Let $C$ be the space of continuous and nondecreasing functions defined on $[0,1]$ and endowed with the sup norm. Let $T:C\rightarrow C$ be a continuous mapping, and consider the following expression: $ U(Tz(x);z)=\int_{0}^{Tz(x)}\left[\int_{0}^{1}F(z(\xi))f(\xi)d\xi+\int_{s\in \Gamma(t,z)}\left\{F(t)-F(z(s))\right\}f(s)ds\right]^{n-1}dt $ where $\Gamma(t,z)=\{s:[0,1]|s\geq z(t)\}$, and $F:[0,1]\rightarrow [0,1]$ is continuously differentiable and increasing; $F'=f$, $f(s)>0, s\in[0,1]$ and $f(s)=0,\,s\notin [0,1]$, $x\in[0,1]$, and $n>2$.
Suppose that I've managed to show that there exists a constant $K$ such that $ ||U(Tz(x);z)-U(Ty(x);y)||\leq K||z-y|| \qquad z,y\in C $ holds true.
What I want is to make a claim about $T$. Particularly, I'd like to show whether is true or not that the following hold:
$ ||Tz(x)-Ty(x)||\leq C||z-y|| \quad z,y\in C $ where $C$ is a constant independent of $z$ and $y$.
Does anyone has any idea how I can prove/disprove this? I have no clue whatsoever on how to proceed, and I do need some help to get through it. Can someone help me or point me the right direction on how to tackle this problem?
Any help/suggestion/insight/reference is greatly appreciated it!