The Picard-Lindelöf theorem on Wikipedia












0












$begingroup$


On the Wikipedia entry of Picard-Lindelöf theorem for the local existence and uniqueness of ODE's, there is a section on the optimization of the solution's interval.



There is a lemma used in this section which says
$$
|| Gamma^m varphi_1 - Gamma^m varphi_2 || leq dfrac{L^m alpha^m}{m!} ||varphi_1 - varphi_2 ||
$$
Please see the article for the definition of symbols above.



I can not get the $m!$ term in the denominator and believe that the lemma is wrong. Am I right?






ordinary-differential-equations







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$endgroup$












  • $begingroup$
    There is also a proof given in the section you cite. Have you read it? Where do you disagree?
    $endgroup$
    – martini
    Jul 15 '14 at 10:03










  • $begingroup$
    m! term should not appear in the denominator.
    $endgroup$
    – pitonist
    Jul 15 '14 at 10:07






  • 1




    $begingroup$
    I think one should replace the claim $$ |Gamma^m phi_1 - Gamma^m phi_2 | le frac{ L^m alpha^m}{m!}|phi_1-phi_2| $$ by $$ |Gamma^m phi_1(s) - Gamma^m phi_2(s) | le frac{ L^m}{m!}(s-t_0)^m|phi_1-phi_2| quad forall sin(t_0,t_0+a). $$ Very sloppy proof there.
    $endgroup$
    – daw
    Jul 15 '14 at 10:20
















0












$begingroup$


On the Wikipedia entry of Picard-Lindelöf theorem for the local existence and uniqueness of ODE's, there is a section on the optimization of the solution's interval.



There is a lemma used in this section which says
$$
|| Gamma^m varphi_1 - Gamma^m varphi_2 || leq dfrac{L^m alpha^m}{m!} ||varphi_1 - varphi_2 ||
$$
Please see the article for the definition of symbols above.



I can not get the $m!$ term in the denominator and believe that the lemma is wrong. Am I right?






ordinary-differential-equations







share|cite|improve this question











$endgroup$












  • $begingroup$
    There is also a proof given in the section you cite. Have you read it? Where do you disagree?
    $endgroup$
    – martini
    Jul 15 '14 at 10:03










  • $begingroup$
    m! term should not appear in the denominator.
    $endgroup$
    – pitonist
    Jul 15 '14 at 10:07






  • 1




    $begingroup$
    I think one should replace the claim $$ |Gamma^m phi_1 - Gamma^m phi_2 | le frac{ L^m alpha^m}{m!}|phi_1-phi_2| $$ by $$ |Gamma^m phi_1(s) - Gamma^m phi_2(s) | le frac{ L^m}{m!}(s-t_0)^m|phi_1-phi_2| quad forall sin(t_0,t_0+a). $$ Very sloppy proof there.
    $endgroup$
    – daw
    Jul 15 '14 at 10:20














0












0








0





$begingroup$


On the Wikipedia entry of Picard-Lindelöf theorem for the local existence and uniqueness of ODE's, there is a section on the optimization of the solution's interval.



There is a lemma used in this section which says
$$
|| Gamma^m varphi_1 - Gamma^m varphi_2 || leq dfrac{L^m alpha^m}{m!} ||varphi_1 - varphi_2 ||
$$
Please see the article for the definition of symbols above.



I can not get the $m!$ term in the denominator and believe that the lemma is wrong. Am I right?






ordinary-differential-equations







share|cite|improve this question











$endgroup$




On the Wikipedia entry of Picard-Lindelöf theorem for the local existence and uniqueness of ODE's, there is a section on the optimization of the solution's interval.



There is a lemma used in this section which says
$$
|| Gamma^m varphi_1 - Gamma^m varphi_2 || leq dfrac{L^m alpha^m}{m!} ||varphi_1 - varphi_2 ||
$$
Please see the article for the definition of symbols above.



I can not get the $m!$ term in the denominator and believe that the lemma is wrong. Am I right?






ordinary-differential-equations




ordinary-differential-equations






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jul 30 '15 at 11:49









Jyrki Lahtonen

110k13171386




110k13171386










asked Jul 15 '14 at 9:56









pitonistpitonist

7918




7918












  • $begingroup$
    There is also a proof given in the section you cite. Have you read it? Where do you disagree?
    $endgroup$
    – martini
    Jul 15 '14 at 10:03










  • $begingroup$
    m! term should not appear in the denominator.
    $endgroup$
    – pitonist
    Jul 15 '14 at 10:07






  • 1




    $begingroup$
    I think one should replace the claim $$ |Gamma^m phi_1 - Gamma^m phi_2 | le frac{ L^m alpha^m}{m!}|phi_1-phi_2| $$ by $$ |Gamma^m phi_1(s) - Gamma^m phi_2(s) | le frac{ L^m}{m!}(s-t_0)^m|phi_1-phi_2| quad forall sin(t_0,t_0+a). $$ Very sloppy proof there.
    $endgroup$
    – daw
    Jul 15 '14 at 10:20


















  • $begingroup$
    There is also a proof given in the section you cite. Have you read it? Where do you disagree?
    $endgroup$
    – martini
    Jul 15 '14 at 10:03










  • $begingroup$
    m! term should not appear in the denominator.
    $endgroup$
    – pitonist
    Jul 15 '14 at 10:07






  • 1




    $begingroup$
    I think one should replace the claim $$ |Gamma^m phi_1 - Gamma^m phi_2 | le frac{ L^m alpha^m}{m!}|phi_1-phi_2| $$ by $$ |Gamma^m phi_1(s) - Gamma^m phi_2(s) | le frac{ L^m}{m!}(s-t_0)^m|phi_1-phi_2| quad forall sin(t_0,t_0+a). $$ Very sloppy proof there.
    $endgroup$
    – daw
    Jul 15 '14 at 10:20
















$begingroup$
There is also a proof given in the section you cite. Have you read it? Where do you disagree?
$endgroup$
– martini
Jul 15 '14 at 10:03




$begingroup$
There is also a proof given in the section you cite. Have you read it? Where do you disagree?
$endgroup$
– martini
Jul 15 '14 at 10:03












$begingroup$
m! term should not appear in the denominator.
$endgroup$
– pitonist
Jul 15 '14 at 10:07




$begingroup$
m! term should not appear in the denominator.
$endgroup$
– pitonist
Jul 15 '14 at 10:07




1




1




$begingroup$
I think one should replace the claim $$ |Gamma^m phi_1 - Gamma^m phi_2 | le frac{ L^m alpha^m}{m!}|phi_1-phi_2| $$ by $$ |Gamma^m phi_1(s) - Gamma^m phi_2(s) | le frac{ L^m}{m!}(s-t_0)^m|phi_1-phi_2| quad forall sin(t_0,t_0+a). $$ Very sloppy proof there.
$endgroup$
– daw
Jul 15 '14 at 10:20




$begingroup$
I think one should replace the claim $$ |Gamma^m phi_1 - Gamma^m phi_2 | le frac{ L^m alpha^m}{m!}|phi_1-phi_2| $$ by $$ |Gamma^m phi_1(s) - Gamma^m phi_2(s) | le frac{ L^m}{m!}(s-t_0)^m|phi_1-phi_2| quad forall sin(t_0,t_0+a). $$ Very sloppy proof there.
$endgroup$
– daw
Jul 15 '14 at 10:20










1 Answer
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$begingroup$

First, let's prove this:
$$
|| Gamma^{m} varphi_1(t) - Gamma^{m} varphi_2(t) || leq dfrac{L^{m} |t-t_0|^{m}}{m!} ||varphi_1 - varphi_2 ||, forall t in I_a
$$



With $m = 1$, is trivial. If the induction hypothesis is:



$$|| Gamma^{m-1} varphi_1(t) - Gamma^{m-1} varphi_2(t) || leq dfrac{L^{m-1} |t-t_0|^{m-1}}{(m-1)!} ||varphi_1 - varphi_2 ||, forall t in I_a
$$



Then $forall t in I_a$:



begin{align}
|| Gamma^{m} varphi_1(t) - Gamma^{m} varphi_2(t) || = left|left| GammaGamma^{m-1} varphi_1(t) - Gamma Gamma^{m-1} varphi_2(t) right|right| &= left|left| int_{t_0}^{t} fleft(s,Gamma^{m-1} varphi_1(s) right) - fleft(s,Gamma^{m-1} varphi_2(s) right) ds right|right| \
&leq left| int_{t_0}^{t} left|left| fleft(s,Gamma^{m-1} varphi_1(s) right) - fleft(s,Gamma^{m-1} varphi_2(s) right) right|right| ds right|
end{align}
But $f$ is Lipschitz in $I_a times B_b$ in the second variable with $L$ as Lipschitz constant. Then:



begin{align}
left| int_{t_0}^{t} left|left| fleft(s,Gamma^{m-1} varphi_1(s) right) - fleft(s,Gamma^{m-1} varphi_2(s) right) right|right| ds right|
&leq
L left| int_{t_0}^{t} left|left| Gamma^{m-1} varphi_1(s) - Gamma^{m-1} varphi_2(s) right|right| ds right| \
end{align}



And by induction hypothesis:



begin{align}
left| int_{t_0}^{t} left|left| fleft(s,Gamma^{m-1} varphi_1(s) right) - fleft(s,Gamma^{m-1} varphi_2(s) right) right|right| ds right|
&leq
L left| int_{t_0}^{t} dfrac{L^{m-1} |s-t_0|^{m-1}}{(m-1)!} ||varphi_1 - varphi_2 || ds right| \
&leq
dfrac{L^m}{(m-1)!} left| int_{t_0}^{t} |s-t_0|^{m-1} ||varphi_1 - varphi_2 || ds right| \
&leq
frac{L^m|t-t_0|^m}{m!} ||varphi_1 - varphi_2 || \
end{align}
Then the previous proposition is right, now:
begin{align}
|| Gamma^{m} varphi_1(t) - Gamma^{m} varphi_2(t) || leq dfrac{L^{m} |t-t_0|^{m}}{m!} ||varphi_1 - varphi_2 || leq dfrac{L^{m} a^{m}}{m!} ||varphi_1 - varphi_2 ||, forall t in I_a
end{align}
Then:
begin{align}
textrm{sup} left{ || Gamma^{m} varphi_1(t) - Gamma^{m} varphi_2(t) || : t in I_a right} &leq dfrac{L^{m} a^{m}}{m!} ||varphi_1 - varphi_2 ||, forall t in I_a \
|| Gamma^{m} varphi_1 - Gamma^{m} varphi_2 || &leq dfrac{L^{m} a^{m}}{m!} ||varphi_1 - varphi_2 ||
end{align}






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    $begingroup$

    First, let's prove this:
    $$
    || Gamma^{m} varphi_1(t) - Gamma^{m} varphi_2(t) || leq dfrac{L^{m} |t-t_0|^{m}}{m!} ||varphi_1 - varphi_2 ||, forall t in I_a
    $$



    With $m = 1$, is trivial. If the induction hypothesis is:



    $$|| Gamma^{m-1} varphi_1(t) - Gamma^{m-1} varphi_2(t) || leq dfrac{L^{m-1} |t-t_0|^{m-1}}{(m-1)!} ||varphi_1 - varphi_2 ||, forall t in I_a
    $$



    Then $forall t in I_a$:



    begin{align}
    || Gamma^{m} varphi_1(t) - Gamma^{m} varphi_2(t) || = left|left| GammaGamma^{m-1} varphi_1(t) - Gamma Gamma^{m-1} varphi_2(t) right|right| &= left|left| int_{t_0}^{t} fleft(s,Gamma^{m-1} varphi_1(s) right) - fleft(s,Gamma^{m-1} varphi_2(s) right) ds right|right| \
    &leq left| int_{t_0}^{t} left|left| fleft(s,Gamma^{m-1} varphi_1(s) right) - fleft(s,Gamma^{m-1} varphi_2(s) right) right|right| ds right|
    end{align}
    But $f$ is Lipschitz in $I_a times B_b$ in the second variable with $L$ as Lipschitz constant. Then:



    begin{align}
    left| int_{t_0}^{t} left|left| fleft(s,Gamma^{m-1} varphi_1(s) right) - fleft(s,Gamma^{m-1} varphi_2(s) right) right|right| ds right|
    &leq
    L left| int_{t_0}^{t} left|left| Gamma^{m-1} varphi_1(s) - Gamma^{m-1} varphi_2(s) right|right| ds right| \
    end{align}



    And by induction hypothesis:



    begin{align}
    left| int_{t_0}^{t} left|left| fleft(s,Gamma^{m-1} varphi_1(s) right) - fleft(s,Gamma^{m-1} varphi_2(s) right) right|right| ds right|
    &leq
    L left| int_{t_0}^{t} dfrac{L^{m-1} |s-t_0|^{m-1}}{(m-1)!} ||varphi_1 - varphi_2 || ds right| \
    &leq
    dfrac{L^m}{(m-1)!} left| int_{t_0}^{t} |s-t_0|^{m-1} ||varphi_1 - varphi_2 || ds right| \
    &leq
    frac{L^m|t-t_0|^m}{m!} ||varphi_1 - varphi_2 || \
    end{align}
    Then the previous proposition is right, now:
    begin{align}
    || Gamma^{m} varphi_1(t) - Gamma^{m} varphi_2(t) || leq dfrac{L^{m} |t-t_0|^{m}}{m!} ||varphi_1 - varphi_2 || leq dfrac{L^{m} a^{m}}{m!} ||varphi_1 - varphi_2 ||, forall t in I_a
    end{align}
    Then:
    begin{align}
    textrm{sup} left{ || Gamma^{m} varphi_1(t) - Gamma^{m} varphi_2(t) || : t in I_a right} &leq dfrac{L^{m} a^{m}}{m!} ||varphi_1 - varphi_2 ||, forall t in I_a \
    || Gamma^{m} varphi_1 - Gamma^{m} varphi_2 || &leq dfrac{L^{m} a^{m}}{m!} ||varphi_1 - varphi_2 ||
    end{align}






    share|cite|improve this answer









    $endgroup$


















      0












      $begingroup$

      First, let's prove this:
      $$
      || Gamma^{m} varphi_1(t) - Gamma^{m} varphi_2(t) || leq dfrac{L^{m} |t-t_0|^{m}}{m!} ||varphi_1 - varphi_2 ||, forall t in I_a
      $$



      With $m = 1$, is trivial. If the induction hypothesis is:



      $$|| Gamma^{m-1} varphi_1(t) - Gamma^{m-1} varphi_2(t) || leq dfrac{L^{m-1} |t-t_0|^{m-1}}{(m-1)!} ||varphi_1 - varphi_2 ||, forall t in I_a
      $$



      Then $forall t in I_a$:



      begin{align}
      || Gamma^{m} varphi_1(t) - Gamma^{m} varphi_2(t) || = left|left| GammaGamma^{m-1} varphi_1(t) - Gamma Gamma^{m-1} varphi_2(t) right|right| &= left|left| int_{t_0}^{t} fleft(s,Gamma^{m-1} varphi_1(s) right) - fleft(s,Gamma^{m-1} varphi_2(s) right) ds right|right| \
      &leq left| int_{t_0}^{t} left|left| fleft(s,Gamma^{m-1} varphi_1(s) right) - fleft(s,Gamma^{m-1} varphi_2(s) right) right|right| ds right|
      end{align}
      But $f$ is Lipschitz in $I_a times B_b$ in the second variable with $L$ as Lipschitz constant. Then:



      begin{align}
      left| int_{t_0}^{t} left|left| fleft(s,Gamma^{m-1} varphi_1(s) right) - fleft(s,Gamma^{m-1} varphi_2(s) right) right|right| ds right|
      &leq
      L left| int_{t_0}^{t} left|left| Gamma^{m-1} varphi_1(s) - Gamma^{m-1} varphi_2(s) right|right| ds right| \
      end{align}



      And by induction hypothesis:



      begin{align}
      left| int_{t_0}^{t} left|left| fleft(s,Gamma^{m-1} varphi_1(s) right) - fleft(s,Gamma^{m-1} varphi_2(s) right) right|right| ds right|
      &leq
      L left| int_{t_0}^{t} dfrac{L^{m-1} |s-t_0|^{m-1}}{(m-1)!} ||varphi_1 - varphi_2 || ds right| \
      &leq
      dfrac{L^m}{(m-1)!} left| int_{t_0}^{t} |s-t_0|^{m-1} ||varphi_1 - varphi_2 || ds right| \
      &leq
      frac{L^m|t-t_0|^m}{m!} ||varphi_1 - varphi_2 || \
      end{align}
      Then the previous proposition is right, now:
      begin{align}
      || Gamma^{m} varphi_1(t) - Gamma^{m} varphi_2(t) || leq dfrac{L^{m} |t-t_0|^{m}}{m!} ||varphi_1 - varphi_2 || leq dfrac{L^{m} a^{m}}{m!} ||varphi_1 - varphi_2 ||, forall t in I_a
      end{align}
      Then:
      begin{align}
      textrm{sup} left{ || Gamma^{m} varphi_1(t) - Gamma^{m} varphi_2(t) || : t in I_a right} &leq dfrac{L^{m} a^{m}}{m!} ||varphi_1 - varphi_2 ||, forall t in I_a \
      || Gamma^{m} varphi_1 - Gamma^{m} varphi_2 || &leq dfrac{L^{m} a^{m}}{m!} ||varphi_1 - varphi_2 ||
      end{align}






      share|cite|improve this answer









      $endgroup$
















        0












        0








        0





        $begingroup$

        First, let's prove this:
        $$
        || Gamma^{m} varphi_1(t) - Gamma^{m} varphi_2(t) || leq dfrac{L^{m} |t-t_0|^{m}}{m!} ||varphi_1 - varphi_2 ||, forall t in I_a
        $$



        With $m = 1$, is trivial. If the induction hypothesis is:



        $$|| Gamma^{m-1} varphi_1(t) - Gamma^{m-1} varphi_2(t) || leq dfrac{L^{m-1} |t-t_0|^{m-1}}{(m-1)!} ||varphi_1 - varphi_2 ||, forall t in I_a
        $$



        Then $forall t in I_a$:



        begin{align}
        || Gamma^{m} varphi_1(t) - Gamma^{m} varphi_2(t) || = left|left| GammaGamma^{m-1} varphi_1(t) - Gamma Gamma^{m-1} varphi_2(t) right|right| &= left|left| int_{t_0}^{t} fleft(s,Gamma^{m-1} varphi_1(s) right) - fleft(s,Gamma^{m-1} varphi_2(s) right) ds right|right| \
        &leq left| int_{t_0}^{t} left|left| fleft(s,Gamma^{m-1} varphi_1(s) right) - fleft(s,Gamma^{m-1} varphi_2(s) right) right|right| ds right|
        end{align}
        But $f$ is Lipschitz in $I_a times B_b$ in the second variable with $L$ as Lipschitz constant. Then:



        begin{align}
        left| int_{t_0}^{t} left|left| fleft(s,Gamma^{m-1} varphi_1(s) right) - fleft(s,Gamma^{m-1} varphi_2(s) right) right|right| ds right|
        &leq
        L left| int_{t_0}^{t} left|left| Gamma^{m-1} varphi_1(s) - Gamma^{m-1} varphi_2(s) right|right| ds right| \
        end{align}



        And by induction hypothesis:



        begin{align}
        left| int_{t_0}^{t} left|left| fleft(s,Gamma^{m-1} varphi_1(s) right) - fleft(s,Gamma^{m-1} varphi_2(s) right) right|right| ds right|
        &leq
        L left| int_{t_0}^{t} dfrac{L^{m-1} |s-t_0|^{m-1}}{(m-1)!} ||varphi_1 - varphi_2 || ds right| \
        &leq
        dfrac{L^m}{(m-1)!} left| int_{t_0}^{t} |s-t_0|^{m-1} ||varphi_1 - varphi_2 || ds right| \
        &leq
        frac{L^m|t-t_0|^m}{m!} ||varphi_1 - varphi_2 || \
        end{align}
        Then the previous proposition is right, now:
        begin{align}
        || Gamma^{m} varphi_1(t) - Gamma^{m} varphi_2(t) || leq dfrac{L^{m} |t-t_0|^{m}}{m!} ||varphi_1 - varphi_2 || leq dfrac{L^{m} a^{m}}{m!} ||varphi_1 - varphi_2 ||, forall t in I_a
        end{align}
        Then:
        begin{align}
        textrm{sup} left{ || Gamma^{m} varphi_1(t) - Gamma^{m} varphi_2(t) || : t in I_a right} &leq dfrac{L^{m} a^{m}}{m!} ||varphi_1 - varphi_2 ||, forall t in I_a \
        || Gamma^{m} varphi_1 - Gamma^{m} varphi_2 || &leq dfrac{L^{m} a^{m}}{m!} ||varphi_1 - varphi_2 ||
        end{align}






        share|cite|improve this answer









        $endgroup$



        First, let's prove this:
        $$
        || Gamma^{m} varphi_1(t) - Gamma^{m} varphi_2(t) || leq dfrac{L^{m} |t-t_0|^{m}}{m!} ||varphi_1 - varphi_2 ||, forall t in I_a
        $$



        With $m = 1$, is trivial. If the induction hypothesis is:



        $$|| Gamma^{m-1} varphi_1(t) - Gamma^{m-1} varphi_2(t) || leq dfrac{L^{m-1} |t-t_0|^{m-1}}{(m-1)!} ||varphi_1 - varphi_2 ||, forall t in I_a
        $$



        Then $forall t in I_a$:



        begin{align}
        || Gamma^{m} varphi_1(t) - Gamma^{m} varphi_2(t) || = left|left| GammaGamma^{m-1} varphi_1(t) - Gamma Gamma^{m-1} varphi_2(t) right|right| &= left|left| int_{t_0}^{t} fleft(s,Gamma^{m-1} varphi_1(s) right) - fleft(s,Gamma^{m-1} varphi_2(s) right) ds right|right| \
        &leq left| int_{t_0}^{t} left|left| fleft(s,Gamma^{m-1} varphi_1(s) right) - fleft(s,Gamma^{m-1} varphi_2(s) right) right|right| ds right|
        end{align}
        But $f$ is Lipschitz in $I_a times B_b$ in the second variable with $L$ as Lipschitz constant. Then:



        begin{align}
        left| int_{t_0}^{t} left|left| fleft(s,Gamma^{m-1} varphi_1(s) right) - fleft(s,Gamma^{m-1} varphi_2(s) right) right|right| ds right|
        &leq
        L left| int_{t_0}^{t} left|left| Gamma^{m-1} varphi_1(s) - Gamma^{m-1} varphi_2(s) right|right| ds right| \
        end{align}



        And by induction hypothesis:



        begin{align}
        left| int_{t_0}^{t} left|left| fleft(s,Gamma^{m-1} varphi_1(s) right) - fleft(s,Gamma^{m-1} varphi_2(s) right) right|right| ds right|
        &leq
        L left| int_{t_0}^{t} dfrac{L^{m-1} |s-t_0|^{m-1}}{(m-1)!} ||varphi_1 - varphi_2 || ds right| \
        &leq
        dfrac{L^m}{(m-1)!} left| int_{t_0}^{t} |s-t_0|^{m-1} ||varphi_1 - varphi_2 || ds right| \
        &leq
        frac{L^m|t-t_0|^m}{m!} ||varphi_1 - varphi_2 || \
        end{align}
        Then the previous proposition is right, now:
        begin{align}
        || Gamma^{m} varphi_1(t) - Gamma^{m} varphi_2(t) || leq dfrac{L^{m} |t-t_0|^{m}}{m!} ||varphi_1 - varphi_2 || leq dfrac{L^{m} a^{m}}{m!} ||varphi_1 - varphi_2 ||, forall t in I_a
        end{align}
        Then:
        begin{align}
        textrm{sup} left{ || Gamma^{m} varphi_1(t) - Gamma^{m} varphi_2(t) || : t in I_a right} &leq dfrac{L^{m} a^{m}}{m!} ||varphi_1 - varphi_2 ||, forall t in I_a \
        || Gamma^{m} varphi_1 - Gamma^{m} varphi_2 || &leq dfrac{L^{m} a^{m}}{m!} ||varphi_1 - varphi_2 ||
        end{align}







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        answered Dec 30 '18 at 7:31









        El boritoEl borito

        666216




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