Convergence of the Neumann series for Volterra kernels

Remarks

The bound in item 1 is proved by induction: $\abs{K_{n+1}(t, s)} \le \int_s^t M \cdot M^n \frac{(u - s)^{n - 1}}{(n - 1)!} \, du = M^{n+1} \frac{(t - s)^n}{n!}$. The factorial makes the series of item 2 converge like the exponential series, by the Weierstrass M-test, whatever the size of $\lambda$. The function $R$ is the resolvent kernel of $K$. Example: for $K = 1$ on $[0, b]$, $K_n(t, s) = (t - s)^{n-1}/(n-1)!$ and $R(t, s; \lambda) = e^{\lambda (t - s)}$. Then $x = 1 + \lambda \int_0^t x$ has the solution $x(t) = 1 + \lambda \int_0^t e^{\lambda(t - s)} \, ds = e^{\lambda t}$. For a Fredholm kernel on $[a, b] \times [a, b]$ the same series converges only for $\abs{\lambda}$ small; the Volterra structure, with integration over $[s, t]$ only, is what produces the factorial.

Going deeper

For a student

The theorem shows that the Volterra integral operator has spectral radius $0$: it is quasinilpotent. Consequently $I - \lambda V$ is invertible for every $\lambda$, which gives the unique solvability of Volterra equations of the second kind. The same estimate, applied to a nonlinearity that is Lipschitz in $x$ uniformly on $[a, b]$, gives a global solution of the Picard–Lindelöf type on all of $[a, b]$. The bound in item 3 is of Grönwall type, and it controls how solutions depend on $f$.