## 2 Basic Notions  
**Definition 1**. : _A binary relation \(\mathcal{R}\) over a set \(X\
eq\emptyset\) defines a partial order if it satisfies the following axioms:_  
1. \(\mathcal{R}\) _is reflexive._
2. \(\mathcal{R}\) _is antisymmetric._
3. \(\mathcal{R}\) _is transitive._  
A set with partial order \(\mathcal{R}\) is known as a partially ordered set denoted by \((X,\mathcal{R})\). Let \((E,\|.\|)\) be a real Banach space, and \(\mathcal{C}\subseteq E\) satisfies the following axioms:  
1. \(\mathcal{C}\) is non-empty closed set such that \(\mathcal{C}\
eq\{\emptyset\}\). 2. \(\forall\ \ell_{1},\ell_{2}\in\mathbb{R}\) such that \(\ell_{1},\ell_{2}\geq 0\) along with \(\theta_{1},\theta_{2}\in\mathcal{C}\), we have \(\ell_{1}\theta_{1}+\ell_{2}\theta_{2}\in\mathcal{C}\). 3. \(\mathcal{C}\cap(-C)=\{\mathbf{0}\}\). Then, \(\mathcal{C}\) defines a cone in \(E\). Given \(\mathcal{C}\subset E\), we define the partial order \(\preceq_{\epsilon}\) on \(\mathcal{C}\) as follows:  
\[\eta\preceq_{\epsilon}\varrho\Leftrightarrow\varrho-\eta\in\mathcal{C}\text{ for all }\eta,\zeta\in E.\]  
Note that \(\eta\prec_{\epsilon}\zeta\) represents \(\eta\preceq_{\epsilon}\zeta\) but \(\eta\
eq\zeta\) and \(\eta\ll\zeta\) shows that \(\zeta-\eta\in\mathcal{C}^{\circ}\) (the interior of \(\mathcal{C}\)). **Definition 2** ([18]). : _The cone \(\aleph\subseteq E\) is called normal if, for all \(\chi,\zeta\in\aleph\), there exists a number \(\mathcal{K}>0\) so that,_  
\[0\preceq_{\epsilon}\chi\preceq_{\epsilon}\xi\Rightarrow\|\chi\|\leq\mathcal{ K}\|_{\zeta}\|.\]  
Throughout this paper, we use the notation \(\Re\) as a partial order in any ordinary set \(P\) and \(\preceq_{\epsilon}\) as a partial order in \(\mathcal{C}\subseteq E\). If \(P\subseteq E\), then \(\Re\) and \(\preceq_{\epsilon}\) would be considered as identical. **Definition 3** ([18]). : _Let \(P\
eq\emptyset\), and if the mapping \(d_{c}:P\times P\mapsto E\) satisfies the following axioms:_  
1. \(\mathbf{0}\preceq_{\epsilon}d_{c}(\ell_{1},\ell_{2})\) _and_ \(d_{c}(\ell_{1},\ell_{2})=\mathbf{0}\) _if and only if_ \(\ell_{1}=\ell_{2}\)_;_
2. \(d_{c}(\ell_{1},\ell_{2})=d_{c}(\ell_{2},\ell_{1})\)_; and_
3. \(d_{c}(\ell_{1},\ell_{3})\preceq_{\epsilon}d_{c}(\ell_{1},\ell_{2})+d_{c}(\ell_{ 2},\ell_{3})\)_,_ \(\forall\ \ell_{1},\ell_{2},\ell_{3}\in P\)_,_  
_then \(P\) is called a cone metric, and the pair \((P,d_{c})\) represents a cone metric space._The generalizations of a metric space enriched the fixed-point theory and its applications in various contemporary fields. [PERSON] [27; 28; 29] (1963), coined the idea of a 2-metric space as a generalization of a metric space. **Definition 4** ([27]). : _Let \(P\
eq\emptyset\), and if a mapping \(d_{2}:P\times P\times P\mapsto[0,\infty)\) satisfies the following axioms:_  
1. _for distinct points_ \(\ell_{1},\ell_{2}\in P\)_,_ \(\exists\ \ell_{3}\in P\) _such that_ \(d_{2}(\ell_{1},\ell_{2},\ell_{3})\
eq\emptyset\)_;_
2. \(d_{2}(\ell_{1},\ell_{2},\ell_{3})=\mathbf{0}\)_, if any two elements of the set_ \(\{\ell_{1},\ell_{2},\ell_{3}\}\) _in_ \(P\) _are equal;_
3. \(d_{2}(\ell_{1},\ell_{2},\ell_{3})=d_{2}(\ell_{1},\ell_{3},\ell_{2})=d_{2}(\ell_ {2},\ell_{1},\ell_{3})\)__ \(=d_{2}(\ell_{3},\ell_{1},\ell_{2})=d_{2}(\ell_{2},\ell_{3},\ell_{1})\)_; and_
4. \(d_{2}(\ell_{1},\ell_{2},\ell_{3})\leq d_{2}(\ell_{1},\ell_{2},\ell)+d_{2}(\ell_ {1},\ell,\ell_{3})+d_{2}(\ell,\ell_{2},\ell_{3})\)_, for all_ \(\ell_{1},\ell_{2},\ell_{3},\ell\in P\)_,_  
_then \(d_{2}\) is called a 2-metric, and \((P,d_{2})\) is then called a 2-metric space._  
The distance \(d_{2}(\ell_{1},\ell_{2},\ell_{3})\) represents the area of a triangle formed by the points \(\ell_{1},\ell_{2},\ell_{3}\in P\). A 2-metric, in contrast to an ordinary metric, is not a continuous function of its variables. As a result, [PERSON] introduced the concept of a \(D\)-metric space in [30] as follows:  
**Definition 5** ([30]). : _Let \(P\
eq\emptyset\), and if a mapping \(D:P\times P\times P\mapsto[0,\infty)\) satisfies the following axioms:_  
1.