From e19da00a10336afd7ffc8a0429dd272af572a96c Mon Sep 17 00:00:00 2001 From: salagne Date: Wed, 1 Jul 2020 15:05:40 +0200 Subject: [PATCH] minor corrections --- ana18.tex | 12 ++++++------ analysisII.pdf | Bin 862805 -> 863389 bytes 2 files changed, 6 insertions(+), 6 deletions(-) diff --git a/ana18.tex b/ana18.tex index 99191d8..560eaa4 100644 --- a/ana18.tex +++ b/ana18.tex @@ -73,7 +73,7 @@ .\end{salign*} \end{proof} -\begin{satz}[Stabilitäts und Eindeutigkeitssatz] +\begin{satz}[Stabilitäts- und Eindeutigkeitssatz] Sei $f(t,x)$ stetig auf $D \subseteq \R \times \R^{n}$ und lokal Lipschitz-stetig bezüglich $x$. Seien $y, v$ zwei beliebige Lösungen der Differentialgleichung \[ @@ -81,10 +81,10 @@ \] auf einem gemeinsamen Existenzintervall $I$. Dann gilt \[ \Vert y(t) - v(t) \Vert \le e^{L(t-t_0)} \Vert y(t_0) - v(t_0)\Vert, \quad t \in I, t \ge t_0 - ,\] wobei $t_0 \in I$ und $L$ die Lipschitz-konstante $L = L_K$ von $f(t,x)$ auf + ,\] wobei $t_0 \in I$ und $L$ die Lipschitz-Konstante $L = L_K$ von $f(t,x)$ auf $K \subseteq D$ mit $K$ beschränkt und $\text{Graph}(y) \subseteq K$, $\text{Graph}(v) \subseteq K$. - Weiterhin: Falls $y(t_0) = v(t_0)$ (d.h. $y$ und $v$ lösen AWA $y' = f(t,y), y(t_0) = y_0)$, dann + Weiterhin: Falls $y(t_0) = v(t_0)$ (d.h. $y$ und $v$ lösen AWA $y' = f(t,y),\ y(t_0) = y_0)$, dann gilt $y(t) = v(t)$, $\forall t \in I$, d.h. die lokale Lösung der AWA (nach Peano) ist eindeutig bestimmt. \end{satz} @@ -92,7 +92,7 @@ Sei $K \subseteq D$ eine beschränkte Teilmenge und $\text{Graph}(y) \subseteq K$, $\text{Graph}(v) \subseteq K$. Betrachte \begin{salign*} - y(t) - v(t) &\stackrel{\text{Integralform}}{=} + h(t) \coloneqq y(t) - v(t) &\stackrel{\text{Integralform}}{=} y(t_0) + \int_{t_0}^{t} f(s, y(s)) \d s - v(t_0) - \int_{t_0}^{t} f(s, v(s)) \d s \\ &= \int_{t_0}^{t} \left( f(s, y(s)) - f(s, v(s)) \right) \d s + y(t_0) - v(t_0) \intertext{Dann folgt} @@ -154,7 +154,7 @@ \Vert g(y)(t) - y_0 \Vert &= \left\Vert \int_{t_0}^{t} f(s, y(s)) \d s \right\Vert \\ &\le \int_{t_0}^{t} \left\Vert f(s, y(s))\right\Vert \d s \\ &\stackrel{f \text{ beschr.}}{\le } M \int_{t_0}^{t} \d s \\ - &= M|t-t_0 + &= M|t-t_0| \le M\epsilon \le \delta .\end{salign*} @@ -184,7 +184,7 @@ y^{k}(t) \coloneqq y_0 + \int_{t_0}^{t} f(s, y^{k-1}(s)) \d s \xrightarrow{k \to \infty} \text{Lösung der AWA} .\end{align*} - \item Ohne Die Forderung der Lipschitz-stetigkeit geht die Eindeutigkeit der Lösung verloren + \item Ohne die Forderung der Lipschitz-stetigkeit geht die Eindeutigkeit der Lösung verloren (siehe Beispiel \ref{bsp:dgl-uneindeutig}), aber die Existenz gilt nach dem Satz von Peano immer noch. \end{enumerate} diff --git a/analysisII.pdf b/analysisII.pdf index 9156729593d57ffc88256bdba8429a8c76dacd85..86a3b1924e24ae25eee84e8c155989a05e878a71 100644 GIT binary patch delta 458611 zcmV)aK&rph)HI#hG>|2ivC{`Ce~pt#Ps1<}h41|pJ|+hbu^q?Bjje1Is4BT2;1F6T zH55n6g4BP{*hVT9>m{)#^WHapNe;@zoROxVB%~n;Isfy1Tuu(oct)54&SXl;ns8YF zOA8{UfPxjIT#@_e$X1(eZ~B-DCZfxj@(iz!xl>cYA;Tq=e;kAmK|Sx< z8c>R&)6OqcG<}JsjH*F*##W&;i+LWk{a|9sA{(=)as(n$$TLAG&w+6k9IwsLFJsiK zJ5vwA`pxZI>ot@mcp{$SA_JB4pgO_serW5^f39)c1oT59lcmFdbbuS7@ePOlFs^VI zUmD$5em=U^W~Dbz7=?;-f7ifDqY3kLu0W~+*_dFO2lkCwwt;(5RrQW)u?}l2c@B)$ z{N_xV1m)uO)+$(Ewf$HH@7dUFbhmSVc8}o2SN{j#41|*VvxnSFWdr0|cZ0Uy1&1y0 z>eBHYhnUqTrmO$RO%urQR`zep18Iz|HrhPyaVjj;Hs1F0V$VZ7y>DiK>`$iS0l} zMWX3NM6g0ckVtZhq~XzY9-YOn@@1Npn~xJ}Vq08U_n&`S=58~2Km8MiAdx{vQt3y) zQKBfqemMVsE}0~TiumQk$atMCY(7btX7M(khlf+^mQCZw38V44`oY5++x)V0>)dys z6l)ywFe4S683{+Q(u889iK%BrFfYNB!Z6ir^)MIB5?HeYcvxiBj&!_SoionQnNn_H 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