Nonlinear F-Contractions in Relational Metric Space and Applications to Fractional Differential Equations

Abstract

During the last decade, F-contraction has been a widely investigated problem in the fixed point theory. There are various outcomes regarding the extensions and generalizations of F-contraction in different perspectives, along with the findings concerning the application of those ideas, mostly in the area of differential and difference equations, fractional calculus, etc. The present article concludes some existence and uniqueness outcomes on fixed points for $(\phi ,\mathrm{F})$–contractions in the context of a metric space endowed with a local class of transitive binary relations. Some illustrative examples are furnished to justify that our contraction conditions are more general than many others in this area. The findings presented herein are used to obtain a unique solution to certain fractional boundary value problems.

1. Introduction

Fractional derivatives provide a few advances over usual derivatives, especially in the area of control theory and modeling. The term FDE is an extension of differential equations exhibiting fractional-order derivatives. Nowadays, FDEs have gained recognition for their remarkable efficiency and advances in the discipline of fractional calculus. For an exhaustive review of fractional calculus and FDE, we consult the classical books of Podlubny [1], Daftardar-Gejji [2], and Kilbas et al. [3]. Zhou et al. [4] and Zhai and Hao [5] studied the solvability of FDE employing fixed point theorems in partially ordered MS. Nevertheless, the unique positive solution for a three-point BVP of FDE was explored by Liang and Zhang [6]. Cabrera et al. [7] exploited order-theoretic fixed point theorems to establish the unique three-point BVP related to FDE. For recent study on FDE, we refer the works of Karapınar et al. [8], Cevikel and Aksoy [9], Khatoon et al. [10], Uddin et al. [11], Aljethi and Kiliçman [12], Abdou [13], Sevinik et al. [14] and Ahmad et al. [15,16].

The BCP serves as a vital and critical conclusion from fixed-point theory. Actually, BCP confirms the occurrence of a unique fixed point of a contraction on a complete MS. A strategy for approximating the unique fixed point is additionally offered by this finding. The research at hand comprises a plethora of generalizations of BCP. In 2012, Wardowski [17] invented the proposal of F-contraction to examine an innovative category of nonlinear contractions that eventually propagate BCP. Subsequently, Turinici [18] gave the some critical observations on F-contractions. Piri and Kumam [19] provided certain analogues of the results of Wardowski [17], assuming the continuity requirement of the auxiliary function F. Vetro [20] initiated the term of $(\phi ,\mathrm{F})$-contractions, which was later followed by Wardowski [21] and Arif and Imdad [22]. Several authors investigated the fixed point outcomes in relational MS, e.g., see [23,24,25,26]. For further details regarding F-contractions, we suggest the readers study a nice treatment due to Karapinar et al. [27].

The relation-theoretic model of the BCP was first proposed by Alam and Imdad [28] in 2015, and it has now drawn interest from many scholars, e.g., [29,30,31,32,33,34]. The occurrence of fixed points in a nonlinear contraction is additionally dependent on the transitiveness of the underlying BR. As the transitivity criterion is quite restrictive, multiple authors (e.g., [30]) adopted locally transitive BR in an attempt to employ an ideal condition of transitivity. A key trait of relational contractions is that no element pairing is needed; the contraction paradox is enough to prevail for comparable elements. Thus, relational contractions are weaker than normal contractions. For this reason, the outcomes regarding relational contractions are readily utilized in solving certain problems of nonlinear analysis, whereas ordinary fixed-point results fail.

The present article aims to demonstrate the occurrence and uniqueness findings on fixed points of $(\phi ,\mathrm{F})$-contractions via a locally $\mathcal{S}$-transitive BR by extending the outcomes of Wardowski [21] to relational MS and to address the occurrence of a unique solution for a specific BVP for a nonlinear FDE utilizing our fixed point outcomes.

2. Preliminaries

A subset of ${\mathit{P}}^2$ is termed a BR on the set $\mathit{P}$. Let $\mathit{P}$ be a set, $\mathcal{S}$ a self-map on $\mathit{P}$; $\varrho$ and R serves as a metric and a BR respectively, on $\mathit{P}$. We say that

Definition 1 

([35]). ${\mathrm{R}}^{-1}:=\{(\mathtt{p},q)\in {\mathbf{P}}^2:(q,\mathtt{p})\in \mathrm{R}\}$ retains inverse of R; while ${\mathrm{R}}^s:=\mathrm{R}\cup {\mathrm{R}}^{-1}$ retains symmetric closure of R.

Definition 2 

([28]). A pair $\mathtt{p},q\in \mathbf{P}$ retains R-comparative if $(\mathtt{p},q)\in \mathrm{R}$ or $(q,\mathtt{p})\in \mathrm{R}$. This sort of pair is depicted by $[\mathtt{p},q]\in \mathrm{R}$.

Remark 1 

([28]). $(\mathtt{p},q)\in {\mathrm{R}}^s⟺[\mathtt{p},q]\in \mathrm{R}.$

Definition 3 

([28]). R retains $\mathcal{S}$-closed BR whenever

$$(\mathtt{p},q)\in \mathrm{R}\Rightarrow (\mathcal{S}\mathtt{p},\mathcal{S}q)\in \mathrm{R}.$$

Proposition 1 

([30]). If R retains $\mathcal{S}$-closed, then R is ${\mathcal{S}}^n$-closed, for every $n\in {\mathbb{N}}_0$.

Definition 4 

([28]). A sequence $\left\{{\mathtt{p}}_n\right\}\subset \mathbf{P}$ along-with $({\mathtt{p}}_n,{\mathtt{p}}_{n+1})\in \mathrm{R}$, $\forall n\in \mathbb{N}$, remains R-preserving.

Definition 5 

([36]). Given $\mathtt{p},q\in \mathbf{P}$, a finite set $\{{\omega }_0,{\omega }_1,\dots ,{\omega }_r\}$ ⊂ $\mathbf{P}$ remains a path of length $r\in \mathbb{N}$ in R from $\mathtt{p}$ to q, whenever

(i)

${\omega }_0=\mathtt{p}$ and ${\omega }_r=q$,

(ii)

$({\omega }_i,{\omega }_{i+1})\in \mathrm{R}$, for every $0\le i\le r-1$.

Definition 6 

([30]). A subset $\mathbf{Q}\subseteq \mathbf{P}$ whose any two elements are joined by a path is R-connected set.

Definition 7 

([28]). R retains ϱ-self-closed if every R-preserving convergent sequence in $\mathbf{P}$ deduces a subsequence each term of which retains R-comparative to the limit of the sequence.

Definition 8 

([29]). $(\mathbf{P},\varrho )$ is the R-complete MS if every Cauchy sequence provided it remains R-preserving must be convergent.

Obviously, a complete MS must be R-complete. Furthermore, for $\mathrm{R}={\mathbf{P}}^2$ both notions coincide.

Definition 9 

([29]). $\mathcal{S}$ is the R-continuous map if for every $\mathtt{p}\in \mathbf{P}$ and for any R-preserving sequence $\left\{{\mathtt{p}}_n\right\}\subset \mathbf{P}$ with ${\mathtt{p}}_n\stackrel{\varrho }{⟶}\mathtt{p}$,

$$\mathcal{S}\left({\mathtt{p}}_n\right)\stackrel{\varrho }{⟶}\mathcal{S}\left(\mathtt{p}\right).$$

Obviously, a continuous map must be R-continuous. Furthermore, for $\mathrm{R}={\mathbf{P}}^2$ both notions coincide.

Definition 10 

([36]). Given $\mathbf{Q}\subseteq \mathbf{P}$, the BR

$${\mathrm{R}|}_{\mathbf{Q}}:=\mathrm{R}\cap {\mathbf{Q}}^2,$$

(on $\mathbf{Q}$), is restriction of R in $\mathbf{Q}$.

Definition 11 

([30]). R retains locally $\mathcal{S}$-transitive if for any R-preserving sequence $\left\{{\mathtt{p}}_n\right\}\subset \mathcal{S}\left(\mathbf{P}\right)$, ${\mathrm{R}|}_{\mathbf{Q}}$ forms a transitive BR, wherein $\mathbf{Q}=\{{\mathtt{p}}_n:n\in \mathbb{N}\}$.

Definition 12 

([37]). A sequence $\left\{{\mathtt{p}}_n\right\}$ in an MS $(\mathbf{P},\varrho )$ remains semi-Cauchy whenever

$$\underset{n\to \infty }{lim}\varrho ({\mathtt{p}}_n,{\mathtt{p}}_{n+1})=0.$$

A Cauchy sequence is obviously semi-Cauchy.

Remark 2. 

To prove the Cauchyness of Picard sequence over F-contractions, the traditional method does not work, which is often proved through the method of contradiction, wherein the following lemma is needed. This lemma is valid only for semi-Cauchy sequences.

Lemma 1 

([18]). Let $(\mathbf{P},\varrho )$ be an MS and $\left\{{\mathtt{p}}_n\right\}$ a sequence in $\mathbf{P}$. Let $\left\{{\mathtt{p}}_n\right\}$ be semi-Cauchy but is not Cauchy. If $▵$ is a countable part of $(0,\infty )$, then $\exists ϵ\in (0,\infty )\backslash ▵$ and two subsequences $\left\{{\mathtt{p}}_{n_{\kappa }}\right\}$ and $\left\{{\mathtt{p}}_{m_{\kappa }}\right\}$ of $\left\{{\mathtt{p}}_n\right\}$ verifying

1. 

$\kappa \le m_{\kappa }<n_{\kappa }$, $\forall \kappa \in \mathbb{N}$,

2. 

$\varrho ({\mathtt{p}}_{m_{\kappa }},{\mathtt{p}}_{n_{\kappa }})>ϵ$, $\forall \kappa \in \mathbb{N}$,

3. 

$\varrho ({\mathtt{p}}_{m_{\kappa }},{\mathtt{p}}_{n_{\kappa }-1})\le ϵ$, $\forall \kappa \in \mathbb{N}$,

4. 

$\underset{\kappa \to \infty }{lim}\varrho ({\mathtt{p}}_{m_{\kappa }+p},{\mathtt{p}}_{n_{\kappa }+q})=ϵ$, $\forall p,q\in \{0,1\}$.

In what follows, $\mathfrak{F}$ refers the collection of the functions $\mathrm{F}:(0,\infty )\to \mathbb{R}$ verifying the items below:

(F₁):

F is strictly increasing,

(F₂):

$\underset{n\to \infty }{lim}\mathrm{F}\left(t_n\right)=-\infty \iff \underset{n\to \infty }{lim}{t}_n=0$.

Also, $\mathsf{\Phi }$ refers to the collection of the functions $\phi :(0,\infty )\to (0,\infty )$ verifying the item below:

$$\underset{t\to r+}{lim\; inf}\phi \left(t\right)>0,\forall r>0.$$

A typical example of the function of $\mathsf{\Phi }$ is the constant function $\phi \left(t\right)=\tau$, where $\tau$ is a positive real number.

We are now entitled to present our main results on fixed points.

Theorem 1. 

Let $(\mathbf{P},\varrho )$ be an MS outfitted with a BR R and $\mathcal{S}$ a self-map on $\mathbf{P}$. Moreover,

(a) 

$(\mathbf{P},\varrho )$ is R-complete,

(b) 

$\exists {\mathtt{p}}_0\in \mathbf{P}$ verifying $({\mathtt{p}}_0,\mathcal{S}{\mathtt{p}}_0)\in \mathrm{R}$,

(c) 

R is $\mathcal{S}$-closed and locally $\mathcal{S}$-transitive,

(d) 

$\mathcal{S}$ is R-continuous, or R is ϱ-self-closed,

(e) 

$\exists \mathrm{F}\in \mathfrak{F}$ and $\phi \in \mathsf{\Phi }$ verifying

$$(\mathtt{p},q)\in \mathrm{R}\mathrm{with}\mathcal{S}\left(\mathtt{p}\right)\ne \mathcal{S}\left(q\right)\Rightarrow \phi \left(\varrho \right(\mathtt{p},q\left)\right)+\mathrm{F}\left(\varrho \right(\mathcal{S}\mathtt{p},\mathcal{S}q\left)\right)\le \mathrm{F}\left(\varrho \right(\mathtt{p},q\left)\right),$$

then $\mathcal{S}$ has a fixed point.

Proof. 

Our strategy is to conclude the proof in the following six phases.

• Step–1. Define a sequence $\left\{{\mathtt{p}}_n\right\}\subset \mathit{P}$ as under

  $${\mathtt{p}}_{\mathrm{n}}:={\mathcal{S}}^{\mathrm{n}}\left({\mathtt{p}}_0\right)=\mathcal{S}\left({\mathtt{p}}_{\mathrm{n}-1}\right),\forall \mathrm{n}\in \mathbb{N}.$$

  (1)
• Step–2. We exhibit that $\left\{{\mathtt{p}}_{\mathrm{n}}\right\}$ retains R-preserving sequence. Through item $\left(b\right)$, $\mathcal{S}$-closedness of R, and Proposition 1, we achieve

  $$({\mathcal{S}}^{\mathrm{n}}{\mathtt{p}}_0,{\mathcal{S}}^{\mathrm{n}+1}{\mathtt{p}}_0)\in \mathrm{R},$$

  which, using (1), becomes

  $$({\mathtt{p}}_{\mathrm{n}},{\mathtt{p}}_{\mathrm{n}+1})\in \mathrm{R},\forall \mathrm{n}\in {\mathbb{N}}_0.$$

  (2)
• Step–3. Define ${\rho }_{\mathrm{n}}:=\varrho ({\mathtt{p}}_n,{\mathtt{p}}_{\mathrm{n}+1})$. If $\exists {\mathrm{n}}_0\in {\mathbb{N}}_0$ verifying ${\rho }_{{\mathrm{n}}_0}=0$, then through (1) we achieve ${\mathtt{p}}_{{\mathrm{n}}_0}={\mathtt{p}}_{{\mathrm{n}}_0+1}=\mathcal{S}\left({\mathtt{p}}_{{\mathrm{n}}_0}\right)$; therefore ${\mathtt{p}}_{{\mathrm{n}}_0}\in Fix\left(\mathcal{S}\right)$, and that is why we wrapped up. If failing that, we arrive at ${\rho }_{\mathrm{n}}>0$, ∀$\mathrm{n}\in {\mathbb{N}}_0$, then we are required to move forward to Step 4.
• Step–4. We shall explore that $\left\{{\mathtt{p}}_{\mathrm{n}}\right\}$ is semi-Cauchy. Consider the case ${\rho }_{\mathrm{n}}>0$, $\forall \mathrm{n}\in \mathbb{N}$. From the data $\phi \left({\rho }_{\mathrm{n}}\right)>0$ and assumption $\left(e\right)$, we get

  $$\begin{array}{ccc}\hfill \mathrm{F}\left({\rho }_{\mathrm{n}+1}\right)& <& \phi \left({\rho }_{\mathrm{n}}\right)+\mathrm{F}\left({\rho }_{\mathrm{n}+1}\right)\hfill \\ & =& \phi \left(\varrho ({\mathtt{p}}_{\mathrm{n}},{\mathtt{p}}_{\mathrm{n}+1})\right)+\mathrm{F}\left(\varrho (\mathcal{S}{\mathtt{p}}_{\mathrm{n}},\mathcal{S}{\mathtt{p}}_{\mathrm{n}+1})\right)\hfill \\ & \le & \mathrm{F}\left(\varrho ({\mathtt{p}}_{\mathrm{n}},{\mathtt{p}}_{\mathrm{n}+1})\right)=\mathrm{F}\left({\rho }_{\mathrm{n}}\right),\hfill \end{array}$$

  (3)

  which gives the following:

  $$\mathrm{F}\left({\rho }_{\mathrm{n}+1}\right)<\mathrm{F}\left({\rho }_{\mathrm{n}}\right).$$

  Axiom $\left({\mathrm{F}}_1\right)$ provides

  $$\begin{array}{c}\hfill {\rho }_{\mathrm{n}+1}<{\rho }_{\mathrm{n}},\end{array}$$

  i.e., $\left\{{\rho }_{\mathrm{n}}\right\}$ is decreasing and hence, $\exists \delta \ge 0$, along with the following:

  $$\underset{\mathrm{n}\to \infty }{lim}{\rho }_{\mathrm{n}}=\delta .$$

  Now, we reveal that $\delta =0$. Assuming, in contrary, that $\delta >0$, then $\exists c>0$ and ${\mathrm{n}}_0\in \mathbb{N}$ verifying $\phi \left({\rho }_{\mathrm{n}}\right)>c$, $\forall \mathrm{n}\ge n_0$. From (3), we attain the following:

  $$\mathrm{F}\left({\rho }_{i+1}\right)-\mathrm{F}\left({\rho }_i\right)<-\phi \left({\rho }_i\right),\forall i\in \mathbb{N}.$$

  Thus, for every $\mathrm{n}\in \mathbb{N}$, we conclude the following:

  $$\begin{array}{ccc}\hfill \mathrm{F}\left({\rho }_{\mathrm{n}}\right)& =& \mathrm{F}\left({\rho }_{{\mathrm{n}}_0}\right)+\sum _{i=n_0}^{\mathrm{n}-1}[\mathrm{F}\left({\rho }_{i+1}\right)-\mathrm{F}\left({\rho }_i\right)]\hfill \\ & <& \mathrm{F}\left({\rho }_{{\mathrm{n}}_0}\right)-\sum _{i=n_0}^{\mathrm{n}-1}\phi \left({\rho }_i\right)\hfill \\ & =& \mathrm{F}\left({\rho }_{{\mathrm{n}}_0}\right)-(\mathrm{n}-{\mathrm{n}}_0)c.\hfill \end{array}$$

  Taking the limit in (4), we get the following:

  $$\begin{array}{c}\hfill \underset{\mathrm{n}\to \infty }{lim}\mathrm{F}\left({\rho }_{\mathrm{n}}\right)=-\infty .\end{array}$$

  Therefore, we have the following:

  $$\begin{array}{c}\hfill \underset{\mathrm{n}\to \infty }{lim}\varrho ({\mathtt{p}}_{\mathrm{n}},{\mathtt{p}}_{\mathrm{n}+1})=0.\end{array}$$

  (4)
• Step–5. We shall conclude that $\left\{{\mathtt{p}}_{\mathrm{n}}\right\}$ is Cauchy. Assuming, in contrary, that $\left\{{\mathtt{p}}_{\mathrm{n}}\right\}$ is not Cauchy. From $\left({\mathrm{F}}_1\right)$, set $▵$ of the points of discontinuity of F retains at most countable. Therefore, in view of (4) and by Lemma 1, there exists $ϵ>0$, $ϵ\in ▵$ and a couple of subsequences $\left\{{\mathtt{p}}_{{\mathrm{n}}_{\kappa }}\right\}$ and $\left\{{\mathtt{p}}_{{\mathrm{m}}_{\kappa }}\right\}$ of $\left\{{\mathtt{p}}_{\mathrm{n}}\right\}$ verifying $\kappa \le {\mathrm{m}}_{\kappa }<{\mathrm{n}}_{\kappa }$, $\varrho ({\mathtt{p}}_{{\mathrm{m}}_{\kappa }},{\mathtt{p}}_{{\mathrm{n}}_{\kappa }})>ϵ$ and $\varrho ({\mathtt{p}}_{{\mathrm{m}}_{\kappa }},{\mathtt{p}}_{{\mathrm{n}}_{\kappa }})\le ϵ$. Further, we have the following:

  $$\underset{\mathrm{n}\to \infty }{lim}\varrho ({\mathtt{p}}_{{\mathrm{m}}_{\kappa }},{\mathtt{p}}_{{\mathrm{n}}_{\kappa }})=ϵ$$

  (5)

  and

  $$\underset{\kappa \to \infty }{lim}\varrho ({\mathtt{p}}_{{\mathrm{m}}_{\kappa }+1},{\mathtt{p}}_{{\mathrm{n}}_{\kappa }+1})=ϵ.$$

  (6)

  Using local transitivity, we have $\varrho ({\mathtt{p}}_{{\mathrm{m}}_{\kappa }},{\mathtt{p}}_{{\mathrm{n}}_{\kappa }})\in \mathrm{R}$. Now, using assumption $\left(e\right)$, we get

  $$\phi \left(\varrho ({\mathtt{p}}_{{\mathrm{m}}_{\kappa }},{\mathtt{p}}_{{\mathrm{n}}_{\kappa }})\right)+\mathrm{F}\left(\varrho ({\mathtt{p}}_{{\mathrm{m}}_{\kappa }+1},{\mathtt{p}}_{{\mathrm{n}}_{\kappa }+1})\right)\le \mathrm{F}\left(\varrho ({\mathtt{p}}_{{\mathrm{m}}_{\kappa }},{\mathtt{p}}_{{\mathrm{n}}_{\kappa }})\right).$$

  Taking lower limit in the above inequality, we get the following:

  $$\underset{\kappa \to \infty }{lim\; inf}\phi \left(\varrho ({\mathtt{p}}_{{\mathrm{m}}_{\kappa }},{\mathtt{p}}_{{\mathrm{n}}_{\kappa }})\right)+\underset{\kappa \to \infty }{lim\; inf}\mathrm{F}\left(\varrho ({\mathtt{p}}_{m_{\kappa }+1},{\mathtt{p}}_{{\mathrm{n}}_{\kappa }+1})\right)\le \underset{\kappa \to \infty }{lim\; inf}\mathrm{F}\left(\varrho ({\mathtt{p}}_{{\mathrm{m}}_{\kappa }},{\mathtt{p}}_{{\mathrm{n}}_{\kappa }})\right).$$

  As F remains continuous at $ϵ$, above inequality reduces to the following:

  $$\underset{\kappa \to \infty }{lim\; inf}\phi \left(\varrho ({\mathtt{p}}_{{\mathrm{m}}_{\kappa }},{\mathtt{p}}_{{\mathrm{n}}_{\kappa }})\right)+\mathrm{F}\left(ϵ\right)\le \mathrm{F}\left(ϵ\right)$$

  thereby yielding the following:

  $$\begin{array}{ccc}\hfill \underset{t\to ϵ+}{lim\; inf}\phi \left(t\right)=\underset{\kappa \to \infty }{lim\; inf}\phi \left(\varrho ({\mathtt{p}}_{{\mathrm{m}}_{\kappa }},{\mathtt{p}}_{{\mathrm{n}}_{\kappa }})\right)& \le & 0,\hfill \end{array}$$

  which is a contradiction. Thereby, $\left\{{\mathtt{p}}_{\mathrm{n}}\right\}$ remains Cauchy. Since $(\mathit{P},\varrho )$ is R-complete, $\exists {\mathtt{p}}^{*}\in \mathcal{S}$ verifying ${\mathtt{p}}_{\mathrm{n}}\stackrel{\varrho }{\to }{\mathtt{p}}^{*}$.
• Step–6. We shall confirm that ${\mathtt{p}}^{*}\in Fix\left(\mathit{P}\right)$ through assumption $\left(d\right)$. If $\mathit{P}$ retains R-continuous, then ${\mathtt{p}}_{\mathrm{n}+1}=\mathbf{P}\left({\mathtt{p}}_{\mathrm{n}}\right)\stackrel{\varrho }{⟶}\mathbf{P}\left({\mathtt{p}}^{*}\right)$; consequently, we achieve $\mathbf{P}\left({\mathtt{p}}^{*}\right)={\mathtt{p}}^{*}$.
  Alternately, assume that R retains $\varrho$-self-closed. Then we deduce a subsequence of $\left\{{\mathtt{p}}_{{\mathrm{n}}_{\kappa }}\right\}$ of $\left\{{\mathtt{p}}_{\mathrm{n}}\right\}$ verifying $[{\mathtt{p}}_{{\mathrm{n}}_{\kappa }},{\mathtt{p}}^{*}]\in \mathrm{R}$, $\forall \kappa \in {\mathbb{N}}_0$.

• We claim that

  $$\varrho ({\mathtt{p}}_{{\mathrm{n}}_{\kappa }+1},\mathcal{S}\mathtt{p})\le \varrho ({\mathtt{p}}_{{\mathrm{n}}_{\kappa }},{\mathtt{p}}^{*})\forall \kappa \in \mathbb{N}.$$

  (7)

  Take a partition $\{{\mathbb{N}}^0,{\mathbb{N}}^{+}\}$ of $\mathbb{N}$ i.e., ${\mathbb{N}}^0\cup {\mathbb{N}}^{+}=\mathbb{N}$ and ${\mathbb{N}}^0\cap {\mathbb{N}}^{+}=\varnothing$ that verify

(1)

$\varrho (\mathcal{S}{\mathtt{p}}_{{\mathrm{n}}_{\kappa }},\mathcal{S}{\mathtt{p}}^{*})=0,\forall \kappa \in {\mathbb{N}}^0,$

(2)

$\varrho (\mathcal{S}{\mathtt{p}}_{{\mathrm{n}}_{\kappa }},\mathcal{S}{\mathtt{p}}^{*})>0,\forall \kappa \in {\mathbb{N}}^{+}.$

• For case (1), we arrive at $\varrho (\mathcal{S}{\mathtt{p}}_{{\mathrm{n}}_{\kappa }},\mathcal{S}{\mathtt{p}}^{*})=0\forall \kappa \in {\mathbb{N}}^0$, that yields that $\varrho ({\mathtt{p}}_{{\mathrm{n}}_{\kappa }+1},\mathcal{S}{\mathtt{p}}^{*})=0,\forall \kappa \in {\mathbb{N}}^0$ and thereby, (7) prevails for every $\kappa \in {\mathbb{N}}^0.$
• If case (2) occurs, then by using assumption $\left(e\right)$, we get the following:

  $$\begin{array}{c}\hfill \phi \left(\varrho ({\mathtt{p}}_{{\mathrm{n}}_{\kappa }},\mathtt{p})\right)+\mathrm{F}\left(\varrho ({\mathtt{p}}_{{\mathrm{n}}_{\kappa }+1},\mathcal{S}{\mathtt{p}}^{*})\right)\le \mathrm{F}\left(\varrho ({\mathtt{p}}_{{\mathrm{n}}_{\kappa }},{\mathtt{p}}^{*})\right),\forall \kappa \in {\mathbb{N}}^{+}\end{array}$$

  implying thereby

  $$\phi \left(\varrho ({\mathtt{p}}_{{\mathrm{n}}_{\kappa }},\mathtt{p})\right)+F\left(\varrho ({\mathtt{p}}_{{\mathrm{n}}_{\kappa }+1},\mathcal{S}{\mathtt{p}}^{*})\right)\le \mathrm{F}\left(\varrho ({\mathtt{p}}_{{\mathrm{n}}_{\kappa }},{\mathtt{p}}^{*})\right),\forall \kappa \in {\mathbb{N}}^{+}.$$

  By $\left({\mathrm{F}}_1\right)$, we arrive at $\varrho ({\mathtt{p}}_{{\mathrm{n}}_{\kappa }+1},\mathcal{S}{\mathtt{p}}^{*})\le \varrho ({\mathtt{p}}_{{\mathrm{n}}_{\kappa }},{\mathtt{p}}^{*}),\forall \kappa \in {\mathbb{N}}^{+}$ and thereby, (7) prevails for every $\kappa \in {\mathbb{N}}^{+}$. Thus, (7) is thereby accurate for every $\kappa \in \mathbb{N}$. Letting the limit of (7) as $n\to \infty$ and from ${\mathtt{p}}_{{\mathrm{n}}_{\kappa }}\stackrel{\varrho }{⟶}{\mathtt{p}}^{*}$, we conclude ${\mathtt{p}}_{{\mathrm{n}}_{\kappa }+1}\stackrel{\varrho }{⟶}\mathcal{S}\left({\mathtt{p}}^{*}\right)$, and thereby $\mathcal{S}\left(\mathtt{p}\right)={\mathtt{p}}^{*}$, i.e., ${\mathtt{p}}^{*}$ retains a fixed point of $\mathcal{S}$.

□

Theorem 2. 

In alliance with the arguments of Theorem 1, if $\mathcal{S}\left(\mathbf{P}\right)$ retains ${\mathrm{R}}^s$-connected, then $\mathcal{S}$ owns a unique fixed point.

Proof. 

In view of Theorem 1, $Fix\left(\mathcal{S}\right)\ne \varnothing$. If ${\mathtt{p}}^{*},q^{*}\in Fix\left(\mathcal{S}\right)$, then

$${\mathcal{S}}^{\mathrm{n}}\left({\mathtt{p}}^{*}\right)={\mathtt{p}}^{*}\mathrm{and}{\mathcal{S}}^{\mathrm{n}}\left(q^{*}\right)=q^{*},\forall \mathrm{n}\in {\mathbb{N}}_0.$$

Trivially, ${\mathtt{p}}^{*},q^{*}\in \mathcal{S}\left(\mathbf{P}\right)$. From ${\mathrm{R}}^s$-connectedness of $\mathcal{S}\left(\mathbf{P}\right)$, we determine a path $\{{\omega }_0,{\omega }_1,{\omega }_2,\dots ,{\omega }_{\kappa }\}$ in ${\mathrm{R}}^s$ from $\mathtt{p}$ to q verifying

$$\begin{array}{c}\hfill {\omega }_0=\mathtt{p},{\omega }_{\kappa }=q\mathrm{and}[{\omega }_i,{\omega }_{i+1}]\in \mathrm{R}\mathrm{for}\mathrm{each}i(0\le i\le \kappa -1).\end{array}$$

(8)

By $\mathcal{S}$-closedness of R, we attain the following:

$$\begin{array}{c}\hfill [{\mathcal{S}}^{\mathrm{n}}{\omega }_i,{\mathcal{S}}^{\mathrm{n}}{\omega }_{i+1}]\in \mathrm{R}\forall i\mathrm{and}\mathrm{n}\in {\mathbb{N}}_0.\end{array}$$

(9)

For every $\mathrm{n}\in {\mathbb{N}}_0$ and for each i, denote the following:

$${\varpi }_{\mathrm{n}}^i=\varrho ({\mathcal{S}}^{\mathrm{n}}{\omega }_i,{\mathcal{S}}^{\mathrm{n}}{\omega }_{i+1}).$$

We show that

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

(10)

Fix i and formulate about two scenarios. In the first scenario, we attain

$${\varpi }_{{\mathrm{n}}_0}^i=\varrho ({\mathcal{S}}^{{\mathrm{n}}_0}{\omega }_i,{\mathcal{S}}^{{\mathrm{n}}_0}{\omega }_{i+1})=0,\mathrm{for} \mathrm{some} {\mathrm{n}}_0\in {\mathbb{N}}_0,$$

which leads to ${\mathcal{S}}^{{\mathrm{n}}_0}\left({\omega }_i\right)={\mathcal{S}}^{{\mathrm{n}}_0}\left({\omega }_{i+1}\right)$. Now applying (1), we have ${\mathcal{S}}^{{\mathrm{n}}_0+1}\left({\omega }_i\right)={\mathcal{S}}^{{\mathrm{n}}_0+1}\left({\omega }_{i+1}\right)$. Hence, ${\varpi }_{{\mathrm{n}}_0+1}^i=0$. Employing induction, we conclude ${\varpi }_{\mathrm{n}}^i=0$∀$\mathrm{n}\ge {\mathrm{n}}_0$, thereby yielding $\underset{\mathrm{n}\to \infty }{lim}{\varpi }_{\mathrm{n}}^i=0$.

Otherwise, assuming that ${\varpi }_{\mathrm{n}}^i>0$ ∀ $\mathrm{n}\in {\mathbb{N}}_0$. Employing the contraction-condition $\left(e\right)$ to (9), we arrive at

$$\begin{array}{ccc}\hfill \mathrm{F}\left({\varpi }_{\mathrm{n}+1}^i\right)& =& \mathrm{F}\left(\varrho ({\mathcal{S}}^{\mathrm{n}+1}{\omega }_i,{\mathcal{S}}^{\mathrm{n}+1}{\omega }_{i+1})\right)\hfill \\ & =& \mathrm{F}\left(\varrho (\mathcal{S}\left({\mathcal{S}}^{\mathrm{n}}{\omega }_i\right),\mathcal{S}\left({\mathcal{S}}^n{\omega }_{i+1}\right))\right)\hfill \\ & \le & \mathrm{F}\left(\varrho ({\mathcal{S}}^{\mathrm{n}}{\omega }_i,{\mathcal{S}}^{\mathrm{n}}{\omega }_{i+1})\right)-\phi \left(\varrho ({\mathcal{S}}^{\mathrm{n}}{\omega }_i,{\mathcal{S}}^{\mathrm{n}}{\omega }_{i+1})\right)\hfill \end{array}$$

so that

$$\begin{array}{ccc}\hfill \mathrm{F}\left({\varpi }_{\mathrm{n}+1}^i\right)& \le & \mathrm{F}\left({\varpi }_{\mathrm{n}}^i\right)-\phi \left({\varpi }_{\mathrm{n}}^i\right).\hfill \end{array}$$

(11)

From (11), we have the following:

$$\begin{array}{ccc}\hfill \mathrm{F}\left({\varpi }_{\mathrm{n}+1}^i\right)& \le & \mathrm{F}\left({\varpi }_{\mathrm{n}}^i\right).\hfill \end{array}$$

By axioms $\left({\mathrm{F}}_1\right)$, we have the following:

$$\begin{array}{ccc}\hfill {\varpi }_{\mathrm{n}+1}^i& \le & {\varpi }_{\mathrm{n}}^i.\hfill \end{array}$$

Therefore, $\left\{{\varpi }_{\mathrm{n}}^i\right\}$ is decreasing sequence in $(0,\infty )$. Thereby ∃ $r\ge 0$ verifying

$$\underset{\mathrm{n}\to \infty }{lim}{\varpi }_{\mathrm{n}}^i=r.$$

Assuming that $r>0$. Using lower limit in inequality (11), we attain the following:

$$\begin{array}{ccc}\hfill \underset{\mathrm{n}\to \infty }{lim\; inf}\mathrm{F}\left({\varpi }_{\mathrm{n}+1}^i\right)& \le & \underset{\mathrm{n}\to \infty }{lim\; inf}\mathrm{F}\left({\varpi }_{\mathrm{n}}^i\right)-\underset{\mathrm{n}\to \infty }{lim\; inf}\phi \left({\varpi }_{\mathrm{n}}^i\right).\hfill \end{array}$$

From axiom $({\mathrm{F}}_2$), we get the following:

$$\underset{\mathrm{n}\to \infty }{lim\; inf}\mathrm{F}\left({\varpi }_{\mathrm{n}}^i\right)=\ell \ne -\infty$$

and so the last inequality becomes $\underset{\mathrm{n}\to \infty }{lim\; inf}\phi \left({\varpi }_{\mathrm{n}}^i\right)\le 0,$ which is a contradiction. Hence, we have $\underset{\mathrm{n}\to \infty }{lim}{\varpi }_{\mathrm{n}}^i=0.$ Thereby, (10) is established for each i$(0\le i\le \kappa -1)$. Now, by the triangle inequality, we find

$$\begin{array}{ccc}\hfill \varrho ({\mathtt{p}}^{*},q^{*})& =& \varrho ({\mathcal{S}}^{\mathrm{n}}{\omega }_0,{\mathcal{S}}^{\mathrm{n}}{\omega }_r)\hfill \\ & \le & {\varpi }_{\mathrm{n}}^0+{\varpi }_{\mathrm{n}}^1+\dots +{\varpi }_{\mathrm{n}}^{r-1}\hfill \\ & \to & 0as\mathrm{n}\to \infty \hfill \end{array}$$

which yields ${\mathtt{p}}^{*}=q^{*}$. Hence, $\mathcal{S}$ possesses a unique fixed point. □

Remark 3. 

If φ is taken as a constant function, then our findings deduce the improved version of the findings of Sawangsup et al. [24], which under the restriction $\mathrm{R}:=⪯$ (a partial ordered relation) further reduce to the outcomes of Durmaz et al. [23].

The following outcome of Wardowski [21] is inferred by Theorem 2, taking $\mathrm{R}={\mathit{P}}^2$.

Corollary 1 

([21]). Let $(\mathbf{P},\varrho )$ be a complete MS and $\mathcal{S}$ a self-map on $\mathbf{P}$. If $\exists F\in \mathfrak{F}$ and $\phi \in \mathsf{\Phi }$, verifying

$$\forall \mathtt{p},q\in \mathbf{P}\mathrm{with}\mathcal{S}\left(\mathtt{p}\right)\ne \mathcal{S}\left(q\right)\Rightarrow \phi \left(\varrho \right(\mathtt{p},q\left)\right)+F\left(\varrho \right(\mathcal{S}\mathtt{p},\mathcal{S}q\left)\right)\le \mathrm{F}\left(\varrho \right(\mathtt{p},q\left)\right),$$

then $\mathcal{S}$ has a unique fixed point.

If we take $\phi \left(t\right)=\vartheta$ (a constant function), then Corollary 1 further reduces to the main finding of Wardowski [17].

3. Illustrative Examples

We will offer some examples that show the validity and utility of our findings.

Example 1. 

Let $A=\left\{{\displaystyle \frac{1}{n^2}}:n\in \mathbb{N}\right\}\cup \left\{0\right\}$, $B=\{2,3\}$ and $\mathbf{P}=A\cup B$. Define a BR R on $\mathbf{P}$ as

$$\mathrm{R}:=\{(\mathtt{p},q)\in {\mathbf{P}}^2:\mathtt{p}=q\mathrm{or}\mathtt{p},q\in A\mathrm{with}\mathtt{p}\le q\}.$$

Obviously, $\mathbf{P}$ is R-complete MS with the usual metric ϱ. Let $\mathcal{S}:\mathbf{P}\to \mathbf{P}$ be a map given by the following:

$$\mathcal{S}\left(\mathtt{p}\right)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{{(n+1)}^2}},\hfill & \mathrm{if}\mathtt{p}={\displaystyle \frac{1}{n^2}}\hfill \\ \mathtt{p},\hfill & \mathrm{if}\mathtt{p}\in \{0,2,3\}.\hfill \end{array}\right.$$

It is easy to see that R is $\mathcal{S}$-closed and locally $\mathcal{S}$-transitive; and $\mathcal{S}$ is R-continuous. Also, for ${\mathtt{p}}_0=0$ we have $({\mathtt{p}}_0,\mathcal{S}{\mathtt{p}}_0)\in \mathrm{R}$.

Define the auxiliary function $\mathrm{F}:(0,\infty )\to \mathbb{R}$ by the following:

$$\mathrm{F}\left(t\right)=\left\{\begin{array}{cc}{\displaystyle \frac{lnt}{\sqrt{t}}},& \mathrm{if}0<t<e^2\\ t-e^2+{\displaystyle \frac{2}{e}},& \mathrm{if}t\ge e^2.\end{array}\right.$$

On $(0,e^2)$, we have

$$\mathrm{F}\left(t\right)={\displaystyle \frac{lnt}{\sqrt{t}}}$$

so that

$${\mathrm{F}}^{\prime }\left(t\right)=t^{-3/2}\left(1-{\displaystyle \frac{1}{2}}lnt\right)>0.$$

Therefore, F is strictly increasing on the interval $(0,e^2)$.

For $e^2\le t<s$, we have $\mathrm{F}\left(t\right)=t-e^2+{\displaystyle \frac{2}{e}}<s-e^2+{\displaystyle \frac{2}{e}}=\mathrm{F}\left(s\right)$. Thus, F is strictly increasing on the interval $[e^2,\infty )$.

Finally, in case $0<t<e^2\le s$, we attain $\mathrm{F}\left(t\right)<F\left(s\right)$ yielding thereby F is strictly increasing.

Thus in all, F verifies axiom $\left({\mathrm{F}}_1\right)$. Also, F satisfies $\left({\mathrm{F}}_2\right)$. It follows that $\mathrm{F}\in \mathfrak{F}$.

Also, define $\phi :(0,\infty )\to (0,\infty )$ by

$$\phi \left(t\right)=ln2,\forall t\in (0,\infty ).$$

Then $\phi \in \mathsf{\Phi }$. The verification of the contraction condition $\left(e\right)$ for this pair is evident. Since all of the necessities of Theorem 1 are realized, $\mathcal{S}$ possesses a fixed point.

Otherwise, since

$$\varrho (\mathcal{S}2,\mathcal{S}3)=\varrho (2,3)=1,$$

then for all $\mathrm{F}\in \mathfrak{F}$ and $\phi \in \mathsf{\Phi }$, we have the following:

$$\phi \left(\varrho \right(2,3\left)\right)+\mathrm{F}\left(\varrho \right(\mathcal{S}2,\mathcal{S}3\left)\right)>\mathrm{F}\left(\varrho \right(2,3\left)\right).$$

Therefore, this example is not applicable to the main result of Wardowski [20].

Example 2. 

Let $\mathbf{P}=(-1,1]$. Define a BR R on $\mathbf{P}$ as

$$\mathrm{R}=\{(\mathtt{p},q)\in {\mathbf{P}}^2:\mathtt{p}>q\ge 0\}.$$

Obviously, $\mathbf{P}$ is R-complete MS with the usual metric ϱ. Let $\mathcal{S}:\mathbf{P}\to \mathbf{P}$ be a map given by the following:

$$\mathcal{S}\left(\mathtt{p}\right)=\left\{\begin{array}{cc}\mathtt{p}+1,\hfill & \mathrm{if}-1<\mathtt{p}<0,\hfill \\ \mathtt{p}-{\mathtt{p}}^2/2,\hfill & \mathrm{if}0\le \mathtt{p}\le 1.\hfill \end{array}\right.$$

It is easy to see that R is $\mathcal{S}$-closed and locally $\mathcal{S}$-transitive; and $\mathcal{S}$ is R-continuous.

Define the auxiliary functions $F:(0,\infty )\to \mathbb{R}$ and $\phi :(0,\infty )\to (0,\infty )$ by

$$\mathrm{F}\left(t\right)=\left\{\begin{array}{cc}t,\hfill & \mathrm{if}0\le t\le 1\hfill \\ t^2,\hfill & \mathrm{if}t>1\hfill \end{array}\right.\mathrm{and}\phi \left(t\right)=\left\{\begin{array}{cc}{\displaystyle \frac{t^2}{2}},\hfill & \mathrm{if}0\le t\le 1,\hfill \\ 4,\hfill & \mathrm{if}t>1.\hfill \end{array}\right.$$

It is easy to verify that $\mathrm{F}\in \mathfrak{F}$ and $\phi \in \mathsf{\Phi }$. Take $\mathtt{p},q\in \mathbf{P}$ with $(\mathtt{p},q)\in \mathrm{R}$ and $\mathcal{S}\left(\mathtt{p}\right)\ne \mathcal{S}\left(q\right)$. Then $\mathtt{p}>q\ge 0$. Thus, we have

$$\begin{array}{ccc}\hfill \mathrm{F}\left(\varrho \right(\mathcal{S}\mathtt{p},\mathcal{S}q\left)\right)& =& (\mathtt{p}-{\displaystyle \frac{1}{2}}{\mathtt{p}}^2)-(q-{\displaystyle \frac{1}{2}}{q}^2)\hfill \\ & =& (\mathtt{p}-q)-{\displaystyle \frac{1}{2}}(\mathtt{p}-q)(\mathtt{p}+q)\hfill \\ & \le & (\mathtt{p}-q)-{\displaystyle \frac{1}{2}}{(\mathtt{p}-q)}^2\hfill \\ & =& \varrho (\mathtt{p},q)-{\displaystyle \frac{1}{2}}{\left(\varrho (\mathtt{p},q)\right)}^2\hfill \\ & =& \mathrm{F}\left(\varrho \right(\mathtt{p},q\left)\right)-\phi \left(\varrho \right(\mathtt{p},q\left)\right)\hfill \end{array}$$

so that

$$\phi \left(\varrho \right(\mathtt{p},q\left)\right)+F\left(\varrho \right(\mathcal{S}\mathtt{p},\mathcal{S}q\left)\right)\le \mathrm{F}\left(\varrho \right(\mathtt{p},q\left)\right).$$

Thus, the contraction-condition $\left(e\right)$ is verified. Since all of the necessities of Theorems 1 and 2 are realized, $\mathcal{S}$ possesses a unique fixed point, namely: $\mathtt{p}=0$.

Example 3. 

Let $\mathbf{P}=[0,1]\cup \{2,3,4,\dots \}$ be a MS with the metric:

$$\varrho (\mathtt{p},q)=\left\{\begin{array}{cc}\mid \mathtt{p}-q\mid ,\hfill & \mathrm{if}\mathtt{p},q\in [0,1]\mathrm{and}\mathtt{p}\ne q,\hfill \\ \mathtt{p}+q,\hfill & \mathrm{if}\mathrm{at}\mathrm{least}\mathrm{one}\mathrm{of}\mathtt{p}\mathrm{or}q\mathrm{does}\mathrm{not}\mathrm{belong}\mathrm{to}[0,1]\mathrm{and}\mathtt{p}\ne q,\hfill \\ 0,\hfill & \mathrm{if}\mathtt{p}=q.\hfill \end{array}\right.$$

Define a BR R on $\mathbf{P}$ as the following:

$$\mathrm{R}=\{(\mathtt{p},q)\in {\mathbf{P}}^2:\mathtt{p}>q\mathrm{and}\mathtt{p}\in \{3,4,5,..\},q\ne 2\}.$$

Obviously, $(\mathbf{P},\varrho )$ is R-complete MS. Let $\mathcal{S}:\mathbf{P}\to \mathbf{P}$ be a map given by the following:

$$\mathcal{S}\left(\mathtt{p}\right)=\left\{\begin{array}{cc}\mathtt{p}-{\mathtt{p}}^3/4,\hfill & \mathrm{if}0\le \mathtt{p}\le 1,\hfill \\ \mathtt{p}-1,\hfill & \mathrm{if}\mathtt{p}\in \{2,3,4,\dots \}.\hfill \end{array}\right.$$

It is easy to see that R is $\mathcal{S}$-closed and locally $\mathcal{S}$-transitive; and $\mathcal{S}$ is R-continuous.

Define the auxiliary functions $\mathrm{F}:(0,\infty )\to \mathbb{R}$ and $\phi :(0,\infty )\to (0,\infty )$ by

$$\mathrm{F}\left(t\right)=\left\{\begin{array}{cc}t+1,\hfill & \mathrm{if}0\le t<1\hfill \\ t^2,\hfill & \mathrm{if}t\ge 1\hfill \end{array}\right.\mathrm{and}\phi \left(t\right)=\left\{\begin{array}{cc}{\displaystyle \frac{t^2}{4}},\hfill & \mathrm{if}0\le t\le 1,\hfill \\ {\displaystyle \frac{1}{5}},\hfill & \mathrm{if}t>1.\hfill \end{array}\right.$$

It is easy to verify that $\mathrm{F}\in \mathfrak{F}$ and $\phi \in \mathsf{\Phi }$.

Take $\mathtt{p},q\in \mathbf{P}$ with $(\mathtt{p},q)\in \mathrm{R}$ and $\mathcal{S}\left(\mathtt{p}\right)\ne \mathcal{S}\left(q\right)$. When $\mathtt{p}\in \{3,4,\dots \}$, then there are two possibilities of choosing q. Firstly, we take $q\in [0,1]$, then we have the following:

$$\begin{array}{ccc}\hfill \varrho (\mathcal{S}\mathtt{p},\mathcal{S}q)& =& \varrho \left(\mathtt{p}-1,q-{\displaystyle \frac{1}{4}}{q}^3\right)=\mathtt{p}-1+q-{\displaystyle \frac{q^3}{4}}\hfill \\ & \le & \mathtt{p}+q-1.\hfill \end{array}$$

Otherwise, if $q\in \{3,4,\dots \}$, then we have the following:

$$\begin{array}{ccc}\hfill \varrho (\mathcal{S}\mathtt{p},\mathcal{S}q)& =& \varrho (\mathtt{p}-1,q-1)=\mathtt{p}+q-2\hfill \\ & <& \mathtt{p}+q-1.\hfill \end{array}$$

Therefore, in both cases, we have

$$\begin{array}{ccc}\hfill \mathrm{F}\left(\varrho \right(\mathcal{S}\mathtt{p},\mathcal{S}q\left)\right)& =& {\left(\varrho (\mathcal{S}\mathtt{p},\mathcal{S}q)\right)}^2<{(\mathtt{p}+q-1)}^2\hfill \\ & <& (\mathtt{p}+q-1)(\mathtt{p}+q+1)={(\mathtt{p}+q)}^2-1\hfill \\ & <& {(\mathtt{p}+q)}^2-{\displaystyle \frac{1}{5}}\hfill \\ & =& \mathrm{F}\left(\varrho \right(\mathtt{p},q\left)\right)-\phi \left(\varrho \right(\mathtt{p},q\left)\right)\hfill \end{array}$$

so that

$$\phi \left(\varrho \right(\mathtt{p},q\left)\right)+\mathrm{F}\left(\varrho \right(\mathcal{S}\mathtt{p},\mathcal{S}q\left)\right)<\mathrm{F}\left(\varrho \right(\mathtt{p},q\left)\right).$$

Thus, the contraction condition $\left(e\right)$ is verified. In view of all the possible cases, we conclude that all the conditions of Theorem 1 are satisfied. Therefore, by Theorem 1, $\mathcal{S}$ has a fixed point, namely: $\mathtt{p}=0$.

Remark 4. 

In Examples 2 and 3, the auxiliary function φ is not constant functions. Henceforth, these examples can not be covered by existing fixed point theorems over F-contractions. This substantiates the utility of Theorems 1 and 2.

4. Applications to Fractional Differential Equations

Recall that given a continuous function $\hslash :[0,\infty )\to \mathbb{R}$, the Caputo derivative of fractional order $\mathrm{e}$ is defined as follows:

$${}^{\mathrm{c}}\Delta ^{\mathrm{e}}\hslash \left(\vartheta \right)={\displaystyle \frac{1}{\mathsf{\Gamma }(\eta -\mathrm{e})}}{\int }_0^{\vartheta }{(\vartheta -\mathsf{s})}^{\eta -\mathrm{e}-1}{\hslash }^{\left(\eta \right)}\left(\mathsf{s}\right)d\mathsf{s},where\eta -1<\mathrm{e}<\eta ,\eta =\left[\mathrm{e}\right]+1.$$

Here $\left[\mathrm{e}\right]$ denotes the integer part of the positive real number $\mathrm{e}$ and $\mathsf{\Gamma }(·)$ indicates the classical Gamma function.

Examine the nonlinear FDE that appears below:

$${}^{\mathrm{c}}\Delta ^{\mathrm{e}}w\left(\vartheta \right)=\mathsf{\Psi }(\vartheta ,w\left(\vartheta \right)),$$

(12)

with the following integral boundary conditions

$$w\left(0\right)=0,w\left(1\right)={\int }_0^{\ell }w\left(\mathsf{s}\right)d\mathsf{s}$$

under the following axioms:

• $1<\mathrm{e}\le 2$,
• $0<\vartheta <1$,
• $0<\ell <1$,
• $\mathsf{\Psi }:[0,1]\times [0,\infty )\to [0,\infty )$ retains continuity.

We arrive at the main finding of this part.

Theorem 3. 

Suppose that the BVP (12) fulfills the foregoing presumptions. If ∃ a monotonically increasing function $\theta :[0,\infty )\to [0,\infty )$ verifying $\theta \left(t\right)\le t{e}^{-t}$, for each $t>0$ such that

$${\mathsf{s}}_1\ge {\mathsf{s}}_2\ge 0⟹0\le [\mathsf{\Psi }(\vartheta ,{\mathsf{s}}_1)-\mathsf{\Psi }(\vartheta ,{\mathsf{s}}_2)]\le {\displaystyle \frac{\mathsf{\Gamma }(\mathrm{e}+1)}{5}}\theta ({\mathsf{s}}_1-{\mathsf{s}}_2),$$

(13)

then BVP (12) possesses a unique solution.

Proof. 

Clearly, the BVP (12) reduces to the integral equation:

$$\begin{array}{cc}\hfill w\left(\vartheta \right)& ={\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathrm{e}\right)}}{\int }_0^{\vartheta }{(\vartheta -\mathsf{s})}^{\mathrm{e}-1}\mathsf{\Psi }(\mathsf{s},w\left(\mathsf{s}\right))d\mathsf{s}-{\displaystyle \frac{2\vartheta }{(2-{\ell }^2)\mathsf{\Gamma }\left(\mathrm{e}\right)}}{\int }_0^1{(1-\mathsf{s})}^{\mathrm{e}-1}\mathsf{\Psi }(\mathsf{s},w\left(\mathsf{s}\right))d\mathsf{s}\hfill \\ & +{\displaystyle \frac{2\vartheta }{(2-{\ell }^2)\mathsf{\Gamma }\left(\mathrm{e}\right)}}{\int }_0^{\ell }\left({\int }_0^{\mathsf{s}}{(\mathsf{s}-\tau )}^{\mathrm{e}-1}\mathsf{\Psi }(\tau ,w\left(\tau \right))d\tau \right)d\mathsf{s}.\hfill \end{array}$$

(14)

Denote $\mathit{P}:=C[0,1]$. Assign the metric on $\mathit{P}$ as follows:

$$\varrho (w,z)=\underset{0\le \vartheta \le 1}{sup}|w\left(\vartheta \right)-z\left(\vartheta \right)|.$$

On $\mathit{P}$, formulate a BR R and a self-map $\mathcal{S}$ given as under

$$\mathrm{R}=\{(w,z)\in {\mathit{P}}^2:w\left(\vartheta \right)\le z\left(\vartheta \right),\mathrm{for} \mathrm{each}\vartheta \in [0,1]\};$$

and

$$\begin{array}{ccc}\hfill \left(\mathcal{S}w\right)\left(\vartheta \right)& =& {\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathrm{e}\right)}}{\int }_0^{\vartheta }{(\vartheta -\mathsf{s})}^{\mathrm{e}-1}\mathsf{\Psi }(\mathsf{s},w\left(\mathsf{s}\right))d\mathsf{s}-{\displaystyle \frac{2\vartheta }{(2-{\ell }^2)\mathsf{\Gamma }\left(\mathrm{e}\right)}}{\int }_0^1{(1-\mathsf{s})}^{\mathrm{e}-1}\mathsf{\Psi }(\mathsf{s},w\left(\mathsf{s}\right))d\mathsf{s}\hfill \\ & & +{\displaystyle \frac{2\vartheta }{(2-{\ell }^2)\mathsf{\Gamma }\left(\mathrm{e}\right)}}{\int }_0^{\ell }\left({\int }_0^{\mathsf{s}}{(\mathsf{s}-\tau )}^{\mathrm{e}-1}\mathsf{\Psi }(\tau ,w\left(\tau \right))d\tau \right)d\mathsf{s}.\hfill \end{array}$$

(15)

We shall execute all of the premises of Theorems 1 and 2.

(a) 

Obviously, $(\mathit{P},\varrho )$ retains R-complete MS.

(b) 

Zero function $\mathbf{0}\in \mathbf{V}$ verifies for every $\vartheta \in [0,1]$, $\mathbf{0}\left(\vartheta \right)\le \left(\mathcal{S}\mathbf{0}\right)\left(\vartheta \right)$ yielding thereby $(\mathbf{0},\mathcal{S}\mathbf{0})\in \mathrm{R}$.

(c) 

Obviously, R being transitive serves as a locally $\mathcal{S}$-transitive BR. Let $(w,z)\in \mathrm{R}$, verifying $w\left(\vartheta \right)\le z\left(\vartheta \right)$, for every $\vartheta \in [0,1]$. Then, we attain

$$\begin{array}{cc}\hfill \left(\mathcal{S}w\right)\left(\vartheta \right)& ={\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathrm{e}\right)}}{\int }_0^{\vartheta }{(\vartheta -\mathsf{s})}^{\mathrm{e}-1}\mathsf{\Psi }(\mathsf{s},w\left(\mathsf{s}\right))d\mathsf{s}-{\displaystyle \frac{2\vartheta }{(2-{\ell }^2)\mathsf{\Gamma }\left(\mathrm{e}\right)}}{\int }_0^1{(1-\mathsf{s})}^{\mathrm{e}-1}\mathsf{\Psi }(\mathsf{s},w\left(\mathsf{s}\right))d\mathsf{s}\hfill \\ \hfill & +{\displaystyle \frac{2\vartheta }{(2-{\ell }^2)\mathsf{\Gamma }\left(\mathrm{e}\right)}}{\int }_0^{\ell }\left({\int }_0^{\mathsf{s}}{(\mathsf{s}-\tau )}^{\mathrm{e}-1}\mathsf{\Psi }(\tau ,w\left(\tau \right))d\tau \right)d\mathsf{s}\hfill \\ \hfill & \le {\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathrm{e}\right)}}{\int }_0^{\vartheta }{(\vartheta -\mathsf{s})}^{\mathrm{e}-1}\mathsf{\Psi }(\mathsf{s},z\left(\mathsf{s}\right))d\mathsf{s}-{\displaystyle \frac{2\vartheta }{(2-{\ell }^2)\mathsf{\Gamma }\left(\mathrm{e}\right)}}{\int }_0^1{(1-\mathsf{s})}^{\mathrm{e}-1}\mathsf{\Psi }(\mathsf{s},z\left(\mathsf{s}\right))d\mathsf{s}\hfill \\ \hfill & +{\displaystyle \frac{2\vartheta }{(2-{\ell }^2)\mathsf{\Gamma }\left(\mathrm{e}\right)}}{\int }_0^{\ell }\left({\int }_0^{\mathsf{s}}{(\mathsf{s}-\tau )}^{\mathrm{e}-1}\mathsf{\Psi }(\tau ,z\left(\tau \right))d\tau \right)d\mathsf{s}\hfill \\ \hfill & =\left(\mathcal{S}z\right)\left(\vartheta \right)\hfill \end{array}$$

so that $(\mathcal{S}w,\mathcal{S}z)\in \mathrm{R}$. Thus, R remains $\mathcal{S}$-closed.

(d) 

To verify that R is $\varrho$-self-closed, let $\left\{w_n\right\}\subset \mathbf{V}$ be a sequence verifying $(w_n,w_{n+1})\in \mathrm{R},\forall n\in \mathbb{N}$ and $w_n\to w$. Then $\left\{w_n\left(\vartheta \right)\right\}$ (where $\vartheta \in [0,1]$,) is monotonically increasing (in the real line), converging to $w\left(\vartheta \right)$; so for each $n\in \mathbb{N}$, we conclude $w_n\left(\vartheta \right)\le w\left(\vartheta \right)$. Hence, $(w_n,w)\in \mathrm{R},\forall n\in \mathbb{N}$.

(e) 

For $(w,z)\in \mathrm{R}$ such that $\mathcal{S}\left(w\right)\ne \mathcal{S}\left(z\right)$, we attain

$$\begin{array}{cc}\hfill \varrho (\mathcal{S}w,\mathcal{S}z)& =\underset{0\le \vartheta \le 1}{sup}|\left(\mathcal{S}w\right)\left(\vartheta \right)-\left(\mathcal{S}z\right)\left(\vartheta \right)|=\underset{0\le \vartheta \le 1}{sup}[\left(\mathcal{S}z\right)\left(\vartheta \right)-\left(\mathcal{S}w\right)\left(\vartheta \right)]\hfill \\ \hfill & =\underset{0\le \vartheta \le 1}{sup}\left[{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathrm{e}\right)}}{\int }_0^{\vartheta }{(\vartheta -\mathsf{s})}^{\mathrm{e}-1}(\mathsf{\Psi }(\mathsf{s},z\left(\mathsf{s}\right))-\mathsf{\Psi }(\mathsf{s},w\left(\mathsf{s}\right)))\right.d\mathsf{s}\hfill \\ \hfill & -{\displaystyle \frac{2\vartheta }{(2-{\ell }^2)\mathsf{\Gamma }\left(\mathrm{e}\right)}}{\int }_0^1{(1-\mathsf{s})}^{\mathrm{e}-1}(\mathsf{\Psi }(\mathsf{s},z\left(\mathsf{s}\right))-\mathsf{\Psi }(\mathsf{s},w\left(\mathsf{s}\right)))d\mathsf{s}\hfill \\ \hfill & \left.+{\displaystyle \frac{2\vartheta }{(2-{\ell }^2)\mathsf{\Gamma }\left(\mathrm{e}\right)}}{\int }_0^{\ell }\left({\int }_0^{\mathsf{s}}{(\mathsf{s}-\tau )}^{\mathrm{e}-1}(\mathsf{\Psi }(\tau ,z\left(\tau \right))-\mathsf{\Psi }(\tau ,w\left(\tau \right)))d\tau \right)d\mathsf{s}\right]\hfill \\ \hfill & \le {\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathrm{e}\right)}}\underset{0\le \vartheta \le 1}{sup}{\int }_0^{\vartheta }{(\vartheta -\mathsf{s})}^{\mathrm{e}-1}(\mathsf{\Psi }(\mathsf{s},z\left(\mathsf{s}\right))-\mathsf{\Psi }(\mathsf{s},w\left(\mathsf{s}\right)))d\mathsf{s}\hfill \\ \hfill & +{\displaystyle \frac{2}{(2-{\ell }^2)\mathsf{\Gamma }\left(\mathrm{e}\right)}}{\int }_0^{\ell }\left({\int }_0^{\mathsf{s}}{(\mathsf{s}-\tau )}^{\mathrm{e}-1}(\mathsf{\Psi }(\tau ,z\left(\tau \right))-\mathsf{\Psi }(\tau ,w\left(\tau \right)))d\tau \right)d\mathsf{s}\hfill \\ \hfill & \le {\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathrm{e}\right)}}\underset{0\le \vartheta \le 1}{sup}{\int }_0^{\vartheta }{(\vartheta -\mathsf{s})}^{\mathrm{e}-1}{\displaystyle \frac{\mathsf{\Gamma }(\mathrm{e}+1)}{5}}\theta (z\left(\mathsf{s}\right)-w\left(\mathsf{s}\right))d\mathsf{s}\hfill \\ \hfill & +{\displaystyle \frac{2}{(2-{\ell }^2)\mathsf{\Gamma }\left(\mathrm{e}\right)}}{\int }_0^{\ell }\left({\int }_0^{\mathsf{s}}{(\mathsf{s}-\tau )}^{\mathrm{e}-1}{\displaystyle \frac{\mathsf{\Gamma }(\mathrm{e}+1)}{5}}\theta (z\left(\tau \right)-w\left(\tau \right))d\tau \right)d\mathsf{s}.\hfill \end{array}$$

By monotonicity of $\theta$, above inequality becomes the following:

$$\begin{array}{cc}\hfill \varrho (\mathcal{S}w,\mathcal{S}z)& \le \theta \left(\varrho (w,z)\right){\displaystyle \frac{\mathsf{\Gamma }(\mathrm{e}+1)}{5}}{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathrm{e}\right)}}\left[\underset{0\le \vartheta \le 1}{sup}{\int }_0^{\vartheta }{(\vartheta -\mathsf{s})}^{\mathrm{e}-1}d\mathsf{s}+{\displaystyle \frac{2}{(2-{\ell }^2)}}{\int }_0^{\ell }{\int }_0^{\mathsf{s}}{(\mathsf{s}-\tau )}^{\mathrm{e}-1}d\tau d\mathsf{s}\right]\hfill \\ \hfill & =\theta \left(\varrho (w,z)\right){\displaystyle \frac{\mathrm{e}+1}{5}}\left[\underset{0\le \vartheta \le 1}{sup}{\int }_0^{\vartheta }{(\vartheta -\mathsf{s})}^{\mathrm{e}-1}d\mathsf{s}+{\displaystyle \frac{2}{(2-{\ell }^2)}}{\int }_0^{\ell }{\int }_0^{\mathsf{s}}{(\mathsf{s}-\tau )}^{\mathrm{e}-1}d\tau d\mathsf{s}\right]\hfill \\ \hfill & \le \theta \left(\varrho \right(w,z\left)\right).\hfill \end{array}$$

As $\theta \left(t\right)\le t{e}^{-t}$, the above inequality reduces to the following:

$$\varrho (\mathcal{S}w,\mathcal{S}z)\le \varrho (w,z)·e^{-\varrho (w,z)}.$$

Taking logarithm on both sides, we get the following:

$$\varrho (w,z)+ln\varrho (\mathcal{S}w,\mathcal{S}z)\le ln\left(\varrho \right(w,z\left)\right).$$

(16)

Define the auxiliary functions $\mathrm{F}\in \mathfrak{F}$ and $\phi \in \mathsf{\Phi }$ by the following:

$$\mathrm{F}\left(t\right)=lnt\mathrm{and}\phi \left(t\right)=t.$$

Then inequality (16) reduces to the following:

$$\phi \left(\varrho \right(w,z\left)\right)+F\left(\varrho \right(\mathcal{S}w,\mathcal{S}z\left)\right)\le \mathrm{F}\left(\varrho \right(w,z\left)\right).$$

The aforementioned assertions of Theorem 1 are thus met. This entails that $\mathcal{S}$ incorporates a fixed point.

For each pair $w,z\in \mathcal{S}\left(\mathit{P}\right)$, define $u:=max\{w,z\}\in \mathbf{V}$. Hence, we obtain $(w,u)\in \mathrm{R}$ and $(z,u)\in \mathrm{R}$. Therefore, $\mathcal{S}\left(\mathit{P}\right)$ is ${\mathrm{R}}^s$-connected. Therefore, owing to Theorem 2, $\mathcal{S}$ enjoys a unique fixed point, which (due to (14) and (15)) serves as the unique solution of (12). □

5. Conclusions

We have established the accuracy of fixed-points and their uniqueness for a relational $(\phi ,\mathrm{F})$–contractions. A selection of prior fixed-point findings was consolidated and expanded by our outcomes, namely, Wardowski [20,21], Sawangsup et al. [24], Durmaz et al. [23], among others. Note that Theorems 1 and 2 are different from those of Imdad et al. [25] as we assumed that F remains strictly increasing while Imdad et al. [25] considered continuity requirement on F. Merely the comparative elements are subjected to the contraction conditions in our investigations. We present a few examples to reinforce these conclusions. To emphasize the value of the theory and the range of our findings, we carried out an application to a specific nonlinear FDE associated with the integral boundary conditions.

The readers can extrapolate our findings in the subsequent aspects as possible future research:

• To alter the attributes of the auxiliary functions F and $\phi$;
• To strengthen our findings for a couple of maps by drilling into common fixed point outcomes;
• To broaden our assessments over quasimetric space, semi-MS, fuzzy MS, multiplicative MS, etc., composed with a BR;
• Applying our research to nonlinear matrix equations or nonlinear integral equations in place of FDE.