Do the following functions $ f: \mathbb{R}^{2} \rightarrow \mathbb{R} $ satisfy a local or global Lipschitz condition with respect to $ x $ ? Justify the answer!
(a) $ f(t, x)=\exp \left(t^{2}\right) x $
(b) $ f(t, x)=\arctan \left(t^{2}+x\right) $
Problem/approach:
I've now added the two "conditions" addressed above below here. In the script we only discussed three other lemmas, but these were more for being able to show the uniqueness theorem for solving a DGL.
Therefore probably only these two definitions are meant (if not, please tell me, then I can look again in the script).
Now I have the problem that I do not know how to check my given functions for the Lipschitz conditions or fulfill them at all. Can someone help me there?
My attempt for (a): Let $ x, \bar{x} \in \mathbb{R}$. $$||f(t,x)-f(t,\bar{x})||=||e^{t^2}x-e^{t^2}\bar{x}||=||e^{t^2}(x-\bar{x})||\leq||e^{t^2}||\cdot||x-\bar{x}||$$
But what should I be able to say from this now? For (b) I don't really have an idea yet, because I don't really understand how to check the conditions.
- A continuous function $ f: I × G \rightarrow \mathbb{R}^{n} $ is called locally Lipschitz regarding $ x $ ($\in G$) if to every point $ \left(t_{1}, x_{1}\right) \in I ×G $ there is a ball $ \bar{B}_{r}\left(x_{1}\right) $ with $r > 0$ and a $ \alpha>0 $ with $ \left[t_{1}-\alpha, t_{1}+\alpha\right] \times \bar{B}_{r}\left(x_{1}\right) \subset I ×G $, and a constant $ L=L\left(t_{1}, x_{1}\right)\geq0 $ exist such that holds:
$ ||f(t, x)-f(t, \bar{x})|| \leq L\left(t_{1}, x_{1}\right)||x-\bar{x}||, \quad \text { if }\left|t-t_{1}\right| \leq \alpha; x, \bar{x} \in \bar{B}_{r}\left(x_{1}\right) . $
- $ f $ is globally Lipschitz with respect to $ x $ if the constant $ L>0 $ is independent of $ \left(t_{1}, x_{1}\right) $, i.e.:
$ ||f(t, x)-f(t, \bar{x})|| \leq L||x-\bar{x}||, \quad $ ∀$t\in I$, ∀$x,\bar{x} \in G$.