Dears,
Let $I:=[0,1]$ and $f:I^{2} \longrightarrow I^{2}$ continuous and such that $f(I^{2})=I^{2}$. For instance, $f(x,y):=(1-x,y)$ or $f(x,y):=(1-x,1-y)$. I am looking for a contiunous $g:I^{2}\times I^{2} \longrightarrow I^{2}$ satisfying the following property:
$d(g((x,y),(u,v)), g((x',y'),(u',v')))\leq \frac{1}{2} [d(f(x,y),f(x',y'))+d(f(u,v),f(u',v'))]$
for each $x,y,u,v,x',y',u',v'\in I$. By $d$, we mean the Euclidean distance.
Somebody can give a (non-trivial) example of such mapping $g$?
Thank very much for your time!!