By Lamberto Cesari
In the previous couple of a long time the speculation of normal differential equations has grown quickly lower than the motion of forces that have been operating either from inside and with out: from inside, as a improvement and deepen ing of the innovations and of the topological and analytical tools led to through LYAPUNOV, POINCARE, BENDIXSON, and some others on the flip of the century; from with no, within the wake of the technological improvement, quite in communications, servomechanisms, vehicle matic controls, and electronics. The early study of the authors simply pointed out lay in hard difficulties of astronomy, however the line of concept therefore produced came upon the main extraordinary functions within the new fields. The physique of study now collected is overwhelming, and plenty of books and experiences have seemed on one or one other of the a number of facets of the recent line of analysis which a few authors name" qualitative concept of differential equations". the aim of the current quantity is to give some of the view issues and questions in a readable brief record for which completeness isn't claimed. The bibliographical notes in each one part are meant to be a advisor to extra distinctive expositions and to the unique papers. a few conventional issues comparable to the Sturm comparability concept were passed over. additionally excluded have been all these papers, facing certain differential equations encouraged via and meant for the applications.
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Additional resources for Asymptotic Behavior and Stability Problems in Ordinary Differential Equations
GHIZZETTI , I. M. SOBOL . For a further investigation on the subject see A. \VINTNER . 8. Linear asymptotic equilibrium. We shall first state and prove the following statement whose interest goes beyond the application we will give below. 3. 8. i) If A(t), ~t< 00, is a matrix whose elements are integrable functions in every finite interval, then every A C solution x (t) of the system x' = A (f) x verifies the relation t IIx (f)1I ;;;;: IIx (0)11 exp J IIA (oc)1I doc. 1) o Proof. 2. 1).
I mention below some of the most recent results, as recently obtained by N. LEVINSON [6, 9] and R. BELLMAN  by means of variants of the process of reduction to L-diagonal form. + + II. 7. i) If A is a constant matrix with real distinct characteristic roots AI' ... , An' if +00 [A f II C(t) II d t < + 00, then there are + C (t)J x such that n solutions x(I), ... , x(n) of the system x' = where the ck are constants, k = 1, ... , n. 7. ii) If A is a constant matrix with distinct characteristic roots AI' "', An, if II B (t) II ~ 0 as t ~ + 00, then there are n solutions x(I), ...
The equation e 2 - (sin 2t/t 2 ) = 0 has the roots e = ± sint/t, none of which is of bounded variation in any [12' + (0), while sin 2 t/t 2 is of bounded variation and absolutely continuous in [1, (0). The limiting equation A2 = 0 has the double root A= O. Now let us proceed to outline the process of reduction to L-diagonal form of a given matrix A + B (t), t:2: to' with B (t) - ? 0 as t -+ + 00, under the following simplifying assumptions: 1. The elements of the matrix B are absolutely continuous and of bounded variation in [to, (0); 2.
Asymptotic Behavior and Stability Problems in Ordinary Differential Equations by Lamberto Cesari