$f命题：$$(f{Bellman -Gronwall不等式})$设常数$k > 0$，函数$f,g$
在$left[ {a,b}
ight]$上非负连续，且对任意$x in left[ {a,b}
ight]$满足[fleft( x
ight) ge k + int_a^x {gleft( t
ight)fleft( t
ight)dt} ]
证明：对任意$x in left[ {a,b}
ight]$，有$fleft( x
ight) ge k{e^{int_a^x {gleft( t
ight)dt} }}$
证明：设[hleft( x
ight) = k + int_a^x {gleft( t
ight)fleft( t
ight)dt} ]则[fleft( x
ight) ge hleft( x
ight) > 0,h'left( x
ight) = gleft( x
ight)fleft( x
ight)]
所以[frac{{h'left( x
ight)}}{{hleft( x
ight)}} ge frac{{h'left( x
ight)}}{{fleft( x
ight)}} = gleft( x
ight)]从而对上式两边积分即得