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  "body": "\n## Summary\n\nA nonlinear Volterra equation is an integral equation $\\varphi(t) = f(t) + \\int_a^t K(t, s, \\varphi(s))\\, ds$.\nThe upper limit of integration is the variable itself, and the unknown may enter the integrand nonlinearly.\nIt extends the linear [[Volterra integral equation|DND8CBEVXT]] and contains every initial value problem $y' = F(t, y)$, $y(a) = y_0$, in integral form.\nWhen $K$ is [[Lipschitz|0AW1E6YTJY]] in its last variable, successive approximations give a unique solution on the whole interval.\nWhen the Lipschitz bound holds only on bounded sets, a solution may exist only on a short interval.\n\n## Explanations\n\n### L1 human\n\nThe value of the unknown at time $t$ depends on its whole past, through a rule that may be nonlinear.\nBecause only the past matters, the solution can be built forward step by step.\nFor example, $\\varphi(t) = 1 + \\int_0^t \\varphi(s)^2\\, ds$ is solved by $\\varphi(t) = 1/(1 - t)$, which blows up at $t = 1$.\nSo, unlike the linear case, a solution need not last for all time.\n",
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