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 [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.

[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

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