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  "body": "\n## Statement\n\nLet $J = [t_{\\min}, t_{\\max}]$ and $t_0 \\in J$, let $x_0 \\in \\mathbb{R}^n$, and let $f : J \\times \\mathbb{R}^n \\to \\mathbb{R}^n$ satisfy the [[Picard–Lindelöf hypotheses|0J7HF2PF1H]] at $(t_0, x_0)$ with constants $a, L, K$.\nLet $B$ be the closed ball of radius $a$ around $x_0$.\nThen there is exactly one continuous curve $\\alpha : J \\to B$ such that\n$$\\alpha(t) = x_0 + \\int_{t_0}^{t} f\\left(s, \\alpha(s)\\right) ds \\quad \\text{for each } t \\in J.$$\n\n## Remarks\n\nThe curve $\\alpha$ is the unique fixed point of the [[Picard operator|00WAVB68F9]] on the curves with values in $B$.\nBy the [[integral equation form|015KK993C5]], this is the same curve as the solution of the [[Picard–Lindelöf theorem|0P6KRD5B7A]].\n\n## Explanations\n\n### L1 human\n\nThis is the Picard–Lindelöf theorem written without derivatives.\nIt says that the integral equation has exactly one continuous solution that stays in the ball.\n",
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  "name": "Picard–Lindelöf theorem in integral form",
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