Let $\beta \in (0, 1)$. We assume $f : [0, 1] \to [0, \infty)$ is a measurable and bounded function such that
$$
f(t) \le \int_0^t (t-s)^{-\frac{1}{2}} [f(s) + |f(s)|^{\beta}] \, \mathrm d s,
\quad \forall t \in [0, 1].
$$

>I would like to ask if $f=0$?

Thank you so much for your elaboration!


  [1]: https://mathoverflow.net/questions/467786/gr%C3%B6nwall-type-inequality-for-ft-le-alpha-int-0t-t-s-frac12-f