Timeline for Simple example of Hammerstein integral equation
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 3, 2022 at 15:42 | comment | added | Martin Väth | It is one variant of a straightforward condition ensuring that the operator $A$ on the right-hand side maps $S=\{x\in C(I,E):\lVert x(t)\rVert\le r(t)\}$ into itself, and so Schauder/Darbo/whatever fixed point theorems are applied in many papers for $A\colon S\to S$. A trivial artificial example is for a measurable kernel satisfying $\lVert k\rVert_\infty\le C$ a function $f$ satisfying $\lVert f(t,u)\rVert\le M$ for $\lVert u\rVert\le CM$: Put $r(t)=CM$, $a(s)=1$, and $b(u)=M$ for $\lVert u\rVert\le CM$. Thus, formally no growth hypotheses needs to be satisfied for $f$ w.r.t. $u$. | |
| Jan 3, 2022 at 9:45 | comment | added | Motaka | Thank you for your comment, can you elaborate more on the "artificial example". I need this because I didn't find such hypotheses in the literature. If you have something to suggest, I will be very grateful. | |
| Jan 3, 2022 at 9:40 | history | bounty ended | Motaka | ||
| Jan 3, 2022 at 9:40 | vote | accept | Motaka | ||
| Dec 30, 2021 at 8:54 | history | answered | Martin Väth | CC BY-SA 4.0 |