Uniqueness of ODE solutions

Description

Eine überraschende Tatsache der Differentialrechnung ist, dass Lösungen nicht notwendigerweise eindeutig sind. Contrary to its name this article will also cover theorems on existence, since uniqueness results often imply existence too.

Uniqueness of ODE solutions (Picard-Lindelöf)

Let $D\subseteq \mathbb{R}\times {\mathbb{R}}^n$ be open and $f:D\to {\mathbb{R}}^n$ be continuous and with respect to $x$ lokal lipschitzstetige function. Then the Initial value problem $x^{\prime }=f(t,x),x(t_0)=x_0$ has a unique local solution on an interval $[t_0-ϵ,t_0+ϵ]$ for some $ϵ>0$.

Be piecing together pieces of the above we obtain the following:

Global existence and uniqueness theorem

Let $D\subseteq \mathbb{R}\times {\mathbb{R}}^n$ be a Gebiet and $f:D\to \mathbb{R}$ continuous and locally lipschitz with respect to $x$. Then for any Initial value problem there is a uniquely determined “maximal” interval $I=]a,b[$ s.t.

1. The Initial value problem has exactly one solution on $I$.
2. Every other solution on a different interval $\lambda :J\to {\mathbb{R}}^n$ must be a restriction of $I$ to $J$

From this follows that the Solution curves are disjoint when the conditions apply.

Existence interval

Die maximale Definitionsmenge $I=]a,b[$, die durch den Globalen Existenz- und Eindeutigkeitssatz gegeben ist nennt man das Lösungs- oder Existenzintervall von $\lambda$.

Existence and Uniqueness for linearly bounded functions

Let $f:]a,b[\times {\mathbb{R}}^n\to \mathbb{R}$ be a continuous and with respect to $x$ lipschitz-continuous function. If $f$ is also linearly bounded, i.e. there are nonnegative, cpntinuous functions $\rho (t),\sigma (t)$ in $t$ $\Vert f(t,x)\Vert \le \rho (t)\Vert x\Vert +\sigma (t)$ for all $(t,x)\in ]a,b[\times {\mathbb{R}}^n$ then all initial value problems have solution defined on all of $]a,b[$

Properties

Boundary behaviour of the existence interval

Let $D$ and $f$ be like in the Global existence and uniqueness theorem. Some solution $\lambda :I=]a,b[\to \mathbb{R}$ is exactly a maximal solution to an Initial value problem if for the boundary point $a$ one of the following applies:

1. The interval is unbounded $a=\infty$
2. The Solution explodes at the boundary: $a>\infty$ and ${\lim }_{t\searrow a}\Vert \lambda (t)\Vert =\infty$
3. The solution hits the boundary of the domain: $a>-\infty ,\partial D\ne \varnothing$ and ${\lim }_{t\searrow a}dist(\partial ,(t,\lambda (t)))=0$

and for the boundary point $b$ one of three analogous conditions apply.

The distance function simply measures the minimal distance between points of the two sets.

Examples

Uniqueness failing

Betrachte die Differentialgleichung $\dot{x}=x^{1/3}$. Die Lösung, die bei $0$ beginnt, ist nicht eindeutig.

• $0$ ist ein Fixpunkt also ist $x(t)=0$ eine Lösung
• $x(t)=(\frac{2}{3}t{)}^{3/2}$ ist auch eine Lösung Man erhält sich durch Separation of variables. Der Grund hierfür ist, dass obere Gleichung in $0$ sehr instabil ist. $(x^{1/3}{)}^{\prime }(0)=\infty$.

Applications

Bound solutions

We want to find an upper of lower bound for a solution $\lambda (t)$ to some first-order one-dimensional ODE. One idea is to use the fact, that Solution curves are disjoint, to follow that the solution curve of $\lambda$ can never cross the solution curve of constant solutions.

Bound solutions 2

A second possibility to bound solutions $x(t)$ is to use the fundamental theorem of calculus: $x(t)-x(t_0)={\int }_{t_0}^t{x}^{\prime }(s)ds={\int }_{t_0}^{t}f(s,x(s))ds$ Even if $x$ is not known, $f$ might allow upper or lower bounds.