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  "body": "\n## Statement\n\nLet $a \u003c b$, let $u, f : [a, b] \\to [0, \\infty)$ be continuous, and let $c \u003e 0$.\nLet $w : [0, \\infty) \\to [0, \\infty)$ be continuous and non-decreasing, with $w(r) \u003e 0$ for $r \u003e 0$.\nDefine $G(r) = \\int_c^r \\frac{ds}{w(s)}$ for $r \u003e 0$, and let $G(\\infty) = \\lim_{r \\to \\infty} G(r) \\in (0, \\infty]$.\nAssume that\n$$u(t) \\le c + \\int_a^t f(s)\\, w(u(s))\\, ds \\quad \\text{for each } t \\in [a, b].$$\nThen for each $t \\in [a, b]$ with $\\int_a^t f(s)\\, ds \u003c G(\\infty)$,\n$$u(t) \\le G^{-1}\\left( \\int_a^t f(s)\\, ds \\right).$$\n\n## Remarks\n\nThe function $G$ is continuous and strictly increasing with $G(c) = 0$, so $G^{-1}$ is defined on $[0, G(\\infty))$.\nProof idea: let $v(t) = c + \\int_a^t f(s)\\, w(u(s))\\, ds$.\nThen $v \\ge c \u003e 0$, $u \\le v$, and $v' = f\\, w(u) \\le f\\, w(v)$ since $w$ is non-decreasing.\nSo $\\frac{d}{dt} G(v(t)) = v'(t) / w(v(t)) \\le f(t)$, and integration gives $G(v(t)) \\le \\int_a^t f$.\nFor $w(r) = r$ one gets $G(r) = \\ln(r/c)$ and the bound $u(t) \\le c\\, \\exp\\left(\\int_a^t f\\right)$, which is [[Grönwall's inequality|07EJHQYR9A]] with constant $\\alpha = c$.\nFor $w(r) = r^2$ and $f = 1$ one gets $u(t) \\le c / (1 - c(t - a))$ while $c(t - a) \u003c 1$; the restriction on $t$ matches the blow-up of the solution of $u' = u^2$.\n\n## Explanations\n\n### L1 human\n\nGrönwall's inequality controls a quantity whose growth rate is at most proportional to its size.\nThe Bihari–LaSalle inequality does the same when the growth rate is a more general increasing function of the size.\nThe bound is the solution of the comparison equation $v' = f\\, w(v)$, $v(a) = c$.\n\n### L2 human\n\nThe inequality is the nonlinear version of the integral form of [[Grönwall's inequality|07EJHQYR9A]].\nIt gives a priori bounds for solutions of nonlinear equations and estimates of the time of existence.\nWith $c \\to 0$ it gives the [[Osgood uniqueness criterion|2AHNVSH76M]], which relaxes the Lipschitz condition in uniqueness theorems.\n",
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