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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.

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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| \\ &=2\left| \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}

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