The boundary-value problem for the linear degenerated singularly perturbed system of differential equations of the second order (Q2414355)

Consider in \(\mathbb{R}^n\) the singularly perturbed and degenerate linear differential equation \[\varepsilon^{2h} A(t)\frac{d^2x}{dt^2}+ \varepsilon^h B(t,\varepsilon)\frac{dx}{dt}+ C(t,\varepsilon)x= f(t,x),\ 0<t<\tau,\] where \(\varepsilon\) is a small positive parameter, \(h\in\mathbb{N}\), \(\det A(t)\equiv 0\) for \(0\le t\le T\) together with the boundary conditions \[\begin{aligned} M_1x(0,\varepsilon)+ N_1x(T,\varepsilon) &= d_1(\varepsilon),\\ M_2\frac{dx}{dt} (0,\varepsilon)+ N_2\frac{dx}{dt} (T,\varepsilon) &= d_2(\varepsilon).\end{aligned}\] The author presents conditions on the matrices \(B\) and \(C\) and on the function \(f\) such that the boundary value problem has a unique solution. Its asymptotic expansion in \(\varepsilon\) is constructed.