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
be two homogeneous linear second order differential equations in self-adjoint form with
and
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),C. Sturm, Mémoire sur les équations différentielles linéaires du second ordre, J. Math. Pures Appl. 1 (1836), 106–186 while the second (alternative) part of the theorem is due to Picone (1910)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. 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
Notes
References
Diaz, J. B.; McLaughlin, Joyce R. Sturm comparison theorems for ordinary and partial differential equations. Bull. Amer. Math. Soc. 75 1969 335–339 Heinrich Guggenheimer (1977) Applicable Geometry, page 79, Krieger, Huntington .