From eddafd9ef3681c3eeee952084c55c72339862cce Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Petr=20Veli=C4=8Dka?= Date: Thu, 27 Feb 2025 12:21:05 +0100 Subject: [PATCH] formatovani --- lokalni-existence.tex | 61 +++++++++++++++++++++++------------------- skripta.pdf | Bin 60620 -> 60681 bytes 2 files changed, 33 insertions(+), 28 deletions(-) diff --git a/lokalni-existence.tex b/lokalni-existence.tex index d1cc912..d0cd93c 100644 --- a/lokalni-existence.tex +++ b/lokalni-existence.tex @@ -28,14 +28,15 @@ Takto definované řešení je nutně spojité a má spojitou derivaci (je tří \item $x$ je řešení (\ref{eq-ode}) splňující $x(t_0) = x_0$, \item pro každé $t \in I$ platí $x(t) = x_0 + \int^t_{t_0} f(x(s), s)ds$. \end{enumerate} - \begin{proof} - Víme, že platí $x'(s) = f(x(s), s)$ pro všechna $s \in I$, což je spojitá funkce, kterou můžeme zintegrovat na $[t_0, t]$. - Potom z Newtonova-Leibnizova vzorce máme $x(t) - x(t_0) = \int_{t_0}^t x'(s) ds = \int_{t_0}^t f(x(s), s) ds$. Tedy $x(t) = x_0 + \int_{t_0}^t f(x(s), s) ds$. - - Pro důkaz opačné strany si uvědomíme, ze pro každé $t\in I$ je pravá strana diferencovatelná, tedy $x'(t) = f(x(t), t)$ a po dosazení $t = t_0$ dostáváme $x(t_0) = x_0$. - \end{proof} \end{lemma} +\begin{proof} + Víme, že platí $x'(s) = f(x(s), s)$ pro všechna $s \in I$, což je spojitá funkce, kterou můžeme zintegrovat na $[t_0, t]$. + Potom z Newtonova-Leibnizova vzorce máme $x(t) - x(t_0) = \int_{t_0}^t x'(s) ds = \int_{t_0}^t f(x(s), s) ds$. Tedy $x(t) = x_0 + \int_{t_0}^t f(x(s), s) ds$. + + Pro důkaz opačné strany si uvědomíme, ze pro každé $t\in I$ je pravá strana diferencovatelná, tedy $x'(t) = f(x(t), t)$ a po dosazení $t = t_0$ dostáváme $x(t_0) = x_0$. +\end{proof} + Teď si zadefinujeme několik pojmů, které charakterizují množiny funkcí, které se chovají jistým způsobem podobně nebo stejně. \begin{definition} @@ -56,27 +57,31 @@ Následující věta nám říká, že na nějakém okolí libovolného bodu exi \begin{theorem}{\textbf{(Peano)}} \label{thm-peano} Nechť $(x_0, t_0) \in \Omega$. Pak existuje $\delta > 0$ a funkce $x(t): (t_0 - \delta, t_0 + \delta) \rightarrow \mathbb{R}^n$, která je řešením (\ref{eq-ode}) a splňuje $x(t_0) = x_0$. - \begin{proof} - Nejdříve dokážeme pomocné tvrzení: - \begin{lemma} - Pokud $\Omega = \mathbb{R}^{n+1}$ a $f$ je omezená na $\Omega$, pak pro každé $T > 0$ existuje řešení (\ref{eq-ode}) na $(t_0 - T, t_0 + T)$ splňující $x(t_0) = x_0$. - \begin{proof} - Řešme ``porušenou" úlohu $(P_\lambda)$: $x(t) = x_0 + \int_{t_0}^t f(x(s - \lambda), s) ds$ pro $ t > t_0$ a $x(t) = x_0$ pro $t \in [t_0 - \lambda, t_0]$. - Na $I_1 := (t_0, t_0 + \lambda]$ definujeme $x(t) = x_0 + \int_{t_0}^t f(x(s - \lambda, s) ds$. - Na $I_2 := (t_0 + \lambda, t_0 + 2\lambda]$ definujeme $x(t)$ obdobně a indukcí pokračujeme až do nekonečna. - Tímto je ``porušená" úloha vyřešena na $[t_0-\lambda, t_0 + T]$. - - Položme $\lambda = \frac{1}{n}$ pro $n = 1,2,\dots$. Pišme dále jen $x_n$ namísto $x_{1/n}$, tedy máme posloupnost funkcí. - Ukážeme, že jsou stejně spojité a stejně omezené. - Stejná omezenost plyne z toho, že $\| x_n (t) \| = \| x_0 + \int_{t_0}^t f(x(s - \frac{1}{n}), s) ds \| \leq \| x_0 \| + \int_{t_0}^t f(x(s - \frac{1}{n}) \| ds$. Ale funkce $f$ je omezená, tedy máme $\| x_n(t) \| \leq \| x_0 \| + (T - t_0) \cdot K$, kde $K$ je příslušná konstanta omezenosti $f$. - Stejnou spojitost máme z odhadu $\| x_n(t) - x_n(r) \| = \| \int_r^t f(x(s - \frac{1}{n}), s) ds \| \leq |t - r| \cdot K$. V poslední nerovnosti jsme odhadli integrál součinem délky intervalu a konstantou omezenosti funkce $f$. Stačí položit $\delta = \frac{\varepsilon}{K}$, potom $\|x_n(t) - x_r(t)\| < \delta K = \varepsilon$. - - Tedy dle Věty \ref{thm-arzela} můžeme z posloupnosti $x_n$ vybrat stejnoměrně konvergentní podposloupnost. Zbývá dokázat, že její limita řeší naši rovnici. - - \hfill \textit{konec 1. přednášky (21.2.2025)} - - \end{proof} - \end{lemma} - \end{proof} \end{theorem} +K důkazu této věty budeme potřebovat pomocné lemma: + +\begin{lemma} + Pokud $\Omega = \mathbb{R}^{n+1}$ a $f$ je omezená na $\Omega$, pak pro každé $T > 0$ existuje řešení (\ref{eq-ode}) na $(t_0 - T, t_0 + T)$ splňující $x(t_0) = x_0$. +\end{lemma} + +\begin{proof} + Řešme ``porušenou" úlohu: $x(t) = x_0 + \int_{t_0}^t f(x(s - \lambda), s) ds$ pro $ t > t_0$ a $x(t) = x_0$ pro $t \in [t_0 - \lambda, t_0]$. + Na $I_1 := (t_0, t_0 + \lambda]$ definujeme $x(t) = x_0 + \int_{t_0}^t f(x(s - \lambda, s) ds$. + Na $I_2 := (t_0 + \lambda, t_0 + 2\lambda]$ definujeme $x(t)$ obdobně a indukcí pokračujeme dokud $t_0 + k\lambda$ nebude větší než $T$. + Tímto je ``porušená" úloha vyřešena na $[t_0-\lambda, t_0 + T]$. + + Položme $\lambda = \frac{1}{n}$ pro $n = 1,2,\dots$. Pišme dále jen $x_n$ namísto $x_{1/n}$, tedy máme posloupnost funkcí. + Ukážeme, že jsou stejně spojité a stejně omezené. + Stejná omezenost plyne z toho, že $\| x_n (t) \| = \| x_0 + \int_{t_0}^t f(x(s - \frac{1}{n}), s) ds \| \leq \| x_0 \| + \int_{t_0}^t f(x(s - \frac{1}{n}) \| ds$. Ale funkce $f$ je omezená, tedy máme $\| x_n(t) \| \leq \| x_0 \| + (T - t_0) \cdot K$, kde $K$ je příslušná konstanta omezenosti $f$. + Stejnou spojitost máme z odhadu $\| x_n(t) - x_n(r) \| = \| \int_r^t f(x(s - \frac{1}{n}), s) ds \| \leq |t - r| \cdot K$. V poslední nerovnosti jsme odhadli integrál součinem délky intervalu a konstantou omezenosti funkce $f$. Stačí položit $\delta = \frac{\varepsilon}{K}$, potom $\|x_n(t) - x_r(t)\| < \delta K = \varepsilon$. + + Tedy dle Věty \ref{thm-arzela} můžeme z posloupnosti $x_n$ vybrat stejnoměrně konvergentní podposloupnost. Zbývá dokázat, že její limita řeší naši rovnici. + + \hfill \textit{konec 1. přednášky (21.2.2025)} + +\end{proof} + +% \begin{proof}[Důkaz Věty \ref{thm-peano}] +% \end{proof} + diff --git a/skripta.pdf b/skripta.pdf index c1d01b8858a9cae98aac7cf66a66664887e2792b..db9a52f0b61dc0039a40587630899ee238066b1f 100644 GIT binary patch delta 20971 zcmZ6xQ*fYN)3zPk#>AS~b~3ST+cvI=ZQHgv@x;!=wr&6S_ijD)*V^c+-dG!5=jvL0 z9zD4V9#%s!?x+JIdhly#rq5@?0H0QJ4*L2o|BZQ0U? zN8P`{!KqxTBLl>n6qC2bS3YU+ANLngkXpQ;0ktmspHsPkg2fq!TX$?zE>@~1``Koy zzLBxL{}P^hHgfVt@7vfMlVF9CbG;tJ&i`;+ogMMSwOncz!|ExZOCbqs zhG^fXSjm!&$4B@$1^yw9ClbX@aZI@14eIeWxF;c*vhLPx=|8&{mF)zufYuS#`k>?L z8^PT3`P@C7#fWU?pe~Y=7Sj6h_&}xW-Z1==!!K=9b85<9jrD*{=L#PA1?Mf<2;!kP zeUb3DKX}Km?U)L}#FJ>>?sY6cG;cX#b1GP>3M2#o&acGefN51U!b(D4|11m`OOpC- zm-Dg9Ct(G3{6qM4B72zNKgePbnq+L_zCT^=uKcP@WRO6s0z9Vh5M~^^nMI@uY!=D$ zLS#y|MVBNdw~L-AO!>g9`!JUi++8_Cdh;jRY9ZWc2q0fSu7;|Gb+~; z@4?q;KJN{o1&l1pc1CUT$mgNphHTAImocn7AN4dK5oO34^=u@1I?hx5WCTnif&m$%@}9#RP)n(UzOKb1uGR{CWdoB6@AKt(C( 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