Find domain where function is Lipschitz

[psie]

Homework Statement

Reduce the ODEs $x'''+x^2=1, x''=x^{-1/2}$ and $x''=\sqrt{1+(x')^2}$ to a system of first order and find a naturally defined region $\Omega$ where the right hand side satisfies a Lipschitz condition.

Relevant Equations

$f$ satisfies a Lipschitz condition in the $x$-variable in a set $\Omega$ if $\lVert f(t,x)-f(t,y)\rVert\leq L\lVert x-y\rVert$, for some positive constant $L$.

The reduction is simple in all cases. For the first one, put $x_1=x, x_2=x'$ and $x_3=x''$. Let $\pmb{x}=(x_1,x_2,x_3)$. Then we get $$\pmb{x}'= \begin{pmatrix}x_1' \\ x_2' \\ x_3' \end{pmatrix}=\begin{pmatrix}x_2 \\ x_3 \\ 1-x_1^2 \end{pmatrix}=\pmb{f}(\pmb{x}),$$ where $\pmb{f}(\pmb{x})=(f_1(\pmb{x}),f_2(\pmb{x}),f_3(\pmb{x}))=(x_2,x_3,1-x_1^2)$.

Similarly, $\pmb{g}(\pmb{x})=(x_2,x_1^{-1/2})$ and $\pmb{h}(\pmb{x})=(x_2,\sqrt{1+(x_2)^2})$ for the other two ODEs.

In the first case, I'm interested in finding a subset of $\mathbb R^3$ such that I can bound $\lVert \pmb{f}(\pmb{x})-\pmb{f}(\pmb{x})\rVert$. I'm unsure how to approach this in all cases, whether to use the definition of some norm directly or the mean value theorem. In the latter case, I'm unsure how the mean value theorem applies to vector-valued functions of a vector. Anyway, grateful for any help.

 

[pasmith]

Perhaps start by calculating $f(x) - f(y)$, and see if you can write $$\|f(x) - f(y)\|^2 = A(x,y)|x_1 - y_1|^2 + B(x,y)|x_2 - y_2|^2 + C(x,y)|x_3 - y_3|^2$$ for positive functions $A$, $B$ and $C$. How can you then guarantee that $$\|f(x) - f(y)\|^2 \leq L^2\|x - y\|^2$$ for some $L > 0$?

 

[psie]

Good idea, however, when it comes to $\pmb{h}(\pmb{x})=(x_2,\sqrt{1+(x_2)^2})$, there is no $x_1$ coordinate included in the components. Maybe this is not a problem. Using the Euclidean norm: $$\lVert \pmb{h}(\pmb{x})-\pmb{h}(\pmb{y})\rVert^2=(x_2-y_2)^2+\left(\sqrt{1+x_2^2}-\sqrt{1+y_2^2}\right)^2 $$ How can I handle the second term, i.e. $\left(\sqrt{1+x_2^2}-\sqrt{1+y_2^2}\right)^2$, so that it potentially doesn't cause any trouble?

As an alternative approach, I think a vector-valued function is Lipschitz iff all of its components are. Therefor we can apply the mean value theorem to the components, which is fairly simple in this case.