# Lemma of Gronwall

Let $T \in \mathbb R^+\cup \{\infty\}$, $t_o \in [0,T)$, $a,b \in L^\infty(t_0,T)$ and $\lambda \in L^1(T_0,t)$, $\lambda(t) \geq 0$ for almost all $t \in (t_o,T)$. From the inequality $$a(t) \leq b(t) + \int_{t_0}^t\lambda(s)a(s)ds \,\,\,\,\,$$ a.e. in $(t_o,T)$ it follows $$a(t) \leq b(t) + \int_{t_0}^t e^{\phi(t)-\phi(s)}\lambda(s)b(s) \,ds$$ for almost all $t\in (t_0,T)$ where $\phi(s):=\int_{t_o}^s \lambda(\tau)\,d\tau$.

Is there a counterexample for the case that $\lambda$ is negative? And what changes about the implication if $t<t_o$?

• Related – Giuseppe Negro Jan 16 '17 at 16:33
• (-1) downvote because this is directly from a homework problem and you haven't shown any effort of thinking of a solution yourself. – Viktor Glombik Jan 22 at 20:48

Here's a simple example of what happens when $\lambda$ is not non-negative.
Take $t_0 =0$, $T=\infty$, and $b(t) =1$ and $\lambda(t) =-1$ for all $t \ge 0$. Further assume we have the equality $$a(t) = b(t) + \int_0^t \lambda(s) a(s) ds = 1 - \int_0^t a(s) ds$$ for $t \ge 0$. Write $F(t) = \int_0^t a(s) ds$. Then the equality reads $$F'(t) = 1 - F(t) \Rightarrow (e^t F(t))' = e^t \Rightarrow F(t) = 1- e^{-t}.$$ Plugging back in shows that $a(t) = e^{-t}$.
Now let's consider the RHS of the Gronwall inequality. We compute $$b(t) + \int_0^t e^{\phi(t)-\phi(s)} \lambda(s) b(s) ds = 1 - e^t \int_0^t e^{-s} ds = 2 - e^{t}.$$ The inequality then holds if and only if $$e^{-t} = a(t) \le b(t) + \int_0^t e^{\phi(t)-\phi(s)} \lambda(s) b(s) ds = 2- e^{t}$$ for all $t \ge 0$, which is equivalent to $$\cosh(t) \le 1 \text{ for all }t \ge 0,$$ a contradiction.
For your second question I can't really give a good answer, as it's not clear to me what you want to assume. Are we supposed to assume that the first inequality also holds for $t < t_0$? If not, then we have no information about $a$ before $t_0$, so how could we say anything about the function?
• That is very elegant. How to get these perfect contradictions with $\cosh$etc? :D – Tesla Jan 16 '17 at 19:20
• For the first task, I'd choose $t_0 = 0$ und $T = \infty$, $a(t) \equiv -1$, $b(t) \equiv 1$ und $\lambda(t) \equiv - 10 < 0$ and for the second $a(t) \equiv 0$, $b(t) \equiv$\lambda(t) \equiv 1\$. – Viktor Glombik Jan 22 at 22:16