Fredholm equation

A Fredholm equation of the first kind is an integral equation of the form

$${\int }_a^{b}K(x,y)f(y)𝑑y=g(x),\forall x\in [a,b],$$

and a Fredholm equation of the second kind is an integral equation of the form

$$f(x)-\lambda {\int }_a^{b}K(x,y)f(y)𝑑y=g(x),\forall x\in [a,b],$$

where

$$Kf(x)={\int }_a^{b}K(x,y)f(y)𝑑y,\forall x\in [a,b]$$

is a compact operator in some function space.