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

1. $(p_1(x)y^{\prime }{)}^{\prime }+q_1(x)y=0$
2. $(p_2(x)y^{\prime }{)}^{\prime }+q_2(x)y=0$

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

$0<p_2(x)\le p_1(x)$

and

$q_1(x)\le q_2(x).$

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

• There exists an Template:Mvar in (z₁, z₂) such that v(x) = 0; or
• there exists a Template:Mvar in R such that v(x) = λ u(x).

The first part of the conclusion is due to Sturm (1836), while the second (alternative) part of the theorem is due to Picone (1910) 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.

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 Template:Isbn .