Supriya Ghosh (Editor)

Kneser's theorem (differential equations)

Updated on
Edit
Like
Comment
Share on FacebookTweet on TwitterShare on LinkedInShare on Reddit

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