Timeline for Existence theorems Volterra Equation of second kind on unbounded domains
Current License: CC BY-SA 3.0
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| Jul 16, 2017 at 16:33 | comment | added | fedja | --I just wanted to be sure there were no overarching theorems to cover my problem.-- You never know that much (after all, what you are going to create yourself is exactly a theorem of that type), but even if there are such theorems, I don't think that should bother you in the slightest. Just use the tools you have in any way you can and try to keep the arguments reasonably clear and elegant. After all, mathematics, as I see it, is about being able to create what you want with what is at your disposal, not about searching old manuscripts for universal recipes. Otherwise, you are welcome :-) | |
| Jul 16, 2017 at 15:25 | comment | added | AD500712838 | In a research problem, I came across a class of kernels that are useful for a different reason. However, I'm trying to show that those same kernels produce solutions to this equation. Fedja: I think you pretty much answered my question. If I am understanding your answer, you are saying this problem definitely depends on the kernels themselves, given $f(x) = e^{-x}$. I am aware of techniques to answer my question for specific kernels. I just wanted to be sure there were no overarching theorems to cover my problem. Thank you! | |
| Jul 16, 2017 at 12:17 | comment | added | fedja | The only general idea I can come up with is to honestly go to the Fourier side and try to see if $\widehat f/(1-\widehat K)$ has a bounded inverse Fourier transform (so $1-\widehat K$ should have no zeroes in the upper half-plane and everything should behave nicely in general). This is, of course, just a restatement of the question. On the other hand, if you have some particular kernel and particular function (class of functions) in mind, it may be much more productive to just tell us what they are. | |
| Jul 16, 2017 at 2:03 | comment | added | AD500712838 | Great point. Assume f is something like e^(-x). | |
| Jul 16, 2017 at 0:50 | comment | added | fedja | Do you want the existence of a bounded solution for every bounded $f$ or something else? | |
| Jul 15, 2017 at 20:31 | history | edited | AD500712838 | CC BY-SA 3.0 |
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| Jul 15, 2017 at 20:21 | history | asked | AD500712838 | CC BY-SA 3.0 |