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Linear Differential Equations with Variable Coefficients in Regular and Some Singular Cases
Author(s) -
Lantsman M. H.
Publication year - 1997
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.3211860112
Subject(s) - mathematics , differential equation , polynomial , asymptotic analysis , formal power series , independence (probability theory) , pure mathematics , linear differential equation , order (exchange) , mathematical analysis , discrete mathematics , combinatorics , power series , statistics , finance , economics
In this paper the asymptotic properties as t → + ∞ for a single linear differential equation of the form x (n) + a 1 ( t ) x (n−1) +…. + a n ( t ) x = 0, where the coefficients aj ( z ) are supposed to be of the power order of growth, are considered. The results obtained in the previous publications of the author were related to the so called regular case when a complete set of roots {λ,( t )}, j = 1, 2, …, n of the characteristic polynomial y n + a 1 ( t )y n−1 + … + a n (t ) possesses the property of asymptotic separability. One of the main restrictions of the regular case consists of the demand that the roots of the set {λ,( t )} have not to be equivalent in pairs for t → + ∞. In this paper we consider the some more general case when the set of characteristic roots possesses the property of asymptotic independence which includes the case when the roots may be equivdent in pairs. But some restrictions on the asymptotic behaviour of their differences λ i ( t )→ λ j ( t ) are preserved. This case demands more complicated technique of investigation. For this purpose the so called asymptotic spaces were introduced. The theory of asymptotic spaces is used for formal solution of an operator equation of the form x = A (x) and has the analogous meaning as the classical theory of solving this equation in Band spaces. For the considered differential equation, the main asymptotic terms of a fundamental system of solution is given in a simple explicit form and the asymptotic fundamental system is represented in the form of asymptotic Emits for several iterate sequences.