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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 $t\in [t_0,t_1]$.

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    $\begingroup$ You can always reduce one case to the other with a change of variable $t \mapsto t_0 + t_1 - t$. $\endgroup$
    – Martin R
    Feb 12 at 9:30
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    $\begingroup$ 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. $\endgroup$ Feb 12 at 10:53

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