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This problem is from Advanced Calculus 3rd by R. Buck.

Let $D$ be the unit disk, $x^2+y^2\leq1$. Consider a transformation $T$ of class $C^1$ on an open set containing $D$,

$$ T:\begin{cases} u=f(x,y),\\ v=g(x,y), \end{cases} $$

whose Jacobian is never $0$ in $D$. Suppose that $T$ is near the identity map in the sense that $|T(p)-p|\leq\frac{1}{3}$ for all $p\in D$. Prove that there is a point $p_0$ with $T(p_0)=(0,0)^T$.

Thanks a lot.

Update: I have figured it out. Seems the problem does not involve too much "inverse function theorem". At first, I have no clue at all. Here is the idea.

Define a function $\varphi(p)=\|T(p)\|_2^2$. $\varphi(p)$ is in fact the distance function between $T(p)$ and $(0,0)^T$. Then we need to show $\min_{p\in D}\varphi(p)=0$. If this is true, then the claim is proved. Next, we show $p^*=\arg\min_{p\in D}\varphi(p)$ is an interior point of $D$. Then we can use the Fermat's condition and this gives $T(p^*)=0$ which completes the proof.

I deeply appreciate if someone can provide me examples of interesting applications of inverse/implicit function theorems.

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  • $\begingroup$ Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be (temporarily) closed. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers. $\endgroup$ – Lord_Farin Jun 13 '13 at 7:09

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