Integral equation of the third kind

Remarks

Example: $x \varphi(x) = x + \int_{-1}^{1} s \, \varphi(s) \, ds$ on $[-1, 1]$, with $a_0(x) = x$, $f(x) = x$, $K(x, s) = s$ and $\lambda = 1$. The coefficient $a_0$ is continuous, vanishes at $0$, and is not identically zero. The function $\varphi = 1$ is a solution, since $\int_{-1}^{1} s \, ds = 0$.

Non-example: $(2 + x) \varphi(x) = f(x) + \int_{-1}^{1} \varphi(s) \, ds$ on $[-1, 1]$ is not of the third kind, since $2 + x$ has no zero there. Dividing by $2 + x$ turns it into an equation of the second kind.

Some authors also allow the upper limit $x$ in place of $b$.