Timeline for Is this a contraction mapping for small $T$?
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| Dec 29, 2021 at 0:46 | comment | added | Iosif Pinelis | @GJC20 : A reason why the above argument is insufficient is that we also need to bound the derivative of $F[h]'(s)$ in $h$ -- that is, the derivative in $h$ of the derivative of $F[h](s)$ in $s$. Also, having looked at your newer question at mathoverflow.net/questions/412652/…, I think you can use here a slightly simpler norm, as suggested in my comment to that question. | |
| Dec 28, 2021 at 20:15 | comment | added | GJC20 | Sure. There is no rush. Anyway I get the idea and succeed in adopting your arguments to calculate the derivative $F[h]'$. However, I am thinking whether there exists a suitable subset s.t. the fixed point theorem. Could you please take a look at my question at mathoverflow.net/questions/412652/… ? Many thanks | |
| Dec 28, 2021 at 20:06 | comment | added | Iosif Pinelis | @GJC20 : Indeed, I will have to revisit this answer -- will try to find time for that. | |
| Dec 28, 2021 at 16:13 | comment | added | GJC20 | Just a typo above equation (2): I think it should be $B_T$ or even $H_T$ instead of $L_T$ (which is not defined) | |
| Dec 28, 2021 at 9:12 | comment | added | GJC20 | Amazing solution! Thanks so much Iosif. As I look for the fixed points of the operator $F$ for small $T$, it suffices to show the existence of some suitable subspace $H$ s.t. $F(H)\subset H$. Do you think it is possible? I post this question independently here mathoverflow.net/questions/412652/… | |
| Dec 28, 2021 at 9:01 | vote | accept | GJC20 | ||
| Dec 28, 2021 at 0:23 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Dec 27, 2021 at 22:13 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Dec 27, 2021 at 21:40 | history | answered | Iosif Pinelis | CC BY-SA 4.0 |