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 $[a, b]$:
 * $$(p_i(t) x_i^\prime)^\prime + q_i(t) x_i = 0, \,\,\,\,\,\,\,\,\,\, x_i(a)=1,\,\, x_i^\prime(a)=R_i$$ 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 $(a, b]$, there holds the identity,

\begin{align} x_{n-1}^2\Delta^{n-1}(p_1r_1)]_a^b = &\int_a^b (x^\prime_{n-1})^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{align} $$

where When $n = 2$ this equality reduces to the Picone identity.
 * $$r_i = x^\prime_i/x_i$$ is the logarithmic derivative,
 * $$W(x_i, x_j) = x^\prime_ix_j - x_ix^\prime_j$$, is the Wronskian determinant,
 * $$C(n-1,k)$$ are binomial coefficients.

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 $[a, b]$ and let be three homogeneous linear second order differential equations in self-adjoint form, where
 * 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) $$(p_3(t) x_3^\prime)^\prime + q_3(t) x_3 = 0, \,\,\,\,\,\,\,\,\,\, x_3(a)=1,\,\, x_3^\prime(a)=R_3$$
 * $pi(t) > 0$ for each $i$ and for all $t$ in $[a, b]$, and
 * the $Ri$ are arbitrary real numbers.

Assume that for all $t$ in $[a, b]$ we have,
 * $$\Delta^2(q_1) \ge 0 $$,
 * $$\Delta^2(p_1) \le 0 $$,
 * $$\Delta^2(p_1(a)R_1) \le 0 $$.

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