Certain Fixed-Point Results for ($\mathfrak{e}$,ψ,Φ)-Enriched Weak Contractions via Theoretic Order with Applications

Abstract

This paper aims to establish several fixed-point theorems within the framework of Banach spaces endowed with a binary relation. By utilizing enriched contraction principles involving two classes of altering-distance functions, the study encompasses various types of contractive mappings, including theoretic-order contractions, Picard–Banach contractions, weak contractions, and non-expansive contractions. A suitable Krasnoselskij iterative scheme is employed to derive the results. Many well-known fixed-point theorems (FPTs) can be obtained as special cases of these findings by assigning specific control functions in the main definitions or selecting an appropriate binary relation. To validate the theoretical results, numerous illustrative examples are provided. Furthermore, the paper demonstrates the applicability of the findings through applications to ordinary differential equations.

1. Introduction and Preliminaries

The Banach theorem [1] remains a foundational tool and a cornerstone in both theoretical and applied mathematics, demonstrating its versatility and effectiveness. This foundational principle establishes both the uniqueness and existence of fixed points for Banach contraction in complete metric spaces, while also presenting a systematic method for their calculation. In recent years, numerous researchers have generalized this theorem by introducing broader classes of contractive mappings and applying it to a variety of spaces.

Alam and Imdad, in [2], presented fixed-point results by employing a weaker contractive condition compared to the classical Banach contraction condition. The condition is required to be satisfied only for points connected by a specific binary relation, rather than universally across the entire space. Later on, many researchers transform famous fixed-point results in the setting of binary relation, sometimes called relational metric spaces. For instance, the author of this paper, Din, along with other collaborators in [3,4,5], explored the classical Perov fixed-point theorems within the framework of binary relations. They also provided illustrative examples and discussed applications in the field of vector-valued metric spaces, demonstrating the versatility of their approach. Alam et al. in [6] extended and refined the results of [2] by employing weaker conditions to establish the fixed-point results for self-mappings equipped with binary relations.

Not long ago, Berinde and Păcurar [7] pioneered the concept of enriched contractions, offering a refined perspective in fixed-point theory, a novel class of contractive operators that extend the framework of Banach contractions and encompass a variety of non-expansive mappings. They showed that enriched contractions guarantee a unique fixed point in Banach spaces, which can be efficiently approximated using Krasnoselskii’s successive sequence. In the framework of a normed space $(P,\parallel ·\parallel )$, a function $T:P\to P$ is defined as an enriched contraction operator if $\exists \mathfrak{e}\ge 0$ and $\mathring{\beta }\in [0,\mathfrak{e}+1)$, such that the following condition holds:

$$\parallel \mathfrak{e}(\mathring{\varkappa }-\mathring{\kappa })+T\left(\mathring{\varkappa }\right)-T\left(\mathring{\kappa }\right)\parallel \le \mathring{\beta }\parallel \mathring{\varkappa }-\mathring{\kappa }\parallel ,\mathrm{for}\mathrm{all}\mathring{\varkappa },\mathring{\kappa }\in P.$$

This concept unifies non-expansive and contractive mappings while consistently guaranteeing unique fixed points, establishing enriched contractions as a significant tool in fixed-point theory. Consider $\mathfrak{D}\subseteq P$ as a convex set with $\mu \in (0,1]$ and $T:\mathfrak{D}\to \mathfrak{D}$. The operator $T_{\mu }:\mathfrak{D}\to \mathfrak{D}$, defined by $T_{\mu }\left(\mathring{\varkappa }\right)=(1-\mu )\mathring{\varkappa }+\mu T\left(\mathring{\varkappa }\right)$, is referred to as the averaged mapping of T. Importantly, the fixed points of the averaged mapping $T_{\mu }$ are identical to the fixed points of the original mapping T. Further research in this direction has been conducted by numerous researchers; for instance, Berinde and Păcurar [8] presented the approximation concepts for enriched contractions in Banach spaces. Anjum et al. [9] explored the fractal structures endowed with enriched $(q,\theta )$-Hutchinson–Barnsley operators. Alraddadi et al. [10] developed fixed-point results and demonstrated their applications to Iterated Function Systems (IFS) for enriched Z-contractions. Meanwhile, Berinde and Păcurar [11] introduced the enriched Ćirić–Reich–Rus contractions. Ali et al. [12] developed numerical approximations for the fixed points and solutions of nonlinear integral equations, while Özger et al. [13] investigated the solutions of Fredholm integral equations using fixed-point results.

Let us revisit some fundamental concepts from the metric fixed-point theory for binary relations.

Definition 1

([14]). Assume that $P\ne \varphi$, and let $Q\subseteq P\times P$ denote a binary relation on P. For any two elements $\mathring{\varkappa },\mathring{\kappa }\in P$, exactly one of the following conditions holds:

(i)

$(\mathring{\varkappa },\mathring{\kappa })\in Q$: this indicates that $\mathring{\varkappa }$ is related to $\mathring{\kappa }$ under Q, commonly written as $\mathring{\varkappa }Q\mathring{\kappa }$.

(ii)

$(\mathring{\varkappa },\mathring{\kappa })\notin Q$: this signifies that $\mathring{\varkappa }$ is not related to $\mathring{\kappa }$ under Q.

By definition, $P\times P$, Ø and ${\Delta }_P=\{(\mathring{\varkappa },\mathring{\varkappa }):\mathring{\varkappa }\in P\}$ qualify as binary relations on P, referred to as the universal relation, empty relation, and diagonal relation, respectively.

In this paper, Q represents the “nonempty binary relation”; we will simply use the term “binary relation”. In the following, we introduce a series of fundamental notions, including key definitions and propositions, that form the basis for metrical theoretic-order fixed-point theorems. These concepts are essential for understanding the underlying structure and properties of such theorems. By establishing these foundational ideas, we provide the necessary groundwork for further exploration and results.

Definition 2

([14]). Consider $P\ne \varphi$ with a binary relation denoted by Q. Let $\mathring{\varkappa },\mathring{\kappa }\in P$, then $\mathring{\varkappa }$ and $\mathring{\kappa }$ are said to be Q-comparative if either $(\mathring{\varkappa },\mathring{\kappa })\in Q$ or $(\mathring{\kappa },\mathring{\varkappa })\in Q$. This relationship is denoted as $[\mathring{\varkappa },\mathring{\kappa }]\in Q$.

Proposition 1

([2]). Take a metric space $(P,\rho )$, with $Q\subseteq P\times P$, where T is a mapping from P into itself, and $\mathfrak{B}$ is a constant belonging to the interval $[0,1)$. Then, the following are equivalent for all $\mathring{\varkappa },\mathring{\kappa }\in P$:

(I)

$(\mathring{\varkappa },\mathring{\kappa })\in Q\Rightarrow \rho (T\mathring{\varkappa },T\mathring{\kappa })\le \mathfrak{B}\rho (\mathring{\varkappa },\mathring{\kappa })$,

(II)

$[\mathring{\varkappa },\mathring{\kappa }]\in Q\Rightarrow \rho (T\mathring{\varkappa },T\mathring{\kappa })\le \mathfrak{B}\rho (\mathring{\varkappa },\mathring{\kappa })$.

The following classification highlights key properties of binary relations on a nonempty set. These properties, such as reflexivity, symmetry, and transitivity, are essential for understanding the structure and behavior of relations in both theoretical studies and practical applications.

Definition 3

([14,15]). A binary relation Q can be categorized as follows:

1.

If $\forall \mathring{\varkappa }\in P,(\mathring{\varkappa },\mathring{\varkappa })\in Q$, then Q is reflexive.

2.

If $\forall \mathring{\varkappa },\mathring{\kappa },\mathring{g}\in P,(\mathring{\varkappa },\mathring{\kappa })\in Q\mathrm{and}(\mathring{\kappa },\mathring{g})\in Q\Rightarrow (\mathring{\varkappa },\mathring{g})\in Q$, then Q is transitive.

3.

If $\forall \mathring{\varkappa },\mathring{\kappa }\in P,[\mathring{\varkappa },\mathring{\kappa }]\in Q$, then Q is complete.

4.

If $\forall \mathring{\varkappa },\mathring{\kappa }\in P,[\mathring{\varkappa },\mathring{\kappa }]\in Q\mathrm{or}\mathring{\varkappa }=\mathring{\kappa }$, then Q is weakly complete.

Definition 4

([14]). Let $P\ne \varphi$ with Q be a binary relation. The following operations are introduced:

1.

The transpose of Q, denoted by $Q^{-1}$, is defined as

$$Q^{-1}=\{(\mathring{\varkappa },\mathring{\kappa })\in P\times P:(\mathring{\kappa },\mathring{\varkappa })\in Q\}.$$

2.

$Q^r$, the reflexive closure of Q, is characterized as:

$$Q^r=Q\cup {\Delta }_P.$$

3.

$Q^s$, the symmetric closure of Q, is defined as:

$$Q^s=Q\cup Q^{-1}.$$

Proposition 2

([2,14]). Let $P\ne \varphi$ with Q be a binary relation, then

$$[\mathring{\varkappa },\mathring{\kappa }]\in Q\iff (\mathring{\varkappa },\mathring{\kappa })\in Q^s.$$

Definition 5

([2]). Assume that $P\ne \varphi$, on which a binary relation Q is defined. A sequence $\left\{{\mathring{\varkappa }}_j\right\}\subset P$ is said to be Q-preserving if it meets the following condition:

$$({\mathring{\varkappa }}_j,{\mathring{\varkappa }}_{j+1})\in Q,\forall j\in \mathbb{N}\cup \left\{0\right\}.$$

Definition 6

([2]). Let $P\ne \varphi$ be a metric space with Q as a binary relation. Then, Q is known as ρ-self-closed if, for any Q-preserving sequence $\left\{{\mathring{\varkappa }}_j\right\}\subset P$ such that ${\mathring{\varkappa }}_j\stackrel{\rho }{\to }\mathring{\varkappa }$, there exists a subsequence $\left\{{\mathring{\varkappa }}_{j_m}\right\}\subset \left\{{\mathring{\varkappa }}_j\right\}$ such that $[{\mathring{\varkappa }}_{j_m},\mathring{\varkappa }]\in Q$ for all $m\in \mathbb{N}\cup \left\{0\right\}$.

Definition 7

([2]). Assume that $P\ne \varphi$ and $T:P\to P$. Then, $Q\subseteq P\times P$ is called T-closed if, for any $\mathring{\varkappa },\mathring{\kappa }\in P$, the following condition holds:

$$(\mathring{\varkappa },\mathring{\kappa })\in Q\Rightarrow (T\mathring{\varkappa },T\mathring{\kappa })\in Q.$$

Proposition 3

([16]). If the binary relation Q is closed under the operator T, then it remains closed under every iterate $T^j$ for all $j\in \mathbb{N}\cup \left\{0\right\}$, where $T^j$ represents the j-th iterates of T.

Proposition 4

([2]). Let P, T, and Q be as described in Definition 7. If Q is T-closed, then its symmetric closure, $Q^s$, is also T-closed.

Definition 8

([17]). A metric space $(P,\rho )$ is known as Q-completeness if every Cauchy sequence in P that respects the binary relation Q has a limit that also belongs to P.

Clearly, any complete metric space is Q-complete, irrespective of the choice of the binary relation Q. Specifically, when Q is the universal relation, Q-completeness aligns with the conventional definition of completeness in metric spaces.

Definition 9

([18]). Assume that $P\ne \varphi$ with $Q\subseteq P\times P$. A subset $\mathfrak{E}\subseteq P$ is considered Q-directed if, for any $\mathring{\varkappa },\mathring{\kappa }\in \mathfrak{E}$, $\exists \mathring{g}\in P$ such that

$$(\mathring{\varkappa },\mathring{g})\in Q\mathrm{and}(\mathring{\kappa },\mathring{g})\in Q.$$

Definition 10

([19]). Assume that $P\ne \varphi$ with $Q\subseteq P\times P$. For $\mathring{\varkappa },\mathring{\kappa }\in P$, a path of length s (where $s\in \mathbb{N}$) in Q from $\mathring{\varkappa }$ to $\mathring{\kappa }$ is defined as a finite sequence

$$\{{\mathring{g}}_0,{\mathring{g}}_1,{\mathring{g}}_2,\dots ,{\mathring{g}}_s\}\subseteq P$$

such that

(i)

${\mathring{g}}_0=\mathring{\varkappa }$ and ${\mathring{g}}_s=\mathring{\kappa },$

(ii)

$({\mathring{g}}_j,{\mathring{g}}_{j+1})\in Q$ for each j with $0\le j\le s-1$.

In this paper, we adopt the following notations:

1.

$\mathrm{Fix}\left(T\right)$: The set of all fixed points of T,

2.

$Q_T:=\{\mathring{\varkappa }\in P$: $(\mathring{\varkappa },T\left(\mathring{\varkappa }\right))\in Q\}$,

3.

$\mathrm{Paths}(\mathring{\varkappa },\mathring{\kappa };Q)$: The collection of all paths in Q from $\mathring{\varkappa }$ to $\mathring{\kappa }$

4.

for $\mathfrak{e}\ge 0$, we define $\mu ={\displaystyle \frac{1}{\mathfrak{e}+1}}\in (0,1]$.

Before moving forward, we provide the definitions of $\parallel ·\parallel$-self-closedness and Q-completeness within the framework of normed spaces. Additionally, we outline several key results that are instrumental in our study.

Definition 11.

A normed space $(P,\parallel ·\parallel )$ is known to be Q-complete if every Cauchy sequence in P that respects the relation Q converges under the norm $\parallel ·\parallel$.

Definition 12.

Let T be a mapping defined on a normed space. The mapping is known to be Q-continuous at a point $\mathring{\varkappa }\in P$ if, for any sequence $\left\{{\mathring{\varkappa }}_n\right\}$ in P that preserves the relation Q and satisfies $\parallel {\mathring{\varkappa }}_n-\mathring{\varkappa }\parallel \to 0$, it holds that $\parallel T\left({\mathring{\varkappa }}_n\right)-T\left(\mathring{\varkappa }\right)\parallel \to 0$. Furthermore, T is known as Q-continuous if this property holds at every point in P.

Definition 13.

Assume that P is a normed space with the norm $\parallel ·\parallel$. A binary relation Q on P is said to be $\parallel ·\parallel$-self-closed if, for any Q-preserving sequence $\left\{{\mathring{\mathfrak{m}}}_j\right\}$ satisfying $\parallel {\mathring{\mathfrak{m}}}_j-\mathring{\mathfrak{m}}\parallel \to 0$, there exists a subsequence $\left\{{\mathring{\mathfrak{m}}}_{j_l}\right\}\subset \left\{{\mathring{\mathfrak{m}}}_l\right\}$ such that $[{\mathring{\mathfrak{m}}}_{j_l},\mathring{\mathfrak{m}}]\in Q$ for all $l\in \mathbb{N}\cup \left\{0\right\}$.

Before proceeding, we revisit the concept of altering-distance functions, which are fundamental in the context of our results. These functions are instrumental in generalizing and extending classical contraction principles.

Definition 14.

A function $\psi :{\mathbb{R}}_{\ge 0}\to {\mathbb{R}}_{\ge 0}$ is known as an altering-distance function that fulfills the given conditions:

(${\psi }_1$)

ψ is non-decreasing and continuous.

(${\psi }_2$)

$\psi \left(c\right)=0$⟺$c=0$.

Dutta and Choudhury in [20] utilized these altering functions to establish a generalized fixed-point theorem, which is presented below.

Theorem 1.

Consider a complete metric space $(P,\rho )$, with $T:P\to P$ that satisfies the inequality:

$$\psi \left(\rho (T\mathring{\varkappa },T\mathring{\kappa })\right)\le \psi \left(\rho (\mathring{\varkappa },\mathring{\kappa })\right)-\mathrm{\Phi }\left(\rho (\mathring{\varkappa },\mathring{\kappa })\right),\mathit{for}\mathit{all}\mathring{\varkappa },\mathring{\kappa }\in P,$$

where ψ and Φ are altering-distance functions; it follows that T has a unique fixed point.

Building on the foundational ideas of Alam and Imdad and their collaborators [2,6], as well as the contributions of Harjani and Sadarangani [21], Salma and Din [22], and Berinde and Păcurar [8,11], we present the concept of enriched Q-preserving weak and almost contractions. Furthermore, we establish the associated fixed-point theorems within a linear normed space with an arbitrary binary relation. We present illustrative examples to demonstrate the practical relevance and applicability of our results. Additionally, the paper emphasizes the importance of these findings by applying them to ordinary differential equations. The existence conditions presented in this work can be implemented computationally using numerical methods such as iterative approximation schemes and fixed-point algorithms. These techniques enable the practical computation of solutions for first- and second-order ODEs by generating successive approximations that converge under the given conditions, bridging the gap between theoretical results and numerical applications.

2. Main Results

To begin, we introduce our primary definition, which serves as the foundation for the results discussed in this paper. This definition encapsulates the key concepts and forms the basis for exploring the enriched contraction principles and their applications.

Definition 15.

Assume that $(P,\parallel ·\parallel )$ is any normed space and $Q\subseteq P\times P$. An operator $T:P\to \mathcal{Y}$, where $P\subseteq \mathcal{Y}$, is called an enriched weak Q-preserving contraction if there exists some non-negative $\mathfrak{e}$ such that

$$\psi [{\displaystyle \frac{\parallel \mathfrak{e}(\mathring{\varkappa }-\mathring{\kappa })+T\mathring{\varkappa }-T\mathring{\kappa }\parallel }{\mathfrak{e}+1}}]\le \psi [\parallel \mathring{\varkappa }-\mathring{\kappa }\parallel ]-\mathrm{\Phi }[\parallel \mathring{\varkappa }-\mathring{\kappa }\parallel ],\forall \mathring{\varkappa },\mathring{\kappa }\in P\mathrm{with}(\mathring{\varkappa },\mathring{\kappa })\in Q,$$

(1)

where ψ and Φ are altering-distance functions.

To account for the binary relation and control functions appearing in the above definition, we establish T as a $(\mathfrak{e},\psi ,\mathrm{\Phi })$-enriched weak Q-preserving contraction.

By leveraging the symmetry of the norm $\parallel ·\parallel$, the following proposition is immediate:

Proposition 5.

A mapping T is a $(\mathfrak{e},\psi ,\mathrm{\Phi })$-enriched weak Q-preserving contraction if and only if T is a $(\mathfrak{e},\psi ,\mathrm{\Phi })$-enriched weak $Q^s$-preserving contraction.

The associated fixed-point theorem for the above-mentioned $(\mathfrak{e},\psi ,\mathrm{\Phi })$-enriched weak $Q^s$-preserving contraction is given below.

Theorem 2.

Assume that $(P,\parallel ·\parallel )$ be any normed space with $Q\subseteq P\times P$. Also, suppose that the following holds:

1.

$(P,\parallel ·\parallel )$ is Q-complete;

2.

T is a $(\mathfrak{e},\psi ,\mathrm{\Phi })$-enriched weak Q-preserving contraction;

3.

Q is $T_{\mu }$-closed, where $\mu ={\displaystyle \frac{1}{\mathfrak{e}+1}}$;

4.

either Q is $\parallel ·\parallel$-self-closed or $T_{\mu }$ is Q-continuous;

5.

$Q_{T_{\mu }}\ne \varphi$.

Consequently, T contains at least one fixed point within P.

Proof. 

By definition of the enriched weak Q-preserving contraction condition, (1) transforms into

$$\psi [\mu \parallel ({\displaystyle \frac{1}{\mu }}-1)(\mathring{\varkappa }-\mathring{\kappa })+T\mathring{\varkappa }-T\mathring{\kappa }\parallel ]\le \psi [\parallel \mathring{\varkappa }-\mathring{\kappa }\parallel ]-\mathrm{\Phi }[\parallel \mathring{\varkappa }-\mathring{\kappa }\parallel ],\forall \mathring{\varkappa },\mathring{\kappa }\in P\mathrm{with}(\mathring{\varkappa },\mathring{\kappa })\in Q,$$

equivalently,

$$\psi [\parallel (1-\mu )(\mathring{\varkappa }-\mathring{\kappa })+\mu (T\mathring{\varkappa }-T\mathring{\kappa })\parallel ]\le \psi [\parallel \mathring{\varkappa }-\mathring{\kappa }\parallel ]-\mathrm{\Phi }[\parallel \mathring{\varkappa }-\mathring{\kappa }\parallel ],\forall \mathring{\varkappa },\mathring{\kappa }\in P\mathrm{with}(\mathring{\varkappa },\mathring{\kappa })\in Q.$$

Utilizing the definition of the average operator, we derive that

$$\psi [\parallel T_{\mu }\mathring{\varkappa }-T_{\mu }\mathring{\kappa }\parallel ]\le \psi [\parallel \mathring{\varkappa }-\mathring{\kappa }\parallel ]-\mathrm{\Phi }[\parallel \mathring{\varkappa }-\mathring{\kappa }\parallel ],\forall \mathring{\varkappa },\mathring{\kappa }\in P\mathrm{with}(\mathring{\varkappa },\mathring{\kappa })\in Q.$$

(2)

There exists an element ${\mathring{\varkappa }}_0\in Q_{T_{\mu }}$. Define a Kransnoselskij iterative scheme $\left\{{\mathring{\varkappa }}_j\right\}$ at the initial guess ${\mathring{\varkappa }}_0$, that is, ${\mathring{\varkappa }}_{j+1}=(1-\mu ){\mathring{\varkappa }}_j+\mu T{\mathring{\varkappa }}_j$, $\forall j\in \mathbb{N}$. As $\left({\mathring{\varkappa }}_0,T{\mathring{\varkappa }}_0\right)\in Q$, using the $T_{\mu }$-closedness of Q and Proposition 3, we yield

$$\left(T^j{\mathring{\varkappa }}_0,T^{j+1}{\mathring{\varkappa }}_0\right)\in Q,\forall j\in \mathbb{N},$$

(3)

which further implies that

$$\left({\mathring{\varkappa }}_j,{\mathring{\varkappa }}_{j+1}\right)\in Q,\forall j\in \mathbb{N}.$$

(4)

It shows that the Kransnoselskij iterative scheme $\left\{{\mathring{\varkappa }}_j\right\}$ is Q-preserving. If there is some $j_1\in \mathbb{N}$ such that ${\mathring{\varkappa }}_{j_1}={\mathring{\varkappa }}_{j_1+1}$, it shows that $T{\mathring{\varkappa }}_{j_1}={\mathring{\varkappa }}_{j_1}$, indicating that T has a fixed point. So, in the remaining part of the proof, we suppose that ${\mathring{\varkappa }}_j\ne {\mathring{\varkappa }}_{j+1}$ for each $j\in \mathbb{N}$.

Using the Q-preserving weak contractivity condition (2)–(4), we deduce for all $j\in \mathbb{N}$ that

$$\begin{array}{ccc}\hfill \psi [\parallel {\mathring{\varkappa }}_j-{\mathring{\varkappa }}_{j+1}\parallel ]& =& \psi [\parallel T{\mathring{\varkappa }}_{j-1}-T{\mathring{\varkappa }}_j\parallel ]\hfill \\ & \le & \psi [\parallel {\mathring{\varkappa }}_{j-1}-{\mathring{\varkappa }}_j\parallel ]-\mathrm{\Phi }[\parallel {\mathring{\varkappa }}_{j-1}-{\mathring{\varkappa }}_j\parallel ]\hfill \\ & \le & \psi [\parallel {\mathring{\varkappa }}_{j-1}-{\mathring{\varkappa }}_j\parallel ].\hfill \end{array}$$

(5)

By leveraging the property that $\psi$ is non-decreasing, we obtain

$$\begin{array}{c}\hfill \parallel {\mathring{\varkappa }}_j-{\mathring{\varkappa }}_{j+1}\parallel \le \parallel {\mathring{\varkappa }}_{j-1}-{\mathring{\varkappa }}_j\parallel .\end{array}$$

This shows that $\{\parallel {\mathring{\varkappa }}_j-{\mathring{\varkappa }}_{j+1}\parallel \}$ is a non-decreasing sequnece of non-negative numbers and is bounded by 0, which implies ${lim}_{j\to \infty }\parallel {\mathring{\varkappa }}_j-{\mathring{\varkappa }}_{j+1}\parallel =w\ge 0$. If $w>0$ then passing to the limit as $j\to \infty$ in (5), we obtain

$$\begin{array}{c}\hfill \psi \left(w\right)\le \psi \left(w\right)-\mathrm{\Phi }\left(w\right)\le \psi \left(w\right),\end{array}$$

which yields $\mathrm{\Phi }\left(w\right)=0$, that is, $w=0$, which is a contradiction. Hence, $w>0$ is not possible, which further yields

$$w=\underset{j\to \infty }{lim}\parallel {\mathring{\varkappa }}_j-{\mathring{\varkappa }}_{j+1}\parallel =0.$$

(6)

Next, we will illustrate that $\left\{{\mathring{\varkappa }}_j\right\}$ forms a Cauchy sequence. Suppose, for the sake of contradiction, that $\left\{{\mathring{\varkappa }}_j\right\}$ is not a Cauchy sequence. Then, for some $\mathfrak{y}>0$, one can take two subsequences $\left\{{\mathring{\varkappa }}_{m\left(j\right)}\right\}$ and $\left\{{\mathring{\varkappa }}_{n\left(j\right)}\right\}$ of $\left\{{\mathring{\varkappa }}_j\right\}$, where $n\left(j\right)>m\left(j\right)>j$, satisfying:

$$\parallel {\mathring{\varkappa }}_{n\left(j\right)}-{\mathring{\varkappa }}_{m\left(j\right)}\parallel \ge \mathfrak{y}.$$

(7)

Furthermore, for each $m\left(j\right)$, $n\left(j\right)$ can be selected as the smallest integer greater than $m\left(j\right)$ that satisfies the above condition. Then,

$$\parallel {\mathring{\varkappa }}_{n\left(j\right)-1}-{\mathring{\varkappa }}_{m\left(j\right)}\parallel <\mathfrak{y}.$$

(8)

By applying the triangle inequality along with (7) and (8), we derive

$$\begin{array}{cc}\hfill \mathfrak{y}& \le \parallel {\mathring{\varkappa }}_{n\left(j\right)}-{\mathring{\varkappa }}_{m\left(j\right)}\parallel \hfill \\ & \le \parallel {\mathring{\varkappa }}_{n\left(j\right)}-{\mathring{\varkappa }}_{n\left(j\right)-1}\parallel +\parallel {\mathring{\varkappa }}_{n\left(j\right)-1}-{\mathring{\varkappa }}_{m\left(j\right)}\parallel \hfill \\ & <\parallel {\mathring{\varkappa }}_{n\left(j\right)}-{\mathring{\varkappa }}_{n\left(j\right)-1}\parallel +\mathfrak{y}.\hfill \end{array}$$

Letting $j\to \infty$, and using the fact that $\parallel {\mathring{\varkappa }}_{j+1}-{\mathring{\varkappa }}_j\parallel \to 0$, we obtain

$$\underset{j\to \infty }{lim}\parallel {\mathring{\varkappa }}_{n\left(j\right)}-{\mathring{\varkappa }}_{m\left(j\right)}\parallel =\mathfrak{y}.$$

(9)

Again, using the triangle inequality, we can write

$$\begin{array}{cc}\hfill \parallel {\mathring{\varkappa }}_{n\left(j\right)}-{\mathring{\varkappa }}_{m\left(j\right)}\parallel & \le \parallel {\mathring{\varkappa }}_{n\left(j\right)}-{\mathring{\varkappa }}_{n\left(j\right)-1}\parallel +\parallel {\mathring{\varkappa }}_{n\left(j\right)-1}-{\mathring{\varkappa }}_{m\left(j\right)-1}\parallel + \parallel {\mathring{\varkappa }}_{m\left(j\right)-1}-{\mathring{\varkappa }}_{m\left(j\right)}\parallel ,\hfill \\ \hfill \parallel {\mathring{\varkappa }}_{n\left(j\right)-1}-{\mathring{\varkappa }}_{m\left(j\right)-1}\parallel & \le \parallel {\mathring{\varkappa }}_{n\left(j\right)-1}-{\mathring{\varkappa }}_{n\left(j\right)}\parallel +\parallel {\mathring{\varkappa }}_{n\left(j\right)}-{\mathring{\varkappa }}_{m\left(j\right)}\parallel +\parallel {\mathring{\varkappa }}_{m\left(j\right)}-{\mathring{\varkappa }}_{m\left(j\right)-1}\parallel .\hfill \end{array}$$

Letting $j\to \infty$ in the above inequalities and using the convergence of $\parallel {\mathring{\varkappa }}_{j+1}-{\mathring{\varkappa }}_j\parallel \to 0$, we find

$$\underset{j\to \infty }{lim}\parallel {\mathring{\varkappa }}_{n\left(j\right)-1}-{\mathring{\varkappa }}_{m\left(j\right)-1}\parallel =\mathfrak{y}.$$

(10)

Finally, applying the definition of the altering-distance function $\psi$, we have

$$\psi \left(\parallel {\mathring{\varkappa }}_{n\left(j\right)}-{\mathring{\varkappa }}_{m\left(j\right)}\parallel \right)\le \psi \left(\parallel {\mathring{\varkappa }}_{n\left(j\right)-1}-{\mathring{\varkappa }}_{m\left(j\right)-1}\parallel \right)-\mathrm{\Phi }\left(\parallel {\mathring{\varkappa }}_{n\left(j\right)-1}-{\mathring{\varkappa }}_{m\left(j\right)-1}\parallel \right)$$

$$\le \psi \left(\parallel {\mathring{\varkappa }}_{n\left(j\right)-1}-{\mathring{\varkappa }}_{m\left(j\right)-1}\parallel \right).$$

Letting $j\to \infty$, and using the results from (9) and (10), we obtain

$$\psi \left(\mathfrak{y}\right)\le \psi \left(\mathfrak{y}\right)-\mathrm{\Phi }\left(\mathfrak{y}\right)\le \psi \left(\mathfrak{y}\right).$$

This yields $\mathrm{\Phi }\left(\mathfrak{y}\right)=0$. Given that $\psi$ is an altering-distance function, $\mathrm{\Phi }\left(\mathfrak{y}\right)=0$ implies $\mathfrak{y}=0$, which contradicts the premise. So, $\left\{{\mathring{\varkappa }}_j\right\}$ is proven to be indeed a Cauchy sequence, which further implies $\left\{{\mathring{\varkappa }}_j\right\}$ is an Q-preserving Cauchy sequence. By the Q-completeness of P, $\exists \mathring{g}\in P$ such that ${\mathring{\varkappa }}_j\stackrel{\parallel ·\parallel }{\to }\mathring{g}$. Finally, we demonstrate that $\mathring{g}$ is a fixed point of T. Assume that $T_{\mu }$ is Q-continuous. Since $\left\{{\mathring{\varkappa }}_j\right\}$ is an Q-preserving sequence and ${\mathring{\varkappa }}_j\stackrel{\parallel ·\parallel }{\to }\mathring{g}$, the Q-continuity of $T_{\mu }$ implies that ${\mathring{\varkappa }}_{j+1}=T_{\mu }\left({\mathring{\varkappa }}_j\right)\stackrel{\parallel ·\parallel }{\to }{T}_{\mu }\left(\mathring{g}\right)$. By the uniqueness of the limit, we conclude that $T_{\mu }\left(\mathring{g}\right)=\mathring{g}$, meaning $\mathring{g}$ is a fixed point of $T_{\mu }$. Consequently, $\mathring{g}$ is also a fixed point of T.

On the other hand, assume that Q is $\parallel ·\parallel$-self-closed. Since $\left\{{\mathring{\varkappa }}_j\right\}$ is a Q-preserving sequence and ${\mathring{\varkappa }}_j\stackrel{\parallel ·\parallel }{\to }\mathring{g}$, there exists a subsequence $\left\{{\mathring{\varkappa }}_{j_n}\right\}\subseteq \left\{{\mathring{\varkappa }}_j\right\}$ such that $\left[{\mathring{\varkappa }}_{j_n},\mathring{g}\right]\in Q$ for all $n\in \mathbb{N}$. By utilizing Proposition 5, the fact that $\left[{\mathring{\varkappa }}_{j_n},\mathring{g}\right]\in Q$, and ${\mathring{\varkappa }}_{j_n}\stackrel{\parallel ·\parallel }{\to }\mathring{g}$, we deduce:

$$\begin{array}{ccc}\hfill \psi [\parallel {\mathring{\varkappa }}_{j_n+1}-T_{\mu }\mathring{g}\parallel ]& =& \psi [\parallel T_{\mu }{\mathring{\varkappa }}_{j_n}-T_{\mu }\mathring{g}\parallel ]\hfill \\ & \le & \psi [\parallel {\mathring{\varkappa }}_{j_n}-\mathring{g}\parallel ]-\mathrm{\Phi }[\parallel {\mathring{\varkappa }}_{j_n}-\mathring{g}\parallel ]\hfill \\ & \le & \psi [\parallel {\mathring{\varkappa }}_{j_n}-\mathring{g}\parallel ].\hfill \end{array}$$

Taking the limit as $j\to \infty$, we arrive at $\psi (\parallel \mathring{g}-T_{\mu }\mathring{g}\parallel )=0$, which leads to $\parallel \mathring{g}-T_{\mu }\mathring{g}\parallel =0$. Consequently, $T_{\mu }\mathring{g}=\mathring{g}$, proving that $\mathring{g}$ is a fixed point of $T_{\mu }$. Consequently, T has a fixed point. □

We provide an example demonstrating that the assumptions of Theorem 2 do not validate the uniqueness of fixed points. Consider $P={\mathbb{R}}^2$ with the binary relation $Q=\left\{\right(1,0),(0,1\left)\right\}$. In this scenario, the distinct elements are not comparable. Moreover, P is a complete metric space under the Euclidean distance. The identity mapping $T(\mathring{\varkappa },\mathring{\kappa })=(\mathring{\varkappa },\mathring{\kappa })$ is continuous, and all other assumptions of Theorem 2 are fulfilled. However, T has two fixed points: $(1,0)$ and $(0,1)$. This demonstrates that the conditions in Theorem 2 do not ensure uniqueness of fixed points.

To establish the uniqueness, in Theorem 2, we present a result that provides a sufficient condition for ensuring uniqueness. This key result not only highlights the interplay between the assumptions and the structure of the mapping but also demonstrates how the given conditions guarantee that the fixed point is unique.

Theorem 3.

If, for all $\mathring{\varkappa },\mathring{\kappa }\in P$, $\exists \mathring{g}\in P$ such that $[\mathring{g},\mathring{\varkappa }]$, $[\mathring{g},\mathring{\kappa }]\in Q$, then T possesses a unique fixed point within P.

Proof. 

Suppose $\exists \mathfrak{w},\mathring{g}\in P$ are fixed points of T. The discussion is then divided into the following two cases for further analysis.

Case 1: Suppose $\mathfrak{w}$ is comparable to $\mathring{g}$. Then, for every $j\in \{0,1,2,\dots \}$, the iterates $T^j\left(\mathfrak{w}\right)=\mathfrak{w}$ and $T^j\left(\mathring{g}\right)=\mathring{g}$ continue to be comparable. Consequently,

$$\begin{array}{cc}\hfill \psi \left(\parallel \mathring{g}-\mathfrak{w}\parallel \right)& =\psi \left(\parallel T^j\left(\mathring{g}\right)-T^j\left(\mathfrak{w}\right)\parallel \right)\hfill \\ & \le \psi \left(\parallel T^{j-1}\left(\mathring{g}\right)-T^{j-1}\left(\mathfrak{w}\right)\parallel \right)-\mathrm{\Phi }\left(\parallel T^{j-1}\left(\mathring{g}\right)-T^{j-1}\left(\mathfrak{w}\right)\parallel \right)\hfill \\ & \le \psi \left(\parallel \mathring{g}-\mathfrak{w}\parallel \right)-\mathrm{\Phi }\left(\parallel \mathring{g}-\mathfrak{w}\parallel \right).\hfill \end{array}$$

Since $\psi$ and $\mathrm{\Phi }$ are altering-distance functions, the final inequality yields $\mathrm{\Phi }\left(\parallel \mathring{g}-\mathfrak{w}\parallel \right)=0$. Given the properties of $\mathrm{\Phi }$, this implies that $\parallel \mathring{g}-\mathfrak{w}\parallel =0$, and hence $\mathring{g}=\mathfrak{w}$.

Case 2: Assume that $\mathfrak{w}$ is not comparable to $\mathring{g}$; the assumption ensures the existence of $\mathring{\varkappa }\in P$ that is comparable to both $\mathfrak{w}$ and $\mathring{g}$. By the monotonicity property of T, the iterates $T^j\left(\mathring{\varkappa }\right)$ remain comparable to both $T^j\left(\mathfrak{w}\right)=\mathfrak{w}$ and $T^j\left(\mathring{g}\right)=\mathring{g}$ for every $j\ge 0$. Furthermore, we obtain

$$\begin{array}{cc}\hfill \psi \left(\parallel \mathring{g}-T^j\left(\mathring{\varkappa }\right)\parallel \right)& =\psi \left(\parallel T^j\left(\mathring{g}\right)-T^j\left(\mathring{\varkappa }\right)\parallel \right)\hfill \\ & \le \psi \left(\parallel T^{j-1}\left(\mathring{g}\right)-T^{j-1}\left(\mathring{\varkappa }\right)\parallel \right)-\mathrm{\Phi }\left(\parallel T^{j-1}\left(\mathring{g}\right)-T^{j-1}\left(\mathring{\varkappa }\right)\parallel \right)\hfill \\ & \le \psi \left(\parallel T^{j-1}\left(\mathring{g}\right)-T^{j-1}\left(\mathring{\varkappa }\right)\parallel \right)\hfill \\ & =\psi \left(\parallel \mathring{g}-T^j\left(\mathring{\varkappa }\right)\parallel \right).\hfill \end{array}$$

Therefore, the final inequality establishes that the sequence $\left\{\psi \left(\parallel \mathring{g}-T^j\left(\mathring{\varkappa }\right)\parallel \right)\right\}$ is non-negative and monotonically decreasing. Due to the monotonicity of $\psi$, the sequence $\left\{\parallel \mathring{g}-T^j\left(\mathring{\varkappa }\right)\parallel \right\}$ is also non-negative and decreasing, ensuring the existence of $\gamma$ such that

$$\underset{j\to \infty }{lim}\parallel \mathring{g}-T^j\left(\mathring{\varkappa }\right)\parallel =\gamma .$$

As $j\to \infty$, in the inequality above and leveraging the characteristics of $\psi$ and $\mathrm{\Phi }$, we obtain:

$$\psi \left(\gamma \right)\le \psi \left(\gamma \right)-\mathrm{\Phi }\left(\gamma \right).$$

This implies that $\mathrm{\Phi }\left(\gamma \right)=0$, and consequently, $\gamma =0$. In a similar fashion, one can demonstrate that

$$\underset{j\to \infty }{lim}\parallel \mathfrak{w}-T^j\left(\mathring{\varkappa }\right)\parallel =0.$$

Finally, since

$$\underset{j\to \infty }{lim}\parallel \mathring{g}-T^j\left(\mathring{\varkappa }\right)\parallel =\underset{j\to \infty }{lim}\parallel \mathfrak{w}-T^j\left(\mathring{\varkappa }\right)\parallel =0.$$

Since the limit is unique, it follows that $\mathring{g}=\mathfrak{w}$. □

Remark 1.

Under the conditions stated in Theorem 3, it can be proven that for any $\mathring{\varkappa }\in P$, the iterative sequence $T^j\left(\mathring{\varkappa }\right)$ converges to the fixed point $\mathring{g}$ as j approaches infinity. This establishes that the operator T is Picard-convergent. If $\mathring{\varkappa }$ is comparable to $\mathring{g}$, then, following the same reasoning as in Theorem 3, it holds that

$$\underset{j\to \infty }{lim}\parallel \mathring{g}-T^j\left(\mathring{\varkappa }\right)\parallel =0,$$

and consequently

$$\underset{j\to \infty }{lim}{T}^j\left(\mathring{\varkappa }\right)=\mathring{g}.$$

If $\mathring{\varkappa }$ is not comparable to $\mathring{g}$, let $\mathring{\kappa }\in P$ be an element that is comparable to both $\mathring{\varkappa }$ and $\mathring{g}$. Using the same reasoning as in Theorem 3, it follows that

$$\underset{j\to \infty }{lim}\parallel \mathring{g}-T^j\left(\mathring{\kappa }\right)\parallel =0,\mathit{and}\underset{j\to \infty }{lim}\parallel T^j\left(\mathring{g}\right)-T^j\left(\mathring{\kappa }\right)\parallel =0,$$

as well as

$$\underset{j\to \infty }{lim}\parallel T^j\left(\mathring{\varkappa }\right)-T^j\left(\mathring{\kappa }\right)\parallel =0.$$

Finally, using the triangle inequality

$$\parallel \mathring{g}-T^j\left(\mathring{\varkappa }\right)\parallel \le \parallel \mathring{g}-T^j\left(\mathring{\kappa }\right)\parallel +\parallel T^j\left(\mathring{\kappa }\right)-T^j\left(\mathring{\varkappa }\right)\parallel ,$$

and taking the limit as $j\to \infty$, we have

$$\underset{j\to \infty }{lim}\parallel \mathring{g}-T^j\left(\mathring{\varkappa }\right)\parallel =0,$$

or equivalently

$$\underset{j\to \infty }{lim}{T}^j\left(\mathring{\varkappa }\right)=\mathring{g}.$$

Remark 2.

Observe that when Q forms a totally ordered binary relation, Theorem 3 establishes that the fixed point is unique.

By setting $\mathfrak{e}=0$, we obtain $\mu =1$ in Theorem 2. This leads to a theoretical fixed-point result for $(\psi ,\mathrm{\Phi })$-weak Q-preserving contractions within the framework of normed spaces.

Corollary 1

([21]). Suppose a normed space $(P,\parallel ·\parallel )$ with $Q\subseteq P\times P$. Assume the following conditions are satisfied:

1.

$(P,\parallel ·\parallel )$ is Q-complete;

2.

T is a $(\psi ,\mathrm{\Phi })$-weak Q-preserving contraction;

3.

Q is T-closed;

4.

either Q is $\parallel ·\parallel$-self-closed or T is Q-continuous;

5.

$Q_T\ne \varphi$.

Consequently, T possesses a fixed point in P.

By defining $Q=P\times P$ in Theorem 2, we obtain the corresponding result for $(\mathfrak{e},\psi ,\mathrm{\Phi })$-enriched weak contractions within the context of normed spaces.

Corollary 2.

Assume that $(P,\parallel ·\parallel )$ is a normed space and a mapping T: $P\to P$. Further, assume that the following assumptions hold:

1.

$(P,\parallel ·\parallel )$ is complete;

2.

T is a $(\mathfrak{e},\psi ,\mathrm{\Phi })$-weak enriched contraction;

Consequently, T has exactly one fixed point in P.

By taking the binary relation $Q=P\times P$ and $\mathfrak{e}=0$ in Theorem 2, we recover the fixed-point results for T as a $(\psi ,\mathrm{\Phi })$-weak contraction, as established by Dutta and Choudhary [20] in the context of normed spaces.

Corollary 3.

Suppose that $(P,\parallel ·\parallel )$ is a normed space. Further, assume that the following assumptions hold:

1.

$(P,\parallel ·\parallel )$ is complete;

2.

T is a $(\psi ,\mathrm{\Phi })$-weak contraction;

Consequently, T has exactly one fixed point in P.

By setting the control function $\psi ={\mathrm{Id}}_P$ in Theorem 2, we obtain a broader version of the main result presented by Prasad and Dimri [23], tailored to the setting of normed spaces.

Corollary 4.

Suppose that $(P,\parallel ·\parallel )$ is a normed space with $Q\subseteq P$. Further, assume that following assumptions hold:

1.

$(P,\parallel ·\parallel )$ is Q-complete;

2.

T is a $(\mathfrak{e},\mathrm{\Phi })$-enriched weak Q-preserving contraction;

3.

Q is $T_{\mu }$-closed;

4.

either Q is $\parallel ·\parallel$-self-closed or $T_{\mu }$ is Q-continuous;

5.

$Q_{T_{\mu }}\ne \varphi$.

Consequently, T has at least one fixed point in P.

By setting the control function $\psi ={\mathrm{Id}}_P$ and $\mathfrak{b}=0$ in Theorem 2, we deduce the main result of Prasad and Dimri [23] in the context of normed spaces.

Corollary 5.

Assume that $(P,\parallel ·\parallel )$ be a normed space with $Q\subseteq P$. Further, assume that the following assumptions hold:

1.

$(P,\parallel ·\parallel )$ is Q-complete;

2.

T is a Φ-weak Q-preserving contraction;

3.

Q is T-closed;

4.

either Q is $\parallel ·\parallel$-self-closed or T is Q-continuous;

5.

$Q_T\ne \varphi$.

Consequently, T has at least one fixed point in P.

By putting the binary relation $Q=P\times P$ and $\psi =I{d}_P$ in Theorem 2, we recover the fixed-point results for enriched weak contraction in the following manner.

Corollary 6.

Suppose that $(P,\parallel ·\parallel )$ is a normed space. Further, assume that the following assumptions hold:

1.

$(P,\parallel ·\parallel )$ is complete;

2.

T is a weak enriched contraction;

Consequently, T possesses a unique fixed point in P.

By setting the binary relation $Q=P\times P$, $\psi =I{d}_P$ and $\mathfrak{e}=0$ in Theorem 2, we recover the fixed-point results for weak contraction in the context of normed spaces.

Corollary 7.

Suppose that $(P,\parallel ·\parallel )$ is a normed space. Further, assume that the following assumptions hold:

1.

$(P,\parallel ·\parallel )$ is complete;

2.

T is a weak contraction;

Consequently, T possesses a unique fixed point in P.

Next, we present a generalized version of Definition 15 and Theorem 2, which can be proven in a manner similar to Theorem 2. Therefore, we leave the proof for the readers. This generalization allows each of the corollaries mentioned earlier to be restated in this broader context. The definition and fixed-point result is stated as follows.

Definition 16.

Assume that $(P,\parallel ·\parallel )$ is a normed space with $Q\subseteq P$. A mapping $T:P\to \mathcal{Y}$, where $P\subseteq \mathcal{Y}$, is called a generalized enriched weak Q-preserving contraction if there exists some non-negative $\mathfrak{e}$ such that

$$\psi \left[{\displaystyle \frac{\parallel \mathfrak{e}(\mathring{\varkappa }-\mathring{\kappa })+T\mathring{\varkappa }-T\mathring{\kappa }\parallel }{\mathfrak{e}+1}}\right]\le \psi \left[{\mathcal{M}}_T(\mathring{\varkappa },\mathring{\kappa })\right]-\mathrm{\Phi }\left[{\mathcal{M}}_T(\mathring{\varkappa },\mathring{\kappa })\right],\forall \mathring{\varkappa },\mathring{\kappa }\in P\mathit{with}(\mathring{\varkappa },\mathring{\kappa })\in Q,$$

(11)

where ${\mathcal{M}}_T(\mathring{\varkappa },\mathring{\kappa })=max\{\parallel \mathring{\varkappa }-\mathring{\kappa }\parallel ,\parallel T\mathring{\varkappa }-\mathring{\kappa }\parallel ,\parallel \mathring{\varkappa }-T\mathring{\kappa }\parallel \}$, and ψ and Φ are altering- distance functions

Theorem 4.

Assume that $(P,\parallel ·\parallel )$ is a normed space with $Q\subseteq P$. Further, assume that the following assumptions hold:

1.

$(P,\parallel ·\parallel )$ is Q-complete;

2.

T is a $(\mathfrak{e},\psi ,\mathrm{\Phi })$-enriched weak Q-preserving contraction;

3.

Q is generalized $T_{\mu }$-closed, where $\mu ={\displaystyle \frac{1}{\mathfrak{e}+1}}$;

4.

either Q is $\parallel ·\parallel$-self-closed or $T_{\mu }$ is Q-continuous;

5.

$Q_{T_{\mu }}\ne \varphi$.

Then, T has at least one fixed point in P.

3. Application to the Differential Equations

Tunç [24,25] investigated the global existence and uniqueness of solutions for integral equations with multiple variable delays and integro-differential equations, and also pioneered the study of existence and uniqueness for solutions of Hammerstein-type functional integral equations. While in [26], Burton explains the stability of functional differential equations using fixed-point theory. In this part, we present two results to demonstrate the practical application of Theorem 2. The first example focuses on investigating the existence of solutions to a first-order periodic problem:

$$\left\{\begin{array}{c}{\mathring{v}}^{\prime }\left(\vartheta \right)=\mathring{\mathfrak{f}}(\vartheta ,\mathring{v}\left(\vartheta \right)),\vartheta \in [0,\phi ],\hfill \\ \mathring{v}\left(0\right)=\mathring{v}\left(\phi \right),\hfill \end{array}\right.$$

(12)

here, $\phi >0$, and $\mathring{\mathfrak{f}}:I\times \mathbb{R}\to \mathbb{R}$ is a continuous function. In the periodic-boundary-value problem given in Equation (12), the condition $\mathring{v}\left(0\right)=\mathring{v}\left(\phi \right)$ ensures continuity and periodicity of the solution over the interval $[0,\phi ]$. This structure is crucial for modeling systems with recurrent behavior, where the solution repeats after each cycle. Such periodic conditions provide a natural framework for analyzing stability and long-term behavior in dynamic systems. In earlier discussions, we considered the space $\mathfrak{C}\left(I\right)$, where $I=[0,\phi ]$, consisting of all continuous functions defined on I. This space is equipped with a norm, defined as:

$$\parallel \mathring{\varkappa }-\mathring{\kappa }\parallel =sup\left\{\right|\mathring{\varkappa }\left(\vartheta \right)-\mathring{\kappa }\left(\vartheta \right)|:\vartheta \in I\},\mathrm{for}\mathring{\varkappa },\mathring{\kappa }\in \mathfrak{C}\left(I\right),$$

which is a Banach space. Additionally, we take the binary relation Q on $\mathfrak{C}\left(I\right)$ as:

$$\mathring{\varkappa },\mathring{\kappa }\in \mathfrak{C}\left(I\right),(\mathring{\varkappa },\mathring{\kappa })\in Q\iff \mathring{\varkappa }\left(\vartheta \right)\ge \mathring{\kappa }\left(\vartheta \right)\mathrm{for}\mathrm{all}\vartheta \in I.$$

(13)

Clearly, the binary relation Q on $\mathfrak{C}\left(I\right)$ is an $Q^s$-directed. It is clear that for any sequence $\left\{{\mathring{\varkappa }}_j\right\}$ satisfying $({\mathring{\varkappa }}_j,{\mathring{\varkappa }}_{j+1})\in Q$ for all $j\in \mathbb{N}$ and ${\mathring{\varkappa }}_j\to \mathring{g}$, we have $({\mathring{\varkappa }}_j,\mathring{g})\in Q$ for all $j\in \mathbb{N}$.

Definition 17.

A lower solution for problem (12) is a function $\mathfrak{O}\in {\mathfrak{C}}^1\left(I\right)$ such that

$$\left\{\begin{array}{c}{\mathfrak{O}}^{\prime }\left(\vartheta \right)\le \mathring{\mathfrak{f}}(\vartheta ,\mathfrak{O}\left(\vartheta \right)),\mathit{for}\vartheta \in I,\hfill \\ \mathfrak{O}\left(0\right)\le \mathfrak{O}\left(\phi \right).\hfill \end{array}\right.$$

Theorem 5.

Consider the problem (12), where $\mathring{\mathfrak{f}}:I\times \mathbb{R}\to \mathbb{R}$ is a continuous function. Assume the existence of constants $\delta >0$ and $\mathfrak{O}>0$ such that:

$$\mathfrak{O}\le {\left({\displaystyle \frac{2\delta \left({\mathrm{e}}^{\delta \phi }-1\right)}{\phi \left({\mathrm{e}}^{\delta \phi }+1\right)}}\right)}^{{\displaystyle \frac{1}{2}}},$$

(14)

and for all $\mathring{\varkappa },\mathring{\kappa }\in \mathbb{R}$ with $\mathring{\varkappa }\ge \mathring{\kappa }$:

$$0\le \mathring{\mathfrak{f}}(\vartheta ,\mathring{\kappa })+\delta \mathring{\kappa }-\left[\mathring{\mathfrak{f}}(\vartheta ,\mathring{\varkappa })+\delta \mathring{\varkappa }\right]\le \mathfrak{O}\sqrt{ln\left[{(\mathring{\kappa }-\mathring{\varkappa })}^2+1\right]}-(1-\mu )(\mathring{\kappa }-\mathring{\varkappa }).$$

(15)

Under these conditions, the presence of a lower solution for (12) ensures the existence of a unique solution to (12).

Proof. 

Problem (12) can be expressed as:

$$\left\{\begin{array}{c}{\mathring{v}}^{\prime }\left(\vartheta \right)+\delta \mathring{v}\left(\vartheta \right)=\mathring{\mathfrak{f}}(\vartheta ,\mathring{v}\left(\vartheta \right))+\delta \mathring{v}\left(\vartheta \right),\mathrm{for}t\in I=[0,\phi ],\hfill \\ \mathring{v}\left(0\right)=\mathring{v}\left(\phi \right).\hfill \end{array}\right.$$

The problem under discussion can be reformulated as the equivalent integral equation as

$$\mathring{v}\left(\vartheta \right)={\int }_0^{\phi }{\mathring{\varpi }}_c(\vartheta ,s)\left[\mathring{\mathfrak{f}}(s,\mathring{v}\left(s\right))+\delta \mathring{v}\left(s\right)\right]\mathrm{d}s,$$

where ${\mathring{\varpi }}_c(\vartheta ,s)$ is the Green’s function given by

$${\mathring{\varpi }}_c(\vartheta ,s)=\left\{\begin{array}{cc}{\displaystyle \frac{{\mathrm{e}}^{\delta (\phi +s-\vartheta )}}{{\mathrm{e}}^{\delta \phi }-1}},\hfill & 0\le s<\vartheta \le \phi ,\hfill \\ {\displaystyle \frac{{\mathrm{e}}^{\delta (s-\vartheta )}}{{\mathrm{e}}^{\delta \phi }-1}},\hfill & 0\le \vartheta <s\le \phi .\hfill \end{array}\right.$$

Define the operator $T:\mathfrak{C}\left(I\right)\to \mathfrak{C}\left(I\right)$ by

$$\left(T\mathring{v}\right)\left(\vartheta \right)={\int }_0^{\phi }{\mathring{\varpi }}_c(\vartheta ,s)\left[\mathring{\mathfrak{f}}(s,\mathring{v}\left(s\right))+\delta \mathring{v}\left(s\right)\right]\mathrm{d}s,$$

which for some $\mu \in (0,1]$, we obtain the average operator $T_{\mu }$ as

$$T_{\mu }\left(\mathring{v}\right)\left(\vartheta \right))={\int }_0^{\phi }{\mathring{\varpi }}_c(\vartheta ,s)\left[\mathring{\mathfrak{f}}(s,\mathring{v}\left(s\right))+\delta \mathring{v}\left(s\right)\right]\mathrm{d}s+(1-\mu )\mathring{v})\left(\vartheta \right),\forall \vartheta \in [0,1].$$

It is important to note that if $\mathring{v}\in \mathfrak{C}\left(I\right)$ is a fixed point of the operator T, then $\mathring{v}\in {\mathfrak{C}}^1\left(I\right)$ serves as a solution to (12). In the following, we verify that the conditions of Theorem 2 are met. It is evident that the mapping $T_{\mu }$ is non-decreasing. Specifically, for $\mathring{v}\ge \sigma$, and utilizing our assumptions, we observe:

$$\mathring{\mathfrak{f}}(\vartheta ,\mathring{v})+\delta \mathring{v}\ge \mathring{\mathfrak{f}}(\vartheta ,\sigma )+\delta \sigma ,$$

which demonstrates, since ${\mathring{\varpi }}_c(\vartheta ,s)>0$, that for $\vartheta \in I$:

$$\begin{array}{ccc}\hfill {\int }_0^{\phi }{\mathring{\varpi }}_c(\vartheta ,s)\left[\mathring{\mathfrak{f}}(s,\mathring{v}\left(s\right))+\delta \mathring{v}\left(s\right)\right]\mathrm{d}s+(1-\mu )\mathring{v}\left(\vartheta \right)& \ge & {\int }_0^{\phi }{\mathring{\varpi }}_c(\vartheta ,s)\left[\mathring{\mathfrak{f}}(s,\sigma \left(s\right))+\delta \sigma \left(s\right)\right]\mathrm{d}s+(1-\mu )\sigma \left(\vartheta \right)\hfill \\ \hfill \Rightarrow T_{\mu }\left(\mathring{v}\right)\left(\vartheta \right))& \ge & T_{\mu }\left(\sigma \right)\left(\vartheta \right)).\hfill \end{array}$$

Equivalently, we obtain $(\mathring{v},\sigma )\in Q\Rightarrow (T_{\mu }\mathring{v},T_{\mu }\sigma )\in Q$, which ensures that Assumption (3), stating that Q is $T_{\mu }$-closed, is satisfied as required by Theorem 2.

In addition, for $(\mathring{v},\sigma )\in Q$, that is, $\mathring{v}\ge \sigma$, we obtain

$$\begin{array}{cc}\hfill \parallel T_{\mu }\mathring{v}-T_{\mu }\sigma \parallel & =\underset{\vartheta \in I}{sup}|T_{\mu }\left(\mathring{v}\right)\left(\vartheta \right))-T_{\mu }\left(\sigma \right)\left(\vartheta \right)\left)\right|\hfill \\ & =\underset{\vartheta \in I}{sup}\left[(T_{\mu }\left(\mathring{v}\right)\left(\vartheta \right)-T_{\mu }\left(\sigma \right)\left(\vartheta \right))\right]\hfill \\ & =\underset{\vartheta \in I}{sup}\left[{\int }_0^{\phi }{\mathring{\varpi }}_c(\vartheta ,s)\left[\mathring{\mathfrak{f}}(s,\mathring{v}\left(s\right))+\delta \mathring{v}\left(s\right)-\mathring{\mathfrak{f}}(s,\sigma \left(s\right))-\delta \sigma \left(s\right)\right]\mathrm{d}s+(1-\mu )(\mathring{v}\left(\vartheta \right)-\sigma \left(\vartheta \right))\right]\hfill \\ & \le \underset{\vartheta \in I}{sup}{\int }_0^{\phi }{\mathring{\varpi }}_c(\vartheta ,s)\mathfrak{O}\sqrt{ln\left[{(\mathring{v}\left(s\right)-\sigma \left(s\right))}^2+1\right]}\mathrm{d}s.\hfill \end{array}$$

Applying the Cauchy–Schwarz inequality to the last integral, we obtain:

$${\int }_0^{\phi }{\mathring{\varpi }}_c(\vartheta ,s)\mathfrak{O}\sqrt{ln\left[{(\mathring{v}\left(s\right)-\sigma \left(s\right))}^2+1\right]}\mathrm{d}s\le {\left({\int }_0^{\phi }{\mathring{\varpi }}_c{(\vartheta ,s)}^2\mathrm{d}s\right)}^{{\displaystyle \frac{1}{2}}}{\left({\int }_0^{\phi }{\mathfrak{O}}^{2}ln\left[{(\mathring{v}\left(s\right)-\sigma \left(s\right))}^2+1\right]\mathrm{d}s\right)}^{{\displaystyle \frac{1}{2}}}.$$

(16)

The first integral in (16) is evaluated as

$$\begin{array}{cc}\hfill {\int }_0^{\phi }{\mathring{\varpi }}_c{(\vartheta ,s)}^2\mathrm{d}s& ={\int }_0^{\vartheta }{\mathring{\varpi }}_c{(\vartheta ,s)}^2\mathrm{d}s+{\int }_{\vartheta }^{\phi }{\mathring{\varpi }}_c{(\vartheta ,s)}^2\mathrm{d}s\hfill \\ & ={\int }_0^{\vartheta }{\displaystyle \frac{{\mathrm{e}}^{2\delta (\phi +s-\vartheta )}}{{\left({\mathrm{e}}^{\delta \phi }-1\right)}^2}}\mathrm{d}s+{\int }_{\vartheta }^{\phi }{\displaystyle \frac{{\mathrm{e}}^{2\delta (s-\vartheta )}}{{\left({\mathrm{e}}^{\delta \phi }-1\right)}^2}}\mathrm{d}s\hfill \\ & ={\displaystyle \frac{1}{2\delta {\left({\mathrm{e}}^{\delta \phi }-1\right)}^2}}\left({\mathrm{e}}^{2\delta \phi }-1\right)\hfill \\ & ={\displaystyle \frac{{\mathrm{e}}^{\delta \phi }+1}{2\delta \left({\mathrm{e}}^{\delta \phi }-1\right)}}.\hfill \end{array}$$

The second integral in (16) provides the following estimate:

$$\begin{array}{cc}\hfill {\int }_0^{\phi }{\mathfrak{O}}^{2}ln\left[{(\mathring{v}\left(s\right)-\sigma \left(s\right))}^2+1\right]\mathrm{d}s& \le {\mathfrak{O}}^{2}ln\left[\parallel \mathring{v}{-\sigma \parallel }^2+1\right]·\phi \hfill \\ & ={\mathfrak{O}}^{2}ln\left[\parallel \mathring{v}{-\sigma \parallel }^2+1\right]·\phi .\hfill \end{array}$$

Taking into account both integrals in (16), we have

$$\begin{array}{cc}\hfill \parallel T_{\mu }\mathring{v}-T_{\mu }\sigma \parallel & \le \underset{\vartheta \in I}{sup}{\left({\displaystyle \frac{{\mathrm{e}}^{\delta \phi }+1}{2\delta \left({\mathrm{e}}^{\delta \phi }-1\right)}}\right)}^{{\displaystyle \frac{1}{2}}}·{\left({\mathfrak{O}}^{2}ln\left[\parallel \mathring{v}{-\sigma \parallel }^2+1\right]·T\right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ & ={\left({\displaystyle \frac{{\mathrm{e}}^{\delta \phi }+1}{2\delta \left({\mathrm{e}}^{\delta \phi }-1\right)}}\right)}^{{\displaystyle \frac{1}{2}}}·\mathfrak{O}·\sqrt{\phi }·{\left(ln\left[\parallel \mathring{v}{-\sigma \parallel }^2+1\right]\right)}^{{\displaystyle \frac{1}{2}}}.\hfill \end{array}$$

From the last inequality, we obtain

$$\parallel T_{\mu }\mathring{v}-T_{\mu }{\sigma \parallel }^2\le {\displaystyle \frac{{\mathrm{e}}^{\delta \phi }+1}{2\delta \left({\mathrm{e}}^{\delta \phi }-1\right)}}·{\mathfrak{O}}^2·\phi ·ln\left[\parallel \mathring{v}{-\sigma \parallel }^2+1\right].$$

Equivalently, we have

$$2\delta \left({\mathrm{e}}^{\delta \phi }-1\right){\parallel T_{\mu }\mathring{v}-T_{\mu }\sigma \parallel }^2\le \left({\mathrm{e}}^{\delta \phi }+1\right)·{\mathfrak{O}}^2·\phi ·ln\left[\parallel \mathring{v}{-\sigma \parallel }^2+1\right].$$

(17)

By our assumption that

$$\mathfrak{O}\le {\left({\displaystyle \frac{2\delta \left({\mathrm{e}}^{\delta \phi }-1\right)}{\phi \left({\mathrm{e}}^{\delta \phi }+1\right)}}\right)}^{{\displaystyle \frac{1}{2}}},$$

therefore, the inequality (17) gives us

$$2\delta \left({\mathrm{e}}^{\delta \phi }-1\right){\parallel T_{\mu }\mathring{v}-T_{\mu }\sigma \parallel }^2\le 2\delta \left({\mathrm{e}}^{\delta \phi }-1\right)·ln\left[\parallel \mathring{v}{-\sigma \parallel }^2+1\right],$$

and hence

$$\begin{array}{cc}\hfill \parallel T_{\mu }\mathring{v}-T_{\mu }{\sigma \parallel }^2& \le ln\left[\parallel \mathring{v}{-\sigma \parallel }^2+1\right]\hfill \\ & =\parallel \mathring{v}{-\sigma \parallel }^2-\left(\parallel \mathring{v}{-\sigma \parallel }^2-ln\left[\parallel \mathring{v}{-\sigma \parallel }^2+1\right]\right).\hfill \end{array}$$

Define $\psi \left(\mathring{\varkappa }\right)={\mathring{\varkappa }}^2$ and $\mathrm{\Phi }\left(\mathring{\varkappa }\right)={\mathring{\varkappa }}^2-ln\left({\mathring{\varkappa }}^2+1\right)$. Clearly, $\psi$ and $\mathrm{\Phi }$ are altering-distance functions. This shows that T is a $(\mathfrak{e},\psi ,\mathrm{\Phi })$-enriched weak Q-preserving contraction.

Finally, let $\mathfrak{O}\left(\vartheta \right)$ be a lower solution for (12). We claim that $\mathfrak{O}\le T\left(\mathfrak{O}\right)$. In fact

$${\mathfrak{O}}^{\prime }\left(\vartheta \right)+\delta \mathfrak{O}\left(\vartheta \right)\le \mathring{\mathfrak{f}}(\vartheta ,\mathfrak{O}\left(\vartheta \right))+\delta \mathfrak{O}\left(\vartheta \right),\mathrm{for}\vartheta \in I.$$

Multiplying by ${\mathrm{e}}^{\delta \vartheta }$:

$${\left(\mathfrak{O}\left(\vartheta \right){\mathrm{e}}^{\delta \vartheta }\right)}^{\prime }\le \left[\mathring{\mathfrak{f}}(\vartheta ,\mathfrak{O}\left(\vartheta \right))+\delta \mathfrak{O}\left(\vartheta \right)\right]{\mathrm{e}}^{\delta \vartheta },\mathrm{for}\vartheta \in I,$$

and this gives:

$$\mathfrak{O}\left(\vartheta \right){\mathrm{e}}^{\delta \vartheta }\le \mathfrak{O}\left(0\right)+{\int }_0^{\vartheta }\left[\mathring{\mathfrak{f}}(s,\mathfrak{O}\left(s\right))+\delta \mathfrak{O}\left(s\right)\right]{\mathrm{e}}^{\delta s}\mathrm{d}s,\mathrm{for}\vartheta \in I.$$

As $\mathfrak{O}\left(0\right)\le \mathfrak{O}\left(\phi \right)$, the last inequality gives:

$$\mathfrak{O}\left(\vartheta \right)\le \left(T\mathfrak{O}\right)\left(\vartheta \right),\mathrm{for}\vartheta \in I.$$

By Theorems 2 and 3, T has a unique fixed point. Consequently, the problem (12) has a unique solution. □

4. Application to Second-Order Ordinary Differential Equations

Consider the problem:

$$\begin{array}{cc}\hfill & -{\displaystyle \frac{{\mathrm{d}}^2\mathring{\varkappa }}{\mathrm{d}{\vartheta }^2}}=\mathring{\mathfrak{f}}(\vartheta ,\mathring{\varkappa }),\mathring{\varkappa }\in [0,\infty ),\vartheta \in [0,1],\hfill \\ \hfill & \mathring{\varkappa }\left(0\right)=\mathring{\varkappa }\left(1\right)=0.\hfill \end{array}$$

(18)

Since the problem includes the boundary conditions $\mathring{\varkappa }\left(0\right)=\mathring{\varkappa }\left(1\right)=0$ and $\mathring{\varkappa }\left(\vartheta \right)$ is a continuous function, it is essential to define the independent variable $\vartheta$ over the interval $[0,1]$ instead of some interval $J\subseteq [0,1]$. This ensures that the function remains well-defined and satisfies the given constraints across the entire domain. The continuity of $\mathring{\varkappa }\left(\vartheta \right)$ guarantees a smooth transition between the boundary points, making $[0,1]$ the natural choice for analysis. Additionally, this interval allows for the direct application of fixed-point theorems and functional analysis techniques, ensuring a rigorous mathematical foundation. Further, the solution $\mathring{\varkappa }\in {\mathfrak{C}}^2[0,1]$ for the problem (18) is equivalent to the solution of the following integral equation:

$$\mathring{\varkappa }\left(\vartheta \right)={\int }_0^1{\mathring{\varpi }}_c(\vartheta ,s)\mathring{\mathfrak{f}}(s,\mathring{\varkappa }\left(s\right))\mathrm{d}s,\mathrm{for}\vartheta \in [0,1],$$

where ${\mathring{\varpi }}_c(\vartheta ,s)$ is the Green function:

$${\mathring{\varpi }}_c(\vartheta ,s)=\left\{\begin{array}{cc}\vartheta (1-s),\hfill & 0\le \vartheta \le s\le 1,\hfill \\ s(1-\vartheta ),\hfill & 0\le s\le \vartheta \le 1.\hfill \end{array}\right.$$

(19)

Next, we present the following result.

Theorem 6.

Examine the problem (18), where $\mathring{\mathfrak{f}}:I\times \mathbb{R}\to [0,\infty )$ is a continuous function that increases monotonically with respect to its second variable. Assume there exists a constant $0<\mathfrak{O}\le 8$ such that the following condition holds: for all $\mathring{\varkappa },\mathring{\kappa }\in \mathbb{R}$ with $\mathring{\kappa }\ge \mathring{\varkappa }$,

$$\mathring{\mathfrak{f}}(\vartheta ,\mathring{\kappa })-\mathring{\mathfrak{f}}(\vartheta ,\mathring{\varkappa })\le \mathfrak{O}\sqrt{ln\left[{(\mathring{\kappa }-\mathring{\varkappa })}^2+1\right]}-(1-\mu )(\mathring{\varkappa }-\mathring{\kappa }).$$

Consequently, the problem (18) admits a unique non-negative solution.

Proof. 

Define the set $P:=\{\mathring{\varkappa }\in \mathfrak{C}[0,1]:\mathring{\varkappa }\left(\vartheta \right)\ge 0\}.$ Clearly, P with usual norm $\parallel \mathring{\varkappa }-\mathring{\kappa }\parallel :=sup\left\{\right|\mathring{\varkappa }\left(\vartheta \right)-\mathring{\kappa }\left(\vartheta \right)|:\vartheta \in [0,1]\}$ is a Banach space. Define the operator

$$\left(T\mathring{\varkappa }\right)\left(\vartheta \right)={\int }_0^1{\mathring{\varpi }}_c(\vartheta ,s)\mathring{\mathfrak{f}}(s,\mathring{\varkappa }\left(s\right))\mathrm{d}s,\mathrm{for}\mathring{\varkappa }\in P,$$

where ${\mathring{\varpi }}_c(\vartheta ,s)$ is the Green function from (19). It is clear that T maps P into itself since $\mathring{\mathfrak{f}}(\vartheta ,\mathring{\varkappa })$ and ${\mathring{\varpi }}_c(\vartheta ,s)$ are non-negative continuous functions. Also, for some $\mu \in (0,1]$, we define the average operator as

$$T_{\mu }\left(\mathring{\varkappa }\right)\left(\vartheta \right))={\int }_0^1{\mathring{\varpi }}_c(\vartheta ,s)\mathring{\mathfrak{f}}(s,\mathring{\varkappa }\left(s\right))\mathrm{d}s+(1-\mu )\mathring{\varkappa })\left(\vartheta \right),\mathrm{for}\mathring{\varkappa }\in P,$$

(20)

Since $\mathring{\mathfrak{f}}$ is non-decreasing with respect to its second variable, for any $\mathring{\varkappa },\mathring{\kappa }\in P$ satisfying $\mathring{\kappa }\ge \mathring{\varkappa }$ and for all $\vartheta \in [0,1]$, it follows that:

$$T_{\mu }\mathring{\kappa }\left(\vartheta \right)={\int }_0^1{\mathring{\varpi }}_c(\vartheta ,s)\mathring{\mathfrak{f}}(s,\mathring{\kappa }\left(s\right))\mathrm{d}s+(1-\mu )\mathring{\kappa }\left(\vartheta \right)\ge {\int }_0^1{\mathring{\varpi }}_c(\vartheta ,s)\mathring{\mathfrak{f}}(s,\mathring{\varkappa }\left(s\right))\mathrm{d}s+(1-\mu )\mathring{\varkappa }\left(\vartheta \right)$$

$$=T_{\mu }\mathring{\varkappa }\left(\vartheta \right).$$

Thus, $T_{\mu }$ is a non-decreasing operator. Additionally, for $\mathring{\kappa }\ge \mathring{\varkappa }$, we compute:

$$\parallel T_{\mu }\mathring{\kappa }-T_{\mu }\mathring{\varkappa }\parallel =\underset{\vartheta \in [0,1]}{sup}|T_{\mu }\mathring{\kappa }\left(\vartheta \right)-T_{\mu }\mathring{\varkappa }\left(\vartheta \right)|.$$

Substituting the definition of $T_{\mu }$, we have

$$\begin{array}{cc}\hfill \parallel T_{\mu }\mathring{\kappa }-T_{\mu }\mathring{\varkappa }\parallel & =\underset{\vartheta \in [0,1]}{sup}\left[{\int }_0^1{\mathring{\varpi }}_c(\vartheta ,s)\left[\mathring{\mathfrak{f}}(s,\mathring{\kappa }\left(s\right))-\mathring{\mathfrak{f}}(s,\mathring{\varkappa }\left(s\right))\right]\mathrm{d}s+(1-\mu )(\mathring{\varkappa }\left(\vartheta \right)-\mathring{\kappa }\left(\vartheta \right))\right]\hfill \\ \hfill & \le \underset{\vartheta \in [0,1]}{sup}{\int }_0^1{\mathring{\varpi }}_c(\vartheta ,s)\mathfrak{O}\sqrt{ln\left[{(\mathring{\kappa }\left(s\right)-\mathring{\varkappa }\left(s\right))}^2+1\right]}\mathrm{d}s\hfill \\ \hfill & \le \mathfrak{O}\sqrt{ln\left[\parallel \mathring{\kappa }-\mathring{\varkappa }{\parallel }^2+1\right]}·\underset{\vartheta \in [0,1]}{sup}{\int }_0^1{\mathring{\varpi }}_c(\vartheta ,s)\mathrm{d}s.\hfill \end{array}$$

(21)

It is easy to check that

$${\int }_0^1{\mathring{\varpi }}_c(\vartheta ,s)\mathrm{d}s={\displaystyle \frac{-{\vartheta }^2}{2}}+{\displaystyle \frac{\vartheta }{2}},$$

and thus

$$\underset{\vartheta \in [0,1]}{sup}{\int }_0^1{\mathring{\varpi }}_c(\vartheta ,s)\mathrm{d}s={\displaystyle \frac{1}{8}}.$$

Using above integrals, (21) and the assumption $0<\mathfrak{O}\le 8$, we have

$$\parallel T_{\mu }\mathring{\kappa }-T_{\mu }\mathring{\varkappa }\parallel \le \sqrt{ln\left[\parallel \mathring{\kappa }-\mathring{\varkappa }{\parallel }^2+1\right]}.$$

Hence, we obtain

$$\parallel T_{\mu }\mathring{\kappa }-T_{\mu }\mathring{\varkappa }{\parallel }^2\le ln\left[\parallel \mathring{\varkappa }-\mathring{\kappa }{\parallel }^2+1\right].$$

Define $\psi \left(\mathring{\varkappa }\right)={\mathring{\varkappa }}^2$ and $\mathrm{\Phi }\left(\mathring{\varkappa }\right)={\mathring{\varkappa }}^2-ln\left({\mathring{\varkappa }}^2+1\right)$. Clearly, $\psi$ and $\mathrm{\Phi }$ are altering-distance functions. From the last inequality, we obtain

$$\psi (\parallel T_{\mu }\mathring{\varkappa }-T_{\mu }\mathring{\kappa })\le \psi (\parallel \mathring{\varkappa }-\mathring{\kappa }\parallel )-\mathrm{\Phi }(\parallel \mathring{\varkappa }-\mathring{\kappa }\parallel ).$$

Equivalently, we obtain

$$\psi [{\displaystyle \frac{\parallel \mathfrak{e}(\mathring{\varkappa }-\mathring{\kappa })+T\mathring{\varkappa }-T\mathring{\kappa }\parallel }{\mathfrak{e}+1}}]\le \psi [\parallel \mathring{\varkappa }-\mathring{\kappa }\parallel ]-\mathrm{\Phi }[\parallel \mathring{\varkappa }-\mathring{\kappa }\parallel ],\forall \mathring{\varkappa },\mathring{\kappa }\in P\mathrm{with}[\mathring{\varkappa },\mathring{\kappa }]\in Q.$$

Finally, due to the non-negative behavior of $\mathring{\mathfrak{f}}$ and ${\mathring{\varpi }}_c$, we have

$$T\left(0\right)={\int }_0^1{\mathring{\varpi }}_c(\vartheta ,s)\mathring{\mathfrak{f}}(s,0)\mathrm{d}s\ge 0.$$

By Theorem 2, T has a unique non-negative fixed point. Consequently, the problem (18) has a unique solution. □

Remark 3.

If we assume $\mathring{\mathfrak{f}}(\vartheta ,\mathring{\varkappa })\ne 0$ for $\vartheta \in (0,1)$, the zero function is not a fixed point of T. Instead, the solution $\mathring{\varkappa }\left(\vartheta \right)$ satisfies $0<\mathring{\varkappa }\left(\vartheta \right)$ for $0<\vartheta <1$.

The shift from weak to enriched weak contractions extends the fixed-point theory by relaxing the contractive assumptions, enabling broader applicability to differential and integral equations. This framework enhances stability, convergence, and iterative methods while unifying existing contraction principles. Its flexibility improves both theoretical guarantees and computational feasibility, making it a valuable tool for addressing complex problems beyond classical contraction conditions. The classical results for applications to first- and second-order ODEs are derived as corollaries of our findings, specifically by setting $\mu =1$, within the framework of normed spaces.

Corollary 8

(Theorem 3.1 [21]). Consider the problem (12), where $\mathring{\mathfrak{f}}:I\times \mathbb{R}\to \mathbb{R}$ is a continuous function. Assume the existence of constants $\delta >0$ and $\mathfrak{O}>0$ such that:

$$\mathfrak{O}\le {\left({\displaystyle \frac{2\delta \left({\mathrm{e}}^{\delta \phi }-1\right)}{\phi \left({\mathrm{e}}^{\delta \phi }+1\right)}}\right)}^{{\displaystyle \frac{1}{2}}},$$

(22)

and for all $\mathring{\varkappa },\mathring{\kappa }\in \mathbb{R}$ with $\mathring{\varkappa }\ge \mathring{\kappa }$:

$$0\le \mathring{\mathfrak{f}}(\vartheta ,\mathring{\kappa })+\delta \mathring{\kappa }-\left[\mathring{\mathfrak{f}}(\vartheta ,\mathring{\varkappa })+\delta \mathring{\varkappa }\right]\le \mathfrak{O}\sqrt{ln\left[{(\mathring{\kappa }-\mathring{\varkappa })}^2+1\right]}.$$

(23)

Under these conditions, a lower solution of (12) ensures a unique solution.

Corollary 9

(Theorem 3.2 [21]). Consider the problem (18), where $\mathring{\mathfrak{f}}:I\times \mathbb{R}\to [0,\infty )$ is a continuous function that is monotonically increasing with respect to its second variable. Suppose there exists a constant $0<\mathfrak{O}\le 8$ satisfying the condition that for all $\mathring{\varkappa },\mathring{\kappa }\in \mathbb{R}$ with $\mathring{\kappa }\ge \mathring{\varkappa }$,

$$\mathring{\mathfrak{f}}(\vartheta ,\mathring{\kappa })-\mathring{\mathfrak{f}}(\vartheta ,\mathring{\varkappa })\le \mathfrak{O}\sqrt{ln\left[{(\mathring{\kappa }-\mathring{\varkappa })}^2+1\right]}.$$

Under these conditions, the problem (18) has a unique non-negative solution.

5. Conclusions and Future Directions

In this study, we have established several fixed-point theorems within the framework of Banach spaces endowed with a binary relation. Utilizing enriched contraction principles involving two distinct classes of altering-distance functions, the results encompass various types of contractive mappings, such as theoretic-order contractions, Picard-Banach contractions, weak contractions, and non-expansive contractions. By employing a suitable Krasnoselskij iterative scheme, we demonstrated the generality of these results, as many well-known fixed-point theorems (FPTs) emerge as special cases through specific control functions or appropriate binary relations. The theoretical findings were supported by illustrative examples, and their applicability was demonstrated through applications to first-order and second-order ordinary differential equations. This work provides a unified approach to fixed-point theory and highlights its potential in addressing practical problems in differential equations.

A potential avenue for future research is to explore whether some of the imposed conditions can be relaxed while still obtaining the same results, thereby broadening the applicability of enriched Q-preserving contractions. Another intriguing direction would be to study the fractal properties of enriched contractions by incorporating the idea of binary relations. This could open new pathways for understanding the dynamics and structure of such mappings in both theoretical and applied contexts.