An investigation of a linear Hamiltonian singularly perturbed boundary value problem (Q2387934)

The author considers the problem of the solvability and the dependence on a parameter of a linear singularly perturbed boundary value problem \[ \dot z= (N(\varepsilon) z+ \ell(\varepsilon)\alpha(\varepsilon))/\gamma(\varepsilon), \] \[ (E,0) z(\tau_1)= x_*,\quad (0,E)z(\tau_2)= \varphi_*, \] where \(x_*,\varphi_*\in\mathbb{R}^n\) are given vectors, \(0,E\in\mathbb{R}^{n\times n}\) are the zero and unit matrices, \[ N(\varepsilon)= \gamma(\varepsilon)\cdot A(\varepsilon)- \ell(\varepsilon) p'(\varepsilon), \] \[ p(\varepsilon)= {d(\varepsilon)\choose q(\varepsilon)},\quad \ell(\varepsilon)= {q(\varepsilon)\choose -d(\varepsilon)};\quad{\mathcal A}(\varepsilon)= {A(\varepsilon)\;M(\varepsilon)\choose D(\varepsilon)- A'(\varepsilon)}, \] where \(A(\varepsilon)\), \(M(\varepsilon)\), \(D(\varepsilon)\in \mathbb{R}^{n\times n}\), \(d(\varepsilon)\), \(q(\varepsilon)\), \(\gamma(\varepsilon)\), \(\alpha(\varepsilon)\in \mathbb{R}^n\) are given sufficiently smooth functions of the parameter \(\varepsilon\in (0,\delta]\), and \(\delta> 0\) is a sufficiently small number.