| Title:
|
Periodic problems for ODEs via multivalued Poincaré operators (English) |
| Author:
|
Górniewicz, Lech |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
34 |
| Issue:
|
1 |
| Year:
|
1998 |
| Pages:
|
93-104 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We shall consider periodic problems for ordinary differential equations of the form \[ {\left\lbrace \begin{array}{ll} x^{\prime }(t)= f(t,x(t)),\\ x(0) = x(a), \end{array}\right.} \] where $ f:[0,a] \times R^n \rightarrow R^n$ satisfies suitable assumptions. To study the above problem we shall follow an approach based on the topological degree theory. Roughly speaking, if on some ball of $R^n$, the topological degree of, associated to (), multivalued Poincaré operator $P$ turns out to be different from zero, then problem () has solutions. Next by using the multivalued version of the classical Liapunov-Krasnoselskǐ guiding potential method we calculate the topological degree of the Poincaré operator $P$. To do it we associate with $f$ a guiding potential $V$ which is assumed to be locally Lipschitzean (instead of $C^1$) and hence, by using Clarke generalized gradient calculus we are able to prove existence results for (), of the classical type, obtained earlier under the assumption that $V$ is $C^1$. Note that using a technique of the same type (adopting to the random case) we are able to obtain all of above mentioned results for the following random periodic problem: \[ {\left\lbrace \begin{array}{ll} x^{\prime }(\xi , t) = f(\xi , t, x(\xi ,t)),\\ x(\xi ,0) = x(\xi , a), \end{array}\right.} \] where $f:\Omega \times [0,a]\times R^n\rightarrow R^n$ is a random operator satisfying suitable assumptions. This paper stands a simplification of earlier works of F. S. De Blasi, G. Pianigiani and L. Górniewicz (see: [gor7], [gor8]), where the case of differential inclusions is considered. (English) |
| Keyword:
|
Periodic processes |
| Keyword:
|
topological degree |
| Keyword:
|
Poincaré translation operator |
| MSC:
|
34B15 |
| MSC:
|
34C25 |
| MSC:
|
34F05 |
| MSC:
|
34G20 |
| MSC:
|
47H10 |
| MSC:
|
47H15 |
| MSC:
|
55M20 |
| idZBL:
|
Zbl 0915.34029 |
| idMR:
|
MR1629672 |
| . |
| Date available:
|
2009-02-17T10:10:32Z |
| Last updated:
|
2012-05-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/107636 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
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| Reference:
|
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| Reference:
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| Reference:
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