In mathematics, in the field of

ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contras ...

s, the Sturm–Picone comparison theorem, named after Jacques Charles François Sturm and Mauro Picone, is a classical theorem which provides criteria for the

oscillation Oscillation is the repetitive or Periodic function, periodic variation, typically in time, of some measure about a central value (often a point of Mechanical equilibrium, equilibrium) or between two or more different states. Familiar examples o ...

and non-oscillation of solutions of certain

linear differential equation In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form :a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^ ...

s in the real domain. Let , for be real-valued continuous functions on the interval 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 In mathematics, and more specifically in abstract algebra, an element ''x'' of a *-algebra is self-adjoint if x^*=x. A self-adjoint element is also Hermitian, though the reverse doesn't necessarily hold. A collection ''C'' of elements of a sta ...

with :

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

and :

q_1(x) \le q_2(x).

Let be a non-trivial solution of (1) with successive roots at and and let be a non-trivial solution of (2). Then one of the following properties holds. *There exists an in such that or *there exists a in such that . 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 field of ordinary differential equations, the Picone identity, named after Mauro Picone, is a classical result about homogeneous linear second order differential equations. Since its inception in 1910 it has been used with tremendous success ...

. In the special case where both equations are identical one obtains the Sturm separation theorem.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 In the field of ordinary differential equations, the Mingarelli identityThe locution was coined by Philip Hartman, according to is a theorem that provides criteria for the oscillation and non-oscillation of solutions of some linear differential ...

Notes

References

*Diaz, J. B.; McLaughlin, Joyce R. ''Sturm comparison theorems for ordinary and partial differential equations''. Bull. Amer. Math. Soc. 75 1969 335–33

*

Heinrich Guggenheimer Heinrich Walter Guggenheimer (July 21, 1924 – March 4, 2021) was a German-born Swiss-American mathematician who has contributed to knowledge in differential geometry, topology, algebraic geometry, and convexity. He has also contributed volumes o ...

(1977) ''Applicable Geometry'', page 79, Krieger, Huntington . * {{DEFAULTSORT:Sturm-Picone comparison theorem Ordinary differential equations Theorems in analysis