I've been trying to prove that the following equation has a unique solution in interval 0 < x < 1 :

$$ x = \Big(\frac{1 - (1-x^2)^K}{1 - (1-x)^K}\Big)^2 $$

Where K is an integer greater than 1. I have checked it numerically and, in addition to x=0 and x=1, there is always a solution in interval (0,1). Does anyone has any idea of how I can prove the existence and uniqueness of such a fixed point?