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  "body": "\n## Definition\n\nLet $a \u003c b$, let $\\Delta = \\set{(t, s) : a \\le s \\le t \\le b}$, let $K \\colon \\Delta \\to \\R$ be continuous, and let $f \\colon [a, b] \\to \\R$ be continuous.\nA *linear Volterra integral equation of the first kind* is\n$$f(t) = \\int_a^t K(t, s) \\, x(s) \\, ds, \\quad t \\in [a, b].$$\nA *linear Volterra integral equation of the second kind* is\n$$x(t) = f(t) + \\lambda \\int_a^t K(t, s) \\, x(s) \\, ds, \\quad t \\in [a, b],$$\nwith a constant $\\lambda$.\nIn both, $x \\colon [a, b] \\to \\R$ is the unknown continuous function and $K$ is the *kernel*.\nThe upper limit of the [[integral|0W29TBTY12]] is the variable $t$.\n\n## Remarks\n\nExample: $x(t) = 1 + \\int_0^t x(s) \\, ds$ on $[0, b]$, second kind with $K = 1$ and $\\lambda = 1$.\nThe function $x = e^t$ solves it, since $1 + \\int_0^t e^s \\, ds = e^t$.\n\nNon-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.\n\nFor the second kind, the solution exists and is unique for every $\\lambda$.\nIt 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.\nThe 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.\n\n## Explanations\n\n### L1 human\n\nThe unknown appears under an integral whose upper limit is the current time.\nSo the equation can be solved step by step forward in time.\n",
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