A Class of φ-Contractions in Orthogonal Metric Spaces with an Application

Abstract

This article deals with certain outcomes on fixed points of an orthogonal nonlinear contraction map in the framework of O-complete metric spaces. The findings investigated herein enhance and sharpen a few outcomes on fixed points. In order to demonstrate our outcomes, we provide a number of illustrative examples. Finally, via our findings, we discuss the existence and uniqueness of solutions to a periodic boundary value problem.

1. Introduction

Within this text, the following notations and abbreviations are adopted:

• $\mathbb{N}$: The set of natural numbers.
• ${\mathbb{N}}_0:=\mathbb{N}\cup \left\{0\right\}$.
• $\mathbb{R}$: The set of real numbers.
• ${\mathbb{R}}^{+}:=[0,\infty )$.
• BCP: Banach contraction principle.
• MS: Metric space.
• BVP: Boundary value problem(s).
• $Fix\left(\zeta \right)$: Fixed-point set of self-map $\zeta$.
• $\mathcal{C}\left(A\right)$: The class of continuous real-valued-functions on a set A.
• ${\mathcal{C}}^{\prime }\left(A\right)$: The class of continuously differentiable real-valued-functions on a set A.

The conception of symmetry has played a fundamental role in Hilbert and Banach spaces. A distance function d expressed as $d(u,v)=d(v,u),\forall u,v\in S$ is known as symmetric. A set that has such a symmetric distance function is known as a symmetric space. Since Banach spaces and metric spaces inherently reflect this symmetry, different studies including symmetric operators have been explored in Banach spaces. The classical BCP is the most paramount and conventional approach in nonlinear functional analysis. In conjunction with ensuring the existence of a unique fixed point, the BCP delivers a practical method for estimating the fixed point. The accessibility of the BCP has contributed to its appeal from an applications perspective. Numerous authors have used contraction mappings to demonstrate the existence of solutions to integral equations, matrix equations, BVP, etc.

A nonlinear contraction is a natural generalization of contraction-inequality by altering a specific auxiliary function $\phi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ in place of the Lipschitz constant $r\in (0,1)$. Nonlinear contractions are also known as $\phi$-contractions. Browder [1] established a first generalization of BCP under $\phi$-contraction, assuming certain conditions on the auxiliary function $\phi$. Soon after, Boyd and Wong [2] slightly enhanced the findings of Browder [1]. Indeed, the Boyd–Wong theorem [2] has garnered a lot of attention within the last fifty years. Matkowski [3] investigated yet another generalization of the Browder theorem [1], which is independent of the Boyd–Wong Theorem. We indicate the classical Boyd–Wong theorem as under the following:

Theorem 1

([2]). Assuming that $(\mathbf{V},\delta )$ is a complete MS and $\zeta :\mathbf{V}\to \mathbf{V}$ is a map. If there exists a right upper semi-continuous function $\phi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ verifying $\phi \left(t\right)<t$ $\forall t>0$ with

$$\delta (\zeta \mathsf{v},\zeta \mathsf{u})\le \phi \left(\delta \right(\mathsf{v},\mathsf{u}\left)\right),\forall \mathsf{v},\mathsf{u}\in \mathbf{V},$$

then ζ enjoys a unique fixed point.

With the constraint $\phi \left(t\right)=rt,where0<r<1$, the $\phi$-contraction reduces to the usual contraction, transforming Theorem 1 into the BCP. Recently, Filali et al. [4] expanded upon the concept of Boyd–Wong contractions on directed graphs by proving new fixed point theorems and applying these to boundary value problems.

In this direction, Mukherjea [5] simplified Theorem 1 by replacing the right upper semi-continuity of $\phi$ with right continuity. Later on, Lakshmikantham and Ćirić [6] expanded the class of auxiliary functions introduced by Mukherjea [5] by considering the following family:

$$\Phi =\left\{\phi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}:\phi \left(t\right)<t,\mathrm{for}\mathrm{all}t>0\mathrm{and}\underset{s\to t+}{lim}\phi \left(s\right)<t,\mathrm{for}\mathrm{all}t>0\right\}.$$

If $\phi$ is a Mukherjea function (i.e., it is right continuous and satisfies $\phi \left(t\right)<t$ for all $t>0$), then $\phi \in \Phi$ as $\underset{s\to t+}{lim}\phi \left(s\right)=\phi \left(t\right)<t$ whenever $t>0$. For elementary properties of the auxiliary functions of the family $\Phi$, readers are advised to consult subsection 2.3.3 of the monograph by Agarwal et al. [7].

Over the past several years, the BCP has been expanded and strengthened to ordered MSs by Ran and Reurings [8] and Nieto and Rodríguez-Loṕez [9]. Such findings are applicable to solving certain typical nonlinear matrix equations and periodic BVPs, whereas ordinary fixed-point findings cannot be employed. The order-theoretic versions of the outcomes by Boyd and Wong [2] and Matkowski [3] were developed by Agarwal et al. [10], O’Regan and Petruşel [11], and Ćirić et al. [12]. On the other hand, Wu and Liu [13] and Karapinar et al. [14] independently proved an outcome on a fixed point of the $\phi$-contraction map relative to $\Phi$ within the framework of ordered MS. In 2017, Gordji et al. [15] recognized the idea of orthogonality in MS and employed it to enhance the Ran–Reurings fixed point theorem. Very recently, Singh et al. [16] established several outcomes on fixed points of orthogonal $\phi$-contraction maps due to Boyd and Wong [2] and Matkowski [3].

Section 2 presents basic notions and auxiliary results needed for our main results. In Section 3, we investigate the outcomes on fixed points of a $\phi$-contraction map based on the class $\Phi$ in the framework of orthogonally complete MS. Section 4 presents multiple examples to support the legitimacy of our findings. In Section 5, we exhibit the reliability of our outcomes by applying them to a BVP, which satisfies certain presumptions. In the last section, we present our conclusions. Our findings are extensions of the outcomes of Gordji et al. [15] from usual contraction to $\phi$-contraction in orthogonal MSs. The findings established herewith are indeed analogs of the outcomes of Singh et al. [16] as we used Lakshmikantham-Ćirić-type contractions to prove fixed-point results in orthogonally complete MSs, while Singh et al. [16] utilized Boyd–Wong-type contractions. Our findings improve and enrich the outcomes of Wu and Liu [13] and Karapinar et al. [14] which are framed within ordered MSs. In contrast, our results are proven in an orthogonal MS (a more general setting).

2. Preliminaries

Given a set $\mathbf{V}$, any subset of ${\mathbf{V}}^2$ is named as a relation on $\mathbf{V}$.

Definition 1

([15]). A relation ⊥ on a set $\mathbf{V}\ne \varnothing$ is denoted as an O-set (or, more precisely, an orthogonal set), denoted by the pair $(\mathbf{V},\perp )$, if $\exists {\mathsf{v}}_0\in \mathbf{V}$ with

$$(\forall \mathsf{u}\in \mathbf{V},\mathsf{u}\perp {\mathsf{v}}_0)or(\forall \mathsf{u}\in \mathbf{V},{\mathsf{v}}_0\perp \mathsf{u}).$$

Such ${\mathsf{v}}_0$ is referred to as an orthogonal element.

Definition 2

([15]). A self-map ζ in an O-set $(\mathbf{V},\perp )$ is referred to as ⊥-preserving if

$$\mathsf{v}\perp \mathsf{u}\Rightarrow \zeta \left(\mathsf{v}\right)\perp \zeta \left(\mathsf{u}\right).$$

Also, $\zeta :\mathbf{V}\to \mathbf{V}$ is called weakly ⊥-preserving if

$$\mathsf{v}\perp \mathsf{u}\Rightarrow \zeta \left(\mathsf{v}\right)\perp \zeta \left(\mathsf{u}\right) or \zeta \left(\mathsf{u}\right)\perp \zeta \left(\mathsf{v}\right).$$

Definition 3

([15]). A sequence $\left\{{\mathsf{v}}_n\right\}$ in an O-set $(\mathbf{V},\perp )$ is denoted as an O-sequence (or, more precisely, an orthogonal sequence) if

$$(\forall n\in \mathbb{N},{\mathsf{v}}_n\perp {\mathsf{v}}_{n+1}) or (\forall n\in \mathbb{N},{\mathsf{v}}_{n+1}\perp {\mathsf{v}}_n).$$

Definition 4

([15]). $(\mathbf{V},\perp ,\delta )$ is referred to as an orthogonal MS if $(\mathbf{V},\perp )$ is an O-set and $(\mathbf{V},\delta )$ is an MS.

Definition 5

([15]). An orthogonal MS $(\mathbf{V},\perp ,\delta )$ is denoted as O-complete (or, more precisely, orthogonally complete) if every Cauchy O-sequence is convergent.

Definition 6

([15]). A self-map ζ in an orthogonal MS $(\mathbf{V},\perp ,\delta )$ is denoted as ⊥-continuous or orthogonally continuous if for all $\mathsf{v}\in \mathbf{V}$ and any O-sequence $\left\{{\mathsf{v}}_n\right\}$ in $\mathbf{V}$ with ${\mathsf{v}}_n\stackrel{\delta }{⟶}\mathsf{v}$, we have

$$\zeta \left({\mathsf{v}}_n\right)\stackrel{\delta }{⟶}\zeta \left(\mathsf{v}\right).$$

Remark 1

([15]). Clearly, every complete MS is O-complete and every continuous map is ⊥-continuous. But converses of both inclusions need not be true. To substantiate it, consider $\mathbf{V}=(0,1]$ equipped with a usual metric δ. We define a relation ⊥ on $\mathbf{V}$ by

$$\mathsf{v}\perp \mathsf{u}⟺{\displaystyle \frac{1}{4}}\le \mathsf{v}\le \mathsf{u}\le {\displaystyle \frac{1}{3}}\mathrm{or}{\displaystyle \frac{1}{2}}\le \mathsf{v}\le \mathsf{u}\le 1.$$

Then, $(\mathbf{V},\delta )$ is an O-complete MS, although it is not complete. Consider the map $\zeta :\mathbf{V}\to \mathbf{V}$ defined by

$$\zeta \left(\mathsf{v}\right)=\left\{\begin{array}{c}{\displaystyle \frac{1}{4}}\mathrm{if}0\le \mathsf{v}<{\displaystyle \frac{1}{2}}\hfill \\ 1\mathrm{if}{\displaystyle \frac{1}{2}}\le \mathsf{v}\le 1.\hfill \end{array}\right.$$

Clearly, ζ is O-continuous but not continuous.

Definition 7

([17]). A O-set $(\mathbf{V},\perp )$ is denoted as transitive if ⊥ is a transitive relation, i.e.,

$$\mathsf{v},\mathsf{u},\mathsf{w}\in \mathbf{V};\mathsf{v}\perp \mathsf{u}\mathrm{and}\mathsf{u}\perp \mathsf{w}⟹\mathsf{v}\perp \mathsf{w}.$$

Recall that a self-map $\zeta$ on an MS $(\mathbf{V},\delta )$ is referred to as a Picard operator (in short: PO) if $Fix\left(\zeta \right)=\left\{{\mathsf{v}}^{*}\right\}$ and ${\zeta }^n\left(\mathsf{v}\right)\stackrel{\delta }{⟶}{\mathsf{v}}^{*}$, $\forall \mathsf{v}\in \mathbf{V}$. A variant of Theorem 1 in the framework of an orthogonal MS that appeared in work by Singh et al. [16] is indicated as follows:

Theorem 2

([16]). Let $(\mathbf{V},\perp ,\delta )$ be an O-complete MS. Suppose that $\zeta :\mathbf{V}\to \mathbf{V}$ is a ⊥-continuous and ⊥-preserving mapping. If there exists a right upper semi-continuous function $\phi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ verifying $\phi \left(t\right)<t$ for all $t>0$ with

$$\delta (\zeta \mathsf{v},\zeta \mathsf{u})\le \phi \left(\delta \right(\mathsf{v},\mathsf{u}\left)\right),\forall \mathsf{v},\mathsf{u}\in \mathbf{V}with\mathsf{v}\perp \mathsf{u},$$

(1)

then ζ enjoys a unique fixed point. Also, ζ is a PO.

Lemma 1

([18]). In an MS $(\mathbf{V},\delta )$, if $\left\{{\mathsf{v}}_n\right\}$ is a non-Cauchy sequence such that $\underset{n\to \infty }{lim}\delta ({\mathsf{v}}_n,{\mathsf{v}}_{n+1})=0$, then there ∃ a positive real ${ϵ}_0$ and the subsequences $\left\{{\mathsf{v}}_{n_k}\right\}$ and $\left\{{\mathsf{v}}_{l_k}\right\}$ that verify the following:

(i)

$k\le l_k<n_{k}foreveryk\in \mathbb{N}$,

(ii)

$\delta ({\mathsf{v}}_{l_k},{\mathsf{v}}_{n_k})>{ϵ}_{0}foreveryk\in \mathbb{N}$,

(iii)

$\delta ({\mathsf{v}}_{l_k},{\mathsf{v}}_{n_{k-1}})\le {ϵ}_{0}foreveryk\in \mathbb{N}$,

(iv)

$\underset{k\to \infty }{lim}\delta ({\mathsf{v}}_{l_k},{\mathsf{v}}_{n_k})={ϵ}_0,$

(v)

$\underset{k\to \infty }{lim}\delta ({\mathsf{v}}_{l_k},{\mathsf{v}}_{n_k+1})={ϵ}_0,$

(vi)

$\underset{k\to \infty }{lim}\delta ({\mathsf{v}}_{l_k+1},{\mathsf{v}}_{n_k})={ϵ}_0,$

(vii)

$\underset{k\to \infty }{lim}\delta ({\mathsf{v}}_{l_k+1},{\mathsf{v}}_{n_k+1})={ϵ}_0.$

3. Main Results

When proving Theorem 2, the authors of [16] utilized the transitivity of a O-set when they applied contractivity conditions on the pair ${\varsigma }_{m_k}$ and ${\varsigma }_{n_k}$, but failed to mention it in the hypotheses.

We now present the corrected version and an analog of Theorem 2 as follows:

Theorem 3.

Let $(\mathbf{V},\perp )$ be a transitive O-set with an orthogonal element ${\mathsf{v}}_0\in \mathbf{V}$ and δ be a metric on $\mathbf{V}$ such that $(\mathbf{V},\perp ,\delta )$ be an O-complete MS. Let $\zeta :\mathbf{V}\to \mathbf{V}$ be a ⊥-continuous and ⊥-preserving map such that ∃$\phi \in \Phi$ verifying the following:

$$\delta (\zeta \mathsf{v},\zeta \mathsf{u})\le \phi \left(\delta \right(\mathsf{v},\mathsf{u}\left)\right),\forall \mathsf{v},\mathsf{u}\in \mathbf{V}with\mathsf{v}\perp \mathsf{u}.$$

(2)

Then, ζ has a unique fixed point. Also, ζ is a PO.

Proof. 

Employing the definition of orthogonality, we conclude the following:

$$(\forall \mathsf{u}\in \mathbf{V},\mathsf{u}\perp {\mathsf{v}}_0)\mathrm{or}(\forall \mathsf{u}\in \mathbf{V},{\mathsf{v}}_0\perp \mathsf{u}).$$

(3)

It follows that

$${\mathsf{v}}_0\perp \zeta \left({\mathsf{v}}_0\right)\mathrm{or}\zeta \left({\mathsf{v}}_0\right)\perp {\mathsf{v}}_0.$$

(4)

Let us define the sequence $\left\{{\mathsf{v}}_n\right\}\subset \mathbf{V}$ as follows:

$${\mathsf{v}}_1=\zeta \left({\mathsf{v}}_0\right),{\mathsf{v}}_2=\zeta \left({\mathsf{v}}_1\right)={\zeta }^2\left({\mathsf{v}}_0\right),{\mathsf{v}}_3=\zeta \left({\mathsf{v}}_2\right)={\zeta }^3\left({\mathsf{v}}_0\right),\cdots$$

so that

$${\mathsf{v}}_n=\zeta \left({\mathsf{v}}_{n-1}\right)={\zeta }^n\left({\mathsf{v}}_0\right),\forall n\in \mathbb{N}.$$

(5)

As $\zeta$ is ⊥-preserving, (4) and (5) imply that

$${\mathsf{v}}_n\perp {\mathsf{v}}_{n+1}or{\mathsf{v}}_{n+1}\perp {\mathsf{v}}_n,\forall n\in \mathbb{N}.$$

(6)

It yields that $\left\{{\mathsf{v}}_n\right\}$ is an O-sequence.

We define ${\delta }_n:=\delta ({\mathsf{v}}_n,{\mathsf{v}}_{n+1})$. In the case where ${\delta }_{n_0}=\delta ({\mathsf{v}}_{n_0},{\mathsf{v}}_{n_0+1})=0$ (where $n_0\in {\mathbb{N}}_0$), the use of (5) tells us that $\zeta \left({\mathsf{v}}_{n_0}\right)={\mathsf{v}}_{n_0}$ and, hence, we have achieved our goal. Elsewhere, if ${\delta }_n>0,\mathrm{for}\mathrm{every}n\in {\mathbb{N}}_0$, then applying contraction-inequality (2) on (6), we obtain the following:

$$\begin{array}{ccc}\hfill {\delta }_n& =& \delta ({\mathsf{v}}_n,{\mathsf{v}}_{n+1})=\delta (\zeta {\mathsf{v}}_{n-1},\zeta {\mathsf{v}}_n)\hfill \\ & \le & \phi \left(\delta ({\mathsf{v}}_{n-1},{\mathsf{v}}_n)\right),\hfill \end{array}$$

so that

$$\begin{array}{c}\hfill {\delta }_n\le \phi \left({\delta }_{n-1}\right)\forall n\in {\mathbb{N}}_0.\end{array}$$

(7)

Made from (7) and the characteristic of $\phi$, we attain

$${\delta }_n\le \phi \left({\delta }_{n-1}\right)<{\delta }_{n-1},\forall n\in \mathbb{N}.$$

It follows that $\left\{{\delta }_n\right\}$ is a decreasing sequence in ${\mathbb{R}}^{+}\setminus \left\{0\right\}$; consequently, we find $l\ge 0$ that enjoys the following:

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

(8)

If possible, we assume that $l>0$. Considering the limits in (7) and (8), along with the characteristic of $\phi$, we attain the following:

$$l=\underset{n\to \infty }{lim}{\delta }_n\le \underset{n\to \infty }{lim}\phi \left({\delta }_{n-1}\right)=\underset{{\delta }_n\to l^{+}}{lim}\phi \left({\delta }_{n-1}\right)<l,$$

which leads to a contradiction. Thus, we conclude $l=0$, i.e.,

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}{\delta }_n=0.\end{array}$$

(9)

We shall show that $\left\{{\mathsf{v}}_n\right\}$ is Cauchy. On the contrary, if $\left\{{\mathsf{v}}_n\right\}$ is not Cauchy. Then, by Lemma 1, there exist there exist ${ϵ}_0>0$ and subsequences $\left\{{\mathsf{v}}_{n_k}\right\}$ and $\left\{{\mathsf{v}}_{l_k}\right\}$ of $\left\{{\mathsf{v}}_n\right\}$ satisfying the conclusions (i)–(vii). Let ${\delta }_k:=\delta ({\mathsf{v}}_{l_k},{\mathsf{v}}_{n_k})$. As $\left\{{\mathsf{v}}_n\right\}$ is a O-sequence and the O-set $(\mathbf{V},\perp )$ is a transitive, we conclude ${\mathsf{v}}_{l_k}\perp {\mathsf{v}}_{n_k}$. Utilizing the contraction-inequality (2), we attain the following:

$$\begin{array}{ccc}\hfill \delta ({\mathsf{v}}_{l_k+1},{\mathsf{v}}_{n_k+1})& =& \delta (\zeta {\mathsf{v}}_{l_k},\zeta {\mathsf{v}}_{n_k})\hfill \\ & \le & \phi \left(\delta ({\mathsf{v}}_{l_k},{\mathsf{v}}_{n_k})\right)\hfill \end{array}$$

so that

$$\delta ({\mathsf{v}}_{l_k+1},{\mathsf{v}}_{n_k+1})\le \phi \left({\delta }_k\right).$$

(10)

Proceeding with the limit in (10) and applying Lemma 1 along with the characteristics of $\phi$, we attain the following:

$${\displaystyle {ϵ}_0=\underset{k\to \infty }{lim}\delta ({\mathsf{v}}_{l_k+1},{\mathsf{v}}_{n_k+1})\le \underset{k\to \infty }{lim}\phi \left({\delta }_k\right)=\underset{s\to {ϵ}_0^{+}}{lim}\phi \left(s\right)<{ϵ}_0,}$$

which leads to a contradiction. Thus, $\left\{{\mathsf{v}}_n\right\}$ remains Cauchy and, hence, by the O-completeness of $\mathbf{V}$, there ∃ $\overline{\mathsf{v}}\in \mathbf{V}$ verifying ${\mathsf{v}}_n\stackrel{\delta }{⟶}\overline{\mathsf{v}}$.

Since $\zeta$ is ⊥-continuous, we have $\zeta \left({\mathsf{v}}_n\right)\stackrel{\delta }{⟶}\zeta \left(\overline{\mathsf{v}}\right)$. By (5), we conclude the following:

$$\zeta \left(\overline{\mathsf{v}}\right)=\underset{n\to \infty }{lim}\zeta \left({\mathsf{v}}_n\right)=\underset{n\to \infty }{lim}{\mathsf{v}}_{n+1}=\overline{\mathsf{v}}.$$

Hence, $\overline{\mathsf{v}}\in Fix\left(\zeta \right)$.

To prove the uniqueness portion, let $\overline{\mathsf{u}}\in Fix\left(\zeta \right)$. Thus, we attain

$$\begin{array}{c}\hfill {\zeta }^n\left(\overline{\mathsf{v}}\right)=\overline{\mathsf{v}}\mathrm{and}{\zeta }^n\left(\overline{\mathsf{u}}\right)=\overline{\mathsf{u}},\forall n\in \mathbb{N}.\end{array}$$

(11)

In view of (3), we obtain the following:

$$[{\mathsf{v}}_0\perp \overline{\mathsf{v}}\mathrm{and}{\mathsf{v}}_0\perp \overline{\mathsf{u}}]\mathrm{or}[\overline{\mathsf{v}}\perp {\mathsf{v}}_0\mathrm{and}\overline{\mathsf{u}}\perp {\mathsf{v}}_0].$$

As $\zeta$ is ⊥-preserving, for all $n\in \mathbb{N}$, we obtain the following:

$$[{\zeta }^n\left({\mathsf{v}}_0\right)\perp {\zeta }^n\left(\overline{\mathsf{v}}\right)\mathrm{and}{\zeta }^n\left({\mathsf{v}}_0\right)\perp {\zeta }^n\left(\overline{\mathsf{u}}\right)]$$

or

$$[{\zeta }^n\left(\overline{\mathsf{v}}\right)\perp {\zeta }^n\left({\mathsf{v}}_0\right)\mathrm{and}{\zeta }^n\left(\overline{\mathsf{u}}\right)\perp {\zeta }^n\left({\mathsf{v}}_0\right)].$$

We denote

$$P_n:=\delta ({\zeta }^n\overline{\mathsf{v}},{\zeta }^n{\mathsf{v}}_0)\mathrm{and}{Q}_n:=\delta ({\zeta }^n\overline{\mathsf{u}},{\zeta }^n{\mathsf{v}}_0).$$

We shall show that

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}{P}_n=\underset{n\to \infty }{lim}{Q}_n=0.\end{array}$$

(12)

If for some $n_0\in {\mathbb{N}}_0$, $P_{n_0}=\delta ({\zeta }^{n_0}\overline{\mathsf{v}},{\zeta }^{n_0}{\mathsf{v}}_0)=0$, we have ${\zeta }^{n_0}\left(\overline{\mathsf{v}}\right)={\zeta }^{n_0}\left({\mathsf{v}}_0\right)$, which yields that ${\zeta }^{n_0+1}\left(\overline{\mathsf{v}}\right)={\zeta }^{n_0+1}\left({\mathsf{v}}_0\right)$ and, hence, we obtain $P_{n_0+1}=\delta ({\zeta }^{n_0+1}\overline{\mathsf{v}},{\zeta }^{n_0+1}{\mathsf{v}}_0)=0$. Using induction, we attain $P_n=0$ for every $n\ge n_0,$ implying $\underset{n\to \infty }{lim}{P}_n=0$. Otherwise, we have the following: $P_n>0$, for all $n\in {\mathbb{N}}_0$. In this case, using the contraction-inequality (2), we obtain the following:

$$\begin{array}{ccc}\hfill P_{n+1}& =& \delta ({\zeta }^{n+1}\overline{\mathsf{v}},{\zeta }^{n+1}{\mathsf{v}}_0)\hfill \\ \hfill & \le & \phi \left(\delta ({\zeta }^n\overline{\mathsf{v}},{\zeta }^n{\mathsf{v}}_0)\right)\hfill \\ & =& \phi \left(P_n\right)\hfill \\ \hfill & <& P_n.\hfill \end{array}$$

(13)

Proceeding the earlier approach, the above inequality leads to the following:

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

Similarly, we can show that

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

Hence, (12) is proved. Using the triangle inequality along with (11) and (12), we obtain the following:

$$\begin{array}{ccc}\hfill \delta (\overline{\mathsf{v}},\overline{\mathsf{u}})& =& \delta ({\zeta }^n\overline{\mathsf{v}},{\zeta }^n\overline{\mathsf{u}})\hfill \\ & \le & P_n+Q_n\hfill \\ & \to & 0\mathrm{as}n\to \infty \hfill \end{array}$$

implying $\overline{\mathsf{u}}=\overline{\mathsf{v}}$; so $Fix\left(\zeta \right)=\left\{\overline{\mathsf{v}}\right\}$.

Finally, let $\mathsf{v}\in \mathbf{V}$ be arbitrary. Similar to earlier, we have the following:

$$[{\mathsf{v}}_0\perp \overline{\mathsf{v}}\mathrm{and}{\mathsf{v}}_0\perp \mathsf{v}]\mathrm{or}[\overline{\mathsf{v}}\perp {\mathsf{v}}_0\mathrm{and}\mathsf{v}\perp {\mathsf{v}}_0]$$

so that we have the following:

$$[{\zeta }^n\left({\mathsf{v}}_0\right)\perp {\zeta }^n\left(\overline{\mathsf{v}}\right)\mathrm{and}{\zeta }^n\left({\mathsf{v}}_0\right)\perp {\zeta }^n\left(\mathsf{v}\right)]$$

(14)

or

$$[{\zeta }^n\left(\overline{\mathsf{v}}\right)\perp {\zeta }^n\left({\mathsf{v}}_0\right)\mathrm{and}{\zeta }^n\left(\mathsf{v}\right)\perp {\zeta }^n\left({\mathsf{v}}_0\right)].$$

(15)

Now, for each $n\in \mathbb{N}$, we claim the following:

$$\begin{array}{ccc}\hfill \delta ({\zeta }^{n+1}\overline{\mathsf{v}},{\zeta }^{n+1}{\mathsf{v}}_0)& \le & \delta ({\zeta }^n\overline{\mathsf{v}},{\zeta }^n{\mathsf{v}}_0)\hfill \\ \hfill \mathrm{and}\end{array}$$

(16)

$$\begin{array}{ccc}\hfill \delta ({\zeta }^{n+1}\mathsf{v},{\zeta }^{n+1}{\mathsf{v}}_0)& \le & \delta ({\zeta }^n\mathsf{v},{\zeta }^n{\mathsf{v}}_0).\hfill \end{array}$$

(17)

If $\delta ({\zeta }^k\mathsf{v},{\zeta }^k{\mathsf{v}}_0)=0$ (for some $k\in \mathbb{N}$), then we have ${\zeta }^k\left(\mathsf{v}\right)={\zeta }^k\left({\mathsf{v}}_0\right)$, implying ${\zeta }^{k+1}\left(\mathsf{v}\right)={\zeta }^{k+1}\left({\mathsf{v}}_0\right)$ so that we have the following: $\delta ({\zeta }^{k+1}\overline{\mathsf{v}},{\zeta }^{k+1}{\mathsf{v}}_0)=0$ and, hence, (16) (similarly (17) also) holds for such $k\in \mathbb{N}$. In either case, employing contraction conditions (2) to (14) and (15), and by the characteristic of $\phi$, we obtain the following:

$$\begin{array}{c}\hfill \delta ({\zeta }^{n+1}\overline{\mathsf{v}},{\zeta }^{n+1}{\mathsf{v}}_0)\le \phi \left(\delta ({\zeta }^n\overline{\mathsf{v}},{\zeta }^n{\mathsf{v}}_0)\right)<\delta ({\zeta }^n\overline{\mathsf{v}},{\zeta }^n{\mathsf{v}}_0)\end{array}$$

and

$$\begin{array}{c}\hfill \delta ({\zeta }^{n+1}\mathsf{v},{\zeta }^{n+1}{\mathsf{v}}_0)\le \phi \left(\delta ({\zeta }^n\mathsf{v},{\zeta }^n{\mathsf{v}}_0)\right)<\delta ({\zeta }^n\mathsf{v},{\zeta }^n{\mathsf{v}}_0).\end{array}$$

Thus, (16) and (17) hold. It follows that $\left\{\delta ({\zeta }^n\overline{\mathsf{v}},{\zeta }^n{\mathsf{v}}_0)\right\}$ and $\left\{\delta ({\zeta }^n\mathsf{v},{\zeta }^n{\mathsf{v}}_0)\right\}$ are decreasing sequences of positive reals, which are also bounded below. Proceeding on the lines as described earlier, we conclude the following:

$$\underset{n\to \infty }{lim}\delta ({\zeta }^n\overline{\mathsf{v}},{\zeta }^n{\mathsf{v}}_0)=\underset{n\to \infty }{lim}\delta ({\zeta }^n\mathsf{v},{\zeta }^n{\mathsf{v}}_0)=0.$$

Using the above and the triangle inequality, we obtain the following:

$$\begin{array}{ccc}\hfill \delta (\overline{\mathsf{v}},{\zeta }^{n+1}\mathsf{v})& =& \delta ({\zeta }^{n+1}\overline{\mathsf{v}},{\zeta }^{n+1}\mathsf{v})\hfill \\ \hfill & \le & \delta ({\zeta }^{n+1}\overline{\mathsf{v}},{\zeta }^{n+1}{\mathsf{v}}_0)+\delta ({\zeta }^{n+1}{\mathsf{v}}_0,{\zeta }^{n+1}\mathsf{v})\hfill \\ \hfill & \to & 0\mathrm{as}n\to \infty \hfill \end{array}$$

so that ${\zeta }^n\left(\mathsf{v}\right)\stackrel{\delta }{⟶}\overline{\mathsf{v}}$. It follows that $\zeta$ is a Picard operator. □

Remark 2.

If ⊥ is taken as a partially ordered relation, then Theorem 3 deduces Theorem 2.1 of Wu and Liu [13] and Theorem 10 of Karapinar et al. [14].

Remark 3.

Under the constraint $\phi \left(t\right)=rt,0<r<1$, Theorem 3 deduces Theorem 3.11 of Gordji et al. [15]. However, in this case, we do not need the transitivity requirement on the O-set.

4. Illustrative Examples

To demonstrate Theorem 3, we take into account the subsequent examples.

Example 1.

Let $\mathbf{V}={\mathbb{R}}^{+}$ with the usual metric δ. Let $\zeta :\mathbf{V}\to \mathbf{V}$ be a map defined by $\zeta \left(\mathsf{v}\right)={\displaystyle \frac{\mathsf{v}}{\mathsf{v}+1}}$. We define a binary relation, such that $\mathsf{v}\perp \mathsf{u}⟺\mathsf{v}-\mathsf{u}>0$. Then, $(\mathbf{V},\perp )$ is a transitive O-set, ζ is ⊥-continuous, and $(\mathbf{V},\perp ,\delta )$ is an O-complete MS. We define a function $\phi \in \Phi$ by $\phi \left(t\right)={\displaystyle \frac{t}{t+1}}$. Now, for $\mathsf{v}\perp \mathsf{u}$, we conclude the following:

$$\begin{array}{ccc}\hfill \delta (\zeta \mathsf{v},\zeta \mathsf{u})& =& |{\displaystyle \frac{\mathsf{v}}{\mathsf{v}+1}}-{\displaystyle \frac{\mathsf{u}}{\mathsf{u}+1}}|=\left|{\displaystyle \frac{\mathsf{v}-\mathsf{u}}{1+\mathsf{v}+\mathsf{u}+\mathsf{v}\mathsf{u}}}\right|\hfill \\ & \le & {\displaystyle \frac{\mathsf{v}-\mathsf{u}}{1+(\mathsf{v}-\mathsf{u})}}={\displaystyle \frac{\delta (\mathsf{v},\mathsf{u})}{1+\delta (\mathsf{v},\mathsf{u})}}\hfill \\ & =& \phi \left(\delta \right(\mathsf{v},\mathsf{u}\left)\right).\hfill \end{array}$$

Therefore, the contraction-inequality (2) is satisfied. Consequently, in lieu of Theorem 3, ζ has a unique fixed point, i.e., $\overline{\mathsf{v}}=0$.

Remark 4.

In Example 1, the relation ⊥ is irreflexive and, hence, it is not a partial ordering. Therefore, Example 1 cannot be covered by the outcomes of Wu and Liu [13] and Karapinar et al. [14].

Example 2.

Consider $\mathbf{V}=[0,1]$ with Euclidean metric δ. Let $\zeta :\mathbf{V}\to \mathbf{V}$ be a map defined by

$$\zeta \left(\mathsf{v}\right)=\left\{\begin{array}{c}{\displaystyle \frac{1}{2}},\mathrm{if}0\le \mathsf{v}<{\displaystyle \frac{1}{2}}\hfill \\ 1,\mathrm{if}{\displaystyle \frac{1}{2}}\le \mathsf{v}\le 1.\hfill \end{array}\right.$$

We define a binary relation, i.e., $\mathsf{v}\perp \mathsf{u}⟺{\displaystyle \frac{3}{4}}\le \mathsf{v}\le \mathsf{u}\le 1$. Then, $(\mathbf{V},\perp )$ is a transitive O-set, ζ is ⊥-continuous, and $(\mathbf{V},\perp ,\delta )$ is an O-complete MS. Define a function $\phi \in \Phi$ by

$$\phi \left(t\right)=\left\{\begin{array}{c}{\displaystyle \frac{t}{2}},\mathrm{if}0\le t<1\hfill \\ 0,\mathrm{if}t=1\hfill \\ t-{\displaystyle \frac{1}{2}}\mathrm{if}t>1.\hfill \end{array}\right.$$

It can be easily verified that the contraction inequality (2) is satisfied for φ. Consequently, ζ enjoys a unique fixed point, i.e., $\overline{\mathsf{v}}=1$.

Remark 5.

In Example 2, φ is not right upper semi-continuous at $t=1$ as

$$\underset{s\to 1+}{lim}\phi \left(s\right)=1/2>0=\phi \left(1\right).$$

Therefore, Example 2 cannot be covered by Theorem 2. This demonstrates the utility of our outcomes over the corresponding findings of Wu and Liu [13], Karapinar et al. [14], and Singh et al. [16].

Example 3.

Let $\mathbf{V}=[0,1)$ with Euclidean metric δ. Let $\zeta :\mathbf{V}\to \mathbf{V}$ be a map defined by

$$\zeta \left(\mathsf{v}\right)=\left\{\begin{array}{c}{\displaystyle \frac{\mathsf{v}}{2}},\mathrm{if}\mathsf{v}\in \mathbb{Q}\cap \mathbf{V}\hfill \\ 0,\mathrm{if}\mathsf{v}\in {\mathbb{Q}}^c\cap \mathbf{V}.\hfill \end{array}\right.$$

We define a binary relation: $\mathsf{v}\perp \mathsf{u}⟺\mathsf{v}\mathsf{u}\in \{\mathsf{v},\mathsf{u}\}$. Then $(\mathbf{V},\perp )$ is a transitive O-set, ζ is ⊥-continuous, and $(\mathbf{V},\perp ,\delta )$ is an O-complete MS. Define a function $\phi \in \Phi$ by $\phi \left(t\right)={\displaystyle \frac{1}{2}}t$. Then, it can be easily verified that ζ is an orthogonal φ-contraction map. Consequently, in lieu of Theorem 3, ζ has a unique fixed point: $\overline{\mathsf{v}}=0$.

Remark 6.

The map involved in Example 3 is not a φ-contraction. Specifically, for the pair $\mathsf{v}=0$ and $\mathsf{u}={\displaystyle \frac{1}{\sqrt{2}}}$, we have

$$\delta (\zeta \mathsf{v},\zeta \mathsf{u})=0>{\displaystyle \frac{1}{\sqrt{2}}}=\delta (\mathsf{v},\mathsf{u}).$$

Thus far, Example 3 demonstrates that it does not function within the context of ordinary MS, which substantiates the utility of fixed-point outcomes in an orthogonal MS over the corresponding outcomes in an ordinary MS.

5. An Application to BVP

Consider the BVP:

$$\left\{\begin{array}{c}{\mathsf{v}}^{\prime }\left(\theta \right)=\mathsf{ϝ}(\theta ,\mathsf{v}\left(\theta \right)),\theta \in [0,L]\hfill \\ \mathsf{v}\left(0\right)=\mathsf{v}\left(L\right)\hfill \end{array}\right.$$

(18)

where $\mathsf{ϝ}\in \mathcal{C}\left(\right[0,L]\times \mathbb{R})$. Recall that a function $\tilde{\mathsf{v}}\in {\mathcal{C}}^{\prime }[0,L]$ is said to form a lower solution of (18) (c.f. [9]) if

$$\left\{\begin{array}{c}{\tilde{\mathsf{v}}}^{\prime }\left(\theta \right)\le \mathsf{ϝ}(\theta ,\tilde{\mathsf{v}}\left(\theta \right)),\theta \in [0,L]\hfill \\ \tilde{\mathsf{v}}\left(0\right)\le \tilde{\mathsf{v}}\left(L\right).\hfill \end{array}\right.$$

In the sequel, we consider the following subfamily of $\Phi$:

$$\tilde{\Phi }=\{\phi \in \Phi :\phi \mathrm{is}\mathrm{monotone}-\mathrm{increasing}\}.$$

In the following lines, one establishes the existence and uniqueness theorem to determine a solution of Problem (18).

Theorem 4.

Along with Problem (18), if there exist $\lambda >0$ and $\phi \in \tilde{\Phi }$ satisfying $\forall a,b\in \mathbb{R}$ with $a\le b$ that

$$0\le \left[\mathsf{ϝ}\right(\theta ,b)+\lambda b]-\left[\mathsf{ϝ}\right(\theta ,a)+\lambda a]\le \lambda \phi (b-a),$$

(19)

then the existence of a lower solution of Problem (18) ensures the existence of the unique solution of the problem.

Proof. 

Problem (18) is as follows:

$$\left\{\begin{array}{c}{\mathsf{v}}^{\prime }\left(\theta \right)+\lambda \mathsf{v}\left(\theta \right)=\mathsf{ϝ}(\theta ,\mathsf{v}\left(\theta \right))+\lambda \mathsf{v}\left(\theta \right),\forall \theta \in [0,L]\hfill \\ \mathsf{v}\left(0\right)=\mathsf{v}\left(L\right).\hfill \end{array}\right.$$

(20)

Clearly, (20) is equivalent to the following integral equation:

$$\mathsf{v}\left(\theta \right)={\int }_0^{L}G(\theta ,\xi )[\mathsf{ϝ}(\xi ,\mathsf{v}\left(\xi \right))+\lambda \mathsf{v}\left(\xi \right)]d\xi$$

(21)

where $G(\theta ,\xi )$ remains the Green function given by the following:

$$G(\theta ,\xi )=\left\{\begin{array}{c}{\displaystyle \frac{e^{\lambda (L+\xi -\theta )}}{e^{\lambda L}-1}},0\le \xi <\theta \le L\hfill \\ {\displaystyle \frac{e^{\lambda (\xi -\theta )}}{e^{\lambda L}-1}},0\le \theta <\xi \le L.\hfill \end{array}\right.$$

Denote $\mathbf{V}:=\mathcal{C}[0,L]$. We define a map $\zeta :\mathbf{V}\to \mathbf{V}$ by

$$\left(\zeta \mathsf{v}\right)\left(\theta \right)={\int }_0^{L}G(\theta ,\xi )[\mathsf{ϝ}(\xi ,\mathsf{v}\left(\xi \right))+\lambda \mathsf{v}\left(\xi \right)]d\xi ,\forall \theta \in [0,L].$$

(22)

On $\mathbf{V}$, we define a metric $\delta$ by the following:

$$\delta (\mathsf{v},\mathsf{u})=\underset{\theta \in [0,L]}{sup}|\mathsf{v}\left(\theta \right)-\mathsf{u}\left(\theta \right)|,\forall \mathsf{v},\mathsf{u}\in \mathbf{V}.$$

(23)

On $\mathbf{V}$, we comprise a relation ⊥ by the following:

$$\mathsf{v}\perp \mathsf{u}⟺\mathsf{v}\left(\theta \right)\le \mathsf{u}\left(\theta \right),\forall \theta \in [0,L].$$

(24)

Clearly, $(\mathbf{V},\perp )$ is a transitive O-set and $(\mathbf{V},\perp ,\delta )$ is an O-complete MS. Also, $\zeta$ is a ⊥-continuous map. Take $\mathsf{v}\perp \mathsf{u}$. By (19), we obtain the following:

$$\mathsf{ϝ}(\theta ,\mathsf{v}(\theta \left)\right)+\lambda \mathsf{v}\left(\theta \right)\le \mathsf{ϝ}(\theta ,\mathsf{u}(\theta \left)\right)+\lambda \mathsf{u}\left(\theta \right),\forall \theta \in [0,L].$$

(25)

Using (22), (25), and $G(\theta ,\xi )>0$, $\forall \theta ,\xi \in [0,L]$, we find the following:

$$\begin{array}{ccc}\hfill \left(\zeta \mathsf{v}\right)\left(\theta \right)& =& {\int }_0^{L}G(\theta ,\xi )[\mathsf{ϝ}(\xi ,\mathsf{v}\left(\xi \right))+\lambda \mathsf{v}\left(\xi \right)]d\xi \hfill \\ & \le & {\int }_0^{L}G(\theta ,\xi )[\mathsf{ϝ}(\xi ,\mathsf{u}\left(\xi \right))+\lambda \mathsf{u}\left(\xi \right)]d\xi \hfill \\ & =& \left(\zeta \mathsf{u}\right)\left(\theta \right),\forall \theta \in [0,L],\hfill \end{array}$$

where using (24) implies that $\zeta \left(\mathsf{v}\right)\perp \zeta \left(\mathsf{u}\right)$ and, hence, $\zeta$ is a ⊥-preserving map.

If $\tilde{\mathsf{v}}\in {\mathcal{C}}^{\prime }[0,L]$ remains a lower solution of (18), then we have the following:

$${\tilde{\mathsf{v}}}^{\prime }\left(\theta \right)+\lambda \tilde{\mathsf{v}}\left(\theta \right)\le \mathsf{ϝ}(\theta ,\tilde{\mathsf{v}}\left(\theta \right))+\lambda \tilde{\mathsf{v}}\left(\theta \right),\forall \theta \in [0,L].$$

Taking the product with $e^{\lambda \theta }$, we attain the following:

$${\left(\tilde{\mathsf{v}}\left(\theta \right)e^{\lambda \theta }\right)}^{\prime }\le [\mathsf{ϝ}(\theta ,\tilde{\mathsf{v}}\left(\theta \right))+\lambda \tilde{\mathsf{v}}\left(\theta \right)]e^{\lambda \theta },\forall \theta \in [0,L]$$

implying the following:

$$\tilde{\mathsf{v}}\left(\theta \right)e^{\lambda \theta }\le \tilde{\mathsf{v}}\left(0\right)+{\int }_0^{\theta }[\mathsf{ϝ}(\xi ,\tilde{\mathsf{v}}\left(\xi \right))+\lambda \tilde{\mathsf{v}}\left(\xi \right)]e^{\lambda \xi }d\xi ,\forall \theta \in [0,L].$$

(26)

Owing to $\tilde{\mathsf{v}}\left(0\right)\le \tilde{\mathsf{v}}\left(L\right)$, one obtains the following:

$$\tilde{\mathsf{v}}\left(0\right)e^{\lambda L}\le \tilde{\mathsf{v}}\left(L\right)e^{\lambda L}\le \tilde{\mathsf{v}}\left(0\right)+{\int }_0^L[\mathsf{ϝ}(\xi ,\tilde{\mathsf{v}}\left(\xi \right))+\lambda \tilde{\mathsf{v}}\left(\xi \right)]e^{\lambda \xi }d\xi$$

so that we have the following:

$$\tilde{\mathsf{v}}\left(0\right)\le {\int }_0^L{\displaystyle \frac{e^{\lambda \xi }}{e^{\lambda L}-1}}[\mathsf{ϝ}(\xi ,\tilde{\mathsf{v}}\left(\xi \right))+\lambda \tilde{\mathsf{v}}\left(\xi \right)]d\xi .$$

(27)

By (26) and (27), one finds the following:

$$\begin{array}{ccc}\hfill \tilde{\mathsf{v}}\left(\theta \right)e^{k\theta }& \le & {\int }_0^L{\displaystyle \frac{e^{\lambda \xi }}{e^{\lambda L}-1}}[\mathsf{ϝ}(\xi ,\tilde{\mathsf{v}}\left(\xi \right))+\lambda \tilde{\mathsf{v}}\left(\xi \right)]d\xi +{\int }_0^{\theta }{e}^{\lambda \xi }[\mathsf{ϝ}(\xi ,\tilde{\mathsf{v}}\left(\xi \right))+\lambda \tilde{\mathsf{v}}\left(\xi \right)]d\xi \hfill \\ & =& {\int }_0^{\theta }{\displaystyle \frac{e^{\lambda (L+\xi )}}{e^{\lambda L}-1}}[\mathsf{ϝ}(\xi ,\tilde{\mathsf{v}}\left(\xi \right))+\lambda \tilde{\mathsf{v}}\left(\xi \right)]d\xi +{\int }_{\theta }^L{\displaystyle \frac{e^{\lambda \xi }}{e^{\lambda L}-1}}[\mathsf{ϝ}(\xi ,\tilde{\mathsf{v}}\left(\xi \right))+\lambda \tilde{\mathsf{v}}\left(\xi \right)]d\xi \hfill \end{array}$$

so that we have the following:

$$\begin{array}{ccc}\hfill \tilde{\mathsf{v}}\left(\theta \right)& \le & {\int }_0^{\theta }{\displaystyle \frac{e^{\lambda (L+\xi -\theta )}}{e^{\lambda L}-1}}[\mathsf{ϝ}(\xi ,\tilde{\mathsf{v}}\left(\xi \right))+\lambda \tilde{\mathsf{v}}\left(\xi \right)]d\xi +{\int }_{\theta }^L{\displaystyle \frac{e^{\lambda (\xi -\theta )}}{e^{\lambda L}-1}}[\mathsf{ϝ}(\xi ,\tilde{\mathsf{v}}\left(\xi \right))+\lambda \tilde{\mathsf{v}}\left(\xi \right)]d\xi \hfill \\ & =& {\int }_0^{L}G(\theta ,\xi )[\mathsf{ϝ}(\xi ,\tilde{\mathsf{v}}\left(\xi \right))+\lambda \tilde{\mathsf{v}}\left(\xi \right)]d\xi \hfill \\ & =& \left(\zeta \tilde{\mathsf{v}}\right)\left(\theta \right),\forall \theta \in [0,L]\hfill \end{array}$$

which implies that $\tilde{\mathsf{v}}\perp \zeta \left(\tilde{\mathsf{v}}\right)$. Hence, $\tilde{\mathsf{v}}\in \mathbf{V}$ is an orthogonal element.

Take $\mathsf{v}\perp \mathsf{u}$. Then, by (19), (22), and (23), one has the following:

$$\begin{array}{ccc}\hfill \delta (\zeta \mathsf{v},\zeta \mathsf{u})& =& \underset{\theta \in [0,L]}{sup}|\left(\zeta \mathsf{v}\right)\left(\theta \right)-\left(\zeta \mathsf{u}\right)\left(\theta \right)|=\underset{\theta \in [0,L]}{sup}\left(\left(\zeta \mathsf{u}\right)\left(\theta \right)-\left(\zeta \mathsf{v}\right)\left(\theta \right)\right)\hfill \\ \hfill & \le & \underset{\theta \in [0,L]}{sup}{\int }_0^{L}G(\theta ,\xi )[\mathsf{ϝ}(\xi ,\mathsf{u}\left(\xi \right))+\lambda \mathsf{u}\left(\xi \right)-\mathsf{ϝ}(\xi ,\mathsf{v}\left(\xi \right))-\lambda \mathsf{v}\left(\xi \right)]d\xi \hfill \\ & \le & \underset{\theta \in [0,L]}{sup}{\int }_0^{L}G(\theta ,\xi )\lambda \phi (\mathsf{u}\left(\xi \right)-\mathsf{v}\left(\xi \right))d\xi .\hfill \end{array}$$

(28)

Now, $0\le \mathsf{u}\left(\xi \right)-\mathsf{v}\left(\xi \right)\le \delta (\mathsf{v},\mathsf{u})$. Employing the monotonic property of $\phi$, we obtain $\phi \left(\mathsf{u}\right(\xi )-\mathsf{v}(\xi \left)\right)\le \phi \left(\delta \right(\mathsf{v},\mathsf{u}\left)\right)$ and, hence, (28) is as follows:

$$\begin{array}{ccc}\hfill \delta (\zeta \mathsf{v},\zeta \mathsf{u})& \le & \lambda \phi \left(\delta (\mathsf{v},\mathsf{u})\right)\underset{\theta \in [0,L]}{sup}{\int }_0^{L}G(\theta ,\xi )d\xi \hfill \\ & =& \lambda \phi \left(\delta (\mathsf{v},\mathsf{u})\right)\underset{\theta \in [0,L]}{sup}{\displaystyle \frac{1}{e^{\lambda L}-1}}\left[{\displaystyle \frac{1}{\lambda }}{e}^{\lambda (L+\xi -\theta )}{|}_0^{\theta }+{\displaystyle \frac{1}{\lambda }}{e}^{\lambda (\xi -\theta )}{|}_{\theta }^L\right]\hfill \\ & =& \lambda \phi \left(\delta (\mathsf{v},\mathsf{u})\right){\displaystyle \frac{1}{\lambda (e^{\lambda L}-1)}}(e^{\lambda L}-1)\hfill \\ & =& \phi \left(\delta \right(\mathsf{v},\mathsf{u}\left)\right)\hfill \end{array}$$

so that

$$\delta (\zeta \mathsf{v},\zeta \mathsf{u})\le \phi \left(\delta \right(\mathsf{v},\mathsf{u}\left)\right),\forall \mathsf{v}\perp \mathsf{u}.$$

Therefore, by Theorem 3, $\zeta$ has a unique fixed point, which forms the unique solution to Problem (18). □

6. Conclusions

In the foregoing text, we have addressed certain outcomes on fixed points of Boyd–Wong-type contractions in the framework of orthogonally complete MSs. The findings presented here improve upon several existing results, particularly those by Boyd and Wong [2], Gordji et al. [15], Wu and Liu [13], Karapinar et al. [14] and Singh et al. [16]. We have also introduced a few illustrative examples that demonstrate the superiority of our findings over the corresponding existing ones.

The concept of $\phi$-contractions was further generalized and developed by Dutta and Choudhury [19], Đorić [20], Popescu [21], Fallahi et al. [22], and others, who introduced the notion of $(\psi ,\varphi )$-contractions that depend on two auxiliary functions. In future work, we can further extend Theorem 3 to accommodate $(\psi ,\varphi )$-contractions within orthogonally complete MSs. We exhibited the existence of a unique solution for a BVP in cases where there is evidence of a lower solution by applying Theorem 3. Likewise, an analogous outcome may be demonstrated when an upper solution is available.