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| In [[mathematics]], in the field of [[ordinary differential equation]]s, the '''Kneser theorem''', named after [[Adolf Kneser]], provides criteria to decide whether a differential equation is [[Oscillation theory|oscillating]] or not.
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| == Statement of the theorem ==
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| Consider an ordinary linear homogenous differential equation of the form
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| :<math>y'' + q(x)y = 0\,</math> | |
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| with
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| :<math>q: [0,+\infty) \to \mathbb{R}</math>
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| [[continuous function|continuous]].
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| We say this equation is ''oscillating'' if it has a solution ''y'' with infinitely many zeros, and ''non-oscillating'' otherwise.
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| The theorem states<ref>{{cite book| last = Teschl| given = Gerald|authorlink=Gerald Teschl| title = Ordinary Differential Equations and Dynamical Systems| publisher=[[American Mathematical Society]]| place = [[Providence, Rhode Island|Providence]]| year = 2012| isbn= 978-0-8218-8328-0| url = http://www.mat.univie.ac.at/~gerald/ftp/book-ode/}}</ref> that the equation is non-oscillating if
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| :<math>\limsup_{x \to +\infty} x^2 q(x) < \tfrac{1}{4}</math>
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| and oscillating if
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| :<math>\liminf_{x \to +\infty} x^2 q(x) > \tfrac{1}{4}.</math>
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| == Example ==
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| To illustrate the theorem consider
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| :<math>q(x) = \left(\frac{1}{4} - a\right) x^{-2} \quad\text{for}\quad x > 0</math>
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| where <math>a</math> is real and non-zero. According to the theorem, solutions will be oscillating or not depending on whether <math>a</math> is positive (non-oscillating) or negative (oscillating) because
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| :<math>\limsup_{x \to +\infty} x^2 q(x) = \liminf_{x \to +\infty} x^2 q(x) = \frac{1}{4} - a</math>
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| To find the solutions for this choice of <math>q(x)</math>, and verify the theorem for this example, substitute the 'Ansatz'
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| :<math>y(x) = x^n \, </math>
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| which gives
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| :<math>n(n-1) + \frac{1}{4} - a = \left(n-\frac{1}{2}\right)^2 - a = 0</math>
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| This means that (for non-zero <math>a</math>) the general solution is
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| :<math>y(x) = A x^{\frac{1}{2} + \sqrt{a}} + B x^{\frac{1}{2} - \sqrt{a}}</math> | |
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| where <math>A</math> and <math>B</math> are arbitrary constants.
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| It is not hard to see that for positive <math>a</math> the solutions do not oscillate while for negative <math>a = -\omega^2</math> the identity
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| :<math>x^{\frac{1}{2} \pm i \omega} = \sqrt{x}\ e^{\pm (i\omega) \ln{x}} = \sqrt{x}\ (\cos{(\omega \ln x)} \pm i \sin{(\omega \ln x)})</math>
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| shows that they do.
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| The general result follows from this example by the [[Sturm–Picone comparison theorem]].
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| ==Extensions==
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| There are many extensions to this result. For a recent account see.<ref>Helge Krüger and Gerald Teschl, ''Effective Prüfer angles and relative oscillation criteria'', J. Diff. Eq. 245 (2008), 3823–3848 [http://dx.doi.org/10.1016/j.jde.2008.06.004]</ref>
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| ==References==
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| {{reflist}}
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| {{DEFAULTSORT:Kneser Theorem}}
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| [[Category:Ordinary differential equations]]
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| [[Category:Theorems in analysis]]
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| [[Category:Oscillation]]
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