| Title:
|
Existence of mild solutions for semilinear equation of evolution (English) |
| Author:
|
Karczewska, Anna |
| Author:
|
Wędrychowicz, Stanisław |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
37 |
| Issue:
|
4 |
| Year:
|
1996 |
| Pages:
|
695-706 |
| . |
| Category:
|
math |
| . |
| Summary:
|
The aim of this paper is to give an existence theorem for a semilinear equation of evolution in the case when the generator of semigroup of operators depends on time parameter. The paper is a generalization of [2]. Basing on the notion of a measure of noncompactness in Banach space, we prove the existence of mild solutions of the equation considered. Additionally, the applicability of the results obtained to control theory is also shown. The main theorem of the paper allows to characterize the set of controls providing solutions of the system considered. Moreover, the application of the main theorem for elliptic equations is given. (English) |
| Keyword:
|
semilinear equation of evolution |
| Keyword:
|
mild solutions |
| Keyword:
|
measure of noncompactness |
| Keyword:
|
sublinear measure |
| MSC:
|
34A10 |
| MSC:
|
34G20 |
| MSC:
|
49E30 |
| MSC:
|
49J24 |
| MSC:
|
49J27 |
| idZBL:
|
Zbl 0893.34057 |
| idMR:
|
MR1440702 |
| . |
| Date available:
|
2009-01-08T18:27:25Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/118879 |
| . |
| Reference:
|
[1] Banaś J., Goebel K.: Measures of Noncompactness in Banach Spaces.Lect. Notes Pure Appl. Math. Marcel Dekker New York and Basel (1980). MR 0591679 |
| Reference:
|
[2] Banaś J., Hajnosz A., Wȩdrychowicz S.: Some generalization of Szufla's theorem for ordinary differential equations in Banach space.Bull. Pol. Acad. Sci., Math. XXIX , No 9-10 (1981), 459-464. MR 0646334 |
| Reference:
|
[3] Coddington E.A., Levinson N.: Theory of Ordinary Differential Equations.Mc Graw-Hill New York-Toronto-London (1955). Zbl 0064.33002, MR 0069338 |
| Reference:
|
[4] Friedman A.: Partial Differential Equations.Krieger Publishing Company Huntington New York (1976). MR 0454266 |
| Reference:
|
[5] Kato T.: Quasi-linear equations of evolution with application to partial differential equations.Lect. Notes Math. Springer Verlag 448 (1975), 25-70. MR 0407477 |
| Reference:
|
[6] Kuratowski K.: Topologie.Volume II PWN Warszawa (1961). Zbl 0102.37602, MR 0133124 |
| Reference:
|
[7] Pazy A.: A class of semi-linear equations of evolution.Isr. J. Math. 20 (1975), 22-36. Zbl 0305.47022, MR 0374996 |
| Reference:
|
[8] Pazy A.: Semigroup of Linear Operators and Applications to Partial Differential Equations.Springer-Verlag New York (1983). MR 0710486 |
| Reference:
|
[9] Rolewicz S.: Functional Analysis and Control Theory Linear Systems.Reidel Publishing Company Dordrecht (1987). Zbl 0633.93002, MR 0920371 |
| . |
