## 4 Main Results
### Result for Increasing Self-Mapping
 \tag{5}\]  
By (\(F_{2}\)), there exists \(S\in B(E,E)\) verifying \(\|S\|_{1}<1\) and  
\[\mathbf{d}_{CA}(\zeta_{\ell_{1}}^{\ell_{2}})\preceq_{\epsilon}\frac{1}{n-1}S( \mathbf{d}_{CA}(\zeta_{\ell_{0}}^{\ell_{1}})).\]  
By (2), we have \(g(\ell_{0})\mathcal{R}g(\ell_{1})\), that is \(\ell_{1}\mathcal{R}\ell_{2}\), so, putting \(\ell=\ell_{1}\) in (3), we have  
\[(I-S)\mathbf{d}_{CA}\big{(}\zeta_{\ell_{1}}^{g(\ell_{1})}\big{)}=(I-S)\mathbf{ d}_{CA}\big{(}\zeta_{\ell_{1}}^{\ell_{2}}\big{)}\preceq_{\epsilon}\mathbf{d}_{ CA}(\zeta_{\ell_{1}}^{\ell_{2}}).\]  
This implies that  
\[\digamma\Big{(}\ \mathbf{d}_{CA}\big{(}\zeta_{g(\ell_{1})}^{g(\ell_{2})} \big{)},\mathbf{d}_{CA}(\zeta_{\ell_{1}}^{\ell_{2}}),\mathbf{d}_{CA}\big{(} \zeta_{\ell_{1}}^{g(\ell_{1})}\big{)},\mathbf{d}_{CA}\big{(}\zeta_{\ell_{2}}^{ g(\ell_{2})}\big{)},\mathbf{d}_{CA}\big{(}\zeta_{\ell_{1}}^{g(\ell_{2})}\big{)}, \mathbf{d}_{CA}\big{(}\zeta_{\ell_{2}}^{g(\ell_{1})}\big{)}\ \ \Big{)}\preceq_{\epsilon}\mathbf{0}.\]  
By (d3), we have  
\[\mathbf{d}_{CA}(\zeta_{\ell_{1}}^{\ell_{2}})\preceq_{\epsilon}(n-1)\mathbf{d} _{CA}(\zeta_{\ell_{1}}^{\ell_{2}})+\mathbf{d}_{CA}(\zeta_{\ell_{2}}^{\ell_{3}}),\]  
and (\(F_{1}\)) implies  
\[\digamma\Big{(}\ \mathbf{d}_{CA}(\zeta_{\ell_{2}}^{\ell_{3}}),\mathbf{d}_{CA}( \zeta_{\ell_{1}}^{\ell_{2}}),\mathbf{d}_{CA}(\zeta_{\ell_{1}}^{\ell_{2}}), \mathbf{d}_{CA}(\zeta_{\ell_{2}}^{\ell_{3}}),(n-1)\mathbf{d}_{CA}(\zeta_{\ell_ {1}}^{\ell_{2}})+\mathbf{d}_{CA}(\zeta_{\ell_{2}}^{\ell_{3}}),\mathbf{0}\ \ \Big{)}\preceq_{\epsilon}\mathbf{0}. \tag{6}\]  
By (\(F_{2}\)), we have  
\[\mathbf{d}_{CA}(\zeta_{\ell_{2}}^{\ell_{3}})\preceq_{\epsilon}\frac{1}{n-1}S( \mathbf{d}_{CA}(\zeta_{\ell_{1}}^{\ell_{2}}))\preceq_{\epsilon}\frac{1}{(n-1 )^{2}}S^{2}(\mathbf{d}_{CA}(\zeta_{\ell_{0}}^{\ell_{1}})).\]  
The same reasoning leads to a sequence \(\{\ell_{n}\}\) such that  
\(\ell_{n}\mathcal{R}\ell_{n+1}\) and \(\ell_{n+1}=g(\ell_{n})\). Furthermore, observing that  
\[(I-S)\mathbf{d}_{CA}\Big{(}\zeta_{\ell_{n-1}}^{g(\ell_{n-1})}\Big{)}=(I-S) \mathbf{d}_{CA}(\zeta_{\ell_{n-1}}^{\ell_{n}})\preceq_{\epsilon}\mathbf{d}_{ CA}(\zeta_{\ell_{n-1}}^{\ell_{n}}),\]  
we have  
\[\mathbf{d}_{CA}(\zeta_{\ell_{n}}^{\ell_{n+1}})\preceq_{\epsilon}\frac{1}{n-1}S (\mathbf{d}_{CA}(\zeta_{\ell_{n-1}}^{\ell_{n}}))\preceq_{\epsilon}\frac{1}{(n-1 )^{2}}S^{2}(\mathbf{d}_{CA}(\zeta_{\ell_{n-2}}^{\ell_{n-1}}))\preceq_{\epsilon }\cdots\preceq_{\epsilon}\frac{1}{(n-1)^{n}}S^{n}(\mathbf{d}_{CA}(\zeta_{\ell _{0}}^{\ell_{1}})).\]  
For \(m,n\in\mathbb{N}\) such that \(m>n\), we note that  
\[\mathbf{d}_{CA}(\zeta_{\ell_{n}}^{\ell_{m}}) \preceq_{\epsilon} (n-1)[\mathbf{d}_{CA}(\zeta_{\ell_{n}}^{\ell_{n+1}})+\mathbf{d}_ {CA}(\zeta_{\ell_{n+1}}^{\ell_{n+2}})+\cdots+\mathbf{d}_{CA}(\zeta_{\ell_{m-1}}^ {\ell_{m}})]\] \[\preceq_{\epsilon} \frac{1}{(n-1)^{n-1}}[S^{n}+\frac{1}{(n-1)}S^{n+1}+......+\frac{ 1}{(n-1)^{m-n-2}}S^{n-1}+..](\mathbf{d}_{CA}(\zeta_{\ell_{0}}^{\ell_{1}}))\] \[(I-S)\mathbf{d}_{CA}\big{(}\zeta_{\ell_{1}}^{g(\ell_{1})}\big{)}=(I-S)\mathbf{ d}_{CA}\big{(}\zeta_{\ell_{1}}^{\ell_{2}}\big{)}\preceq_{\epsilon}\mathbf{d}_{ CA}(\zeta_{\ell_{1}}^{\ell_{2}}).\]0  
We deduce the following information from the last inequality. \[(I-S)\mathbf{d}_{CA}\big{(}\zeta_{\ell_{1}}^{g(\ell_{1})}\big{)}=(I-S)\mathbf{ d}_{CA}\big{(}\zeta_{\ell_{1}}^{\ell_{2}}\big{)}\preceq_{\epsilon}\mathbf{d}_{ CA}(\zeta_{\ell_{1}}^{\ell_{2}}).\]1  
By Remark 1 and the assumption that \(\|S\|_{1}<1\), we infer (using the facts given in [18; 19]) that \(\lim_{n\to\infty}\mathbf{d}_{CA}(\zeta_{\ell_{n}}^{\ell_{m}})=\mathbf{0}\), or equivalently, \(\mathbf{d}_{CA}(\zeta_{\ell_{n}}^{\ell_{m}})\ll_{\epsilon}C_{A}\) for some \(\mathbf{0}\ll_{\epsilon}C_{A}\). By using the technique appeared in [19], we can obtain that \(\mathbf{d}_{CA}(\zeta_{\ell_{n}}^{\ell_{m}})\ll_{\epsilon}C_{A}\). Thus, \(\{\ell_{n}\}\) is a Cauchysequence in \(P\).