On the semilinear multi-valued flow under constraints and the periodic problem

Ralf Bader

Commentationes Mathematicae Universitatis Carolinae (2000)

  • Volume: 41, Issue: 4, page 719-734
  • ISSN: 0010-2628

Abstract

top
* * 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 δ -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.

How to cite

top

Bader, Ralf. "On the semilinear multi-valued flow under constraints and the periodic problem." Commentationes Mathematicae Universitatis Carolinae 41.4 (2000): 719-734. <http://eudml.org/doc/248592>.

@article{Bader2000,
abstract = {$^\{**\}$ 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.},
author = {Bader, Ralf},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {multi-valued maps; $C_0$-semigroup; initial value problem under constraints; $R_\delta $-sets; periodic solutions; equilibria; control problem; multi-valued maps; -sets; periodic solutions; semilinear evolution inclusions},
language = {eng},
number = {4},
pages = {719-734},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On the semilinear multi-valued flow under constraints and the periodic problem},
url = {http://eudml.org/doc/248592},
volume = {41},
year = {2000},
}

TY - JOUR
AU - Bader, Ralf
TI - On the semilinear multi-valued flow under constraints and the periodic problem
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2000
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 41
IS - 4
SP - 719
EP - 734
AB - $^{**}$ 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.
LA - eng
KW - multi-valued maps; $C_0$-semigroup; initial value problem under constraints; $R_\delta $-sets; periodic solutions; equilibria; control problem; multi-valued maps; -sets; periodic solutions; semilinear evolution inclusions
UR - http://eudml.org/doc/248592
ER -

References

top
  1. Aubin J.-P., Frankowska H., Set-valued Analysis, Birkhäuser, 1990. Zbl1168.49014MR1048347
  2. Anichini G., Zecca P., Multivalued differential equations in Banach spaces. An application in control theory, J. Optim. Theory and Appl. 21 (1977), 477-486. (1977) MR0440144
  3. Bader R., 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. Zbl1012.47043MR1666164
  4. Bader R., The periodic problem for semilinear differential inclusions in Banach spaces, Comment. Math. Univ. Carolinae 39 (1998), 671-684. (1998) Zbl1060.34508MR1715457
  5. Ben-El-Mechaiekh H., Kryszewski W., Equilibria of set-valued maps on nonconvex domains, Trans. Amer. Math. Soc. 349 (1997), 4159-4179. (1997) Zbl0887.47040MR1401763
  6. Bothe D., Multivalued differential equations on graphs and applications, Ph. D. dissertation, Universität Paderborn, 1992. Zbl0789.34013MR1148288
  7. Conti G., Obukhovskii V., Zecca P., 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. MR1448435
  8. Deimling K., Periodic solutions of differential equations in Banach spaces, Manuscripta Math. 24 (1978), 31-44. (1978) Zbl0373.34032MR0499551
  9. Deimling K., Multivalued Differential Equations, de Gruyter, Berlin-New York, 1992. Zbl0820.34009MR1189795
  10. Diestel J., Remarks on weak compactness in L 1 ( μ , X ) , Glasgow Math. J. 18 (1977), 87-91. (1977) Zbl0342.46020
  11. Górniewicz L., 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. MR1368672
  12. Hu S., Papageorgiou N.S., On the topological regularity of the solution set of differential inclusions with constraints, J. Differential Equations 107 (1994), 280-290. (1994) Zbl0796.34017MR1264523
  13. Hyman D.M., On decreasing sequences of compact absolute retracts, Fund. Math. 64 (1969), 91-97. (1969) Zbl0174.25804MR0253303
  14. Kamenskii M., Obukhovskii V., Zecca P., Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, de Gruyter, to appear. Zbl0988.34001MR1831201
  15. Martin R., Nonlinear Operators and Differential Equations in Banach Spaces, Wiley, New York, 1976. Zbl0333.47023MR0492671
  16. Pavel N., Invariant sets for a class of semi-linear equations of evolution, Nonlinear Anal. 1 (1977), 187-196. (1977) Zbl0344.45001MR0637080
  17. Prüss, J., Periodic solutions of semilinear evolution equations, Nonlinear Anal. 3 (1979), 601-612. (1979) MR0541871
  18. Shuzhong Shi., Viability theorems for a class of differential-operator inclusions, J. Differential Equations 79 (1989), 232-257. (1989) Zbl0694.34011MR1000688

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.