Supriya Ghosh (Editor)

Sturm–Picone comparison theorem

Updated on
Share on FacebookTweet on TwitterShare on LinkedInShare on Reddit
Covid-19

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 pi, qi i = 1, 2, be real-valued continuous functions on the interval [ab] and let

  1. ( p 1 ( x ) y ) + q 1 ( x ) y = 0
  2. ( p 2 ( x ) y ) + q 2 ( x ) y = 0

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

0 < p 2 ( x ) p 1 ( x )

and

q 1 ( x ) 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 (z1z2) 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), 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

    Sturm–Picone comparison theorem Wikipedia


    Similar Topics
    Captain John Smith and Pocahontas
    James Abrahart
    Patricia Claxton
    Topics
     
    B
    i
    Link
    H2
    L