From 6f71aefdec8cd87e1eddde43f2bda96366b84cef Mon Sep 17 00:00:00 2001 From: JosuaKugler Date: Wed, 8 Jul 2020 21:34:35 +0200 Subject: [PATCH] minor corrections --- ana19.tex | 22 +++++++++++++--------- analysisII.pdf | Bin 1062206 -> 1004407 bytes 2 files changed, 13 insertions(+), 9 deletions(-) diff --git a/ana19.tex b/ana19.tex index 1f8dead..00cb5d5 100644 --- a/ana19.tex +++ b/ana19.tex @@ -88,19 +88,23 @@ \[ y'(t) = f(t,y(t)) \ \ \ \ \ \implies \ \ \ \ \ y'(t) - f(t,y(t)) + f(t,0) = f(t,0). \] - Womit wir erhalten: + Indem wir das Skalarprodukt mit $y(t)$ bilden, erhalten wir daraus: \[ \left(y(t), y'(t)\right) - \left(f(t,y(t)) - f(t,0), y(t) - 0\right) = \left(f(t,0),y(t)\right). \] - Wegen Monotonie gilt: + Außerdem gilt \[ - \left(y(t), y'(t)\right) = \frac{1}{2}\frac{\d{}}{\d{t}} \norm{y(t)}^{2} + \left(f(t,y(t)) - f(t,0), y(t)\right) \leq -\lambda\norm{y(t)}^{2}. + \left(y(t), y'(t)\right) = \frac{1}{2}\frac{\d{}}{\d{t}} \norm{y(t)}^{2} + \] + und aufgrund der Monotonie: + \[ + \left(f(t,y(t)) - f(t,0), y(t)\right) \leq -\lambda\norm{y(t)}^{2}. \] Somit können wir folgern: \begin{salign*} - \frac{1}{2} \cdot \frac{\d{}}{\d{t}} \norm{y(t)}^{2} + 2\lambda \norm{y(t)}^{2} &\leq \frac{1}{2} \cdot \frac{\d{}}{\d{t}} \norm{y(t)}^{2} - \left( f(t,y(t)) - f(t,0), y(t)\right)\\ + \frac{1}{2} \cdot \frac{\d{}}{\d{t}} \norm{y(t)}^{2} + \lambda \norm{y(t)}^{2} &\leq \frac{1}{2} \cdot \frac{\d{}}{\d{t}} \norm{y(t)}^{2} - \left( f(t,y(t)) - f(t,0), y(t)\right)\\ &= \left(f(t,0), y(t)\right) \oldstackrel{\text{CSU}}{\leq} \norm{f(t,0)}\cdot \norm{y(t)} \\ - &\leq \frac{1}{2\lambda} \norm{f(t,0)}^{2} + \frac{\lambda}{2} \norm{y(t)}^{2} + &\stackrel{2ab \le a^2 + b^2}{\le} \frac{1}{2\lambda} \norm{f(t,0)}^{2} + \frac{\lambda}{2} \norm{y(t)}^{2} \end{salign*} woraus folgt \begin{salign*} @@ -249,7 +253,7 @@ \begin{enumerate}[1)] \item Sei $\psi \coloneqq \int_{t_{0}}^{t} \phi^{-1}(s)b(s) \d{s} + c$. Dann gilt für $t\geq t_{0}$: $\psi' = \phi^{-1}(t)b(t)$. Für $y_{b} = \phi\psi$ gilt dann: \[ - y_{b}' = \phi'\psi + \phi\psi' = A\phi\psi = \phi\phi^{-1}b = Ay_{b} + b. + y_{b}' = \phi'\psi + \phi\psi' = A\phi\psi + \phi\phi^{-1}b = Ay_{b} + b. \] Also ist $y_{b}$ eine Lösung des inhomogenen Systems und gilt $c=y_{0}$, dann löst $y_{b}$ die AWA $y' = Ay + b$, $y(t_{0}) = y_{0}$. Daraus folgt 1) und 3). \item Sei $y$ eine zweite Lösung des inhomogenen Systems. Für $w=y-y_{b}$ gilt dann: @@ -282,7 +286,7 @@ & y'(t) = f(t,y(t)), && t \in I = [a,b] \\ & r(y(a),y(b)) = 0 \end{salign*} - Gesucht ist eine stetig differenzierbare Lösung $y \colon I \to \R^{n}$ die beide Bedingungne erfüllt. Die zweite Bedingung lässt sich verallgemeinern zu einer Mehrpunkt RWA: + Gesucht ist eine stetig differenzierbare Lösung $y \colon I \to \R^{n}$ die beide Bedingungen erfüllt. Die zweite Bedingung lässt sich verallgemeinern zu einer Mehrpunkt RWA: \[ r\left( y(t_{1}),...,y(t_{k}) \right) = 0. \] @@ -326,7 +330,7 @@ \end{definition} \begin{bem} - Lösung des inhomogenen DLG-System ist der Form: + Eine Lösung des inhomogenen DLG-System ist von der Form: \[ y(t,s) = \varphi^{0}(t) + \sum_{i=1}^{n} s_{i}\varphi^{i}(t) = \varphi^{0}(t) + \phi(t)s. \] @@ -377,7 +381,7 @@ weil die Funktionen $\{ \varphi^{1},...,\varphi^{n}\}$ eine Basis des Lösungsraum des assoziierten homogenen DGL bilden. \\ Für homogene RWA gilt $g - B_{b}\varphi^{0}(b) = 0$ und die Gleichung für $s$ lautet \[ - \left( B_{a} +B_{b}\phi(b)\right)s = 0- + \left( B_{a} +B_{b}\phi(b)\right)s = 0 \] Daraus folgen alle Behauptungen. \end{proof} diff --git a/analysisII.pdf b/analysisII.pdf index 3c95f8cc7e75ec46507e42c31ee175c2662267aa..0a506e3e33b6126207a813c0d5e73160b468a06a 100644 GIT binary patch delta 772155 zcmV)cK&ZdI>Ol9VJrE^OL`E$!E;R}z(A34)3Ne>q9|;oyG?$T{3M+q&Q%!H3Fc7@I 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