View source for Oscillating differential equation

An ordinary differential equation which has at least one [[Oscillating solution|oscillating solution]]. There are different concepts of the oscillation of a solution. The most widespread are oscillation at a point (usually taken to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070470/o0704701.png" />) and oscillation on an interval. A non-zero solution of the equation

<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/o/o070/o070470/o0704702.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070470/o0704703.png" />, is called oscillating at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070470/o0704705.png" /> (or on an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070470/o0704706.png" />) if it has a sequence of zeros which converges to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070470/o0704707.png" /> (respectively, there are at least <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070470/o0704708.png" /> zeros in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070470/o0704709.png" /> counted according to their multiplicity). Equation (1) is oscillating at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070470/o07047010.png" /> or on an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070470/o07047011.png" /> if its solutions are oscillating (at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070470/o07047012.png" />, respectively, on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070470/o07047013.png" />).

Among equations which are oscillatory at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070470/o07047014.png" /> the equations which possess the properties <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070470/o07047015.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070470/o07047016.png" />, i.e. which are compatible in a specific sense with one of the equations

<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/o/o070/o070470/o07047017.png" /></td> </tr></table>

are distinguished. Equation (1) is said to possess property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070470/o07047019.png" /> if all its solutions defined in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070470/o07047020.png" /> are oscillating when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070470/o07047021.png" /> is even; when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070470/o07047022.png" /> is odd, they should either be oscillating or satisfy the condition

<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/o/o070/o070470/o07047023.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

If every solution of equation (1) defined in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070470/o07047024.png" />, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070470/o07047025.png" /> is even, is either oscillating, or satisfies condition (2) or

<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/o/o070/o070470/o07047026.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

while when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070470/o07047027.png" /> is odd, it is either oscillating or satisfies condition (3), then the equation possesses property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070470/o07047029.png" />.

The linear equation

<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/o/o070/o070470/o07047030.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>

with a locally summable coefficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070470/o07047031.png" /> possesses property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070470/o07047032.png" /> (property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070470/o07047033.png" />) if

<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/o/o070/o070470/o07047034.png" /></td> </tr></table>

and either

<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/o/o070/o070470/o07047035.png" /></td> </tr></table>

or

<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/o/o070/o070470/o07047036.png" /></td> </tr></table>

when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070470/o07047037.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070470/o07047038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070470/o07047039.png" /> is the smallest (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070470/o07047040.png" /> is the largest) of the local minima (maxima) of the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070470/o07047041.png" /> (see [[#References|[1]]]–[[#References|[5]]]).

An equation of Emden–Fowler type

<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/o/o070/o070470/o07047042.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>

with a locally summable non-positive (non-negative) coefficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070470/o07047043.png" /> possesses property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070470/o07047044.png" /> (property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070470/o07047045.png" />) if and only if

<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/o/o070/o070470/o07047046.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070470/o07047047.png" /> (see [[#References|[4]]], [[#References|[6]]], [[#References|[7]]]).

In a number of cases the question of the oscillation of equation (1) can be reduced to the same question for the standard equations of the form (4) and (5) using a [[Comparison theorem|comparison theorem]] (see [[#References|[11]]]).

In studying the oscillatory properties of equations with deviating argument, certain specific features arise. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070470/o07047048.png" /> is odd, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070470/o07047049.png" />, and if for large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070470/o07047050.png" /> the inequality

<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/o/o070/o070470/o07047051.png" /></td> </tr></table>

is fulfilled, then all non-zero solutions of the equation

<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/o/o070/o070470/o07047052.png" /></td> </tr></table>

are oscillatory at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070470/o07047053.png" /> (see [[#References|[10]]], [[#References|[11]]]). At the same time, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070470/o07047054.png" /> is non-positive and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070470/o07047055.png" /> is odd, the non-retarded equation (4) always has a non-oscillating solution.

The concepts of oscillation and non-oscillation on an interval are generally studied for linear homogeneous equations. They are of fundamental value in the theory of boundary value problems (see [[#References|[12]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Kneser,   "Untersuchungen über die reellen Nullstellen der Integrale linearer Integralgleichungen"  ''Math. Ann.'' , '''42'''  (1893)  pp. 409–435</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.G. Mikusinksi,   "On Fite's oscillation theorems"  ''Colloq. Math.'' , '''2'''  (1951)  pp. 34–39</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.A. Kondrat'ev,   "The oscillatory character of solutions of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070470/o07047056.png" />"  ''Trudy Moskov. Mat. Obshch.'' , '''10'''  (1961)  pp. 419–436  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  I.T. Kiguradze,   "On the oscillatory character of solutions of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070470/o07047057.png" />"  ''Mat. Sb.'' , '''65''' :  2  (1964)  pp. 172–187  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  T.A. Chanturiya,   "On a comparison theorem for linear differential equations"  ''Math. USSR Izv.'' , '''10''' :  5  (1976)  pp. 1075–1088  ''Izv. Akad. Nauk. SSSR Ser. Mat.'' , '''40''' :  5  (1976)  pp. 1128–1142</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  I. Ličko,   M. Švec,   "La charactère oscillatoire des solutions de l'équation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070470/o07047058.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070470/o07047059.png" />"  ''Chekhosl. Mat. Zh.'' , '''13'''  (1963)  pp. 481–491</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  I.T. Kiguradze,   "On the oscillatory and monotone solutions of ordinary differential equations"  ''Arch. Math.'' , '''14''' :  1  (1978)  pp. 21–44</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  C.A. Swanson,   "Comparison and oscillation theory of linear differential equations" , Acad. Press  (1968)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  P. Hartman,   "Ordinary differential equations" , Birkhäuser  (1982)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  A.D. Myshkis,   "Linear differential equations with retarded argument" , Moscow  (1972)  (In Russian)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  R.G. Koplatadze,   T.A. Chanturiya,   "On the oscillatory properties of differential equations with deviating argument" , Tbilisi  (1977)  (In Russian)</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  A.Yu. Levin,   "Non-oscillation of the solutions of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070470/o07047060.png" />"  ''Russian Math. Surveys'' , '''24''' :  2  (1969)  pp. 43–99  ''Uspekhi Mat. Nauk'' , '''24''' :  2  (1969)  pp. 43–96</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.K. Hale,   "Ordinary differential equations" , Wiley  (1969)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W.T. Reid,   "Sturmian theory for ordinary differential equations" , Springer  (1980)</TD></TR></table>