# Trying to understand Lipschitz condition and some examples

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.

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