Existence and Uniqueness

In summary, it seems that if you can show that f(x) is Lipschitz with respect to q(x), then you can get a unique x.
  • #1
LeBrad
214
0
I am familiar with the existence and uniqueness of solutions to the system

[tex] \dot{x} = f(x) [/tex]

requiring [tex]f(x)[/tex] to be Lipschitz continuous, but I am wondering what the conditions are for the system

[tex] \dot{q}(x) = f(x) [/tex].

It seems like I could make the same argument for there existing a unique [tex]q(x)[/tex] provided [tex]f(x)[/tex] is Lipschitz with respect to [tex]q(x)[/tex]. Then if [tex]q(x)[/tex] is invertible or one-to-one or whatever the proper math term is, then I can get a unique [tex]x[/tex]. Is that correct? If that's the case it looks like I'm just doing a nonlinear change of coordinates, showing uniqueness in that coordinate system, and then having a unique map back to the original coordinate system.
 
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  • #2
Ok, since nobody complained I'm going to assume what I said is correct. In that case, I want to show Lipschitzness of f(x) with respect to q(x). If I define [itex]\tilde{x} = q(x)[/itex] and assume f is Lipschitz with respect to x, then

[tex]||f(x_1)-f(x_2)||\leq L ||x_1 - x_2|| = L ||q^{-1}(\tilde{x}_1) - q^{-1}(\tilde{x}_2)|| [/tex].

So if [itex]q^{-1}(\tilde{x})[/itex] is Lipschitz with respect to [tex]\tilde{x}[/itex],

[tex] ||q^{-1}(\tilde{x}_1) - q^{-1}(\tilde{x}_2)|| \leq M||\tilde{x}_1 - \tilde{x}_2||[/tex],

then

[tex]||f(x_1)-f(x_2)||\leq LM ||\tilde{x}_1 - \tilde{x}_2||[/tex].

So it seems it is sufficient to show that f(x) is Lipschitz with respect to x and that [itex]q^{-1}(\tilde{x})[/itex] is Lipschitz with respect to [tex]\tilde{x}[/itex]. Does that look correct?
 
  • #3
Keep in mind that

[tex]\dot{q}(x) = \frac{\partial q}{\partial x}\dot{x}= f(x) [/tex]

If nonzero or invertible in general (otherwise you have what is called a singular or descriptor system), [tex]\frac{\partial q}{\partial x}[/tex] is also a function of [tex]x[/tex] might be carried to the other side and you have another [tex]\dot{x} = \hat{f}(x)[/tex]
 
  • #4
Yeah, I know I can do that, but I was trying to keep that as a last resort. I have reason to keep it in the form
[tex] \dot{q}(x) = f(x) [/tex]
if possible.
 
  • #5
Yes, but proving if the [tex] \hat{f}(x) [/tex] is Lipschitz, is much more easier. Then you can say, OK now we multiply the differential equation from the left with some non-vanishing function [tex]h(x)[/tex] and then take
[tex]h(x)=\frac{\partial g}{\partial x}[/tex]

What I am trying to say is you have a point there, but it does not bring much difference into the problem nature. But, if you can prove that without inverting the function, then you have a nice result. Such as analyzing the properties of the linear singular system

[tex] E \dot{x} = Ax[/tex]

where E is not invertible. People usually dive into the problem by saying that the pencil [tex]\lambda E - A[/tex] is regular, does not have impulsive modes etc. You will definitely need some more assumptions to handle that issue when it becomes a general nonlinear differential system.
 

What is the concept of existence and uniqueness?

The concept of existence and uniqueness refers to the properties of solutions to a mathematical equation or system of equations. It states that for a given set of conditions or constraints, there exists only one solution that satisfies all the conditions.

Why is existence and uniqueness important in scientific research?

Existence and uniqueness is important in scientific research because it ensures that the solutions obtained from mathematical models or equations are accurate and reliable. It also helps in verifying the validity of the assumptions made in the model.

What are the conditions for existence and uniqueness to hold?

The conditions for existence and uniqueness to hold are typically related to the continuity, differentiability, and boundedness of the equations or functions involved in the problem. In general, the equations must be well-defined and have a unique solution for each set of initial conditions.

Can existence and uniqueness be proven for all mathematical problems?

No, existence and uniqueness cannot be proven for all mathematical problems. There are some cases where multiple solutions may exist or where the conditions for existence and uniqueness cannot be satisfied. In such cases, alternative methods or assumptions may be needed to find a solution.

How does the concept of existence and uniqueness apply to real-world problems?

In real-world problems, existence and uniqueness can help in finding the most appropriate solution that satisfies all the constraints and conditions. It also allows for the comparison of different solutions and helps in determining the most optimal one.

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