From 01ec815e4d37921d6ffc52aef6ec1915caa63163 Mon Sep 17 00:00:00 2001 From: Petr Velycko Date: Thu, 15 May 2025 17:28:33 +0200 Subject: [PATCH] opravy preklepu apod. --- jednoznacnost-reseni.tex | 7 ++++--- lokalni-existence.tex | 34 ++++++++++++++++------------------ maximalni-reseni.tex | 4 ++-- skripta.pdf | Bin 226953 -> 227172 bytes zavislost-na-podmince.tex | 12 +++++++----- 5 files changed, 29 insertions(+), 28 deletions(-) diff --git a/jednoznacnost-reseni.tex b/jednoznacnost-reseni.tex index 40993d9..1e32806 100644 --- a/jednoznacnost-reseni.tex +++ b/jednoznacnost-reseni.tex @@ -4,7 +4,8 @@ V této kapitole se budeme věnovat otázce jednoznačnosti řešení diferenci \begin{definition} Řekneme, že rovnice \eqref{eq-ode} má v $\Omega$ vlastnost \textit{globální jednoznačnosti}, jestliže pro libovolná řešení $(x, I), (y, J)$ splňující $x(t_0) = y(t_0)$ pro nějaké $t_0 \in I \cup J$, potom $x(t) = y(t)$ pro všechna $t \in I \cup J$. - Řekneme, že rovnice \eqref{eq-ode} má v $\Omega$ vlastnost \textit{lokální jednoznačnosti}, jestliže pro libovolná řešení $(x, I), (y, J)$ splňující $x(t_0) = y(t_0)$ pro nějaké $t_0 \in I \cup J$, potom existuje $\delta$ takové, že $x(t) = y(t)$ pro všechna $t \in (t_0 - \delta, t_0 + \delta)$. + + Řekneme, že rovnice \eqref{eq-ode} má v $\Omega$ vlastnost \textit{lokální jednoznačnosti}, jestliže pro libovolná řešení $(x, I), (y, J)$ splňující $x(t_0) = y(t_0)$ pro nějaké $t_0 \in I \cup J$ existuje $\delta > 0$ takové, že $x(t) = y(t)$ pro všechna $t \in (t_0 - \delta, t_0 + \delta)$. \end{definition} \begin{theorem} @@ -15,7 +16,7 @@ V této kapitole se budeme věnovat otázce jednoznačnosti řešení diferenci Pro důkaz opačné implikace nechť máme dvě řešení $(x, I), (y, J)$ splňující $x(t_0) = y(t_0) = x_0$ pro nějaké $t_0 \in I \cap J$. Bez újmy na obecnosti nechť $I \cup J = (a, b)$. Položme $M = \{t : x(t) = y(t)\}$. Tato množina je díky předpokladu neprázdná, nechť $c := \sup M$. - Pro spor předpokládejme, že $c < b$. Potom platí $x(c) = \lim_{t\rightarrow c^-} x(t) = \lim_{t\rightarrow c^-} y(t)$, což se díky spojitosti $y$ rovná $y(c)$. Tedy $c$ je maximum $M$. Ale díky lokální jednoznačnosti existuje okolí $(c, x(c)$, na kterém platí $x = y$. Tedy $x(c + \delta) = y(c + \delta)$ pro nějaké $\delta > 0$, což je spor s tím, že $c = \sup M$. + Pro spor předpokládejme, že $c < b$. Potom platí $x(c) = \lim_{t\rightarrow c^-} x(t) = \lim_{t\rightarrow c^-} y(t)$, což se díky spojitosti $y$ rovná $y(c)$. Tedy $c$ je maximum $M$. Ale díky lokální jednoznačnosti existuje okolí bodu $(x(c), c)$, na kterém platí $x = y$. Tedy $x(c + \delta) = y(c + \delta)$ pro nějaké $\delta > 0$, což je spor s tím, že $c = \sup M$. \end{proof} \begin{definition} @@ -51,7 +52,7 @@ Zavedeme značení $f \in C_x^1(\Omega)$, jestliže $\pdv{f}{x_i}$ existují a j $$n K \max |y_i - x_i| \leq nK | y - x |,$$ kde poslední nerovnost plyne z faktu, že $|y -x| = \sqrt{\sum_{i = 1}^n |y_i - x_i|^2}$. - Tedy $f$ je lokálně Lipschitzovská s konstantou $n \cdot K$. + Tedy $f$ je lokálně lipschitzovská s konstantou $n \cdot K$. \end{proof} \textit{Rule of thumb (just for fun)}: platí $f$ spojitá $\Rightarrow$ existuje řešení, $f \in C^1 \Rightarrow$ řešení je určeno jednoznačně. diff --git a/lokalni-existence.tex b/lokalni-existence.tex index fc0dd50..3a6af0b 100644 --- a/lokalni-existence.tex +++ b/lokalni-existence.tex @@ -42,7 +42,7 @@ Teď si zadefinujeme několik pojmů, které charakterizují množiny funkcí, k \begin{definition} Řekneme, že funkce množiny $M \subset C(K, \mathbb{R}^n)$ jsou \begin{enumerate} - \item \textit{stejně spojité}, jestliže pro každé $x \in K$ a každé $\varepsilon > 0$ existuje $\delta > 0$ takové, že $\| f(x) - f(y) \| < \epsilon$ pro všechna $y \in (x - \delta, x + \delta)$ a všechny $f \in M$. + \item \textit{stejně spojité}, jestliže pro každé $x \in K$ a každé $\varepsilon > 0$ existuje $\delta > 0$ (společné pro všechny funkce) takové, že $\| f(x) - f(y) \| < \varepsilon$ pro všechna $y \in (x - \delta, x + \delta)$ a všechny $f \in M$. \item \textit{stejně omezené}, jestliže existuje $C > 0$ takové, že $\|f\| \leq C$ pro všechna $f \in M$. \end{enumerate} \end{definition} @@ -67,41 +67,39 @@ K důkazu této věty budeme potřebovat pomocné lemma: \end{lemma} \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 $x(t) = x_0 + \int_{t_0}^t f(x(s - \lambda, s) ds$. + Řešme ``porušenou" úlohu $P_\lambda$: $x(t) = x_0 + \int_{t_0}^t f(x(s - \lambda), s) ds$ pro $ t > t_0$ kde dodefinováváme funkci $x(t) = x_0$ na intervalu $t \in [t_0 - \lambda, t_0]$. + 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 řešení úloh $P_\frac{1}{n}$ tvoří posloupnost funkcí. + Ukážeme, že jsou stejně spojité a stejně omezené. - Stejná omezenost plyne z toho, že $\| x_n (t) \| = \| x_0 + \int_{t_0}^t f(x(s - \frac{1}{n}), s) ds \| \leq \| x_0 \| + \int_{t_0}^t f(x(s - \frac{1}{n}), s) \| ds$. 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 $\|x_n(t) - x_r(t)\| < \delta K = \varepsilon$. + Stejná omezenost plyne z toho, že $\| x_n (t) \| = \| x_0 + \int_{t_0}^t f(x(s - \frac{1}{n}), s) ds \| \leq \| x_0 \| + \| \int_{t_0}^t f(x(s - \frac{1}{n}), s) \| ds$. 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 $\|x_n(t) - x_n(r)\| < \delta K = \varepsilon$. Tedy dle Věty \ref{thm-arzela} můžeme z posloupnosti $x_n$ vybrat stejnoměrně konvergentní podposloupnost $x_{n_k}$. Zbývá dokázat, že její limita řeší naši rovnici. \hfill \textit{konec 1. přednášky (21.2.2025)} - Zřejmě pro $k \rightarrow \infty$ platí $x_{n_k} \rightarrow x(t)$ a pokud $\int_{t_0}^t f(x_{n_k}(s - \frac{1}{n}), s) ds$ konverguje k $\int_{t_0}^t f(x(s - \frac{1}{n}),s)ds$, máme hotovo. - Tato vlastnost plyne z toho, že $\| \int_{t_0}^t f(x_{n_k}(s - \frac{1}{n_k}), s) - f(x(s), s) ds\| \leq \int_{t_0}^t \| f(x_{n_k}(s - \frac{1}{n_k}), s) - f(x(s - \frac{1}{n_k}), s) \| + \| f(x(s - \frac{1}{n_k}), s) - f(x(s), s \| ds$. + Zřejmě pro $k \rightarrow \infty$ platí $x_{n_k} \rightarrow x(t)$ a pokud $\int_{t_0}^t f(x_{n_k}(s - \frac{1}{n_k}), s) ds$ konverguje k $\int_{t_0}^t f(x(s),s)ds$, máme hotovo. + Skutečně, $\| \int_{t_0}^t f(x_{n_k}(s - \frac{1}{n_k}), s) - f(x(s), s) ds\| \leq \int_{t_0}^t \| f(x_{n_k}(s - \frac{1}{n_k}), s) - f(x(s - \frac{1}{n_k}), s) \| + \| f(x(s - \frac{1}{n_k}), s) - f(x(s), s) \| ds$. - Jelikož $f$ je spojitá, musí být stejnoměrně spojitá na kompaktní množině $[t_0, t_0 + T] \times \overline{B(0, r) \cap \Omega}$, jinými slovy platí, že pro $\varepsilon > 0$ existuje $\delta$ takové, že pro každé dva body $x, y$ takové, že $\|x - y\| < \delta$ máme, že $f(x, s) - f(y, \hat(s))$. - - Ze stejnoměrné konvergence $x_{n_k}$ máme, že pro $\delta > 0$ existuje $k_0$ takové, že pro všechna $k \geq k_0$ platí $\|x_{n_k}(s - \frac{1}{n_k}) - x(s - \frac{1}{n_k})\|<\delta$. - - Jelikož $x$ je spojitá, na kompaktním intervalu $[t_0, t_0 + T]$ je také stejnoměrně spojitá. Potom pro $\delta> 0$ existuje $k_1$ takové, že pro všechna $k \geq k_1$ platí $\| x(s - \frac{1}{n_k} - x(s) \| < \delta$. + Jelikož $f$ je spojitá, musí být stejnoměrně spojitá na kompaktní množině $[t_0, t_0 + T] \times \overline{B(0, r) \cap \Omega}$, jinými slovy platí, že pro $\varepsilon > 0$ existuje $\delta$ takové, že pro každé dva body $x, y$ takové, že $\|x - y\| < \delta$ máme, že $|f(x, s) - f(y, s)| < \varepsilon$. Ze stejnoměrné konvergence $x_{n_k}$ máme, že pro $\delta > 0$ existuje $k_0$ takové, že pro všechna $k \geq k_0$ platí $\|x_{n_k}(s - \frac{1}{n_k}) - x(s - \frac{1}{n_k})\|<\delta$. Jelikož $x$ je spojitá, na kompaktním intervalu $[t_0, t_0 + T]$ je také stejnoměrně spojitá. Potom pro $\delta> 0$ existuje $k_1$ takové, že pro všechna $k \geq k_1$ platí $\| x(s - \frac{1}{n_k}) - x(s) \| < \delta$. Potom pro všechna $k \geq \max\{k_0, k_1\}$ platí, že náš integrál je menší nebo roven $\int_{t_0}^t \varepsilon + \varepsilon ds \leq T\cdot 2\varepsilon$, tedy jsme opravdu nalezli požadované řešení. - Existence řešení na $[t_0- T, t_0]$ se ukáže podobně. + Existence řešení na $[t_0 - T, t_0]$ se ukáže podobně. \end{proof} \begin{proof}[Důkaz Věty \ref{thm-peano}] - Uvažujme dvě koule kolem bodu $(x_0, t_0)$ takové, že $K_1 \subset K_2 \subset \Omega$. + Uvažujme dvě (uzavřené) koule kolem bodu $(x_0, t_0)$ takové, že $K_1 \subsetneq K_2 \subset \Omega$. Definujeme $\tilde{f}(x, t) = \begin{cases} f(x, t) \text{ v } K_1,\\ - \text{spojitě v } K_2 \setminus K_1\\ - 0, (x, t) \in \mathbb{R}^{n+1} \setminus K_2 - \end{cases}$. + \text{spojitě v } K_2 \setminus K_1,\\ + 0, (x, t) \in \mathbb{R}^{n+1} \setminus K_2. + \end{cases}$ + Tato funkce je na $\Omega$ spojitá a omezená. - Z Lemmatu \ref{lemma-special-solution} máme, že rovnice $x' = \tilde{f}(x, t)$ má řešení $x$ splňující počáteční podmínku $x(t_0) = x_0$. Nazveme toto řešení $\tilde{x}$. Potom ze spojitosti $\tilde{x}$. Tedy existuje $\delta > 0$ takové, že graf $\tilde{x}$ na $(t_0 - \delta, t_0 + \delta)$ leží v $K_1$. Restrikce $\tilde{x}$ na tento interval nám tedy dává řešení původní rovnice. + Z Lemmatu \ref{lemma-special-solution} máme, že rovnice $x' = \tilde{f}(x, t)$ má řešení $x$ splňující počáteční podmínku $x(t_0) = x_0$. Nazveme toto řešení $\tilde{x}$. Potom ze spojitosti $\tilde{x}$ existuje $\delta > 0$ takové, že graf $\tilde{x}$ na $(t_0 - \delta, t_0 + \delta)$ leží v $K_1$. Restrikce $\tilde{x}$ na tento interval nám tedy dává řešení původní rovnice. \end{proof} diff --git a/maximalni-reseni.tex b/maximalni-reseni.tex index a61cbcb..c360e3f 100644 --- a/maximalni-reseni.tex +++ b/maximalni-reseni.tex @@ -14,7 +14,7 @@ V této kapitole se budeme věnovat otázce rozšíření řešení na co nejvě \begin{proof} Mějme řešení $(x, I)$ takové, že $I = (a, b)$. Budeme induktivně prodlužovat za bod $b$ (na druhou stranu se to pak udělá analogicky). Položme $x_0 = x$, $b_0 = b$, $I_0 = I$. V $n$-tém kroku dostaneme řešení $(x_n, I_n)$, kde $I_n = (a, b_n)$. Dále definujeme $\omega_n = \sup \{z > b_n; (x_n, I_n) \text{ lze prodloužit na } (a, z) \}$. Pokud příslušná množina je prázdná, jsme hotovi, neboť řešení již nejde prodloužit, tedy je maximální. - V opačném případě můžeme definovat $b_{n + 1} = \frac{b_n + \omega_n}{2}$ (pokud $\omega_n < \infty$), případně $b_{n + 1} = b_n + 1$. Tímto postupem získám rostoucí posloupnost $b_n$, která musí mít limitu. Označme tuto limitu $\beta$. Dále položme $\tilde{I} = (a, \beta)$, $\tilde{x} = x_n(t)$, pro všechna $t \in \tilde{I}$ zvolím $n$ tak, aby $t \in I_n$. Na volbě $n$ nezávisí, neboť na příslušných intervalech jsou funkce $x_n$ stejné. + V opačném případě můžeme definovat $b_{n + 1} = \frac{b_n + \omega_n}{2}$ (pokud $\omega_n < \infty$), případně $b_{n + 1} = b_n + 1$. Tímto postupem získáme rostoucí posloupnost $b_n$, která musí mít limitu. Označme tuto limitu $\beta$. Dále položme $\tilde{I} = (a, \beta)$, $\tilde{x} = x_n(t)$, pro všechna $t \in \tilde{I}$ zvolím $n$ tak, aby $t \in I_n$. Na volbě $n$ nezávisí, neboť na příslušných intervalech jsou funkce $x_n$ stejné. Dokážeme, že takto definované řešení $(\tilde{x}, \tilde{I})$ je maximální. Pro spor budeme předpokládat, že existuje rozšíření na $(a, \hat{\beta})$ takové, že $\hat{\beta}$. Okamžitě vidíme, že $\beta < \infty$. Vezmeme $n$ takové, aby $\beta - b_n < \hat{\beta} - \beta$ a $\beta - b_n < 1$ (existuje díky tomu, že $b_n$ konvergují k $\beta$). V tom případě $(x_n, I_n)$ má prodloužení až do $\hat{\beta}$, tedy $\omega_n \geq \hat{\beta}$. Pak ale (pokud $\omega_n = \infty$) $b_{n + 1} = b_n + 1 > \beta$, máme spor, případně pro $\omega_n$ konečné máme $b_{n + 1} = \frac{b_n + \omega_n}{2} > \frac{2\beta - \hat{\beta} + \hat{\beta}}{2} = \beta$, opět jsme došli ke sporu. \end{proof} @@ -41,7 +41,7 @@ V případě $f$ lipschitzovské se důkaz dá výrazně zjednodušit. Budeme uv \end{lemma} \begin{proof} - Nutnost těchto podmínek plyne triviálně z podstaty prodloužení (cvičení). Dokážeme, že jde o podmínky postačující. Nechť tedy máme $(x_0, b)$ jakou novou počáteční podmínku, dle Peanovy vety existuje řešení $\hat{x}$ na $(b - \delta, b + \delta)$ splňující tuto počáteční podmínku. Definujeme $\hat{x}(t) = \begin{cases} x(t), t < b \\ \tilde{x}(t), t \geq b \end{cases}$. Potom $\hat{x}$ je řešení (díky principu nalepování) a navíc prodlužuje $x$ za bod $b$, což jsme chtěli dokázat. + Nutnost těchto podmínek plyne triviálně z podstaty prodloužení (cvičení). Dokážeme, že jde o podmínky postačující. Nechť tedy máme $(x_0, b)$ jako novou počáteční podmínku, dle Peanovy vety existuje řešení $\hat{x}$ na $(b - \delta, b + \delta)$ splňující tuto počáteční podmínku. Definujeme $\hat{x}(t) = \begin{cases} x(t), t < b \\ \tilde{x}(t), t \geq b \end{cases}$. Potom $\hat{x}$ je řešení (díky principu nalepování) a navíc prodlužuje $x$ za bod $b$, což jsme chtěli dokázat. \end{proof} Na závěr si uvedeme jednu důležitou větu, která nám poskytne představu o tom, jak vypadají maximální řešení diferenciálních rovnic. diff --git a/skripta.pdf b/skripta.pdf index f065be7f96c0441b7274a7a3992a0617d0694bf7..db966f42fa123231b2dfef184e090a361d4821e4 100644 GIT binary patch delta 54724 zcmZ6SV?&?~xP)_Uwl~|hZQI;z*VdEUYlOF#hD#nsu|*bdfn?NU!Vmb3+<`&g5T+id_@RNzXX1Bp3-i21i83+H;+?C4mZ z6UYD*%h$74EU$K9O3r2u#D0nl9p>xOHGSn~EN|WZ3sBck+tAOd_`Tls0$THl_x)-2 zd~DGn__cT)Y@&j*J9)KgWdYGJEZwn=_YdHL&i9y(F%)#H-gRQ+I(|{s{)os;-rd+G z5BXM>WN7r7CF@frKgPA?Da9$9jav-V~6&J#L%Dl)R*gl5yB9|ORJ zD`|DB8s;Wo-xMyWvdInZDf9`~+k=EY-s39~B@kr|j5;4g2{9|@1}0I`|;; zDAdG9)*w(BMCms>YDysN`mexJtP@xk3@#&eL(s27kX*>0P^{n&e%;?fqa>xjmm4&- z@pgjqdGaC{uR!i67D%P4h>11!a6=dj$}D22+zVWwr}B%!3;(TJ8j`+L@?x5I!R)Ll zBVCX)LOZIAg3ex{_p&JTUM1svAfmMQBHP9~>YA_V9Xi&*pgAY9--0FXu?BQZDT6F4 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zGV%3Jbop>yzTxt6@udX2TveA#UCtF>>Pwfm;SSt|dvG5fz(ZJ$$FKrVVzv14{xs%) XoT6#>mpgX@9||%yFbX9lMNdWweTzQ1 diff --git a/zavislost-na-podmince.tex b/zavislost-na-podmince.tex index e533bb6..8bbfb7f 100644 --- a/zavislost-na-podmince.tex +++ b/zavislost-na-podmince.tex @@ -27,13 +27,15 @@ \begin{proof} Můžeme psát - $$ \| x(t) - y(t) \| = \| x(t_0) + \int_{t_0}^t f(x(s), s) ds - (y(t_0) + \int_{t_0}^t f(y(s), s) ds) \| \leq $$ - $$ \| x(t_0) - y(t_0) \| + \| \int_{t_0}^t \left| f(x(s), s) - f(y(s), s) \right| ds \| \leq $$ - $$ \| x(t_0) - y(t_0) \| + \| \int_{t_0}^t L | x(s) - y(s) | ds.$$ + \begin{align*} + \| x(t) - y(t) \| &= \left\| x(t_0) + \int_{t_0}^t f(x(s), s) ds - (y(t_0) + \int_{t_0}^t f(y(s), s) ds) \right\| \leq \\ + &\leq \left\| x(t_0) - y(t_0) \right\| + \left\| \int_{t_0}^t \left| f(x(s), s) - f(y(s), s) \right| ds \right\| \leq \\ + &\leq \left\| x(t_0) - y(t_0) \right\| + \left\| \int_{t_0}^t L | x(s) - y(s) | ds \right\|. + \end{align*} Poté z Gronwallova lemmatu dostáváme, že - $$ \| x(t) - y(t) \| \leq K e^{|\int_{t_0}^t L ds|} = K e^{|t - t_0| L}, $$ - kde funkci $w(s)$ ze znění lemmatu odpovídá výraz $\|x(s) - y(s)\|$. + $$ \| x(t) - y(t) \| \leq \| x(t_0) - y(t_0) \| e^{|\int_{t_0}^t L ds|} = \| x(t_0) - y(t_0) \| e^{|t - t_0| L}, $$ + kde funkci $w(s)$ ze znění lemmatu odpovídá výraz $\|x(s) - y(s)\|$ a $K = \| x(t_0) - y(t_0) \|$. \end{proof} Jednoduchým důsledkem tohoto lemmatu je mj. jednoznačnost řešení (stačí uvažovat řešení s $x(t_0) = y(t_0)$).