Gronwall-like inequality with double integral

Question

Assume that a non-negative function $f(t)$ on $[0,\infty)$ satisfies $f(0)=0$ and for certain $0<\alpha<1$, $$ f(t)\leq \int_0^t\int_0^se^{-(s-u)}|f(u)|^\alpha \, du \, ds. $$ Can we get $f(t)=0$ for every $t>0$?? Or is there exists a counter example? Thanks for your help!

Answer 1

The answer is no. It is possible that $f(t) > 0$ for $t > 0$. As a counterexample, consider $f(t) = c_0t^{2/(1-\alpha)}$ for suitable $c_0$, to be chosen later. Then $$ \frac{1}{2}\int_0^t(t-u)f(t)^\alpha \, du = c_0^\alpha C_1 t^{2/(1-\alpha)} = c_0^{\alpha - 1}C_1 f(t) $$ for a suitable $C_1 > 0$. Choose $c_0$ such that $c_0^{\alpha-1}C_1 = 1$. Therefore for all $t$ $$ f(t) = \frac{1}{2}\int_0^t(t-u)f(t)^\alpha \, du \,. $$ Now for $0 \le t-u \le \frac{3}{2}$ $1 - e^{u-t} \ge 1/2 (t-u)$ and therefore $$ f(t) = \frac{1}{2}\int_0^t(t-u)f(t)^\alpha \, du \le \int_0^t(1-e^{u-t}) f^\alpha(u) \, du \quad \text{on} \quad [0,3/2] \, . $$