Timeline for Convergence of the solutions of a ODE system
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Jun 27, 2021 at 10:08 | answer | added | leo monsaingeon | timeline score: 3 | |
| Jun 27, 2021 at 10:00 | comment | added | Jochen Glueck | Since $F$ and $G$ don't depend on $n$, there should be an $n$-independent time interval on which the solution exists for each $n$; doesn't this follow readily from the typical proof of uniqueness and existence via fixed point iteration? | |
| Jun 26, 2021 at 23:53 | comment | added | moonlight | @leo monsaingeon I have edited the question. Please take a look and let me know if the conditions are enough. I also posted in math.SE but no answer there. My question is not general but for a particular problem I have encountered. With more details I have provided including Lipschitz continuity on some interval, still the answer is not trivial for me. I am not an expert in ODE. | |
| Jun 26, 2021 at 23:49 | history | edited | moonlight | CC BY-SA 4.0 |
added 246 characters in body
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| Jun 26, 2021 at 13:20 | comment | added | leo monsaingeon | well, as far as I am concerned Caratheodory's theorem requires some Lipschitz regularity to guarantee uniqueness (which the OP seems to take for granted, as "the" solution indicates). At first I thought the OP didn't want to assume Lipschitz regularity, this is why I "raised a flag". With Lipschitz regularity the question becomes almost trivial, so I suggest migrating to math.SE | |
| Jun 25, 2021 at 22:49 | comment | added | Jaap Eldering | @leomonsaingeon for 1) existence follows from Caratheodory's theorem: en.wikipedia.org/wiki/Carath%C3%A9odory%27s_existence_theorem. If F,G are moreover Lipschitz in x,y then also uniqueness follows. And for convergence of the solutions you likely at least need F,G to be Lipschitz. | |
| Jun 25, 2021 at 21:00 | review | Close votes | |||
| Jun 30, 2021 at 3:02 | |||||
| Jun 25, 2021 at 20:43 | comment | added | leo monsaingeon | For 1): what do you mean by "the" solution $x(t),y(t)$ to the original ODE? Do you even have well-posedness for your ODE? I don't think so, given that you have discontinuous coefficients... For 2): actually the convergence $x_n(t),y_n(t)\to x(t),y(t)$ may presumably answer to my problem regarding 1) (at least for the existence) but proving such a convergence is far from being trivial. I suggest you start first by thinking about the well-posedness issue, and then post again | |
| Jun 25, 2021 at 20:40 | history | edited | moonlight | CC BY-SA 4.0 |
edited title
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| Jun 25, 2021 at 20:32 | history | asked | moonlight | CC BY-SA 4.0 |