Difference between revisions of "Disconjugacy"
(Importing text file) |
m (Richard Pinch moved page Disconcugacy to Disconjugacy: Correct spelling) |
(No difference)
|
Revision as of 07:35, 23 November 2014
An th order homogeneous linear differential operator (equation)
![]() | (a1) |
is called disconjugate on an interval if no non-trivial solution has
zeros on
, multiple zeros being counted according to their multiplicity. (In the Russian literature this is called non-oscillation on
; cf. also Oscillating solution; Oscillating differential equation.) If (a1) has a solution with
zeros on an interval, then the infimum of all values
,
, such that some solution has
zeros on
is called the conjugate point of
and is denoted by
. This infimum is achieved by a solution which has a total of at least
zeros at
and
and is positive on
. If the equation has continuous coefficients, the conjugate point
is a strictly increasing, continuous function of
. 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
![]() |
where is some norm of
,
and
are suitable constants. These are "smallness conditions" which express the proximity of (a1) to the disconjugate equation
. See [a12].
is disconjugate on
if and only if it has there the Pólya factorization
![]() |
or the equivalent Mammana factorization
![]() |
Among the various Pólya factorizations, the most important is the Trench canonical form [a11]: If is disconjugate on
,
, then there is essentially one factorization such that
,
.
Disconjugacy is closely related to solvability of the de la Vallée-Poussin multiple-point problem ,
,
,
. The Green's function of a disconjugate operator
and the related homogeneous boundary value problem satisfies
![]() |
for ,
[a7]. Another interesting boundary value problem is the focal boundary value problem
,
,
,
.
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 ) implies that there exists a point
such that (a1) is disconjugate on
. For equations of order
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)
![]() | (a2) |
has been studied using the Sturm (and Sturm–Picone) comparison theorem, the Prüfer transformation and the Riccati equation . It is also closely related to the positive definiteness of the quadratic functional
. See [a10], [a1], [a5]. For example, (a2) is disconjugate on
if
.
2) Third-order equations are studied in [a3].
3) For a self-adjoint differential equation , the existence of a solution with two zeros of multiplicity
has been studied. Their absence is called
-disconjugacy.
4) Disconjugacy of the analytic equation 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 and their generalizations
[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 ![]() |
[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 |
Disconjugacy. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Disconjugacy&oldid=34874