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 $t\in [t_0,t_1]$.