lancer6238
New member
- Joined
- Apr 4, 2007
- Messages
- 3
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
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