A class of singularly perturbed boundary value problems with unstable spectrum of the limit operator (Q5951379)

For a class of singularly perturbed multipoint linear boundary value problems with an unstable spectrum of the limit operator, conditions for the existence of a unique solution bounded as \(\varepsilon\to 0\) are stated. The boundary value problems under consideration in the paper are described by \[ \varepsilon^m dy/dx= \delta(x, \varepsilon) A(x)y,\quad\sum^n_{k=1} F_k y(x_,\varepsilon)= \alpha,\;a= x_1< x_2<\cdots< x_n= b, \] with \(\delta(x,\varepsilon)= \delta_0(x)+ \varepsilon\), \(\delta_0(x)\equiv 0\), \(x\in\Omega\subset [a,b]\). The problems considered in the present paper are, in a sense, similar to the well-known class of problems with a turning point. For two-point boundary value problems of this class, a new type of asymptotic expansions of solutions is constructed. Also, it is indicated the possibility to construct (by an iterative method) another class of asymptotic representations of solutions to two- and multipoint singularly perturbed boundary value problems under less restrictive conditions (than in the first algorithm) imposed on the spectrum of the limit operator.