upper-bound of a Volterra-type integral?

Question

Let $f:\mathbb{R}\mapsto [0,+\infty)$ be a normalized (i.e. integral is 1) delay kernel, $t>0$, and $y\in\mathbb{R}$ is a solution (unknown) of a differential equation. I want to find an upper bound (supremum) of the integral $$ \int\limits_{0}^{\infty}|y(t-u)|f(u)du. $$ In particular, what I really want is to have $$ \int\limits_{0}^{\infty}|y(t-u)|f(u)du \leq |y(t)|. $$ Can this inequality be justified? Under what conditions?