Specifically, $Tx(t)=\int_0^1k(s,t)x(s)\mathbb{d}s$, where $k$ is a continuous function on $[0,1]^2$ and $x,y \in C[0,1]$, $x(t),y(t)>0$ for all $t \in [0,1]$
I know a mathematical analysis way to solve this, I will appreciate it if there is a functional analysis way. I've tried the following but stuck:
If we can prove $x$ and $y$ are linearly dependent, there exists $\lambda$ such that $y=\lambda x$, and as the condition indicates $T^2x=x$, we conclude $y=x $ or $y=-x$,the latter can be excluded by $x(t)>0$ and $y(t)>0$.
To prove the two are linearly dependent, assume the opposite, then there must be some controdiction, I stuck at here.
I tried to apply fix point theorem on this problem, but in this case $T$ is linear and $0$ is a fix point, we also know that the norm of $T$, $\|T\|=\|\int_0^1|k(s,t)|\mathbb{d}s\|$ is no less than $1$ , in fact, if $\|T\|<1,$ we can conclude $x=0$ which is controdict with $x(t)>0$