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.

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

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