Gronwall Inequality

Let $f, g$ be two continuous functions , non negative in $[a, b]$ and let $c>0$. If $\displaystyle f(x) \leq c + \int_a^b f(t) g(t) \, {\rm d}t$ then prove that:

$$f(x) \leq ce^{\displaystyle \int_a^x g(t)\, {\rm d}t}$$

Solution

Let $\displaystyle h(x)= c + \int_a^x f(t) g(t) \, {\rm d}t \geq c$. Differentiating we get:

$$h'(x) = f(x) g(x) \leq g(x) \left ( c + \int_a^x f(t) g(t) \, {\rm d}t \right) \leq g(x) h(x)$$

But since $h(x)>0$ then this implies $(\ln h (x))' \leq g(x)$ or$(\ln h(x))' \leq G'(x)$ where $\displaystyle G(x) = \int_a^x g(t)\, {\rm d}t$.

Let $K(x)= \ln \dfrac{h(x)}{c}- G(x), \;\; x \in [a, b]$ . We easily deduce that $K$ is stricly decreasing and that $K(a)=0$. The inequality follows.