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Let $K=[0,1]\times [0,1]$. Find a continuous mapping $F:K\rightarrow \mathbb{R}^2$ satisfying:

  • $\|F(x)-F(y)\|\leq \|x-y\| \quad\forall x,y\in K,$

  • There exists $\gamma>0$ such that for all $x,y\in K$ $$\left<F(x),y-x\right>\geq 0\Longrightarrow\left<F(y), y-x\right>\geq \gamma\|x-y\|^2,$$

  • There exist $u,v\in K$ such that $\left<F(u)-F(v), u-v\right><0$.

Here, $\|.\|$ is the Euclidean norm and $\left<,.,\right>$ is the scalar product in $\mathbb{R}^2$.

Thank you for all comments and helping.

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Could you tell us how this question arose, and what your progress is on it? For instance, have you ruled out certain classes of maps, like restriction of linear or affine maps? –  Olivier Bégassat Jul 23 '12 at 3:59
    
@Olivier Bégassat: I have tried it. It is easily to construct a mapping satisfying all conditions in the case $\mathbb{R}$. It is very difficult to to construct such mapping in $\mathbb{R}^2$. It is interesting to construct an $\textbf{affine mapping}$ satisfying all above conditions. Thank you for your comments. –  blindman Jul 23 '12 at 4:04
    
How did you come to consider this qestion? Can you motivate the two final conditions? It might help others to think about your problem to know its context and why the conditions are what they are. I don't understand your last comment fully, do you have a solution to this problem? –  Olivier Bégassat Jul 23 '12 at 4:07
    
@OlivierBégassat: Two final conditions relate to some kind of monotonicity conception. The last condition means that $F$ is not monotone on $K$. Thank you for your consideration. –  blindman Jul 23 '12 at 4:11
    
Could you expand on your comment, it is unfortunately too vague to be of any help for me ^^, also, please tell us your progress. –  Olivier Bégassat Jul 23 '12 at 4:12
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