Asymptotics of solutions of a linear singularity perturbed system with degeneracy

Abstract

Asymptotic expansions with respect to a small parameter e are constructed for a fundamental system of solutions of a linear singularity perturbed system of equations of the form

$$\varepsilon ^h A(t,\varepsilon )\frac{{dx}}{{dt}} = B(t,\varepsilon )x$$

on a finite interval of variation of the independent variable. It is assumed here that the (n×n) -matricesA (t, ɛ) and B(t,ɛ) can be expanded in series in powers of$A(t,\varepsilon ) = \sum\limits_{k = 0}^\infty {A_k } (t)\varepsilon ^k , B(t,\varepsilon ) = \sum\limits_{k = 0}^\infty {B_k } (t)\varepsilon ^k ,$ where the matrix A₀(t)is degenerate on the given interval and the pencil of matrices B∩(t)−ΩA₀ (t) has several “finite” and “infinite” elementary divisors of the same multiplicity.