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In the field of [[ordinary differential equation]]s, the '''Mingarelli identity''' (coined by [[Philip Hartman]]<ref name="Hartman">{{harvnb|Clark D.N., G. Pecelli, and R. Sacksteder|1981}}</ref>) is a theorem that provides criteria for the [[oscillation theory|oscillation]] and [[oscillation theory|non-oscillation]] of solutions of some [[linear differential equation]]s in the real domain. It extends the [[Picone identity]] from two to three or more differential equations of the second order. Its most basic form appears here.
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== The identity ==
Consider the <math>n</math> solutions of the following (uncoupled) system of second order linear differential equations over the ''t''-interval [''a'',&nbsp;''b''].
<math>(p_i(t) x_i^\prime)^\prime + q_i(t) x_i = 0, \,\,\,\,\,\,\,\,\,\, x_i(a)=1,\,\, x_i^\prime(a)=R_i\,</math>
where <math>i=1,2, \ldots, n</math>. Let <math>\Delta</math> denote the forward difference operator, i.e., <math>\Delta x_i = x_{i+1}-x_i.</math> The second order difference operator is found by iterating the first order operator as in <math>\Delta^2 (x_i) = \Delta(\Delta x_i) = x_{i+2}-2x_{i+1}+x_{i}</math>, with a similar definition for the higher iterates.  
 
Leaving out the independent variable ''t'' for convenience, and assuming the <math>x_i(t) \ne 0</math> on (''a'',&nbsp;''b''], there holds the identity,<ref name="Mingarelli">{{harvnb|Mingarelli|1979|p=223}}</ref>
: <math>
\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}
</math>
 
where <math>r_i = x^\prime_i/x_i</math> is a [[logarithmic derivative]], <math>W(x_i, x_j) = x^\prime_ix_j - x_ix^\prime_j</math>, is a [[Wronskian]] and the <math>C(n-1,k)</math> are [[binomial coefficients]]. When <math>n=2</math> this reduces to the [[Picone identity]].
 
The above identity leads quickly to the following comparison theorem for three linear differential equations,<ref name="Mingarelli">{{harv|Mingarelli|1979|Theorem 2}}</ref> extending the [[Sturm–Picone comparison theorem]].
 
Let <math>p_i,\, q_i,\,</math> ''i''&nbsp;=&nbsp;1,&nbsp;2, 3 be real-valued continuous functions on the interval [''a'',&nbsp;''b''] and let
#<math>(p_1(t) x_1^\prime)^\prime + q_1(t) x_1 = 0, \,\,\,\,\,\,\,\,\,\, x_1(a)=1,\,\, x_1^\prime(a)=R_1\,</math>
#<math>(p_2(t) x_2^\prime)^\prime + q_2(t) x_2 = 0, \,\,\,\,\,\,\,\,\,\, x_2(a)=1,\,\,  x_2^\prime(a)=R_2\,</math>
#<math>(p_3(t) x_3^\prime)^\prime + q_3(t) x_3 = 0, \,\,\,\,\,\,\,\,\,\, x_3(a)=1,\,\,  x_3^\prime(a)=R_3\,</math>
be three homogeneous linear second order differential equations in [[self-adjoint form]] with
:<math>p_i(t)>0\,</math> for each i and for all ''t'' in [''a'',&nbsp;''b''], and where the <math>R_i</math> are arbitrary real numbers.
 
Assume that for all ''t'' in [''a'',&nbsp;''b''] we have,
:<math>\Delta^2(q_1) \ge 0 </math>,
:<math>\Delta^2(p_1) \le 0 </math>,
:<math>\Delta^2(p_1(a)R_1) \le 0 </math>.
If <math>x_1(t) > 0</math> on [''a'',&nbsp;''b''], and <math>x_2(b)=0</math>, then any solution <math>x_3(t)</math> has at least one zero in [''a'',&nbsp;''b''].
 
== References ==
 
{{Reflist}}
 
* {{cite book
|last=Clark D.N., G. Pecelli, and R. Sacksteder
|first=
|year=1981
|author-link=
|author2-link= G. Pecelli
|author3-link=R. Sacksteder
|language=
|title=Contributions to Analysis and Geometry
|journal=
|series=
|volume=
|pages=
|location= Baltimore, USA
|publisher= Johns Hopkins University Press
|isbn=
|url=
|ref=harv}}
 
* {{cite journal
|last=Mingarelli
|first= Angelo B.
|year=1979
|author-link=
|author2-link=
|language=
|title= Some extensions of the Sturm–Picone theorem
|journal= Comptes Rendus Math. Rep. Acad. Sci. Canada
|series=
|volume=1 (4)
|pages=223–226
|location= Toronto, Ontario, Canada
|publisher= The Royal Society of Canada
|isbn=
|url=
|ref=harv}}
 
[[Category:Ordinary differential equations]]
[[Category:Mathematical identities]]

Revision as of 00:38, 5 March 2014

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