Fixed-Point Results for Krasnoselskii, Meir–Keeler, and Boyd–Wong-Type Mappings with Applications to Dynamic Market Equilibrium

Abstract

This paper introduces the idea of a cone m-hemi metric space, which extends the idea of an m-hemi metric space. By presenting non-trivial examples, we demonstrate the superiority of cone m-hemi metric spaces over m-hemi metric spaces. Further, we extend the Banach contraction principle and Krasnoselskii, Meir–Keeler, Boyd–Wong, and some other fixed-point results in the setting of complete and compact cone m-hemi metric spaces. Furthermore, we provide several non-trivial examples and applications to the Fredholm integral equation and dynamic market equilibrium to demonstrate the validity of the main results.

1. Introduction

Mathematics is a fundamental field of scientific inquiry that influences all areas of life. It combines various sub-fields, with fixed-point theory as a key area in pure and applied mathematics. This theory is broad and adaptable, with applications in numerous disciplines, including mathematical economics, approximation theory, variational inequalities, management, game theory, social sciences, and optimization theory. Over the last 50 to 60 years, fixed-point theory has rapidly become a prominent dynamic and intriguing field of mathematical research. The foundational work in this area began with Poincare [1], who first explored fixed-point theory in 1886. Subsequently, Banach [2] proved in 1922 that contraction mappings in a complete metric space have a unique fixed point. The following is the formulation of the Banach fixed-point theorem.

Consider a metric space $(\mathfrak{D},\Delta )$ and a contraction mapping $T:\mathfrak{D}\to \mathfrak{D}$. For any $\varkappa ,y\in \mathfrak{D},$ there exist $\beta \in [0,1)$ such that

$$\Delta (T\varkappa ,Ty)\le \beta \Delta (\varkappa ,y)\mathrm{for}\mathrm{all}\varkappa ,y\in \mathfrak{D}.$$

Under these conditions, T possesses a unique fixed point. In the early phases of fixed-point theory, the Banach contraction principle is acknowledged as a fundamental concept, advancing many subsequent fixed-point theorems.

A significant branch of fixed-point theory is metric fixed-point theory, with applications extending beyond mathematics into other fields. The origins of metric fixed-point theory can be traced back to Banach’s contraction principle introduced in 1922. When this principle falls short, researchers have developed various generalized metric spaces and contraction principles, leading to the establishment of numerous fixed-point theorems. Examples include quasi-metric spaces, which omit the symmetry axioms [3], and semi-metric spaces, which do not require the triangle inequality [4]. Further, partial metric spaces allow for the possibility that the distance between the point and itself does not necessarily need to be zero [5]. In the academic literature, several generalizations of metric spaces have been introduced, including F-metric, b-metric, and modular metric spaces, where numerous theorems concerning fixed points and common fixed points have been formulated [6,7,8,9,10,11,12,13]. The idea of a 2-metric was presented by Gahler [14], derived from geometry involving more than two points, and, subsequently, the topological properties of 2-metric spaces were explored by Lahiri et al. [15]. The idea of G-metric, a generalized metric space based on the geometry of three points, was proposed by Mustafa and Sims [16]. Branciari [17] extended the traditional metric by replacing the standard triangle inequality with four four-point conditions, leading to the notion of a rectangular metric. Additionally, Choi et al. [18] introduced the g-metric with degree n, which characterizes a distance measure involving $n+1$ points. This advancement expands on the traditional distance metrics that involve only two points, and the g-metric incorporates three points.

The concept of an m-hemi metric defined on a set containing at least $m+2$ elements, where m is an integer, was proposed by Deza and Rosenberg [19]. This innovative concept has garnered significant attention from researchers. Studies have explored various topological properties and fixed-point theorems connected to Banach’s contraction principle and its generalizations for mappings defined on m-hemi metric spaces [20].

The idea of cone metric spaces has also garnered considerable interest from researchers. Huang and Zhang [21] were the pioneers in defining cone metric spaces. Rzepecki [22] offered a similar interpretation and, after a detailed exploration of convergence and completeness in these spaces, established multiple fixed-point outcomes for contractive mappings. Later, Liu and Xu [23] extended the idea of cone metric spaces to Banach algebras and applied the Banach contraction principle within this framework. See [24,25,26,27] for more related studies. Ali et al. [28] worked on the approximation of fixed points and the solution of a nonlinear integral equation. Özger et al. [29] proved several fixed-point results and determined the existence and uniqueness of Fredholm-type integral equations by utilizing fixed-point theorems. Firozjah et al. [30] worked on the concept of cone b-metric spaces over Banach algebras and obtained several fixed-point results without the condition of normality of cones. Huang et al. [31] discussed some topological properties and fixed-point results in cone metric spaces over Banach algebras. Firozjah et al. [32] proved several fixed-point results under generalized c-distance in cone b-metric spaces over Banach algebras. Huang et al. [33] generalized a famous result for a Banach-type contractive mapping from $p\left(k\right)\in [0,{\displaystyle \frac{1}{s}})$ to $p\left(k\right)\in [0,1)$ in a cone b-metric space over Banach algebra with coefficient $s\ge 1$, where $p\left(k\right)$ is the spectral radius of the generalized Lipshitz constant k. Du and Karapınar [34] investigated the answer to the question regarding whether the results in cone b-metric spaces generalize the existing ones or are equivalent to them. Janković et al. [35] shortened the proofs of fixed-point results in cone metric spaces when the cone is normal and solid. Zabrejko [36] provided a brief overview of fixed-point theorems that extend the Banach–Caccioppoli principle for contractive mapping. Further, they established the presence and uniqueness of fixed points for operators in K-metric or K-normed linear spaces, including local convex spaces and Banach space scales.

In this paper, we present the idea of a cone m-hemi metric space as a generalization of the m-hemi metric space and explore its topological properties. Our main goal is to use complete and compact cone m-hemi metric spaces to prove fixed-point theorems.

The structure of this paper is organized as follows:

• Section 2 introduces the fundamental definitions and examples related to the Banach algebras, metric spaces, cones, cone metric spaces, and m-hemi metric spaces, along with relevant examples for the subsequent sections.
• Section 3 focuses on proving fixed points by utilizing the Banach contraction principle and Krasnoselskii, Meir–Keeler, and Boyd–Wong contraction mapping principles in a complete normal cone m-hemi metric space. This section also includes essential definitions, lemmas, and examples pertinent to our study.
• Section 4 applies the results to Fredholm integral equations and integral equations in dynamic market equilibrium economics to demonstrate the applicability of our main findings. In conclusion, we summarize the key points of the paper.

2. Preliminaries

First, we review some fundamental concepts related to Banach algebra and cone m-hemi metric spaces that are required for the subsequent parts. While researching function spaces, Frechet [24] was the pioneer to introduce the concept of metric space. A function that describes the idea of distance between any two nonempty sets is called a metric.

Definition 1 ([25]).

In metric spaces, an ordered pair $(\mathfrak{D},\Delta )$ represent a set $\mathfrak{D}$, and Δ is a metric on $\mathfrak{D}$, often known as a distance function on $\mathfrak{D}$. This function is defined regarding $\mathfrak{D}\times \mathfrak{D}$, and, for every $\varkappa ,y,z\in \mathfrak{D},$ we have

(S1) 

Δ is finite, non-negative, and real-valued;

(S2) 

$\Delta (\varkappa ,y)=0$ if and only if $\varkappa =y$;

(S3) 

$\Delta (\varkappa ,y)=\Delta (y,\varkappa )$ (symmetry);

(S4) 

$\Delta (\varkappa ,y)\le \Delta (\varkappa ,z)+\Delta (z,y)$ (triangular inequality).

The sign × stands for the Cartesian product of sets. For the sets $W\mathrm{and}Y$, the Cartesian product $W\times Y$ consists of all the possible ordered pairs $(\delta ,L)$, where $\delta \in W\mathrm{and}L\in Y.$ Therefore, $\mathfrak{D}\times \mathfrak{D}$ refer to the set of all the ordered pairs formed by elements from $\mathfrak{D}$.

Example 1 ([25]).

Define the function $\Delta :\mathbb{R}\times \mathbb{R}\to \mathbb{R}$ as follows:

$$\Delta (\varpi ,\phi )=|\varpi -\phi |\mathit{for}\mathit{all}\varpi ,\phi \in \mathbb{R}.$$

Therefore, Δ is a metric on the real numbers set $\mathbb{R}.$

To demonstrate that Δ is indeed a metric on $\mathbb{R},$ the conditions $\left(S1\right)$ to $\left(S3\right)$ are straightforward. We only need to verify the condition $\left(S4\right).$ Let

$$\begin{array}{ccc}\hfill \Delta (\varpi ,\nu )& =& |\varpi -\nu |,\hfill \\ & =& |\varpi -\phi +\phi -\nu |,\hfill \\ & \le & |\varpi ,\phi |+|\phi -\nu |,\hfill \\ \hfill \Delta (\varpi ,\nu )& \le & \Delta (\varpi ,\phi )+\Delta (\phi ,\nu ).\hfill \end{array}$$

This demonstrates that Δ is a metric on the set of $\mathbb{R}.$

Definition 2 ([26]).

If a Banach space ϝ has a multiplication $\mathsf{ϝ}\times \mathsf{ϝ}\to \mathsf{ϝ}$ that meets certain properties, it is called a Banach algebra. These properties are as follows: for all $\varpi ,\phi ,\nu \in \mathsf{ϝ}$, $\delta \in \mathbb{R},$

(1) 

$\left(\varpi \phi \right)\nu =\varpi \left(\phi \nu \right);$

(2) 

$\varpi (\phi +\nu )=\varpi \phi +\phi \nu \mathit{and}(\varpi +\phi )\nu =\varpi \nu +\phi \nu ;$

(3) 

$\delta \left(\varpi \phi \right)=\left(\delta \varpi \right)\phi =\varpi \left(\delta \phi \right);$

(4) 

Existence of $\mathsf{ϝ}\in \mathsf{ϝ}$ such that $\varpi \mathsf{ϝ}=\mathsf{ϝ}\varpi =\varpi ;$

(5) 

$∥\mathsf{ϝ}∥=1;$

(6) 

$∥\varpi \phi ∥\le ∥\varpi ∥.∥\phi ∥.$

When there is ${\varpi }^{-1}\in \mathsf{ϝ}$ in such a manner that $\varpi {\varpi }^{-1}={\varpi }^{-1}\varpi =\mathsf{ϝ},$ then an element $\varpi \in \mathsf{ϝ}$ is considered to be invertible.

Example 2. 

The set of complex numbers $\mathbb{C}$ and the set of real numbers $\mathbb{R}$, each equipped with their respective norms defined by absolute values, are both Banach algebras.

Definition 3 ([27]).

A subset ρ of real Banach algebra ϝ is called a cone if it satisfies the following conditions:

(i) 

$\{\theta ,e\}\subset \rho$;

(ii) 

${\rho }^2=\rho \rho \subset \rho ,\rho \cap (-\rho )=\left\{\theta \right\};$

(iii) 

$p,q\in \mathbb{R},p,q\ge 0,c,d\in \rho \Rightarrow pc+qd\subset \rho .$

For a cone $\rho \subset \mathsf{ϝ},$ we establish a partial order ≤ on $\mathsf{ϝ}$ relative to $\rho$ as follows: $\varpi \le \phi$ if and only if $\phi -\varpi \in \rho .$ We use $\varpi <\phi$ to denote $\varpi \le \phi$ with $\varpi \ne \phi$ and $\varpi \ll \phi$ to indicate that $\phi -\varpi \in int\rho$; $int\rho$ indicates the interior of $\rho .$ The cone $\rho$ is referred to as normal if there exists a number $k\ge 1$ such that, for all $\varpi ,\phi \in \mathsf{ϝ},$

$$0\le \varpi \le \phi \Rightarrow ∥\varpi ∥\le k∥\phi ∥.$$

The least positive number that meets the condition above is referred to as the normal constant of $\rho$. In the subsequent discussion, we assume that $\mathsf{ϝ}$ is a real Banach algebra, $\rho$ is a cone in $\mathsf{ϝ}$ with $int\rho \ne 0$, and ≤ denotes the partial ordering associated with $\rho$.

If every increasing sequence with upper bounds converges, the cone $\rho$ is referred to as regular. Specifically, if $\left\{{\varkappa }_u\right\}$ is a sequence such that

$${\varkappa }_1\le {\varkappa }_2\le \cdots \le {\varkappa }_u\le \cdots \le y,$$

there are $\varkappa \in \mathsf{ϝ}$ such that $∥{\varkappa }_u-\varkappa ∥\to 0(u\to \infty )$ for some $y\in \mathsf{ϝ}.$ Equivalently, if every decreasing sequence with lower bounds converges, the cone $\rho$ is referred to as regular. It is widely recognized that a regular cone is also a normal cone.

Definition 4 ([23]).

Assume that $\mathfrak{D}$ is a nonempty set. The mapping $\Delta :\mathfrak{D}\times \mathfrak{D}\to \mathsf{ϝ}$ must meet the following conditions:

(1) 

$\theta <\Delta (\nu ,\phi ),$ for all $\nu ,\phi \in \mathfrak{D}$ with $\nu \ne \phi$ and $\Delta (\nu ,\phi )=\theta$ if and only if $\nu =\phi ;$

(2) 

$\Delta (\nu ,\phi )=\Delta (\phi ,\nu )$$\mathit{for}\mathit{all}$$\nu ,\phi \in \mathfrak{D};$

(3) 

$\Delta (\nu ,\phi )\le \Delta (\nu ,\delta )+\Delta (\delta ,\phi )$$\mathit{for}\mathit{all}$$\nu ,\phi ,\delta \in \mathfrak{D}.$

Then, Δ is considered to be a cone metric on $\mathfrak{D},$ along with the pair $(\mathfrak{D},\Delta )$, known as a cone metric space over Banach algebra $\mathsf{ϝ}.$

Example 3 ([21]).

Suppose $\mathsf{ϝ}={\mathbb{R}}^2,$ $\Omega =\{(\varpi ,\phi )\in \mathsf{ϝ}:\varpi ,\phi \ge 0\}\subset {\mathbb{R}}^2,$ $\mathfrak{D}=\mathbb{R}\mathit{and}$ $\Delta :\mathfrak{D}\times \mathfrak{D}\to \mathsf{ϝ}\mathit{such}\mathit{that}\Delta (\varpi ,\phi )=\left(\right|\varpi -\phi |,\delta |\varpi -\phi \left|\right),\mathit{where}\delta \ge 0$ represents a constant. Therefore, $(\mathfrak{D},\Delta )$ is said to be a cone metric space.

Definition 5 ([19]).

Assume $m\in {\mathbb{Z}}^{+}$ and E is a set containing at least $m+2$ elements. Δ: $E^{m+1}\to \mathbb{R}$ is known m-hemi metric if at all possible $x_1,x_2,\cdots ,x_{m+2}\in E,$

1. 

$\Delta (x_1,x_2,\cdots ,x_{m+1})\ge 0$ (non-negativity);

2. 

Δ$(x_1,x_2,\cdots ,x_{m+1})=0\iff$ for any $x_i,x_k\in E,x_i=x_k$ (zero-conditioned);

3. 

$\Delta (x_1,x_2,\cdots ,x_{m+1})=\Delta (x_{\pi \left(1\right)},x_{\pi \left(2\right)},\cdots ,x_{\pi (m+1)})$ for any rearrangements π of the set $\{1,2,\cdots ,m+1\}$ (totally symmetric);

4. 

$\Delta (x_1,x_2,\cdots ,x_{m+1})\le {\sum }_{\iota =1}^{\iota =m+1}\Delta (x_1,\cdots ,x_{\iota -1},x_{\iota +1},\cdots ,x_{m+2})$ (m-simplex inequality).

Then, $(E,\Delta )$ called an m-hemi metric space.

The notion of the m-hemi metric serves as a generalization to extend the idea of the semi-metric for the m parameters. An important specific instance of the m-hemi metric is the scenario in which $m=$ 2.

A function $\Delta :E^3\to \mathbb{R}$ is referred to as a 2-metric if $\Delta$ meets the conditions $\left(1\right),\left(2\right),\left(3\right)$ of Definition 5 and the following tetrahedron inequality,

$$\Delta (x_1,x_2,x_3)\le \Delta (x_1,x_2,x_4)+\Delta (x_1,x_3,x_4)+\Delta (x_2,x_3,x_4).$$

3. Main Results

In cone m-hemi metric space, we formulate a unique fixed-point theorem for contraction mappings and a generalized contraction principle. In addition, we provide a few non-trivial examples to validate our primary findings. Further, we provide some definitions and prove the necessary lemmas.

Definition 6. 

Assume $m\in {\mathbb{Z}}^{+}$ and E is set to contain at least $m+2$ elements. Δ: $E^{m+1}\to \mathsf{ϝ}$ (Banach algebra) is called cone m-hemi metric if at all possible $x_1,x_2,\cdots ,x_{m+2}\in E,$

1. 

$\Delta (x_1,x_2,\cdots ,x_{m+1})\ge 0$ (non-negativity);

2. 

Δ$(x_1,x_2,\cdots ,x_{m+1})=0\iff$ for any $x_i,x_k\in E,x_i=x_k$ (zero-conditioned);

3. 

$\Delta (x_1,x_2,\cdots ,x_{m+1})=\Delta (x_{\pi \left(1\right)},x_{\pi \left(2\right)},\cdots ,x_{\pi (m+1)})$ for any rearrangements π of the set $\{1,2,\cdots ,m+1\}$ (totally symmetric);

4. 

$\Delta (x_1,x_2,\cdots ,x_{m+1})\le {\sum }_{\iota =1}^{\iota =m+1}\Delta (x_1,\cdots ,x_{\iota -1},x_{\iota +1},\cdots ,x_{m+2})$ (m-simplex inequality).

Then, $(E,\Delta )$ is called a cone m-hemi metric space.

Now, we discuss that, if $\Delta$ is a cone m-hemi metric, then ${\displaystyle \frac{\Delta }{1+\Delta }}$ is also a cone m-hemi metric on E.

Lemma 1. 

Let Δ be a cone m-hemi metric on E and then ${\displaystyle \frac{\Delta }{1+\Delta }}$ be a cone m-hemi metric on E.

Proof. 

The non-negativity of $\Delta$ ensures the non-negativity ${\displaystyle \frac{\Delta }{1+\Delta }}$. Furthermore, the identity implies axiom $\left(2\right)$ and total symmetry

$${\displaystyle \frac{\Delta (x_1,\cdots ,x_{m+1})}{1+\Delta (x_1,\cdots ,x_{m+1})}}=1-{\displaystyle \frac{1}{1+\Delta (x_1,\cdots ,x_{m+1})}}$$

(1)

and the fact that $\Delta$ exhibits zero-conditioned and totally symmetric aspects. Therefore, we need to demonstrate that ${\displaystyle \frac{\Delta }{1+\Delta }}$ satisfies axiom $\left(4\right)$.

Because ${\displaystyle \frac{\Delta }{1+\Delta }}$ is strictly increasing in $\Delta$, and since $\Delta$ axiom $\left(4\right)$, we have

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\Delta (x_1,\cdots ,x_{m+1})}{1+\Delta (x_1,\cdots ,x_{m+1})}}& \le & {\displaystyle \frac{{\sum }_{\iota =1}^{m+1}\Delta (x_1,\cdots ,x_{\iota -1},x_{\iota +1},\cdots ,x_{m+2})}{1+{\sum }_{\iota =1}^{m+1}\Delta (x_1,\cdots ,x_{\iota -1},x_{\iota +1},\cdots ,x_{m+2})}}\hfill \\ & =& \sum _{\iota =1}^m{\displaystyle \frac{\Delta (x_1,\cdots ,x_{\iota -1},x_{\iota +1},\cdots ,x_{m+2})}{1+{\sum }_{j=1}^{m+1}\Delta (x_1,\cdots ,x_{j-1},x_{j+1},\cdots ,x_{m+2})}}.\hfill \end{array}$$

(2)

Moreover, for each $\iota \in \{1,\cdots ,m+1\}$, the following inequality holds

$${\displaystyle \frac{\Delta (x_1,\cdots ,x_{\iota -1},x_{\iota +1},\cdots ,x_{m+2})}{1+{\sum }_{j=1}^{m+1}\Delta (x_1,\cdots ,x_{j-1},x_{j+1},\cdots ,x_{m+2})}}\le {\displaystyle \frac{\Delta (x_1,\cdots ,x_{\iota -1},x_{\iota +1},\cdots ,x_{m+2})}{1+\Delta (x_1,\cdots ,x_{\iota -1},x_{\iota +1},\cdots ,x_{m+2})}}.$$

(3)

By adding (3) to all $\iota \in \{1,\cdots ,m+1\}$ and then combining the result with inequality (2), the proof is concluded. □

Now, we examine if $\Delta$ is a cone m-hemi metric, and then $min\{1,\Delta \}$ is also a cone m-hemi metric on E.

Lemma 2. 

Let Δ be a cone m-hemi metric on E, and then $min\{1,\Delta \}$ is a cone m-hemi metric on E.

Proof. 

The properties of non-negativity, axiom $\left(2\right)$, and symmetry for $min\{1,\Delta \}$ can be derived from the corresponding properties $\Delta$. Therefore, we need to verify that $min\{1,\Delta \}$ satisfies axiom $\left(4\right)$.

We will examine the different scenarios.

Assume there exists $j\in \{1,\cdots ,m+1\}$ such that

$$\Delta (x_1,\cdots ,x_{m+1})\le \Delta (x_1,\cdots ,x_{j-1},x_{j+1},\cdots ,x_{m+2}).$$

In this case, we have

$$\begin{array}{ccc}\hfill min\{1,\Delta (x_1,\cdots ,x_{m+1})\}& \le & min\{1,\Delta (x_1,\cdots ,x_{\iota -1},x_{j+1},\cdots ,x_{m+2})\}\hfill \\ & \le & \sum _{\iota =1}^{m+1}min\{1,\Delta (x_1,\cdots ,x_{\iota -1},x_{\iota +1},\cdots ,x_{m+2})\}.\hfill \end{array}$$

(4)

Therefore, we can assume that, for every $\iota \in \{1,\cdots ,m+1\}$, the following hold:

$$\Delta (x_1,\cdots ,x_{m+1})\ge \Delta (x_1,\cdots ,x_{\iota -1},x_{\iota +1},\cdots ,x_{m+2}).$$

(5)

Suppose $\Delta (x_1,\cdots ,x_{m+1})\le 1$. In this case, we have for all $\iota \in \{1,\cdots ,m+2\},$

$$min\{1,\Delta (x_1,\cdots ,x_{\iota -1},x_{\iota +1},\cdots ,x_{m+2})\}=\Delta (x_1,\cdots ,x_{\iota -1},x_{j+1},\cdots ,x_{m+2}),$$

and this implies that $min\{1,\Delta \}$ satisfies axiom $\left(4\right)$.

Next, suppose $\Delta (x_1,\cdots ,x_{m+1})>1$. Moreover, suppose there is an $j\in \{1,\cdots ,m+1\}$ such that $\Delta (x_1,\cdots ,x_{j-1},x_{j+1},\cdots ,x_{m+2})\ge 1$. In this case, we have

$$\begin{array}{c}\hfill min\{1,\Delta (x_1,\cdots ,x_{m+1}\}=1=min\{1,\Delta (x_1,\cdots ,x_{j-1},x_{j+1},\cdots ,x_{m+2})\}\\ \hfill \le \sum _{\iota =1}^{m+1}min\{1,\Delta (x_1,\cdots ,x_{\iota -1},x_{\iota +1},\cdots ,x_{m+2})\}.\end{array}$$

Therefore, suppose that $\Delta (x_1,\cdots ,x_{\iota -1},x_{j+1},\cdots ,x_{m+2})\le 1$ for all $\iota \in \{1,\cdots ,m+1\}$. Finally, since $\Delta$ satisfies axiom $\left(4\right)$, we have

$$\begin{array}{ccc}\hfill min\{1,\Delta (x_1,\cdots ,x_{m+1})\}& =& 1<\Delta (x_1,\cdots ,x_{m+1})\hfill \\ & \le & \sum _{\iota =1}^{m+1}\Delta (x_1,\cdots ,x_{\iota -1},x_{\iota +1},\cdots ,x_{m+2})\hfill \\ & =& \sum _{\iota =1}^{m+1}min\{1,\Delta (x_1,\cdots ,x_{\iota -1},x_{\iota +1},\cdots ,x_{m+2})\}.\hfill \end{array}$$

This concludes the proof. □

Example 4. 

Let $\mathsf{ϝ}={\mathbb{R}}^2$, $E=\mathbb{N}$ and

$$\rho =\{(\varpi ,\phi ):\varpi ,\phi \ge 0\}\subseteq {\mathbb{R}}^2.$$

Define $\Delta :{\mathbb{N}}^{m+1}\to \mathsf{ϝ}$

$$\Delta (x_1,x_2,\cdots ,x_{m+1})=\left\{\begin{array}{cc}(0,0)\hfill & \mathit{if}{x}_i=x_j,\mathit{for}\mathit{any}i,j\hfill \\ (1,1)\hfill & \mathit{if}{x}_i\ne x_j,\mathit{for}\mathit{all}i,j.\hfill \end{array}\right.$$

Here, $(E,\Delta )$ is cone m-hemi metric space but not m-hemi metric space because in a standard cone m-hemi metric space distance function Δ required to map $E^{m+1}$ to $\mathbb{R}$.

Definition 7. 

Consider $(E,\Delta )$ to be a cone m-hemi metric space. For $x_0,x_1,\cdots ,x_m\in E$ and $\epsilon >0,$

$$Y(x_0,x_1,\cdots ,x_{m-1},\epsilon )=\{y\in E:\Delta (x_0,x_1,\cdots ,x_{m-1},y)<\epsilon \}$$

is known as x-open ball with radius ε and centers at $x_0,x_1,\cdots ,x_{m-1}$.

The topology on E defined by using all x-balls as a sub-basis is referred to as the m-hemi metric topology and is indicated by $\tau$. Elements of $\tau$ are termed x-open sets, while their complements are called x-closed sets.

Lemma 3. 

Let $W\subseteq E$ and let $(E,\Delta )$ be a cone m-hemi metric space. If and only if $\delta \in Y(\delta ,x_1,x_2,\cdots ,x_{m-1},{\epsilon }_1)\cap Y(\delta ,x_1^{\prime },x_2^{\prime },\cdots ,x_{m-1}^{\prime },{\epsilon }_2)\subset W$, then W is an x-open set. There are finite numbers of points $x_1,x_2,\cdots ,x_1^{\prime },x_2^{\prime },\cdots ,x_m^{\prime }$ and ${\epsilon }_1,{\epsilon }_2>0$ for every $\delta \in W.$

Proof. 

The claim’s appropriateness is apparent as the intersection of x-balls $(\delta ,x_1,x_2,\cdots ,x_{m-1},{\epsilon }_1)\cap Y(\delta ,x_1^{\prime },x_2^{\prime },\cdots ,x_{m-1}^{\prime },{\epsilon }_2)$ is x-open, indicating that the condition is instantly satisfied.

Conversely, let $\delta \in W$ and W be an x-open set. Subsequently, x-open balls $Y(x_1,x_2,\cdots ,x_m,{\epsilon }_1)\mathrm{and}Y(x_1^{\prime },x_2^{\prime },\cdots ,x_m^{\prime },{\epsilon }_2)$ exist such that $\delta \in Y(x_1,x_2,\cdots ,x_m,{\epsilon }_1)\cap Y(x_1^{\prime },$ $x_2^{\prime },\cdots ,x_m^{\prime },{\epsilon }_2)\subset W$. Given that $\delta \in Y(x_1,x_2,\cdots ,x_m,{\epsilon }_1)$ and $\delta \in Y(x_1^{\prime },x_2^{\prime },\cdots ,x_m^{\prime },{\epsilon }_2)$, then $\Delta (x_1,x_2,\cdots ,x_m,\delta )=p_1<{\epsilon }_1$ and $\Delta (x_1^{\prime },x_2^{\prime },\cdots ,x_m^{\prime },\delta )=p_2<{\epsilon }_2$. For $\iota =1,2$, choose $T_{\iota }<{\displaystyle \frac{{\epsilon }_{\iota }-p_{\iota }}{2}}.$ Afterward, we possess

$$\begin{array}{ccc}\hfill \delta & \in & Y(\delta ,x_1,x_2,\cdots ,x_{m-1},T_1)\cap Y(\delta ,x_1^{\prime },x_2^{\prime },\cdots ,x_{m-1}^{\prime },T_2)\hfill \\ & \subset & Y(x_1,x_2,\cdots ,x_m,{\epsilon }_1)\cap Y(x_1^{\prime },x_2^{\prime },\cdots ,x_m^{\prime },{\epsilon }_2)\subset W.\hfill \end{array}$$

This concludes the proof. □

Definition 8. 

Given a cone m-hemi metric space $(E,\Delta )$ and a subset W, its x-closure, represented by $W^{-}$, is defined as the x-closure of W with regard to the topology τ.

Definition 9. 

Consider a cone m-hemi metric space $(E,\Delta )$, and ρ denotes a normal cone, where the normal constant is k. In E, let $\left\{x_u\right\}$ be the sequence.

1. 

$\left\{x_u\right\}$ is said to be x-convergent to a $y\in E$ if and only if

$$\underset{u_1,u_2,\cdots ,u_m\to \infty }{lim}\Delta (x_{u_1},x_{u_2},\cdots ,x_{u_m},y)=0.$$

Therefore, if and only if $\left\{x_u\right\}$ converges to y with regard to topology τ, it is x-convergent to y.

2. 

A sequence $\left\{x_u\right\}$ is called x-Cauchy if and only if

$$\underset{u,m\to \infty }{lim}\Delta (x_{u_0},x_{u_1},\cdots ,x_{u_m})=0.$$

3. 

$(E,\Delta )$ is called x-complete if each x-Cauchy sequence in E is x-convergent.

4. 

A mapping T is considered to be x-continuous on E if $T{x}_u\to Ty$ when $x_u\to y$.

Proposition 1. 

Let $\left\{x_u\right\}$ be sequence in E and $(E,\Delta )$ be a cone m-hemi metric space.

(i). 

$\left\{x_u\right\}$ converges to $y\in E\mathit{if}\mathit{for}\mathit{all}\epsilon >0\mathit{there}\mathit{exists}{u}_0\in \mathbb{N}$ such that

$$u_1,u_2,\cdots ,u_m\ge u_0\Rightarrow \Delta (y,x_{u_1},\cdots ,x_{u_m})<\epsilon .$$

(ii). 

For each $\epsilon >0$, $\mathit{there}\mathit{exists}{u}_0\in \mathbb{N}$ so that $u_0,u_1,\cdots ,u_m\ge u_0\Rightarrow \Delta (x_{u_0},x_{u_1},\cdots ,x_{u_m})<\epsilon$ if $\left\{x_u\right\}$ is x-Cauchy sequence.

Now, we provide the lemma on the uniqueness of the limit in cone m-hemi metric space.

Lemma 4. 

The limit in cone m-hemi metric space is unique.

Proof. 

Consider $\rho$ denotes a normal cone, where the normal constant is k, and let $(E,\Delta )$ be a cone m-hemi metric space. In E, let $\left\{x_u\right\}$ be an x-convergent sequence. Assume that $\varkappa ,y\in E$ are both the limits of the sequence $\left\{x_u\right\}$. Thus, we obtain for all $\epsilon >0$ that there exist $u_0\in \mathbb{N}$ such that $u_1,\cdots ,u_m\ge u_0\Rightarrow {\Delta }^s(\varkappa ,x_{u_1},\cdots ,x_{u_m})<{\displaystyle \frac{\epsilon }{m+1}}$ and there exist $u_0^{\prime }\in \mathbb{N}$ such that $u_1,\cdots ,u_m\ge u_0^{\prime }\Rightarrow \Delta (y,x_{u_1},\cdots ,x_{u_m})<{\displaystyle \frac{\epsilon }{m+1}}$. Let $N=max\{u_0,u_0^{\prime }\}$.

Using m-simplex inequality for all different elements $x_{u_1},\cdots ,x_{u_{m-1}},x_{u_m}\in E$, for $m>N$,

$$\begin{array}{ccc}\hfill \Delta (\varkappa ,y,x_{u_1},...,x_{u_{m-1}})& \le & [\Delta (y,x_{u_1},...,x_{u_{m-1}},x_{u_m})\hfill \\ & +& \Delta (\varkappa ,x_{u_1},...,x_{u_{m-1}},x_{u_m})\hfill \\ & +& \Delta (\varkappa ,y,x_{u_2},...,x_{u_{m-1}},x_{u_m})\hfill \\ & ⋮& \\ & +& \Delta (\varkappa ,y,x_{u_1},...,x_{u_{m-2}},x_{u_m})]\hfill \\ & <& (m+1){\displaystyle \frac{\epsilon }{m+1}}=\epsilon .\hfill \end{array}$$

Hence, $∥\Delta (\varkappa ,y,x_{u_1},\cdots ,x_{u_{m-1}})∥<k∥\epsilon ∥$. Since $\epsilon$ is arbitrary, $\Delta (\varkappa ,y,x_{u_1},\cdots ,x_{u_{m-1}})=0$. Thus, $\varkappa =y$. □

Definition 10. 

A cone m-hemi metric space $(E,\Delta )$ is considered to be compact if each sequence in E contains a convergent subsequence.

Now, we prove the Banach contraction principle in the context of cone m-hemi-metric spaces.

Theorem 1. 

Let $(E,\Delta )$ be an x-complete cone m-hemi-metric space and ρ denote a normal cone, where the normal constant is k and $T:E\to E$ represent an x-continuous self mapping that satisfies

$$\Delta (T{x}_m,\dots ,T{x}_0)\le \lambda \Delta (x_m,\dots ,x_0)$$

(6)

for all $x_m,\dots ,x_0\in E$, $\lambda \in [0,1)$. Then, T has a unique fixed point.

Proof. 

Let $m\in {\mathbb{Z}}^{+}$ and $x_0\in E$ be any point. We define a sequence $\mathrm{for}\mathrm{all}u\ge 1,$ $x_m^{u+1}=T{x}_m^u$. From (6), we have

$$\begin{array}{ccc}\hfill \Delta (x_{m+1}^{\left(u\right)},x_m^{\left(u\right)},\cdots ,x_1^{\left(u\right)})& =& \Delta (T{x}_{m+1}^{(u-1)},T{x}_m^{(u-1)},\cdots ,T{x}_1^{(u-1)})\hfill \\ & \le & \lambda \Delta (x_{m+1}^{(u-1)},x_m^{(u-1)},...,x_1^{(u-1)})\hfill \\ & \le & \cdots \hfill \\ & \le & {\lambda }^u\Delta (x_{m+1}^{\left(0\right)},x_m^{\left(0\right)},...,x_1^{\left(0\right)}).\hfill \end{array}$$

Given that $\rho$ is a normal cone with a normal constant k, it follows that

$$\parallel \Delta (x_{m+1}^{\left(u\right)},x_m^{\left(u\right)},\dots ,x_1^{\left(u\right)})\parallel \le {\lambda }^{u}k\parallel \Delta (x_{m+1}^{\left(0\right)},x_m^{\left(0\right)},...,x_1^{\left(0\right)})\parallel .$$

This implies

$$∥\Delta (x_{m+1}^{\left(u\right)},x_m^{\left(u\right)},\dots ,x_1^{\left(u\right)})∥\to 0\mathrm{as}m,u\to \infty .$$

Hence, $\left\{x_m^{\left(u\right)}\right\}$ is an x-Cauchy sequence. As $(E,\Delta )$ is x-complete, there exist a $y\in E$ with $x_m^{\left(u\right)}\to y$. By x-continuity of T,

$$\underset{u\to \infty }{lim}\Delta \parallel T{x}_m^{\left(u\right)},\dots ,T{x}_1^{\left(u\right)},Ty\parallel =0$$

and by

$$\underset{u\to \infty }{lim}\Delta \parallel (T{x}_m^{\left(u\right)},...,T{x}_1^{\left(u\right)},Ty)\parallel =0$$

and limit’s uniqueness $Ty=y$.

We now demonstrate the fixed point’s uniqueness. Assume that T has distinct fixed point $x_1,x_2,...,x_{m+1}$ with $x_i\ne x_j$ for $1\le i,j\le m+1$. By (6), we have

$$\begin{array}{ccc}\hfill \Delta (x_1,x_2,...,x_{m+1})& =& \Delta (T{x}_1,T{x}_2,...,T{x}_{m+1})\hfill \\ & \le & \lambda \Delta (x_1,x_2,...,x_{m+1}),\hfill \end{array}$$

which contradicts $\lambda \in [0,1)$. As a result, T possesses a unique fixed point. □

Corollary 1. 

Consider $(E,\Delta )$ to be a complete cone m-hemi metric space, and ρ represents a normal cone, where the normal constant is k. For $\epsilon \in \mathsf{ϝ}$ with $0\ll \epsilon$ and $x_0,x_1,x_2,\cdots ,x_{m-1}\in E,$ set $B(x_0,x_1,x_2,\cdots ,x_{m-1},\epsilon )$ = $\{y\in E$ $|\Delta (x_0,x_1,x_2,\cdots ,x_{m-1},y)\le \epsilon \}.$ Suppose $T:E\to E$ is a mapping that satisfies the contractive condition

$$\Delta (T{x}_m,\dots ,T{x}_0)\le \lambda \Delta (x_m,\dots ,x_0)$$

for all $x_m,\dots ,x_0\in E$, $\lambda \in [0,1)$, which is a constant. Moreover, assume that $\Delta (x_0,x_1,x_2,\cdots ,x_{m-1},Ty)\le (1-\lambda )\epsilon .$ Then, T possesses a unique fixed point in $B(x_0,x_1,x_2,\cdots ,x_{m-1},\epsilon ).$

Proof. 

We just have to demonstrate that $B(x_0,x_1,x_2,\cdots ,x_{m-1},\epsilon )$ is complete and $Ty\in B(x_0,x_1,x_2,\cdots ,x_{m-1},\epsilon )$ for all y in $B(x_0,x_1,x_2,\cdots ,x_m,\epsilon )$.

Suppose $\left\{x_u\right\}$ is a Cauchy sequence in $B(x_0,x_1,x_2,\cdots ,x_{m-1},\epsilon ).$ Then, $\left\{x_u\right\}$ is also a Cauchy sequence in E. Due to the completeness of E, there is $y\in E$ so that $\left\{x_u\right\}\to y(u\to \infty ).$ We have

$$\begin{array}{ccc}\hfill \Delta (x_0,x_1,x_2,\cdots ,x_{m-1},y)& \le & [\Delta (x_u,x_0,x_1,x_2,\cdots ,x_{m-1})\hfill \\ & +& \Delta (x_u,y,x_2,x_3,\cdots ,x_{m-1})\hfill \\ & +& \Delta (x_u,y,x_1,x_3,\cdots ,x_{m-1})\hfill \\ & ⋮& \\ & +& \Delta (x_u,y,x_1,x_2,\cdots ,x_{m-2})]\hfill \\ & \le & \epsilon .\hfill \end{array}$$

Hence, $\Delta (x_0,x_1,x_2,\cdots ,x_{m-1},y)\le \epsilon ,$ and $y\in B(x_0,x_1,x_2,\cdots ,x_{m-1},\epsilon ).$

Therefore, $B(x_0,x_1,x_2,\cdots ,x_{m-1},\epsilon )$ is complete. For every $y\in B(x_0,x_1,x_2,\cdots ,x_{m-1},\epsilon ),$ we have

$$\begin{array}{ccc}\hfill \Delta (x_0,x_1,\cdots ,x_{m-1},Ty)& \le & [\Delta (T{x}_u,x_1,x_2,\cdots ,Ty)\hfill \\ & +& \Delta (x_0,T{x}_u,x_2,\cdots ,x_{m-1},Ty)\hfill \\ & ⋮& \\ & +& \Delta (x_0,x_1,x_2,\cdots ,T{x}_u,Ty)\hfill \\ & +& \Delta (x_0,x_1,x_2,\cdots ,x_{m-1},T{x}_u)]\hfill \\ & \le & (1-\lambda )\epsilon .\hfill \end{array}$$

Since $\lambda <1,$ it implies $\Delta (x_0,x_1,x_2,\cdots ,x_{m-1},Ty)<\epsilon .$ Hence, $Ty\in B(x_0,x_1,x_2,\cdots ,$ $x_{m-1},\epsilon ).$ □

Corollary 2. 

Let $(E,\Delta )$ be a complete cone m-hemi metric space. Let ρ represent a normal cone, where the normal constant is $k.$ Consider a mapping $T:E\to E$ x-continuous that satisfies for some positive integer u

$$\Delta (T^u{x}_m,\cdots ,T^u{x}_0)\le \lambda \Delta (x_m,\cdots ,x_0),\mathit{for}\mathit{all}{x}_m,\cdots ,x_0\in E.$$

Where $\lambda \in [0,1)$ is a constant. Then, T possesses a unique fixed point in E.

Proof. 

According to Theorem 1$,$ $T^u$ possesses a unique fixed point, which we denote as $x^{\ast }.$ However, $T^u\left(T{x}^{\ast }\right)=T\left(T^u{x}^{\ast }\right)=T{x}^{\ast },$ so $T{x}^{\ast }$ is a fixed point of $T^u$ as well. Therefore, $T{x}^{\ast }=x^{\ast }$, indicating that $x^{\ast }$ is a fixed point of $T.$ As the fixed point of T is also a fixed point of $T^u$, the fixed point T is unique. □

Example 5. 

Let $E=[0,{\displaystyle \frac{3}{2}})$. Define $\Delta :E^{m+1}\to {\mathbb{R}}^2,\Delta (x_1,x_2,...,x_{m+1})=min\{(1,1),\left(\right|x_i-x_j|,\beta |x_i-x_j\left|\right\},\beta >0$ and $T:E\to E,$

$$T\left(x\right)=\left\{\begin{array}{cc}{\displaystyle \frac{x^2}{2}},\hfill & \mathit{if}x\in [0,1)\hfill \\ {\displaystyle \frac{2x-1}{2}},\hfill & \mathit{if}x\in [1,{\displaystyle \frac{3}{2}}).\hfill \end{array}\right.$$

Then, Δ is an x-complete cone m-hemi metric space on E. Then, (6) is satisfied and T possesses a unique fixed point, which is 0. We depict the graphical behavior of Δ in Figure 1.

Figure 1. Depicts the graphical behavior of $\Delta$ in the above example.

Now, we prove the fixed-point theorem for compact cone m-hemi metric spaces.

Theorem 2. 

Consider $(E,\Delta )$ as a sequentially compact cone m-hemi metric space, and ρ represents a regular cone. Assume that the mapping $T:E\to E$ represents an x-continuous self mapping that satisfies the contractive condition

$$\Delta (Tm,\cdots ,T{x}_0)<\Delta (x_m,\cdots ,x_0),\mathit{for}\mathit{all}{x}_m,\cdots ,x_0\in E.$$

(7)

Then, T possesses a unique fixed point in E.

Proof. 

Let $m\in {\mathbb{Z}}^{+}$ and $x_0\in E$ be any point. We define a sequence $\mathrm{for}\mathrm{all}u\ge 1,$ $x_m^{(u+1)}=T{x}_m^{\left(u\right)}.$ In the case that, for some u, $x_m^{u+1}=x_m^u,$ $x_m^u$ serves as a fixed point of $T,$ concluding the proof. Accordingly, we assume for all u that $x_m^{u+1}\ne x_m^u.$ Define ${\Delta }_u=\Delta (x_{m+1}^{u+1},\cdots ,x_1^{u+1}),$ and then

$$\begin{array}{ccc}\hfill {\Delta }_{u+1}& =& \Delta (x_{m+1}^{u+2},\cdots ,x_1^{u+2})\hfill \\ & =& \Delta (T{x}_{m+1}^{u+1},\cdots ,T{x}_1^{u+1})\hfill \\ & <& \Delta (x_{m+1}^{u+1},\cdots ,x_1^{u+1})\hfill \\ & =& {\Delta }_u^x.\hfill \end{array}$$

Therefore, ${\Delta }_u$ is a decreasing sequence with a lower bound of 0. Since $\rho$ is a regular, there exists is ${\Delta }^{\ast }\in \mathsf{ϝ}$ such that ${\Delta }_u\to {\Delta }^{\ast }(u\to \infty ).$ Due to the sequence compactness of E, there is a sub-sequence $\left\{x_{u_i}\right\}$ of $\left\{x_u\right\}$ and $x^{\ast }\in E$ such that $x_{u_i}\to x^{\ast }(i\to \infty )$. It follows that

$$\begin{array}{ccc}\hfill \Delta (T{x}_{m+1}^{u_i},\cdots ,T{x}_1^{u_i},T{x}^{\ast })& <& \Delta (x_{m+1}^{u_i},\cdots ,x_1^{u_i},x^{\ast }),i=1,2,\cdots .\hfill \end{array}$$

So,

$$\begin{array}{ccc}\hfill ∥\Delta (T{x}_{m+1}^{u_i},\cdots ,T{x}_1^{u_i},T{x}^{\ast })∥& <& k∥\Delta (x_{m+1}^{u_i},\cdots ,x_1^{u_i},x^{\ast })∥\to 0(i\to \infty ),\hfill \end{array}$$

where k denotes the normal constant of $\rho$. By x-continuity of $T,$

$$\underset{u\to \infty }{lim}∥\Delta (T{x}_{m+1}^{u_i},\cdots ,T{x}_1^{u_i},T{x}^{\ast })∥=0.$$

Moreover, by

$$\underset{u\to \infty }{lim}∥\Delta (T{x}_{m+1}^{u_i},\cdots ,T{x}_1^{u_i},T{x}^{\ast })∥=0$$

and limit’s uniqueness, $T{x}^{\ast }=x^{\ast }$.

We now demonstrate the fixed point’s uniqueness. Suppose that T has the distinct fixed point $x_1,x_2,\cdots ,x_{m+1}$ with $x_i\ne x_j$ for $1\le i,j\le m+1.$ From (7), we have

$$\begin{array}{ccc}\hfill \Delta (x_1,x_2,\cdots ,x_{m+1})& =& \Delta (T{x}_1,T{x}_2,\cdots ,T{x}_{m+1})\hfill \\ & <& \Delta (x_1,x_2,\cdots ,x_{m+1}),\hfill \end{array}$$

which is a contradiction. As a result, T possesses a unique fixed point. □

Example 6. 

Consider $E=[0,1]$. Define $\Delta :E^{m+1}\to {\mathbb{R}}^2,$ $\Delta (x_1,x_2,...,x_{m+1})=({max}_{1\le i,j\le m+1}$ $|x_i-x_j|,{max}_{1\le i,j\le m+1}\beta |x_i-x_j\left|\right)$ and $T:E\to E,$

$$T\left(x\right)={\displaystyle \frac{x^2}{2}}.$$

This example demonstrates that the mapping $T\left(x\right)={\displaystyle \frac{x^2}{2}}$ on the sequentially compact cone m-hemi metric space $(E,\Delta )$ with $E=[0,1]$ satisfies the contractive condition, and T possesses a unique fixed point at $x=0.$

Definition 11. 

Consider a metric space $(E,\Delta )$. A mapping $T:E\to E$ is referred to as a Krasnosalskii mapping if given $0<\delta <L$ there exists $0<\alpha (\delta ,L)<1$ such that

$$\Delta (T{x}_m,\cdots ,T{x}_0)\le \alpha (\delta ,L)\Delta (x_m,\cdots ,x_0)$$

whenever $\delta <\Delta (x_m,\cdots ,x_0)<L.$

Now, we prove the Krasnosalskii fixed-point theorem in the setting of cone m-hemi metric spaces.

Theorem 3. 

Let $(E,\Delta )$ be a complete cone m-hemi metric space and ρ denote a normal cone, where the normal constant is $k.$ A mapping $T:E\to E$ represent an x-continuous self-mapping that satisfies a Krasnosalskii condition if given $0<\delta <L$ there exist $0<\alpha (\delta ,L)<1$ such that

$$\Delta (T{x}_m,\cdots ,T{x}_0)\le \alpha (\delta ,L)\Delta (x_m,\cdots ,x_0)$$

(8)

whenever $\delta <\Delta (x_m,\cdots ,x_0)<L\mathrm{for}\mathrm{all}{x}_m,\cdots ,x_0\in E.$ Then, T possesses a unique fixed point.

Proof. 

We show that T is contractive. Let $x_m,\cdots ,x_1\in E;$ $x_i\ne x_j$ and let

$$\begin{array}{ccc}\hfill C& =& \Delta (x_m,\cdots ,x_1)>0\hfill \\ & \Rightarrow & 0<C/2<C=\Delta (x_m,\cdots ,x_1)<C+1\hfill \\ & \Rightarrow & C/2<\Delta (x_m,\cdots ,x_1)<C+1.\hfill \end{array}$$

Since T is Krasnosalskii, there exists

$$0<\alpha (C/2,C+1)<1\mathrm{such}\mathrm{that}$$

$$\Delta (T{x}_m,\cdots ,T{x}_1)\le \alpha (C/2,C+1)\Delta (x_m,\cdots ,x_1)<\Delta (x_m,\cdots ,x_1)$$

(9)

$$\Rightarrow \Delta (x_m,\cdots ,x_1)<\Delta (x_m,\cdots ,x_1)\mathrm{for}\mathrm{all}{x}_m,\cdots ,x_1\in E;x_i\ne x_j.$$

$\Rightarrow T$ is contractive. Now, let $m\in {\mathbb{Z}}^{+}$ and $x_0\in E$ be any point. We define a sequence $\mathrm{for}\mathrm{all}u\ge 1,$ $x_m^{u+1}=T{x}_m^u$.

From (8), we obtain

$$\begin{array}{ccc}\hfill \Delta (x_{m+1}^{\left(u\right)},x_m^{\left(u\right)},\cdots ,x_1^{\left(u\right)})& =& \Delta (T{x}_{m+1}^{(u-1)},T{x}_m^{(u-1)},\cdots ,T{x}_1^{(u-1)})\hfill \\ & \le & \alpha (\delta ,L)\Delta (x_{m+1}^{(u-1)},x_m^{(u-1)},...,x_1^{(u-1)})\hfill \\ & \le & \cdots \hfill \\ & \le & {\left[\alpha (\delta ,L)\right]}^u\Delta (x_{m+1}^{\left(0\right)},x_m^{\left(0\right)},...,x_1^{\left(0\right)}).\hfill \end{array}$$

Given a normal cone $\rho$ with normal constant k, it follows that

$$\parallel \Delta (x_{m+1}^{\left(u\right)},x_m^{\left(u\right)},\dots ,x_1^{\left(u\right)})\parallel <{\left[\alpha (\delta ,L)\right]}^{u}k\parallel \Delta (x_{m+1}^{\left(0\right)},x_m^{\left(0\right)},...,x_1^{\left(0\right)})\parallel .$$

This implies

$$∥\Delta (x_{m+1}^{\left(u\right)},x_m^{\left(u\right)},\dots ,x_1^{\left(u\right)})∥\to 0\mathrm{as}m,u\to \infty .$$

Hence, $\left\{x_m^{\left(u\right)}\right\}$ is an x-Cauchy sequence. Since $(E,\Delta )$ is x-complete, there exist a $y\in E$ with $x_m^{\left(u\right)}\to y$. By x-continuity of T,

$$\underset{u\to \infty }{lim}\Delta \parallel T{x}_m^{\left(u\right)},\dots ,T{x}_1^{\left(u\right)},Ty\parallel =0$$

and by

$$\underset{u\to \infty }{lim}\Delta \parallel (T{x}_m^{\left(u\right)},...,T{x}_1^{\left(u\right)},Ty)\parallel =0$$

and limit’s uniqueness $Ty=y$.

Now, we demonstrate the fixed point’s uniqueness. Assume that T has distinct fixed points $x_1,x_2,...,x_{m+1}$ with $x_i\ne x_j$ for $1\le i,j\le m+1$. By using (8), we deduce

$$\begin{array}{ccc}\hfill \Delta (x_1,x_2,...,x_{m+1})& =& \Delta (T{x}_1,T{x}_2,...,T{x}_{m+1})\hfill \\ & \le & \alpha (\delta ,L)\Delta (x_1,x_2,...,x_{m+1}),\hfill \end{array}$$

which contradicts $\alpha (\delta ,L)\in (0,1)$. As a result, T has a unique fixed point. □

Example 7. 

Consider ϝ to be a Banach algebra $\mathfrak{C}\left(K\right),$ which consists of all continuous real-valued functions. Let $\rho =\{\mathfrak{f}\in \mathsf{ϝ}$ $\left|\mathfrak{f}\right(\mu )\ge 0\mathit{for}\mathit{all}\mu \in K\}$. Then, $\rho \subset \mathsf{ϝ}$ represents a normal cone, where the normal constant $k=1.$ Consider $E=\mathfrak{C}\left(K\right)$, define $\Delta :E^{m+1}\to \mathsf{ϝ},$ $\Delta ({\mathfrak{f}}_1,{\mathfrak{f}}_2,...,{\mathfrak{f}}_{m+1})={max}_{1\le i,j\le m+1}|{\mathfrak{f}}_i\left(\mu \right)-{\mathfrak{f}}_j\left(\mu \right)|$, and $T:E\to E$,

$$T\left(\mathfrak{f}\left(\mu \right)\right)={\displaystyle \frac{\mathfrak{f}\left(\mu \right)}{3}}.$$

This example demonstrates that the mapping $T\left(\mathfrak{f}\left(\mu \right)\right)={\displaystyle \frac{\mathfrak{f}\left(\mu \right)}{3}}$ on the complete cone m-hemi metric space $(E,\Delta )$ satisfies the Krasnosalskii condition with $\alpha (\delta ,L)<{\displaystyle \frac{1}{3}}$, so T possesses a unique fixed point at $x=0$.

Definition 12. 

Consider $(E,\Delta )$ is a metric space and $T:E\to E.$ Thus, T is said to satisfy the Meir–Keeler contractive condition if for given $\epsilon >0$ $\mathit{there}\mathit{exists}$ $\delta >0$ so that

$$\Delta (T{x}_m,\cdots ,T{x}_0)<\epsilon \mathit{whenever}\epsilon \le \Delta (x_m,\cdots ,x_0)<\epsilon +\delta .$$

Now, we prove fixed-point theorem for Meir–Keeler contraction in the setting of cone m-hemi metric spaces.

Theorem 4. 

Let $(E,\Delta )$ be a complete cone m-hemi metric space and ρ represent a normal cone, where the normal constant is $k.$ Let $T:E\to E$ represent a x-continuous self-mapping that fulfills the Meir–Keeler contractive condition if for given $\epsilon >0$ $\mathit{there}\mathit{exists}$ $\delta >0$ so that

$$\Delta (T{x}_m,\cdots ,T{x}_0)<\epsilon \mathit{whenever}\epsilon \le \Delta (x_m,\cdots ,x_0)<\epsilon +\delta .$$

(10)

Then, T possesses a unique fixed point.

Proof. 

We demonstrate that T is contractive. Let $x_m,\cdots ,x_1\in E;$ $x_i\ne x_j$ and then $\epsilon =\Delta (x_m,\cdots ,x_1)>0$. Since T satisfies the Meir–Keeler contractive condition, $\mathrm{there}\mathrm{exists}$ $\delta >0$ such that

$$\begin{array}{ccc}\hfill \Delta (T{x}_m,\cdots ,T{x}_1)<\epsilon & \mathrm{whenever}& \epsilon \le \epsilon =\Delta (x_m,\cdots ,x_1)<\epsilon +\delta \hfill \\ \hfill \Rightarrow \Delta (T{x}_m,\cdots ,T{x}_1)<\epsilon & =& \Delta (x_m,\cdots ,x_1)\hfill \\ \hfill \Rightarrow \Delta (T{x}_m,\cdots ,T{x}_1)& <& \Delta (x_m,\cdots ,x_1).\hfill \end{array}$$

⇒T is contractive.

Let ${\Delta }_u=\Delta (x_{m+1}^{u+1},\cdots ,x_1^{u+1})$ and then

$$\begin{array}{ccc}\hfill {\Delta }_{u+1}& =& \Delta (x_{m+1}^{u+2},\cdots ,x_1^{u+2})\hfill \\ & =& \Delta (T{x}_{m+1}^{u+1},\cdots ,T{x}_1^{u+1})\hfill \\ & <& \Delta (x_{m+1}^{u+1},\cdots ,x_1^{u+1})\hfill \\ & =& {\Delta }_u^x.\hfill \end{array}$$

Therefore, the sequence ${\Delta }_u$ is decreasing and has a lower bound of 0.

This implies that ${\Delta }_u$ is convergent sequence. Let ${\Delta }_u\to \Delta \ge 0\mathrm{as}u\to \infty .$ We claim that $\Delta =0.$ Suppose that $\Delta >0,$ and, taking $\epsilon =\Delta >0$, $\mathrm{there}\mathrm{exists}$ $\delta >0$ so that

$$\Delta (T{x}_m,\cdots ,T{x}_1)<\epsilon \mathrm{whenever}\epsilon \le \Delta (x_m,\cdots ,x_1)<\epsilon +\delta .$$

(11)

Since ${\Delta }_u\to \Delta =\epsilon ,$ for $\delta >0,$ $\mathrm{there}\mathrm{exists}$ $N\left(\delta \right)\in \mathbb{N}$ so that

$$\begin{array}{ccc}\hfill |{\Delta }_u-\epsilon |<\delta & \mathrm{for}\mathrm{all}& u\ge N\hfill \\ \hfill \Rightarrow {\Delta }_u<\epsilon +\delta & \mathrm{for}\mathrm{all}& u\ge N.\hfill \end{array}$$

In particular, for $u=\mathbb{N},$ ${\Delta }_N=\Delta (x_{m+1}^{N+1},\cdots ,x_1^{N+1})<\epsilon +\delta .$ But,

$$\epsilon =\Delta \le {\Delta }_N=\Delta (x_{m+1}^{N+1},\cdots ,x_1^{N+1})$$

$$\because \Delta =inf\left\{{\Delta }_u\right\}$$

$$\Rightarrow \epsilon \le {\Delta }_N=\Delta (x_{m+1}^{N+1},\cdots ,x_1^{N+1})<\epsilon +\delta .$$

Applying (11), we obtain

$$\Rightarrow {\Delta }_{N+1}=\Delta (x_{m+1}^{N+2},\cdots ,x_1^{N+2})<\epsilon =\Delta ,$$

which is a contradiction tot $\Delta$ being infimum of ${\Delta }_u.$ Hence, $\Delta =0.$

Now, let $m\in {\mathbb{Z}}^{+}$ and $x_0\in E$ be any point. We define a sequence $\mathrm{for}\mathrm{all}u\ge 1,$ $x_m^{u+1}=T{x}_m^u$.

$$\begin{array}{ccc}\hfill \Delta (x_{m+1}^{\left(u\right)},x_m^{\left(u\right)},\cdots ,x_1^{\left(u\right)})& =& \Delta (T{x}_{m+1}^{(u-1)},T{x}_m^{(u-1)},\cdots ,T{x}_1^{(u-1)})\hfill \\ & <& \Delta (x_{m+1}^{(u-1)},x_m^{(u-1)},...,x_1^{(u-1)})\hfill \\ & <& \cdots \hfill \\ & <& \Delta (x_{m+1}^{\left(0\right)},x_m^{\left(0\right)},...,x_1^{\left(0\right)}).\hfill \end{array}$$

Given that $\rho$ is a normal cone with normal constant k, it follows that,

$$\parallel \Delta (x_{m+1}^{\left(u\right)},x_m^{\left(u\right)},\dots ,x_1^{\left(u\right)})\parallel <k\parallel \Delta (x_{m+1}^{\left(0\right)},x_m^{\left(0\right)},\dots ,x_1^{\left(0\right)})\parallel .$$

This implies

$$∥\Delta (x_{m+1}^{\left(u\right)},x_m^{\left(u\right)},\dots ,x_1^{\left(u\right)})∥\to 0\mathrm{as}m,u\to \infty .$$

Hence, $\left\{x_m^{\left(u\right)}\right\}$ is an x-Cauchy sequence. As $(E,\Delta )$ is x-complete, there exist a $y\in E$ with $x_m^{\left(u\right)}\to y$. By x-continuity of T,

$$\underset{u\to \infty }{lim}\Delta \parallel T{x}_m^{\left(u\right)},\dots ,T{x}_1^{\left(u\right)},Ty\parallel =0$$

and by

$$\underset{u\to \infty }{lim}\Delta \parallel (T{x}_m^{\left(u\right)},...,T{x}_1^{\left(u\right)},Ty)\parallel =0$$

and limit’s uniqueness $Ty=y$.

Now, we examine the fixed point’s uniqueness. Assume that T has distinct fixed point $x_1^{\ast },x_2^{\ast },...,x_{m+1}^{\ast }$ with $x_i^{\ast }\ne x_j^{\ast }$ for $1\le i,j\le m+1$

$$\Delta (T{x}_1^{\ast },T{x}_2^{\ast },\dots ,T{x}_{m+1}^{\ast })=\Delta (x_1^{\ast },x_2^{\ast },\dots ,x_{m+1}^{\ast })>0.$$

Now, denote

$$\epsilon =\Delta (x_1^{\ast },\cdots ,x_{m+1}^{\ast }).$$

By Meir–Keeler condition, for $\epsilon =\Delta (x_1^{\ast },\cdots ,x_{m+1}^{\ast })$, there exists $\delta >0$ such that

$$\mathrm{If}\epsilon \le \Delta (x_1,\cdots ,x_m+1)<\epsilon +\delta ,\mathrm{then}\Delta (T{x}_1,\cdots ,T{x}_{m+1})<\epsilon .$$

Applying the condition to $x_1=x_1^{\ast },\cdots ,x_{m+1}=x_{m+1}^{\ast },$ we have

$$\Delta (T{x}_1^{\ast },T{x}_2^{\ast },\dots ,T{x}_{m+1}^{\ast })=\Delta (x_1^{\ast },x_2^{\ast },\dots ,x_{m+1}^{\ast })<\epsilon =\Delta (x_1^{\ast },x_2^{\ast },\dots ,x_{m+1}^{\ast }),$$

which is a contradiction because it implies $\Delta (x_1^{\ast },x_2^{\ast },\dots ,x_{m+1}^{\ast })<\Delta (x_1^{\ast },x_2^{\ast },\dots ,x_{m+1}^{\ast })$. As a result, T possesses a unique fixed point. □

Example 8. 

Let $E=[0,1]$. Define $\Delta :E^{m+1}\to {\mathbb{R}}^2,\Delta (x_1,x_2,...,x_{m+1})=({max}_{1\le i,j\le m+1}|x_i-x_j|,{max}_{1\le i,j\le m+1}\beta |x_i-x_j\left|\right)$ and $T:E\to E,$

$$T\left(x\right)={\displaystyle \frac{x}{3}}+{\displaystyle \frac{1}{3}}.$$

This example demonstrates that the mapping $T\left(x\right)={\displaystyle \frac{x}{3}}+{\displaystyle \frac{1}{3}}$ on the x-complete cone m-hemi metric space $(E,\Delta )$ satisfies the Meir–Keeler condition with $\delta =2\epsilon$, and T possesses a unique fixed point at $x={\displaystyle \frac{1}{2}}$.

Definition 13. 

A function $\mathfrak{f}$ is right-continuous at a point $\delta \in \mathbb{R}$ if $\mathfrak{f}$ is defined on an interval $[\delta ,L]$ lying to the right of δ and if ${lim}_{\varkappa \to {\delta }^{+}}\mathfrak{f}\left(\varkappa \right)=\mathfrak{f}\left(\delta \right).$ The definition for left-continuous is defined similarly, with $[\delta ,L]$ lying on the left and ${lim}_{\varkappa \to {\delta }^{-}}\mathfrak{f}\left(\varkappa \right)=\mathfrak{f}\left(\delta \right).$

Definition 14. 

Consider $(E,\Delta )$ as a metric space. The mapping $T:E\to E$ is referred to as the Boyd–Wong mapping if there exists a function $\phi :[0,\infty )\to [0,\infty )$ so that

(i) 

$\phi \left(\mathfrak{r}\right)<\mathfrak{r}\mathrm{if}\mathfrak{r}>0;$

(ii) 

$\Delta (T{x}_m,\cdots ,T{x}_0)\le \phi (\Delta (x_m,\cdots ,x_0));$

(iii) 

φ is right-continuous.

Now, we prove Boyd–Wong’s fixed-point theorem in the setting of cone m-hemi metric spaces.

Theorem 5. 

Consider $(E,\Delta )$ as a complete metric space and ρ a normal cone with normal constant k. A mapping $T:E\to E$ represents an x-continuous self-mapping that satisfies the Boyd–Wong condition if there exists a right-continuous function $\phi :[0,\infty )\to [0,\infty )$ such that $\phi \left(\mathfrak{r}\right)<\mathfrak{r}$ if $\mathfrak{r}>0$ and

$$\Delta (T{x}_m,\cdots ,T{x}_0)\le \phi (\Delta (x_m,\cdots ,x_0))$$

(12)

for all $x_m,\cdots ,x_0\in E.$ Then, T possesses a unique fixed point.

Proof. 

Let $m\in {\mathbb{Z}}^{+}$ and $x_0\in E$ be any point. We define a sequence $\mathrm{for}\mathrm{all}u\ge 1,$ $x_m^{u+1}=T{x}_m^u$.

Let $\phi$ be the function that satisfies Boyd–Wong map condition. So, we assume that, for all $u,x_m^{u+1}\ne x_m^u.$ Set ${\Delta }_u=\Delta (x_{m+1}^{u+1},\cdots ,x_1^{u+1}),$ and then

$$\begin{array}{ccc}\hfill {\Delta }_{u+1}& =& \Delta (x_{m+1}^{u+2},\cdots ,x_1^{u+2})\hfill \\ & =& \Delta (T{x}_{m+1}^{u+1},\cdots ,T{x}_1^{u+1})\hfill \\ & <& \phi (\Delta (x_{m+1}^{u+1},\cdots ,x_1^{u+1}))\hfill \\ & <& \Delta (x_{m+1}^{u+1},\cdots ,x_1^{u+1})\sin \mathrm{ce}\phi \left(\mathfrak{r}\right)<\mathfrak{r}\hfill \\ & =& {\Delta }_u^x.\hfill \end{array}$$

Therefore, the sequence ${\Delta }_u$ is decreasing and has a lower bound of 0.

This implies that $\left({\Delta }_u\right)$ is a convergent sequence. Let ${\Delta }_u\to \Delta \ge 0\mathrm{as}u\to \infty .$

Suppose ${\Delta }_{u+1}\le \phi \left({\Delta }_u\right)$. Since $\phi$ is right-continuous function, taking the limit of u on both sides yields

$$\begin{array}{ccc}\hfill \Delta & =& \underset{u\to \infty }{lim}{\Delta }_{u+1}\hfill \\ & \le & \underset{u\to \infty }{lim}\phi \left({\Delta }_u^x\right)\hfill \\ & =& \phi \left(\underset{u\to \infty }{lim}{\Delta }_u^x\right)\hfill \\ & =& \phi (\Delta ).\hfill \end{array}$$

Further, $\Delta \le \phi (\Delta ),$ which is contradiction to the fact that $\phi \left(\mathfrak{r}\right)<\mathfrak{r}$ $\mathrm{for}\mathrm{all}$ $\mathfrak{r}>0.$ Hence, $\Delta =0.$ Now, we show that $\left\{x_m^{\left(u\right)}\right\}$ is a Cauchy sequence. On the contrary, we suppose that $\left\{x_m^{\left(u\right)}\right\}$ is not a Cauchy sequence. Then, there exists an $\epsilon >0$ such that, for each $l=1,2,\cdots$, we choose integer $u_l$ with $u_l>l.$ So, we have

$$b_l=\Delta (x_{m+1}^{\left(u_l\right)},\cdots ,x_1^{\left(u_l\right)})\ge \epsilon .$$

(13)

Since ${\Delta }_u$ is a decreasing sequence, let $u_l$ be the smallest integer that is greater than l and satisfies $\left(13\right)$. We can choose the smallest $u_l$ possible, meaning that

$$\Delta (x_{m+1}^{u_l-1},\cdots ,x_1^{u_l-1})<\epsilon .$$

(14)

Now,

$$\begin{array}{ccc}\hfill \epsilon & \le & b_l=\Delta (x_{m+1}^{\left(u_l\right)},\cdots ,x_1^{\left(u_l\right)})\hfill \\ & =& \Delta (T{x}_{m+1}^{(u_l-1)},\cdots ,T{x}_1^{(u_l-1)})\hfill \\ & \le & \phi (\Delta (x_{m+1}^{(u_l-1)},\cdots ,x_1^{(u_l-1)}))\hfill \\ & <& \phi \left(\epsilon \right)\hfill \\ & \le & \epsilon \sin \mathrm{ce}\phi \left(\mathfrak{r}\right)\le \mathfrak{r}\mathrm{for}\mathrm{all}\mathfrak{r}\in [0,\infty ).\hfill \end{array}$$

Taking the limit as $l\to \infty ,$ we obtain

$$\epsilon \le \underset{l\to \infty }{lim}{b}_l\le \epsilon ,$$

implying that $b_l\to \epsilon$ from above as $l\to \infty ,$ and we have

$$\underset{l\to \infty }{lim}{b}_l=\epsilon .$$

Now,

$$\begin{array}{ccc}\hfill b_l& =& \Delta (x_{m+1}^{\left(u_l\right)},\cdots ,x_1^{\left(u_l\right)})\hfill \\ & =& \Delta (T{x}_{m+1}^{(u_l-1)},\cdots ,T{x}_1^{(u_l-1)})\hfill \\ & \le & \phi (\Delta (x_{m+1}^{(u_l-1)},\cdots ,x_1^{(u_l-1)}))\hfill \\ & \le & \phi \left(b_{l_1}\right).\hfill \end{array}$$

Taking the limit $l\to \infty$ yields

$$\epsilon =\underset{l\to \infty }{lim}{b}_l\le \underset{l\to \infty }{lim}\phi \left(b_{l_1}\right)=\phi (\underset{l\to \infty }{lim}\left(b_{l_1}\right)=\phi \left(\epsilon \right),$$

which implies that $\epsilon \le \phi \left(\epsilon \right).$

Based on the $\phi$ property, this implies that $\epsilon =0,$ which is a contradiction. Thus, the sequence $\left(x_m^{\left(u\right)}\right)$ is Cauchy. As $(E,\Delta )$ is x-complete, there exist a $y\in E$ with $x_m^{\left(u\right)}\to y$. By x-continuity of T,

$$\underset{u\to \infty }{lim}\Delta \parallel T{x}_m^{\left(u\right)},\dots ,T{x}_1^{\left(u\right)},Ty\parallel =0$$

and by

$$\underset{u\to \infty }{lim}\Delta \parallel (T{x}_m^{\left(u\right)},...,T{x}_1^{\left(u\right)},Ty)\parallel =0$$

and limit’s uniqueness $Ty=y$.

Now, we examine the fixed point’s uniqueness. Assume that T has distinct fixed point $x_1,x_2,...,x_{m+1}$ with $x_i\ne x_j$ for $1\le i,j\le m+1$. By using (12), we obtain

$$\begin{array}{ccc}\hfill \Delta (x_1,x_2,...,x_{m+1})& =& \Delta (T{x}_1,T{x}_2,...,T{x}_{m+1})\hfill \\ & \le & \phi (\Delta (x_1,x_2,...,x_{m+1})),\hfill \end{array}$$

which contradicts $\phi \left(\mathfrak{r}\right)<\mathfrak{r}$ $\mathrm{for}\mathrm{all}$ $\mathfrak{r}>0$. As a result, T possesses a unique fixed point. □

Example 9. 

Consider $\mathsf{ϝ}=M_u\left(\mathbb{R}\right)=\{A=x_{{ij}_{u\times u}}|x_{ij}\in \mathbb{R}\mathit{for}\mathit{all}1\le i,j\le u\}$ to be the algebra of all the u-square real matrices. Let $\rho =\{A\in \mathsf{ϝ}|x_{ij}\ge 0\mathit{for}\mathit{all}1\le i,j\le u\}$. Then, $\rho \subseteq \mathsf{ϝ}$ represents a normal cone, where the normal constant is $k=1.$ Let $E=M_u\left(\mathbb{R}\right)$ define $\Delta :E^{m+1}\to \mathsf{ϝ},\Delta (A_1,A_2,...,A_{m+1})={max}_{1\le i,j\le m+1}|A_i-A_j|$, and $T:E\to E$,

$$T\left(A\right)={\left({\displaystyle \frac{x_{ij}}{2}}\right)}_{n\times n}.$$

We choose $\phi \left(\mathfrak{r}\right)={\displaystyle \frac{\mathfrak{r}}{2}}$, which satisfies $\phi \left(\mathfrak{r}\right)<\mathfrak{r}$ for $\mathfrak{r}>0.$ The fixed point is the zero matrix $A=0.$

4. Applications

In this part, we provide applications to the Fredholm integral equation and dynamic market equilibrium.

4.1. Application to Fredholm Integral Equation

Now, we examine the solution’s uniqueness and existence in the Freholm integral equation in the framework of cone m-hemi metric space. Consider the Fredholm integral equation

$$x\left(\mu \right)={\int }_0^1\daleth (\mu ,\mathfrak{s},\sigma \left(\mathfrak{s}\right))d\mathfrak{s},\mu \in [\delta ,L]$$

(15)

and $\mathfrak{C}[0,1]$ represent the set of continuous functions with cone m-hemi metric

$$\Delta (x_1,x_2,\cdots ,x_m,x_{m+1})=\left\{\begin{array}{cc}0,\hfill & \mathrm{if}\mathrm{any}{x}_i=x_j\hfill \\ \underset{\mu \in [a,b]}{max}\left\{\right|x_i\left(\mu \right)-x_j\left(\mu \right)\left|\right\},\hfill & \mathrm{if}{x}_{ij}\hfill \end{array}\right.$$

$x_i\in \mathfrak{C}[0,1]$ and for $i,j=1,2,\cdots ,m+1.$

Theorem 6. 

Consider an integral Equation (15) and assume

(i) 

the continuous function $\daleth :[0,1]\times [0,1]\times \mathbb{R}\to {\mathbb{R}}^{+};$

(ii) 

$\lambda \in [0,1)$ and for every $(\mu ,\mathfrak{s})\in [0,1]\times [0,1]$ so that

$$|\daleth (\mu ,\mathfrak{s},x_1\left(\mathfrak{s}\right))-\daleth (\mu ,\mathfrak{s},x_2\left(\mathfrak{s}\right))|\le \lambda |x_1\left(\mathfrak{s}\right)-x_2\left(\mathfrak{s}\right)|.$$

Then, there exists a unique solution to integral Equation (15).

Proof. 

For every $x_i\in \mathfrak{C}[0,1]$, let $x_i\ne x_j$. According to integral equation formulation, $1\le i,j\le m+1$,

$$\begin{array}{ccc}\hfill \Delta (T{x}_1,T{x}_2,\cdots ,T{x}_{m+1})& =& \underset{\mu \in [0,1]}{max}\left\{\right|T{x}_i\left(\mu \right)-T{x}_j\left(\mu \right)\left|\right\}\hfill \\ & =& \underset{\mu \in [0,1]}{max}\left|{\int }_0^1(\daleth (\mu ,\mathfrak{s},x_i\left(\mathfrak{s}\right))-\daleth (\mu ,\mathfrak{s},x_j\left(\mathfrak{s}\right))d\mathfrak{s}\right|\hfill \\ & \le & \underset{\mu \in [0,1]}{max}{\int }_0^1|\daleth (\mu ,\mathfrak{s},x_i\left(\mathfrak{s}\right))-\daleth (\mu ,\mathfrak{s},x_j\left(\mathfrak{s}\right))|d\mathfrak{s}\hfill \\ & \le & \lambda \underset{\mu \in [0,1]}{max}\left|{\int }_0^1(x_i\left(\mathfrak{s}\right)-x_j\left(\mathfrak{s}\right))d\mathfrak{s}\right|\hfill \\ & \le & \lambda \underset{\mu \in [0,1]}{max}\left|(x_i\left(\mathfrak{s}\right)-x_j\left(\mathfrak{s}\right))\right|\hfill \\ & =& \lambda \Delta (x_1,x_2,\cdots ,x_{m+1}).\hfill \end{array}$$

According to Theorem 1, there exists a unique solution to integral Equation (15). □

4.2. Application to Dynamic Market Equilibrium

Now, we demonstrate how to use our proven conclusion to resolve an integral equation uniquely in the context of dynamic market equilibrium economics. Existing prices and pricing patterns (whether prices are rising or declining and whether they are falling or rising at an increasing or decreasing rate) possess an effect on supply $q_{\delta }$ and demand $q_T$ in many marketplaces. Determining the current price $x\left(\beta \right)$, the first derivative is $\beth x\left(\beta \right)/\beth \beta$ and the second derivative ${\beth }^{2}x\left(\beta \right)/\beth {\beta }^2.$ Assume

$$\begin{array}{ccc}\hfill q_{\delta }& =& J_1+{\eta }_{1}x\left(\beta \right)+{\mathsf{ϝ}}_1{\displaystyle \frac{\beth x\left(\beta \right)}{\beth \beta }}+z_1{\displaystyle \frac{{\beth }^{2}x\left(\beta \right)}{\beth {\beta }^2}},\hfill \\ \hfill q_T& =& J_2+{\eta }_{2}x\left(\beta \right)+{\mathsf{ϝ}}_2{\displaystyle \frac{\beth x\left(\beta \right)}{\beth \beta }}+z_2{\displaystyle \frac{{\beth }^{2}x\left(\beta \right)}{\beth {\beta }^2}}.\hfill \end{array}$$

(16)

Here, $J_1,J_2,{\eta }_1,{\eta }_2,{\mathsf{ϝ}}_1,\mathrm{and}{\mathsf{ϝ}}_2$ are constants. If prices equilibrate the market continuously over time, we discuss the market’s dynamic stability. In equilibrium, $q_{\delta }=q_T.$ So,

$$J_1+{\eta }_{1}x\left(\beta \right)+{\mathsf{ϝ}}_1{\displaystyle \frac{\beth x\left(\beta \right)}{\beth \beta }}+z_1{\displaystyle \frac{{\beth }^{2}x\left(\beta \right)}{\beth {\beta }^2}}=J_2+{\eta }_{2}x\left(\beta \right)+{\mathsf{ϝ}}_2{\displaystyle \frac{\beth x\left(\beta \right)}{\beth \beta }}+z_2{\displaystyle \frac{{\beth }^{2}x\left(\beta \right)}{\beth {\beta }^2}}.$$

(17)

Since

$$(z_1-z_2){\displaystyle \frac{{\beth }^{2}x\left(\beta \right)}{\beth {\beta }^2}}+({\mathsf{ϝ}}_1-{\mathsf{ϝ}}_2)\beth {\displaystyle \frac{\beth x\left(\beta \right)}{\beth \beta }}+({\eta }_1-{\eta }_2)x\left(\beta \right)=-(J_1-J_2),$$

(18)

letting $z=z_1-z_2,\mathsf{ϝ}={\mathsf{ϝ}}_1-{\mathsf{ϝ}}_2,\eta ={\eta }_1-{\eta }_2,\mathrm{and}J=J_1-J_2$ in above, we have

$$z{\displaystyle \frac{{\beth }^{2}x\left(\beta \right)}{\beth {\beta }^2}}+\mathsf{ϝ}{\displaystyle \frac{\beth x\left(\beta \right)}{\beth \beta }}+\eta x\left(\beta \right)=-J.$$

(19)

When dividing by z, the function $x\left(\beta \right)$ is determined by the initial value problem that follows

$$\left\{\begin{array}{c}{x}^{″}+{\displaystyle \frac{\mathsf{ϝ}}{z}}{x}^{″}+{\displaystyle \frac{\eta }{z}}x\left(\beta \right)=-{\displaystyle \frac{J}{z}},\hfill \\ x\left(0\right)=0,\hfill \\ x^{\prime }\left(0\right)=0,\hfill \end{array}\right.$$

(20)

where ${\mathsf{ϝ}}^2/z=4\eta /z\mathrm{and}\eta /\mathsf{ϝ}=\nu$ is a continuous function. It can be demonstrated that problem (20) is comparable to the integral equation that follows:

$$x\left(\beta \right)={\int }_0^{\Gamma }\Phi (\beta ,s)T(\beta ,s,x\left(s\right))\beth s,$$

(21)

where $\Phi (\beta ,s)$ represent a Green’s function, defined as follows:

$$\Phi (\beta ,s)=\left\{\begin{array}{cc}s{\mathsf{ϝ}}^{\left(\nu 2\right)(\beta -s)}\hfill & \mathrm{if}0\le s\le \beta \le \Gamma ,\hfill \\ \beta {\mathsf{ϝ}}^{\left(\nu 2\right)(s-\beta )}\hfill & \mathrm{if}0\le \beta \le s\le \beta \le \Gamma .\hfill \end{array}\right.$$

(22)

We will demonstrate that a solution exists for the integral equation as follows:

$$x\left(\beta \right)={\int }_0^{\Gamma }G(\beta ,s,x\left(s\right))\beth s.$$

(23)

Let $\mathfrak{D}=\mathfrak{C}\left(\right[0,\Gamma \left]\right)$ represent the set of continuous real-valued functions defined over the interval $[0,\Gamma ]$. For $\beta >0,$ we define

$$\Delta (x_1,x_2,\cdots ,x_m,x_{m+1})=\left\{\begin{array}{cc}0,\hfill & \mathrm{if}\mathrm{any}{x}_i=x_j\hfill \\ \underset{\beta \in [0,\Gamma ]}{max}\left\{x_i\left(\beta \right)\right\};\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

(24)

$x_i\in [0,\Gamma ]$ and for $i,j=1,2,\cdots ,m+1.$ $T:\mathfrak{D}\to \mathfrak{D}$ is defined by

$$Tx\left(\beta \right)={\int }_0^{\Gamma }G(\beta ,s,x\left(s\right))\beth s.$$

(25)

Theorem 7. 

Consider integral Equation $\left(23\right)$ and assume that

(i)

$\mathit{the}\mathit{continuous}\mathit{function}G:[0,\Gamma ]\times [0,\Gamma ]\to {\mathbb{R}}^{+}$;

(ii)

$\mathit{for}\mathit{all}(\beta ,s)\in [0,\Gamma ]\times [0,\Gamma ]\mathit{and}\lambda \in [0,1)$ such that

$$\left|G\right(\beta ,s,x\left(s\right)\left)\right|\le \lambda \left|x\right(s\left)\right|.$$

Then, integral Equation $\left(23\right)$ possesses a unique solution.

Proof. 

Let all $x_i\in \mathfrak{C}[0,\Gamma ]$ such that $x_i\ne x_j.$ By the use of assumptions $\left(i\right)\mathrm{to}\left(ii\right),$ $1\le i\le m+1,$ we have

$$\begin{array}{ccc}\hfill \Delta (\mathfrak{d}{x}_1,\mathfrak{d}{x}_2,\cdots ,\mathfrak{d}{x}_{m+1})& =& \underset{\beta \in [0,\Gamma ]}{max}\left\{\right|\mathfrak{d}{x}_i\left(\beta \right)\left|\right\}\hfill \\ & =& \underset{\beta \in [0,\Gamma ]}{max}\left|{\int }_0^{\Gamma }G(\beta ,s,x_i\left(s\right)\beth s\right|\hfill \\ & \le & \underset{\beta \in [0,\Gamma ]}{max}{\int }_0^{\Gamma }\left|G(\beta ,s,x_i\left(s\right))\right|\beth s\hfill \\ & \le & \lambda \underset{\beta \in [0,\Gamma ]}{max}\left|{\int }_0^{\Gamma }{x}_i\left(s\right)\beth s\right|\hfill \\ & \le & \lambda \underset{\beta \in [0,\Gamma ]}{max}\left|x_i\left(s\right)\right|\hfill \\ & =& \lambda \Delta (x_1,x_2,\cdots ,x_{m+1}).\hfill \end{array}$$

According to Theorem 1, there exists a unique solution to integral Equation (23). □

5. Conclusions

In this study, we introduced the concept of cone m-hemi metric spaces as a generalization of m-hemi metric spaces. Further, we proved several fixed-point results by utilizing generalized contractive conditions, including the Banach contraction principle and Krasnoselskii, Meir–Keeler, and Boyd–Wong contraction mappings in cone m-hemi metric spaces over Banach algebra. We provided several non-trivial examples that show the validity of our main results. Furthermore, regarding applications, we found the existence and uniqueness of the Fredholm integral equation and solution to an integral equation used in dynamic market equilibrium economics with the help of the main results. Our new findings could inspire other researchers to extend and enhance various results in these spaces and explore applications in related fields.