Proving an inequality w/ locally Lipschitz function

[lancer6238]

Hi all, I need help proving an inequality.

Suppose the function f(t,x) is locally Lipschitz on the domain $\displaystyle G \subset \mathbb{R^2}$, that is, $\displaystyle |f(t,x_1)-f(t,x_2)| \leq k(t) |x_1 - x_2|$ for all $\displaystyle (t, x_1),(t,x_2) \in G$. Define I = (a,b) and $\displaystyle \phi_1(t)$ and $\displaystyle \phi_2(t)$ are 2 continuous functions on I. Assume that, if $\displaystyle (t, \phi_i(t)) \in G$, then the function $\displaystyle f(t, \phi_i(t))$ is an integrable function on I for i = 1, 2. Suppose that for i = 1,2 and $\displaystyle t \in I$,

$\displaystyle \phi_i(t) = \phi_i(t_0) + \int^t_{t_0} f(s, \phi_i(s))\,ds + E_i(t)$

and $\displaystyle |\phi_1(t_0) - \phi_2(t_0)| \leq \delta$

for some constant $\displaystyle \delta$. Show that for all $\displaystyle t \in (t_0, b)$ we have

$\displaystyle |\phi_1(t) - \phi_2(t)| \leq \delta e^{\int^t_{t_0} k(s) \,ds} + E(t) + \int^t_{t_0} E(s) k(s) e^{\int^s_{t_0} k(r) \,dr} \,ds$

where $\displaystyle E(t) = |E_1(t)| + |E_2(t)|$

I managed to get $\displaystyle \delta e^{\int^t_{t_0} k(s) \,ds}$ using triangle inequality and Gronwall's inequality, but I cannot seem to get the last 2 terms in the inequality.

Here's what I did:

$\displaystyle |\phi_1(t) - \phi_2(t)|$
$\displaystyle = |\phi_1(t_0) + \int^t_{t_0} f(s, \phi_1(s)) \,ds + E_1(t) - \phi_2(t_0) - \int^t_{t_0} f(s, \phi_2(s)) \,ds - E_2(t)|$
$\displaystyle = |\phi_1(t_0) - \phi_2(t_0) + \int^t_{t_0} f(s, \phi_1(s)) \,ds - \int^t_{t_0} f(s, \phi_2(s)) \,ds + E_1(t) - E_2(t)|$
\(\displaystyle \leq |\phi_1(t_0) - \phi_2(t_0)| + |\int^t_{t_0} f(s, \phi_1(s)) }ds - \int^t_{t_0} f(s, phi_2(s)) \,ds + E_1(t) - E_2(t)|\)
$\displaystyle \leq \delta + |E_1(t) - E_2(t)| + |\int^t_{t_0} f(s, \phi_1(s)) - f(s, \phi_2(s)) \,ds|$
$\displaystyle \leq \delta + |E_1(t) + E_2(t)| + \int^t_{t_0} |f(s, \phi_1(s)) - f(s, \phi_2(s))| \,ds$
$\displaystyle \leq \delta + |E_1(t)| + |E_2(t)| + \int^t_{t_0} k(s)|\phi_1(s) - \phi_2(s)| \, ds$
$\displaystyle \leq (\delta + E(t)) e^{\int^t_{t_0} k(s)\, ds}$
\(\displaystyle = \delta e^{\int^t_{t_0} k(s) \, ds} + E(t) e^{\int^t_{t_0} k(s) \, ds\)

This is where I got stuck.

Please help.

Thank you.

Regards,
Rayne