If one exists, find a continuous, bounded function $f: \mathbb{R} \to \mathbb{R}$ which is not identically zero and which satisfies$$0 = \int_0^\infty K(t, s)f(s)\,ds$$for all $t \in \mathbb{R}$, where $K(t, s) = 1/(s^{7/4}(1 + e^{t/s}))$.
After attempting to solve this for some time, I'm starting to believe that there is no solution. The condition that the integral has to vanish for any choice of $t$ seems quite strong, my recent attempts have been to assume points where $f \neq 0$ and try to create singularities in $f$ from this, but to not much success.
After some searching online, I think this kind of integral is called a Fredholm integral, which seems to suggest a connection to some kind of physical system. Is there a standard method for dealing with this kind of integral?