\\begin{thm}[Gronwall \u4e0d\u7b49\u5f0f]\u5047\u8bbe $u,v:[a,b]\\to\\R$ \u662f\u8fde\u7eed\u51fd\u6570, \u4e14 $u\\geq0$. \u5982\u679c
\n\\[
\nv(t)\\leq C+\\int_a^t v(s)u(s)\\rd s,\\quad t\\in [a,b],
\n\\]
\n\u8fd9\u91cc $C$ \u662f\u4e00\u4e2a\u5e38\u6570, \u90a3\u4e48
\n\\[
\nv(t)\\leq C\\exp\\left(
\n\\int_a^t u(s)\\rd s
\n\\right).
\n\\]
\n\\end{thm}
\n\\begin{proof}
\n\u4ee4 $\\alpha(t)=C+\\int_a^t v(s)u(s)\\rd s$, $\\beta(t)=C\\exp\\left(\\int_a^t u(s)\\rd s\\right)$, \u5219 $\\alpha(a)=C=\\beta(a)$, \u4e14$$\\left(\\frac{\\alpha(t)}{\\beta(t)}\\right)^\\prime=\\frac{1}{\\beta^2(t)}(\\alpha^\\prime\\beta-\\alpha\\beta^\\prime)=\\frac{1}{\\beta^2(t)}(u v \\beta-\\alpha\\beta u )=\\frac{u}{\\beta}(v-\\alpha),$$ \u6ce8\u610f\u5230 $v-\\alpha\\leq 0$ \u4ee5\u53ca $\\alpha(a)\/\\beta(a)=1$, \u8fd9\u6837, \u82e5$C>0$ \u5219 $\\beta>0$, \u4ece\u800c $\\alpha\/\\beta\\leq 1$, \u5373 $\\alpha\\leq\\beta$; \u82e5 $C<0$, \u5219 $\\beta<0$, \u4ece\u800c $\\alpha\/\\beta\\geq 1$, \u6545\u4ecd\u6709 $\\alpha\\leq \\beta$. \u82e5 $C=0$, \u5219\u53ef\u4ee4 $C_n=1\/n$ \u5229\u7528\u524d\u9762\u7ed3\u679c\u5e76\u4ee4 $n\\to\\infty$ \u5f97\u5230 $v(t)\\leq 0$.
\n\\end{proof}<\/p>\n","protected":false},"excerpt":{"rendered":"
\\begin{thm}[Gronwall \u4e0d\u7b49\u5f0f]\u5047\u8bbe $u,v:[a,b]\\t …<\/p>\n