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{{MSC|34L}}

An  $  n $
th order homogeneous [[Linear differential operator|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 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

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

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]]].

$  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 [[#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]]  $  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} $[[#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  $  + \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.

1) The Sturm–Liouville operator (cf. [[Sturm–Liouville equation|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|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]]].

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  $  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  $  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.

====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>