By Lamberto Cesari
In the previous couple of a long time the idea of standard differential equations has grown speedily lower than the motion of forces that have been operating either from inside and with out: from inside of, as a improvement and deepen ing of the ideas and of the topological and analytical tools caused through LYAPUNOV, POINCARE, BENDIXSON, and some others on the flip of the century; from with no, within the wake of the technological improvement, fairly in communications, servomechanisms, automobile matic controls, and electronics. The early study of the authors simply pointed out lay in demanding difficulties of astronomy, however the line of idea hence produced chanced on the main amazing functions within the new fields. The physique of analysis now accrued is overwhelming, and plenty of books and experiences have seemed on one or one other of the a number of elements of the recent line of study which a few authors name" qualitative conception of differential equations". the aim of the current quantity is to provide a number of the view issues and questions in a readable brief document for which completeness isn't really claimed. The bibliographical notes in each one part are meant to be a advisor to extra specified expositions and to the unique papers. a few conventional subject matters corresponding to the Sturm comparability thought were passed over. additionally excluded have been all these papers, facing exact differential equations influenced via and meant for the applications.
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Additional resources for Asymptotic Behavior and Stability Problems in Ordinary Differential Equations
PERRON [6, 9]. H. SPATH [1,2], N. Y. LVASCENKO , R. BELLMAN [1, 4], S. FAEDO , N. LEVINSON [6, 9], A. WINTNER [1,6, 10]. 3). but they are" asymptotically" equivalent to the solutions of the system y' = A y, at least in the sense that for every x[y] there is a y[x] such that x-y-+o as 1-++00 (N. LEVINSON [6,9]). ) 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.
Therefore, we will have IdetP(I) 1< M, I detp-l (I) 1< M, and P-1(A + B) P =A = diag [('1 (I), "', (t)] for all I;;;: 12 . ° ° + en We shall now denote by L the class of the matrices C(I) whose elements are +00 measurable functions of I with 1 IIC(I) II dl < 00. 2) where A=p-l(A+B)P is the diagonal matrix [('1(1), ... ,(',,(1)]. and D(I)= p-l C P- p-l P' is of class L. 2) with A (I) a diagonal matrix and D (I) of class L is called" of L diagonal form" by 1. M. RAPOPORT. 2). If we now suppose R [('i (I)] $; for all I;;;: 12 and i = 1, 2, .
Iii). 2) we have t x = Y where now II y II;:;;; c1 ' + I Y(t - oc) C (oc) x (oc) doc, II YII::;;; c2 for o o. all t;;;;; Hence t IIxll;;;; lIyl! 2. i) IIxll;;;;M, where M=c1 exp (c +00 2/ II C(oc)lIdOC) for all t;;;;; o. 3. iii). 4. Further conditions for boundedness. 1) is due to A. WIMAN [1, 2J, R. CACCIOPPOLI [1J and G. 4. 1) are bounded in [0, 00). A. 7)]. 4. 3. ii), is not true as the first of the two following examples shows. Consider the two equations (L. CESAR I [3J) [d. 1)J: + + (a) x" - (2ft) x' + x = 0, x" + (2ft) x' + x (b) + = 0, ° having monotone coefficients in [1, 00) approaching, as t---++ 00, the coefficients of the equation with constant coefficients (c) x" x = whose solutions are all bounded in [1, 00).
Asymptotic Behavior and Stability Problems in Ordinary Differential Equations by Lamberto Cesari
Categories: Differential Equations