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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''<sub>1</sub> and ''z''<sub>2</sub> and let ''v'' be a non-trivial solution of (2). Then one of the following properties holds. | |||
*There exists an ''x'' in [''z''<sub>1</sub>, ''z''<sub>2</sub>] 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]] | |||
Revision as of 18:15, 2 December 2013
In mathematics, in the field of ordinary differential equations, the Sturm–Picone comparison theorem, named after Jacques Charles François Sturm and Mauro Picone, is a classical theorem which provides criteria for the oscillation and non-oscillation of solutions of certain linear differential equations in the real domain.
Let i = 1, 2, be real-valued continuous functions on the interval [a, b] and let
be two homogeneous linear second order differential equations in self-adjoint form with
and
Let u be a non-trivial solution of (1) with successive roots at z1 and z2 and let v be a non-trivial solution of (2). Then one of the following properties holds.
- There exists an x in [z1, z2] 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).[1] The second (alternative) part of this theorem is due to Picone (1910)[2][3] 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
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- Diaz, J. B.; McLaughlin, Joyce R. Sturm comparison theorems for ordinary and partial differential equations. Bull. Amer. Math. Soc. 75 1969 335–339 pdf
- Heinrich Guggenheimer (1977) Applicable Geometry, page 79, Krieger, Huntington ISBN 0-88275-368-1 .
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- ↑ C. Sturm, Mémoire sur les équations différentielles linéaires du second ordre, J. Math. Pures Appl. 1 (1836), 106–186
- ↑ 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.
- ↑ Template:Cite doi