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.

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.