
Differential Equations Ordinary and Partial Differential Equations Math Forum 
 LinkBack  Thread Tools  Display Modes 
February 27th, 2018, 03:02 PM  #1 
Newbie Joined: Feb 2018 From: Perú Posts: 3 Thanks: 1  How to prove the inequality of solutions for two Cauchy problems?
View image Thank you for your suggestion. 
March 2nd, 2018, 09:56 PM  #2 
Senior Member Joined: Sep 2016 From: USA Posts: 522 Thanks: 296 Math Focus: Dynamical systems, analytic function theory, numerics 
Just integrate to obtain $\Phi_i$ directly. You get \[ \Phi_i(t,x) = x + \int_0^t g_i(s,\phi_i(s,x)) \ ds \] so clearly $\Phi_1 \leq \Phi_2$ since this obviously holds for these integrals under the assumption that $g_1 \leq g_2$. 
March 3rd, 2018, 08:01 PM  #3 
Newbie Joined: Feb 2018 From: Perú Posts: 3 Thanks: 1  Why? Thank you

March 4th, 2018, 11:20 PM  #4 
Senior Member Joined: Sep 2016 From: USA Posts: 522 Thanks: 296 Math Focus: Dynamical systems, analytic function theory, numerics 
Your integrals make no sense. There should be no $t$ in the integrand. Also, you don't need to prove that \[ \int_0^1 g_1(s,\Phi_1(t,s)) \ ds \leq \int_0^1 g_2(s,\Phi_2(t,s)) \ ds \]. You only need to prove that the difference, $\Phi_2  \Phi_1$ is nonnegative whcih easily follows from the fact that its derivative is nonnegative. Specifically, write $h = \Phi_2  \Phi_1$ and $G = g_2  g_1$, then $\frac{dh}{dt} = G$ and $h(0,x) = 0$ for all $x$. Integrating the differential equation for $h$, you have \[h(t,x) = \int_0^t G(s, h(t,s)) \ ds \] so clearly $h \geq 0$. Is this more clear? 
March 5th, 2018, 02:56 PM  #5 
Newbie Joined: Feb 2018 From: Perú Posts: 3 Thanks: 1  Thank you very much
You're right. I have made a mistake.


Tags 
cauchy, inequality, problems, prove, solutions 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Cauchy Schwarz Inequality  zylo  Real Analysis  2  September 26th, 2017 10:30 PM 
CauchySchwarz inequality  jugger3  Linear Algebra  2  August 21st, 2013 03:00 PM 
CauchyScwarz inequality  fahad nasir  Linear Algebra  1  July 20th, 2013 12:53 AM 
Insight to the cauchyschwarz inequality  aaronmath  Calculus  3  February 5th, 2012 09:50 AM 
Prove it, Without Cauchy's inequality, Liouville's theorem  bigli  Complex Analysis  5  February 1st, 2012 06:51 AM 