TITLE: Nonlinear Grönwall inequality
QUESTION [6 upvotes]: Let $T>0$, $\alpha,\beta>0$ and consider a non-negative continuous function $x$ on $[0,T]$ such that for all $t \in [0,T]$ one has
$$x(t) \leq \alpha+\beta\left(\int_0^t x(s)\,\mathrm ds \right)^{1/2}.$$
Does anyone knows what kind of Grönwall inequality I can get from this ? It would be fantastic if I can get something like $x(t) \leq Ct$.

REPLY [2 votes]: The inequality implies 
$$
x(t) \le \alpha + \epsilon \int_0^tx(s) ds + \frac{\beta}{4 \epsilon}
$$
for any $\epsilon > 0$. By the usual Gronwall inequality, this implies
$$
x(t) \le \left(\alpha + \frac{\beta}{4 \epsilon} \right) e^{\epsilon t}
$$
for any $t > 0,\epsilon > 0$. Now set $\epsilon = t^{-1}$ to obtain
$$
x(t) \le \left(\alpha + \frac{\beta}{4} t \right)e
$$
You may be able to get better constants by choose $\epsilon = \delta t^{-1}$ with the right $\delta > 0$.