Let $f$ be a non-negative function supported and integrable on the positive real axis, such that $$\int_0^\infty f(x+y)p(y) dy = c[p] f(x), $$ where $c[p]$ a number (functional) dependent on function $p$.

1)Suppose the above equation holds for every non-negative function $p$ supported and integrable on the positive real axis. Is $f(x) = a\exp(-kx)$ for some non-negative $k$ and $a$?

2) Suppose the above equation holds for some non-negative function $p$ supported and integrable on the positive real axis. I suppose $f(x)$ is not necessarily an exponential function. How can we characterize $f$.

I tried using Laplace transform but got stuck. Similarly I tried using Fourier transform with the coefficient on the upper half complex plane. The form I obtained is $$\tilde f(k) \tilde p(-k) = c[p]\tilde f(k).$$ How do I proceed from here? Perhaps I should take a completely different route?