## 4 Asymptotic Solution to the Boundary-Value Problem (17)
### Justification of the First-Order Asymptotic Solution to Problems (26) and (27)  
Using the results of Sections 4.2.1-4.2.3 and 4.2.5-4.2.9, we can rewrite the components of the first-order asymptotic solution to Problems (26) and (27) (see Equation (28)) as follows:  
\[z_{1}^{1}(t,\varepsilon)=z_{1,0}(t)+\varepsilon\big{(}z_{1,1}(t)+ z_{1,1}^{0}(t/\varepsilon)\big{)},\] \[z_{2}^{1}(t,\varepsilon)=z_{2,0}(t)+z_{2,0}^{0}(t/\varepsilon)+ \varepsilon\big{(}z_{2,1}(t)+z_{2,1}^{0}(t/\  
Like the matrix-valued function \(\Phi(t,\varepsilon)\), the matrix-valued function \(\Psi(t,\varepsilon)\) is also continuous with respect to \((t,\varepsilon)\in[0,t_{f}]\times(-\infty,+\infty)\). Moreover, for any pair \((t,\varepsilon)\in[0,t_{f}]\times(-\infty,+\infty)\), both matrices \(\Phi(t,\varepsilon)\) and \(\Psi(t,\varepsilon)\) have the same set of eigenvalues \(\mu_{i}(t,\varepsilon)\), \((i=1,\ldots,r,r+1,\ldots,2r)\).. Using Equations (50), (51), and (104), we can represent the matrix \(\Psi(t,\varepsilon)\) in the following block form:  
\[\Psi(t,\varepsilon)=\left( \begin{array}{cc}\Psi_{1}(t,\varepsilon)&\Psi_{2} (t,\varepsilon)\\ \Psi_{3}(t,\varepsilon)&\Psi_{4}(t,\varepsilon)\end{array} \right),\quad(t, \varepsilon)\in[0,t_{f}]\times(-\infty,+\infty),\]  
\[\Psi_{1}(t,\varepsilon)=D_{u,2}^{1/2}(t)+0.5\varepsilon\big{(}A_{4}(t)+D_{u,2 }^{-1/2}(t)A_{4}^{\top}(t)D_{u,2}^{1/2}(t)\big{)},\]  
\[\Psi_{2}(t,\varepsilon)=0.25\varepsilon\big{(}A_{4}(t)D_{u,2}^{-1/2}(t)-D_{u,2 }^{-1/2}(t)A_{4}^{\top}(t)\big{)},\]  
\[\Psi_{3}(t,\varepsilon)=\varepsilon\big{(}D_{u,2}^{1/2}(t)A_{4}(t)-A_{4}^{ \top}(t)D_{u,2}^{1/2}(t)\big{)},\]  
\[\Psi_{4}(t,\varepsilon)=-D_{u,2}^{1/2}(t)+0.5\varepsilon\big{(}D_{u,2}^{1/2}(t )A_{4}(t)D_{u,2}^{-1/2}(t)+A_{4}^{\top}(t)\big{)}. \tag{108}\]  
Setting \(\varepsilon=0\) in (108), we obtain  
\[\Psi(t,0)=\left( \begin{array}{cc}D_{u,2}^{1/2}(t)&0\\ 0&-D_{u,2}^{1/2}(t)\end{array} \right),\quad t\in[0,t_{f}]. \tag{109}\]  
For any \(t\in[0,t_{f}]\), the eigenvalues of \(\Psi(t,0)\) are \(\
u_{i}(t)\), \((i=1,\ldots,r)\) and \(-\
u_{i}(t)\), \((i=1,\ldots,r)\), where \(\
u_{i}(t)\), \((i=1,\ldots,r)\) are the eigenvalues of the matrix \(D_{u,2}^{1/2}(t)\). The latter matrix is symmetric and positive definite for all \(t\in[0,t_{f}]\). Therefore, \(\
u_{i}(t)\), \((i=1,\ldots,r)\) are real and positive for all \(t\in[0,t_{f}]\). Moreover, since the matrix-valued function \(D_{u,2}^{1/2}(t)\) is continuous in the interval \([0,t_{f}]\), then, due to the results of [70], the functions \(\
u_{i}(t)\), \((i=1,\ldots,r)\) also are continuous for \(t\in[0,t_{f}]\). This feature, along with the aforementioned positiveness of these functions directly yields the existence of a positive number \(\alpha\) such that the following inequality is satisfied:  
\[\
u_{i}(t)\geq 2\alpha,\quad t\in[0,t_{f}],\quad i=1,\ldots,r. \tag{110}\]  
Furthermore, since, for any \(t\in[0,t_{f}]\), the matrix-valued function \(\Psi(t,\varepsilon)\) is continuous with respect to \(\varepsilon\in(-\infty,+\infty)\), then  
\[\mu_{i}(t,0)=\
u_{i}(t),\quad\mu_{i+r}(t,0)=-\
u_{i}(t),\quad t\in[0,t_{f}], \quad i=1,\ldots,r,\]  
meaning, along with (110), that  
\[\mu_{i}(t,0)\geq 2\alpha,\quad\mu_{i+r}(t,0)\leq-2\alpha,\quad t\in[0,t_{f}], \quad i=1,\ldots,r. \tag{111}\]  
The matrix-valued function \(\Psi(t,\varepsilon)\), being continuous with respect to \((t,\varepsilon)\in[0,t_{f}]\times(-\infty,+\infty)\), is continuous in \(\varepsilon\in(-\infty,+\infty)\) uniformly with respect to \(t\in[0,t_{f}]\). Therefore, by virtue of [70], the eigenvalues \(\mu_{i}(t,\varepsilon)\), \((i=1,\ldots,r,r+1,\ldots,2r)\) are continuous in \(\varepsilon\in(-\infty,+\infty)\) uniformly with respect to \(t\in[0,t_{f}]\). This observation, along with the inequalities in (111), yields the existence of a number \(\varepsilon>0\) such that, for all \(t\in[0,t_{f}]\) and all \(\varepsilon\in(0,\varepsilon]\), Inequalities (105) and (106) are satisfied.