$u(x)\leq 1+\int_{0}^{x}\frac{\varphi '(t)u(t)dt}{\varphi (t)}$
Dễ thấy $ u(0) \leq 1$
Do $\varphi (t)$ đồng biến và $\varphi(0)=1 \Rightarrow \varphi (t)\geq 1 \forall t\in [0;\infty) $
$\Rightarrow u(x)-\varphi (x) \leq \int_{0}^{x}\left (\varphi '(t)(u(x)-1) \right ) dt=u(x)-\varphi (x)-\int_{0}^{x}\varphi '(x) $
$\Rightarrow \varphi(x)<1-\int_{0}^{x}u'(t)\varphi (t)dt ;\varphi(x)\geq 1 \forall x\in [0;\infty] \Rightarrow u'(t)<0 $
Xét hàm số $g(x)=u(x)-\varphi (x) $
$g'(x)=u'(x)-\varphi' (x) <0 ;g(0)=u(0)-\varphi (0)<0 \Rightarrow g(x)<0 \forall x\in [0;\infty]$