Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

I Finding a Lipschitz Constant

Tags:
  1. Aug 30, 2016 #1
    Compute a Lipschitz constant K as in (3.7) $$f(t, y_2)-f(t, y_1)=K(y_2-y_1) \space\space (3.7)$$, and then show that the function f satisfies the Lipschitz condition in the region indicated:

    $$f(t, y)=p(t)\cos{y}+q(t)\sin{y},\space {(t, y) | \space |t|\leq 100, |y|<\infty}$$ where p,q are continuous functions on $$-100\leq t \leq 100$$

    I honestly have no idea how to even begin this. Other than the definition on Lipschitz continuity (f and df/dy are continuous on the region given) the book being used doesn't really talk about anything Lipschitz.

    And just as disclaimer, this is *technically* homework however its nothing turned in or for a grade. Just something for practice.

    Any help, especially with at least getting started, is much appreciated.
     
  2. jcsd
  3. Aug 30, 2016 #2

    Krylov

    User Avatar
    Science Advisor
    Education Advisor

    I would write (3.7) as
    $$
    |f(t,y_2) - f(t,y_1)| \le K |y_2 - y_1|
    $$
    and I assume that ##K## is supposed to not depend on ##t##.

    Hint: Explain and use the fact that ##\tfrac{df}{dy}## is bounded on the given region, say ##\Omega##. Argue that ##K := \sup_{(t,y) \in \Omega}{\left|\tfrac{df}{dy}(t,y)\right|}## can be taken as a Lipschitz constant.

    Does this come from a text on ODEs?
     
  4. Aug 30, 2016 #3

    fresh_42

    Staff: Mentor

    ##p(t),q(t)## are continuous on the compact set ##[-100,100]##, aren't they? What am I missing?

    Edit: Got it. I've automatically associated a "##\leq##" with Lipschitz, not the actual equality.
     
    Last edited: Aug 30, 2016
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted