I have the following fixed point equation: $$ F(t)= \displaystyle\sum_{r=1}^M \int_{\mathbb{R}} F ((t-c)r) g(r) h(c) \: \mathrm{d}c$$ where: $F$ is a cumulative density function of a certain random variable, $g$ is a probability mass function of a discrete random variable that ranges in $1,2,\dots, M$ and $h$ is a density function of a continous random variable with support in $\mathbb{R}$.
My question: is there a way to prove that a fixed point exists? I was trying to show that the map $\Phi(F)=\displaystyle\sum_{r=1}^M \int_{\mathbb{R}} F ((t-c)r) g(r) h(c) \: \mathrm{d}c$ is a contraction with respect to some distance but I haven't gone too far.