View source for Disconjugacy

An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110220/d1102201.png" />th order homogeneous [[Linear differential operator|linear differential operator]] (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/d/d110/d110220/d1102202.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

is called disconjugate on an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110220/d1102203.png" /> if no non-trivial solution has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110220/d1102204.png" /> zeros on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110220/d1102205.png" />, multiple zeros being counted according to their multiplicity. (In the Russian literature this is called non-oscillation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110220/d1102206.png" />; cf. also [[Oscillating solution|Oscillating solution]]; [[Oscillating differential equation|Oscillating differential equation]].) If (a1) has a solution with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110220/d1102207.png" /> zeros on an interval, then the infimum of all values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110220/d1102208.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110220/d1102209.png" />, such that some solution has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110220/d11022010.png" /> zeros on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110220/d11022011.png" /> is called the conjugate point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110220/d11022012.png" /> and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110220/d11022013.png" />. This infimum is achieved by a solution which has a total of at least <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110220/d11022014.png" /> zeros at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110220/d11022015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110220/d11022016.png" /> and is positive on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110220/d11022017.png" />. If the equation has continuous coefficients, the conjugate point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110220/d11022018.png" /> is a strictly increasing, continuous function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110220/d11022019.png" />. The adjoint equation has the same conjugate point as (a1). For general properties, see [[#References|[a1]]], [[#References|[a7]]].

There are numerous explicit sufficient criteria for the equation (a1) to be disconjugate. Many of them are of the form

<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/d/d110/d110220/d11022020.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110220/d11022021.png" /> is some [[Norm|norm]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110220/d11022022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110220/d11022023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110220/d11022024.png" /> are suitable constants. These are  "smallness conditions"  which express the proximity of (a1) to the disconjugate equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110220/d11022025.png" />. See [[#References|[a12]]].

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110220/d11022026.png" /> is disconjugate on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110220/d11022027.png" /> if and only if it has there the Pólya factorization

<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/d/d110/d110220/d11022028.png" /></td> </tr></table>

or the equivalent Mammana factorization

<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/d/d110/d110220/d11022029.png" /></td> </tr></table>

Among the various Pólya factorizations, the most important is the Trench canonical form [[#References|[a11]]]: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110220/d11022030.png" /> is disconjugate on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110220/d11022031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110220/d11022032.png" />, then there is essentially one factorization such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110220/d11022033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110220/d11022034.png" />.

Disconjugacy is closely related to solvability of the [[De la Vallée-Poussin multiple-point problem|de la Vallée-Poussin multiple-point problem]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110220/d11022035.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110220/d11022036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110220/d11022037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110220/d11022038.png" />. The Green's function of a disconjugate operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110220/d11022039.png" /> and the related homogeneous boundary value problem satisfies

<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/d/d110/d110220/d11022040.png" /></td> </tr></table>

for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110220/d11022041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110220/d11022042.png" /> [[#References|[a7]]]. Another interesting boundary value problem is the focal boundary value problem <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110220/d11022043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110220/d11022044.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110220/d11022045.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110220/d11022046.png" />.

For a second-order equation, the Sturm separation theorem (cf. [[Sturm theorem|Sturm theorem]]) yields that non-oscillation (i.e., no solution has a sequence of zeros converging to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110220/d11022047.png" />) implies that there exists a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110220/d11022048.png" /> such that (a1) is disconjugate on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110220/d11022049.png" />. For equations of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110220/d11022050.png" /> this conclusion holds for a class of equations [[#References|[a2]]] but not for all equations [[#References|[a4]]].

Particular results about disconjugacy exist for various special types of differential equations.

1) The Sturm–Liouville operator (cf. [[Sturm–Liouville equation|Sturm–Liouville 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/d/d110/d110220/d11022051.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>

has been studied using the Sturm (and Sturm–Picone) comparison theorem, the Prüfer transformation and the [[Riccati equation|Riccati equation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110220/d11022052.png" />. It is also closely related to the positive definiteness of the quadratic functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110220/d11022053.png" />. See [[#References|[a10]]], [[#References|[a1]]], [[#References|[a5]]]. For example, (a2) is disconjugate on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110220/d11022054.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110220/d11022055.png" />.

2) Third-order equations are studied in [[#References|[a3]]].

3) For a [[Self-adjoint differential equation|self-adjoint differential equation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110220/d11022056.png" />, the existence of a solution with two zeros of multiplicity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110220/d11022057.png" /> has been studied. Their absence is called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110220/d11022059.png" />-disconjugacy.

4) Disconjugacy of the analytic equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110220/d11022060.png" /> in a complex domain is connected to the theory of univalent functions. See [[#References|[a8]]], [[#References|[a5]]] and [[Univalent function|Univalent function]].

5) Many particularly elegant result are available for two-term equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110220/d11022061.png" /> and their generalizations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110220/d11022062.png" /> [[#References|[a6]]], [[#References|[a2]]].

Disconjugacy has also been studied for certain second-order linear differential systems of higher dimension [[#References|[a1]]], [[#References|[a9]]]. In the historical prologue of [[#References|[a9]]], the connection to the calculus of variations (cf. also [[Variational calculus|Variational calculus]]) is explained. The concepts of disconjugacy and oscillation have also been generalized to non-linear differential equations and functional-differential equations.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W.A. Coppel,   "Disconjugacy" , ''Lecture Notes in Mathematics'' , '''220''' , Springer  (1971)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  U. Elias,   "Oscillation theory of two-term differential equations" , Kluwer Acad. Publ.  (1997)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M. Gregus,   "Third order linear differential equations" , Reidel  (1987)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  Gustafson, G. B.,   "The nonequivalence of oscillation and nondisconjugacy"  ''Proc. Amer. Math. Soc.'' , '''25'''  (1970)  pp. 254–260</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  E. Hille,   "Lectures on ordinary differential equations" , Addison-Wesley  (1968)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  I.T. Kiguradze,   T.A. Chanturia,   "Asymptotic properties of solutions of nonautonomous ordinary differential equations" , Kluwer Acad. Publ.  (1993)  (In Russian)</TD></TR><TR><TD valign="top">[a7]</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/d/d110/d110220/d11022063.png" />"  ''Russian Math. Surveys'' , '''24'''  (1969)  pp. 43–99  (In Russian)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  Z. Nehari,   "The Schwarzian derivative and schlicht functions"  ''Bull. Amer. Math. Soc.'' , '''55'''  (1949)  pp. 545–551</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  W.T. Reid,   "Sturmian theory for ordinary differential equations" , Springer  (1980)</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  C.A. Swanson,   "Comparison and oscillatory theory of linear differential equations" , Acad. Press  (1968)</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  W.F. Trench,   "Canonical forms and principal systems for general disconjugate equation"  ''Trans. Amer. Math. Soc.'' , '''189'''  (1974)  pp. 319–327</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  D. Willet,   "Generalized de la Vallée Poussin disconjugacy tests for linear differential equations"  ''Canadian Math. Bull.'' , '''14'''  (1971)  pp. 419–428</TD></TR></table>