# A special integral equation of Volterra type

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.

• Is your $a$ positive or not necessarily? – fedja Dec 6 '18 at 22:16
• @fedja The $a$ is not necessarily positive. – Gustave Dec 7 '18 at 15:52