In the context of Fredholm integral equation of the first kind: $$ f(x) = \int_0^\infty K(x,y)\phi(y) dy $$ where $f, K$ are given and $\phi$ is unknown, I read somewhere that if the homogeneous equation (i.e. $f(x) := 0$) only has the trivial solution $\phi(y) = 0$ almost everywhere, then for any $f(x)$ the inhomogeneous equation always has at least one solution.

Q1: Is this correct? Are there any particular requirements on the kernel $K$ and/or $f$ to make it correct?

Q2: How do you call K when the homogeneous equation only has the trivial solution? Invertible? Nonsingular? ...

Thanks for your help! p.


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