Fredholm integral equation

Definition

Let $a < b$, let $K \colon [a, b] \times [a, b] \to \C$ be a kernel, let $f \colon [a, b] \to \C$ be given, and let $\lambda \in \C$. A Fredholm integral equation of the first kind is the equation $$\int_a^b K(x, s) \varphi(s) \, ds = f(x) \quad \text{for all } x \in [a, b].$$ A Fredholm integral equation of the second kind is the equation $$\varphi(x) = f(x) + \lambda \int_a^b K(x, s) \varphi(s) \, ds \quad \text{for all } x \in [a, b].$$ In both, $\varphi \colon [a, b] \to \C$ is the unknown function, and both limits of integration are the fixed numbers $a$ and $b$. A Fredholm integral equation is a linear integral equation of one of these two forms.