# 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 |

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Disconjugacy.

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