Existence of mild solutions for semilinear equation of evolution
Karczewska, Anna ; Wędrychowicz, Stanisław
Commentationes Mathematicae Universitatis Carolinae, Tome 37 (1996), p. 695-706 / Harvested from Czech Digital Mathematics Library
Access to full text
Full (PDF)

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.

Publié le : 1996-01-01
Classification:  34A10,  34G20,  49E30,  49J24,  49J27 
@article{118879,
     author = {Anna Karczewska and Stanis\l aw W\k edrychowicz},
     title = {Existence of mild solutions for semilinear equation of evolution},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     volume = {37},
     year = {1996},
     pages = {695-706},
     zbl = {0893.34057},
     mrnumber = {1440702},
     language = {en},
     url = {http://dml.mathdoc.fr/item/118879}
}

Karczewska, Anna; Wędrychowicz, Stanisław. Existence of mild solutions for semilinear equation of evolution. Commentationes Mathematicae Universitatis Carolinae, Tome 37 (1996) pp. 695-706. http://gdmltest.u-ga.fr/item/118879/

Banaś J.; Goebel K. Measures of Noncompactness in Banach Spaces, Lect. Notes Pure Appl. Math. Marcel Dekker New York and Basel (1980). (1980) | MR 0591679

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. (1981) | MR 0646334

Coddington E.A.; Levinson N. Theory of Ordinary Differential Equations, Mc Graw-Hill New York-Toronto-London (1955). (1955) | MR 0069338 | Zbl 0064.33002

Friedman A. Partial Differential Equations, Krieger Publishing Company Huntington New York (1976). (1976) | MR 0454266

Kato T. Quasi-linear equations of evolution with application to partial differential equations, Lect. Notes Math. Springer Verlag 448 (1975), 25-70. (1975) | MR 0407477

Kuratowski K. Topologie, Volume II PWN Warszawa (1961). (1961) | MR 0133124 | Zbl 0102.37602

Pazy A. A class of semi-linear equations of evolution, Isr. J. Math. 20 (1975), 22-36. (1975) | MR 0374996 | Zbl 0305.47022

Pazy A. Semigroup of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag New York (1983). (1983) | MR 0710486

Rolewicz S. Functional Analysis and Control Theory Linear Systems, Reidel Publishing Company Dordrecht (1987). (1987) | MR 0920371 | Zbl 0633.93002