View source for Non-oscillation interval

''interval of disconjugacy''

A connected interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067220/n0672201.png" /> on the real axis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067220/n0672202.png" /> such that any non-trivial solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067220/n0672203.png" /> of a given ordinary linear differential equation of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067220/n0672204.png" /> with real coefficients,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067220/n0672205.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

has on it more than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067220/n0672206.png" /> zeros, an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067220/n0672207.png" />-fold zero counted <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067220/n0672208.png" /> times. Properties of solutions of (*) on a non-oscillation interval have been well studied (see, for example, [[#References|[1]]]–[[#References|[3]]]). There are several generalizations of the concept of a non-oscillation interval, to linear systems of differential equations, to non-linear differential equations, and also to other types of equations (difference, with deviating argument).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P. Hartman,   "Ordinary differential equations" , Birkhäuser  (1982)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.Yu. Levin,   "Non-oscillation of solutions of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067220/n0672209.png" />"  ''Russian Math. Surveys'' , '''24''' :  2  (1969)  pp. 43–99  ''Uspekhi Mat. Nauk'' , '''24''' :  2  (1969)  pp. 43–96</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  W.A. Coppel,   "Disconjugacy" , Springer  (1971)</TD></TR></table>