Let $c>0$ and $f:[0, \infty)\to\mathbb R^+$ a non negative function such that
$$f(t)\leq -c\int_0^tf(s)ds+\int_0^te^{-s}\sqrt{f(s)}ds.$$
Is it possible to prove that $\lim_{t\to +\infty} f(t)=0$?
Without the second term in the sum I would be able to conclude that $f(t)\leq f(0) e^{-ct}$. How can I conclude with this extra term?
Thank you very much.