Let $x,y \in[0,1] $, consider the following system of equations:
$$ ((x+y)/2)^n-x=0 $$ $$ {x^n \over x^n+y^n+1}-y=0 $$
where $ n \in N $
a) Transform the system of equations into equivalent fixed point problem. b) Show that a solution exists by using fixed point theorem.
So my idea was to use $ y=2x^{1/n}-x $ and get $$ f(x) = {{x^n} \over {x^n+(2x^{1/n}-x)^n+1}}-2x^{1/n}+x=0 $$ Then the fix point of $f(x)+x $ will give a solution to the system of equations.
My question is then: Is this the way to proceed? or there is a better way?