Sturm-Picone comparison theorem

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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 certain linear differential equations.

[edit] Sturm-Picone comparison theorem

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 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).