I need to prove the following with the help of Gronwall's inequality:

If, for $t \in [a,b]$, $$\phi(t) \leq \delta_2(t-a) + \delta_1 \int_{a}^{t}\phi(s)ds + \delta_3,$$ where $\phi$ is a nonnegative continuous function on $[a,b]$, and $\delta_1>0, \delta_2 \geq 0$, and $\delta_3 \geq 0$ are constants, then $$\phi(t) \leq (\frac{\delta_2}{\delta_1}+\delta_3)e^{\delta_1(t-a)} - \frac{\delta_2}{\delta_1}.$$

Here is the version of Gronwall's inequality that I am using:

Gronwall: Suppose that $a<b$ and let $\alpha, \phi,$ and $\psi$ be nonnegative continuous functions defined on the interval $[a,b]$. Moreover, suppose that $\alpha$ is differentiable on $(a,b)$ with nonnegative continuous derivative $\dot\alpha$. If, for all $t \in [a,b]$, $$\phi(t) \leq \alpha(t) + \int_{a}^{t}\psi(s)\phi(s)ds,$$ then $$\phi(t) \leq \alpha(t)e^{\int_{a}^{t}\psi(s)ds}$$ for all $t \in [a,b]$.

My attempts:

I've tried everything with around 6 pages of work...

$1.$ I set $\alpha = (\frac{\delta_2}{\delta_1}+\delta_3)$, and $\phi(t) = \delta_1$ as per the formula, and tried to match the formulae, but without success.

$2.$ I used the following link: http://books.google.com/books?id=b93ltUhTEWoC&pg=PA4&lpg=PA4&dq=averaging+methods+in+nonlinear+dynamical+systems+%2B+verhulst+%2B+gronwall&source=bl&ots=1Nij6H8aRX&sig=s6da2CCNEj2SDRzfM4wFmkiPTRM&hl=en&sa=X&ei=yLNOU9iSIZCdyATDsYGACw&ved=0CCwQ6AEwAA#v=onepage&q=gronwall&f=false

and followed their suggestion (Lemma $1.3.3$), but that did not work. It looks like something is messed up in their proof of Lemma $1.3.3$.

$3.$ I set the top two inequalities (in my post) equal to each other and then solved for $\phi(t)$, but that yielded me $\alpha(t) = \frac{\delta_2}{\delta_1}+\delta_3$, which does not work.

$4.$ I set $\phi(t) =$ R.H.S. of the top inequality in my post and then differentiated, and tried to derive the result by an approach similar to the proof of Lemma 1.3.1, but that gave me unsatisfactory results.

What more can I do?


1 Answer 1


[This answer is given at http://mathhelpboards.com/differential-equations-17/need-help-proving-lemma-10176.html#post47171 .]

You have quoted Grönwall's inequality wrongly. The $\phi$ in the exponential should be $\psi$.

You can write the term $\delta_2(t-a)$ as $\displaystyle\delta_1\int_a^t \frac{\delta_2}{\delta_1}\,ds.$ Now replace the function $\phi(t)$ by $\phi(t) + \dfrac{\delta_2}{\delta_1}$. Take $\alpha(t)$ to be the constant $\dfrac{\delta_2}{\delta_1} + \delta_3$ and $\psi(t)$ to be the constant $1$, and apply Grönwall's inequality.


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