Extremal solutions and relaxation for second order vector differential inclusions

Evgenios P. Avgerinos; Nikolaos S. Papageorgiou

Archivum Mathematicum (1998)

  • Volume: 034, Issue: 4, page 427-434
  • ISSN: 0044-8753

Abstract

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In this paper we consider periodic and Dirichlet problems for second order vector differential inclusions. First we show the existence of extremal solutions of the periodic problem (i.e. solutions moving through the extreme points of the multifunction). Then for the Dirichlet problem we show that the extremal solutions are dense in the C 1 ( T , R N ) -norm in the set of solutions of the “convex” problem (relaxation theorem).

How to cite

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Avgerinos, Evgenios P., and Papageorgiou, Nikolaos S.. "Extremal solutions and relaxation for second order vector differential inclusions." Archivum Mathematicum 034.4 (1998): 427-434. <http://eudml.org/doc/248190>.

@article{Avgerinos1998,
abstract = {In this paper we consider periodic and Dirichlet problems for second order vector differential inclusions. First we show the existence of extremal solutions of the periodic problem (i.e. solutions moving through the extreme points of the multifunction). Then for the Dirichlet problem we show that the extremal solutions are dense in the $C^1(T,R^N)$-norm in the set of solutions of the “convex” problem (relaxation theorem).},
author = {Avgerinos, Evgenios P., Papageorgiou, Nikolaos S.},
journal = {Archivum Mathematicum},
keywords = {lower semicontinuous multifunctions; continuous embedding; compact embedding; continuous selector; extremal solution; relaxation theorem; lower semicontinuous multifunctions; continuous embedding; compact embedding; continuous selector; extremal solution; relaxation theorem},
language = {eng},
number = {4},
pages = {427-434},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Extremal solutions and relaxation for second order vector differential inclusions},
url = {http://eudml.org/doc/248190},
volume = {034},
year = {1998},
}

TY - JOUR
AU - Avgerinos, Evgenios P.
AU - Papageorgiou, Nikolaos S.
TI - Extremal solutions and relaxation for second order vector differential inclusions
JO - Archivum Mathematicum
PY - 1998
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 034
IS - 4
SP - 427
EP - 434
AB - In this paper we consider periodic and Dirichlet problems for second order vector differential inclusions. First we show the existence of extremal solutions of the periodic problem (i.e. solutions moving through the extreme points of the multifunction). Then for the Dirichlet problem we show that the extremal solutions are dense in the $C^1(T,R^N)$-norm in the set of solutions of the “convex” problem (relaxation theorem).
LA - eng
KW - lower semicontinuous multifunctions; continuous embedding; compact embedding; continuous selector; extremal solution; relaxation theorem; lower semicontinuous multifunctions; continuous embedding; compact embedding; continuous selector; extremal solution; relaxation theorem
UR - http://eudml.org/doc/248190
ER -

References

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  5. Gutman S., Topological equivalence in the space of integrable vector valued functions, Proc. AMS. 93(1985), 40-42. (1985) Zbl0529.46027MR0766523
  6. Kisielewicz M., Differential Inclusions and Optimal Control, Kluwer, Dordrecht, The Netherlands, (1991). (1991) MR1135796
  7. Klein E., Thompson A., Theory of Correspondences, Wiley, New York, (1984). (1984) Zbl0556.28012MR0752692
  8. Papageorgiou N. S., On measurable multifunctions with applications to random multivalued equations, Math. Japonica, 32, (1987), 437-464. (1987) Zbl0634.28005MR0914749
  9. Šeda V., On some nonlinear boundary value problems for ordinary differential equations, Archivum Math. (Brno) 25(1989), 207-222. (1989) MR1188065
  10. Tolstonogov A. A., Extreme continuous selectors for multivalued maps and the bang-bang principle for evolution equations, Soviet. Math. Doklady 42(1991), 481-485. (1991) MR1121349
  11. Wagner D., Surveys of measurable selection theorems, SIAM J. Control Optim. 15 (1977), 857-903. (1977) MR0486391

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