## 3 Results
#### 3.2.2 Proof of the Inequalities in Conditions (A\({}_{1}\)-A\({}_{4}\))  
In the previous section, we built the functions \(\overline{U}_{n}(\mathbf{x},\varepsilon)\), \(\underline{U}_{\underline{n}}(\mathbf{x},\varepsilon)\), \(\overline{V}_{n}(\mathbf{x},\varepsilon)\), \(\underline{V}_{\underline{n}}(\mathbf{x},\varepsilon)\). Now, we have to prove that these functions satisfy Conditions (A\({}_{1}\)-A\({}_{4}\)) for sufficiently small \(\varepsilon\) values and thus are indeed the upper and lower solutions of Problem (3). We will carry this out using the example of the upper solution. The functions \(\alpha^{(\mp)}(\mathbf{x})\), \(\beta^{(\mp)}(\mathbf{x})\), \(q^{(\mp)}u(\tau,\mathbf{y})\), \(q^{(\mp)}v(\tau,\mathbf{y})\) and \(m^{(\mp)}u(\sigma,\mathbf{y})\) are defined so that they take strictly positive values in their domains. This ensures that Condition (A\({}_{1}\)) is satisfied. We have built the asymptotic approximation in such a way that for any type of quasi-monotonicity, these inequalities are true (see the equalities in Equations (20)-(23)), as well as Equations (31), (32), (37) and (39):  
\[L_{u,\varepsilon}(\overline{U}_{n},v) \leq-\varepsilon^{n}\Big{(}A+d^{u}e^{-s^{u}|\tau|}+D^{u}e^{-K^{u} |\sigma|}\Big{)}+O\Big{(}\varepsilon^{n+1}\Big{)},\] \[L_{u,\varepsilon}(\underline{U}_{\underline{n}},v) \geq\quad\varepsilon^{n}\Big{(}A+d^{u}e^{-s^{u}|\tau|}+D^{u}e^{-K^ {u}|\sigma|}\Big{)}+O\Big{(}\varepsilon^{n+1}\Big{)};\] \[L_{v,\varepsilon}(u,\overline{V}_{n}) \leq-\varepsilon^{n}B+O\Big{(}\varepsilon^{n+1}\Big{)},\quad L_{ v,\varepsilon}(u,\underline{V}_{\underline{n}})\geq\varepsilon^{n}B+O\Big{(} \varepsilon^{n+1}\Big{)}.\]  
This provides fulfillment of the inequalities in Equation (26) for sufficiently small \(\varepsilon\) values and positive constants \(A\), \(B\), \(d^{u}\) and \(D^{u}\). Condition (A\({}_{3}\)) is proven in the same way as in [23]. The differences in the derivatives of the \(U\) and \(V\) components of the upper solution in Condition (A\({}_{4}\)) can be expressed as follows:  
\[\frac{\partial\overline{U}_{n}^{(-)}}{\partial n_{\gamma}^{-}}( \mathbf{y},\varepsilon)-\frac{\partial\overline{U}_{n}^{(+)}}{\partial n_{ \gamma}^{+}}(\mathbf{y},\varepsilon) =\] \[\frac{\partial\overline{V}_{n}^{(-)}}{\partial n_{\gamma}^{-}}( \mathbf{y},\varepsilon)-\frac{\partial\overline{V}_{n}^{(+)}}{\partial n_{ \gamma}^{+}}(\mathbf{y},\varepsilon) = \varepsilon^{n-1}\Bigg{(}\frac{\partial q^{(-)}v}{\partial\tau}( 0,\mathbf{y})-\frac{\partial q^{(+)}v}{\partial\tau}(0,\mathbf{y})\Bigg{)}+O (\varepsilon^{n}),\ \ \mathbf{y}\in\gamma.\]  
Using the explicit form (Equation (38)) of the functions \(m^{(\mp)}u(\sigma,\mathbf{y})\), we can obtain the expression for the jump of the derivatives of the upper solution's \(U\) component on the curve \(\gamma\) that is valid for each type of quasi-monotonicity:  
\[\frac{\partial\overline{U}_{n}^{(-)}}{\partial n_{\gamma}^{-}}( \mathbf{y},\varepsilon)-\frac{\partial\overline{U}_{n}^{(+)}}{\partial n_{ \gamma}^{+}}(\mathbf{y},\varepsilon)=\varepsilon^{n-2}\bigg{(}\delta^{u}\frac {\partial H_{0}^{u}}{\partial p}(p_{0}(\mathbf{y}),q_{0}(\mathbf{y}),\mathbf{y} )+Ju(\mathbf{y})\bigg{)}+O\Big{(}\varepsilon^{n-1}\Big{)},\ \ \mathbf{y}\in\gamma,\]  
where the function \(Ju(\mathbf{y})\) is bounded and independent on \(\delta^{u}\). Here, we use the expression in Equation (10) (the second line). Due to Proposition 4, it is sufficient to choose the value \(\delta^{u}\) to be sufficiently large so that the expression on the right-hand side is positive, and thus Condition (A\({}_{4}\)) for the upper solution's \(U\) component is satisfied.