LINEAR-DIFFERENTIAL EQUATIONS WITH VARIABLE-COEFFICIENTS IN REGULAR AND SOME SINGULAR CASES

In this paper the asymptotic properties as t --> +infinity for a singl e linear differential equation of the form x((n)) + alpha(1)(t)x((n-1) ) +...+ alpha(n)(t)x = 0, where the coefficients alpha(j)(z) are suppo sed to be of the power order of growth, are considered. The results ob tained in the previous publications of the author were related to the so called regular case when a complete set of roots {lambda(j)(t)}, j = 1,2,..., n of the characteristic polynomial y(n) + alpha(1)(t)y(n-1) +...+ alpha(n)(t) possesses the property of asymptotic separability. One of the main restrictions of the regular case consists of the deman d that the roots of the set {lambda(j)(t)} have not to be equivalent i n pairs for t --> +infinity. In this paper we consider the some more g eneral case when the set of characteristic roots possesses the propert y of asymptotic independence which includes the case when the roots ma y be equivalent in pairs. But some restrictions on the asymptotic beha viour of their differences lambda(i)(t) - lambda(j)(t) are preserved. This case demands more complicated technique of investigation. For thi s purpose the so called asymptotic spaces were introduced. The theory of asymptotic spaces is used for formal solution of an operator equati on of the form x = A(x) and has the analogous meaning as the classical theory of solving this equation in Banach spaces. For the considered differential equation, the main asymptotic terms of a fundamental syst em of solution is given in a simple explicit form and the asymptotic f undamental system is represented in the form of asymptotic Limits for several iterate sequences.