From de860606fdde9929a4baeb477596ef612ea5833c Mon Sep 17 00:00:00 2001 From: Robert Date: Thu, 1 Apr 2021 14:23:37 +0200 Subject: [PATCH] =?UTF-8?q?finished=20picard=20lindel=C3=B6f?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- chapters/ode.tex | 1 + chapters/sections/picard_lindelof.tex | 357 ++++++++++++++++++++++++++ script.pdf | Bin 1129939 -> 1159169 bytes 3 files changed, 358 insertions(+) create mode 100644 chapters/sections/picard_lindelof.tex diff --git a/chapters/ode.tex b/chapters/ode.tex index f9a157b..c35ba99 100644 --- a/chapters/ode.tex +++ b/chapters/ode.tex @@ -7,4 +7,5 @@ \pagebreak \subfile{sections/solution_methods.tex} + \subfile{sections/picard_lindelof.tex} \end{document} \ No newline at end of file diff --git a/chapters/sections/picard_lindelof.tex b/chapters/sections/picard_lindelof.tex new file mode 100644 index 0000000..5259a0f --- /dev/null +++ b/chapters/sections/picard_lindelof.tex @@ -0,0 +1,357 @@ +% !TeX root = ../../script.tex +\documentclass[../../script.tex]{subfiles} + +\begin{document} +\section{The Picard-Lindelöf Theorem} + +\begin{eg} + Consider the ODE + \[ + y' = 2\sqrt{\abs{y}} + \] + Possible solutions are + \begin{align*} + y(x) &= (x - c)^2 ~~c > 0 \\ + y(x) &= -(x - c)^2 ~~c < 0 \\ + y(x) = 0 + \end{align*} + Another solution could be + \[ + y(x) = \begin{cases} + -(x - a)^2, & x \in (-\infty, a) \\ + 0, & x \in [a, b] \\ + (x - b)^2, & x \in (b, \infty) + \end{cases} + \] + for $a, b \in \realn$ with $a \le b$ So the IVP $y(0) = 0$ has many solutions. +\end{eg} + +\begin{defi} + Let $D \subset \realn \times \realn^n$ be open, $(x_0, y_0) \in D$ and $f: D \rightarrow \realn^n$. + We say $f$ fulfils a local Lipschitz-condition in the point $(x_0, y_0)$ if there exists a neighbourhood $U$ of $(x_0, y_0)$ such that + \[ + \norm{f(x, y) - f(x, z)} \le L \norm{y - z} ~~\forall (x, y), (x, z) \in U + \] +\end{defi} + +\begin{eg} + Consider + \begin{align*} + f: \realn &\longrightarrow \realn \\ + (x, y) &\longmapsto x^2y^2 + \end{align*} + Then + \begin{align*} + \abs{f(x, y) - f(x, z)} = \abs{x^2(y^2 - z^2)} &= \abs{x^2(y - z)(y + z)} \\ + &= \underbrace{\abs{x^2 (y + z)}}_{\alpha(x, y, z)} \abs{y - z} + \end{align*} + The function $\alpha(x, y, z)$ is unbounded, so the global Lipschitz condition isn't satisfied. + Now choose a fixed $(x_0, y_0) \in \realn \times \realn$, and set + \[ + R > \max\set{\abs{x_0}, \abs{y_0}} + \] + Then $\forall (x, y), (x, z) \in (-R, R) \times (-R, R)$ + \[ + \alpha(x, y, z) \le R^2 \abs{y + z} \le R^2\left(\abs{y} + \abs{z}\right) \le 2R^3 + \] + So $f$ fulfils a local Lipschitz condition in $(x_0, y_0)$. +\end{eg} + +\begin{defi} + Let $\measure$ be a measure space, $f: \Omega \rightarrow \realn^n$ measurable and $f_1, \cdots, f_n$ are the component functions of $f$. So + \[ + f(\omega) = \left(f_1(\omega), f_2(\omega), \cdots, f_n(\omega)\right) + \] + $f$ is said to be integrable if $f_1, \cdots, f_n$ are integrable, and we define + \[ + \int f \dd\mu = \left(\int f_1 \dd\mu, \int f_2 \dd\mu, \cdots, \int f_n \dd\mu \right) + \] +\end{defi} + +\begin{thm} + Let $\measure$ be a measure space, define $\dnorm$ to be the norm on $\realn^n$ and let $f: \Omega \rightarrow \realn^n$ be measurable. + Then $f$ is integrable if and only if $\norm{f}$ is integrable, and + \[ + \norm{\int f \dd\mu} \le \int \norm{f} \dd\mu + \] +\end{thm} +\begin{proof} + Without proof. +\end{proof} + +\begin{lem}\label{lem:picardlem} + Let $D \subset \realn \times \realn^n$, $(x_0, y_0) \in D$ and $f: D \rightarrow \realn^n$ continuous. + Let $I$ be an open interval and $y: I \rightarrow \realn^n$ be continuously differentiable, such that + $(x, y(x)) \in D ~~\forall x \in I$. Then $y$ is a solution of the IVP + \begin{align*} + y' = f(x, y) && y(x_0) = y_0 + \end{align*} + if and only if $y$ satisfies the integral equation + \[ + y(x) = y_0 + \int_{x_0}^x f(t, y(t)) \dd{t} + \] +\end{lem} +\begin{proof} + Let $y$ fulfil the IVP. Then + \[ + y(x) - y_0 = y(x) - y(x_0) = \int_{x_0}^x y'(t) \dd{t} = \int_{x_0}^x f(t, y(t)) \dd{t} + \] + If $y$ fulfils the integral equation, then + \[ + y'(x) = f(x, y(x)) + \] +\end{proof} + +\begin{thm}[Picard-Lindelöf Theorem] + Let $D \subset \realn \times \realn^n$ be open, $(x_0, y_0) \in D$ and $f: D \rightarrow \realn^n$ continuous such that $f$ fulfils a local Lipschitz condition in $y$. + Then $\exists \epsilon > 0$ such that the IVP + \begin{align*} + y' = f(x, y) && y(x_0) = y_0 + \end{align*} + has exactly one solution on $(x_0 - \epsilon, x_0 + \epsilon)$. +\end{thm} +\begin{proof} + Let $U \subset D$ be a neighbourhood of $(x_0, y_0)$, such that + \begin{equation} + \norm{f(x, y) - f(x, z)} \le L \norm{y - z} ~~\forall (x, y), (x, z) \in U + \end{equation} + Choose $a, r > 0$ such that + \begin{equation} + \tilde{D} = [x_0 - a, x_0 + a] \times K_r(y_0) \subset U + \end{equation} + + \begin{center} + \begin{tikzpicture} + \draw[thick, ->] (3, 0) -- (4, 0) node[right] {$x$}; + \draw[thick, ->] (3, 0) -- (3, 1) node[above] {$y$}; + + \draw[thick] (0, 0) ellipse (2.5cm and 4cm); + \node at (1.3, 3) {$D$}; + + \draw[dashed] (2, -1.5) -- (2, 1.5) -- (-2, 1.5) -- (-2, -1.5) -- (2, -1.5); + \node[below left] at (2, 1.5) {$\tilde{D}$}; + \draw[fill] (0, 0) circle[radius=1pt] node[below right] {$(x_0, y_0)$}; + + \node[below] at (0, -1.5) {$a$}; + \node[left] at (-2, 0) {$r$}; + \end{tikzpicture} + \end{center} + $\tilde{D}$ is compact and $f$ is continuous, i.e. $f$ is bounded on $\tilde{D}$ by $M \in (0, \infty)$. + \begin{equation} + \norm{f(x, y)} \le M ~~\forall (x, y) \in \tilde{D} + \end{equation} + Choose an $\epsilon$ such that $0 < \epsilon \le a$ and such that + \begin{align} + \epsilon M < r && \epsilon L < 1 + \end{align} + Set $I := (x_0 - \epsilon, x_0 + \epsilon)$, and + \begin{equation} + X = \set[y \text{ continuous}]{y: I \rightarrow K_r(y_0)} \subset C(I) + \end{equation} + $X$ is closed, and thus complete. + Define $T: X \rightarrow X$ with + \begin{equation} + T(y)(x) := y_0 + \int_{x_0}^x f(t, y(t)) \dd{t} + \end{equation} + We want to show $T(y) \subset X$: + \begin{equation} + \begin{split} + \norm{T(y)(x) - y_0} = \norm{\int_{x_0}^x f(t, y(t)) \dd{t}} &\le \int_{x_0}^x \norm{f(t, y(t))} \dd{t} \\ + &\le M \int_{x_0}^x \dd{t} < \epsilon M < r + \end{split} + \end{equation} + Now consider + \begin{equation} + \begin{split} + \norm{T(y)(x) - T(\tilde{y})(x)} &= \norm{\int_{x_0}^x(f(t, y(t)) - f(t, \tilde{y}(t))) \dd{t}} \\ + &\le \int_{x_0}^x \norm{f(t, y(t)) - f(t, \tilde{y}(t))} \dd{t} \\ + &\le \int_{x_0}^x L \cdot \norm{y(t) - \tilde{y}(t)} \dd{t} \\ + &\le \int_{x_0}^x L \supnorm{y - \tilde{y}} \le L\supnorm{y - \tilde{y}} \cdot \epsilon + \end{split} + \end{equation} + By taking the supremum over all $x \in I$ we get + \begin{equation} + \supnorm{T(y) - T(\tilde{y})} \le \underbrace{\epsilon L}_{< 1} \supnorm{y - \tilde{y}} + \end{equation} + So $T: X \rightarrow X$ is strictly contractive. + According to the Banach fixed point theorem, there exists a unique fixed point of $T$ in $x$, that means + $\exists y \in X$ such that + \begin{equation} + y_0 + \int_{x_0}^x f(t, y(t)) \dd{t} = T(y)(x) = y(x) ~~\forall x \in I + \end{equation} + Due to \Cref{lem:picardlem}, there eixsts a unique solution to the ODE. +\end{proof} + +\begin{rem} + One can approximate a fixed point by repeatedly applying $T$. For example consider + \[ + \phi(x) = y_0 + \] + and define + \begin{align*} + \phi_0 = \phi && \phi_i = T(\phi_{i-1}) = y_0 + \int_{x_0}^x f(t, \phi_{i-1}(t)) \dd{t} + \end{align*} + This process is called Picard iteration, and the $\phi_i$ converge uniformly to the solution. +\end{rem} + +\begin{eg} + Consider + \[ + y' = \sqrt{\norm{y}} + \] + Then + \[ + \limes{y}{0} \left(\frac{\abs{f(x, y) - f(x, 0)}}{\abs{y - 0}}\right) = \limes{y}{0} \rec{\sqrt{\abs{y}}} \conv{} \infty + \] + Which means the local Lipschitz condition is not satisfied. +\end{eg} + +\begin{thm} + Let $D \subset \realn \times \realn^n$ be open, $f: D \rightarrow \realn^n$ continuously differentiable. + Then $f$ satisfies a local Lipschitz condition in terms of $y$. +\end{thm} +\begin{proof} + Let $(x_0, y_0) \in D$. Choose $r > 0$ such that $\cball[r](x_0, y_0) \subset D$. + The total derivative $D_y f$ is continuous and thus bounded on $\cball[r](x_0, y_0)$. + \begin{equation} + \exists L > 0: ~~\norm{D_y f(x, y)} \le L ~~\forall (x, y) \in \cball[r](x_0, y_0) + \end{equation} + Applying the generalized mean value theorem yields + \begin{equation} + \begin{split} + \norm{f(x, y) - f(x, z)} &\le \sup_{t \in [0, 1]} \norm{D_y f(x, y + t(z - y))} \norm{y - z} \\ + &\le L \norm{y - z} + \end{split} + \end{equation} + If $n=1$ we can specify + \begin{equation} + \abs{f(x, y) - f(x, z)} = \abs{\partial_y f(x, \xi) (y - z)} + \end{equation} +\end{proof} + +\begin{eg} + Consider + \[ + y'' = -\frac{y}{\norm{y}^3} + \] + The function + \begin{align*} + f: \underbrace{\realn \times \realn^3 \setminus \set{0} \times \realn^3}_D &\longrightarrow \realn^3 \times \realn^3 \\ + (x, y, z) &\longmapsto \left(z, (y_1^2 + y_2^2 + y_3^2)^{-\frac{3}{2}} \cdot y \right) + \end{align*} + is continuously differentiable. So the IVP for arbitrary initial points in $D$ has a locally unique solution. +\end{eg} + +\begin{defi} + Let $D \subset \realn \times \realn^n$ be open, $(x_0, y_0) \in D$. A solution $\tilde{y}: \tilde{I} \rightarrow \realn^n$ of the IVP + \begin{align*} + y' = f(x, y) && y(x_0) = y_0 + \end{align*} + is said to be a (real) continuation of the solution $y: I \rightarrow \realn^n$ if $I \subset \tilde{I}$ and $y(x) = \tilde{y}(x) ~~\forall x \in I$. + A solution $y$ is said to be a maximal solution if there are no real continuations. +\end{defi} + +\begin{thm} + Let $D \subset \realn \times \realn^n$ be open, $(x_0, y_0) \in D$ and $f: D \rightarrow \realn^n$ continuous and satisfying a local Lipschitz condition in terms of $y$. + Then the IVP + \begin{align*} + y' = f(x, y) && y(x_0) = y_0 + \end{align*} + has a unique solution. +\end{thm} +\begin{proof} + First, let $y: I \rightarrow \realn^n$ and $\tilde{y}: \tilde{I} \rightarrow \realn^n$ be solutions of the IVP. + Then $y = \tilde{y}$ on $I \cap \tilde{I} =: (a, b)$. Let + \begin{equation} + c = \sup\set[{y = \tilde{y} \text{ on } [x_0, \tilde{c}]}]{\tilde{c} \in [x_0, b)} + \end{equation} + According to Picard-Lindelöf, such $\tilde{c}$ exist. Then there exists a sequence $\seq{c} \subset (x_0, c)$ such that $y = \tilde{y}$ on + $[x_0, c_n) ~~\forall n \in \natn$ and $c_n \rightarrow c$. + If $c < b$, then + \begin{equation} + y(c) = \tilde{y}(c) + \end{equation} + because $y(c_n) = \tilde{y}(c_n) ~~\forall n \in \natn$. + The IVP + \begin{align} + u' = f(x, u) && u(c) = y(c) + \end{align} + has a locally unique solution on $(c - \epsilon, c + \epsilon) ~~\epsilon > 0$ according to Picard-Lindelöf. + Since the $y$ and $\tilde{y}$ are both solutions to the IVP, they are identical on $(c - \epsilon, c + \epsilon)$. + However, this contradicts the construction of $c$, so $c = b$. + \begin{equation} + \implies y = \tilde{y} \quad \text{on } [x_0, b) + \end{equation} + Analogously, one can prove $y = \tilde{y}$ on $(a, x_0]$. + Now let $I_{\max}$ be the union of all open intervals that are domains of the solution of the IVP. + For $x \in I_{\max}$ we can choose + \begin{equation} + y_{\max}(x) = y(x) + \end{equation} + for arbitrary solutions $y: I \rightarrow \realn$ with $x \in I$. So + \begin{equation} + y_{\max}: I_{\max} \rightarrow \realn + \end{equation} + is a maximal solution that is unique. +\end{proof} + +\begin{eg} + \begin{enumerate}[(i)] + \item Consider + \begin{align*} + y' = e^{-y} && y(1) = 0 + \end{align*} + The solution to this is + \begin{align*} + y: (0, \infty] &\longrightarrow \realn \\ + x &\longmapsto \ln(x) + \end{align*} + and is maximal. + + \item Consider + \begin{align*} + y' = -i \frac{y}{x^2} && y\left(\rec{2\pi}\right) = 1 + \end{align*} + The solution to this is + \begin{align*} + (0, \infty) &\longrightarrow \cmpln \\ + x &\longmapsto e^{\frac{i}{x}} + \end{align*} + and is maximal. + \end{enumerate} +\end{eg} + +We define $\metric$ to be a metric space, $x \in X$ and $A \subset X$. Then +\[ + d(x, A) = \inf\set[y \in A]{d(x, y)} +\] + +\begin{thm} + Let $D \subset \realn \times \realn^n$ be open, $(x_0, y_0) \subset D$ and $f: D \rightarrow \realn^n$ continuous and satisfying the local Lipschitz condition in terms of $y$. + Let $a, b \in \realn \cup \set{-\infty, \infty}$ such that + \[ + -\infty \le a < x_0 < b \infty \infty + \] + and let + \[ + y: (a, b) \rightarrow \realn + \] + a solution of the IVP + \begin{align*} + y' = f(x, y) && y(x_0) = y_0 + \end{align*} + Then $y$ is the maximal solution of the IVP if and only if one of these conditions + \begin{align*} + \text{(i)} && b = \infty \\ + \text{(ii)} && \limes{x}{b} \norm{y(x)} = \infty \\ + \text{(iii)} && \limes{x}{b} d((x, y(x)), \boundary{D}) = 0 + \end{align*} + and one of these + \begin{align*} + \text{(i)} && a = -\infty \\ + \text{(ii)} && \limes{x}{a} \norm{y(x)} = \infty \\ + \text{(iii)} && \limes{x}{a} d((x, y(x)), \boundary{D}) = 0 + \end{align*} + is fulfilled. +\end{thm} +\end{document} \ 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