# Gronwall inequality for backward (linear) differential equation

In the differential form of the Gronwall's Lemma, we have the following: $$\frac{d}{dt} \phi(t) \leq \psi(t) \phi(t),$$ for all $$t\geq t_0$$. Then, we get $$\phi(t) \leq \phi(t_0) exp\left(\int_{t_0}^t \psi(s)ds \right),$$ for all $$t\geq t_0$$.

My remark is this inequality assumes that we can bound the function $$\phi$$ that is a solution to a linear differential equation with its intial condition $$\phi(t_0)$$. Also, known as "forward" differential equations.

Now, my question is, could we have the same estimate if we have a terminal condition (for a backward differential equation)?

To be precise, do we have the following:

$$\phi(t) \leq \phi(t_0) exp\left(\int_{t_0}^t \psi(s)ds \right),$$ for all $$t\in [t_0,t_1]$$. Could, we have

$$\phi(t) \leq \phi(t_1) exp\left(\int_{t}^{t_1} \psi(s)ds \right),$$ for all $$t\in [t_0,t_1]$$.

• You can always reduce one case to the other with a change of variable $t \mapsto t_0 + t_1 - t$. Feb 12 at 9:30
• And consider $\tilde\phi(t)=-\phi(t_0+t_1-t)$. This restores the original situation, but transforming back it will show that the upper bound in forward direction turns into a lower bound in backward direction. Feb 12 at 10:53