Regularization and asymptotic solutions of singularly perturbed problems with point singularities in the spectrum of the limiting operator (Q1820914)

In the asymptotic analysis of the solutions to singularly perturbed problems of the form \(\epsilon y'=A(x)y+h(x),\) \(y(0,\epsilon)=y^ 0,\) where \(y=\{y_ 1,...,y_ n\}\), \(h(x)=\{h_ 1,...,h_ n\}\) is a known vector-function, \(\epsilon >0\) is a small parameter, and \(x\in [0,a]\), a special role is played by the character of the spectrum \(\{\lambda_ j(x)\), \(j=1,...,n\}\) of the limiting operator A(x). The present paper is concerned with the simplest case of instability of the spectrum of the operator A(x).