Fixed point results for almost ($\zeta -\theta _{\rho }$)-contractions on quasi metric spaces and an application

1.   Introduction

The notion of a quasi metric space represents an intriguing extension of the classical metric space, achieved by relaxing the requirement of symmetry. Among various alternatives to the metric space, the quasi metric space stands out as a particularly intuitive concept that finds direct applicability in real-world scenarios. A simple illustration of a quasi metric arises when considering the distance traveled by a commuter between their home and workplace in a city characterized by one-way streets and two-way roads. For additional and specific instances of quasi metrics, along with compelling fixed-point outcomes within this context, please references ^{[4,5,7,8,12,13,15,24,26,27,31]}. Over the past few decades, numerous papers on the fixed point theory have been published, many of which extend the established fixed point results in various ways: By altering the abstract space, by substituting the contraction condition with a milder one and so on and so forth. As a result, an inherent question arises: Can the existing outcomes be amalgamated in an uncomplicated manner? Several responses have been provided, and among these, a few of the most intriguing answers are related to the '$\theta$-contraction' and the 'simulation function'. The definition of the $\theta$-contraction is given by Jleli et al. ^[20], while the concept of the simulation function is introduced by Khojasteh et al. ^[22]. Furthermore, using these functions, a multitude of single-valued fixed point results have been achieved in the standard metric space. In this work, we will explore the response to the inquiry: How can we amalgamate established fixed point theorems within the framework of a quasi-metric space by employing the $\theta$-contraction and the simulation function? In order to provide the most effective solution, we will additionally make use of another auxiliary function known as an admissible mapping. It is notably intriguing that the admissible mapping possesses the capability to merge the fixed point theorems within a metric space coupled with a partially ordered set, as well as the associated fixed point propositions resulting from cyclic contractions or standard contractions. For a more comprehensive understanding, refer to sources such as ^{[1,2,10,11,17,18,21,23]}. As a result, we have harmonized various fixed point outcomes within the framework of a quasi-metric space, utilizing both simulation functions, $\theta$-contraction and admissible mappings.

2.   Preliminaries

Now, review the definitions and notations related to quasi-metric space: $\Lambda \neq \emptyset$ and $\rho$ are a function $\rho :\Lambda \times \Lambda \rightarrow \mathbb{R}$ such that for each $\omega, \gamma, \eta \in \Lambda$:

(ⅰ) $\rho (\omega, \omega) = 0$,

(ⅱ) $\rho (\omega, \gamma)\leq \rho (\omega, \eta)+\rho (\eta, \gamma)$ (triangle inequality),

(ⅲ) $\rho (\omega, \gamma) = \rho (\gamma, \omega) = 0$ implies $\omega = \gamma$,

(ⅳ) $\rho (\omega, \gamma) = 0$ implies $\omega = \gamma$.

If (ⅰ) and (ⅱ) conditions are satisfied, then $\rho$ is called a quasi-pseudo metric (shortly qpm); if (ⅰ)–(ⅲ) conditions are satisfied, then $\rho$ is called quasi metric (shortly qm); in addition, if a qm $\rho$ satisfies (ⅳ), then $\rho$ is called $T_{1}$-qm. It is evident that every metric is a $T_{1}$-qm, every $T_{1}$-qm is a qm and every qm is a qpm. Then, the pair $(\Lambda, \rho)$ is also said to be a quasi-pseudo metric space (shortly qpms). Moreover, each qpm $\rho$ on $\Lambda$ generates a topology $\tau _{\rho }$ on $\Lambda$ of the family of open balls as a base defined as follows:

where $B_{\rho }(\omega _{0}, \varepsilon) = \{\gamma \in \Lambda :\rho (\omega _{0}, \gamma) < \varepsilon \}$.

If $\rho$ is a qm on $\Lambda$, then $\tau _{\rho }$ is a $T_{0}$ topology, and if $\rho$ is a $T_{1}$-qm, then $\tau _{\rho }$ is a $T_{1}$ topology on $\Lambda$. If $\rho$ is a qm and $\tau _{\rho }$ is a $T_{1}$ topology, then $\rho$ is $T_{1}$-qm.

The mapping $\overline{\rho }$ defines

as a qpm whenever $\rho$ is a qpm on $\Lambda$. To find the fixed point, the most important part is to use the completeness of the metric space. However, since there is no symmetry conditions in a qm, there are many kinds of completeness in these spaces in the literature (see ^[9,28,30]).

Let $(\Lambda, \rho)$ be a qms, then the convergence of a sequence $\{\omega _{n}\}$ to $\omega$ w. r. t. $\tau _{\rho }$ called $\rho$ -convergence is defined as $\omega _{n}\overset{\rho }{\rightarrow }\omega$ if, and only if, $\rho (\omega, \omega _{n})\rightarrow 0$. Similarly, the convergence of a sequence $\{\omega _{n}\}$ to $\omega$ w. r. t. $\tau _{ \overline{\rho }}$ called $\overline{\rho }$-convergence is defined $\omega _{n}\overset{\overline{\rho }}{\rightarrow }\omega$ if, and only if, $\rho (\omega _{n}, \omega)\rightarrow 0$ for $\omega \in \Lambda$. A more detailed explanation of some essential metric properties can be found in ^[26]. Also, a sequence $\{\omega _{n}\}$ in $\Lambda$ is called left (right) $K$-Cauchy if for every $\varepsilon > 0,$ there exists $n_{0}\in \mathbb{N}$ such that for all $n, k\in \mathbb{N}$ with $n\geq k\geq n_{0}$ $(k\geq n\geq n_{0}),$ $\rho (\omega _{k}, \omega _{n}) < \varepsilon.$ The left $K$-Cauchy property under $\rho$ implies the right $K$-Cauchy property under $\overline{\rho }$. Assuming

the sequence $\{\omega _{n}\}$ in the quasi-metric space $(\Lambda, \rho)$ is left $K$-Cauchy.

In a metric space, every convergent sequence is indeed a Cauchy sequence, but since this may not hold true in qms, there have been several definitions of completeness. A qms $(\Lambda, \rho)$ is said to be left (right) $K$ (resp. $M$)-complete if every left (right) $K$-Cauchy sequence is $\rho$ (resp. $\overline{\rho }$)-convergent.

Now, we explain the approach of $\alpha$-admissibility as constructed by Samet et al. ^[29].

Let $\Lambda \neq \emptyset$, $\Upsilon :\Lambda \rightarrow \Lambda$ be a mapping and $\alpha :\Lambda \times \Lambda \rightarrow \lbrack 0, +\infty)$ be a function. In this context, $\Upsilon$ is said to be $\alpha$ -admissible if it satisfies the following condition:

By introducing the approach of $\alpha$-admissibility, Samet et al. ^[29] was able to establish some general fixed point results that encompassed many well-known theorems of complete metric spaces.

In addition to these, in the study conducted by Jleli and Samet in ^[20], they led to the introduction of a new type of contractive mapping known as a $\theta$-contraction. This $\theta$-contraction serves as an attractive generalization within the field. To better understand this approach, let's review some notions and related results concerning $\theta$-contraction.

The family of $\theta :(0, +\infty)\rightarrow (1, +\infty)$ functions that satisfy the following conditions can be denoted by the set $\Theta :$

$(\theta _{1})$ $\theta$ is nondecreasing;

$(\theta _{2})$ Considering every sequence $\left\{ \varkappa _{n}\right\} \subset \left(0, +\infty \right),$ $\lim_{n\rightarrow +\infty }\varkappa _{n} = 0^{+}$ if, and only if, $\lim_{n\rightarrow +\infty }\theta (\varkappa _{n}) = 1$;

$(\theta _{3})$ There exists $0 < p < 1$ and $\beta \in (0, +\infty ]$ such that $\lim_{\varkappa \rightarrow 0^{+}}\frac{\theta (\varkappa)-1}{\varkappa^{p} } = \beta$.

If we define $\theta (\varkappa) = e^{\sqrt{\varkappa }}$ for $\varkappa \leq 1$ and $\theta (\varkappa) = 9$ for $\varkappa > 1$, then $\theta \in \Theta$.

Let $\theta \in \Theta$ and $(\Lambda, \rho)$ be a quasi metric space, then $\Upsilon :\Lambda \rightarrow \Lambda$ is said to be a $\theta$ -contraction if there exists $0 < \delta < 1$ such that

for each $\omega, \gamma \in \Lambda$ with $\rho (\Upsilon \omega, \Upsilon \gamma) > 0$.

By choosing appropriate functions for $\theta$, such as $\theta _{1}(\varkappa) = e^{\sqrt{\varkappa }}$ and $\theta _{2}(\varkappa) = e^{ \sqrt{\varkappa e^{\varkappa }}}$, it is possible to obtain different types of nonequivalent contractions using (2.1).

Jleli and Samet proved that every $\theta$-contraction on a complete metric space possesses a unique fixed point. This result provides a valuable insight into the uniqueness and existence of fixed points for a wide range of contractive mappings. If you are interested in exploring more papers and literature related to $\theta$-contractions, there are several resources available (see ^[3,19]).

On the other hand, Khojasteh et al. ^[22] introduced an innovative category of contractions through the utilization of the following concept of simulation functions. By employing the concept, they ^[22] established numerous fixed point theorems and demonstrated that numerous well-established findings in the literature stem directly from the outcomes they derived. Furthermore, using the simulation function, generalizations of many known theorems have been obtained (see ^[6,25,30]). To better understand this approach, let's review some notions and related results concerning simulation functions.

The function $\zeta :\left[ 0, +\infty \right) \times \left[ 0, +\infty \right) \rightarrow \mathbb{R}$ is said to be a simulation function that satisfies the following conditions and can be denoted by the set $\mathbb{Z}:$

$(\zeta _{1})$ $\zeta \left(0, 0\right) = 0$;

$(\zeta _{2})$ $\zeta \left(t, s\right) < s-t$ for all $t, s > 0$;

$(\zeta _{3})$ if $\left\{ t_{n}\right\}, \left\{ s_{n}\right\}$ are sequence in $(0, +\infty)$ such that

then $\overline{\underset{n\rightarrow \infty }{\lim }}\zeta \left(t_{n}, s_{n}\right) < 0.$

If we define $\zeta _{1}(t, s) = \psi (s)-\varphi (t)$ for all $t, s\geq 0$, where $\psi, \varphi :\left[ 0, +\infty \right) \rightarrow \left[ 0, +\infty \right)$ are two continuous functions such that $\psi (t) = \varphi (t) = 0$ if, only if, $t = 0$ and $\psi (t) < t\leq \varphi (t)$ for all $t > 0$, then $\zeta \in \mathbb{Z}$.

3.   The results

Our results are based on a novel approach that we have developed.

Let $(\Lambda, \rho)$ be a qms, $\Upsilon :\Lambda \rightarrow \Lambda$ be a given mapping and $\alpha :\Lambda \times \Lambda \rightarrow \lbrack 0, +\infty)$ be a function. We will consider the following set

Definition 1. Let $(\Lambda, \rho)$ be a qms and $\Upsilon :\Lambda \rightarrow \Lambda$ be a mapping satisfying

Let $\alpha :\Lambda \times \Lambda \rightarrow \lbrack 0, +\infty)$, $\zeta \in \mathbb{Z}$ and $\theta \in \Theta$ be three functions, then we say that $\Upsilon$ is an almost ($\zeta -\theta _{\rho }$)-contraction if there exists $0 < \delta < 1$ and $L\geq 0$ such that

for each $(\omega, \gamma)\in \Upsilon _{\alpha }$, where

Before presenting our main results, let us recall some important remarks:

● If $(\Lambda, \rho)$ is a $T_{1}$-qms, then every mapping $\Upsilon :\Lambda \rightarrow \Lambda$ satisfies the condition (3.2).

● It is clear from (3.1)–(3.3) that if $\Upsilon$ is an almost ($\zeta -\theta _{\rho }$)-contraction, then

for each $\omega, \gamma \in \Lambda$ with $\alpha (\omega, \gamma)\geq 1.$

By utilizing the approach of the almost ($\zeta -\theta _{\rho }$)-contraction, we will now present the following theorem.

Theorem 1. Let $(\Lambda, \rho)$ be a Hausdorff right $K$-complete $T_{1}$ -qms and let $\Upsilon :\Lambda \rightarrow \Lambda$ be $\tau _{\rho }$ -continuous, $\alpha$-admissible and an almost ($\zeta -\theta _{\rho }$)-contraction. If there exists $\omega _{0}\in \Lambda$ such that $\alpha (\Upsilon \omega _{0}, \omega _{0})\geq 1$, then $\Upsilon$ has a fixed point in $\Lambda$.

Proof. Let $\omega _{0}\in \Lambda$ be such that $\alpha (\Upsilon \omega _{0}, \omega _{0})\geq 1$. Define a sequence $\{\omega _{n}\}$ in $\Lambda$ by $\omega _{n+1} = \Upsilon \omega _{n}$ for each $n$ in $\mathbb{N}$. Since $\Upsilon$ is $\alpha$-admissible, then $\alpha (\omega _{n+1}, \omega _{n})\geq 1$ for each $n$ in $\mathbb{N}$. If there exists $k\in \mathbb{N}$ with $\rho (\Upsilon \omega _{k}, \omega _{k}) = 0$, then $\omega _{k} = \Upsilon \omega _{k}$, since $\rho$ is $T_{1}$-qm. Hence, $\omega _{k}$ is a fixed point of $\Upsilon$. Presume $\rho (\Upsilon \omega _{n}, \omega _{n}) > 0$ for each $n$ in $\mathbb{N}$. In this case, the pair $(\omega _{n+1}, \omega _{n})$ for each $n$ in $\mathbb{N}$ belongs to $\Upsilon _{\alpha }$. Since $\Upsilon$ is an almost ($\zeta -\theta _{\rho }$)-contraction, we have

and so from ($\zeta _{2}$), we have

Hence, from ($\theta _{1}$) we obtain

If $\max \left\{ \rho (\omega _{n+1}, \omega _{n}), \rho (\omega _{n}, \omega _{n-1})\right\} = \rho (\omega _{n+1}, \omega _{n})$, using (3.4), we get

which is a contradiction. Thus, $\max \left\{ \rho (\omega _{n+1}, \omega _{n}), \rho (\omega _{n}, \omega _{n-1})\right\} = \rho (\omega _{n}, \omega _{n-1})$, and then we obtain

for each $n$ in $\mathbb{N}$. Denote $f_{n} = \rho (\omega _{n+1}, \omega _{n})$ for $n$ in $\mathbb{N}$, then $f_{n} > 0$ for each $n$ in $\mathbb{N}$, and repeating this process by using (3.5) we have

for each $n$ in $\mathbb{N}.$ When taking the limit as $n\rightarrow +\infty$ in (3.6), we obtain

Using ($\theta _{2}$), we can deduce that $\lim_{n\rightarrow +\infty }f_{n} = 0^{+};$ thus, using ($\theta _{3}$), there exists $p\in \left(0, 1\right)$ and $\beta \in (0, +\infty ]$ such that

Presume that $\beta < +\infty.$ In this case, let $F = \frac{\beta }{2} > 0.$ Using the definition of the limit, there exists $n_{0}$ in $\mathbb{N}$ such that, for each $n_{0}\leq n$,

This implies that for each $n_{0}\leq n$,

then, for each $n_{0}\leq n$,

where $D = 1/F.$

Presume now that $\beta = +\infty.$ Let $F > 0$ be an arbitrary positive number. Using the definition of the limit, there exists $n_{0}$ in $\mathbb{N}$ such that for each $n_{0}\leq n$,

This implies that for each $n_{0}\leq n$,

where $D = 1/F.$

Thus, in all cases, there exists $D > 0$ and $n_{0}$ in $\mathbb{N}$ such that

for each $n_{0}\leq n$. Using (3.6), we obtain

for each $n_{0}\leq n$. Letting $n\rightarrow +\infty$ from the last inequality, we have

Thus, there exists $n_{1}$ in $\mathbb{N}$ such that $n\left[ f_{n}\right] ^{p}\leq 1$ for each $n\geq n_{1},$ so we have, for each $n\geq n_{1},$

In order to show that $\left\{ \omega _{n}\right\}$ is a right $K$-Cauchy sequence, consider $m, n$ in $\mathbb{N}$ such that $m > n\geq n_{1}.$ Using the triangular inequality for $\rho$ and using (3.8), we have

By the convergence of the series $\sum\limits_{i = 1}^{+\infty }\frac{1}{ i^{1/p}}$, we get $\rho (\omega _{m}, \omega _{n})\ \rightarrow 0$ as $n\rightarrow +\infty$. This yields that $\left\{ \omega _{n}\right\}$ is a right $K$-Cauchy sequence in the qms $\left(\Lambda, \rho \right)$. Since $\left(\Lambda, \rho \right)$ is a right $K$-complete, there exists $\eta \in \Lambda$ such that the sequence $\left\{ \omega _{n}\right\}$ is $\rho$-converges to $\eta \in \Lambda$; that is, $\rho (\eta, \omega _{n})\rightarrow 0$ as $n\rightarrow +\infty$. Since $\Upsilon$ is $\tau _{\rho }$-continuous, then $\rho (\Upsilon \eta, \Upsilon \omega _{n}) = \rho (\Upsilon \eta, \omega _{n+1})\rightarrow 0$ as $n\rightarrow +\infty.$ Since $\Lambda$ is Hausdorff, we get $\Upsilon \eta = \eta$. □

We may use the option to substitute the continuity assumption of $\Upsilon$ in Theorem 1 with the following hypothesis:

($R$) If $\{\omega _{n}\}$ is a sequence in $\Lambda$ such that $\alpha (\omega _{n+1}, \omega _{n})\geq 1$ for all $n$ in $\mathbb{N}$, when the distance $\rho (\omega, \omega _{n})$$\rightarrow$$0$, then $\alpha (\omega, \omega _{n})\geq 1$ for all $n$ in $\mathbb{N}$.

In the theorem below, it is assumed that the space $(\Lambda, \rho)$ is Hausdorff; that is, $\tau _{\rho }$ is a Hausdorff topology, in which case it is clear that the limit of the convergent sequence is unique.

Theorem 2. Let $(\Lambda, \rho)$ be a Hausdorff right $K$-complete $T_{1}$-qms such that ($R$) holds, and let $\Upsilon :\Lambda \rightarrow \Lambda$ be an $\alpha$-admissible and almost ($\zeta -\theta _{\rho }$)-contraction. If $\theta$ is continuous and there exists $\omega _{0}\in \Lambda$ such that $\alpha (\Upsilon \omega _{0}, \omega _{0})\geq 1$, then $\Upsilon$ has a fixed point in $\Lambda$.

Proof. Similar to the proof of Theorem 1, we obtain $\left\{ \omega _{n}\right\}$ as a right $K$-Cauchy sequence in the qms $\left(\Lambda, \rho \right)$. Since $\left(\Lambda, \rho \right)$ is a right $K$ -complete, there exists $\eta \in \Lambda$ such that the sequence $\left\{ \omega _{n}\right\}$ is $\rho$-convergent to $\eta \in \Lambda$; that is, $\rho (\eta, \omega _{n})\rightarrow 0$ as $n\rightarrow +\infty$. Hence, from (3.3), we have

and so

Hence, we have

where

Letting $n\rightarrow +\infty$ from the given inequality, we have

Therefore, if$\ \rho (\Upsilon \eta, \eta)\neq 0$, from (3.9),

which is a contradiction. Hence $\rho (\Upsilon \eta, \eta) = 0$; that is, $\Upsilon \eta = \eta$. □

In Theorem 1, if we consider the approach of $\tau _{\overline{ \rho }}$-continuity, we can derive the following theorem.

Theorem 3. Let $(\Lambda, \rho)$ be a right $M$-complete $T_{1}$-qms such that $(\Lambda, \tau _{\overline{\rho }})$ is Hausdorff and $\Upsilon :\Lambda \rightarrow \Lambda$ is an $\alpha$-admissible and almost ($\zeta -\theta _{\rho }$)-contraction. Presume that $\Upsilon$ is $\tau _{ \overline{\rho }}$-continuous. If there exists $\omega _{0}\in \Lambda$ such that $\alpha (\Upsilon \omega _{0}, \omega _{0})\geq 1$, then $\Upsilon$ has a fixed point in $\Lambda.$

Proof. Similar to the proof of Theorem 1, we can take iterative sequence $\{\omega _{n}\}$ as right $K$-Cauchy. Since $(\Lambda, \rho)$ is right $M$ -complete, there exists $\eta \in \Lambda$ such that $\{\omega _{n}\}$ is $\overline{\rho }$-convergent to $\eta$; that is, $\rho (\omega _{n}, \eta)\rightarrow 0$ as $n\rightarrow +\infty$. Using $\tau _{\overline{\rho }}$ -continuity of $\Upsilon$, we get $\rho (\Upsilon \omega _{n}, \Upsilon \eta) = \rho (\omega _{n+1}, \Upsilon \eta)\rightarrow 0$ as $n\rightarrow +\infty$. Since $(\Lambda, \tau _{\overline{\rho }})$ is Hausdorff, we get $\eta = \Upsilon \eta$. □

Based on the outcomes we have derived, we present diverse fixed point conclusions within the existing literature, and we can derive the following corollaries:

Corollary 1. Let $(\Lambda, \rho)$ be a Hausdorff right $K$-complete $T_{1}$-qms and $\Upsilon :\Lambda \rightarrow \Lambda$ be given a mapping that satisfies

for each $\omega, \gamma \in \Lambda$, where $0 < \delta < 1$ and $L\geq 0$. Presume that $\Upsilon$ is $\alpha$-admissible and $\tau _{\rho }$ -continuous or ($R$) holds. If $\theta$ is continuous and there exists $\omega _{0}\in \Lambda$ such that $\alpha (\Upsilon \omega _{0}, \omega _{0})\geq 1$, then $\Upsilon$ has a fixed point in $\Lambda$.

Proof. It suffices to take a simulation function $\zeta (t, s) = ks-t$ for all $s, t\geq 0$ in Theorem 1, (resp. Theorem 2). □

Corollary 2 (see Durmaz and Altun ^[14]). Let $(\Lambda, \rho)$ be a Hausdorff right $K$-complete $T_{1}$-qms and $\Upsilon :\Lambda \rightarrow \Lambda$ be given a mapping that satisfies

for each $\omega, \gamma \in \Lambda$, where $0 < \delta < 1$. Presume that $\Upsilon$ is $\tau _{\rho }$-continuous or ($R$) holds with $\theta$ as continuous, then $\Upsilon$ has a fixed point in $\Lambda$.

Proof. It suffices to choose the mapping $\alpha :\Lambda$ $\times$ $\Lambda \rightarrow \lbrack 0, +\infty)$ such that $\alpha \left(\omega, \gamma \right) \geq 1$ for all $\omega, \gamma \in \Lambda$ and $L = 0$ with $\zeta (t, s) = ks-t$ for all $s, t\geq 0$ in Theorem 1. □

Remark 1. By considering the notion of left completeness in the sense of $K$, $M$ and $Smyth$, we can extend similar fixed point results to the setting of qms.

4.   Application

In this part, we propose a novel application in which we demonstrate the existence and uniqueness of the solution to a fractional boundary value problem (FBVP) using Theorem 1: Here, for continuous functions $a:\left[ 0, 1\right] \rightarrow \mathbb{R}$ and $f: \mathbb{R} \rightarrow \mathbb{R}$, we consider the FBVP given as

where $\alpha \in (1, 2],$ $\beta \in \lbrack 0, 1]$ and $D_{0^{+}}^{\gamma }$ is Riemann-Liouville derivative of order $\gamma$. It is well known that the operator $D_{0^{+}}^{\gamma }$ is defined as, for positive integer $n$ and $\gamma \in (n-1, n]$,

for a function $\xi :\left[ 0, 1\right] \rightarrow \mathbb{R}$, provided the righthand side exists. It is demonstrated in ^[16] that (4.1) is equivalent to the following integral equation:

where $G(t, s)$ is the associated Green's function defined by

Define an operator $\Upsilon :C[0, 1]\rightarrow C[0, 1]$ by

Hence, $\eta$ is a solution of (4.1) whenever it is a fixed point of $\Upsilon.$

Let $(\Lambda, \rho)$ be the $T_{1}$-qms, where $\Lambda = C[0, 1]$ and $\rho$ is defined by

In this case $(\Lambda, \rho)$ is both Hausdorff and right $K$-complete.

Now we can state the following theorem:

Theorem 4. The FBVP (4.1) has a solution under the following assumption: For all $\xi, \eta \in \Lambda$,

and

where $M = \left\Vert a\right\Vert _{\infty }$.

Proof. First of all, we know by Lemma 3.1 of ^[16] that $G(t, s)\geq 0$ for all $t, s\in \lbrack 0, 1]$ and

Consider the operator $\Upsilon :C[0, 1]\rightarrow C[0, 1]$ defined by

then for any $\xi, \zeta \in C[0, 1]$, we have

Therefore, $\Upsilon$ is an ($\zeta -\theta _{\rho }$)-contraction with the functions $\alpha (\xi, \eta) = 1,$ $\zeta (t, s) = ks-t$ and $\theta (x) = e^{ \sqrt{x}}$. The other conditions of Theorem 1 are clearly satisfied. Consequently, there exists $\zeta \in C\left[ 0, 1\right]$, which is fixed point of the operator $\Upsilon$. Hence, the (4.1) has a solution in $C\left[ 0, 1\right]$. □

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Conflict of interest

The authors declare no conflict of interest.