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

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

and

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),[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

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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

References

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