Sturm–Picone comparison theorem

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 p_i, q_i for i = 1, 2 be real-valued continuous functions on the interval [a, b] and let

1. ${\displaystyle (p_{1}(x)y^{\prime })^{\prime }+q_{1}(x)y=0}$
2. ${\displaystyle (p_{2}(x)y^{\prime })^{\prime }+q_{2}(x)y=0}$

be two homogeneous linear second order differential equations in self-adjoint form with

${\displaystyle 0<p_{2}(x)\leq p_{1}(x)}$

and

${\displaystyle q_{1}(x)\leq q_{2}(x).}$

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).

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]

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 [1]
• Heinrich Guggenheimer (1977) Applicable Geometry, page 79, Krieger, Huntington ISBN 0-88275-368-1 .
• Teschl, G. (2012). Ordinary Differential Equations and Dynamical Systems. Providence: American Mathematical Society. ISBN 978-0-8218-8328-0.

This page was last edited on 29 August 2023, at 12:32