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.