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  "body": "\n## Statement\n\nLet $a \u003c b$, let $\\Delta = \\set{(t, s) : a \\le s \\le t \\le b}$, and let $K \\colon \\Delta \\to \\C$ be continuous with $M = \\max_\\Delta \\abs{K}$.\nDefine the [[iterated kernels|9QNBFERMHD]] on $\\Delta$ by $K_1 = K$ and\n$$K_{n+1}(t, s) = \\int_s^t K(t, u) \\, K_n(u, s) \\, du, \\quad n \\ge 1.$$\nThen the following hold.\n\n1. Each $K_n$ is continuous on $\\Delta$ and $\\abs{K_n(t, s)} \\le M^n \\dfrac{(t - s)^{n - 1}}{(n - 1)!}$ for all $(t, s) \\in \\Delta$.\n2. For every $\\lambda \\in \\C$, the series $R(t, s; \\lambda) = \\sum_{n \\ge 1} \\lambda^{n - 1} K_n(t, s)$ converges absolutely and uniformly on $\\Delta$, and uniformly for $\\lambda$ in bounded sets.\n3. The sum $R$ is continuous on $\\Delta$, entire in $\\lambda$, and $\\abs{R(t, s; \\lambda)} \\le M e^{\\abs{\\lambda} M (t - s)}$.\n4. For every continuous $f \\colon [a, b] \\to \\C$, the [[Liouville–Neumann series|H1HWSWME6B]] $x = f + \\sum_{n \\ge 1} \\lambda^n \\int_a^t K_n(t, s) f(s) \\, ds$ converges uniformly on $[a, b]$, and its sum\n$$x(t) = f(t) + \\lambda \\int_a^t R(t, s; \\lambda) \\, f(s) \\, ds$$\nsolves the [[Volterra equation of the second kind|ATPSAHYG7C]] $x(t) = f(t) + \\lambda \\int_a^t K(t, s) \\, x(s) \\, ds$.\n\n## Remarks\n\nThe bound in item 1 is proved by induction: $\\abs{K_{n+1}(t, s)} \\le \\int_s^t M \\cdot M^n \\frac{(u - s)^{n - 1}}{(n - 1)!} \\, du = M^{n+1} \\frac{(t - s)^n}{n!}$.\nThe factorial makes the series of item 2 converge like the exponential series, by the Weierstrass M-test, whatever the size of $\\lambda$.\nThe function $R$ is the [[resolvent kernel|ARTJV1XCRX]] of $K$.\nExample: for $K = 1$ on $[0, b]$, $K_n(t, s) = (t - s)^{n-1}/(n-1)!$ and $R(t, s; \\lambda) = e^{\\lambda (t - s)}$.\nThen $x = 1 + \\lambda \\int_0^t x$ has the solution $x(t) = 1 + \\lambda \\int_0^t e^{\\lambda(t - s)} \\, ds = e^{\\lambda t}$.\nFor 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.\n\n## Explanations\n\n### L1 human\n\nThe Neumann series solves an equation by substituting it into itself again and again.\nIn general, this only works if the kernel is small.\nFor a Volterra kernel, each substitution integrates over a shrinking triangle of past times, and the size of the $n$-th term drops like $1/(n - 1)!$.\nSo the iteration always converges, as the exponential series does.\n\n### L2 human\n\nThe theorem shows that the Volterra integral operator has spectral radius $0$: it is quasinilpotent.\nConsequently $I - \\lambda V$ is invertible for every $\\lambda$, which gives the [[unique solvability of Volterra equations of the second kind|WG4GVC9HR8]].\nThe 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 $[a, b]$.\nThe bound in item 3 is of [[Grönwall|07EJHQYR9A]] type, and it controls how solutions depend on $f$.\n",
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