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

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

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

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

and

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

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 $v (x) = λ u (x)$ .

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

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.
Hinton, D. (2005). "Sturm's 1836 Oscillation Results Evolution of the Theory". Sturm-Liouville Theory. pp. 1–27. doi:10.1007/3-7643-7359-8_1. ISBN 3-7643-7066-1.
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

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 ISBN 0-88275-368-1 .
Teschl, G. (2012). Ordinary Differential Equations and Dynamical Systems. Providence: American Mathematical Society. ISBN 978-0-8218-8328-0.

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[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] Hinton, D. (2005). "Sturm's 1836 Oscillation Results Evolution of the Theory". Sturm-Liouville Theory. pp. 1–27. doi:10.1007/3-7643-7359-8_1. ISBN 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