I was wondering if anyone might be able to give me an answer to the following, or at least point me to a reference that will allow me to figure it out.
I have an integral equation of the following form:
$g(x,y)=\int K(x,z)f(z,y)dz$
Where $g$ and $K$ are known.
First of all, for this to have a solution do I generally need $y$ to be a scalar (i.e. could there still be a solution when $y$ is a vector of some predetermined length). Secondly, are there simple conditions I can place on $g$ and $K$ that guarantee a solution $f$ to the above exists (no need for uniqueness) that is smooth (e.g. satisfies a Holder condition or something like that).
Any help appreciated, have been searching around the web for answers for days and haven't found what I'm looking for.