Asymptotical Formulas For Solutions Of Linear Differential Systems Of The First Order

Abstract

In this paper a system of differential equations y′ − A(·,λ)y = 0 is considered on the finite interval [a,b] where λ ∈ C, A(·, λ):= λ A1+ A0 +λ −1A−1(·,λ) and A1,A0, A− 1 are n × n matrix-functions. The main assumptions: A1 is absolutely continuous on the interval [a, b], A0 and A- 1(·,λ) are summable on the same interval when ¦λ¦ is sufficiently large; the roots φ1(x),…,φn(x) of the characteristic equation det (φ E — A1) = 0 are different for all x ∈ [a,b] and do not vanish; there exists some unlimited set Ω ⊂ C on which the inequalities Re(λφ1(x)) ≤ … ≤ Re (λφn(x)) are fulfilled for all x ∈ [a,b] and for some numeration of the functions φj(x). The asymptotic formula of the exponential type for a fundamental matrix of solutions of the system is obtained for sufficiently large ¦λ¦. The remainder term of this formula has a new type dependence on properties of the coefficients A1(x), Ao(x) and A- 1(x).