Homogeneous integral equation

Definition

Let $a < b$, let $a_0, f \colon [a, b] \to \C$ and a kernel $K \colon [a, b] \times [a, b] \to \C$ be given, and let $\lambda \in \C$. For each $x \in [a, b]$, let $I_x$ be either $[a, b]$ or $[a, x]$. A linear integral equation $$a_0(x) \varphi(x) = f(x) + \lambda \int_{I_x} K(x, s) \varphi(s) \, ds \quad \text{for all } x \in [a, b]$$ is homogeneous if $f(x) = 0$ for all $x \in [a, b]$, and inhomogeneous otherwise. For an equation of the second kind, the homogeneous equation is $$\varphi(x) = \lambda \int_{I_x} K(x, s) \varphi(s) \, ds \quad \text{for all } x \in [a, b].$$