## 3 Results
### Existence and Stability Results  
**Theorem 1**. : _Assuming Propositions 1-4 and 5 (the latter for quasi-monotonicities of the NP and PN types), for sufficiently small \(\varepsilon>0\) values, there exists a solution \((u_{\varepsilon}(\mathbf{x}),v_{\varepsilon}(\mathbf{x}))\) to Problem (3) in the sense of Definition 2, for which the pair of functions \((U_{n}(\mathbf{x},\varepsilon),V_{n}(\mathbf{x},\varepsilon))\) is an asymptotic approximation uniform in \(\overline{D}\) with an accuracy of \(\mathrm{O}\big{(}\varepsilon^{n+1}\big{)}\); in other words, we have_  
\[|U_{n}(\mathbf{x},\varepsilon)-u_{\varepsilon}(\mathbf{x})|+|V_{n}(\mathbf{x},\varepsilon )-v_{\varepsilon}(\mathbf{x})|\leq C\varepsilon^{n+1},\ \mathbf{x}\in\overline{D}, \tag{40}\]  
_where \(C\) is a positive constant independent on \(\varepsilon\)._  
Proof of Theorem 1.: The proof of this theorem is based on the following statement:  
**Lemma 1**. : _Let there be the pairs of functions \(\big{(}\overline{U}(\mathbf{x}),\overline{V}(\mathbf{x})\big{)}\) and \(\big{(}\underline{U}(\mathbf{x}),\underline{V}(\mathbf{x})\big{)}\), which are the upper and lower solutions in the sense of Definition 3, respectively. Then, there exists at least one solution \((u_{\varepsilon}(\mathbf{x}),v_{\varepsilon}(\mathbf{x}))\) to Problem (3) in the sense of Definition 2 for which the following inequalities hold:_  
\[\underline{U}(\mathbf{x})\leq u_{\varepsilon}(\mathbf{x})\leq\overline{U}(\mathbf{x}),\ \ \underline{V}(\mathbf{x})\leq v_{\varepsilon}(\mathbf{x})\leq\overline{V}(\mathbf{x}),\ \ \mathbf{x}\in\overline{D}. \tag{41}\]  
The proof of Lemma 1 can be carried out using the method of monotone iterations [14]. However, in [14] elliptic systems with continuous right-hand sides are dealt with. In the case of the system in Equation (3) with the right-hand sides of Equation (2), the proof given in [15] for the one-dimensional case is applicable, with reference to [5] regarding the smoothness of the solutions to linear problems for the terms of iterative sequences as well as reference to properties of the volume potential (see [24]) regarding the smoothness of the solution to Problem (3). From Lemma 1, given the upper and lower solutions defined by the Equations (27)-(30), it follows that Problem (3) has a solution \((u_{\varepsilon}(\mathbf{x}),v_{\varepsilon}(\mathbf{x}))\) for which the inequalities in Equation (41) are valid. From here, we have  
\[\underline{U_{n}}(\mathbf{x},\varepsilon)-U_{n-1}(\mathbf{x},\varepsilon) \leq u_{\varepsilon}(\mathbf{x})-U_{n-1}(\mathbf{x},\varepsilon)\leq\overline{U}_{n} (\mathbf{x},\varepsilon)-U_{n-1}(\mathbf{x},\varepsilon),\] \[\underline{V_{n}}(\mathbf{x},\varepsilon)-V_{n-1}(\mathbf{x},\varepsilon )\leq v_{\varepsilon}(\mathbf{x})-V_{n-1}(\mathbf{x},\varepsilon)\leq\overline{V}_{n} (\mathbf{x},\varepsilon)-V_{n-1}(\mathbf{x},\varepsilon);\ \ \mathbf{x}\in\overline{D}.\]  
The differences on the left and right sides of each double inequality have the order \(O(\varepsilon^{n})\) (see Equations (27)-(30)), and the same order is for the differences \(\overline{U}_{n}(\mathbf{x},\varepsilon)-\underline{U_{n}}(\mathbf{x},\varepsilon)\), \(\overline{V}_{n}(\mathbf{x},\varepsilon)-\underline{V_{n}}(\mathbf{x},\varepsilon)\). To obtain the estimate (Equation (40)) from these inequalities, we just have to replace the index \(n\) with \(n+1\). **Theorem 2**.