Volterra integral equation

Remarks

Example: $x(t) = 1 + \int_0^t x(s) \, ds$ on $[0, b]$, second kind with $K = 1$ and $\lambda = 1$. The function $x = e^t$ solves it, since $1 + \int_0^t e^s \, ds = e^t$.

Non-example: $x(t) = f(t) + \int_a^b K(t, s) x(s) \, ds$ has a fixed upper limit $b$; it is a Fredholm integral equation, not a Volterra one.

For the second kind, the solution exists and is unique for every $\lambda$. It is the sum of the Neumann series $x = \sum_{j \ge 0} \lambda^j V^j f$, where $(V x)(t) = \int_a^t K(t, s) x(s) \, ds$; the bound $\abs{(V^j f)(t)} \le \norm{K}_\infty^j (t - a)^j \norm{f}_\infty / j!$ makes it converge. The same scheme gives existence for nonlinear equations $x(t) = f(t) + \int_a^t K(t, s, x(s)) \, ds$ with $K$ Lipschitz in $x$, which include the integral form of initial value problems.