Suppose that $$ex cfrac{ d f}{ d t}+hleq gfquad (f,g,hgeq 0, tin [0,T]). eex$$ Then for $tin [0,T]$, $$ex f(t)+int_0^t h(s) d s leq f(0)sez{ 1+int_0^t g(s) d scdot expsex{int_0^t g(s) d s} }. eex$$
Suppose that $$ex cfrac{ d f}{ d t}+hleq gfquad (f,g,hgeq 0, tin [0,T]). eex$$ Then for $tin [0,T]$, $$ex f(t)+int_0^t h(s) d s leq f(0)sez{ 1+int_0^t g(s) d scdot expsex{int_0^t g(s) d s} }. eex$$