On a two-point boundary-value problem for a linear singularly perturbed system with degenerations (Q2906192)

Consider the linear boundary value problem NEWLINE\[NEWLINE\begin{gathered} \varepsilon^h B(t){dx\over dt}= A(t,\varepsilon) x+ f(t,\varepsilon),\quad 0< t< T,\\ Mx(0,\varepsilon)+ Nx(T,\varepsilon)= d(\varepsilon)\end{gathered}\tag{\(*\)}NEWLINE\]NEWLINE with \(x(t,\varepsilon)\in \mathbb{R}^n\), \(0<\varepsilon< \varepsilon_0\ll 1\), \(d(\varepsilon)\in \mathbb{R}^{n-1}\), \(f(t,\varepsilon)\in \mathbb{R}^n\); \(B(t)\), \(A(t,\varepsilon)\), \(M\) and \(N\) are matrices of corresponding type, where \(\det B(t)\equiv 0\). The author formulates conditions under which \((*)\) has a unique solution \(x(t,\varepsilon)\), additionally its asymptotic representation is given.