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2013-12-02T17:15:56Z

158.39.149.68: /* General Solution of Laplaces Equation */

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 <math>p_i,\, q_i,\,</math> ''i'' = 1, 2, be real-valued continuous functions on the interval [''a'', ''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 ''u'' be a non-trivial solution of (1) with successive roots at ''z''1 and ''z''2 and let ''v'' be a non-trivial solution of (2). Then one of the following properties holds.
*There exists an ''x'' in [''z''1, ''z''2] such that ''v''(''x'') = 0; or
*there exists a λ in '''R''' such that ''v''(''x'') = λ ''u''(''x'').

NOTE: 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> The second (alternative) part of this 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 doi|10.1007/3-7643-7359-8_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]]. 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]].

== References ==

{{Reflist}}
*Diaz, J. B.; [[Joyce McLaughlin|McLaughlin, Joyce R.]] ''Sturm comparison theorems for ordinary and partial differential equations''. Bull. Amer. Math. Soc. 75 1969 335–339 [http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.bams/1183530292 pdf]
* [[Heinrich Guggenheimer]] (1977) ''Applicable Geometry'', page 79, Krieger, Huntington ISBN 0-88275-368-1 .
*{{cite book| last = Teschl| given = G.|authorlink=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 = http://www.mat.univie.ac.at/~gerald/ftp/book-ode/}}

{{DEFAULTSORT:Sturm-Picone comparison theorem}}
[[Category:Ordinary differential equations]]
[[Category:Theorems in analysis]]

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