## 5 Examples and Consequences  
We elaborate our results through the following examples. **Example 5**. : _Let \(E=(\mathbb{R}^{\cdot}\|\cdot\|)\) as a real Banach space, along with a cone \(\mathcal{C}=\{\ell\in\mathbb{R}:\ell\geq 0\}\) in \(E\). Define the cone \(\boldsymbol{A}\)-metric by \(\boldsymbol{d}_{CA}(\ell_{1},\ell_{2},.....,\ell_{n})=\sum_{i=1}^{n}\sum_{i<j} |\ell_{i}-\ell_{j}|\). Let \(P=\{0,\frac{1}{6}\}\subset E\) and \(g:P\to P\) be given by_  
\[g(0  
_Then, \((I-S)(\boldsymbol{A}(\zeta_{\ell}^{g(\ell)}))\preceq_{e}\boldsymbol{A}(\zeta_{ \ell}^{\kappa})\) implies_  
\[\digamma\left(\boldsymbol{d}_{CA}(\zeta_{g(\ell)}^{g(\kappa)}),\boldsymbol{d}_{ CA}(\zeta_{\ell}^{\kappa})\right),\boldsymbol{d}_{CA}(\zeta_{\ell}^{g(\ell)}, \boldsymbol{d}_{CA}(\zeta_{\kappa}^{g(\kappa)}),\boldsymbol{d}_{CA}(\zeta_{ \ell}^{g(\kappa)}),\boldsymbol{d}_{CA}(\zeta_{\kappa}^{g(\ell)})\ \ \right)\preceq_{e} \boldsymbol{0}.\]  
_Use of Theorem 2 leads to have a fixed point of \(g\), such that \(g(\frac{1}{6})=\frac{1}{6}\)._  
**Corollary 1**. : _Let \((P,\boldsymbol{d}_{CA})\) be a complete cone \(\boldsymbol{A}\)-metric space. Let \(S\in B(E,E)\) with \(\left\|S\right\|_{1}<1\) and \(I:E\to E\) be an identity operator. If the mapping \(g:P\to P\), for all comparable elements \(\ell,\kappa\in P\), and \(\digamma\in\mathcal{H}\) satisfies the following condition_  
\[(I-S)(\boldsymbol{d}_{CA}(\zeta_{\ell}^{g(\ell)}))\preceq_{e}\boldsymbol{d}_{ CA}(\zeta_{\ell}^{\kappa})\Rightarrow\boldsymbol{d}_{CA}(\zeta_{g(\ell)}^{g( \kappa)})\preceq_{e}\mathcal{S}(\boldsymbol{d}_{CA}(\zeta_{\ell}^{\kappa})),\]  
_and_  
1. _there exists_ \(\ell_{0}\in P\) _such that either_ \(\ell_{0}\mathcal{R}g(\ell_{0})\) _or_ \(g(\ell_{0})\mathcal{R}\ell_{0}\)_;_
2. _for all_ \(\ell,\kappa\in P\)_,_ \(\ell\mathcal{R}\kappa\) _we have_ \(g(\ell)\mathcal{R}g(\kappa)\) _or_ \(g(\kappa)\mathcal{R}g(\ell)\)_; and_
3. _for a sequence_ \(\{\ell_{n}\}\) _with_ \(\ell_{n}\to c^{*}\) _whose all sequential terms are comparable, we have_ \(\ell_{n}\mathcal{R}c^{*}\) _for all_ \(n\in\mathbb{N}\)_._  
_Then, there exists_ \(c^{*}\in P\) _such that_ \(c^{*}=g(c^{*})\)_._  
Proof. : Define \(\digamma:E^{6}\to E\) by \(\digamma\left(\ell_{1},\ell_{2},\ell_{3},\ell_{4},\ell_{5},\ell_{6}\right)= \omega\ell_{1}-\ell_{2};\omega>1\), and the operator \(S:E\to E\) by \(\mathcal{S}(\ell)=h\ell\) for all \(\ell\in E;h<1\), so that \(S\in B(E,E)\). Now, by Theorem 4, there exists \(c^{*}\in P\) such that \(c^{*}=g(c^{*})\). **Corollary 2**. : _Let \((P,\boldsymbol{d}_{CA})\) be a complete cone \(\boldsymbol{A}\)-metric space. Let \(S\in B(E,E)\) with \(\left\|S\right\|_{1}<1\) and \(I:E\to E\) be an identity operator. If the mapping \(g:P\to P\), for all comparable elements \(\ell,\kappa\in P\), and \(\digamma\in\mathcal{H}\) satisfies the following condition_  
\[(I-S)(\boldsymbol{d}_{CA}(\zeta_{\ell}^{g(\ell)}))\preceq_{e}\boldsymbol{d}_{ CA}(\zeta_{\ell}^{\kappa})\Rightarrow\boldsymbol{d}_{CA}(\zeta_{g(\ell)}^{g( \kappa)})\preceq_{e}\frac{1}{n-1}\mathcal{S}(\boldsymbol{d}_{CA}(\zeta_{\ell }^{\kappa})),\]  
_and_  
1. _there exists_ \(\ell_{0}\in P\) _verifying either_ \(\ell_{0}\mathcal{R}g(\ell_{0})\) _or_ \(g(\ell_{0})\mathcal{R}\ell_{0}\)_;_
2. _for all_ \(\ell,\kappa\in P\)_,_ \(\ell\mathcal{R}\kappa\) _implies either_ \(g(\ell)\mathcal{R}g(\kappa)\) _or_ \(g(\kappa)\mathcal{R}g(\ell)\)_; and_
3. _for a sequence_ \(\{\ell_{n}\}\) _converging to_ \(c^{*}\) _and_ \(\ell_{n}\mathcal{R}\ell_{n-1}\)_, we have_ \(\ell_{n}\mathcal{R}c^{*}\) _for all_ \(n\in\mathbb{N}\)_._  
_Then, there exists a vector \(c^{*}\in P\) such that \(c^{*}=g(c^{*})\)._  
Proof. : Taking \(S:E\to E\) such that \(S(v)=v\), \(\forall\ v\in E\) and following the proof of Corollary 1, we obtain the result. **Example 6**. : _Let \(E=(\mathbb{R}^{3},\|\cdot\|)\) with \(\|\ell\|=\max\{(|\ell_{1}|,|\ell_{2}|,|\ell_{3}|)\}\), then \((E,\|.\|)\) is a real Banach space. Take a cone in \(E\) as \(\mathcal{C}=\{(\ell,\bar{\zeta},\
u)\in\mathbb{R}^{3}:\ell,\bar{\zeta},\
u\geq 0\}\).