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  "body": "\n## Statement\n\nLet $t_0 \\ge 0$ and let $q \\colon [t_0, \\infty) \\to \\R$ be continuous.\nConsider the [[linear differential equation|0QQRP17Z7S]]\n$$y'' + q(t) y = 0 \\quad \\text{on } [t_0, \\infty).$$\nFor $t \u003e t_0$ put\n$$A(t) = \\frac{1}{t} \\int_{t_0}^{t} \\int_{t_0}^{s} q(u) \\, du \\, ds.$$\nAssume that the equation is nonoscillatory in the sense of [[oscillation theory|ZH2S50NSZ6]], and that $\\liminf_{t \\to \\infty} A(t) \u003e -\\infty$, where $\\liminf$ is the [[limit inferior|033RZ32T4Q]].\nThen $\\lim_{t \\to \\infty} A(t)$ exists and is a finite real number.\n\n## Remarks\n\nThe proof goes through the [[Riccati equation|0VT2QK6X6Q]].\nA nonoscillatory equation has a solution $y \u003e 0$ on some interval $[T, \\infty)$, and $w = y'/y$ satisfies $w' + w^2 + q = 0$ there.\nIntegrating gives $w(t) = w(T) - \\int_T^t q - \\int_T^t w^2$.\nAveraging this identity and using the lower bound on $A$, one shows that $\\int_T^\\infty w^2 \u003c \\infty$, and then the average $A(t)$ converges.\n\nRead as a test for oscillation, the theorem says: if $\\liminf A(t) \u003e -\\infty$ and either $\\liminf A(t) \u003c \\limsup A(t)$ or $A(t) \\to \\infty$, then the equation is oscillatory.\nThe case $A(t) \\to \\infty$ is Wintner's oscillation criterion.\n\nExample: for $q = 1$ on $[0, \\infty)$ one gets $A(t) = t/2 \\to \\infty$, so the equation is oscillatory; indeed $\\sin t$ is a solution.\nFor $q(t) = 1/(4t^2)$ on $[1, \\infty)$ the equation has the positive solution $\\sqrt{t}$, so it is nonoscillatory.\nHere $A(t) = (t - 1 - \\ln t)/(4t)$, which tends to the finite limit $1/4$, as the theorem predicts.\n\n## Explanations\n\n### L1 human\n\nThe equation $y'' + q y = 0$ describes a spring whose stiffness $q(t)$ changes with time.\nThe solutions either keep crossing zero forever or eventually stay on one side.\nThe theorem looks at a double average of the stiffness.\nIf the solutions stop crossing zero and this average is bounded below, then the average must settle to a finite value.\nSo an average that grows without bound, or that keeps swinging, forces endless oscillation.\n\n### L2 human\n\nThe Sturm comparison theorem compares $q$ with a constant or an Euler coefficient pointwise.\nThe Hartman–Wintner theorem uses only averages of $q$, so it also applies when $q$ changes sign.\nIt improves Wintner's criterion, which asks that $A(t) \\to \\infty$.\nThe key tool is the Riccati substitution $w = y'/y$, which turns a nonoscillatory linear equation into a first-order nonlinear one with a solution on a half-line.\nLater criteria of Kamenev and Philos replace the weight $1/t$ by other weights in the same Riccati argument.\n",
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