Fixed Point Theorems in Elliptic-Valued Suprametric Spaces and Their Applications

Abstract

The purpose of this article is to introduce the concept of elliptic-valued suprametric spaces and to establish some common fixed point theorems within this newly proposed framework. The development of elliptic-valued suprametric spaces, along with the main results presented here, is illustrated through non-trivial examples. As applications, we employ our fixed point results to study the nonlinear Volterra integral equation of the second kind, demonstrating the existence and uniqueness of solutions under suitable conditions. In particular, we highlight the role of such equations in age-structured population models, where they serve as powerful tools for describing hereditary effects, density-dependent interactions, and delayed responses in population dynamics. This connection bridges the abstract theory with applied contexts in mathematical biology, ecology, and epidemiology, emphasizing the relevance of elliptic-valued suprametric structures in both theoretical analysis and real-world applications. Furthermore, we derive corresponding fixed point results in elliptic-valued suprametric spaces, complex-valued suprametric spaces, elliptic-valued metric spaces, and complex-valued metric spaces as corollaries of our main findings.

1. Introduction

The real number system serves as the cornerstone of classical mathematics, providing a continuous framework that unites rational numbers (fractions) with irrational numbers (non-repeating, non-terminating decimals). This system forms the basis of both everyday measurements and advanced scientific modeling, from determining physical distances to simulating natural phenomena. Yet, as mathematics evolved, new problems demanded extensions beyond the reals, leading to the birth of complex numbers in the 16th century.

The idea, first explored by Gerolamo Cardano and further developed by Rafael Bombelli [1], introduced the imaginary unit i, where $i^2=-1$, to solve equations such as $x^3=15x+4$. This innovation allowed solutions to previously unsolvable algebraic problems and laid the groundwork for broader number systems. During the 19th century, William Kingdon Clifford [2] introduced a new unit j with the property $j^2=1$, giving rise to hyperbolic numbers of the form $a+bj$. Unlike complex numbers, these split into “light-like” components when $a^2-b^2=0$, making them especially relevant in relativity theory and wave mechanics, where their geometry naturally models Lorentz transformations. Later, in the early 20th century, Eduard Study [3] presented dual numbers, defined by a nilpotent unit $ϵ$ with ${ϵ}^2=0$. Numbers of the form $a+bϵ$ became powerful tools in computing derivatives, particularly in robotics and computer graphics, since the infinitesimal unit $ϵ$ effectively encodes variations—a principle now central to automatic differentiation in modern machine learning. A unified perspective arises by defining a unit i such that $i^2=p$, with $p\in \mathbb{R}$; when $p<0$ this yields elliptic numbers (the familiar complex system without zero divisors), when $p=0$ it produces parabolic or dual numbers with nilpotent elements, and when $p>0$ it leads to hyperbolic numbers characterized by the presence of zero divisors, as in split-complex algebra.

Fréchet [4] was the first to formalize the concept of a metric space, which is composed of a set paired with a distance function satisfying three fundamental properties: (i) the distance between any two distinct elements is positive, and the distance from an element to itself is zero; (ii) the distance between two points is symmetric, meaning it does not depend on the order of the points; and (iii) the triangle inequality holds, so the direct distance between two points is never greater than the sum of distances through any intermediate point. These foundational principles provide a rigorous framework for studying continuity, convergence, and geometric structures in both theoretical and applied contexts. Within metric spaces, one of the most notable results is the Banach Contraction Principle (BCP) [5], which guarantees that any contraction mapping on a complete metric space has a unique fixed point. Building on this, Fisher [6] introduced rational contractions, extending the classical fixed point theory. Later, Azam et al. [7] generalized the traditional metric space concept by defining complex-valued metric spaces, where the distance function assumes values in the complex numbers while satisfying appropriately modified metric axioms. This generalization widens the range of applications metric space theory to more complex mathematical structures and practical problems. In complex-valued metric spaces, Azam et al. [7] established common fixed point theorems using Fisher’s rational contraction approach. Further extending this line of work, Rouzkard et al. [8] introduced an additional term into the rational contractive condition, providing a more general fixed point result that builds on the work of Azam et al. [7].

The pioneering work of Ozturk et al. [9] initiated elliptic-valued metric spaces, where the metric is defined using elliptic numbers—a generalized number system inspired by the theory of elliptic functions. These numbers, which have strong connections to analysis of elliptic curves in the complex domain, provide a richer and more intricate mathematical structure than conventional real or complex numbers. In the framework of elliptic-valued metric spaces, distances between points are not simply scalar values; instead, they are influenced by the characteristics of elliptic functions. This framework offers a versatile setting for studying geometric arrangements in spaces with cyclic, periodic, or otherwise complex interactions, making it particularly relevant for applications. Alamri et al. [10,11] have advanced fixed point theory in elliptic-valued metric spaces, establishing existence and uniqueness results for various mappings. Their work demonstrates applications to nonlinear Fredholm integral equations and provides insights into complex systems, including models relevant to climate dynamics.

On the other hand, Berzig [12] extended the classical concept of a metric space by introducing suprametric spaces. In this framework, the standard notion of distance is generalized to include an additional term that accounts for interactions among points through an intermediate element. This modification allows suprametric spaces to capture more complex relationships than classical metric spaces, making them suitable for studying nonlinear systems and generalized distance structures while preserving fundamental properties such as positivity, symmetry, and the uniqueness of zero distance. Subsequently, Panda et al. [13] extended this concept to complex numbers and gave the notion of complex-valued suprametric spaces. Their work not only extends classical fixed point theory but also demonstrates applications to both Barnsley Fern fractal generation and the solution of mixed Volterra–Fredholm integral equations, highlighting the versatility of suprametric spaces in bridging abstract analysis with concrete computational and applied problems. Afterwards, Abdou [14] and Shammaky et al. [15] employed the framework of complex-valued suprametric spaces to establish new fixed point results under generalized contraction conditions. Their work demonstrated the applicability of this setting by solving nonlinear mixed Volterra–Fredholm integral equations arising in predator–prey dynamics and addressing the solutions of Volterra–Hammerstein nonlinear integral equations, thereby highlighting the effectiveness of complex-valued suprametric spaces in tackling diverse classes of nonlinear integral equations.

Age-structured population models play a central role in mathematical biology, as they provide a framework for describing how demographic characteristics such as birth, survival, and mortality rates depend on the age distribution within a population. A powerful tool for analyzing such models is the nonlinear Volterra integral equation of the second kind, which naturally incorporates memory effects and time delays to capture interactions across different age classes. These integral equations arise when reformulating age-dependent partial differential equations into integral forms, allowing one to study the evolution of population density functions over time. The nonlinear nature of these equations reflects the influence of density-dependent feedback mechanisms, competition, and other ecological interactions. By employing fixed point theory, one can rigorously establish the existence and uniqueness of solutions to these nonlinear Volterra equations, ensuring the mathematical validity of the underlying biological models. This approach not only guarantees well-posedness but also provides a systematic method to analyze long-term behavior, stability, and equilibrium states in age-structured populations, offering deep insights into population dynamics and their ecological implications. Further information on this topic can be found in refs. [16,17,18].

In this research work, we introduce the concept of elliptic-valued suprametric spaces and establish several new common fixed point theorems under generalized contractive conditions. The results obtained extend and unify notable contributions in the literature, including the fundamental theorems of Panda et al. [13] and Abdou [14] in the setting of complex-valued suprametric spaces, the main theorems of Alamri [11] and Ozturk et al. [9] in the framework of elliptic-valued metric spaces, as well as the theorems of Azam et al. [7] and Rouzkard et al. [8] in the context of complex-valued metric spaces. To demonstrate the effectiveness of the proposed framework, we also provide a detailed example that illustrates its applicability and advantages. We apply our fixed point results to nonlinear Volterra integral equations of the second kind, establishing existence and uniqueness of solutions. In particular, these equations model age-structured populations, capturing hereditary effects, density dependence, and delayed responses, thereby linking abstract theory to applications in mathematical biology and ecology.

2. Preliminaries

We start by recalling the essential notations and definitions that will be employed throughout this work. For a more detailed exposition, the reader is referred to [1]. Denote by ${\mathbb{E}}_p$ the collection of elliptic numbers, characterized as

$${\mathbb{E}}_p=\left\{z=\nu +i\omega :\nu ,\omega \in \mathbb{R},i^2=p<0\right\}.$$

Given $z\in {\mathbb{E}}_p$ expressed as $\nu +i\omega$, $\nu$ represents the real component of z, and $\omega$ represents the imaginary component of z.

The basic arithmetic operations such that addition, scalar multiplication, elliptic multiplication, and conjugation for elliptic numbers, are established in the subsequent manner.

For, $z_1={\nu }_1+i{\omega }_1$ and $z_2={\nu }_2+i{\omega }_2$, their sum is defined by

$$z_1+z_2={\nu }_1+{\nu }_2+i\left({\omega }_1+{\omega }_2\right).$$

The product of $z_1\in {\mathbb{E}}_p$ and $\lambda \in \mathbb{R}$ is given by

$$\lambda z_1=\lambda \left({\nu }_1+i{\omega }_1\right)=\lambda {\nu }_1+i\lambda {\omega }_1.$$

Furthermore, the elliptic multiplication of $z_1$ and $z_2$ belonging to ${\mathbb{E}}_p$ is defined as

$$z_1{z}_2=\left({\nu }_1+i{\omega }_1\right)\left({\nu }_2+i{\omega }_2\right)=\left({\nu }_1{\nu }_2+p{\omega }_1{\omega }_2\right)+i\left({\nu }_1{\omega }_2+{\nu }_2{\omega }_1\right).$$

The conjugate of z is denoted by $\overline{z}$ and is described by

$$\overline{z}=Re\left(z\right)-iIm\left(z\right)=\nu -i\omega .$$

Finally, the elliptic norm of $z\in {\mathbb{E}}_p$ is defined as

$${\parallel z\parallel }_{\mathbb{E}}=\sqrt{z\overline{z}}=\sqrt{{\nu }^2-p{\omega }^2}.$$

It is readily observed that ${\mathbb{E}}_p$ constitutes a two-dimensional vector space over the real field $\mathbb{R}$, under the operations of addition and scalar multiplication. Furthermore, ${\mathbb{E}}_p$ satisfies the criteria to be considered a field. Therefore, a bijective mapping exists between ${\mathbb{E}}_p$ and ${\mathbb{R}}^2$, enabling a unique planar representation for each elliptic number $z=\nu +i\omega$. This structure, known as the elliptic plane, determines the distance between $z_1=\left({\nu }_1,{\omega }_1\right)$ and $z_2=\left({\nu }_2,{\omega }_2\right)\in {\mathbb{E}}_p$ by

$${∥z_1-z_2∥}_{\mathbb{E}}=\sqrt{{\left({\nu }_1-{\nu }_2\right)}^2-p{\left({\omega }_1-{\omega }_2\right)}^2},$$

subject to the conditions ${\left({\nu }_1-{\nu }_2\right)}^2-p{\left({\omega }_1-{\omega }_2\right)}^2>0$ and $p<0$.

Within this elliptic geometry, the locus of points equidistant from the origin at a unit distance forms an ellipse, as characterized by the equation ${\nu }^2-p{\omega }^2=1$ (see [1]). From this point forward, we denote the zero vector in ${\mathbb{E}}_p$ by $\theta$. A partial order, denoted ⪯, is defined on ${\mathbb{E}}_p$ as follows:

For any $z_1={\nu }_1+i{\omega }_1$, $z_2={\nu }_2+i{\omega }_2\in {\mathbb{E}}_p$, we stipulate that $z_1⪯z_2$ iff

$$Re\left(z_1\right)\le Re\left(z_2\right)\mathrm{and}Im\left(z_1\right)\le Im\left(z_2\right).$$

Thus, $z_1⪯z_2$ holds if and only if any of the conditions listed below are true:

$$\begin{array}{ccc}\hfill \left(\mathrm{i}\right)Re\left(z_1\right)& =& Re\left(z_2\right)\mathrm{and}Im\left(z_1\right)<Im\left(z_2\right),\hfill \\ \hfill \left(\mathrm{ii}\right)Re\left(z_1\right)& <& Re\left(z_2\right)\mathrm{and}Im\left(z_1\right)=Im\left(z_2\right),\hfill \\ \\ \hfill \left(\mathrm{iii}\right)Re\left(z_1\right)& <& Re\left(z_2\right)\mathrm{and}Im\left(z_1\right)<Im\left(z_2\right),\hfill \\ \\ \hfill \left(\mathrm{iv}\right)Re\left(z_1\right)& =& Re\left(z_2\right)\mathrm{and}Im\left(z_1\right)=Im\left(z_2\right).\hfill \end{array}$$

To be precise, $z_1\precsim z_2\left(\mathrm{with}{z}_1\ne z_2\right)$ signifies that at least one of conditions (i), (ii) or (iii) hold, while $z_1\prec z_2$ is used only when condition (iii) is true. The partial order ⪯ on ${\mathbb{E}}_p$ satisfies the following properties:

(i) If $\theta ⪯z_1\precsim z_2$, then the elliptic norm of $z_1$ is strictly less than the elliptic norm of $z_2$, i.e., ${∥z_1∥}_{\mathbb{E}}<{∥z_2∥}_{\mathbb{E}}$.

(ii) The relation $z_1⪯z_2$ holds if and only if $z_1-z_2⪯\theta$,

(iii) If $z_1⪯z_2$ and $z_2⪯z_3$, then $z_1⪯z_3$ (transitivity).

(iv) For $z_1⪯z_2$ and any positive real number $\lambda >0$, we have $\lambda z_1⪯\lambda z_2$.

(v) The conditions $\theta ⪯z_1$ and $\theta ⪯z_2$ do not guarantee that $\theta ⪯z_1{z}_2$.

Although the order ⪯ on ${\mathbb{E}}_p$ is partial and does not ensure comparability of arbitrary elements, all inequalities in this work are imposed within the positive cone $\left\{z\in {\mathbb{E}}_p:\theta ⪯z\right\}$, which is closed under addition and scalar multiplication. Moreover, convergence and completeness are characterized via the elliptic norm ${∥·∥}_{\mathbb{E}}$, thereby avoiding difficulties arising from incomparability.

Ozturk et al. [9] introduced the notion of elliptic-valued metric spaces, extending classical metric spaces by allowing distances to take values in ${\mathbb{E}}_p$ with respect to the above partial order.

Definition 1

([9]). Given a non-empty set Ξ and a mapping $d:\Xi \times \Xi \to {\mathbb{E}}_p$, which satisfies the following axiomatic conditions:

$\left(E_1\right)$: $\theta ⪯d(\varkappa ,\sigma )$, and $d(\varkappa ,\sigma )=\theta$⇔$\varkappa =\sigma$,

$\left(E_2\right)$: $d(\varkappa ,\sigma )=d(\sigma ,\varkappa )$,

$\left(E_3\right)$: $d(\varkappa ,\sigma )⪯d(\varkappa ,\varrho )+d(\varrho ,\sigma )$,

for all $\varkappa ,\sigma ,\varrho \in \Xi$, then $(\Xi ,d)$ is said to be an elliptic-valued metric space.

Example 1

([9]). Let $\Xi ={\mathbb{E}}_p$. We introduce $d:{\mathbb{E}}_p\times {\mathbb{E}}_p\to$${\mathbb{E}}_p$ which is defined as follows:

$$d\left(z_1,z_2\right)=\sqrt{{\left({\nu }_1-{\nu }_2\right)}^2-p{\left({\omega }_1-{\omega }_2\right)}^2,}$$

for ${\left({\nu }_1-{\nu }_2\right)}^2-p{\left({\omega }_1-{\omega }_2\right)}^2>0$ and for $z_1={\nu }_1+i{\omega }_1,z_2={\nu }_2+i{\omega }_2$ in ${\mathbb{E}}_p$ and $i^2=p<0$, then $\left({\mathbb{E}}_p,d\right)$ satisfies the characteristics of an elliptic-valued metric space.

Ozturk et al. [9] formulated the following fixed point theorem.

Theorem 1

([9]). $(\Xi ,d)$ be a complete elliptic-valued metric space, and consider a self-mappings $\mathfrak{F}$: Ξ$\to \Xi$. Suppose there exist the constants $\varpi \in [0,1)$ such that

$$d\left(\mathfrak{F}\varkappa ,\mathfrak{F}\sigma \right)⪯\varpi d\left(\varkappa ,\sigma \right),$$

for all $\varkappa ,\sigma \in$$\Xi ,$ then $\mathfrak{F}$ admits a unique fixed point.

Berzig [12] introduced the concept of suprametric space in this way.

Definition 2

([12]). Let $\Xi \ne \varnothing$ and ϑ be a non-negative real constant. Consider a function $d:\Xi \times \Xi \to {\mathbb{R}}^{+}$ that satisfies the following properties:

(i) 

$0\le d(\varkappa ,\sigma )$ and $d(\varkappa ,\sigma )=0$⟺$\varkappa =\sigma$,

(ii) 

$d(\varkappa ,\sigma )=d(\sigma ,\varkappa )$,

(iii) 

$d(\varkappa ,\sigma )\le d(\varkappa ,\varrho )+d(\varrho ,\sigma )+\vartheta d(\varkappa ,\varrho )d(\varrho ,\sigma )$,

for all $\varkappa ,\sigma ,\varrho \in \Xi$, then $(\Xi ,d)$ is called a suprametric space.

Example 2.

Let $\Xi =\left\{0,1,2\right\}$ and define the distance function $d:\Xi \times \Xi \to {\mathbb{R}}^{+}$ as follows:

$$d(0,1)=d(1,0)=0.5$$

$$d(0,2)=d(2,0)=1$$

$$d(1,2)=d(2,1)=2,$$

and

$$d(0,0)=d(1,1)=d(2,2)=0.$$

Let us choose $⋏=1.5$. Then $\left(\Xi ,d\right)$ is a suprametric space but not a metric space because the triangle inequality of metric space is not satisfied, i.e.,

$$2=d(1,2)>d(1,0)+d(0,2)=0.5+1.$$

In this way, we define the notion of elliptic-valued suprametric spaces as follows:

Definition 3.

Given a non-empty set Ξ, and a mapping $d:\Xi \times \Xi \to {\mathbb{E}}_p$, which satisfies the following axiomatic conditions:

$\left(E_{S_1}\right)$: $\theta ⪯d(\varkappa ,\sigma )$, and $d(\varkappa ,\sigma )=\theta$⇔$\varkappa =\sigma$,

$\left(E_{S_2}\right)$: $d(\varkappa ,\sigma )=d(\sigma ,\varkappa )$,

$\left(E_{S_3}\right)$: $d(\varkappa ,\sigma )⪯d(\varkappa ,\varrho )+d(\varrho ,\sigma )+\vartheta d(\varkappa ,\varrho )d(\varrho ,\sigma )$,

for all $\varkappa ,\sigma ,\varrho \in \Xi$, where $\vartheta \in {\mathbb{E}}_p$ with $\vartheta ⪰\theta$. The order ⪯ is considered with respect to the positive cone of ${\mathbb{E}}_p$, which is closed under addition and multiplication. Then $(\Xi ,d)$ is said to be an elliptic-valued suprametric space.

Remark 1.

The parameter ϑ in the elliptic-valued suprametric inequality $\left(E_{S_3}\right)$ introduces a controlled interaction term between distances, allowing the space to capture interdependencies that cannot be represented by the usual triangle inequality. Geometrically, this means that the “distance” between two points may depend not only on their direct separation but also on their relationships with intermediate points, providing a richer structure than classical elliptic-valued metric spaces. This added flexibility is particularly useful in modeling systems with coupled or nonlinear interactions, such as age-structured populations or other multi-component dynamical systems, where standard metrics fail to reflect these interactions. Therefore, every elliptic-valued metric space is a special case of an elliptic-valued suprametric space (when $\vartheta =\theta$), but the converse does not hold, highlighting the generality and applicability of the suprametric framework. Although the order on ${\mathbb{E}}_p$ is partial and some elements may be incomparable, this does not affect fixed point arguments, since all inequalities are formulated within the positive cone and convergence is deduced via the induced elliptic norm. Consequently, incomparability does not interfere with the validity of the theoretical results.

Example 3.

Let $\Xi =\mathbb{R}$ and

$${\mathbb{E}}_p=\left\{z=\nu +i\omega :\nu ,\omega \in \mathbb{R},i^2=p<0\right\}.$$

Define $d:\Xi \times \Xi \to$${\mathbb{E}}_p$ by

$$d(\varkappa ,\sigma )=(1+i)\left|\varkappa -\sigma \right|,$$

for $\varkappa ,\sigma \in \Xi$.

• $\left(E_{S_1}\right)$ Clearly, $d(\varkappa ,\sigma )⪰\theta$ and $d(\varkappa ,\sigma )=\theta$ if and only if $\varkappa =\sigma$,
• $\left(E_{S_2}\right)$$d(\varkappa ,\sigma )=d(\sigma ,\varkappa )$.
• $\left(E_{S_3}\right)$ For all $\varkappa ,\sigma ,\varrho \in \Xi$, we have

  $$d(\varkappa ,\sigma )⪯d(\varkappa ,\varrho )+d(\varrho ,\sigma )+\vartheta d(\varkappa ,\varrho )d(\varrho ,\sigma ),$$

  holds if we take $\vartheta ⪰\theta$, because

  $$\left|\varkappa -\sigma \right|\le \left|\varkappa -\varrho \right|+\left|\varrho -\sigma \right|+\vartheta \left|\varkappa -\varrho \right|\left|\varrho -\sigma \right|.$$

  Since both real and imaginary parts obey the same inequality, the condition is satisfied in ${\mathbb{E}}_p$. Thus, $\left(\Xi ,d\right)$ is an elliptic-valued suprametric space.

Example 4.

Let ${\mathbb{E}}_p=\left\{z=\nu +i\omega :\nu ,\omega \in \mathbb{R},i^2=p<0\right\}$ and choose the non-empty set $\Xi =\{a,b,c\}$. Define $d:\Xi \times \Xi \to$${\mathbb{E}}_p$ by

$$d(a,a)=d(b,b)=d(c,c)=\theta ,$$

where θ is the zero element in ${\mathbb{E}}_p$ and

$$d(a,b)=d(b,a)=2,$$

$$d(b,c)=d(c,b)=2,$$

$$d(a,c)=d(c,a)=5,$$

where the numbers 2 and 5 are identified with the elliptic numbers $2+0i$ and $5+0i$ in ${\mathbb{E}}_p$.

• $\left(E_{S_1}\right)$ Non-negativity and identity of indiscernibles are clear.
• $\left(E_{S_2}\right)$ Symmetry holds.
• $\left(E_{S_3}\right)$ Choose $\vartheta =1\in {\mathbb{E}}_p$ (indeed $\vartheta ⪰\theta$). For the nontrivial triple ($a,b,c$), we have

  $$d(a,b)+d(b,c)+\vartheta d(a,b)d(b,c)=2+2+1·(2·2)=8,$$

  and $d(a,c)=5⪯8$. The same computation applies to any permutation of {$a,b,c$}. Hence, $\left(E_{S_3}\right)$ is satisfied with $\vartheta =1$. Thus $\left(\Xi ,d\right)$ is an elliptic-valued suprametric space. However, the usual elliptic-valued metric triangle inequality

  $$d(a,c)⪯d(a,b)+d(b,c),$$

  does not hold here because

  $$5\cancel{⪯}2+2=4.$$
  Therefore ($\Xi ,d$) is not an elliptic-valued metric space.

Example 5.

Let $\Xi =\mathbb{C}$. Define $d:\Xi \times \Xi \to$${\mathbb{E}}_p$ by

$$d(\varkappa ,\sigma )=(\alpha +i\beta )\left|\varkappa -\sigma \right|,$$

for $\varkappa ,\sigma \in \Xi$, where $\alpha >0$ and $\beta >0$.

• $\left(E_{S_1}\right)-$$\left(E_{S_2}\right)$ hold trivially.
• $\left(E_{S_3}\right)$ holds because the standard triangle inequality on $\left|·\right|$ gives

  $$\left|\varkappa -\sigma \right|\le \left|\varkappa -\varrho \right|+\left|\varrho -\sigma \right|,$$

  which implies

  $$d(\varkappa ,\sigma )⪯d(\varkappa ,\varrho )+d(\varrho ,\sigma )+\vartheta d(\varkappa ,\varrho )d(\varrho ,\sigma ).$$
  Thus, $\left(\Xi ,d\right)$ is an elliptic-valued suprametric space.

Example 6.

Let $\Xi =C\left(\right[0,\Theta ],\mathbb{R})$ with $\Theta >0$, be the space of continuous real-valued functions on the interval $[0,\Theta ].$ Define a function $d:\Xi \times \Xi \to {\mathbb{E}}_p$ by

$$d(\varkappa ,\sigma )=\left({∥\varkappa -\sigma ∥}_{\infty }\right)e^{i{\displaystyle \frac{\pi }{4}}}=\left(\underset{t\in [0,\Theta ]}{max}∥\varkappa \left(t\right)-\sigma \left(t\right)∥\right)e^{i{\displaystyle \frac{\pi }{4}}},$$

where $i^2=p<0$. We now verify that d satisfies all the conditions of an elliptic-valued suprametric:

• $\left(E_{S_1}\right)$ Since ${∥\varkappa -\sigma ∥}_{\infty }\ge 0$, it follows that $\theta ⪯d(\varkappa ,\sigma )$. Moreover, ${∥\varkappa -\sigma ∥}_{\infty }=0$ if and only if $\varkappa =\sigma$, hence $d(\varkappa ,\sigma )=\theta ⟺\varkappa =\sigma$.
• $\left(E_{S_2}\right)$ Because ${∥\varkappa -\sigma ∥}_{\infty }={∥\sigma -\varkappa ∥}_{\infty }$, we have $d(\varkappa ,\sigma )=d(\sigma ,\varkappa )$.
• $\left(E_{S_3}\right)$ The standard sup-norm inequality

  $${∥\varkappa -\sigma ∥}_{\infty }\le {∥\varkappa -\varrho ∥}_{\infty }+{∥\varrho -\sigma ∥}_{\infty },$$

  implies that the triangle-type condition in an elliptic-valued suprametric space holds, i.e.,

  $$d(\varkappa ,\sigma )⪯d(\varkappa ,\varrho )+d(\varrho ,\sigma )+\vartheta d(\varkappa ,\varrho )d(\varrho ,\sigma ),$$

  for some $\vartheta ⪰\theta$. Hence, $(\Xi ,d)$ is an elliptic-valued suprametric space.

Definition 4.

A point ϰ in the elliptic-valued suprametric space $(\Xi ,d)$ is defined as an e-interior element of a subset $A$ within Ξ if a non-zero elliptic number r, satisfying $\theta \prec r$ for some r in ${\mathbb{E}}_p$ can be found, allowing for

$$B(\varkappa ,r)=\{\sigma \in \Xi :d(\varkappa ,\sigma )\prec r\}\subseteq A.$$

The notation $B(\varkappa ,r)$ represents an open ball in the elliptic-valued suprametric space $(\Xi ,d)$, centered at ϰ and having a radius of r. Consequently, a closed ball, denoted as $\overline{B(\varkappa ,r)}$, is defined by the set

$$\overline{B(\varkappa ,r)}=\{\sigma \in \Xi :d(\varkappa ,\sigma )⪯r\}.$$

A point ϰ in the set Ξ is said to be a limit point of a subset $A\subset$Ξ if, for every nonzero elliptic number r, there exists at least one point in A that lies within the punctured elliptic ball with center ϰ and radius r. Mathematically, this is expressed as

$$\left(B(\varkappa ,r)-\left\{\varkappa \right\}\right)\cap A\ne \varnothing .$$

The characteristic of openness for a subset A in Ξ is determined by whether every point within A is an interior point. A subset A is designated as closed if it encompasses all of its limit points. The set Ω, consisting of all open balls $B(\varkappa ,r)$ where ϰ belongs to Ξ and $\theta \prec r$ for some r in ${\mathbb{E}}_p$, establishes a subbasis that generates a Hausdorff topology on Ξ.

Definition 5.

Let ${\left\{{\varkappa }_{\mathsf{ȷ}}\right\}}_{\mathsf{ȷ}\in \mathbb{N}}$ be a sequence in an elliptic-valued suprametric space $(\Xi ,d)$.

(i) We say that the sequence ${\left\{{\varkappa }_{\mathsf{ȷ}}\right\}}_{\mathsf{ȷ}\in \mathbb{N}}$ converges to $\varkappa \in \Xi$ if, given every non-zero elliptic number $\delta \in {\mathbb{E}}_p$ with $\theta \prec \delta$, there is a ${\mathsf{ȷ}}_0\in \mathbb{N}$ such that $d\left({\varkappa }_{\mathsf{ȷ}},\varkappa \right)\prec \delta$, ∀$\mathsf{ȷ}>{\mathsf{ȷ}}_0$. This is written as ${\varkappa }_{\mathsf{ȷ}}\to \varkappa$ as $\mathsf{ȷ}\to \infty$ or ${lim}_{\mathsf{ȷ}\to \infty }{\varkappa }_{\mathsf{ȷ}}=\varkappa$.

(ii) A sequence ${\left\{{\varkappa }_{\mathsf{ȷ}}\right\}}_{\mathsf{ȷ}\in \mathbb{N}}$ in ($\Xi ,d)$ is a Cauchy sequence if, for every non-zero elliptic number $\delta \in {\mathbb{E}}_p$ with $\theta \prec \delta$, there exists ${\mathsf{ȷ}}_0\in \mathbb{N}$ such that $d\left({\varkappa }_{\mathsf{ȷ}},{\varkappa }_{\mathsf{ȷ}+m}\right)\prec \delta$, for all $\mathsf{ȷ}>{\mathsf{ȷ}}_0$ and $m\in \mathbb{N}$.

(iii) The elliptic-valued suprametric space ($\Xi ,d$) is defined as complete if and only if every Cauchy sequence within Ξ converges to an element of Ξ.

The ensuing lemmas will be utilized.

Lemma 1.

Let $(\Xi ,d)$ be an elliptic-valued suprametric space and ${\left\{{\varkappa }_{\mathsf{ȷ}}\right\}}_{\mathsf{ȷ}\in \mathbb{N}}$ be a sequence in Ξ. The sequence ${\left\{{\varkappa }_{\mathsf{ȷ}}\right\}}_{\mathsf{ȷ}\in \mathbb{N}}$ converges to $\varkappa \in \Xi$, $\mathsf{ȷ}\to \infty$ (or equivalently $d\left({\varkappa }_{\mathsf{ȷ}},\varkappa \right)\to \theta$ as $\mathsf{ȷ}\to \infty$) if and only if

$${∥d\left({\varkappa }_{\mathsf{ȷ}},\varkappa \right)∥}_{\mathbb{E}}\to 0,$$

as $\mathsf{ȷ}\to \infty$.

Proof. 

Suppose that ${\left\{{\varkappa }_{\mathsf{ȷ}}\right\}}_{\mathsf{ȷ}\in \mathbb{N}}$ converges to $\varkappa \in \Xi$ as $\mathsf{ȷ}\to \infty$. Choose

$$\delta ={\displaystyle \frac{\epsilon }{\sqrt{1-p}}}+i{\displaystyle \frac{\epsilon }{\sqrt{1-p}}},$$

for a given $\epsilon >0$ and $p<0$. Then $\theta \prec \delta \in {\mathbb{E}}_p$ and there exists a natural number ${\mathsf{ȷ}}_0$ such that

$$d\left({\varkappa }_{\mathsf{ȷ}},\varkappa \right)\prec \delta ,$$

for all $\mathsf{ȷ}\ge {\mathsf{ȷ}}_0$ and $p<0$. Thus

$${∥d\left({\varkappa }_{\mathsf{ȷ}},\varkappa \right)∥}_{\mathbb{E}}\le {∥\delta ∥}_{\mathbb{E}}=\epsilon ,$$

for all $\mathsf{ȷ}\ge {\mathsf{ȷ}}_0$ and $p<0$. Consequently,

$$\underset{\mathsf{ȷ}\to \infty }{lim}{∥d\left({\varkappa }_{\mathsf{ȷ}},\varkappa \right)∥}_{\mathbb{E}}=0.$$

Conversely, suppose that ${∥d\left({\varkappa }_{\mathsf{ȷ}},\varkappa \right)∥}_{\mathbb{E}}\to 0$ as $\mathsf{ȷ}\to \infty$. Then for $\theta \prec \delta \in {\mathbb{E}}_p$, there exists a real number $\tau >0$, such that for $\epsilon \in {\mathbb{E}}_p$ and ${∥\epsilon ∥}_{\mathbb{E}}<\tau$ implies that $\epsilon \prec \delta$. For this real number, there exists a natural number ${\mathsf{ȷ}}_0$ such that

$${∥d\left({\varkappa }_{\mathsf{ȷ}},\varkappa \right)∥}_{\mathbb{E}}<\tau ,$$

for all $\mathsf{ȷ}\ge {\mathsf{ȷ}}_0$. This concludes that

$$d\left({\varkappa }_{\mathsf{ȷ}},\varkappa \right)\prec \delta ,$$

for all $\mathsf{ȷ}\ge {\mathsf{ȷ}}_0$. Hence, ${\varkappa }_{\mathsf{ȷ}}$ converges to $\varkappa$. □

Lemma 2.

A sequence ${\left\{{\varkappa }_{\mathsf{ȷ}}\right\}}_{\mathsf{ȷ}\in \mathbb{N}}$ is a Cauchy sequence if and only if the elliptic norm of $d\left({\varkappa }_{\mathsf{ȷ}},{\varkappa }_{\mathsf{ȷ}+m}\right)$ tends to zero as j grows infinitely large. This condition is mathematically stated as

$${∥d\left({\varkappa }_{\mathsf{ȷ}},{\varkappa }_{\mathsf{ȷ}+m}\right)∥}_{\mathbb{E}}\to 0,$$

as $\mathsf{ȷ}\to \infty$, with $m\in \mathbb{N}$.

Proof. 

Assume that the sequence ${\left\{{\varkappa }_{\mathsf{ȷ}}\right\}}_{\mathsf{ȷ}\in \mathbb{N}}$ is a Cauchy sequence. Choose

$$\delta ={\displaystyle \frac{\epsilon }{\sqrt{1-p}}}+i{\displaystyle \frac{\epsilon }{\sqrt{1-p}}},$$

for a given $\epsilon >0$ and $p<0$. Then $\theta \prec \delta \in {\mathbb{E}}_p$ and there exists a natural number ${\mathsf{ȷ}}_0$ such that

$$d\left({\varkappa }_{\mathsf{ȷ}},{\varkappa }_{\mathsf{ȷ}+m}\right)\prec \delta ,$$

for all $\mathsf{ȷ}\ge {\mathsf{ȷ}}_0$, $m\in \mathbb{N}$ and $p<0$. Thus

$${∥d\left({\varkappa }_{\mathsf{ȷ}},{\varkappa }_{\mathsf{ȷ}+m}\right)∥}_{\mathbb{E}}<{∥\delta ∥}_{\mathbb{E}}=\epsilon ,$$

for all $\mathsf{ȷ}\ge {\mathsf{ȷ}}_0$, $m\in \mathbb{N}$ and $p<0$. Hence, we arrive at the following statement

$${∥d\left({\varkappa }_{\mathsf{ȷ}},{\varkappa }_{\mathsf{ȷ}+m}\right)∥}_{\mathbb{E}}\to 0,$$

as $\mathsf{ȷ}\to \infty$, with $m\in \mathbb{N}$. For the converse part, let us assume that ${∥d\left({\varkappa }_{\mathsf{ȷ}},{\varkappa }_{\mathsf{ȷ}+m}\right)∥}_{\mathbb{E}}\to 0$ as $\mathsf{ȷ}\to \infty$ and $m\in \mathbb{N}.$ Then given $\theta \prec \delta \in {\mathbb{E}}_p$, there exists a real number $\tau >0$, such that for $\epsilon \in {\mathbb{E}}_p$ and ${∥\epsilon ∥}_{\mathbb{E}}<\tau$ implies that $\epsilon \prec \delta$. For this real number, there exists a natural number ${\mathsf{ȷ}}_0$ such that

$${∥d\left({\varkappa }_{\mathsf{ȷ}},{\varkappa }_{\mathsf{ȷ}+m}\right)∥}_{\mathbb{E}}<\tau ,$$

for all $\mathsf{ȷ}\ge {\mathsf{ȷ}}_0$ and $m\in \mathbb{N}$, i.e.,

$$d\left({\varkappa }_{\mathsf{ȷ}},{\varkappa }_{\mathsf{ȷ}+m}\right)\prec \delta ,$$

for all $\mathsf{ȷ}\ge {\mathsf{ȷ}}_0$ and $m\in \mathbb{N}.$ Hence, ${\left\{{\varkappa }_{\mathsf{ȷ}}\right\}}_{\mathsf{ȷ}\in \mathbb{N}}$ is a Cauchy sequence. □

Remark 2.

It is a fundamental property of convergent sequences within an elliptic-valued suprametric space that they converge to one, and only one, limit.

3. Main Results

In this section, we establish the primary fixed point results that form the foundation of our study. Throughout, we consider $(\Xi ,d)$ as a complete elliptic-valued suprametric space. By leveraging the rich structure of elliptic-valued suprametric spaces, we derive existence and uniqueness results for fixed points under generalized contractive conditions. These results not only extend classical fixed point theorems to the E-VSM setting but also provide the necessary framework for subsequent applications to nonlinear integral equations and related problems.

Theorem 2.

Let $(\Xi ,d)$ be a complete elliptic-valued suprametric space and let $\mathcal{H},\mathfrak{F}:\Xi$$\to \Xi$. Suppose that there exist the constants ${\varpi }_1,{\varpi }_2,{\varpi }_3\in [0,1)$ with ${\varpi }_1+{\varpi }_2+{\varpi }_3<1$ such that

$$d\left(\mathcal{H}\varkappa ,\mathfrak{F}\sigma \right)⪯{\varpi }_{1}d\left(\varkappa ,\sigma \right)+{\varpi }_2{\displaystyle \frac{d\left(\varkappa ,\mathcal{H}\varkappa \right)d\left(\sigma ,\mathfrak{F}\sigma \right)}{1+d\left(\varkappa ,\sigma \right)}}+{\varpi }_3{\displaystyle \frac{d\left(\sigma ,\mathcal{H}\varkappa \right)d\left(\varkappa ,\mathfrak{F}\sigma \right)}{1+d\left(\varkappa ,\sigma \right)}},$$

(1)

for all $\varkappa ,\sigma \in \Xi$ with $\varkappa \ne \sigma$, then there exists a point ${\varkappa }^{*}\in \Xi$ such that ${\varkappa }^{*}=\mathcal{H}{\varkappa }^{*}=\mathfrak{F}{\varkappa }^{*}$.

Proof. 

Let ${\varkappa }_0\in \Xi$ be an arbitrary point. Define a sequence {${\varkappa }_{\mathsf{ȷ}}$} by

$${\varkappa }_{2\mathsf{ȷ}+1}=\mathfrak{F}{\varkappa }_{2\mathsf{ȷ}}\mathrm{and}{\varkappa }_{2\mathsf{ȷ}+2}=\mathcal{H}{\varkappa }_{2\mathsf{ȷ}+1},$$

for all $\mathsf{ȷ}\ge 0$. By (1), we have

$$\begin{array}{ccc}\hfill d({\varkappa }_{2\mathsf{ȷ}},{\varkappa }_{2\mathsf{ȷ}+1})& =& d(\mathcal{H}{\varkappa }_{2\mathsf{ȷ}-1},\mathfrak{F}{\varkappa }_{2\mathsf{ȷ}})⪯{\varpi }_{1}d({\varkappa }_{2\mathsf{ȷ}-1},{\varkappa }_{2\mathsf{ȷ}})+{\varpi }_2{\displaystyle \frac{d({\varkappa }_{2\mathsf{ȷ}-1},\mathcal{H}{\varkappa }_{2\mathsf{ȷ}-1})d({\varkappa }_{2\mathsf{ȷ}},\mathfrak{F}{\varkappa }_{2\mathsf{ȷ}})}{1+d({\varkappa }_{2\mathsf{ȷ}-1},{\varkappa }_{2\mathsf{ȷ}})}}\hfill \\ & & +{\varpi }_3{\displaystyle \frac{d\left({\varkappa }_{2\mathsf{ȷ}},\mathcal{H}{\varkappa }_{2\mathsf{ȷ}-1}\right)d\left({\varkappa }_{2\mathsf{ȷ}-1},\mathfrak{F}{\varkappa }_{2\mathsf{ȷ}}\right)}{1+d\left({\varkappa }_{2\mathsf{ȷ}-1},{\varkappa }_{2\mathsf{ȷ}}\right)}}\hfill \\ & =& {\varpi }_{1}d({\varkappa }_{2\mathsf{ȷ}-1},{\varkappa }_{2\mathsf{ȷ}})+{\varpi }_2{\displaystyle \frac{d({\varkappa }_{2\mathsf{ȷ}-1},{\varkappa }_{2\mathsf{ȷ}})d({\varkappa }_{2\mathsf{ȷ}},{\varkappa }_{2\mathsf{ȷ}+1})}{1+d({\varkappa }_{2\mathsf{ȷ}-1},{\varkappa }_{2\mathsf{ȷ}})}},\hfill \end{array}$$

which implies that

$$\begin{array}{ccc}\hfill {∥d({\varkappa }_{2\mathsf{ȷ}},{\varkappa }_{2\mathsf{ȷ}+1})∥}_{\mathbb{E}}& \le & {\varpi }_1{∥d({\varkappa }_{2\mathsf{ȷ}-1},{\varkappa }_{2\mathsf{ȷ}})∥}_{\mathbb{E}}\hfill \\ & & +{\varpi }_2{\displaystyle \frac{{∥d({\varkappa }_{2\mathsf{ȷ}-1},{\varkappa }_{2\mathsf{ȷ}})∥}_{\mathbb{E}}}{{∥1+d({\varkappa }_{2\mathsf{ȷ}-1},{\varkappa }_{2\mathsf{ȷ}})∥}_{\mathbb{E}}}}{∥d({\varkappa }_{2\mathsf{ȷ}},{\varkappa }_{2\mathsf{ȷ}+1})∥}_{\mathbb{E}}\hfill \\ & \le & {\varpi }_1{∥d({\varkappa }_{2\mathsf{ȷ}-1},{\varkappa }_{2\mathsf{ȷ}})∥}_{\mathbb{E}}+{\varpi }_2{∥d({\varkappa }_{2\mathsf{ȷ}},{\varkappa }_{2\mathsf{ȷ}+1})∥}_{\mathbb{E}}.\hfill \end{array}$$

It yields that

$${∥d({\varkappa }_{2\mathsf{ȷ}},{\varkappa }_{2\mathsf{ȷ}+1})∥}_{\mathbb{E}}\le {\displaystyle \frac{{\varpi }_1}{1-{\varpi }_2}}{∥d({\varkappa }_{2\mathsf{ȷ}-1},{\varkappa }_{2\mathsf{ȷ}})∥}_{\mathbb{E}},$$

(2)

for all j. Similarly, by (1), we have

$$\begin{array}{ccc}\hfill d({\varkappa }_{2\mathsf{ȷ}+1},{\varkappa }_{2\mathsf{ȷ}+2})& =& d(\mathfrak{F}{\varkappa }_{2\mathsf{ȷ}},\mathcal{H}{\varkappa }_{2\mathsf{ȷ}+1})=d(\mathcal{H}{\varkappa }_{2\mathsf{ȷ}+1},\mathfrak{F}{\varkappa }_{2\mathsf{ȷ}})\hfill \\ & ⪯& {\varpi }_{1}d({\varkappa }_{2\mathsf{ȷ}+1},{\varkappa }_{2\mathsf{ȷ}})+{\varpi }_2{\displaystyle \frac{d({\varkappa }_{2\mathsf{ȷ}+1},\mathcal{H}{\varkappa }_{2\mathsf{ȷ}+1})d({\varkappa }_{2\mathsf{ȷ}},\mathfrak{F}{\varkappa }_{2\mathsf{ȷ}})}{1+d({\varkappa }_{2\mathsf{ȷ}+1},{\varkappa }_{2\mathsf{ȷ}})}}\hfill \\ & & +{\varpi }_3{\displaystyle \frac{d\left({\varkappa }_{2\mathsf{ȷ}},\mathcal{H}{\varkappa }_{2\mathsf{ȷ}+1}\right)d\left({\varkappa }_{2\mathsf{ȷ}+1},\mathfrak{F}{\varkappa }_{2\mathsf{ȷ}}\right)}{1+d\left({\varkappa }_{2\mathsf{ȷ}+1},{\varkappa }_{2\mathsf{ȷ}}\right)}}\hfill \\ & =& {\varpi }_{1}d({\varkappa }_{2\mathsf{ȷ}+1},{\varkappa }_{2\mathsf{ȷ}})+{\varpi }_2{\displaystyle \frac{d({\varkappa }_{2\mathsf{ȷ}+1},{\varkappa }_{2\mathsf{ȷ}+2})d({\varkappa }_{2\mathsf{ȷ}},{\varkappa }_{2\mathsf{ȷ}+1})}{1+d({\varkappa }_{2\mathsf{ȷ}+1},{\varkappa }_{2\mathsf{ȷ}})}},\hfill \end{array}$$

which implies that

$$\begin{array}{ccc}\hfill {∥d({\varkappa }_{2\mathsf{ȷ}+1},{\varkappa }_{2\mathsf{ȷ}+2})∥}_{\mathbb{E}}& \le & {\varpi }_1{∥d({\varkappa }_{2\mathsf{ȷ}},{\varkappa }_{2\mathsf{ȷ}+1})∥}_{\mathbb{E}}\hfill \\ & & +{\varpi }_2{∥d({\varkappa }_{2\mathsf{ȷ}+1},{\varkappa }_{2\mathsf{ȷ}+2})∥}_{\mathbb{E}}{\displaystyle \frac{{∥d({\varkappa }_{2\mathsf{ȷ}},{\varkappa }_{2\mathsf{ȷ}+1})∥}_{\mathbb{E}}}{{∥1+d({\varkappa }_{2\mathsf{ȷ}+1},{\varkappa }_{2\mathsf{ȷ}})∥}_{\mathbb{E}}}},\hfill \end{array}$$

for all j. Now since ${\displaystyle \frac{{∥d({\varkappa }_{2\mathsf{ȷ}},{\varkappa }_{2\mathsf{ȷ}+1})∥}_{\mathbb{E}}}{{∥1+d({\varkappa }_{2\mathsf{ȷ}+1},{\varkappa }_{2\mathsf{ȷ}})∥}_{\mathbb{E}}}}<1$, so we have

$$\begin{array}{ccc}\hfill {∥d({\varkappa }_{2\mathsf{ȷ}+1},{\varkappa }_{2\mathsf{ȷ}+2})∥}_{\mathbb{E}}& \le & {\varpi }_1{∥d({\varkappa }_{2\mathsf{ȷ}},{\varkappa }_{2\mathsf{ȷ}+1})∥}_{\mathbb{E}}\hfill \\ & & +{\varpi }_2{∥d({\varkappa }_{2\mathsf{ȷ}+1},{\varkappa }_{2\mathsf{ȷ}+2})∥}_{\mathbb{E}},\hfill \end{array}$$

which implies that

$${∥d({\varkappa }_{2\mathsf{ȷ}+1},{\varkappa }_{2\mathsf{ȷ}+2})∥}_{\mathbb{E}}\le {\displaystyle \frac{{\varpi }_1}{1-{\varpi }_2}}{∥d({\varkappa }_{2\mathsf{ȷ}},{\varkappa }_{2\mathsf{ȷ}+1})∥}_{\mathbb{E}}.$$

(3)

Now let ${\displaystyle \frac{{\varpi }_1}{1-{\varpi }_2}}=\varpi <1$, then from (2) and (3), we conclude that

$${∥d({\varkappa }_{\mathsf{ȷ}},{\varkappa }_{\mathsf{ȷ}+1})∥}_{\mathbb{E}}\le \varpi {∥d({\varkappa }_{\mathsf{ȷ}-1},{\varkappa }_{\mathsf{ȷ}})∥}_{\mathbb{E}},$$

for every ȷ. This further yields that

$$\begin{array}{ccc}\hfill {∥d({\varkappa }_{\mathsf{ȷ}},{\varkappa }_{\mathsf{ȷ}+1})∥}_{\mathbb{E}}& \le & \varpi {∥d({\varkappa }_{\mathsf{ȷ}-1},{\varkappa }_{\mathsf{ȷ}})∥}_{\mathbb{E}}\le {\varpi }^2{∥d({\varkappa }_{\mathsf{ȷ}-2},{\varkappa }_{\mathsf{ȷ}-1})∥}_{\mathbb{E}}\hfill \\ & \le & \dots \le {\varpi }^{\mathsf{ȷ}}{∥d({\varkappa }_0,{\varkappa }_1)∥}_{\mathbb{E}},\hfill \end{array}$$

(4)

for every ȷ. Now, for any natural number m greater than n, we have

$$\begin{array}{ccc}\hfill {∥d\left({\varkappa }_n,{\varkappa }_m\right)∥}_{\mathbb{E}}& \le & {∥d\left({\varkappa }_n,{\varkappa }_{n+1}\right)∥}_{\mathbb{E}}+{∥d\left({\varkappa }_{n+1},{\varkappa }_m\right)∥}_{\mathbb{E}}\hfill \\ & & +{∥\vartheta ∥}_{\mathbb{E}}{∥d\left({\varkappa }_n,{\varkappa }_{n+1}\right)∥}_{\mathbb{E}}{∥d\left({\varkappa }_{n+1},{\varkappa }_m\right)∥}_{\mathbb{E}},\hfill \end{array}$$

which implies

$${∥d\left({\varkappa }_n,{\varkappa }_m\right)∥}_{\mathbb{E}}\le {∥d\left({\varkappa }_n,{\varkappa }_{n+1}\right)∥}_{\mathbb{E}}+\left(1+{∥\vartheta ∥}_{\mathbb{E}}{∥d\left({\varkappa }_n,{\varkappa }_{n+1}\right)∥}_{\mathbb{E}}\right){∥d\left({\varkappa }_{n+1},{\varkappa }_m\right)∥}_{\mathbb{E}}.$$

(5)

Now, we can apply the triangle inequality again to the second term ${∥d\left({\varkappa }_{n+1},{\varkappa }_m\right)∥}_{\mathbb{E}}$, we have

$$\begin{array}{ccc}\hfill {∥d\left({\varkappa }_{n+1},{\varkappa }_m\right)∥}_{\mathbb{E}}& \le & {∥d\left({\varkappa }_{n+1},{\varkappa }_{n+2}\right)∥}_{\mathbb{E}}+{∥d\left({\varkappa }_{n+2},{\varkappa }_m\right)∥}_{\mathbb{E}}\hfill \\ & & +{∥\vartheta ∥}_{\mathbb{E}}{∥d\left({\varkappa }_{n+1},{\varkappa }_{n+2}\right)∥}_{\mathbb{E}}{∥d\left({\varkappa }_{n+2},{\varkappa }_m\right)∥}_{\mathbb{E}}\hfill \\ & =& {∥d\left({\varkappa }_{n+1},{\varkappa }_{n+2}\right)∥}_{\mathbb{E}}\hfill \\ & & +\left(1+{∥\vartheta ∥}_{\mathbb{E}}{∥d\left({\varkappa }_{n+1},{\varkappa }_{n+2}\right)∥}_{\mathbb{E}}\right){∥d\left({\varkappa }_{n+2},{\varkappa }_m\right)∥}_{\mathbb{E}}.\hfill \end{array}$$

Now, we can apply the triangle inequality again to the second term ${∥d\left({\varkappa }_{n+2},{\varkappa }_m\right)∥}_{\mathbb{E}}$, we have

$$\begin{array}{ccc}\hfill {∥d\left({\varkappa }_{n+2},{\varkappa }_m\right)∥}_{\mathbb{E}}& \le & {∥d\left({\varkappa }_{n+2},{\varkappa }_{n+3}\right)∥}_{\mathbb{E}}+{∥d\left({\varkappa }_{n+3},{\varkappa }_m\right)∥}_{\mathbb{E}}\hfill \\ & & +{∥\vartheta ∥}_{\mathbb{E}}{∥d\left({\varkappa }_{n+2},{\varkappa }_{n+3}\right)∥}_{\mathbb{E}}{∥d\left({\varkappa }_{n+3},{\varkappa }_m\right)∥}_{\mathbb{E}}\hfill \\ & =& {∥d\left({\varkappa }_{n+2},{\varkappa }_{n+3}\right)∥}_{\mathbb{E}}\hfill \\ & & +\left(1+{∥\vartheta ∥}_{\mathbb{E}}{∥d\left({\varkappa }_{n+2},{\varkappa }_{n+3}\right)∥}_{\mathbb{E}}\right){∥d\left({\varkappa }_{n+3},{\varkappa }_m\right)∥}_{\mathbb{E}},\hfill \end{array}$$

and so on

$$\begin{array}{ccc}\hfill {∥d\left({\varkappa }_{m-2},{\varkappa }_m\right)∥}_{\mathbb{E}}& \le & {∥d\left({\varkappa }_{m-2},{\varkappa }_{m-1}\right)∥}_{\mathbb{E}}+{∥d\left({\varkappa }_{m-1},{\varkappa }_m\right)∥}_{\mathbb{E}}\hfill \\ & & +{∥\vartheta ∥}_{\mathbb{E}}{∥d\left({\varkappa }_{m-2},{\varkappa }_{m-1}\right)∥}_{\mathbb{E}}{∥d\left({\varkappa }_{m-1},{\varkappa }_m\right)∥}_{\mathbb{E}}\hfill \\ & =& {∥d\left({\varkappa }_{m-2},{\varkappa }_{m-1}\right)∥}_{\mathbb{E}}+\left(1+{∥\vartheta ∥}_{\mathbb{E}}{∥d\left({\varkappa }_{m-2},{\varkappa }_{m-1}\right)∥}_{\mathbb{E}}\right){∥d\left({\varkappa }_{m-1},{\varkappa }_m\right)∥}_{\mathbb{E}}.\hfill \end{array}$$

Recursively substituting each inequality into the preceding one (5) and simplifying, we have

$${∥d\left({\varkappa }_n,{\varkappa }_m\right)∥}_{\mathbb{E}}\le \sum _{k=n}^{m-1}∥d\left({\varkappa }_n,{\varkappa }_{n+1}\right)∥\prod _{i=n}^{k-1}\left(1+{∥\vartheta ∥}_{\mathbb{E}}∥d\left({\varkappa }_i,{\varkappa }_{i+1}\right)∥\right).$$

By the inequality (4), we have

$${∥d\left({\varkappa }_n,{\varkappa }_m\right)∥}_{\mathbb{E}}\le {∥d\left({\varkappa }_0,{\varkappa }_1\right)∥}_{\mathbb{E}}\sum _{k=n}^{m-1}{\varpi }^k\prod _{i=n}^{k-1}\left(1+{\varpi }^i{∥\vartheta ∥}_{\mathbb{E}}{∥d\left({\varkappa }_0,{\varkappa }_1\right)∥}_{\mathbb{E}}\right).$$

(6)

Since $\varpi \in (0,1)$ and norms are nonnegative, we have

$$1+{\varpi }^i{∥\vartheta ∥}_{\mathbb{E}}{∥d\left({\varkappa }_0,{\varkappa }_1\right)∥}_{\mathbb{E}}\le exp\left({\varpi }^i{∥\vartheta ∥}_{\mathbb{E}}{∥d\left({\varkappa }_0,{\varkappa }_1\right)∥}_{\mathbb{E}}\right).$$

Therefore,

$$\prod _{i=n}^{k-1}\left(1+{\varpi }^i{∥\vartheta ∥}_{\mathbb{E}}{∥d\left({\varkappa }_0,{\varkappa }_1\right)∥}_{\mathbb{E}}\right)\le exp\left({∥\vartheta ∥}_{\mathbb{E}}{∥d\left({\varkappa }_0,{\varkappa }_1\right)∥}_{\mathbb{E}}\sum _{i=n}^{k-1}{\varpi }^i\right).$$

Since $\varpi \in (0,1)$,

$$\sum _{i=n}^{k-1}{\varpi }^i\le \sum _{i=n}^{\infty }{\varpi }^i={\displaystyle \frac{{\varpi }^n}{1-\varpi }}\le {\displaystyle \frac{1}{1-\varpi }}.$$

Thus,

$$\prod _{i=n}^{k-1}\left(1+{\varpi }^i{∥\vartheta ∥}_{\mathbb{E}}{∥d\left({\varkappa }_0,{\varkappa }_1\right)∥}_{\mathbb{E}}\right)\le exp\left({\displaystyle \frac{{∥\vartheta ∥}_{\mathbb{E}}{∥d\left({\varkappa }_0,{\varkappa }_1\right)∥}_{\mathbb{E}}}{1-\varpi }}\right)=C,$$

where $C>0$ is a finite constant independent of m and n. This proves uniform boundedness of the product. Substituting this bound into (6), we obtain

$${∥d\left({\varkappa }_n,{\varkappa }_m\right)∥}_{\mathbb{E}}\le C{∥d\left({\varkappa }_0,{\varkappa }_1\right)∥}_{\mathbb{E}}\sum _{k=n}^{m-1}{\varpi }^k.$$

(7)

Since ${\displaystyle \sum _{k=n}^{m-1}}{\varpi }^k$ is a finite geometric series with the first term ${\varpi }^n$ and the common ratio $\varpi$. The sum of a finite geometric series is given by

$$\sum _{k=n}^{m-1}{\varpi }^k={\varpi }^n{\displaystyle \frac{\left(1-{\varpi }^{m-n}\right)}{1-\varpi }}.$$

As $m\to \infty$, ${\varpi }^{m-n}\to 0$. Therefore, the sum converges to ${\displaystyle \frac{{\varpi }^n}{1-\varpi }}$, i.e.,

$$\underset{m\to \infty }{lim}{\varpi }^n{\displaystyle \frac{\left(1-{\varpi }^{m-n}\right)}{1-\varpi }}={\displaystyle \frac{{\varpi }^n}{1-\varpi }}.$$

Since $\varpi <1$, the expression ${\displaystyle \frac{{\varpi }^n}{1-\varpi }}$ approaches 0 as $n\to \infty$. Taking the limit as $n\to \infty$ in the inequality (7) and employing the established facts, we arrive at

$$\underset{n\to \infty }{lim}{∥d\left({\varkappa }_n,{\varkappa }_m\right)∥}_{\mathbb{E}}=0,$$

which implies that $\left\{{\varkappa }_n\right\}$ is Cauchy. As $\Xi$ is complete, so there exists ${\varkappa }^{*}\in \Xi$ such that ${\varkappa }_n\to {\varkappa }_n^{*}$ as $n\to \infty$. Thus,

$$\underset{n\to \infty }{lim}{\varkappa }_n={\varkappa }^{*}.$$

(8)

To show that ${\varkappa }^{*}$ is a fixed point of $\mathcal{H}$, we apply the triangle inequality together with inequality (1) as follows

$$\begin{array}{ccc}\hfill d\left({\varkappa }^{*},\mathcal{H}{\varkappa }^{*}\right)& ⪯& d\left({\varkappa }^{*},{\varkappa }_{2\mathsf{ȷ}+1}\right)+d\left({\varkappa }_{2\mathsf{ȷ}+1},\mathcal{H}{\varkappa }^{*}\right)+\vartheta d\left({\varkappa }^{*},{\varkappa }_{2\mathsf{ȷ}+1}\right)d\left({\varkappa }_{2\mathsf{ȷ}+1},\mathcal{H}{\varkappa }^{*}\right)\hfill \\ & =& d\left({\varkappa }^{*},{\varkappa }_{2\mathsf{ȷ}+1}\right)+d\left(\mathfrak{F}{\varkappa }_{2\mathsf{ȷ}},\mathcal{H}{\varkappa }^{*}\right)+\vartheta d\left({\varkappa }^{*},{\varkappa }_{2\mathsf{ȷ}+1}\right)d\left(\mathfrak{F}{\varkappa }_{2\mathsf{ȷ}},\mathcal{H}{\varkappa }^{*}\right)\hfill \\ & =& d\left({\varkappa }^{*},{\varkappa }_{2\mathsf{ȷ}+1}\right)+d\left(\mathcal{H}{\varkappa }^{*},\mathfrak{F}{\varkappa }_{2\mathsf{ȷ}}\right)+\vartheta d\left({\varkappa }^{*},{\varkappa }_{2\mathsf{ȷ}+1}\right)d\left(\mathcal{H}{\varkappa }^{*},\mathfrak{F}{\varkappa }_{2\mathsf{ȷ}}\right)\hfill \\ & ⪯& d\left({\varkappa }^{*},{\varkappa }_{2\mathsf{ȷ}+1}\right)+{\varpi }_{1}d\left({\varkappa }^{*},{\varkappa }_{2\mathsf{ȷ}}\right)\hfill \\ & & +{\varpi }_2{\displaystyle \frac{d\left({\varkappa }^{*},\mathcal{H}{\varkappa }^{*}\right)·d\left({\varkappa }_{2\mathsf{ȷ}},\mathfrak{F}{\varkappa }_{2\mathsf{ȷ}}\right)}{1+d\left({\varkappa }^{*},{\varkappa }_{2\mathsf{ȷ}}\right)}}\hfill \\ & & +{\varpi }_3{\displaystyle \frac{d\left({\varkappa }_{2\mathsf{ȷ}},\mathcal{H}{\varkappa }^{*}\right)d\left({\varkappa }^{*},\mathfrak{F}{\varkappa }_{2\mathsf{ȷ}}\right)}{1+d\left({\varkappa }^{*},{\varkappa }_{2\mathsf{ȷ}}\right)}}\hfill \\ & & +\vartheta d\left({\varkappa }^{*},{\varkappa }_{2\mathsf{ȷ}+1}\right)\left(\begin{array}{c}{\varpi }_{1}d\left({\varkappa }^{*},{\varkappa }_{2\mathsf{ȷ}}\right)+{\varpi }_2{\displaystyle \frac{d\left({\varkappa }^{*},\mathcal{H}{\varkappa }^{*}\right)·d\left({\varkappa }_{2\mathsf{ȷ}},\mathfrak{F}{\varkappa }_{2\mathsf{ȷ}}\right)}{1+d\left({\varkappa }^{*},{\varkappa }_{2\mathsf{ȷ}}\right)}}\\ +{\varpi }_3{\displaystyle \frac{d\left({\varkappa }_{2\mathsf{ȷ}},\mathcal{H}{\varkappa }^{*}\right)d\left({\varkappa }^{*},\mathfrak{F}{\varkappa }_{2\mathsf{ȷ}}\right)}{1+d\left({\varkappa }^{*},{\varkappa }_{2\mathsf{ȷ}}\right)}}\end{array}\right)\hfill \\ & =& d\left({\varkappa }^{*},{\varkappa }_{2\mathsf{ȷ}+1}\right)+{\varpi }_{1}d\left({\varkappa }^{*},{\varkappa }_{2\mathsf{ȷ}}\right)\hfill \\ & & +{\varpi }_2{\displaystyle \frac{d\left({\varkappa }^{*},\mathcal{H}{\varkappa }^{*}\right)·d\left({\varkappa }_{2\mathsf{ȷ}},{\varkappa }_{2\mathsf{ȷ}}\right)}{1+d\left({\varkappa }^{*},{\varkappa }_{2\mathsf{ȷ}}\right)}}+{\varpi }_3{\displaystyle \frac{d\left({\varkappa }_{2\mathsf{ȷ}},\mathcal{H}{\varkappa }^{*}\right)d\left({\varkappa }^{*},{\varkappa }_{2\mathsf{ȷ}+1}\right)}{1+d\left({\varkappa }^{*},{\varkappa }_{2\mathsf{ȷ}}\right)}}\hfill \\ & & +\vartheta d\left({\varkappa }^{*},{\varkappa }_{2\mathsf{ȷ}+1}\right)\left(\begin{array}{c}{\varpi }_{1}d\left({\varkappa }^{*},{\varkappa }_{2\mathsf{ȷ}}\right)+{\varpi }_2{\displaystyle \frac{d\left({\varkappa }^{*},\mathcal{H}{\varkappa }^{*}\right)·d\left({\varkappa }_{2\mathsf{ȷ}},{\varkappa }_{2\mathsf{ȷ}}\right)}{1+d\left({\varkappa }^{*},{\varkappa }_{2\mathsf{ȷ}}\right)}}\\ +{\varpi }_3{\displaystyle \frac{d\left({\varkappa }_{2\mathsf{ȷ}},\mathcal{H}{\varkappa }^{*}\right)d\left({\varkappa }^{*},{\varkappa }_{2\mathsf{ȷ}+1}\right)}{1+d\left({\varkappa }^{*},{\varkappa }_{2\mathsf{ȷ}}\right)}}\end{array}\right),\hfill \end{array}$$

which implies that

$$\begin{array}{ccc}\hfill {∥d\left({\varkappa }^{*},\mathcal{H}{\varkappa }^{*}\right)∥}_{\mathbb{E}}& \le & {∥d\left({\varkappa }^{*},{\varkappa }_{2\mathsf{ȷ}+1}\right)∥}_{\mathbb{E}}+{\varpi }_1{∥d\left({\varkappa }^{*},{\varkappa }_{2\mathsf{ȷ}}\right)∥}_{\mathbb{E}}\hfill \\ & & +{\varpi }_2{\displaystyle \frac{{∥d\left({\varkappa }^{*},\mathcal{H}{\varkappa }^{*}\right)∥}_{\mathbb{E}}·{∥d\left({\varkappa }_{2\mathsf{ȷ}},{\varkappa }_{2\mathsf{ȷ}+1}\right)∥}_{\mathbb{E}}}{{∥1+d\left({\varkappa }^{*},{\varkappa }_{2\mathsf{ȷ}}\right)∥}_{\mathbb{E}}}}\hfill \\ & & +{\varpi }_3{\displaystyle \frac{{∥d\left({\varkappa }_{2\mathsf{ȷ}},\mathcal{H}{\varkappa }^{*}\right)∥}_{\mathbb{E}}{∥d\left({\varkappa }^{*},{\varkappa }_{2\mathsf{ȷ}+1}\right)∥}_{\mathbb{E}}}{{∥1+d\left({\varkappa }^{*},{\varkappa }_{2\mathsf{ȷ}}\right)∥}_{\mathbb{E}}}}\hfill \\ & & +{∥\vartheta ∥}_{\mathbb{E}}{∥d\left({\varkappa }^{*},{\varkappa }_{2\mathsf{ȷ}+1}\right)∥}_{\mathbb{E}}\left(\begin{array}{c}{\varpi }_1{∥d\left({\varkappa }^{*},{\varkappa }_{2\mathsf{ȷ}}\right)∥}_{\mathbb{E}}+{\varpi }_2{\displaystyle \frac{{∥d\left({\varkappa }^{*},\mathcal{H}{\varkappa }^{*}\right)∥}_{\mathbb{E}}·{∥d\left({\varkappa }_{2\mathsf{ȷ}},{\varkappa }_{2\mathsf{ȷ}+1}\right)∥}_{\mathbb{E}}}{{∥1+d\left({\varkappa }^{*},{\varkappa }_{2\mathsf{ȷ}}\right)∥}_{\mathbb{E}}}}\\ +{\varpi }_3{\displaystyle \frac{{∥d\left({\varkappa }_{2\mathsf{ȷ}},\mathcal{H}{\varkappa }^{*}\right)∥}_{\mathbb{E}}{∥d\left({\varkappa }^{*},{\varkappa }_{2\mathsf{ȷ}+1}\right)∥}_{\mathbb{E}}}{{∥1+d\left({\varkappa }^{*},{\varkappa }_{2\mathsf{ȷ}}\right)∥}_{\mathbb{E}}}}\end{array}\right).\hfill \end{array}$$

Letting $\mathsf{ȷ}\to \infty$ in the above inequality and using the fact that ${\varkappa }_{\mathsf{ȷ}}\to {\varkappa }^{*}\in \Xi$ as $\mathsf{ȷ}\to \infty$, we get

$${∥d\left({\varkappa }^{*},\mathcal{H}{\varkappa }^{*}\right)∥}_{\mathbb{E}}\le 0,$$

so ${∥d\left({\varkappa }^{*},\mathcal{H}{\varkappa }^{*}\right)∥}_{\mathbb{E}}=0$, implies ${\varkappa }^{*}=\mathcal{H}{\varkappa }^{*}$. Thus ${\varkappa }^{*}$ is a fixed point of $\mathcal{H}$. Similarly, we prove that ${\varkappa }^{*}$ is fixed point of $\mathfrak{F}$. By (1), we have

$$\begin{array}{ccc}\hfill d\left({\varkappa }^{*},\mathfrak{F}{\varkappa }^{*}\right)& ⪯& d\left({\varkappa }^{*},{\varkappa }_{2\mathsf{ȷ}+2}\right)+d\left({\varkappa }_{2\mathsf{ȷ}+2},\mathfrak{F}{\varkappa }^{*}\right)+\vartheta d\left({\varkappa }^{*},{\varkappa }_{2\mathsf{ȷ}+2}\right)d\left({\varkappa }_{2\mathsf{ȷ}+2},\mathfrak{F}{\varkappa }^{*}\right)\hfill \\ & =& d\left({\varkappa }^{*},{\varkappa }_{2\mathsf{ȷ}+2}\right)+d\left(\mathcal{H}{\varkappa }_{2\mathsf{ȷ}+1},\mathfrak{F}{\varkappa }^{*}\right)+\vartheta d\left({\varkappa }^{*},{\varkappa }_{2\mathsf{ȷ}+2}\right)d\left(\mathcal{H}{\varkappa }_{2\mathsf{ȷ}+1},\mathfrak{F}{\varkappa }^{*}\right)\hfill \\ & ⪯& d\left({\varkappa }^{*},{\varkappa }_{2\mathsf{ȷ}+2}\right)+{\varpi }_{1}d\left({\varkappa }_{2\mathsf{ȷ}+1},{\varkappa }^{*}\right)+{\varpi }_2{\displaystyle \frac{d\left({\varkappa }_{2\mathsf{ȷ}+1},\mathcal{H}{\varkappa }_{2\mathsf{ȷ}+1}\right)·d\left({\varkappa }^{*},\mathfrak{F}{\varkappa }^{*}\right)}{1+d\left({\varkappa }_{2\mathsf{ȷ}+1},{\varkappa }^{*}\right)}}\hfill \\ & & +{\varpi }_3{\displaystyle \frac{d\left({\varkappa }^{*},\mathcal{H}{\varkappa }_{2\mathsf{ȷ}+1}\right)d\left({\varkappa }_{2\mathsf{ȷ}+1},\mathfrak{F}{\varkappa }^{*}\right)}{1+d\left({\varkappa }_{2\mathsf{ȷ}+1},{\varkappa }^{*}\right)}}\hfill \\ & & +\vartheta d\left({\varkappa }^{*},{\varkappa }_{2\mathsf{ȷ}+2}\right)\left(\begin{array}{c}{\varpi }_{1}d\left({\varkappa }_{2\mathsf{ȷ}+1},{\varkappa }^{*}\right)+{\varpi }_2{\displaystyle \frac{d\left({\varkappa }_{2\mathsf{ȷ}+1},\mathcal{H}{\varkappa }_{2\mathsf{ȷ}+1}\right)·d\left({\varkappa }^{*},\mathfrak{F}{\varkappa }^{*}\right)}{1+d\left({\varkappa }_{2\mathsf{ȷ}+1},{\varkappa }^{*}\right)}}\\ +{\varpi }_3{\displaystyle \frac{d\left({\varkappa }^{*},\mathcal{H}{\varkappa }_{2\mathsf{ȷ}+1}\right)d\left({\varkappa }_{2\mathsf{ȷ}+1},\mathfrak{F}{\varkappa }^{*}\right)}{1+d\left({\varkappa }_{2\mathsf{ȷ}+1},{\varkappa }^{*}\right)}}\end{array}\right)\hfill \\ & =& d\left({\varkappa }^{*},{\varkappa }_{2\mathsf{ȷ}+2}\right)+{\varpi }_{1}d\left({\varkappa }_{2\mathsf{ȷ}+1},{\varkappa }^{*}\right)+{\varpi }_2{\displaystyle \frac{d\left({\varkappa }_{2\mathsf{ȷ}+1},{\varkappa }_{2\mathsf{ȷ}+2}\right)·d\left({\varkappa }^{*},\mathfrak{F}{\varkappa }^{*}\right)}{1+d\left({\varkappa }_{2\mathsf{ȷ}+1},{\varkappa }^{*}\right)}}\hfill \\ & & +{\varpi }_3{\displaystyle \frac{d\left({\varkappa }^{*},{\varkappa }_{2\mathsf{ȷ}+2}\right)d\left({\varkappa }_{2\mathsf{ȷ}+1},\mathfrak{F}{\varkappa }^{*}\right)}{1+d\left({\varkappa }_{2\mathsf{ȷ}+1},{\varkappa }^{*}\right)}}\hfill \\ & & +\vartheta d\left({\varkappa }^{*},{\varkappa }_{2\mathsf{ȷ}+2}\right)\left(\begin{array}{c}{\varpi }_{1}d\left({\varkappa }_{2\mathsf{ȷ}+1},{\varkappa }^{*}\right)+{\varpi }_2{\displaystyle \frac{d\left({\varkappa }_{2\mathsf{ȷ}+1},{\varkappa }_{2\mathsf{ȷ}+2}\right)·d\left({\varkappa }^{*},\mathfrak{F}{\varkappa }^{*}\right)}{1+d\left({\varkappa }_{2\mathsf{ȷ}+1},{\varkappa }^{*}\right)}}\\ +{\varpi }_3{\displaystyle \frac{d\left({\varkappa }^{*},{\varkappa }_{2\mathsf{ȷ}+2}\right)d\left({\varkappa }_{2\mathsf{ȷ}+1},\mathfrak{F}{\varkappa }^{*}\right)}{1+d\left({\varkappa }_{2\mathsf{ȷ}+1},{\varkappa }^{*}\right)}}\end{array}\right),\hfill \end{array}$$

which implies that

$$\begin{array}{ccc}\hfill {∥d\left({\varkappa }^{*},\mathfrak{F}{\varkappa }^{*}\right)∥}_{\mathbb{E}}& \le & {∥d\left({\varkappa }^{*},{\varkappa }_{2\mathsf{ȷ}+2}\right)∥}_{\mathbb{E}}+{∥d\left({\varkappa }_{2\mathsf{ȷ}+2},\mathfrak{F}{\varkappa }^{*}\right)∥}_{\mathbb{E}}\hfill \\ & \le & {∥d\left({\varkappa }^{*},{\varkappa }_{2\mathsf{ȷ}+2}\right)∥}_{\mathbb{E}}+{∥d\left(\mathcal{H}{\varkappa }_{2\mathsf{ȷ}+1},\mathfrak{F}{\varkappa }^{*}\right)∥}_{\mathbb{E}}\hfill \\ & \le & {∥d\left({\varkappa }^{*},{\varkappa }_{2\mathsf{ȷ}+2}\right)∥}_{\mathbb{E}}+{\varpi }_1{∥d\left({\varkappa }_{2\mathsf{ȷ}+1},{\varkappa }^{*}\right)∥}_{\mathbb{E}}\hfill \\ & & +{\varpi }_2{\displaystyle \frac{{∥d\left({\varkappa }_{2\mathsf{ȷ}+1},{\varkappa }_{2\mathsf{ȷ}+2}\right)∥}_{\mathbb{E}}·{∥d\left({\varkappa }^{*},\mathfrak{F}{\varkappa }^{*}\right)∥}_{\mathbb{E}}}{{∥1+d\left({\varkappa }_{2\mathsf{ȷ}+1},{\varkappa }^{*}\right)∥}_{\mathbb{E}}}}\hfill \\ & & +{\varpi }_3{\displaystyle \frac{{∥d\left({\varkappa }^{*},{\varkappa }_{2\mathsf{ȷ}+2}\right)∥}_{\mathbb{E}}{∥d\left({\varkappa }_{2\mathsf{ȷ}+1},\mathfrak{F}{\varkappa }^{*}\right)∥}_{\mathbb{E}}}{{∥1+d\left({\varkappa }_{2\mathsf{ȷ}+1},{\varkappa }^{*}\right)∥}_{\mathbb{E}}}}\hfill \\ & & +{∥\vartheta ∥}_{\mathbb{E}}{∥d\left({\varkappa }^{*},{\varkappa }_{2\mathsf{ȷ}+2}\right)∥}_{\mathbb{E}}\left(\begin{array}{c}{\varpi }_1{∥d\left({\varkappa }_{2\mathsf{ȷ}+1},{\varkappa }^{*}\right)∥}_{\mathbb{E}}\\ +{\varpi }_2{\displaystyle \frac{{∥d\left({\varkappa }_{2\mathsf{ȷ}+1},{\varkappa }_{2\mathsf{ȷ}+2}\right)∥}_{\mathbb{E}}·{∥d\left({\varkappa }^{*},\mathfrak{F}{\varkappa }^{*}\right)∥}_{\mathbb{E}}}{{∥1+d\left({\varkappa }_{2\mathsf{ȷ}+1},{\varkappa }^{*}\right)∥}_{\mathbb{E}}}}\\ +{\varpi }_3{\displaystyle \frac{{∥d\left({\varkappa }^{*},{\varkappa }_{2\mathsf{ȷ}+2}\right)∥}_{\mathbb{E}}{∥d\left({\varkappa }_{2\mathsf{ȷ}+1},\mathfrak{F}{\varkappa }^{*}\right)∥}_{\mathbb{E}}}{{∥1+d\left({\varkappa }_{2\mathsf{ȷ}+1},{\varkappa }^{*}\right)∥}_{\mathbb{E}}}}\end{array}\right).\hfill \end{array}$$

Letting $\mathsf{ȷ}\to \infty$ in the above inequality and using the fact that ${\varkappa }_{\mathsf{ȷ}}\to {\varkappa }^{*}\in \Xi$ as $\mathsf{ȷ}\to \infty$, we get

$${∥d\left({\varkappa }^{*},\mathfrak{F}{\varkappa }^{*}\right)∥}_{\mathbb{E}}=0.$$

Hence, ${\varkappa }^{*}=\mathfrak{F}{\varkappa }^{*}$. Thus ${\varkappa }^{*}$ is a fixed point of $\mathfrak{F}$. Therefore, ${\varkappa }^{*}$ is common fixed point of $\mathcal{H}$ and $\mathfrak{F}.$ For the purpose of proving the uniqueness of ${\varkappa }^{*}$, let us presume that $\mathfrak{F}$ and $\mathcal{H}$ possess a distinct common fixed point, denoted as ${\varkappa }^{/}$. Then

$${\varkappa }^{/}=\mathcal{H}{\varkappa }^{/}=\mathfrak{F}{\varkappa }^{/},$$

but ${\varkappa }^{*}\ne {\varkappa }^{/}$. Now from (1), we have

$$\begin{array}{ccc}\hfill d\left({\varkappa }^{*},{\varkappa }^{/}\right)& =& d\left(\mathcal{H}{\varkappa }^{*},\mathfrak{F}{\varkappa }^{/}\right)\hfill \\ & ⪯& {\varpi }_{1}d\left({\varkappa }^{*},{\varkappa }^{/}\right)+{\varpi }_2{\displaystyle \frac{d\left({\varkappa }^{*},\mathcal{H}{\varkappa }^{*}\right)d\left({\varkappa }^{/},\mathfrak{F}{\varkappa }^{/}\right)}{1+d\left(\varkappa ,\sigma \right)}}\hfill \\ & & +{\varpi }_3{\displaystyle \frac{d\left({\varkappa }^{/},\mathcal{H}{\varkappa }^{*}\right)d\left({\varkappa }^{*},\mathfrak{F}{\varkappa }^{/}\right)}{1+d\left({\varkappa }^{*},{\varkappa }^{/}\right)}}\hfill \\ & =& {\varpi }_{1}d\left({\varkappa }^{*},{\varkappa }^{/}\right)+{\varpi }_3{\displaystyle \frac{d\left({\varkappa }^{/},\mathcal{H}{\varkappa }^{*}\right)d\left({\varkappa }^{*},\mathfrak{F}{\varkappa }^{/}\right)}{1+d\left({\varkappa }^{*},{\varkappa }^{/}\right)}},\hfill \end{array}$$

which implies that

$$\begin{array}{ccc}\hfill {∥d\left({\varkappa }^{*},{\varkappa }^{/}\right)∥}_{\mathbb{E}}& \le & {\varpi }_1{∥d\left({\varkappa }^{*},{\varkappa }^{/}\right)∥}_{\mathbb{E}}+{\varpi }_3{\displaystyle \frac{{∥d\left({\varkappa }^{/},{\varkappa }^{*}\right)∥}_{\mathbb{E}}{∥d\left({\varkappa }^{*},{\varkappa }^{/}\right)∥}_{\mathbb{E}}}{{∥1+d\left({\varkappa }^{*},{\varkappa }^{/}\right)∥}_{\mathbb{E}}}}\hfill \\ & \le & {\varpi }_1{∥d\left({\varkappa }^{*},{\varkappa }^{/}\right)∥}_{\mathbb{E}}+{\varpi }_3{∥d\left({\varkappa }^{/},{\varkappa }^{*}\right)∥}_{\mathbb{E}},\hfill \end{array}$$

that is,

$$(1-{\varpi }_1-{\varpi }_3){∥d\left({\varkappa }^{*},{\varkappa }^{/}\right)∥}_{\mathbb{E}}\le 0.$$

Since $(1-{\varpi }_1-{\varpi }_3)\ne 0$, so ${∥d\left({\varkappa }^{*},{\varkappa }^{/}\right)∥}_{\mathbb{E}}=0$, implies that ${\varkappa }^{*}={\varkappa }^{/}$. Hence, the common fixed point of $\mathfrak{F}$ and $\mathcal{H}$ is unique. □

Example 7.

Let $\Xi =[0,1]$ and

$${\mathbb{E}}_p=\left\{z=\nu +i\omega :\nu ,\omega \in \mathbb{R},i^2=p=-0.5<0\right\}.$$

Define $d:\Xi \times \Xi \to$${\mathbb{E}}_p$ by

$$d(\varkappa ,\sigma )=(1+i)\left|\varkappa -\sigma \right|,$$

for $\varkappa ,\sigma \in \Xi$. Then $\left(\Xi ,d\right)$ is a complete elliptic-valued suprametric space. Define $\mathcal{H},\mathfrak{F}:\Xi$$\to \Xi$ as follows

$$\mathcal{H}\varkappa ={\displaystyle \frac{1}{5}}\varkappa \mathrm{and}\mathfrak{F}\varkappa ={\displaystyle \frac{1}{10}}\varkappa ,$$

for all $\varkappa \in \Xi$. Then

$$d\left(\varkappa ,\sigma \right)=(1+i)\left|\varkappa -\sigma \right|,$$

$$d\left(\varkappa ,\mathcal{H}\varkappa \right)={\displaystyle \frac{4}{5}}(1+i)\left|\varkappa \right|,$$

$$d\left(\sigma ,\mathfrak{F}\sigma \right)={\displaystyle \frac{9}{10}}(1+i)\left|\sigma \right|,$$

$$d\left(\sigma ,\mathcal{H}\varkappa \right)=(1+i)\left|\sigma -{\displaystyle \frac{1}{5}}\varkappa \right|,$$

$$d\left(\varkappa ,\mathfrak{F}\sigma \right)=(1+i)\left|\varkappa -{\displaystyle \frac{1}{10}}\sigma \right|.$$

Then for all $\varkappa ,\sigma \in \Xi$ with $\varkappa \ne \sigma$, we have

$$\begin{array}{ccc}\hfill d\left(\mathcal{H}\varkappa ,\mathfrak{F}\sigma \right)& =& (1+i)\left|{\displaystyle \frac{1}{5}}\varkappa -{\displaystyle \frac{1}{10}}\sigma \right|⪯(1+i)max\left\{{\displaystyle \frac{1}{5}},{\displaystyle \frac{1}{10}}\right\}\left|\varkappa -\sigma \right|\hfill \\ & =& (1+i){\displaystyle \frac{1}{5}}\left|\varkappa -\sigma \right|\hfill \\ & \prec & (1+i){\displaystyle \frac{1}{5}}\left|\varkappa -\sigma \right|+{\displaystyle \frac{3}{10}}{\displaystyle \frac{\frac{18}{25}{(1+i)}^2\left|\varkappa \right|\left|\sigma \right|}{1+(1+i)\left|\varkappa -\sigma \right|}}+{\displaystyle \frac{2}{5}}{\displaystyle \frac{{(1+i)}^2\left|\sigma -\frac{1}{5}\varkappa \right|\left|\varkappa -\frac{1}{10}\sigma \right|}{1+(1+i)\left|\varkappa -\sigma \right|}}\hfill \\ & =& {\displaystyle \frac{1}{5}}d\left(\varkappa ,\sigma \right)+{\displaystyle \frac{3}{10}}{\displaystyle \frac{d\left(\varkappa ,\mathcal{H}\varkappa \right)d\left(\sigma ,\mathfrak{F}\sigma \right)}{1+d\left(\varkappa ,\sigma \right)}}+{\displaystyle \frac{2}{5}}{\displaystyle \frac{d\left(\sigma ,\mathcal{H}\varkappa \right)d\left(\varkappa ,\mathfrak{F}\sigma \right)}{1+d\left(\varkappa ,\sigma \right)}}\hfill \\ & =& {\varpi }_{1}d\left(\varkappa ,\sigma \right)+{\varpi }_2{\displaystyle \frac{d\left(\varkappa ,\mathcal{H}\varkappa \right)d\left(\sigma ,\mathfrak{F}\sigma \right)}{1+d\left(\varkappa ,\sigma \right)}}+{\varpi }_3{\displaystyle \frac{d\left(\sigma ,\mathcal{H}\varkappa \right)d\left(\varkappa ,\mathfrak{F}\sigma \right)}{1+d\left(\varkappa ,\sigma \right)}}.\hfill \end{array}$$

Thus the contractive condition (1) of Theorem 2 is fulfilled with ${\varpi }_1={\displaystyle \frac{1}{5}}$ and ${\varpi }_2={\displaystyle \frac{3}{10}},{\varpi }_3={\displaystyle \frac{2}{5}}$. Consequently, ${\varpi }_1+{\varpi }_2+{\varpi }_3=0.9<1$. Hence, 0 is the unique fp of mappings $\mathcal{H}$ and $\mathfrak{F}.$

Example 8.

Let $\Xi =\mathbb{R}$ and

$${\mathbb{E}}_p=\left\{z=\nu +i\omega :\nu ,\omega \in \mathbb{R},i^2=p<0\right\}.$$

Define $d:\Xi \times \Xi \to$${\mathbb{E}}_p$ by

$$d(\varkappa ,\sigma )=(1+i)\left|\varkappa -\sigma \right|,$$

for $\varkappa ,\sigma \in \Xi$. Then $\left(\Xi ,d\right)$ is a complete elliptic-valued suprametric space. Fix $c\in [0,1)$ and $b\in \mathbb{R}$. Define

$$\mathcal{T}\left(\varkappa \right)=c\varkappa +b,\mathcal{H}=\mathcal{T},\mathfrak{F}=\mathcal{T}.$$

Take ${\varpi }_1=c$, ${\varpi }_2={\varpi }_3=0$. Then

$$d\left(\mathcal{H}\varkappa ,\mathfrak{F}\sigma \right)=d\left(\mathcal{T}\varkappa ,\mathcal{T}\sigma \right)=(1+i)\left|c\varkappa -c\sigma \right|=cd\left(\varkappa ,\sigma \right).$$

So the inequality (1) of Theorem 2 is satisfied. The common fixed point ${\varkappa }^{*}$ satisfies ${\varkappa }^{*}=c{\varkappa }^{*}+b⟹{\varkappa }^{*}={\displaystyle \frac{b}{1-c}}$ (unique because $c<1$). This gives a nonzero fixed point whenever $b\ne 0$.

Corollary 1.

Let $\mathfrak{F}:\Xi$$\to \Xi$. Suppose that there exist the constants ${\varpi }_1,{\varpi }_2,{\varpi }_3\in [0,1)$ with ${\varpi }_1+{\varpi }_2+{\varpi }_3<1$ such that

$$d\left(\mathfrak{F}\varkappa ,\mathfrak{F}\sigma \right)⪯{\varpi }_{1}d\left(\varkappa ,\sigma \right)+{\varpi }_2{\displaystyle \frac{d\left(\varkappa ,\mathfrak{F}\varkappa \right)d\left(\sigma ,\mathfrak{F}\sigma \right)}{1+d\left(\varkappa ,\sigma \right)}}+{\varpi }_3{\displaystyle \frac{d\left(\sigma ,\mathfrak{F}\varkappa \right)d\left(\varkappa ,\mathfrak{F}\sigma \right)}{1+d\left(\varkappa ,\sigma \right)}},$$

for all $\varkappa ,\sigma \in \Xi$ with $\varkappa \ne \sigma$, then there exists a point ${\varkappa }^{*}\in \Xi$ such that ${\varkappa }^{*}=\mathfrak{F}{\varkappa }^{*}$.

Proof. 

Take $\mathcal{H}=\mathfrak{F}$ in Theorem 2. □

Example 9.

Let $\Xi =[0,1]$ and

$${\mathbb{E}}_p=\left\{z=\nu +i\omega :\nu ,\omega \in \mathbb{R},i^2=p<0\right\}.$$

Define $d:\Xi \times \Xi \to$${\mathbb{E}}_p$ by

$$d(\varkappa ,\sigma )=(1+i)\left|\varkappa -\sigma \right|,$$

for $\varkappa ,\sigma \in \Xi$. Then $\left(\Xi ,d\right)$ is a complete elliptic-valued suprametric space. Define the mappings $\mathfrak{F}:\Xi$$\to \Xi$ as follows

$$\mathfrak{F}\left(\varkappa \right)={\displaystyle \frac{1}{2}}\varkappa$$

for $\varkappa \in \Xi$. Let ${\varpi }_1={\displaystyle \frac{1}{2}},{\varpi }_2={\displaystyle \frac{1}{4}},{\varpi }_3={\displaystyle \frac{1}{6}}$. Then ${\varpi }_1+{\varpi }_2+{\varpi }_3=0.9166<1$. For any $\varkappa ,\sigma \in [0,1]$, we have

$$\begin{array}{ccc}\hfill \left|\mathfrak{F}\left(\varkappa \right)-\mathfrak{F}\left(\sigma \right)\right|& =& \left|{\displaystyle \frac{1}{2}}\varkappa -{\displaystyle \frac{1}{2}}\sigma \right|\le {\displaystyle \frac{1}{2}}\left|\varkappa -\sigma \right|\hfill \\ & & +{\displaystyle \frac{1}{4}}{\displaystyle \frac{\left|\varkappa -\frac{1}{2}\varkappa \right|\left|\sigma -\frac{1}{2}\sigma \right|}{1+\left|\varkappa -\sigma \right|}}\hfill \\ & & +{\displaystyle \frac{1}{6}}{\displaystyle \frac{\left|\sigma -\frac{1}{2}\varkappa \right|\left|\varkappa -\frac{1}{2}\sigma \right|}{1+\left|\varkappa -\sigma \right|}}.\hfill \end{array}$$

Then

$$\begin{array}{ccc}\hfill d\left(\mathfrak{F}\right(\varkappa ),\mathfrak{F}(\sigma \left)\right)& =& \left|\mathfrak{F}\left(\varkappa \right)-\mathfrak{F}\left(\sigma \right)\right|(1+i)=\left|{\displaystyle \frac{1}{2}}\varkappa -{\displaystyle \frac{1}{2}}\sigma \right|(1+i)\le {\displaystyle \frac{1}{2}}\left|\varkappa -\sigma \right|(1+i)\hfill \\ & & +{\displaystyle \frac{1}{4}}{\displaystyle \frac{\left|\varkappa -\frac{1}{2}\varkappa \right|(1+i)\left|\sigma -\frac{1}{2}\sigma \right|(1+i)}{1+\left|\varkappa -\sigma \right|(1+i)}}\hfill \\ & & +{\displaystyle \frac{1}{6}}{\displaystyle \frac{\left|\sigma -\frac{1}{2}\varkappa \right|(1+i)\left|\varkappa -\frac{1}{2}\sigma \right|(1+i)}{1+\left|\varkappa -\sigma \right|(1+i)}}\hfill \\ & =& {\displaystyle \frac{1}{2}}d(\varkappa ,\sigma )+{\displaystyle \frac{1}{4}}{\displaystyle \frac{d\left(\varkappa ,\mathfrak{F}\varkappa \right)d\left(\sigma ,\mathfrak{F}\sigma \right)}{1+d\left(\varkappa ,\sigma \right)}}+{\displaystyle \frac{1}{6}}{\displaystyle \frac{d\left(\sigma ,\mathfrak{F}\varkappa \right)d\left(\varkappa ,\mathfrak{F}\sigma \right)}{1+d\left(\varkappa ,\sigma \right)}}.\hfill \end{array}$$

Then all the conditions of Corollary 1 are satisfied and 0 is fixed point of $\mathfrak{F}.$

Corollary 2.

Let $\mathcal{H},\mathfrak{F}:\Xi$$\to \Xi$. Suppose that there exist the constants ${\varpi }_1,{\varpi }_2\in [0,1)$ with ${\varpi }_1+{\varpi }_2<1$ such that

$$d\left(\mathcal{H}\varkappa ,\mathfrak{F}\sigma \right)⪯{\varpi }_{1}d\left(\varkappa ,\sigma \right)+{\varpi }_2{\displaystyle \frac{d\left(\varkappa ,\mathcal{H}\varkappa \right)d\left(\sigma ,\mathfrak{F}\sigma \right)}{1+d\left(\varkappa ,\sigma \right)}},$$

for all $\varkappa ,\sigma \in \Xi$ with $\varkappa \ne \sigma$, then there exists a point ${\varkappa }^{*}\in \Xi$ such that ${\varkappa }^{*}=\mathcal{H}{\varkappa }^{*}=\mathfrak{F}{\varkappa }^{*}$.

Proof. 

Take ${\varpi }_3=0$ in Theorem 2. □

Corollary 3.

Let $\mathcal{H},\mathfrak{F}:\Xi$$\to \Xi$. Suppose that there exist the constants ${\varpi }_1,{\varpi }_3\in [0,1)$ with ${\varpi }_1+{\varpi }_3<1$ such that

$$d\left(\mathcal{H}\varkappa ,\mathfrak{F}\sigma \right)⪯{\varpi }_{1}d\left(\varkappa ,\sigma \right)+{\varpi }_3{\displaystyle \frac{d\left(\sigma ,\mathcal{H}\varkappa \right)d\left(\varkappa ,\mathfrak{F}\sigma \right)}{1+d\left(\varkappa ,\sigma \right)}},$$

for all $\varkappa ,\sigma \in \Xi$ with $\varkappa \ne \sigma$, then there exists a point ${\varkappa }^{*}\in \Xi$ such that ${\varkappa }^{*}=\mathcal{H}{\varkappa }^{*}=\mathfrak{F}{\varkappa }^{*}$.

Proof. 

Take ${\varpi }_2=0$ in Theorem 2. □

Corollary 4.

Let $\mathfrak{F}:\Xi$$\to \Xi$. Suppose that there exist the constants ${\varpi }_1\in [0,1)$ such that

$$d\left(\mathfrak{F}\varkappa ,\mathfrak{F}\sigma \right)⪯{\varpi }_{1}d\left(\varkappa ,\sigma \right),$$

for all $\varkappa ,\sigma \in \Xi$ with $\varkappa \ne \sigma$, then there exists a point ${\varkappa }^{*}\in \Xi$ such that ${\varkappa }^{*}=\mathfrak{F}{\varkappa }^{*}$.

Proof. 

Take ${\varpi }_2={\varpi }_3=0$ and $\mathcal{H}=\mathfrak{F}$ in Theorem 2. □

Remark 3.

(i) By setting $p=-1$ in Definition 3, the concept of an elliptic-valued suprametric space reduces to a complex-valued suprametric space, and Theorem 2 consequently recovers the main result of Panda et al. [13].

(ii) If we set $p=-1$ in Definition 3 and take ${\varpi }_3=0$, then Theorem 2 corresponds to the principal result of Abdou [14].

(iii) When $\vartheta =\theta$ in Definition 3, the structure of an elliptic-valued suprametric space simplifies to that of an elliptic-valued metric space, and Theorem 2 directly recovers the main theorem of Alamri [11].

(iv) If we take $p=-1$ and $\vartheta =\theta$ in Definition 3, then the elliptic-valued suprametric space reduces to a complex-valued metric space, and Theorem 2 yields the main result of Rouzkard et al. [8].

(v) Furthermore, if ${\varpi }_2=0$ under the assumptions of (iv), then Theorem 2 reduces to the fundamental result of Azam et al. [7].

4. Applications

4.1. Fixed Point Approach for Solving the Nonlinear Volterra Integral Equations

The nonlinear Volterra integral equation of the second kind is an essential mathematical model that arises in a wide range of scientific and engineering problems, including physics, biology, control theory, and climate modeling. Such equations typically take the form:

$$\varkappa \left(t\right)=f\left(t\right)+{\int }_a^{t}K(t,s,\varkappa \left(s\right))ds,t\in [a,b],$$

where $f\left(t\right)$ is a given function, $K(t,s,\varkappa (s\left)\right)$ is a possibly nonlinear kernel, and the unknown function $\varkappa \left(t\right)$ appears both inside the integral and outside. The nonlinearity of the kernel K often reflects complex interactions, memory effects, or feedback mechanisms present in the system being modeled.

Solving these equations is a fundamental task because they describe processes with history-dependent dynamics, such as population growth in biology, viscoelastic materials in physics, or non-local heat transport in climate studies. The second kind, in particular, is distinguished by the presence of the unknown function both inside and outside the integral, which introduces additional analytical and computational challenges compared to linear or first-kind equations.

One of the most effective methods for addressing nonlinear Volterra integral equations of the second kind is the fixed point approach. This method transforms the integral equation into an equivalent operator equation of the form:

$$\left(\mathfrak{F}\varkappa \right)\left(t\right)=f\left(t\right)+{\int }_a^{t}K(t,s,\varkappa \left(s\right))ds,$$

and seeks a solution $\varkappa$ such that $\mathfrak{F}\varkappa =\varkappa$. The approach leverages contraction mappings, ensuring that iterative approximations converge to a unique solution under appropriate conditions. The Banach contraction principle and its generalizations provide a rigorous theoretical framework for establishing the existence and uniqueness of solutions. Specifically, if the operator $\mathfrak{F}$ is a contraction in a suitable function space, the Banach theorem guarantees that repeated application of $\mathfrak{F}$ starting from any initial guess produces a sequence converging to the exact solution.

This methodology is particularly advantageous when dealing with complex kernels and nonlinear dependencies, which are common in real-world applications. In addition to theoretical guarantees, the fixed point approach naturally leads to iterative numerical schemes that can be implemented efficiently on computers, making it a practical tool for simulation and modeling purposes.

In this section, we employ a fixed point result in elliptic-valued suprametric spaces to solve nonlinear Volterra integral equations of the second kind. This approach effectively handles the complex and nonlinear nature of the integral kernel and provides a rigorous foundation for existence and uniqueness of solutions, offering a powerful tool for both theoretical analysis and iterative computation.

Let $\Xi =C\left(\right[0,\Theta ],\mathbb{R})$ with $\Theta >0$, be the space of continuous real-valued functions on the interval $[0,\Theta ].$ Define a function $d:\Xi \times \Xi \to {\mathbb{E}}_p$ by

$$d(\varkappa ,\sigma )=\left({∥\varkappa -\sigma ∥}_{\infty }\right)e^{i{\displaystyle \frac{\pi }{4}}}=\left(\underset{t\in [0,\Theta ]}{max}∥\varkappa \left(t\right)-\sigma \left(t\right)∥\right)e^{i{\displaystyle \frac{\pi }{4}}},$$

where $i^2=p<0$. Then $\left(C([0,\Theta ],\mathbb{R}),∥·∥\right)$ is complete elliptic-valued suprametric space.

Let us consider the nonlinear Volterra integral equation of the second kind

$$\varkappa \left(t\right)=f\left(t\right)+{\int }_0^{t}K(t,s,\varkappa \left(s\right))ds,t\in [0,\Theta ]$$

(9)

where

• $\varkappa \left(t\right)$ is the unknown function to be solved.
• $f\left(t\right)$ is a given function, often called the free term.
• $K(t,s,\varkappa (s\left)\right)$ is the kernel of the integral equation, which may depend nonlinearly on $\varkappa \left(s\right)$.

Theorem 3.

Let $\left(C\left(\right[0,\Theta ],\mathbb{R}),d\right)$ be a complete elliptic-valued suprametric space and $\Theta >0.$ Assume

(a) 

$K:[0,\Theta ]\times [0,\Theta ]\times \mathbb{R}\to \mathbb{R}$ and $f:[0,\Theta ]\to \mathbb{R}$ are continuous;

(b) 

there exists a constant $L\ge 0$ such that

$$∥K(t,s,\varkappa )-K(t,s,\sigma )∥\le L∥\varkappa -\sigma ∥,$$

for all $t,s\in [0,\Theta ]$, $\varkappa ,\sigma \in \mathbb{R}$;

(c) 

the constant $L\ge 0$ satisfies the smallness condition

$$L\Theta <1.$$

Then, the nonlinear Volterra integral equation of the second kind (9) has a unique solution in $C\left(\right[0,\Theta ],\mathbb{R})$.

Proof. 

Define the operator $\mathfrak{F}:\Xi \to \Xi$ by

$$\left(\mathfrak{F}\varkappa \right)\left(t\right)=f\left(t\right)+{\int }_0^{t}K(t,s,\varkappa \left(s\right))ds,t\in [0,\Theta ].$$

Continuity of f and K implies that for each fixed $\varkappa \in \Xi$, the integrand $K(t,s,\varkappa (s\left)\right)$ is continuous in s for each fixed t and then ${\int }_0^{t}K(t,s,\varkappa \left(s\right))ds$ is continuous (standard parameter dependent integral continuity). Hence, $\mathfrak{F}\varkappa \in \Xi$, so $\mathfrak{F}$ maps $\Xi$ into itself. For $\varkappa ,\sigma \in C\left(\right[0,\Theta ],\mathbb{R})$ and each $t\in [0,\Theta ]$, we compute

$$\begin{array}{ccc}\hfill \left|\left(\mathfrak{F}\varkappa \right)\left(t\right)-\left(\mathfrak{F}\sigma \right)\left(t\right)\right|& =& \left|{\int }_0^t\left(K(t,s,\varkappa (s\left)\right)-K(t,s,\sigma (s\left)\right)\right)ds\right|\hfill \\ & \le & {\int }_0^t\left|\left(K(t,s,\varkappa (s\left)\right)-K(t,s,\sigma (s\left)\right)\right)\right|ds\hfill \\ & \le & {\int }_0^{t}L\left|\varkappa \left(s\right)-\sigma \left(s\right)\right|\hfill \\ & \le & L{\int }_0^t{∥\varkappa -\sigma ∥}_{\infty }ds=Lt{∥\varkappa -\sigma ∥}_{\infty }.\hfill \end{array}$$

Taking the supremum over $t\in [0,\Theta ]$, gives the supremum norm estimate

$${∥\mathfrak{F}\varkappa -\mathfrak{F}\sigma ∥}_{\infty }\le L\Theta {∥\varkappa -\sigma ∥}_{\infty }.$$

By hypothesis $L\Theta <1$. Set ${\varpi }_1=L\Theta \in [0,1)$. Then

$$d(\mathfrak{F}\varkappa ,\mathfrak{F}\sigma )=\left({∥\mathfrak{F}\varkappa -\mathfrak{F}\sigma ∥}_{\infty }\right)e^{i{\displaystyle \frac{\pi }{4}}}⪯L\Theta \left({∥\varkappa -\sigma ∥}_{\infty }\right)e^{i{\displaystyle \frac{\pi }{4}}}={\varpi }_{1}d(\varkappa ,\sigma ),$$

for every $\varkappa ,\sigma \in \Xi$. Consequently, all the hypotheses of Corollary 4 are fulfilled, and $\mathfrak{F}$ possesses a unique fixed point, i.e., $\exists \varkappa \in \Xi$ such that

$$\varkappa =\mathfrak{F}\varkappa .$$

Hence, the nonlinear Volterra integral equation of the second kind (9) has a unique solution. □

Example 10.

Consider the nonlinear Volterra integral equation of the second kind

$$\varkappa \left(t\right)=f\left(t\right)+{\int }_0^{t}a(t,s)\varkappa \left(s\right)ds,t\in [0,\Theta ],$$

(10)

take $f\in C\left(\right[0,\Theta ],\mathbb{R})$ and $K(t,s,\varkappa (s\left)\right)=a(t,s)\varkappa \left(s\right)$, where $a\in C\left(\right[0,\Theta ]\times [0,\Theta ],\mathbb{R})$.

• Lipschitz in x:

  $$\left|K(t,s,\varkappa )-K(t,s,\sigma )\right|=\left|a(t,s)\right|\left|\varkappa -\sigma \right|.$$
• Set $L={sup}_{(t,s)\in {[0,\Theta ]}^2}\left|a(t,s)\right|$ (finite by continuity).
• Condition (c): choose Θ with $L\Theta <1$ (e.g., if $L=2$, take any $0<\Theta <{\displaystyle \frac{1}{2}}$).
• Condition (a) holds, since the functions a and f are continuous.

Therefore, all the conditions of aforementioned Theorem are satisfied. Thus the nonlinear Volterra integral equation of the second kind has a unique solution.

4.2. Applications of Nonlinear Volterra Integral Equations of the Second, Kind in Age-Structured Population Models

Nonlinear Volterra integral equations of the second kind play a fundamental role in age-structured population dynamics, where demographic processes such as reproduction and survival depend explicitly on age. In this context, the population density $\varkappa \left(t\right)$ represents the density of individuals of age t and is influenced by the cumulative contribution of individuals at earlier ages. Consider the nonlinear Volterra integral equation

$$\varkappa \left(t\right)=f\left(t\right)+{\int }_0^{t}a(t,s)\varkappa \left(s\right)ds,t\in [0,\Theta ]$$

where $f\left(t\right)$ denotes the initial population input or baseline distribution, and the kernel, and the kernel

$$a(t,s)=\beta \left(s\right)e^{-\mu (t-s)}$$

models age-dependent fertility and survival mechanisms. Here, $\beta \left(s\right)$ is the fertility rate at age s, while the exponential term represents the probability of survival from age s to age t, with $\mu >0$ denoting the mortality coefficient.

Such kernels naturally arise in biological systems where reproduction at earlier ages contributes to later population levels, weighted by survival probabilities. The continuity of $a(t,s)$ guarantees that

$$L=\underset{(t,s)\in {[0,\Theta ]}^2}{sup}\left|a(t,s)\right|<\infty ,$$

and hence the Lipschitz condition with respect to $\varkappa$,

$$\left|K(t,s,\varkappa )-K(t,s,\sigma )\right|=L\left|\varkappa -\sigma \right|,$$

is satisfied. By choosing a sufficiently small time horizon $\Theta >0$ such that $L\Theta <1$, all the assumptions of the fixed point theorem in the elliptic-valued suprametric space are fulfilled. Therefore, the considered age-structured population model admits a unique solution on $[0,\Theta ]$. This existence result ensures the mathematical well-posedness of the model, while uniqueness guarantees the predictability of the population evolution over the prescribed time interval. Biologically, the restriction $L\Theta <1$ corresponds to analyzing short-term population dynamics, which is reasonable for populations with rapidly varying birth or death rates.

Hence, nonlinear Volterra integral equations of the second kind provide a rigorous and flexible framework for age-structured population models, effectively bridging biological realism with fixed point theory in generalized metric settings.

5. Conclusions

In this work, the notion of elliptic-valued suprametric spaces was introduced, and several new common fixed point theorems were established under generalized contractive mappings. The results achieved broaden and unify significant contributions in the existing literature, including the fundamental theorems of Panda et al. [13] and Abdou [14] in complex-valued suprametric spaces, the principal theorems of Alamri [11] and Ozturk et al. [9] in the setting of elliptic-valued metric space, as well as the results of Azam et al. [7] and Rouzkard et al. [8] in the context of complex-valued metric space. We applied the developed fixed point results to nonlinear Volterra integral equations of the second kind, demonstrating the existence and uniqueness of solutions. They specifically illustrated the use of these equations in age-structured population models, capturing hereditary effects, density-dependent interactions, and delayed responses, thereby linking the theoretical framework to practical applications in mathematical biology and ecology. Furthermore, the applicability and effectiveness of the proposed framework were demonstrated through a comprehensive example.