Disconjugacy

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 " 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