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I need help to understand this question, it's not that clear for me:

Using the norm $|x|+|y|$ and the Mean value inequality, give a condition over the partial derivatives of $f(x,y)$ and $g(x,y)$ for the Fixed point theorem may be applied to the transformation:

$$T(x,y)=(f(x,y),g(x,y))$$

I'm not sure what exactly I should do, how to start with this one. Thanks a lot!

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Do you want conditions for applying the Brower fixed-point theorem? In that case, the theorem applies as soon as the transformation $T$ is continuous and maps, say, the region $\{|x|+|y| \leq 1\}$ back into itself. The Mean value theorem gives you estimates, for instance, on how much $T(x,y)$ can deviate from $T(0,0)$. How big can the partials of $f,g$ be if you want this estimate? – A Blumenthal Jan 25 at 20:27

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