# Mingarelli identity

*https://en.wikipedia.org/wiki/Mingarelli_identity*

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 [*a*, *b*]:

- 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 *x _{i}*(

*t*) ≠ 0 on (

*a*,

*b*], there holds the identity,

^{ [2]}

where

- is the logarithmic derivative,
- , is the Wronskian determinant,
- are binomial coefficients.

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 p_{i}, q_{i} *i* = 1, 2, 3, be real-valued continuous functions on the interval [*a*, *b*] and let

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

*p*(_{i}*t*) > 0 for each i and for all t in [*a*,*b*] , and- the R
_{i}are arbitrary real numbers.

Assume that for all *t* in [*a*, *b*] we have,

- ,
- ,
- .

Then, if *x*_{1}(*t*) > 0 on [*a*, *b*] and *x*_{2}(*b*) = 0, then any solution *x*_{3}(*t*) has at least one zero in [*a*, *b*].

## Notes

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