# Mingarelli identity

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

In the field of ordinary differential equations, the Mingarelli identity  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]:

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

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

$\Delta x_{i}=x_{i+1}-x_{i}.$ The second order difference operator is found by iterating the first order operator as in

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

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

{\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

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

1. $(p_{1}(t)x_{1}^{\prime })^{\prime }+q_{1}(t)x_{1}=0,\,\,\,\,\,\,\,\,\,\,x_{1}(a)=1,\,\,x_{1}^{\prime }(a)=R_{1}$ 2. $(p_{2}(t)x_{2}^{\prime })^{\prime }+q_{2}(t)x_{2}=0,\,\,\,\,\,\,\,\,\,\,x_{2}(a)=1,\,\,x_{2}^{\prime }(a)=R_{2}$ 3. $(p_{3}(t)x_{3}^{\prime })^{\prime }+q_{3}(t)x_{3}=0,\,\,\,\,\,\,\,\,\,\,x_{3}(a)=1,\,\,x_{3}^{\prime }(a)=R_{3}$ be three homogeneous linear second order differential equations in self-adjoint form, where

• 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,

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

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