Mingarelli identity

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In the field of ordinary differential equations, the Mingarelli identity [1] is a theorem that provides criteria for the oscillation and non-oscillation of solutions of some linear differential equations in the real domain. It extends the Picone identity from two to three or more differential equations of the second order.

The identity

Consider the n solutions of the following (uncoupled) system of second order linear differential equations over the t–interval [ab]:

where .

Let denote the forward difference operator, i.e.

The second order difference operator is found by iterating the first order operator as in

,

with a similar definition for the higher iterates. Leaving out the independent variable t for convenience, and assuming the xi(t) ≠ 0 on (ab], there holds the identity, [2]

where

When n = 2 this equality reduces to the Picone identity.

An application

The above identity leads quickly to the following comparison theorem for three linear differential equations, [3] which extends the classical Sturm–Picone comparison theorem.

Let pi, qi i = 1, 2, 3, be real-valued continuous functions on the interval [ab] and let

be three homogeneous linear second order differential equations in self-adjoint form, where

  • pi(t) > 0 for each i and for all t in [ab] , and
  • the Ri are arbitrary real numbers.

Assume that for all t in [ab] we have,

,
,
.

Then, if x1(t) > 0 on [ab] and x2(b) = 0, then any solution x3(t) has at least one zero in [ab].

Notes

  1. ^ The locution was coined by Philip Hartman, according to Clark D.N., G. Pecelli, and R. Sacksteder (1981)
  2. ^ ( Mingarelli 1979, p. 223).
  3. ^ ( Mingarelli 1979, Theorem 2).

References

  • Clark D.N.; G. Pecelli & R. Sacksteder (1981). Contributions to Analysis and Geometry. Baltimore, USA: Johns Hopkins University Press. pp. ix+357. ISBN  0-80182-779-5.
  • Mingarelli, Angelo B. (1979). "Some extensions of the Sturm–Picone theorem". Comptes Rendus Mathématique. Toronto, Ontario, Canada: The Royal Society of Canada. 1 (4): 223–226.