Relational Strict Almost Contractions Employing Test Functions and an Application to Nonlinear Integral Equations

Abstract

The present study deals with some fixed-point outcomes under a nonlinear formulation of strict almost contractions in a metric space endued with an arbitrary relation. The outcomes established herein enhance and develop various existing outcomes. To convince you of the infallibility of our outcomes, a few examples are presented. We apply our findings to investigate the validity of the unique solution of a nonlinear integral problem.

1. Introduction

An essential and significant outcome of the theory of metrical fixed points is the Banach contraction principle (BCP). Indeed, BCP guarantees that a contraction on a complete metric space (MS) owns a unique fixed point. This outcome additionally offers a way to estimate the unique fixed point. Many generalizations of the BCP can be found in the literature. BCP and its generalizations are applicable for solving many real world problems (c.f. [1,2]).

A novel and straightforward version of BCP was presented in 2015 by Alam and Imdad [3]. In this version, the metric space is supplied with a relation, and the map preserves this relation. Because of creativity, the findings of Alam and Imdad [3] has been extended and refined by plenty of researchers, e.g., [4,5,6,7,8,9,10,11,12,13,14]. One of the main characteristics of relational contractions is that the contraction-inequality must be satisfied for comparative elements only, not for every pair of elements. It turns out that relational contractions turn out to be weaker than corresponding ordinary contractions and, therefore, they can be used to solve certain types of integral equations and boundary value problems; meanwhile, the outcomes on fixed points of ordinary metric space are not.

In 2004, Berinde [15] implemented a new generalization of BCP, referred to as almost contraction.

Definition 1

([15]). A self-mapping $\mathbf{P}$ on a MS $(\mathbf{Z},\varrho )$ is named an almost contraction if there exist $0<\gamma <1$ and $K\ge 0$, enjoying

$\varrho (\mathbf{P}z,\mathbf{P}\mathsf{w})\le \gamma \varrho (z,\mathsf{w})+K\varrho (z,\mathbf{P}\mathsf{w}),\mathrm{for}\mathrm{all}z,\mathsf{w}\in \mathbf{Z}.$

Owing to symmetry of $\varrho$, the preceding condition is identical to the following one:

$\varrho (\mathbf{P}z,\mathbf{P}\mathsf{w})\le \gamma \varrho (z,\mathsf{w})+K\varrho (w,\mathbf{P}\mathsf{z}),\mathrm{for}\mathrm{all}z,\mathsf{w}\in \mathbf{Z}.$

Theorem 1

([15]). An almost contraction on a complete MS possesses a fixed point.

The concept of almost contractions has been developed by a number of researchers; for instance, see [16,17,18,19,20,21,22,23,24]. The almost contraction is not uniquely fixed, but Picard’s iterative sequence continues to converge to a fixed point. To formulate a uniqueness theorem, Babu et al. [19] formulated a significantly limited class of almost contractions.

Theorem 2

([19]). Let $\mathbf{P}$ be a self-map on a complete MS $(\mathbf{Z},\varrho )$. If there exist $0<\gamma <1$ and $K\ge 0$, enjoying

$\varrho (\mathbf{P}z,\mathbf{P}\mathsf{w})\le \gamma \varrho (z,\mathsf{w})+Kmin\left\{\varrho \right(z,\mathbf{P}\mathsf{w}),\varrho (w,\mathbf{P}\mathsf{z}),\varrho (z,\mathbf{P}\mathsf{z}),\varrho (w,\mathbf{P}\mathsf{w}\left)\right\},\mathrm{for}\mathrm{all}z,\mathsf{w}\in \mathbf{Z},$

then $\mathbf{P}$ possesses a unique fixed point.

With influence from Berinde [15], Turinici [25] proposed the class $\Psi$ of functions $\psi :[0,+\infty )\to [0,+\infty )$ that verifies

$\psi \left(0\right)=0and\underset{r\to 0}{lim}\psi \left(r\right)=0.$

This article deals with the following nonlinear variant of contraction-inequality utilized in Theorem 2 under a binary relation:

$\begin{array}{c}\hfill \varrho (\mathbf{P}z,\mathbf{P}\mathsf{w})\le \gamma \varrho (z,\mathsf{w})+min\left\{\psi \right(\varrho (\mathsf{z},\mathbf{P}w)),\psi (\varrho (\mathsf{w},\mathbf{P}z)),\psi (\varrho (\mathsf{z},\mathbf{P}z)),\psi (\varrho (\mathsf{w},\mathbf{P}w)\left)\right\}.\end{array}$

We incorporate a few examples to demonstrate our findings. By employing our findings, we carry out the unique solution of nonlinear integral equations, verifying some additional hypotheses.

2. Relation-Theoretic Notions

As usual, the sets $\mathbb{R}$, $\mathbb{N}$, and ${\mathbb{N}}_0$ will denote, respectively, the sets of: real numbers, natural numbers, and whole numbers. A subset of ${\mathbf{Z}}^2$ is defined as a relation ℜ on a set $\mathbf{Z}$. In the concepts that follow, $\mathbf{Z}$ will be considered the ambient set, $\varrho$ will be a metric on $\mathbf{Z}$, ℜ will be a relation on $\mathbf{Z}$, and $\mathbf{P}:\mathbf{Z}\to \mathbf{Z}$ will be a mapping.

Definition 2

([3]). $\mathsf{z},\mathsf{w}\in \mathbf{Z}$ are referred to as ℜ-comparative, symbolized by $[\mathsf{z},\mathsf{w}]\in \Re$, if

$(\mathsf{z},\mathsf{w})\in \Re or(\mathsf{w},\mathsf{z})\in \Re .$

Definition 3

([26]). ${\Re }^{-1}:=\{(\mathsf{z},\mathsf{w})\in {\mathbf{Z}}^2:(\mathsf{w},\mathsf{z})\in \Re \}$ is referred to as the inverse of ℜ.

Definition 4

([26]). ${\Re }^s:=\Re \cup {\Re }^{-1}$ is referred to as the symmetric closure of ℜ.

Proposition 1

([3]). $(\mathsf{z},\mathsf{w})\in {\Re }^s\iff [\mathsf{z},\mathsf{w}]\in \Re$.

Definition 5

([3]). ℜ is referred to as $\mathbf{P}$-closed if

$(\mathsf{z},\mathsf{w})\in \Re \Rightarrow (\mathbf{P}\mathsf{z},\mathbf{P}\mathsf{w})\in \Re .$

For example, if we consider a self-map $\mathbf{P}$ on the partially ordered set $(\mathbf{Z},⪯)$, then the ‘$\mathbf{P}$-closedness of relation ⪯’ is equivalent to saying that $\mathbf{P}$ is ⪯-increasing.

Proposition 2

( [5]). ℜ is ${\mathbf{P}}^n$-closed for every $n\in \mathbb{N}$, whenever ℜ is $\mathbf{P}$-closed.

Definition 6

([27]). $\mathbf{W}\subseteq \mathbf{Z}$ is referred to as ℜ-directed set if, for every $\mathsf{z},\mathsf{w}\in \mathbf{W}$, $\mathrm{there}\mathrm{exists}$$\mathsf{u}\in \mathbf{Z}$ with $(\mathsf{z},\mathsf{u})\in \Re$ and $(\mathsf{w},\mathsf{u})\in \Re$.

Definition 7

([3]). A sequence $\left\{{\mathsf{z}}_n\right\}\subset \mathbf{Z}$ is referred to as ℜ-preserving if $({\mathsf{z}}_n,{\mathsf{z}}_{n+1})\in \Re$, $\mathrm{for}\mathrm{all}$$n\in {\mathbb{N}}_0$.

Definition 8

([3]). ℜ is referred to as ϱ-self-closed if convergence limit of any ℜ-preserving convergent sequence in $(\mathbf{Z},\varrho )$ is ℜ-comparative with each term of a subsequence.

Definition 9

([4]). $\mathbf{P}$ is referred to as ℜ-continuous if, for all $\mathsf{z}\in \mathbf{Z}$, and for any ℜ-preserving sequence $\left\{{\mathsf{z}}_n\right\}\subset \mathbf{Z}$, we verify ${\mathsf{z}}_n\stackrel{\varrho }{⟶}\mathsf{z}$,

$\mathbf{P}\left({\mathsf{z}}_n\right)\stackrel{\varrho }{⟶}\mathbf{P}\left(\mathsf{z}\right).$

Definition 10

([4]). A MS $(\mathbf{Z},\varrho )$ is referred to as ℜ-complete if each ℜ-preserving Cauchy sequence converges.

Proposition 3

([5]). If ℜ is $\mathbf{P}$-closed, then for each $n\in {\mathbb{N}}_0$, ℜ is ${\mathbf{P}}^n$-closed.

Proposition 4.

For any $0<\gamma <1$ and $\psi \in \Psi$, (I) and (II) are equivalent:

• (I). $$\begin{gathered} \varrho (\mathbf{P}z,\mathbf{P}w)\le \gamma \varrho (z,w)+min\left\{\psi \right(\varrho (z,\mathbf{P}z)),\psi (\varrho (w,\mathbf{P}w)),\psi (\varrho (z,\mathbf{P}w)),\psi (\varrho (w,\mathbf{P}z)\left)\right\}, \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \mathrm{for}\mathrm{all}(z,w)\in \Re \end{gathered}$$;

• (II). $$\begin{gathered} \varrho (\mathbf{P}z,\mathbf{P}w)\le \gamma \varrho (z,w)+min\left\{\psi \right(\varrho (z,\mathbf{P}z)),\psi (\varrho (w,\mathbf{P}w)),\psi (\varrho (z,\mathbf{P}w)),\psi (\varrho (w,\mathbf{P}z)\left)\right\}, \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \mathrm{for}\mathrm{all}[z,w]\in \Re \end{gathered}$$.

Proof. 

The conclusion is followed using the symmetry of $\varrho$. □

3. Main Results

We are now going to demonstrate the following outcomes, ensuring the fixed-point of certain class of nonlinear almost contractions in relational MS (i.e., MS comprised with a relation).

Theorem 3.

Assume that $(\mathbf{Z},\varrho )$ continues to be a MS comprised with a relation ℜ, while $\mathbf{P}:\mathbf{Z}\to \mathbf{Z}$ remains a map. Also,

• (a). ℜ is $\mathbf{P}$-closed;

• (b). $(\mathbf{Z},\varrho )$ is ℜ-complete;

• (c). $\mathrm{There}\mathrm{exists}$$z_0\in \mathbf{Z}$ for which $(z_0,\mathbf{P}{z}_0)\in \Re$;

• (d). $\mathbf{P}$ remains ℜ-continuous or ℜ is ϱ-self-closed;

• (e). $\mathrm{There}\mathrm{exists}$$0\le \gamma <1$ and $\psi \in \Psi$, verifying for all $(z,\mathsf{w})\in \Re$ that

  $\begin{array}{c}\hfill \varrho (\mathbf{P}z,\mathbf{P}\mathsf{w})\le \gamma \varrho (z,\mathsf{w})+min\left\{\psi \right(\varrho (\mathsf{z},\mathbf{P}w)),\psi (\varrho (\mathsf{w},\mathbf{P}z)),\psi (\varrho (\mathsf{z},\mathbf{P}z)),\psi (\varrho (\mathsf{w},\mathbf{P}w)\left)\right\}.\end{array}$

Then, $\mathbf{P}$ possesses a fixed point.

Proof. 

Starting with $z_0\in \mathbf{Z}$, define the sequence $\left\{z_n\right\}\subset \mathbf{Z}$, such that

(1)$z_n={\mathbf{P}}^n\left(z_0\right)=\mathbf{P}\left(z_{n-1}\right),\mathrm{for}\mathrm{all}n\in \mathbb{N}.$

Using (a) and Proposition 2, we find

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

which, utilizing (1), becomes

(2)$(z_n,z_{n+1})\in \Re ,\mathrm{for}\mathrm{all}n\in \mathbb{N}.$

Thus, $\left\{z_n\right\}$ is a ℜ-preserving sequence.

Define ${\varrho }_n:=\varrho (z_n,z_{n+1})$. Employing the contraction condition (e) to (2), and by (1), we arrive at

$\begin{array}{ccc}\hfill \varrho (z_n,z_{n+1})& \le & \gamma {\varrho }_{n-1}+min\{\psi \left(\varrho (z_{n-1},\mathbf{P}{z}_n)\right),\psi \left(\varrho (z_n,\mathbf{P}{z}_{n-1})\right),\psi \left(\varrho (z_{n-1},\mathbf{P}{z}_{n-1})\right),\psi \left(\varrho (z_n,\mathbf{P}{z}_n)\right)\}\hfill \\ & =& \gamma {\varrho }_{n-1}+min\{\psi \left(\varrho (z_{n-1},z_{n+1})\right),\psi \left(0\right),\psi \left({\varrho }_{n-1}\right),\psi \left({\varrho }_n\right)\}\hfill \end{array}$

so that

${\varrho }_n\le \gamma {\varrho }_{n-1},\mathrm{for}\mathrm{all}n\in \mathbb{N},$

which, by easy induction, provides

(3)${\varrho }_n\le {\gamma }^n{\varrho }_0,\mathrm{for}\mathrm{all}n\in \mathbb{N}.$

For all $n<m$, by (3) and triangular inequality, we arrive at

$\begin{array}{ccc}\hfill \varrho (z_n,z_m)& \le & {\varrho }_n+{\varrho }_{n+1}+{\varrho }_{n+2}+\cdots +{\varrho }_{m-1}\hfill \\ & \le & ({\gamma }^n+{\gamma }^{n+1}+{\gamma }^{n+2}+\cdots +{\gamma }^{m-1}){\varrho }_0\hfill \\ & =& {\gamma }^n(1+\gamma +{\gamma }^2+\cdots +{\gamma }^{m-n-1}){\varrho }_0\hfill \\ & \le & {\displaystyle \frac{{\gamma }^n}{1-\gamma }}{\varrho }_0\hfill \\ & \to & 0\mathrm{as}n\left(\mathrm{and}\mathrm{hence}m\right)\to +\infty .\hfill \end{array}$

Hence, $\left\{z_n\right\}$ is Cauchy. Using ℜ-completeness of $\mathbf{Z}$, $\mathrm{there}\mathrm{exists}$$z^{*}\in \mathbf{Z}$ with $z_n\stackrel{\varrho }{⟶}{z}^{*}$.

For the rest of the proof, we need to verify that $z^{*}$ retains a fixed point of $\mathbf{P}$. Assuming $\mathbf{P}$ is ℜ-continuous. Owing to the accessibility of (2) and $z_n\stackrel{\varrho }{⟶}{z}^{*}$, we have

$z_{n+1}=\mathbf{P}\left(z_n\right)\stackrel{\varrho }{⟶}\mathbf{P}\left(z^{*}\right).$

By uniqueness of limit, we find $\mathbf{P}\left(z^{*}\right)=z^{*}$. As another option, assume that ℜ is $\varrho$-self-closed. Owing to the accessibility of (2) and $z_n\stackrel{\varrho }{⟶}{z}^{*}$, we can determine a subsequence $\left\{z_{n_k}\right\}$ of $\left\{z_n\right\}$, verifying $[z_{n_k},z^{*}]\in \Re ,\mathrm{for}\mathrm{all}k\in \mathbb{N}$. Set ${\delta }_n:=\varrho (z^{*},z_n)$. By contraction condition (e) and Proposition 4, we conclude that

(4)$\begin{array}{ccc}\hfill \varrho (z_{n_k+1},\mathbf{P}{z}^{*})& =& \varrho (\mathbf{P}{z}_{n_k},\mathbf{P}{z}^{*})\hfill \\ \hfill & \le & \gamma \varrho (z_{n_k},z^{*})+min\{\psi \left(\varrho (z_{n_k},\mathbf{P}{z}^{*})\right),\psi \left(\varrho (z^{*},\mathbf{P}{z}_{n_k})\right),\psi \left(\varrho (z_{n_k},\mathbf{P}{z}_{n_k})\right),\psi \left(\varrho (z^{*},\mathbf{P}{z}^{*})\right)\}\hfill \\ & =& \gamma {\delta }_{n_k}+min\{\psi \left(\varrho (z_{n_k},\mathbf{P}{z}^{*})\right),\psi \left({\delta }_{n_k+1}\right),\psi \left({\varrho }_{n_k}\right),\psi \left(\varrho (z^{*},\mathbf{P}{z}^{*})\right)\}.\hfill \end{array}$

Now, $z_{n_k}\stackrel{\varrho }{⟶}{z}^{*}$ yields that ${\delta }_{n_k}⟶0$ in $[0,+\infty )$, if $k\to +\infty$. Thus, allowing $k\to +\infty$ in (4), we arrive at

$\begin{array}{c}\hfill \underset{k\to +\infty }{lim}\varrho (z_{n_k+1},\mathbf{P}{z}^{*})=0\end{array}$

so that $z_{n_k+1}\stackrel{\varrho }{⟶}\mathbf{P}\left(z^{*}\right)$ thereby yielding $\mathbf{P}\left(z^{*}\right)=z^{*}$. Therefore, $z^{*}$ retains a fixed point of $\mathbf{P}$ in each case. □

Theorem 4.

Alongside Theorem 3, if $\mathbf{P}\left(\mathbf{Z}\right)$ is ${\Re }^s$-directed, then $\mathbf{P}$ enjoys a unique fixed point.

Proof. 

In lieu of Theorem 3, $F\left(\mathbf{P}\right)\ne \varnothing$, take $z^{*},w^{*}\in F\left(\mathbf{P}\right)$. We conclude that

(5)${\mathbf{P}}^{\mathsf{n}}\left(z^{*}\right)=z^{*}and{\mathbf{P}}^{\mathsf{n}}\left(w^{*}\right)=w^{*},\mathrm{for}\mathrm{all}\mathsf{n}\in {\mathbb{N}}_0.$

As $z^{*},w^{*}\in \mathbf{P}\left(\mathbf{Z}\right)$ and $\mathbf{P}\left(\mathbf{Z}\right)$ is ${\Re }^s$-directed, $\mathrm{there}\mathrm{is}$${\omega }_0\in \mathbb{P}$, that verifies $[z^{*},{\omega }_0]\in \Re$ and $[w^{*},{\omega }_0]\in \Re$. Consider the following sequence:

(6)${\omega }_{\mathsf{n}}={\mathbf{P}}^{\mathsf{n}}\left({\omega }_0\right)=\mathbf{P}\left({\omega }_{\mathsf{n}-1}\right),\mathrm{for}\mathrm{all}\mathsf{n}\in {\mathbb{N}}_0.$

Using (5), (6), assumption (a), and Proposition 2, we obtain

(7)$[z^{*},{\omega }_{\mathsf{n}}]\in \Re and[w^{*},{\omega }_{\mathsf{n}}]\in \Re ,\mathrm{for}\mathrm{all}\mathsf{n}\in {\mathbb{N}}_0.$

Employing (6), (7), contraction condition (e), and Proposition 4, we conclude that

$\begin{array}{ccc}\hfill \varrho (z^{*},{\omega }_{\mathsf{n}+1})& =& \varrho (\mathbf{P}{z}^{*},\mathbf{P}{\omega }_{\mathsf{n}})\hfill \\ & \le & \gamma \varrho (z^{*},{\omega }_{\mathsf{n}})+min\{\psi \left(\varrho (z^{*},\mathbf{P}{\omega }_{\mathsf{n}})\right),\psi \left(\varrho ({\omega }_{\mathsf{n}},\mathbf{P}{z}^{*})\right),\psi \left(\varrho (z^{*},\mathbf{P}{z}^{*})\right),\psi \left(\varrho ({\omega }_{\mathsf{n}},\mathbf{P}{\omega }_{\mathsf{n}})\right)\}\hfill \\ & =& \gamma \varrho (z^{*},{\omega }_{\mathsf{n}})+min\{\psi \left(\varrho (z^{*},{\omega }_{\mathsf{n}})\right),\psi \left(\varrho ({\omega }_{\mathsf{n}},z^{*})\right),0,\psi \left(\varrho ({\omega }_{\mathsf{n}},{\omega }_{\mathsf{n}+1})\right))\}\hfill \end{array}$

so that $\varrho (z^{*},{\omega }_{\mathsf{n}+1})\le \gamma \varrho (z^{*},{\omega }_{\mathsf{n}})$, which by easy induction gives rise to

(8)$\varrho (z^{*},{\omega }_{\mathsf{n}+1})\le {\gamma }^{\mathsf{n}+1}\varrho (z^{*},{\omega }_0).$

Similarly, we can show that

(9)$\varrho (w^{*},{\omega }_{\mathsf{n}+1})\le {\gamma }^{\mathsf{n}+1}\varrho (w^{*},{\omega }_0).$

From (8) and (9), we find

$\begin{array}{ccc}\hfill \varrho (z^{*},w^{*})& \le & \varrho (z^{*},{\omega }_{\mathsf{n}+1})+\varrho ({\omega }_{\mathsf{n}+1},w^{*})\hfill \\ & \le & {\gamma }^{\mathsf{n}+1}[\varrho (z^{*},{\omega }_0)+\varrho (w^{*},{\omega }_0)]\hfill \\ & \to & 0,asn\to +\infty \hfill \end{array}$

thereby implying $z=\mathsf{w}$. This concludes the proof. □

4. Illustrative Examples

The following examples are offered to corroborate the findings explored in the preceding section.

Example 1.

Take $\mathbf{Z}=[-1,4]$ endued with Euclidean metric ϱ. On $\mathbf{Z}$, define a relation $\Re =\{(z,w)\in {\mathbf{Z}}^2:z-w\ge 0\}$. Consider the map $\mathbf{P}:\mathbf{Z}\to \mathbf{Z}$, such that

$\mathbf{P}\left(z\right)=\left\{\begin{array}{c}z/2\mathrm{if}-1\le z\le 2\hfill \\ 1\mathrm{if}2<z\le 4.\hfill \end{array}\right.$

Then, $(\mathbf{Z},\varrho )$ is a ℜ-complete MS, and ℜ is $\mathbf{P}$-closed. It is straightforward to verify that the contraction-inequality of Theorem 3 holds for $\gamma =1/2$ and $\psi \left(t\right)=ln(1+t)$. The rest of the hypotheses of Theorems 3 and 4 also hold and, hence, $\mathbf{P}$ admits a unique fixed point: $z^{*}=0$.

Example 2.

Take $\mathbf{Z}=[1,3]$ endued with Euclidean metric ϱ. On $\mathbf{Z}$, define a relation $\Re =\left\{\right(1,1),(1,2),(2,1),(2,2),(1,3\left)\right\}$. Consider the map $\mathbf{P}:\mathbf{Z}\to \mathbf{Z}$, such that

$\mathbf{P}\left(z\right)=\left\{\begin{array}{c}1\mathrm{if}1\le z\le 2\hfill \\ 2\mathrm{if}2<z\le 3.\hfill \end{array}\right.$

Then, $(\mathbf{Z},\varrho )$ is a ℜ-complete MS, and ℜ is $\mathbf{P}$-closed. It is straightforward to verify that the contraction-inequality of Theorem 3 holds for $\gamma =1/3$ and $\psi \left(t\right)=3t$.

Let $\left\{z_n\right\}\subset \mathbf{Z}$ be an ℜ-preserving sequence satisfying $z_n\stackrel{\varrho }{⟶}\omega$, so that $(z_n,z_{n+1})\in \Re ,\mathrm{for}\mathrm{all}n\in \mathbb{N}$. Here, $(z_n,z_{n+1})\notin \left\{(1,3)\right\}$, thereby implying $(z_n,z_{n+1})\in \{(1,1),(1,2),(2,1),(2,2)\}$, $\mathrm{for}\mathrm{all}n\in \mathbb{N}$ so that $\left\{z_n\right\}\subset \{1,2\}$. As $\{1,2\}$ is closed, we have $[z_n,\omega ]\in \Re$. Therefore, ℜ is ϱ-self-closed. The rest of hypotheses of Theorems 3 and 4 also hold and, hence, $\mathbf{P}$ admits a unique fixed point: $z^{*}=1$.

5. Applications

The present section implements our findings to arrive at a (unique) solution of the nonlinear integral equation:

(10)$\xi \left(\tau \right)=\theta \left(\tau \right)+{\int }_a^b\Phi (\tau ,\omega )ϝ(\omega ,\xi \left(\omega \right))d\omega ,\tau \in I:=[a,b],$

where $\theta :I\to \mathbb{R}$, $\Phi :I\times I\to \mathbb{R}$, and $ϝ:I\times \mathbb{R}\to \mathbb{R}$ remain functions. As usual, $\mathcal{C}\left(I\right)$ will denote the family of the continuous real valued functions on I.

Definition 11.

$\underline{\xi }\in \mathcal{C}\left(I\right)$ is said to form a lower solution of (10) if $\mathrm{for}\mathrm{all}\tau \in I$

$\underline{\xi }\left(\tau \right)\le \theta \left(\tau \right)+{\int }_a^b\Phi (\tau ,\omega )ϝ(\omega ,\underline{\xi }\left(\omega \right))d\omega .$

Definition 12.

$\overline{\xi }\in \mathcal{C}\left(I\right)$ is said to form an upper solution of (10) if $\mathrm{for}\mathrm{all}\tau \in I$

$\overline{\xi }\left(\tau \right)\ge \theta \left(\tau \right)+{\int }_a^b\Phi (\tau ,\omega )ϝ(\omega ,\overline{\xi }\left(\omega \right))d\omega .$

Finally, we present the main outcomes of this section.

Theorem 5.

Along with Problem (10), we assume that

• (i). θ, ϝ and Φ remain continuous;

• (ii). $\Phi (\tau ,\omega )>0,\mathrm{for}\mathrm{all}\tau ,\omega \in I$;

• (iii). $\mathrm{There}\mathrm{exists}$$\alpha >0$ and $\beta \ge 0$, with $\alpha >\beta$ verifying

  $0\le \alpha \left[ϝ\right(\tau ,x)-ϝ(\tau ,y\left)\right]\le \beta (x-y),\mathrm{for}\mathrm{all}\tau \in Iand\mathrm{for}\mathrm{all}x,y\in \mathbb{R}withx\ge y,$

• (iv). $\underset{\tau \in I}{sup}{\int }_a^b\Phi (\tau ,\omega )d\omega \le 1$.

Then, Equation (10) has a unique solution, provided it enjoys a lower solution.

Proof. 

On $\mathbf{Z}:=\mathcal{C}\left(I\right)$, consider the metric:

(11)$\varrho (\xi ,\eta )=\underset{\tau \in I}{sup}|\xi \left(\tau \right)-\eta \left(\tau \right)|,\mathrm{for}\mathrm{all}\xi ,\eta \in \mathbf{Z}.$

On $\mathbf{Z}$, define the following relation:

(12)$\Re =\{(\xi ,\eta )\in {\mathbf{Z}}^2:\xi \left(\tau \right)\le \eta \left(\tau \right),\mathrm{for}\mathrm{all}\tau \in I\}.$

Define a map $\mathbf{P}:\mathbf{Z}\to \mathbf{Z}$ by

(13)$\left(\mathbf{P}\xi \right)\left(\tau \right)=\theta \left(\tau \right)+{\int }_a^b\Phi (\tau ,\omega )ϝ(\omega ,\xi \left(\omega \right))d\omega ,\mathrm{for}\mathrm{all}\tau \in I.$

Thus, $\xi \in \mathbf{Z}$ remains a solution of Problem (10), if and only if $\xi \in F\left(\mathbf{P}\right)$.

We will validate all hypotheses of Theorems 3 and 4.

• (a). Clearly, the MS $(\mathbf{Z},\varrho )$ is ℜ-complete.

• (b). Take $\xi ,\eta \in \mathbf{Z}$ verifying $(\xi ,\eta )\in \Re$. From (iii), we find

  (14)$ϝ(\tau ,\xi (\omega \left)\right)-ϝ(\tau ,\eta (\omega \left)\right)\le 0,\mathrm{for}\mathrm{all}\tau ,\omega \in I.$

  By (13), (14) and Assumption (ii), we obtain

  $\begin{array}{c}\hfill \left(\mathbf{P}\xi \right)\left(\tau \right)-\left(\mathbf{P}\eta \right)\left(\tau \right)={\int }_a^b\Phi (\tau ,\omega )[ϝ(\omega ,\xi \left(\omega \right))-ϝ(\omega ,\eta \left(\omega \right))]d\omega \le 0,\end{array}$

  so that $\left(\mathbf{P}\xi \right)\left(\tau \right)\le \left(\mathbf{P}\eta \right)\left(\tau \right)$ which, using (12), yields that $(\mathbf{P}\xi ,\mathbf{P}\eta )\in \Re$, and hence ℜ is $\mathbf{P}$-closed.

• (c). If $\underline{\xi }\in \mathbf{Z}$ forms a lower solution of (10), then

  $\underline{\xi }\left(\tau \right)\le \theta \left(\tau \right)+{\int }_a^b\Phi (\tau ,\omega )ϝ(\omega ,\underline{\xi }\left(\omega \right))d\omega =\left(\mathbf{P}\underline{\xi }\right)\left(\tau \right)$

  thereby implying $(\underline{\xi },\mathbf{P}\underline{\xi })\in \Re$.

• (d). Consider an ℜ-preserving sequence $\left\{{\xi }_n\right\}\subset \mathbf{Z}$ that converges to $\omega \in \mathbf{Z}$. Then, for every $\tau \in I$, $\left\{{\xi }_n\left(\tau \right)\right\}$ is increasing in $\mathbb{R}$, converging to $\omega \left(\tau \right)$. Thus far, ${\xi }_n\left(\tau \right)\le \omega \left(\tau \right),\mathrm{for}\mathrm{all}n\in \mathbb{N}$ and $\mathrm{for}\mathrm{all}\tau \in I$. From (12), we have $({\xi }_n,\omega )\in \Re ,\mathrm{for}\mathrm{all}n\in \mathbb{N}$. Hence, ℜ is $\varrho$-self-closed.

• (e). Let $\xi ,\eta \in \mathbf{Z}$, such that $(\xi ,\eta )\in \Re$. Using (iii), (11), and (13), we obtain

  (15)$\begin{array}{ccc}\hfill \varrho (\mathbf{P}\xi ,\mathbf{P}\eta )& =& \underset{\tau \in I}{sup}|\left(\mathbf{P}\xi \right)\left(\tau \right)-\left(\mathbf{P}\eta \right)\left(\tau \right)|=\underset{\tau \in I}{sup}[\left(\mathbf{P}\eta \right)\left(\tau \right)-\left(\mathbf{P}\xi \right)\left(\tau \right)]\hfill \\ \hfill & =& \underset{\tau \in I}{sup}{\int }_a^b\Phi (\tau ,\omega )[ϝ(\omega ,\eta \left(\omega \right))-ϝ(\omega ,\xi \left(\omega \right))]d\omega \hfill \\ & \le & \underset{\tau \in I}{sup}{\int }_a^b\Phi (\tau ,\omega ){\displaystyle \frac{\beta }{\alpha }}[\eta \left(\omega \right)-\xi \left(\omega \right)]d\omega .\hfill \end{array}$

  As $0\le \eta \left(\omega \right)-\xi \left(\omega \right)\le \varrho (\xi ,\eta )$, so (15) reduces to

  $\begin{array}{ccc}\hfill \varrho (\mathbf{P}\xi ,\mathbf{P}\eta )& \le & {\displaystyle \frac{\beta }{\alpha }}\varrho (\xi ,\eta )\underset{\tau \in I}{sup}{\int }_a^b\Phi (\tau ,\omega )d\omega \hfill \\ & \le & {\displaystyle \frac{\beta }{\alpha }}\varrho (\xi ,\eta ).1\hfill \\ & =& \gamma \varrho (\xi ,\eta )),where\gamma =\beta /\alpha \hfill \end{array}$

  so that

  $\begin{array}{ccc}\hfill \varrho (\mathbf{P}\xi ,\mathbf{P}\eta )& \le & \gamma \varrho (\xi ,\eta )+min\left\{\psi \right(\varrho (\xi ,\mathbf{P}\xi )),\psi (\varrho (\eta ,\mathbf{P}\eta )),\psi (\varrho (\xi ,\mathbf{P}\eta )),\psi (\varrho (\eta ,\mathbf{P}\xi )\left)\right\},\hfill \\ & \mathrm{for}\mathrm{all}& \xi ,\eta \in \mathbf{Z}\mathrm{such}\mathrm{that}(\xi ,\eta )\in \Re ,\hfill \end{array}$

  where $\psi \in \Psi$ is arbitrary.

Take $\xi ,\eta \in \mathbf{P}\left(\mathbf{Z}\right)$, and define $\varpi \left(\tau \right):=max\left\{\xi \right(\tau ),\eta (\tau \left)\right\}$. Then, $(\xi ,\varpi )\in \Re$ and $(\eta ,\varpi )\in \Re$. It follows that $\mathbf{P}\left(\mathbf{Z}\right)$ is ${\Re }^s$-directed. Henceforth, by applying Theorem 4, $\mathbf{P}$ admits a unique fixed point, which is a unique solution of (10). □

Theorem 6.

Along with conditions (i)–(iv) of Theorem 5, Problem (10) has a unique solution, provided that it has an upper solution.

Proof. 

Consider a metric $\varrho$ on $\mathbf{Z}:=\mathcal{C}\left(I\right)$, and a map $\mathbf{P}:\mathbf{Z}\to \mathbf{Z}$, the same as in the proof of Theorem 5. On $\mathbf{Z}$, take a relation

(16)${\Re }^{\prime }=\{(\xi ,\eta )\in {\mathbf{Z}}^2:\xi \left(\tau \right)\ge \eta \left(\tau \right),\mathrm{for}\mathrm{all}\tau \in I\}.$

Now, we will check all hypotheses of Theorems 3 and 4.

• (a). Clearly, the MS $(\mathbf{Z},\varrho )$ is ${\Re }^{\prime }$-complete.

• (b). Take $\xi ,\eta \in \mathbf{Z}$ verifying $(\xi ,\eta )\in {\Re }^{\prime }$. By (iii), we find

  (17)$ϝ(\tau ,\xi (\omega \left)\right)-ϝ(\tau ,\eta (\omega \left)\right)\ge 0,\mathrm{for}\mathrm{all}\tau ,\omega \in I.$

  By (13), (17) and (ii), we find

  $\begin{array}{c}\hfill \left(\mathbf{P}\xi \right)\left(\tau \right)-\left(\mathbf{P}\eta \right)\left(\tau \right)={\int }_a^b\Phi (\tau ,\omega )[ϝ(\omega ,\xi \left(\omega \right))-ϝ(\omega ,\eta \left(\omega \right))]d\omega \ge 0,\end{array}$

  so that $\left(\mathbf{P}\xi \right)\left(\tau \right)\ge \left(\mathbf{P}\eta \right)\left(\tau \right)$ which, using (16), yields that $(\mathbf{P}\xi ,\mathbf{P}\eta )\in {\Re }^{\prime }$, and hence ${\Re }^{\prime }$ is $\mathbf{P}$-closed.

• (c). If $\overline{\xi }\in \mathbf{Z}$ is an upper solution of (10), then we have

  $\overline{\xi }\left(\tau \right)\ge \theta \left(\tau \right)+{\int }_a^b\Phi (\tau ,\omega )ϝ(\omega ,\overline{\xi }\left(\omega \right))d\omega =\left(\mathbf{P}\overline{\xi }\right)\left(\tau \right)$

  thereby implying $(\overline{\xi },\mathbf{P}\overline{\xi })\in {\Re }^{\prime }$.

• (d). Consider an ${\Re }^{\prime }$-preserving sequence $\left\{{\xi }_n\right\}\subset \mathbf{Z}$ that converges to $\omega \in \mathbf{Z}$. Then, for every $\tau \in I$, $\left\{{\xi }_n\left(\tau \right)\right\}$ is decreasing in $\mathbb{R}$, converging to $\omega \left(\tau \right)$. Thus far, ${\xi }_n\left(\tau \right)\ge \omega \left(\tau \right),\mathrm{for}\mathrm{all}n\in \mathbb{N}$ and $\mathrm{for}\mathrm{all}\tau \in I$. From (16), we have $({\xi }_n,\omega )\in {\Re }^{\prime },\mathrm{for}\mathrm{all}n\in \mathbb{N}$. Hence, ${\Re }^{\prime }$ is $\varrho$-self-closed.

• (e). Let $\xi ,\eta \in \mathbf{Z}$, such that $(\xi ,\eta )\in {\Re }^{\prime }$. By (iii), (11) and (13), we obtain

  (18)$\begin{array}{ccc}\hfill \varrho (\mathbf{P}\xi ,\mathbf{P}\eta )& =& \underset{\tau \in I}{sup}|\left(\mathbf{P}\xi \right)\left(\tau \right)-\left(\mathbf{P}\eta \right)\left(\tau \right)|=\underset{\tau \in I}{sup}[\left(\mathbf{P}\xi \right)\left(\tau \right)-\left(\mathbf{P}\eta \right)\left(\tau \right)]\hfill \\ \hfill & =& \underset{\tau \in I}{sup}{\int }_a^b\Phi (\tau ,\omega )[ϝ(\omega ,\xi \left(\omega \right))-ϝ(\omega ,\eta \left(\omega \right))]d\omega \hfill \\ & \le & \underset{\tau \in I}{sup}{\int }_a^b\Phi (\tau ,\omega ){\displaystyle \frac{\beta }{\alpha }}[\xi \left(\omega \right)-\eta \left(\omega \right)]d\omega .\hfill \end{array}$

  As $0\le \xi \left(\omega \right)-\eta \left(\omega \right)\le \varrho (\xi ,\eta )$, (18) reduces to

  $\begin{array}{ccc}\hfill \varrho (\mathbf{P}\xi ,\mathbf{P}\eta )& \le & {\displaystyle \frac{\beta }{\alpha }}\varrho (\xi ,\eta )\underset{\tau \in I}{sup}{\int }_a^b\Phi (\tau ,\omega )d\omega \hfill \\ & \le & {\displaystyle \frac{\beta }{\alpha }}\varrho (\xi ,\eta ).1\hfill \\ & =& \gamma \varrho (\xi ,\eta )),where\gamma =\beta /\alpha \hfill \end{array}$

  so that

  $\begin{array}{ccc}\hfill \varrho (\mathbf{P}\xi ,\mathbf{P}\eta )& \le & \gamma \varrho (\xi ,\eta )+min\left\{\psi \right(\varrho (\xi ,\mathbf{P}\xi )),\psi (\varrho (\eta ,\mathbf{P}\eta )),\psi (\varrho (\xi ,\mathbf{P}\eta )),\psi (\varrho (\eta ,\mathbf{P}\xi )\left)\right\},\hfill \\ & \mathrm{for}\mathrm{all}& \xi ,\eta \in \mathbf{Z}\mathrm{such}\mathrm{that}(\xi ,\eta )\in {\Re }^{\prime },\hfill \end{array}$

  where $\psi \in \Psi$ is arbitrary.

Take $\xi ,\eta \in \mathbf{P}\left(\mathbf{Z}\right)$, and define $\varpi \left(\tau \right):=min\left\{\xi \right(\tau ),\eta (\tau \left)\right\}$. Then, $(\xi ,\varpi )\in {\Re }^{\prime }$ and $(\eta ,\varpi )\in {\Re }^{\prime }$. It follows that $\mathbf{P}\left(\mathbf{Z}\right)$ is ${\Re }^s$-directed. Thus, by applying Theorem 4, $\mathbf{P}$ admits a unique fixed point, which is a unique solution of (10). □

6. Conclusions

In this study, our investigation examined findings on fixed-points under a nonlinear formulation of almost contractions on in relational MS. A number of examples were also conducted to illustrate these findings. Our findings extend the corresponding findings of Babu et al. [19] under the restriction $\Re ={\mathbf{Z}}^2$ and $\psi \left(t\right)=K·t$ ($K\ge 0$), and Theorem 4 reduces to Theorem 2. Our result can be applied in the ares of nonlinear matrix equations, periodic boundary value problems, and nonlinear integral equations.

In particular, we applied our findings to a special types of nonlinear integral equations. The general formulations of such nonlinear integral equations can be solved directly by employing the BCP (see Theorem 5.2.2 of [1]). However, the nonlinear integral equations investigated herewith require additional conditions (especially the existence of lower or upper solutions) that are solvable using the fixed point findings involving relational-theoretic contractions.

For future works, the outcomes investigated herewith can be further expanded to relatively generalized contraction conditions, on replacing the term $\gamma \varrho (z,\mathsf{w})$ by the terms involved in $\phi$-contractions, or weak contractions. Following similar lines of research, we can extend some recent outcomes of graph MS (e.g., [28,29,30]) to relational MS.

Author Contributions

Conceptualization, D.F. and F.A.K.; methodology, F.A.K.; formal analysis, F.A.K.; resources, D.F. and F.A.K.; writing—original draft, F.A.K.; writing—review and editing, D.F.; funding acquisition, D.F. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.

Conflicts of Interest

All authors declare no conflicts of interest.

Footnotes

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