Unstable and multiple elements of the spectrum in a system of singularly perturbed differential equations (Q1589384)

The following problem is considered \[ \varepsilon^2 w'= A(x)w+ h(x),\quad Bw(0,\varepsilon)+ Cw(x_0, \varepsilon)+ Dw(a,\varepsilon)= \varepsilon^{-1}\alpha+ w^0,\quad \varepsilon\to 0, \] on \(I= [0,a]\). It is required to construct a sufficiently smooth solution for all \(x\in I\) and sufficiently small \(\varepsilon: 0<\varepsilon< \varepsilon_0\ll 1\), also to prove of such solution that on some compact subinterval of \(I\) which does not contain \(x=0\), \(x= x_0\), \(x= a\), \(\lim_{\varepsilon\to 0_+} w(x,\varepsilon)= \omega(x)\), where \(\omega(x)= -A^{-1}(x) h(x)\). The problem is studied under the assumption that some of the eigenvalues of \(A(x)\) are in the right half-plane and nonsimple.