(a) Suppose that u(t) is a continuous solution of

u(t)\leq f(t)+\int_{t_0}^t h(s)u(s)ds, t\geq t_0,

where f,h are continuous and h(t)\geq 0 for t\geq t_0. Show that

u(t)\leq f(t)+\int_{t_0}^t f(s)h(s)\exp\left[\int_s^t h(\sigma)d\sigma\right]ds, t\geq t_0.


This I have done! But then we have...


(b) Moreover, if f(t) is increasing for t\geq t_0, show that

u(t)\leq f(t)\exp\left[\int_{t_0}^t h(s)ds\right], t\geq t_0.


In order to do the exercise, we are given as a theorem the following formulation of the Gronwall inequality:


If u(t)\leq M+\int_{t_0}^t h(s)u(s)ds, t\geq t_0,

where M\in\mathbb{R}, h(t)\geq 0 for all t\geq t_0 and h is continuous, and u(t) is a continuous solution, then

u(t)\leq M\exp\left[\int_{t_0}^t h(s)ds\right], t\geq t_0.


However, I believe this theorem may only be useful for part (a), which, as I said, I have already done. I provide it just in case.

Thanks in advance!