Denseness of the set of Cauchy problems with nonunique solutions in the set of all Cauchy problems
We prove the following theorem: Let $E$ be an arbitrary Banach space, $G$ be an open set in the space $R×E$, and $f : G → E$ be an arbitrary continuous mapping. Then, for an arbitrary point $(t_0, x_0) ∈ G$ and an arbitrary number $ε > 0$, there exists a continuous mapping $g : G → E$ such th...
Saved in:
| Date: | 2012 |
|---|---|
| Main Authors: | , |
| Format: | Article |
| Language: | Ukrainian English |
| Published: |
Institute of Mathematics, NAS of Ukraine
2012
|
| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/2635 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
| Download file: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| Summary: | We prove the following theorem: Let $E$ be an arbitrary Banach space, $G$ be an open set in the space $R×E$, and $f : G → E$ be an arbitrary continuous mapping. Then, for an arbitrary point $(t_0, x_0) ∈ G$ and an arbitrary number $ε > 0$, there exists
a continuous mapping $g : G → E$ such that
$$\sup_{(t,x)∈G}||g(t, x) − f(t, x)|| \leq \varepsilon$$
and the Cauchy problem
$$\frac{dz(t)}{dt} = g(t, z(t)), z(t0) = x_0$$
has more than one solution. |
|---|