View source for Sturm–Picone comparison theorem

In [[mathematics]], in the field of [[ordinary differential equation]]s, the '''Sturm–Picone comparison theorem''', named after [[Jacques Charles François Sturm]] and [[Mauro Picone]], is a classical theorem which provides criteria for the [[oscillation theory|oscillation]] and [[oscillation theory|non-oscillation]] of solutions of certain [[linear differential equation]]s in the real domain.

Let {{mvar|p<sub>i</sub>}}, {{mvar|q<sub>i</sub>}} for {{math|''i''&nbsp;{{=}}&nbsp;1,&nbsp;2}} be real-valued continuous functions on the interval {{math|[''a'',&nbsp;''b'']}} and let
#<math>(p_1(x) y^\prime)^\prime + q_1(x) y = 0 </math> 
#<math>(p_2(x) y^\prime)^\prime + q_2(x) y = 0 </math>
be two homogeneous linear second order differential equations in [[self-adjoint form]] with
:<math>0 < p_2(x) \le p_1(x)</math>
and
:<math>q_1(x) \le q_2(x).</math>

Let {{mvar|u}} be a non-trivial solution of (1) with successive roots at {{mvar|z<sub>1</sub>}} and {{mvar|''z''<sub>2</sub>}} and let {{mvar|v}} be a non-trivial solution of (2). Then one of the following properties holds.
*There exists an {{mvar|x}} in {{math|(''z''<sub>1</sub>,&nbsp;''z''<sub>2</sub>)}} such that {{math|''v''(''x'')&nbsp;{{=}}&nbsp;0;}} or
*there exists a {{mvar|λ}} in '''R''' such that {{math|''v''(''x'')&nbsp;{{=}}&nbsp;λ&thinsp;''u''(''x'')}}.

The first part of the conclusion is due to Sturm (1836),<ref>C. Sturm, Mémoire sur les équations différentielles linéaires du second ordre, J. Math. Pures Appl. 1 (1836), 106–186</ref> while the second (alternative) part of the theorem is due to Picone (1910)<ref>M. Picone, Sui valori eccezionali di un parametro da cui dipende un'equazione differenziale lineare ordinaria del second'ordine, Ann. Scuola Norm. Pisa 11 (1909), 1–141.</ref><ref>{{Cite book | last1 = Hinton | first1 = D. | chapter = Sturm’s 1836 Oscillation Results Evolution of the Theory | doi = 10.1007/3-7643-7359-8_1 | title = Sturm-Liouville Theory | pages = 1–27 | year = 2005 | isbn = 3-7643-7066-1 }}</ref> whose simple proof was given using his now famous [[Picone identity]]. In the special case where both equations are identical one obtains the [[Sturm separation theorem]].<ref>For an extension of this important theorem to a comparison theorem involving three or more real second order equations see the [[Hartman–Mingarelli comparison theorem]] where a simple proof was given using the [[Mingarelli identity]]</ref>

==Notes==
{{Reflist|30em}}

== References ==

*Diaz, J. B.; [[Joyce McLaughlin|McLaughlin, Joyce R.]] ''Sturm comparison theorems for ordinary and partial differential equations''. Bull. Amer. Math. Soc. 75 1969 335&ndash;339 [https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-75/issue-2/Sturm-comparison-theorems-for-ordinary-and-partial-differential-equations/bams/1183530292.full]
* [[Heinrich Guggenheimer]] (1977) ''Applicable Geometry'', page 79, Krieger, Huntington {{isbn|0-88275-368-1}} .
*{{cite book| last = Teschl| given = G.|author-link=Gerald Teschl| title = Ordinary Differential Equations and Dynamical Systems| publisher=[[American Mathematical Society]]| place = [[Providence, Rhode Island|Providence]]| year = 2012| isbn= 978-0-8218-8328-0| url = https://www.mat.univie.ac.at/~gerald/ftp/book-ode/}}

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[[Category:Ordinary differential equations]]
[[Category:Theorems in analysis]]