Volterra equation of the first kind

Remarks

Example: $\int_0^t (t - s) \, x(s) \, ds = t^3$ on $[0, 1]$ is of the first kind with $K(t, s) = t - s$. Differentiating twice gives $x(t) = 6t$, and indeed $\int_0^t (t - s) \, 6s \, ds = 3t^3 - 2t^3 = t^3$. Non-example: $x(t) - \int_0^t x(s) \, ds = 1$ is not of the first kind, since $x$ also appears outside the integral; it is a Volterra equation of the second kind. A necessary condition for a continuous solution when $K$ is continuous is $f(a) = 0$, since the integral vanishes at $t = a$. Often $K$ is required to be continuous on $\Delta$, but weakly singular kernels such as $(t - s)^{-1/2}$ are also allowed, as in Abel's integral equation.