Kneser's theorem (differential equations)

Consider an ordinary linear homogeneous differential equation of the form

:$y\text{'}\text{'}\; +\; q(x)y\; =\; 0$

with

:$q:\; [0,+\backslash infty)\; \backslash to\; \backslash mathbb\{R\}$

continuous.

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

The theorem states{{cite book| last = Teschl| given = Gerald|authorlink=Gerald Teschl| title = Ordinary Differential Equations and Dynamical Systems| publisher=American Mathematical Society| place = Providence| year = 2012| isbn= 978-0-8218-8328-0| url = https://www.mat.univie.ac.at/~gerald/ftp/book-ode/}} that the equation is non-oscillating if

:

and oscillating if

:$\backslash liminf\_\{x\; \backslash to\; +\backslash infty\}\; x^2\; q(x)\; >\; \backslash tfrac\{1\}\{4\}.$

To illustrate the theorem consider

:$q(x)\; =\; \backslash left(\backslash frac\{1\}\{4\}\; -\; a\backslash right)\; x^\{-2\}\; \backslash quad\backslash text\{for\}\backslash quad\; x\; >\; 0$

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

:$\backslash limsup\_\{x\; \backslash to\; +\backslash infty\}\; x^2\; q(x)\; =\; \backslash liminf\_\{x\; \backslash to\; +\backslash infty\}\; x^2\; q(x)\; =\; \backslash frac\{1\}\{4\}\; -\; a$

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

:$n(n-1)\; +\; \backslash frac\{1\}\{4\}\; -\; a\; =\; \backslash left(n-\backslash frac\{1\}\{2\}\backslash right)^2\; -\; a\; =\; 0$

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

:$y(x)\; =\; A\; x^\{\backslash frac\{1\}\{2\}\; +\; \backslash sqrt\{a\}\}\; +\; B\; x^\{\backslash frac\{1\}\{2\}\; -\; \backslash sqrt\{a\}\}$

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 $a\; =\; -\backslash omega^2$ the identity

:$x^\{\backslash frac\{1\}\{2\}\; \backslash pm\; i\; \backslash omega\}\; =\; \backslash sqrt\{x\}\backslash \; e^\{\backslash pm\; (i\backslash omega)\; \backslash ln\{x\}\}\; =\; \backslash sqrt\{x\}\backslash \; (\backslash cos\{(\backslash omega\; \backslash ln\; x)\}\; \backslash pm\; i\; \backslash sin\{(\backslash omega\; \backslash ln\; x)\})$

shows that they do.

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

There are many extensions to this result, such as the Gesztesy–Ünal criterion.{{Cite journal |last1=Krüger |first1=Helge |last2=Teschl |first2=Gerald |date=2008 |title=Effective Prüfer angles and relative oscillation criteria |url=https://linkinghub.elsevier.com/retrieve/pii/S0022039608002635 |journal=Journal of Differential Equations |language=en |volume=245 |issue=12 |pages=3823–3848 |doi=10.1016/j.jde.2008.06.004|arxiv=0709.0127 |bibcode=2008JDE...245.3823K |s2cid=6693175 }}

While Peano's existence theorem guarantees the existence of solutions of certain initial values problems with continuous right hand side, H. Kneser's theorem deals with the topology of the set of those solutions. Precisely, H. Kneser's theorem states the following:{{Citation |title=Über die Lösungen eines Systems gewöhnlicher Differentialgleichungen, das der Lipschitzschen Bedingung nicht genügt [7–23] |date=2005-01-31 |url=https://www.degruyter.com/document/doi/10.1515/9783110894516.58/html |work=Gesammelte Abhandlungen / Collected Papers |pages=58–61 |editor-last=Hofmann |editor-first=Karl H. |place=Berlin, New York |publisher=DE GRUYTER |doi=10.1515/9783110894516.58 |isbn=978-3-11-089451-6 |access-date=2023-01-21 |editor2-last=Betsch |editor2-first=Gerhard}}{{Cite book |last=Hartman |first=Philip |url=http://epubs.siam.org/doi/book/10.1137/1.9780898719222 |title=Ordinary Differential Equations |date=2002 |publisher=Society for Industrial and Applied Mathematics |isbn=978-0-89871-510-1 |edition=Second |language=en |doi=10.1137/1.9780898719222.ch2}}

Let $f\backslash colon\; \backslash R\backslash times\; \backslash R^n\; \backslash rightarrow\; \backslash R^n$ be a continuous function on the region $\backslash mathcal\{R\}:=[t\_0,\; t\_0+a]\; \backslash times\; \backslash \{x\; \backslash in\; \backslash mathbb\{R\}^n:\; \backslash Vert\; x-x\_0\backslash Vert\; \backslash le\; b\backslash \}$, and such that $|f(t,\; x)|\; \backslash le\; M$ for all $(t,x)\; \backslash in\; \backslash mathcal\{R\}$.

Given a real number $c$ satisfying

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Category:Ordinary differential equations

Category:Theorems in mathematical analysis

Category:Oscillation