## 4 Main Results
### Result for Monotone Self-Mapping  
The following result encapsulates the statements of Theorems 2 and 3. **Theorem 4**. : _Let \((P,\mathbf{d}_{cA})\) be a complete cone \(\mathbf{A}\)-metric space and \(\mathcal{C}\subset E\) be a cone. Let \(S\in B(E,E)\) so that \(\left\|S\right\|_{1}<1\), and \(1:E\to E\) is an identity operator. If the mapping \(g:P\to P\), for all comparable elements \(\ell,\kappa\in P\), and \(\Gamma\in\mathcal{H}\) satisfies the following condition_  
\[(I-S)\big{(}\mathbf{d}_{cA}(\zeta_{\ell}^{g(\ell)})\preceq_{e}\mathbf{d}_{cA }(\zeta_{\ell}^{\kappa})\text{ implies}\]  
\[\Gamma\left( \begin{array}{c}\mathbf{d}_{cA}(\zeta_{g(\ell)}^{g(\kappa)}), \mathbf{d}_{cA}(\zeta_{\ell}^{\kappa}),\mathbf{d}_{cA}(\zeta_{\ell}^{g(\ell)}), \mathbf{d}_{cA}(\zeta_{\ell}^{g(\kappa)}),\mathbf{d}_{cA}(\zeta_{\ell}^{g( \kappa)}),\mathbf{d}_{cA}(\zeta_{\ell}^{g(\ell)})\end{array} \right)\preceq_{e} \mathbf{0} \tag{11}\]  
_and_  
1. _there exists_ \(\ell_{0}\in P\) _such that_ \(\ell_{0}\mathcal{R}g(\ell_{0})\) _or_ \(g(\ell_{0})\mathcal{R}\ell_{0}\)_;_
2. _the mapping_ \(g\) _is monotone; and_
3. _for a sequence_ \(\{\ell_{n}\}\) _with all comparable sequential terms such that_ \(\ell_{n}\to c^{*}\)_, we have_ \(\ell_{n}\mathcal{R}c^{*}\)__\(\forall\ n\in\mathbb{N}\)_._  
_Then, there exists \(c^{*}\in P\) such that \(c^{*}=g(c^{*})\)._  
Proof. : Proceeding with an initial guess \(\ell_{0}\in P\), we construct a sequence \(\{\ell_{n}\}\) so that \(\ell_{n}=g(\ell_{n-1})\)\(\forall\ n\in\mathbb{N}\). By assumption (1), we have \(\ell_{0}\mathcal{R}g(\ell_{0})=\ell_{1}\) and by (2), \(\ell_{2}\mathcal{R}\ell_{1}\). Using (11), we obtain  
\[(I-S)\big{(}\mathbf{d}_{cA}(\zeta_{g(\ell_{0})}^{\ell_{0}})\big{)}=(I-S)\big{(} \mathbf{d}_{cA}(\zeta_{\ell_{1}}^{\ell_{0}})\big{)}\preceq_{e}\mathbf{d}_{cA }(\zeta_{\ell_{1}}^{\ell_{0}})\text{ implies}\]  
\[\Gamma\left( \begin{array}{c}\mathbf{d}_{cA}(\zeta_{g(\ell_{0})}^{g(\ell_{1})} ),\mathbf{d}_{cA}(\zeta_{\ell_{0}}^{\ell_{1}}),\mathbf{d}_{cA}(\zeta_{\ell_{0 }}^{g(\ell_{0})}),\mathbf{d}_{cA}(\zeta_{\ell_{1}}^{g(\ell_{1})}),\mathbf{d}_{ cA}(\zeta_{\ell_{0}}^{g(\ell_{1})}),\mathbf{d}_{cA}(\zeta_{\ell_{1}}^{g(\ell_{0})}) \end{array} \right)\preceq_{e}\mathbf{0},\]  
that is,  
\[\Gamma\left( \begin{array}{c}\mathbf{d}_{cA}(\zeta_{\ell_{1}}^{\ell_{2}}), \mathbf{d}_{cA}(\zeta_{\ell_{0}}^{\ell_{1}}),\mathbf{d}_{cA}(\zeta_{\ell_{0}}^ {\ell_{1}}),\mathbf{d}_{cA}(\zeta_{\ell_{1}}^{\ell_{2}}),\mathbf{d}_{cA}(\zeta _{\ell_{0}}^{\ell_{2}}),\mathbf{d}_{cA}(\zeta_{\ell_{1}}^{\ell_{1}})\end{array}  \right)\preceq_{e}\mathbf{0}. \tag{12}\]  
By (d3) and \((F_{1})\),  
\[\Gamma\left( \begin{array}{c}\mathbf{d}_{cA}(\zeta_{\ell_{1}}^{\ell_{2}}), \mathbf{d}_{cA}(\zeta_{\ell_{0}}^{\ell_{1}}),\mathbf{d}_{cA}(\zeta_{\ell_{0}}^ {\ell_{1}}),\mathbf{d}_{cA}(\zeta_{\ell_{1}}^{\ell_{2}}),(n-1)\mathbf{d}_{cA}( \zeta_{\ell_{0}}^{\ell_{1}})+\mathbf{d}_{cA}(\zeta_{\ell_{1}}^{\ell_{2}}),0) \end{array} \right)\preceq_{e}\mathbf{0}. \tag{13}\]  
Using \((F_{2})\), we have  
\[\mathbf{d}_{cA}(\zeta_{\ell_{1}}^{\ell_{2}})\preceq_{e}\frac{1}{n-1}S(\mathbf{ d}_{cA}(\zeta_{\ell_{0}}^{\ell_{1}})).\]  
Since \(\ell_{2}\mathcal{R}\ell_{1}\), by (11), we have  
\[(I-S)\big{(}\mathbf{d}_{cA}(\zeta_{g(\ell_{0})}^{g(\ell_{1})})\big{)}=(I-S) \big{(}\mathbf{d}_{cA}(\zeta_{\ell_{1}}^{\ell_{2}})\big{)}\preceq_{e}\mathbf{d}_{ cA}(\zeta_{\ell_{1}}^{\ell_{2}})\text{ implies}\]  
\[\Gamma\left( \begin{array}{c}\mathbf{d}_{cA}(\zeta_{g(\ell_{1})}^{g(\ell_{2})}), \mathbf{d}_{cA}(\zeta_{\ell_{1}}^{\ell_{2}}),\mathbf{d}_{cA}(\zeta_{\ell_{1}}^ {g(\ell_{1})}),\mathbf{d}_{cA}(\zeta_{\ell_{2}}^{g(\ell_{2})}),\mathbf{d}_{cA}( \zeta_{\ell_{1}}^{g(\ell_{2})}),\mathbf{d}_{cA}(\zeta_{\ell_{2}}^{g(\ell_{1})}) \end{array}