Trying to understand Lipschitz condition and some examples












1














I'm really new with ODE's and I need your help to understand the Lipschitz function and some examples.



First, the theory concepts:




A function $f(t, y)$ is said to satisfy a Lipschitz condition in the variable $y$ on a set $D$ in $R^2$ if there exists a constant $L > 0$ such that $|f(t, y_1)− f(t, y_2)| ≤ L |y_1 − y_2| $ (1),
whenever both points $(t,y_1)$ and $(t, y_2)$ are in $D$. The constant $L$ is called a Lipschitz constant for f.




First example

Let $f(t,y) = t|y|$, in $D=[1,2] times [-3,4]$. Does f satisfy a Lipschitz
condition on D?



Ok the solution is:



$|f(t, y_1)− f(t, y_2)| = bigl|t|y_1| - t|y_2|bigl| ≤ |t| bigl||y_1| − |y_2|bigl| ≤ 2|y_1 − y_2|$



And L = 2.



So my questions:





  • If:



    $ |f(t, y_1)− f(t, y_2)| = |t| bigl||y_1| − |y_2|bigl|$,



    then the "result" should be:



    $|t| bigl||y_1| − |y_2|bigl| ≤ L|y_1 − y_2|$ for the formula in (1), so how you get L and why is the solution L=2? I'm not understanding how to reach that form.



  • I can imagine that $L=2$ is related with the top $[1,2]$ first interval. Do I need to try the $[-3,4]$ values for the $y_1$ and $y_2$ variables respectively?


  • Why do we need to use absolute values in every part of the function?

  • Whenever I face a problem like this, do I need to reach the $|(some Number)||y_1 - y_2|$ form and pick the $someNumber$ as $L$? It will be always possible to do?


Second example



Let $f(t,y) = frac{2y}{1 + y^2}(1+sin(t))$, in $D=[0,1] times Re$. Does f satisfy a Lipschitz condition on D?



Here I have tried to get the (1) form but I couldn't, I have literally no idea and I would need a step by step resolution to understand this. I would appreciate some resources to learn about this theorem...



Thanks in advance, as you can see I am very lost on this.






differential-equations lipschitz-functions







share|cite|improve this question



























    1














    I'm really new with ODE's and I need your help to understand the Lipschitz function and some examples.



    First, the theory concepts:




    A function $f(t, y)$ is said to satisfy a Lipschitz condition in the variable $y$ on a set $D$ in $R^2$ if there exists a constant $L > 0$ such that $|f(t, y_1)− f(t, y_2)| ≤ L |y_1 − y_2| $ (1),
    whenever both points $(t,y_1)$ and $(t, y_2)$ are in $D$. The constant $L$ is called a Lipschitz constant for f.




    First example

    Let $f(t,y) = t|y|$, in $D=[1,2] times [-3,4]$. Does f satisfy a Lipschitz
    condition on D?



    Ok the solution is:



    $|f(t, y_1)− f(t, y_2)| = bigl|t|y_1| - t|y_2|bigl| ≤ |t| bigl||y_1| − |y_2|bigl| ≤ 2|y_1 − y_2|$



    And L = 2.



    So my questions:





    • If:



      $ |f(t, y_1)− f(t, y_2)| = |t| bigl||y_1| − |y_2|bigl|$,



      then the "result" should be:



      $|t| bigl||y_1| − |y_2|bigl| ≤ L|y_1 − y_2|$ for the formula in (1), so how you get L and why is the solution L=2? I'm not understanding how to reach that form.



    • I can imagine that $L=2$ is related with the top $[1,2]$ first interval. Do I need to try the $[-3,4]$ values for the $y_1$ and $y_2$ variables respectively?


    • Why do we need to use absolute values in every part of the function?

    • Whenever I face a problem like this, do I need to reach the $|(some Number)||y_1 - y_2|$ form and pick the $someNumber$ as $L$? It will be always possible to do?


    Second example



    Let $f(t,y) = frac{2y}{1 + y^2}(1+sin(t))$, in $D=[0,1] times Re$. Does f satisfy a Lipschitz condition on D?



    Here I have tried to get the (1) form but I couldn't, I have literally no idea and I would need a step by step resolution to understand this. I would appreciate some resources to learn about this theorem...



    Thanks in advance, as you can see I am very lost on this.






    differential-equations lipschitz-functions







    share|cite|improve this question

























      1












      1








      1


      1





      I'm really new with ODE's and I need your help to understand the Lipschitz function and some examples.



      First, the theory concepts:




      A function $f(t, y)$ is said to satisfy a Lipschitz condition in the variable $y$ on a set $D$ in $R^2$ if there exists a constant $L > 0$ such that $|f(t, y_1)− f(t, y_2)| ≤ L |y_1 − y_2| $ (1),
      whenever both points $(t,y_1)$ and $(t, y_2)$ are in $D$. The constant $L$ is called a Lipschitz constant for f.




      First example

      Let $f(t,y) = t|y|$, in $D=[1,2] times [-3,4]$. Does f satisfy a Lipschitz
      condition on D?



      Ok the solution is:



      $|f(t, y_1)− f(t, y_2)| = bigl|t|y_1| - t|y_2|bigl| ≤ |t| bigl||y_1| − |y_2|bigl| ≤ 2|y_1 − y_2|$



      And L = 2.



      So my questions:





      • If:



        $ |f(t, y_1)− f(t, y_2)| = |t| bigl||y_1| − |y_2|bigl|$,



        then the "result" should be:



        $|t| bigl||y_1| − |y_2|bigl| ≤ L|y_1 − y_2|$ for the formula in (1), so how you get L and why is the solution L=2? I'm not understanding how to reach that form.



      • I can imagine that $L=2$ is related with the top $[1,2]$ first interval. Do I need to try the $[-3,4]$ values for the $y_1$ and $y_2$ variables respectively?


      • Why do we need to use absolute values in every part of the function?

      • Whenever I face a problem like this, do I need to reach the $|(some Number)||y_1 - y_2|$ form and pick the $someNumber$ as $L$? It will be always possible to do?


      Second example



      Let $f(t,y) = frac{2y}{1 + y^2}(1+sin(t))$, in $D=[0,1] times Re$. Does f satisfy a Lipschitz condition on D?



      Here I have tried to get the (1) form but I couldn't, I have literally no idea and I would need a step by step resolution to understand this. I would appreciate some resources to learn about this theorem...



      Thanks in advance, as you can see I am very lost on this.






      differential-equations lipschitz-functions







      share|cite|improve this question













      I'm really new with ODE's and I need your help to understand the Lipschitz function and some examples.



      First, the theory concepts:




      A function $f(t, y)$ is said to satisfy a Lipschitz condition in the variable $y$ on a set $D$ in $R^2$ if there exists a constant $L > 0$ such that $|f(t, y_1)− f(t, y_2)| ≤ L |y_1 − y_2| $ (1),
      whenever both points $(t,y_1)$ and $(t, y_2)$ are in $D$. The constant $L$ is called a Lipschitz constant for f.




      First example

      Let $f(t,y) = t|y|$, in $D=[1,2] times [-3,4]$. Does f satisfy a Lipschitz
      condition on D?



      Ok the solution is:



      $|f(t, y_1)− f(t, y_2)| = bigl|t|y_1| - t|y_2|bigl| ≤ |t| bigl||y_1| − |y_2|bigl| ≤ 2|y_1 − y_2|$



      And L = 2.



      So my questions:





      • If:



        $ |f(t, y_1)− f(t, y_2)| = |t| bigl||y_1| − |y_2|bigl|$,



        then the "result" should be:



        $|t| bigl||y_1| − |y_2|bigl| ≤ L|y_1 − y_2|$ for the formula in (1), so how you get L and why is the solution L=2? I'm not understanding how to reach that form.



      • I can imagine that $L=2$ is related with the top $[1,2]$ first interval. Do I need to try the $[-3,4]$ values for the $y_1$ and $y_2$ variables respectively?


      • Why do we need to use absolute values in every part of the function?

      • Whenever I face a problem like this, do I need to reach the $|(some Number)||y_1 - y_2|$ form and pick the $someNumber$ as $L$? It will be always possible to do?


      Second example



      Let $f(t,y) = frac{2y}{1 + y^2}(1+sin(t))$, in $D=[0,1] times Re$. Does f satisfy a Lipschitz condition on D?



      Here I have tried to get the (1) form but I couldn't, I have literally no idea and I would need a step by step resolution to understand this. I would appreciate some resources to learn about this theorem...



      Thanks in advance, as you can see I am very lost on this.






      differential-equations lipschitz-functions




      differential-equations lipschitz-functions






      share|cite|improve this question













      share|cite|improve this question











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      asked Nov 26 at 0:48









      MakeItFun

      61




      61






















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          • For your first question, we know that $t in [1,2]$, hence $|t| le 2$.


          • $||y_1|-|y_2|| le |y_1-y_2|$ is due to reversed triangle inequality.


          • Lipchitz condition is phrased in terms of absolute value, hence the absolute values that you see.


          • If it satisfies Lipschitz condition, then such $L$ exists, I am not claiming that it is easy to find the Lipschitz coefficient though.


          • For the second example:



          begin{align}
          |f(t,y_1)-f(t,y_2)| &= |1+sin(t)|left| frac{y_1}{1+y_1^2}-frac{y_2}{1+y_2^2}right|\
          &le2 left| frac{y_1}{1+y_1^2}-frac{y_2}{1+y_2^2}right| \
          &=2left| frac{1-y_3^2}{(1+y_3^2)^2} right||y_1-y_2| text{, by MVT}\
          &le 2 cdot frac{1+y_3^2}{(1+y_3^2)^2} cdot|y_1-y_2|text{, by triangle inequality} \
          &= 2 cdot frac{1}{(1+y_3^2)} cdot|y_1-y_2| \
          &le 2|y_1-y_2|
          end{align}






          share|cite|improve this answer





















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            • For your first question, we know that $t in [1,2]$, hence $|t| le 2$.


            • $||y_1|-|y_2|| le |y_1-y_2|$ is due to reversed triangle inequality.


            • Lipchitz condition is phrased in terms of absolute value, hence the absolute values that you see.


            • If it satisfies Lipschitz condition, then such $L$ exists, I am not claiming that it is easy to find the Lipschitz coefficient though.


            • For the second example:



            begin{align}
            |f(t,y_1)-f(t,y_2)| &= |1+sin(t)|left| frac{y_1}{1+y_1^2}-frac{y_2}{1+y_2^2}right|\
            &le2 left| frac{y_1}{1+y_1^2}-frac{y_2}{1+y_2^2}right| \
            &=2left| frac{1-y_3^2}{(1+y_3^2)^2} right||y_1-y_2| text{, by MVT}\
            &le 2 cdot frac{1+y_3^2}{(1+y_3^2)^2} cdot|y_1-y_2|text{, by triangle inequality} \
            &= 2 cdot frac{1}{(1+y_3^2)} cdot|y_1-y_2| \
            &le 2|y_1-y_2|
            end{align}






            share|cite|improve this answer


























              0















              • For your first question, we know that $t in [1,2]$, hence $|t| le 2$.


              • $||y_1|-|y_2|| le |y_1-y_2|$ is due to reversed triangle inequality.


              • Lipchitz condition is phrased in terms of absolute value, hence the absolute values that you see.


              • If it satisfies Lipschitz condition, then such $L$ exists, I am not claiming that it is easy to find the Lipschitz coefficient though.


              • For the second example:



              begin{align}
              |f(t,y_1)-f(t,y_2)| &= |1+sin(t)|left| frac{y_1}{1+y_1^2}-frac{y_2}{1+y_2^2}right|\
              &le2 left| frac{y_1}{1+y_1^2}-frac{y_2}{1+y_2^2}right| \
              &=2left| frac{1-y_3^2}{(1+y_3^2)^2} right||y_1-y_2| text{, by MVT}\
              &le 2 cdot frac{1+y_3^2}{(1+y_3^2)^2} cdot|y_1-y_2|text{, by triangle inequality} \
              &= 2 cdot frac{1}{(1+y_3^2)} cdot|y_1-y_2| \
              &le 2|y_1-y_2|
              end{align}






              share|cite|improve this answer
























                0












                0








                0







                • For your first question, we know that $t in [1,2]$, hence $|t| le 2$.


                • $||y_1|-|y_2|| le |y_1-y_2|$ is due to reversed triangle inequality.


                • Lipchitz condition is phrased in terms of absolute value, hence the absolute values that you see.


                • If it satisfies Lipschitz condition, then such $L$ exists, I am not claiming that it is easy to find the Lipschitz coefficient though.


                • For the second example:



                begin{align}
                |f(t,y_1)-f(t,y_2)| &= |1+sin(t)|left| frac{y_1}{1+y_1^2}-frac{y_2}{1+y_2^2}right|\
                &le2 left| frac{y_1}{1+y_1^2}-frac{y_2}{1+y_2^2}right| \
                &=2left| frac{1-y_3^2}{(1+y_3^2)^2} right||y_1-y_2| text{, by MVT}\
                &le 2 cdot frac{1+y_3^2}{(1+y_3^2)^2} cdot|y_1-y_2|text{, by triangle inequality} \
                &= 2 cdot frac{1}{(1+y_3^2)} cdot|y_1-y_2| \
                &le 2|y_1-y_2|
                end{align}






                share|cite|improve this answer













                • For your first question, we know that $t in [1,2]$, hence $|t| le 2$.


                • $||y_1|-|y_2|| le |y_1-y_2|$ is due to reversed triangle inequality.


                • Lipchitz condition is phrased in terms of absolute value, hence the absolute values that you see.


                • If it satisfies Lipschitz condition, then such $L$ exists, I am not claiming that it is easy to find the Lipschitz coefficient though.


                • For the second example:



                begin{align}
                |f(t,y_1)-f(t,y_2)| &= |1+sin(t)|left| frac{y_1}{1+y_1^2}-frac{y_2}{1+y_2^2}right|\
                &le2 left| frac{y_1}{1+y_1^2}-frac{y_2}{1+y_2^2}right| \
                &=2left| frac{1-y_3^2}{(1+y_3^2)^2} right||y_1-y_2| text{, by MVT}\
                &le 2 cdot frac{1+y_3^2}{(1+y_3^2)^2} cdot|y_1-y_2|text{, by triangle inequality} \
                &= 2 cdot frac{1}{(1+y_3^2)} cdot|y_1-y_2| \
                &le 2|y_1-y_2|
                end{align}







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Nov 26 at 1:19









                Siong Thye Goh

                98.6k1464116




                98.6k1464116






























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