ,\mathsf{p}-2)-{\displaystyle \frac{{\theta }^{\mathsf{p}-\rho }}{\mathsf{\Gamma }\left(\mathsf{p}\right)}}\beta (1-\rho ,\mathsf{p}).$$

Define

$$\mathcal{ϝ}\left(\theta \right):={\displaystyle \frac{\beta (1-\rho ,\mathsf{p}-2)}{\mathsf{\Gamma }\left(\mathsf{p}\right)}}{\theta }^{\mathsf{p}-1}-{\displaystyle \frac{\beta (1-\rho ,\mathsf{p})}{\mathsf{\Gamma }\left(\mathsf{p}\right)}}{\theta }^{\mathsf{p}-\rho }$$

Finally, $\mathcal{ϝ}\left(\theta \right)$ being increasing on $[0,1]$ yields that

$$\underset{0\le \theta \le 1}{sup}{\int }_0^1\mathbf{G}(\theta ,\tau ){\tau }^{-\rho }d\tau =\underset{0\le \theta \le 1}{sup}\mathcal{ϝ}\left(\theta \right)=\mathcal{ϝ}\left(1\right)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{p}\right)}}[\beta (1-\rho ,\mathsf{p}-2)-\beta (1-\rho ,\mathsf{p})].$$

□

Lemma 2.

If $\rho \in (0,1)$, then

$${\int }_0^1\mathbf{H}(\delta ,\tau ){\tau }^{-\rho }d\tau ={\displaystyle \frac{(\mathsf{p}-1(\mathsf{p}-2)}{\mathsf{\Gamma }\left(\mathsf{p}\right)}}\left({\delta }^{\mathsf{p}-3}-{\delta }^{\mathsf{p}-\rho -2}\beta (1-\rho ,\mathsf{p}-2)\right).$$

Proof. 

Observe that

$$\begin{array}{cc}& {\int }_0^1\mathbf{H}(\delta ,\tau ){\tau }^{-\rho }d\tau ={\int }_0^{\delta }\mathbf{H}(\delta ,\tau ){\tau }^{-\rho }d\tau +{\int }_{\delta }^1\mathbf{H}(\delta ,\tau ){\tau }^{-\rho }d\tau \hfill \\ & ={\int }_0^{\delta }{\displaystyle \frac{(\mathsf{p}-1)(\mathsf{p}-2)}{\mathsf{\Gamma }\left(\mathsf{p}\right)}}\left[{\delta }^{\mathsf{p}-3}{(1-\tau )}^{\mathsf{p}-3}-{(\delta -\tau )}^{\mathsf{p}-3}\right]{\tau }^{-\rho }d\tau +{\int }_{\delta }^1{\displaystyle \frac{(\mathsf{p}-1)(\mathsf{p}-2)}{\mathsf{\Gamma }\left(\mathsf{p}\right)}}{\delta }^{\mathsf{p}-3}{(1-\tau )}^{\mathsf{p}-3}{\tau }^{-\rho }d\tau \hfill \\ \hfill & ={\int }_0^1{\displaystyle \frac{(\mathsf{p}-1)(\mathsf{p}-2)}{\mathsf{\Gamma }\left(\mathsf{p}\right)}}{\delta }^{\mathsf{p}-3}{(1-\tau )}^{\mathsf{p}-3}{\tau }^{-\rho }d\tau -{\int }_0^{\delta }{\displaystyle \frac{(\mathsf{p}-1)(\mathsf{p}-2)}{\mathsf{\Gamma }\left(\mathsf{p}\right)}}{(\delta -\tau )}^{\mathsf{p}-3}{\tau }^{-\rho }d\tau \hfill \\ \hfill & ={\displaystyle \frac{(\mathsf{p}-1)(\mathsf{p}-2)}{\mathsf{\Gamma }\left(\mathsf{p}\right)}}{\delta }^{\mathsf{p}-3}{\int }_0^1{(1-\tau )}^{\mathsf{p}-1}{\tau }^{-\rho }d\tau \hfill \\ \hfill & -{\displaystyle \frac{(\mathsf{p}-1)(\mathsf{p}-2)}{\mathsf{\Gamma }\left(\mathsf{p}\right)}}{\int }_0^{\delta }{(\delta -\tau )}^{\mathsf{p}-3}{\tau }^{-\rho }d\tau \hfill \\ & ={\displaystyle \frac{(\mathsf{p}-1)(\mathsf{p}-2)}{\mathsf{\Gamma }\left(\mathsf{p}\right)}}{\delta }^{\mathsf{p}-3}\beta (1-\rho ,\mathsf{p}-2)-{\displaystyle \frac{(\mathsf{p}-1)(\mathsf{p}-2)}{\mathsf{\Gamma }\left(\mathsf{p}\right)}}{\int }_0^{\delta }{(\delta -\tau )}^{\mathsf{p}-3}{\tau }^{-\rho }d\tau .\hfill \end{array}$$

Like the proof of Lemma 1, we attain

$$\begin{array}{cc}\hfill {\int }_0^1\mathbf{H}(\delta ,\tau ){\tau }^{-\rho }d\tau & ={\displaystyle \frac{(\mathsf{p}-1)(\mathsf{p}-2)}{\mathsf{\Gamma }\left(\mathsf{p}\right)}}{\delta }^{\mathsf{p}-3}\beta (1-\rho ,\mathsf{p}-2)\hfill \\ \hfill & -{\displaystyle \frac{(\mathsf{p}-1)(\mathsf{p}-2)}{\mathsf{\Gamma }\left(\mathsf{p}\right)}}{\delta }^{\mathsf{p}-\rho -2}\beta (1-\rho ,\mathsf{p}-2)\hfill \\ \hfill & ={\displaystyle \frac{(\mathsf{p}-1)(\mathsf{p}-2)}{\mathsf{\Gamma }\left(\mathsf{p}\right)}}\left({\delta }^{\mathsf{p}-3}-{\delta }^{\mathsf{p}-\rho -2}\right)\beta (1-\rho ,\mathsf{p}-2).\hfill \end{array}$$

□

Remark 2.

Define

$$\mu :={\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{p}\right)}}\left[\left(1+{\displaystyle \frac{\beta ({\delta }^{\mathsf{p}-3}-{\delta }^{\mathsf{p}-\rho -2})}{1-\beta {\delta }^{\mathsf{p}-3}}}\right)\beta (1-\rho ,\mathsf{p}-2)-\beta (1-\rho ,\mathsf{p})\right].$$

Lastly, we will prove the prime outcomes.

Theorem 6.

Assume that the BVP (15) verifies above presumptions. Also, let $0<\rho <1$ and ${\theta }^{\rho }\hslash (\theta ,\tau )$ be continuous. If ∃$\lambda \in (0,1/\mu ]$ and $\mathsf{\Phi }\in \mathsf{\Gamma }$ with

$${\tau }_1\ge {\tau }_2\ge 0a\mathsf{n}d0\le \theta \le 1⟹0\le {\theta }^{\rho }[\hslash (\theta ,{\tau }_1)-\hslash (\theta ,{\tau }_2)]\le \lambda \mathsf{\Phi }({\tau }_1-{\tau }_2),$$

(19)

then BVP (15) admits a unique solution.

Proof. 

On $C[0,1]$, equip the following metric:

$$\varpi (\mathsf{v},\mathsf{w})=\underset{0\le \theta \le 1}{sup}|\mathsf{v}\left(\theta \right)-\mathsf{w}\left(\theta \right)|.$$

Let

$$\mathbf{V}=\{\mathsf{v}\in C[0,1]:\mathsf{v}(\theta )\ge 0\}.$$

On $\mathbf{V}$, define a BR $\mathbf{S}$ and a self-map $\mathcal{P}$ given below:

$$\mathbf{S}=\{(\mathsf{v},\mathsf{w})\in {\mathbf{V}}^2:\mathsf{v}\left(\theta \right)\le \mathsf{w}\left(\theta \right),\mathrm{for}\mathrm{each}\theta \in [0,1]\};$$

and

$$\begin{array}{cc}\hfill \left(\mathcal{P}\mathsf{v}\right)\left(\theta \right)=& {\int }_0^1\mathbf{G}(\theta ,\tau )\hslash (\tau ,\mathsf{v}\left(\tau \right))d\tau +{\displaystyle \frac{q{\theta }^{\mathsf{p}-1}}{(\mathsf{p}-1)(\mathsf{p}-2)(1-q{\delta }^{\mathsf{p}-3})}}{\int }_0^1\mathbf{H}(\delta ,\tau )\hslash (\tau ,\mathsf{v}\left(\tau \right))d\tau .\hfill \end{array}$$

(20)

(a)

Clearly, $(\mathbf{V},\varpi )$ remains $\mathbf{S}$-complete MS.

(b)

Assume that $\mathbf{0}\in \mathbf{V}$ is a zero function. Then for each $\theta \in [0,1]$, we conclude $\mathbf{0}\left(\theta \right)\le \left(\mathcal{P}\mathbf{0}\right)\left(\theta \right)$ so that $(\mathbf{0},\mathcal{P}\mathbf{0})\in \mathbf{S}$.

(c)

Clearly $\mathbf{S}$ being transitive is locally $\mathcal{P}$-transitive. Take $(\mathsf{v},\mathsf{w})\in \mathbf{S}$ implying $\mathsf{v}\left(\theta \right)\le \mathsf{w}\left(\theta \right)$, for every $\theta \in [0,1]$. Hence, we conclude

$$\begin{array}{cc}\hfill \left(\mathcal{P}\mathsf{v}\right)\left(\theta \right)& ={\int }_0^1\mathbf{G}(\theta ,\tau )\hslash (\tau ,\mathsf{v}\left(\tau \right))d\tau +{\displaystyle \frac{q{\theta }^{\mathsf{p}-1}}{(\mathsf{p}-1)(\mathsf{p}-2)(1-q{\delta }^{\mathsf{p}-3})}}{\int }_0^1\mathbf{H}(\delta ,\tau )\hslash (\tau ,\mathsf{v}\left(\tau \right))d\tau .\hfill \\ \hfill & ={\int }_0^1\mathbf{G}(\theta ,\tau ){\tau }^{-\rho }{\tau }^{\rho }\hslash (x,\mathsf{v}\left(\tau \right))d\tau \hfill \\ \hfill & +{\displaystyle \frac{q{\theta }^{\mathsf{p}-1}}{(\mathsf{p}-1)(\mathsf{p}-2)(1-q{\delta }^{\mathsf{p}-3})}}{\int }_0^1\mathbf{H}(\delta ,\tau ){\tau }^{-\rho }{\tau }^{\rho }\hslash (\tau ,\mathsf{v}\left(\tau \right))d\tau \hfill \\ \hfill & \le {\int }_0^1\mathbf{G}(\theta ,\tau ){\tau }^{-\rho }{\tau }^{\rho }\hslash (\tau ,\mathsf{w}\left(\tau \right))d\tau \hfill \\ \hfill & +{\displaystyle \frac{q{\theta }^{\mathsf{p}-1}}{(\mathsf{p}-1)(\mathsf{p}-2)(1-q{\delta }^{\mathsf{p}-3})}}{\int }_0^1\mathbf{H}(\delta ,\tau ){\tau }^{-\rho }{\tau }^{\rho }\hslash (\tau ,\mathsf{w}\left(\tau \right))d\tau \hfill \\ \hfill & ={\int }_0^1\mathbf{G}(\theta ,\tau )\hslash (\tau ,\mathsf{w}\left(\tau \right))d\tau +{\displaystyle \frac{q^{\mathsf{p}-1}}{(\mathsf{p}-1)(\mathsf{p}-2)(1-q{\delta }^{\mathsf{p}-3})}}{\int }_0^1\mathbf{H}(\delta ,\tau )\hslash (\tau ,\mathsf{w}\left(\tau \right))d\tau \hfill \\ \hfill & =\left(\mathcal{P}\mathsf{w}\right)\left(\theta \right)\hfill \end{array}$$

yielding $(\mathcal{P}\mathsf{v},\mathcal{P}\mathsf{w})\in \mathbf{S}$. Therefore, $\mathbf{S}$ is $\mathcal{P}$-closed.

(d)

We will confirm that $\mathbf{S}$ is $\varpi$-self-closed. Assume $\left\{{\mathsf{v}}_n\right\}\subset \mathbf{V}$ ensuring ${\mathsf{v}}_{\mathsf{n}}\to \mathsf{v}$ and $({\mathsf{v}}_{\mathsf{n}},{\mathsf{v}}_{\mathsf{n}+1})\in \mathbf{S},\forall \mathsf{n}\in \mathbb{N}$. Then, $\left\{{\mathsf{v}}_n\left(\theta \right)\right\}$ (where $\theta \in [0,1]$, ) is increasing real sequence converging to $\mathsf{v}\left(\theta \right)$; thereby, to each $\mathsf{n}\in \mathbb{N}$, we obtain ${\mathsf{v}}_{\mathsf{n}}\left(\theta \right)\le \mathsf{v}\left(\theta \right)$. Thus, $({\mathsf{v}}_n,\mathsf{v})\in \mathbf{S},\forall n\in \mathbb{N}$.

(f)

For $(\mathsf{v},\mathsf{w})\in \mathbf{S}$, we have

$$\begin{array}{cc}\hfill \varpi (\mathcal{P}\mathsf{v},\mathcal{P}\mathsf{w})& =\underset{0\le \theta \le 1}{sup}|\left(\mathcal{P}\mathsf{v}\right)\left(\theta \right)-\left(\mathcal{P}\mathsf{w}\right)\left(\theta \right)|=\underset{0\le \theta \le 1}{sup}[\left(\mathcal{P}\mathsf{w}\right)\left(\theta \right)-\left(\mathcal{P}\mathsf{v}\right)\left(\theta \right)]\hfill \\ \hfill & =\underset{0\le \theta \le 1}{sup}\left[{\int }_0^1\mathbf{G}(\theta ,\tau )(\hslash (\tau ,\mathsf{w}\left(\tau \right))-\hslash (\tau ,\mathsf{v}\left(\tau \right)))\right.d\tau \hfill \\ \hfill & \left.+{\displaystyle \frac{q{\theta }^{\mathsf{p}-1}}{(\mathsf{p}-1)(\mathsf{p}-2)(1-q{\delta }^{\mathsf{p}-3})}}{\int }_0^1\mathbf{H}(\delta ,\tau )(\hslash (\tau ,\mathsf{w}\left(\tau \right))-\hslash (\tau ,\mathsf{v})\left(\tau \right))d\tau \right]\hfill \\ \hfill & \le \underset{0\le \theta \le 1}{sup}{\int }_0^1\mathbf{G}(\theta ,\tau ){\tau }^{-\rho }{\tau }^{\rho }[\hslash (\tau ,\mathsf{w}\left(\tau \right))-\hslash (\tau ,\mathsf{v}\left(\tau \right))]d\tau \hfill \\ \hfill & +{\displaystyle \frac{q}{(\mathsf{p}-1)(\mathsf{p}-2)(1-q{\delta }^{\mathsf{p}-3})}}{\int }_0^1\mathbf{H}(\delta ,\tau ){\tau }^{-\rho }{\tau }^{\rho }[\hslash (\tau ,\mathsf{w}\left(\tau \right))-\hslash (\tau ,\mathsf{v})\left(\tau \right)]d\tau \hfill \\ \hfill & \le sup{\int }_0^1\mathbf{G}(\theta ,\tau ){\tau }^{-\rho }\lambda \mathsf{\Phi }(\mathsf{w}\left(\tau \right)-\mathsf{v}\left(\tau \right))d\tau \hfill \\ \hfill & +{\displaystyle \frac{q}{(\mathsf{p}-1)(\mathsf{p}-2)(1-q{\delta }^{\mathsf{p}-3})}}{\int }_0^1\mathbf{H}(\delta ,\tau ){\tau }^{-\rho }\lambda \mathsf{\Phi }\left(\mathsf{w}\left(\tau \right)\right)-\mathsf{v}\left(\tau \right)d\tau .\hfill \end{array}$$

By the monotonic property of $\mathsf{\Phi }$, the above inequality becomes

$$\begin{array}{cc}\hfill \varpi (\mathcal{P}\mathsf{v},\mathcal{P}\mathsf{w})& \le \lambda \mathsf{\Phi }\left(\varpi (\mathsf{v},\mathsf{w})\right)\underset{0\le \theta \le 0}{sup}{\int }_0^1\mathbf{G}(\theta ,\tau ){\tau }^{-\rho }d\tau \hfill \\ \hfill & +{\displaystyle \frac{q}{(\mathsf{p}-1)(\mathsf{p}-2)(1-q\delta \mathsf{p}-3)}}\lambda \mathsf{\Phi }\left(\varpi (\mathsf{w},\mathsf{v})\right){\int }_0^1\mathbf{H}(\delta ,\tau ){\tau }^{\rho }d\tau \hfill \\ \hfill & =\lambda \mathsf{\Phi }\left(\varpi (\mathsf{v},\mathsf{w})\right)\left[\underset{0\le \theta \le 0}{sup}{\int }_0^1\mathbf{G}(\theta ,\tau ){\tau }^{-\rho }d\tau \right.\hfill \\ \hfill & \left.+{\displaystyle \frac{q}{(\mathsf{p}-1)(\mathsf{p}-2)(1-q{\delta }^{\mathsf{p}-3})}}{\int }_0^1\mathbf{H}(\delta ,\tau ){\tau }^{-\rho }d\tau \right].\hfill \end{array}$$

(21)

Using Lemmas 1 and 2, (21) reduces to

$$\begin{array}{cc}\hfill \varpi (\mathcal{P}\mathsf{v},\mathcal{P}\mathsf{w})\le & \lambda \mathsf{\Phi }\left(\varpi (\mathsf{v},\mathsf{w})\right)\left[{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{p}\right)}}(\beta (1-\rho ,\mathsf{p}-2)-\beta (1-\rho \mathsf{p}))+{\displaystyle \frac{q}{(\mathsf{p}-1)(\mathsf{p}-2)(1-q{\delta }^{\mathsf{p}-3})}}\right.\hfill \\ \hfill & \left.\times {\displaystyle \frac{(\mathsf{p}-1)(\mathsf{p}-2)}{\mathsf{\Gamma }\left(\mathsf{p}\right)}}\left({\delta }^{\mathsf{p}-3}-{\delta }^{\mathsf{p}-\rho -2}\right)\right]\hfill \\ \hfill & =\lambda \mathsf{\Phi }\left(\varpi (\mathsf{v},\mathsf{w})\right)\left[{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{p}\right)}}(\beta (1-\rho ,\mathsf{p}-2)-\beta (1-\rho ,\mathsf{p}))\right.\hfill \\ \hfill & \left.+{\displaystyle \frac{q({\delta }^{\mathsf{p}-3}-{\delta }^{\mathsf{p}-\rho -2})}{(1-q{\delta }^{\mathsf{p}-3})\mathsf{\Gamma }\left(\mathsf{p}\right)}}\beta (1-\rho ,\mathsf{p}-2)\right]\hfill \\ \hfill & =\lambda \mathsf{\Phi }\left(\varpi (\mathsf{v},\mathsf{w})\right)\left[{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{p}\right)}}\left[\left(1+{\displaystyle \frac{q({\delta }^{\mathsf{p}-3}-{\delta }^{\mathsf{p}-\rho -2})}{1-q-q{\delta }^{\mathsf{p}-3}}}\right)\beta (1-\rho ,\mathsf{p}-2)-\beta (1-\rho ,\mathsf{p})\right]\right]\hfill \\ \hfill & =\lambda \mathsf{\Phi }\left(\varpi \right(\mathsf{v},\mathsf{w}\left)\right)\mu .\hfill \end{array}$$

As $0<\lambda \le 1/\mu$, the above inequality reduces to

$$\varpi (\mathcal{P}\mathsf{v},\mathcal{P}\mathsf{w})\le \lambda \mathsf{\Phi }\left(\varpi \right(\mathsf{v},\mathsf{w}\left)\right)\mu \le \mathsf{\Phi }\left(\varpi \right(\mathsf{v},\mathsf{w}\left)\right)$$

yielding thereby

$$\begin{array}{ccc}\hfill \varpi (\mathcal{P}\mathsf{z},\mathcal{P}\mathsf{w})& \le & \mathsf{\Phi }\left(\varpi \right(\mathsf{z},\mathsf{w}\left)\right)+min\left\{\varrho \right(\varpi (\mathsf{w},\mathcal{P}\mathsf{z})),\varrho (\varpi (\mathsf{z},\mathcal{P}\mathsf{w})),\varrho (\varpi (\mathsf{z},\mathcal{P}\mathsf{z})),\varrho (\varpi (\mathsf{w},\mathcal{P}\mathsf{w})\left)\right\},\hfill \end{array}$$

for every arbitrary choice of $\varrho \in \mathsf{\Omega }$.

(g)

For every pair $\mathsf{v},\mathsf{w}\in \mathcal{P}\left(\mathbf{V}\right)$, set $u:=max\{\mathsf{v},\mathsf{w}\}\in \mathbf{V}$. So, we find $(\mathsf{v},u)\in \mathbf{S}$ and $(\mathsf{w},u)\in \mathbf{S}$. Hence, $\mathcal{P}\left(\mathbf{V}\right)$ is ${\mathbf{S}}^s$-directed.

Therefore, using Theorem 5, $\mathcal{P}$ owns a unique fixed point, which (owing to (16) and (20)) solves (15). □

Theorem 7.

Along-with the conditions of Theorem 6, BVP (15) admits a (unique) positive solution.

Proof. 

Applying Theorem 6, assume that $\widehat{\mathsf{w}}\in \mathbf{V}$ serves as the unique solution of (15). Since $\widehat{\mathsf{w}}\in \mathbf{V}$, therefore, we attain $\widehat{\mathsf{w}}\left(\theta \right)\ge 0$, $\forall \theta \in [0,1]$. It follows that the (unique) solution $\widehat{\mathsf{w}}$ remains non-negative. We will prove that $\widehat{\mathsf{w}}$ is positive, i.e., $\widehat{\mathsf{p}}\left(\mathsf{s}\right)>0$, to each $\mathsf{s}\in (0,1)$. If there is a some ${\theta }^{*}\in (0,1)$ such that $\widehat{\mathsf{w}}\left({\theta }^{*}\right)=0$, then using (16), we obtain

$$\widehat{\mathsf{w}}\left({\theta }^{*}\right)={\int }_0^1\mathbf{G}({\theta }^{*},\tau )\hslash (\tau ,\widehat{\mathsf{w}}\left(\tau \right))d\tau +{\displaystyle \frac{q{\theta }^{*\mathsf{p}-1}}{(\mathsf{p}-1)(\mathsf{p}-2)(1-q{\delta }^{\mathsf{p}-3})}}{\int }_0^1\mathbf{H}(\delta ,\tau )\hslash (\tau ,x\left(\tau \right))d\tau =0.$$

As ℏ is non-negative, owing to Proposition 5, the two terms involved in RHS are non-negative. Thus, we conclude

$$\begin{array}{cc}\hfill & {\int }_0^1\mathbf{G}({\theta }^{*},\tau )\hslash (\tau ,\widehat{\mathsf{w}}\left(\tau \right))d\tau =0,\hfill \\ \hfill & {\int }_0^1\mathbf{H}(\delta ,\tau )\hslash (\tau ,\tau \left(\tau \right))d\tau =0,\hfill \end{array}$$

so that

$$\left\{\begin{array}{c}\mathbf{G}({\theta }^{*},\tau )\hslash (\tau ,\widehat{\mathsf{w}}\left(\tau \right))=0,a.e.\left(\tau \right),\hfill \\ \mathbf{H}(\delta ,\tau )\hslash (\tau ,\widehat{\mathsf{w}}\left(\tau \right))=0,a.e\left(\tau \right).\hfill \end{array}\right.$$

(22)

Let $\kappa >0$ be arbitrary. The singular property of ℏ yields the existence of $R>0$ with $\hslash (\tau ,0)>\kappa$, ∀$\tau \in [0,1]\cap (0,R)$. Now, we have

$$[0,1]\cap (0,R)\subset \{\tau \in [0,1]:\hslash (\tau ,\widehat{\mathsf{w}}\left(\tau \right))>\kappa \},$$

and

$$\mathsf{\Lambda }\left(\right[0,1]\cap (0,R\left)\right)>0,$$

where $\mathsf{\Lambda }$ is a Lebesque measure. Therefore, (22) implies that

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\mathbf{G}({\theta }^{*},\tau )=0,a.e.\left(\tau \right),\hfill \\ \mathbf{H}(\delta ,\tau )=0,a.e.\left(\tau \right),\hfill \end{array}\right.\end{array}$$

which contradicts the rationality of the functions $\mathbf{G}({\theta }^{*},·)$ and $\mathbf{H}(\delta ,·)$. This concludes the proof. □

7. Conclusions and Future Directions

We have demonstrated the validity of fixed points and their uniqueness for a relation-theoretic almost Matkowski contraction of Pant type. Our outcomes expanded and unified a few known fixed-point findings. The contraction conditions in our investigations are imposed to the comparative elements only. To corroborate these findings, we presented a few examples. We also filled out an application to certain singular FDE to emphasize the worth of the theory and the depth of our findings.

As some possible future works, the readers can generalize our outcomes in the following ways:

• To vary the features of auxiliary functions $\mathsf{\Phi }$ and $\varrho$;
• To enhance our findings over symmetric space, quasimetric space, cone MS, fuzzy MS, etc., composed with a BR;
• To improve our finding for two maps by investigating common fixed-point findings;
• To apply our finding in the area of nonlinear integral equations instead of fractional BVP.