I found the following definition in a source while reading Existence and Uniqueness of solution of IVP

While understanding the concept of Existence and Uniqueness Theorem for Initial Value Problems of System of Ordinary Differential Equations, I first came across "vector - valued functions" and the condition for the uniqueness of solution in vector valued function (Lipschitz Condition for Vector Valued Function).

As mentioned in the picture attached above, we have always checked the Lipschitz condition in the vector variable $\mathbf{x}$ for the vector valued function $\mathbf{X}$.

My question here is that can we not show the Lipschitz condition for the scalar variable $t$ instead of the vector variable $\mathbf{x}$?

I also tried one example of a vector valued function $f\left( x, \mathbf{y} \right) = \left( 3x + 2y_1, y_1 - y_2 \right)$ where $\mathbf{y} = \left( y_1, y_2 \right)$.

To check the Lipschitz condition in the scalar variable x, I proceeded as follows:-

$$\left| f\left( x_2, \mathbf{y} \right) - f\left( x_1, \mathbf{y} \right) \right| = \left| \left( 3 \left( x_2 - x_1 \right), 0 \right) \right| \leq 3 \left| x_2 - x_1 \right|$$

So, I would like to ask here that is this approach correct? Or do we have to always check Lipschitz condition for vector valued function in the vector variable?

Note: The picture attached above has a phrase iff, however I think it is because the article talks about system of differential equations and the uniqueness of the solution to the system. But, in this question, I ask in general if we can check Lipschitz condition for the scalar variable.

  • $\begingroup$ this question makes no sense. The definition itself says it involves the vector variable. So you must check it on the vector variable. $\endgroup$ – dezdichado Oct 12 '17 at 6:51
  • $\begingroup$ But what if we check it on the scalar variable? $\endgroup$ – Aniruddha Deshmukh Oct 12 '17 at 6:53
  • $\begingroup$ What do you like to know? If checking the Lipschitz-condition on the scalar variable is sufficient for the function to be Lipschitz on the vector variable? Or do you want to know, if you can prove existence and/or uniqueness, if you just have a Lipschitz condition in the scalar variable? In both cases the answer would be in general no, for example $f(x,t)=\sqrt{x}$ with initial condition $x(0)=0$. (No uniqueness and no Lipschitz condition in $x$ in this case ) $\endgroup$ – humanStampedist Oct 12 '17 at 9:36
  • $\begingroup$ All your views over the question are absolutelt correct. But, as far as I have learnt, Lipschitz condition is to be satisfied in a particular variable. Here in the definition there is an iff condition involving the vector variable. Threfore, I think that it would mean we cannot have Lipschitz condition in scalar variable. And that is my question. Can we have Lipschitz condition in the scalar variable for the vector valued function so given to us? $\endgroup$ – Aniruddha Deshmukh Oct 12 '17 at 11:48
  • $\begingroup$ The definition needs a Lipschitz condition in the vector variable. It does not specify anything about the scalar variable. So it does not matter whether $f$ is Lipschitz in the scalar variable or not. $\endgroup$ – humanStampedist Oct 12 '17 at 14:09

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