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]:

${\displaystyle (p_{i}(t)x_{i}^{\prime })^{\prime }+q_{i}(t)x_{i}=0,\,\,\,\,\,\,\,\,\,\,x_{i}(a)=1,\,\,x_{i}^{\prime }(a)=R_{i}\,}$ where ${\displaystyle i=1,2,\ldots ,n}$.

Let $\Delta$ denote the forward difference operator, i.e.

${\displaystyle \Delta x_{i}=x_{i+1}-x_{i}.}$

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

${\displaystyle \Delta ^{2}(x_{i})=\Delta (\Delta x_{i})=x_{i+2}-2x_{i+1}+x_{i},}$,

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]

${\displaystyle {\begin{aligned}x_{n-1}^{2}\Delta ^{n-1}(p_{1}r_{1})]_{a}^{b}&=\int _{a}^{b}(x_{n-1}^{\prime })^{2}\Delta ^{n-1}(p_{1})-\int _{a}^{b}x_{n-1}^{2}\Delta ^{n-1}(q_{1})-\sum _{k=0}^{n-1}C(n-1,k)(-1)^{n-k-1}\int _{a}^{b}p_{k+1}W^{2}(x_{k+1},x_{n-1})/x_{k+1}^{2},\end{aligned}}}$

where

• ${\displaystyle r_{i}=x_{i}^{\prime }/x_{i}}$ is the logarithmic derivative,
• ${\displaystyle W(x_{i},x_{j})=x_{i}^{\prime }x_{j}-x_{i}x_{j}^{\prime }}$, is the Wronskian determinant,
• ${\displaystyle C(n-1,k)}$ 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

1. ${\displaystyle (p_{1}(t)x_{1}^{\prime })^{\prime }+q_{1}(t)x_{1}=0,\,\,\,\,\,\,\,\,\,\,x_{1}(a)=1,\,\,x_{1}^{\prime }(a)=R_{1}\,}$
2. ${\displaystyle (p_{2}(t)x_{2}^{\prime })^{\prime }+q_{2}(t)x_{2}=0,\,\,\,\,\,\,\,\,\,\,x_{2}(a)=1,\,\,x_{2}^{\prime }(a)=R_{2}\,}$
3. ${\displaystyle (p_{3}(t)x_{3}^{\prime })^{\prime }+q_{3}(t)x_{3}=0,\,\,\,\,\,\,\,\,\,\,x_{3}(a)=1,\,\,x_{3}^{\prime }(a)=R_{3}\,}$

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,

${\displaystyle \Delta ^{2}(q_{1})\geq 0}$,

${\displaystyle \Delta ^{2}(p_{1})\leq 0}$,

${\displaystyle \Delta ^{2}(p_{1}(a)R_{1})\leq 0}$.

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

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 mathématiques de l'Académie des sciences. Toronto, Ontario, Canada: The Royal Society of Canada. 1 (4): 223–226.