I have a Fredholm integral equation of the second kind given as
$$f(x)=g(x)+\lambda\int_{-\infty}^\infty K(x,y)f(y)dy, $$
where $\lambda\in(0,1)$, the kernel $K(x,y)=\phi(x-y)$ is a Gaussian function, and $g(x)$ is a known continuous function. I am interested in solutions $f$ that satisfy this.
I could find a solution by the Neumann series approach (see, for example, Section 5.3 here: http://www.hep.caltech.edu/~fcp/math/integralEquations/integralEquations.pdf).
My question is now: How do I know whether that solution is unique? Could there be other functions that solve this (discontinuous functions $f$ are allowed)?
What I could find is, e.g., that the Neumann series converges to the unique solution if $|\lambda|M(b-a)<1$, where $M$ is an upper bound for $K$ and $a,b$ are the integral bounds. However, since in my case $b-a=\infty$ this is obviously not satisfied. But perhaps weaker conditions exist, especially given I already know that the Neumann series converges to a solution? Is the solution then unique?
Edit: Another condition I found is: The Neumann series converges to the unique solution if $|\lambda|\|K\|_2<1$, where the norm is defined as
$$\|K\|_2=\left(\int_{-\infty}^\infty \int_{-\infty}^\infty |K(x,y)|^2 dxdy\right)^{1/2}.$$
However, I don't think I can evaluate that double integral for a Gaussian function $K(x,y)=\phi(x-x)$, can I?