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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)
+
An $  n $
 +
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>
+
$$ \tag{a1 }
 +
Ly \equiv y ^ {( n ) } + p _ {1} ( x ) y ^ {( n - 1 ) } + \dots + p _ {n} ( x ) y = 0
 +
$$
  
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]]].
+
is called disconjugate on an interval $  I $
 +
if no non-trivial solution has $  n $
 +
zeros on $  I $,  
 +
multiple zeros being counted according to their multiplicity. (In the Russian literature this is called non-oscillation on $  I $;  
 +
cf. also [[Oscillating solution|Oscillating solution]]; [[Oscillating differential equation|Oscillating differential equation]].) If (a1) has a solution with $  n $
 +
zeros on an interval, then the infimum of all values $  c $,  
 +
$  c > a $,  
 +
such that some solution has $  n $
 +
zeros on $  [ a,c ] $
 +
is called the conjugate point of $  a $
 +
and is denoted by $  \eta ( a ) $.  
 +
This infimum is achieved by a solution which has a total of at least $  n $
 +
zeros at $  a $
 +
and $  \eta ( a ) $
 +
and is positive on $  ( a, \eta ( a ) ) $.  
 +
If the equation has continuous coefficients, the conjugate point $  \eta ( a ) $
 +
is a strictly increasing, continuous function of $  a $.  
 +
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
 
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>
+
$$
 +
\sum ^ {n _ {k}  = 1 } c _ {k} ( b - a )  ^ {k} \left \| {p _ {k} } \right \| < 1,
 +
$$
  
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]]].
+
where $  \| {p _ {k} } \| $
 +
is some [[Norm|norm]] of $  p _ {k} $,
 +
$  I = [ a,b ] $
 +
and $  c _ {k} $
 +
are suitable constants. These are  "smallness conditions"  which express the proximity of (a1) to the disconjugate equation $  y ^ {( n ) } = 0 $.  
 +
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
+
$  L $
 +
is disconjugate on $  [ a,b ] $
 +
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>
+
$$
 +
Ly \equiv \rho _ {n} {
 +
\frac{d}{dx }
 +
} \left ( \rho _ {n - 1 }  \dots {
 +
\frac{d}{dx }
 +
} \left ( \rho _ {1} {
 +
\frac{d}{dx }
 +
} ( \rho _ {0} y ) \right ) \dots \right ) ,  \rho _ {i} > 0,
 +
$$
  
 
or the equivalent Mammana factorization
 
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>
+
$$
 +
Ly = \left ( {
 +
\frac{d}{dx }
 +
} + r _ {n} \right ) \dots \left ( {
 +
\frac{d}{dx }
 +
} + r _ {1} \right ) y.
 +
$$
  
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" />.
+
Among the various Pólya factorizations, the most important is the Trench canonical form [[#References|[a11]]]: If $  L $
 +
is disconjugate on $  ( a,b ) $,  
 +
$  b \leq  \infty $,  
 +
then there is essentially one factorization such that $  \int  ^ {b} {\rho _ {i} ^ {- 1 } } = \infty $,  
 +
$  i = 1 \dots n - 1 $.
  
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
+
Disconjugacy is closely related to solvability of the [[De la Vallée-Poussin multiple-point problem|de la Vallée-Poussin multiple-point problem]] $  Ly = g $,  
 +
$  y ^ {( i ) } ( x _ {j} ) = a _ {ij }  $,  
 +
$  i = 0 \dots r _ {j} - 1 $,  
 +
$  \sum _ {1}  ^ {m} r _ {j} = n $.  
 +
The Green's function of a disconjugate operator $  L $
 +
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>
+
$$
 +
{
 +
\frac{G ( x,t ) }{( x - x _ {1} ) ^ {r _ {1} } \dots ( x - x _ {m} ) ^ {r _ {m} } }
 +
} > 0
 +
$$
  
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 $  x _ {1} \leq  x \leq  x _ {m} $,
 +
$  x _ {1} < t < x _ {m} $[[#References|[a7]]]. Another interesting boundary value problem is the focal boundary value problem $  y ^ {( i ) } ( x _ {j} ) = 0 $,
 +
$  i = r _ {j - 1 }  \dots r _ {j} - 1 $,  
 +
$  j = 1 \dots m $,
 +
$  0 = r _ {0} < r _ {1} < \dots < r _ {m} = n - 1 $.
  
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]]].
+
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 $  + \infty $)  
 +
implies that there exists a point $  a $
 +
such that (a1) is disconjugate on $  [ a, \infty ) $.  
 +
For equations of order $  n > 2 $
 +
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.
 
Particular results about disconjugacy exist for various special types of differential equations.
Line 35: Line 110:
 
1) The Sturm–Liouville operator (cf. [[Sturm–Liouville equation|Sturm–Liouville equation]])
 
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>
+
$$ \tag{a2 }
 +
( p y  ^  \prime  )  ^  \prime  + q y = 0,  p > 0,
 +
$$
  
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" />.
+
has been studied using the Sturm (and Sturm–Picone) comparison theorem, the Prüfer transformation and the [[Riccati equation|Riccati equation]] $  z  ^  \prime  + q + { {z  ^ {2} } / p } = 0 $.  
 +
It is also closely related to the positive definiteness of the quadratic functional $  \int _ {a}  ^ {b} {( p y ^ {\prime 2 } - q y  ^ {2} ) } $.  
 +
See [[#References|[a10]]], [[#References|[a1]]], [[#References|[a5]]]. For example, (a2) is disconjugate on $  [ a,b ] $
 +
if $  \int _ {a}  ^ {b} {p ^ {- 1 } } \times \int _ {a}  ^ {b} {| q | } < 4 $.
  
 
2) Third-order equations are studied in [[#References|[a3]]].
 
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.
+
3) For a [[Self-adjoint differential equation|self-adjoint differential equation]] $  \sum _ {i = 0 }  ^ {m} ( p _ {m - i }  y ^ {( i ) } ) ^ {( i ) } = 0 $,  
 +
the existence of a solution with two zeros of multiplicity $  m $
 +
has been studied. Their absence is called $  ( m,m ) $-
 +
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]].
+
4) Disconjugacy of the analytic equation $  w  ^  \prime  + p ( z ) w = 0 $
 +
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]]].
+
5) Many particularly elegant result are available for two-term equations $  y ^ {( n ) } + p ( x ) y = 0 $
 +
and their generalizations $  Ly + p ( x ) y = 0 $[[#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.
 
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====
 
====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>
+
<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 $x ^ { ( n ) } + p _ { 1 } ( t ) x ^ { ( n - 1 ) } + \ldots + p _ { n } ( t ) x = 0$"  ''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>

Latest revision as of 07:36, 23 January 2024


2020 Mathematics Subject Classification: Primary: 34L [MSN][ZBL]

An $ n $ th order homogeneous linear differential operator (equation)

$$ \tag{a1 } Ly \equiv y ^ {( n ) } + p _ {1} ( x ) y ^ {( n - 1 ) } + \dots + p _ {n} ( x ) y = 0 $$

is called disconjugate on an interval $ I $ if no non-trivial solution has $ n $ zeros on $ I $, multiple zeros being counted according to their multiplicity. (In the Russian literature this is called non-oscillation on $ I $; cf. also Oscillating solution; Oscillating differential equation.) If (a1) has a solution with $ n $ zeros on an interval, then the infimum of all values $ c $, $ c > a $, such that some solution has $ n $ zeros on $ [ a,c ] $ is called the conjugate point of $ a $ and is denoted by $ \eta ( a ) $. This infimum is achieved by a solution which has a total of at least $ n $ zeros at $ a $ and $ \eta ( a ) $ and is positive on $ ( a, \eta ( a ) ) $. If the equation has continuous coefficients, the conjugate point $ \eta ( a ) $ is a strictly increasing, continuous function of $ a $. The adjoint equation has the same conjugate point as (a1). For general properties, see [a1], [a7].

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

$$ \sum ^ {n _ {k} = 1 } c _ {k} ( b - a ) ^ {k} \left \| {p _ {k} } \right \| < 1, $$

where $ \| {p _ {k} } \| $ is some norm of $ p _ {k} $, $ I = [ a,b ] $ and $ c _ {k} $ are suitable constants. These are "smallness conditions" which express the proximity of (a1) to the disconjugate equation $ y ^ {( n ) } = 0 $. See [a12].

$ L $ is disconjugate on $ [ a,b ] $ if and only if it has there the Pólya factorization

$$ Ly \equiv \rho _ {n} { \frac{d}{dx } } \left ( \rho _ {n - 1 } \dots { \frac{d}{dx } } \left ( \rho _ {1} { \frac{d}{dx } } ( \rho _ {0} y ) \right ) \dots \right ) , \rho _ {i} > 0, $$

or the equivalent Mammana factorization

$$ Ly = \left ( { \frac{d}{dx } } + r _ {n} \right ) \dots \left ( { \frac{d}{dx } } + r _ {1} \right ) y. $$

Among the various Pólya factorizations, the most important is the Trench canonical form [a11]: If $ L $ is disconjugate on $ ( a,b ) $, $ b \leq \infty $, then there is essentially one factorization such that $ \int ^ {b} {\rho _ {i} ^ {- 1 } } = \infty $, $ i = 1 \dots n - 1 $.

Disconjugacy is closely related to solvability of the de la Vallée-Poussin multiple-point problem $ Ly = g $, $ y ^ {( i ) } ( x _ {j} ) = a _ {ij } $, $ i = 0 \dots r _ {j} - 1 $, $ \sum _ {1} ^ {m} r _ {j} = n $. The Green's function of a disconjugate operator $ L $ and the related homogeneous boundary value problem satisfies

$$ { \frac{G ( x,t ) }{( x - x _ {1} ) ^ {r _ {1} } \dots ( x - x _ {m} ) ^ {r _ {m} } } } > 0 $$

for $ x _ {1} \leq x \leq x _ {m} $, $ x _ {1} < t < x _ {m} $[a7]. Another interesting boundary value problem is the focal boundary value problem $ y ^ {( i ) } ( x _ {j} ) = 0 $, $ i = r _ {j - 1 } \dots r _ {j} - 1 $, $ j = 1 \dots m $, $ 0 = r _ {0} < r _ {1} < \dots < r _ {m} = n - 1 $.

For a second-order equation, the Sturm separation theorem (cf. Sturm theorem) yields that non-oscillation (i.e., no solution has a sequence of zeros converging to $ + \infty $) implies that there exists a point $ a $ such that (a1) is disconjugate on $ [ a, \infty ) $. For equations of order $ n > 2 $ this conclusion holds for a class of equations [a2] but not for all equations [a4].

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

1) The Sturm–Liouville operator (cf. Sturm–Liouville equation)

$$ \tag{a2 } ( p y ^ \prime ) ^ \prime + q y = 0, p > 0, $$

has been studied using the Sturm (and Sturm–Picone) comparison theorem, the Prüfer transformation and the Riccati equation $ z ^ \prime + q + { {z ^ {2} } / p } = 0 $. It is also closely related to the positive definiteness of the quadratic functional $ \int _ {a} ^ {b} {( p y ^ {\prime 2 } - q y ^ {2} ) } $. See [a10], [a1], [a5]. For example, (a2) is disconjugate on $ [ a,b ] $ if $ \int _ {a} ^ {b} {p ^ {- 1 } } \times \int _ {a} ^ {b} {| q | } < 4 $.

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

3) For a self-adjoint differential equation $ \sum _ {i = 0 } ^ {m} ( p _ {m - i } y ^ {( i ) } ) ^ {( i ) } = 0 $, the existence of a solution with two zeros of multiplicity $ m $ has been studied. Their absence is called $ ( m,m ) $- disconjugacy.

4) Disconjugacy of the analytic equation $ w ^ \prime + p ( z ) w = 0 $ in a complex domain is connected to the theory of univalent functions. See [a8], [a5] and Univalent function.

5) Many particularly elegant result are available for two-term equations $ y ^ {( n ) } + p ( x ) y = 0 $ and their generalizations $ Ly + p ( x ) y = 0 $[a6], [a2].

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

References

[a1] W.A. Coppel, "Disconjugacy" , Lecture Notes in Mathematics , 220 , Springer (1971)
[a2] U. Elias, "Oscillation theory of two-term differential equations" , Kluwer Acad. Publ. (1997)
[a3] M. Gregus, "Third order linear differential equations" , Reidel (1987)
[a4] Gustafson, G. B., "The nonequivalence of oscillation and nondisconjugacy" Proc. Amer. Math. Soc. , 25 (1970) pp. 254–260
[a5] E. Hille, "Lectures on ordinary differential equations" , Addison-Wesley (1968)
[a6] I.T. Kiguradze, T.A. Chanturia, "Asymptotic properties of solutions of nonautonomous ordinary differential equations" , Kluwer Acad. Publ. (1993) (In Russian)
[a7] A.Yu. Levin, "Non-oscillation of solutions of the equation $x ^ { ( n ) } + p _ { 1 } ( t ) x ^ { ( n - 1 ) } + \ldots + p _ { n } ( t ) x = 0$" Russian Math. Surveys , 24 (1969) pp. 43–99 (In Russian)
[a8] Z. Nehari, "The Schwarzian derivative and schlicht functions" Bull. Amer. Math. Soc. , 55 (1949) pp. 545–551
[a9] W.T. Reid, "Sturmian theory for ordinary differential equations" , Springer (1980)
[a10] C.A. Swanson, "Comparison and oscillatory theory of linear differential equations" , Acad. Press (1968)
[a11] W.F. Trench, "Canonical forms and principal systems for general disconjugate equation" Trans. Amer. Math. Soc. , 189 (1974) pp. 319–327
[a12] D. Willet, "Generalized de la Vallée Poussin disconjugacy tests for linear differential equations" Canadian Math. Bull. , 14 (1971) pp. 419–428
How to Cite This Entry:
Disconjugacy. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Disconjugacy&oldid=34876
This article was adapted from an original article by U. Elias (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article