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Kneser's theorem (differential equations)

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In mathematics, in the field of ordinary differential equations, the Kneser theorem, named after Adolf Kneser, provides criteria to decide whether a differential equation is oscillating or not.

Contents

Statement of the theorem

Consider an ordinary linear homogeneous differential equation of the form

y ″ + q ( x ) y = 0

with

q : [ 0 , + ∞ ) → R

continuous. We say this equation is oscillating if it has a solution y with infinitely many zeros, and non-oscillating otherwise.

The theorem states that the equation is non-oscillating if

lim sup x → + ∞ x 2 q ( x ) < 1 4

and oscillating if

lim inf x → + ∞ x 2 q ( x ) > 1 4 .

Example

To illustrate the theorem consider

q ( x ) = ( 1 4 − a ) x − 2 for x > 0

where a is real and non-zero. According to the theorem, solutions will be oscillating or not depending on whether a is positive (non-oscillating) or negative (oscillating) because

lim sup x → + ∞ x 2 q ( x ) = lim inf x → + ∞ x 2 q ( x ) = 1 4 − a

To find the solutions for this choice of q ( x ) , and verify the theorem for this example, substitute the 'Ansatz'

y ( x ) = x n

which gives

n ( n − 1 ) + 1 4 − a = ( n − 1 2 ) 2 − a = 0

This means that (for non-zero a ) the general solution is

y ( x ) = A x 1 2 + a + B x 1 2 − a

where A and B are arbitrary constants.

It is not hard to see that for positive a the solutions do not oscillate while for negative a = − ω 2 the identity

x 1 2 ± i ω = x   e ± ( i ω ) ln ⁡ x = x   ( cos ⁡ ( ω ln ⁡ x ) ± i sin ⁡ ( ω ln ⁡ x ) )

shows that they do.

The general result follows from this example by the Sturm–Picone comparison theorem.

Extensions

There are many extensions to this result. For a recent account see.

References

Kneser's theorem (differential equations) Wikipedia
(Text) CC BY-SA


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