05 - Picard's Existence Theorem for Systems

Picard's Existence Theorem for Systems

Let $f:S\to {\mathbb{R}}^2$ be a function defined on the set $S=[a-h,a+h]\times B_k(\underline{b})$ which satisfies

• $H(i)$: $\underline{f}(x,\underline{y})$ is continuous on $S$, $||\underline{f}(x,\underline{y})|{|}_1\le M$ and $Mh\le k$
• $H(ii)$: $f$ is Lipschitz with respect to $\underline{y}$ on $S$ so
  There exists $L$ such that for all $x\in [a-h,a+h]$ and $\underline{y},\tilde{\underline{y}}\in B_k(b)$ then

$$||\underline{f}(x,\underline{y})-\underline{f}(x,\tilde{\underline{y}})|{|}_1\le L||\underline{y}-\tilde{\underline{y}}|{|}_1$$

Then IVP

$${\underline{y}}^{\prime }(x)=\underline{f}(x,\underline{y}(x))$$

Has a unique solution on interval $[a-h,a+h]$

Tips for Showing Conditions

Note that conditions on functions $\underline{f}:S\to {\mathbb{R}}^2$ is a function of 3 variables and NOT $x\mapsto \underline{f}(x,\underline{y}(x))$

Can extend for systems of $n$ ODEs for any $n\in \mathbb{N}$ but we focus on $n=2$

When checking for the Lipschitz Condition for vector valued function $\underline{f}$
Simply check for the components as we are using the $l^1$ norm i.e. there exists $L_1,L_2$

$$\begin{array}{rl}|f_1(x,y_1,y_2)-f_1(x,{\tilde{y}}_1,{\tilde{y}}_2)|& \le L_1(|y_1-{\tilde{y}}_1|+|y_2-{\tilde{y}}_2|)\\ |f_2(x,y_1,y_2)-f_2(x,{\tilde{y}}_1,{\tilde{y}}_2)|& \le L_2(|y_1-{\tilde{y}}_1|+|y_2-{\tilde{y}}_2|)\end{array}$$

then take $L=L_1+L_2$

May also be handy to add in “smart 0s” e.g.

$$f_1(x,y_1,y_2)-f_1(x,{\tilde{y}}_1,{\tilde{y}}_2)=(f_1(x,y_1,y_2)-f_1(x,{\tilde{y}}_1,y_2))+(f_1(x,{\tilde{y}}_1,y_2)-f_1(x,{\tilde{y}}_1,{\tilde{y}}_2))$$

So we can then use MVT so there is some $\xi$ between $y_1$ and ${\tilde{y}}_1$ so that

$$f_1(x,y_1,y_2)-f_1(x,{\tilde{y}}_1,y_2)={\partial }_{y_1}{f}_1(x,\xi ,y_2)(y_1-{\tilde{y}}_1)$$

and similarily for some ${\xi }^{\prime }$ between $y_2$ and ${\tilde{y}}_2$

$$f_1(x,{\tilde{y}}_1,y_2)-f_1(x,{\tilde{y}}_1,{\tilde{y}}_2)={\partial }_{y_2}{f}_1(x,{\tilde{y}}_1,{\xi }^{\prime })(y_2-{\tilde{y}}_2)$$

Solutions will not only exist on interval $[a-h,a+h]$ but on maximal existence interval

$$(T_{-},T_{+})$$

which for continious function $\underline{f}:{\mathbb{R}}^3\to {\mathbb{R}}^2$ which satisfy Lipschitz condition on
every compact subset $S$ of ${\mathbb{R}}^3$ will be given by all of $\mathbb{R}$ (assuming no blow-up)