Proving Existence of Differential System with Unequal Time Variables

[cris(c)]

Hi everyone,

I'm not quite sure how to proceed to show existence (and perhaps uniqueness) of the following system of (first order) differential equations:
$\dot{x}=f(t_1,x,y,z)$
$\dot{y}=g(t_2,x,y,z)$
$\dot{z}=h(t_3,x,y,z)$

where $\dot{x}=\frac{\partial x}{\partial t_1}$, $\dot{y}=\frac{\partial y}{\partial t_2}$, and $\dot{z}=\frac{\partial z}{\partial t_3}$.

All existence theorems I've seen are formulated such that $t_1=t_2=t_3$. I've tried reading the proofs to see if I can figure out a way to apply them to this problem, but I can't see how...Does someone knows whether these theorems hold true when $t_1 \neq t_2 \neq t_3$? Any help/reference where to look for such theorem would be greatly appreciate!

 

[jvicens]

Hi there,

First of all, great question! It's always important to consider the assumptions and limitations of the theorems you are using in your research.

To address your specific question, the existence theorems you have seen are typically formulated for systems of differential equations where the independent variable, in this case t, is the same for all equations. However, there are also existence theorems that allow for different independent variables.

One such theorem is the Picard-Lindelöf theorem, also known as the Cauchy-Lipschitz theorem. This theorem guarantees the existence and uniqueness of solutions to a system of differential equations as long as the functions f, g, and h are continuous and satisfy certain Lipschitz conditions. These conditions ensure that the solutions to the system are well-behaved and do not exhibit any wild behavior.

Another important theorem to consider is the Peano existence theorem, which guarantees the existence of solutions to a system of differential equations as long as the functions f, g, and h are continuous. However, this theorem does not guarantee uniqueness of solutions.

I would recommend looking into these theorems and their proofs to see how they can be applied to your specific problem. Additionally, talking to other researchers in your field or consulting with a mathematics expert may also provide valuable insights and guidance.

Best of luck with your research!