Sturm–Picone comparison theorem explained

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, for be real-valued continuous functions on the interval and let

(p _{1(x)} y ^{\prime)} ^{\prime} + q _{1(x)} y = 0 (p _{2(x)} y ^{\prime)} ^{\prime} + q _{2(x)} y = 0 be two homogeneous linear second order differential equations in self-adjoint form with0 < p _{2(x)} \le p _{1(x)} andq _{1(x)} \le q _{2(x).}

Let be a non-trivial solution of (1) with successive roots at and and let be a non-trivial solution of (2). Then one of the following properties holds.

• There exists an in such that or
• there exists a in R such that .

The first part of the conclusion is due to Sturm (1836),^[1] while the second (alternative) part of the 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.^[4]

References

• Diaz, J. B.; McLaughlin, Joyce R. Sturm comparison theorems for ordinary and partial differential equations. Bull. Amer. Math. Soc. 75 1969 335 - 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 .
• Book: Teschl. G.. Gerald Teschl. Ordinary Differential Equations and Dynamical Systems. American Mathematical Society. Providence. 2012. 978-0-8218-8328-0.

Notes and References

1. C. Sturm, Mémoire sur les équations différentielles linéaires du second ordre, J. Math. Pures Appl. 1 (1836), 106–186
2. 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.
3. Book: Hinton . D. . Sturm’s 1836 Oscillation Results Evolution of the Theory . 10.1007/3-7643-7359-8_1 . Sturm-Liouville Theory . 1–27 . 2005 . 3-7643-7066-1 .
4. 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