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February 27th, 2018, 02:02 PM   #1
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How to prove the inequality of solutions for two Cauchy problems?

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Attached Images Cauchy_problem-min.jpg (17.2 KB, 22 views) March 2nd, 2018, 08:56 PM #2 Senior Member   Joined: Sep 2016 From: USA Posts: 620 Thanks: 391 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, 07:01 PM   #3
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Why?

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 Originally Posted by SDK 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 $g_1 \leq g_2$.
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Attached Images question01-min.jpg (16.4 KB, 14 views) March 4th, 2018, 10:20 PM #4 Senior Member   Joined: Sep 2016 From: USA Posts: 620 Thanks: 391 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 non-negative whcih easily follows from the fact that its derivative is non-negative. 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? Thanks from Mategatico March 5th, 2018, 01:56 PM #5 Newbie   Joined: Feb 2018 From: Perú Posts: 3 Thanks: 1 Thank you very much You're right. I have made a mistake. Thanks from Joppy Tags cauchy, inequality, problems, prove, solutions Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post zylo Real Analysis 2 September 26th, 2017 09:30 PM jugger3 Linear Algebra 2 August 21st, 2013 02:00 PM fahad nasir Linear Algebra 1 July 19th, 2013 11:53 PM aaron-math Calculus 3 February 5th, 2012 08:50 AM bigli Complex Analysis 5 February 1st, 2012 05:51 AM

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