Mingarelli 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 ${\displaystyle \Delta }$ 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.

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].

• 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.