Suppose for two given functions $f_1,f_2 \colon \mathbb{R}^2 \to \mathbb{R}$ there exist unique solutions $y_1$ and $y_2$ with the intersection of their intervals of existence $[0,\epsilon)$ to the integral equations $$y_k(x)=\int_{0}^{x} f_k(t,y_k(t)) dt$$. Moreover, suppose that $f_1(x,y)\leq f_2(x,y)$ (or strictly if that makes things easier), does it follow that $y_1 \leq y_2$ on $[0,\epsilon)$?

It's true if $f_1$ and $f_2$ are Lipschitz in $y$ and continuous in $x$ (i.e., satisfy the hypotheses of the standard uniqueness and existence theorem) or if $f_1$ and $f_2$ are decreasing in $y$ (i.e., satisfy the hypotheses of Peano's uniqueness theorem). The assumption of unique solutions is certainly necessary.