$^{**}$ In the paper we will be concerned with the topological structure of the set of solutions of the initial value problem of a semilinear multi-valued system on a closed and convex set. Assuming that the linear part of the system generates a $C_0$-semigroup we show the $R_\delta $-structure of this set under certain natural boundary conditions. Using this result we obtain several criteria for the existence of periodic solutions for the semilinear system. As an application the problem of controlled heat transfer in an isotropic rigid body is considered.
@article{119204,
author = {Ralf Bader},
title = {On the semilinear multi-valued flow under constraints and the periodic problem},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {41},
year = {2000},
pages = {719-734},
zbl = {1057.34064},
mrnumber = {1800171},
language = {en},
url = {http://dml.mathdoc.fr/item/119204}
}
Bader, Ralf. On the semilinear multi-valued flow under constraints and the periodic problem. Commentationes Mathematicae Universitatis Carolinae, Tome 41 (2000) pp. 719-734. http://gdmltest.u-ga.fr/item/119204/
Set-valued Analysis, Birkhäuser, 1990. | MR 1048347 | Zbl 1168.49014
Multivalued differential equations in Banach spaces. An application in control theory, J. Optim. Theory and Appl. 21 (1977), 477-486. (1977) | MR 0440144
Fixed point theorems for compositions of set-valued maps with single-valued maps, Annales Universitatis Mariae Curie-Skłodowska, Vol. LI.2, Sectio A, Lublin, 1997, pp.29-41. | MR 1666164 | Zbl 1012.47043
The periodic problem for semilinear differential inclusions in Banach spaces, Comment. Math. Univ. Carolinae 39 (1998), 671-684. (1998) | MR 1715457 | Zbl 1060.34508
Equilibria of set-valued maps on nonconvex domains, Trans. Amer. Math. Soc. 349 (1997), 4159-4179. (1997) | MR 1401763 | Zbl 0887.47040
Multivalued differential equations on graphs and applications, Ph. D. dissertation, Universität Paderborn, 1992. | MR 1148288 | Zbl 0789.34013
On the topological structure of the solution set for a semilinear functional-differential inclusion in a Banach space, in: Topology in Nonlinear Analysis, K. Geba and L. Górniewicz (eds.), Polish Academy of Sciences, Institute of Mathematics, Banach Center Publications 35, Warszawa, 1996, pp.159-169. | MR 1448435
Periodic solutions of differential equations in Banach spaces, Manuscripta Math. 24 (1978), 31-44. (1978) | MR 0499551 | Zbl 0373.34032
Multivalued Differential Equations, de Gruyter, Berlin-New York, 1992. | MR 1189795 | Zbl 0820.34009
Remarks on weak compactness in $L_1(\mu,X)$, Glasgow Math. J. 18 (1977), 87-91. (1977) | Zbl 0342.46020
Topological approach to differential inclusions, in: Topological methods in differential equations and inclusions, A. Granas and M. Frigon (eds.), NATO ASI Series C 472, Kluwer Academic Publishers, 1995, pp.129-190. | MR 1368672
On the topological regularity of the solution set of differential inclusions with constraints, J. Differential Equations 107 (1994), 280-290. (1994) | MR 1264523 | Zbl 0796.34017
On decreasing sequences of compact absolute retracts, Fund. Math. 64 (1969), 91-97. (1969) | MR 0253303 | Zbl 0174.25804
Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, de Gruyter, to appear. | MR 1831201 | Zbl 0988.34001
Nonlinear Operators and Differential Equations in Banach Spaces, Wiley, New York, 1976. | MR 0492671 | Zbl 0333.47023
Invariant sets for a class of semi-linear equations of evolution, Nonlinear Anal. 1 (1977), 187-196. (1977) | MR 0637080 | Zbl 0344.45001
Periodic solutions of semilinear evolution equations, Nonlinear Anal. 3 (1979), 601-612. (1979) | MR 0541871
Viability theorems for a class of differential-operator inclusions, J. Differential Equations 79 (1989), 232-257. (1989) | MR 1000688 | Zbl 0694.34011