Let  $a,f \in L^2(0,t)$ (where $t \leqslant 1$), and consider the following integral equation: 
$$
f(t)\int_0^t {a(s)ds}  + \int_0^t {a(t - s)} f(s)ds = 0
$$
>My question is : under what condition on $a$ the unique solution is $f=0$ ?
Thanks.