## 2 Basic Notions
\(D(\ell_{1},\ell_{2},\ell_{3})\geq\mathbf{0}\) _and_ \(D(\ell_{1},\ell_{2},\ell_{3})=\mathbf{0}\) _if and only if_ \(\ell_{1}=\ell_{2}=\ell_{3}\)_;_
2. \(D(\ell_{1},\ell_{2},\ell_{3})=D(\ell_{1},\ell_{3},\ell_{2})=D(\ell_{2},\ell_ {1},\ell_{3})=D(\ell_{3},\ell_{1},\ell_{2})=D(\ell_{2},\ell_{3},\ell_{1})=D( \ell_{3},\ell_{2},\ell_{1})\)_; and_
3. \(D(\ell_{1},\ell_{2},\ell_{3})\leq D(\ell_{1},\ell_{2},\ell)+D(\ell_{1},\ell_ {3})+D(\ell,\ell_{2},\ell_{3})\)_, for all_ \(\ell_{1},\ell_{2},\ell_{3},\ell\in P\)_,_  
_then \((P,D)\) denotes the \(D\)-metric space._  
[PERSON] and [PERSON] [31; 32] (2006) introduced \(G\)-metric space as a generalization of \(D\)-metric space. The deficiency of [PERSON]'s theory of \(D\)-metric is thus corrected. Moreover, [PERSON] et al. [33] (2007) observed that the condition \((D1)\) can be modified as follows:  
**Definition 6** ([33]). : _Let \(P\
eq\emptyset\), and if a mapping \(D^{*}:P\times P\times P\mapsto[0,\infty)\) satisfies the following axioms:_  
1. \(D^{*}(\ell_{1},\ell_{2},\ell_{3})=\mathbf{0}\) _if and only if_ \(\ell_{1}=\ell_{2}=\ell_{3}\)_;_
2. \(D^{*}(\ell_{1},\ell_{2},\ell_{2})\geq\mathbf{0}\)_;_
3. \(D^{*}(\ell_{1},\ell_{2},\ell_{3})=D^{*}(\ell_{1},\ell_{3},\ell_{2})=D^{*}(\ell _{2},\ell_{1},\ell_{3})=D^{*}(\ell_{3},\ell_{1},\ell_{2})=D^{*}(\ell_{2},\ell_ {3},\ell_{1})=D^{*}(\ell_{3},\ell_{2},\ell_{1})\)_; and_
4. \(D^{*}(\ell_{1},\ell_{2},\ell_{3})\leq D^{*}(\ell_{1},\ell_{2},\ell)+D^{*}( \ell,\ell_{3},\ell_{3})\)_,_  
_then \((P,D^{*})\) is called a \(D^{*}\)-metric space. \(D^{*}\)-metric space can be viewed as an improved version of \(D\)-metric space._  
Later, [PERSON] et al. [34] (2012) introduced another generalization of a metric space known as an \(S\)-metric space. [PERSON] et al. [35] (2015), introduced the idea of \(\mathbf{A}\)-metric spaces as follows:  
**Definition 7** ([35]). : _Let \(P\
eq\emptyset\), and if a mapping \(\boldsymbol{d}_{P}:P^{n}\mapsto[0,\infty)\) satisfies the following axioms: (for all \(\ell_{i},\ell\in P\), \(i=1,2,......n\)) \(\boldsymbol{d}_{P}(\ell_{1},\ell_{2},......,\ell_{n-1},\ell_{n})\geq 0\); (A2) \(\boldsymbol{d}_{P}(\ell_{1},\ell_{2},......,\ell_{n-1},\ell_{n})=0\) if and only if \(\ell_{1}=\ell_{2}=......=\ell_{n-1}=\ell_{n}\); and (A3) \(\boldsymbol{d}_{P}(\ell_{1},\ell_{2},......,\ell_{n-1},\ell_{n})\leq\boldsymbol{d }_{P}(\xi_{\ell_{1}}^{\ell})+\boldsymbol{d}_{P}(\xi_{\ell_{2}}^{\ell})+......+ \boldsymbol{d}_{P}(\xi_{\ell_{n-1}}^{\ell})+\boldsymbol{d}_{P}(\xi_{\ell_{n}}^{ \ell})\), then \((P,\boldsymbol{d}_{P})\) is called a \(\mathbf{A}\)-metric space._  
One can check that \(\mathbf{A}\)-metric space is an \(n\)-dimensional \(S\)-metric space. We can have a metric \(d\) and a \(S\)-metric by assuming \(n=2\) and \(n=3\), respectively, in an \(\mathbf{A}\)-metric space. **Example 1** ([35]). : _Let \(P=R\), and define \(\mathbf{d}_{P}:P^{n}\mapsto[0,\infty)\) by_  
\[\mathbf{d}_{P}(\ell_{1},\ell_{2},\cdots,\ell_{n-1},\ell_{n})=\sum_{\omega=1}^{n}\sum _{\omega<\ell}|\ell_{\omega}-\ell_{\ell}|.\]  
_Then, \((P,\mathbf{d}_{P})\) is an \(\mathbf{A}\)-metric space._  
**Example 2**. : _Let \(P=\mathbb{R}\), and define \(\mathbf{d}_{P}:P^{n}\mapsto[0,\infty)\) by_  
\[\mathbf{d}_{P}(\ell_{1},\ell_{2},......,\ell_{n-1},\ell_{n}) = |\ell_{n}+\ell_{n-1}+......+\ell_{2}-(n-1)\ell_{1}|\] \[+ |\ell_{n}+\ell_{n-1}+......+\ell_{3}-(n-2)\ell_{2}|\] \[\vdots\] \[+ |\ell_{n}+\ell_{n-1}+\ell_{n-2}-3\ell_{n-3}|\] \[+ |\ell_{n}+\ell_{n-1}-2\ell_{n-2}|\] \[+ |\ell_{n}-\ell_{n-1}|.\]  
_Then, \((P,\mathbf{d}_{A})\) is an \(\mathbf{A}\)-metric space. For more details on \(\mathbf{A}\)-metric spaces, see [35]._  
[PERSON] et al. [36] (2017) introduced the concept of a cone \(\mathbf{A}\)-metric space. They also established some new results in a cone \(\mathbf{A}\)-metric space. **Definition 8** ([36]). : _Let \(P\
eq\emptyset\), and if an operator \(\mathbf{d}_{CA}:P^{n}\mapsto E\) satisfies the following axioms: (for all \(\ell_{i},\ell\in P\), \(i=1,2,......n\))_  
1.