I do not think this function is contractive.

Assume that it is,

|F(x)-F(y)| <= K*|x-y| for -1/3<=x,y<=1/3

Take y=0 to get,

|F(x)|<= K*|x| for -1/3<=x<=1/3

Then certainly,

|F(x)|<= K*|x| for 0<x<=1/3

Thus,

x^{2/3} <= K*x for 0<x<=1/3

But then,

x^{-1/3} <= K

Then certainly

x^{-1/3} <= K+3

But this is false if you choose,

x>1/(K+3)^3