Existence of a local solution

1. Sep 28, 2006

fortune

Hi,

For a first order Diff Equa. x'=f(x,t) and the IC: x(0)=x_0.
with t from [0 to infinity)
If f(x,t) doesn't satisfy the Lipschitz condition, can I say for sure that there doesn't exist a global unique solution?
I think the answer is "no" but I am not sure. Can you all confirm?

Also, can I use the Lipschitz condition to check the existence of a local solution around the IC? I see somebody often check the continuity of f(x) and df(x)/dx around the IC. Is this equavilent to the Lipschitz?

Thanks

2. Sep 29, 2006

HallsofIvy

Staff Emeritus
No, even if the Lipschitz condition is not satisfied there may still exist a unique solution. You just can't be certain that the solution is unique.

You need Lipschitz to guarentee uniqueness. The fact that the function f(x,y) is continuous is sufficient to give existance.

Showing that $\frac{\partial f}{\partial x}$ is "overkill". You can use the mean value theorem to show that if a function is differentiable on an interval, then it is Lipschitz there so differentiable is sufficient. But there exist Lipschitz functions that are not differentiable so it is not necessary.