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$\bf命题：$$(\bf{Bellman -Gronwall不等式})$设常数$k > 0$，函数$f,g$
在$\left[ {a,b} \right]$上非负连续，且对任意$x \in \left[ {a,b} \right]$满足\[f\left( x \right) \ge k + \int_a^x {g\left( t \right)f\left( t \right)dt} \]
证明：对任意$x \in \left[ {a,b} \right]$，有$f\left( x \right) \ge k{e^{\int_a^x {g\left( t \right)dt} }}$

证明：设\[h\left( x \right) = k + \int_a^x {g\left( t \right)f\left( t \right)dt} \]则\[f\left( x \right) \ge h\left( x \right) > 0,h'\left( x \right) = g\left( x \right)f\left( x \right)\]
所以\[\frac{{h'\left( x \right)}}{{h\left( x \right)}} \ge \frac{{h'\left( x \right)}}{{f\left( x \right)}} = g\left( x \right)\]从而对上式两边积分即得