On existence of equilibria of set-valued maps

Grzegorz Gabor; Marc Quincampoix

Bollettino dell'Unione Matematica Italiana (2003)

  • Volume: 6-B, Issue: 2, page 309-321
  • ISSN: 0392-4033

Abstract

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The present paper is devoted to sufficient conditions for existence of equilibria of Lipschitz multivalued maps in prescribed subsets of finite-dimensional spaces. The main improvement of the present study lies in the fact that we do not suppose any regular assumptions on the boundary of the subset. Our approach is based on behaviour of trajectories to the corresponding differential inclusion.

How to cite

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Gabor, Grzegorz, and Quincampoix, Marc. "On existence of equilibria of set-valued maps." Bollettino dell'Unione Matematica Italiana 6-B.2 (2003): 309-321. <http://eudml.org/doc/194712>.

@article{Gabor2003,
abstract = {The present paper is devoted to sufficient conditions for existence of equilibria of Lipschitz multivalued maps in prescribed subsets of finite-dimensional spaces. The main improvement of the present study lies in the fact that we do not suppose any regular assumptions on the boundary of the subset. Our approach is based on behaviour of trajectories to the corresponding differential inclusion.},
author = {Gabor, Grzegorz, Quincampoix, Marc},
journal = {Bollettino dell'Unione Matematica Italiana},
keywords = {differential inclusions; Marchaud multivalued mappings; existence of equlibria},
language = {eng},
month = {6},
number = {2},
pages = {309-321},
publisher = {Unione Matematica Italiana},
title = {On existence of equilibria of set-valued maps},
url = {http://eudml.org/doc/194712},
volume = {6-B},
year = {2003},
}

TY - JOUR
AU - Gabor, Grzegorz
AU - Quincampoix, Marc
TI - On existence of equilibria of set-valued maps
JO - Bollettino dell'Unione Matematica Italiana
DA - 2003/6//
PB - Unione Matematica Italiana
VL - 6-B
IS - 2
SP - 309
EP - 321
AB - The present paper is devoted to sufficient conditions for existence of equilibria of Lipschitz multivalued maps in prescribed subsets of finite-dimensional spaces. The main improvement of the present study lies in the fact that we do not suppose any regular assumptions on the boundary of the subset. Our approach is based on behaviour of trajectories to the corresponding differential inclusion.
LA - eng
KW - differential inclusions; Marchaud multivalued mappings; existence of equlibria
UR - http://eudml.org/doc/194712
ER -

References

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  1. AUBIN, J.-P.- CELLINA, A., Differential Inclusions, Springer, 1984. Zbl0538.34007MR755330
  2. AUBIN, J.-P., Viability Theory, Birkhäuser, Boston, 1991. Zbl0755.93003MR1134779
  3. BEN-EL-MECHAIEKH, H.- KRYSZEWSKI, W., Équilibres dans les ensembles nonconvexes, C. R. Acad. Sci. Paris Sér. I, 320 (1995), 573-576. Zbl0833.54024MR1322339
  4. BEN-EL-MECHAIEKH, H.- KRYSZEWSKI, W., Equilibria of set-valued maps on nonconvex domains, Trans. Amer. Math. Soc., 349 (1997), 4159-4179. Zbl0887.47040MR1401763
  5. BONY, J. M., Principe du maximum, inégalité de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés, Ann. Institut Fourier, Grenoble, 19, 1 (1969), 277-304. Zbl0176.09703MR262881
  6. BROWDER, F., The fixed point theory of multivalued mappings in topological vector spaces, Math. Ann., 177 (1968), 283-301. Zbl0176.45204MR229101
  7. BROWN, R., The Lefschetz Fized Point Theorem, Scott, Foresman and Comp., Glenview Ill., London1971. Zbl0216.19601MR283793
  8. CARDALIAGUET, P., Sufficient conditions of nonemptiness of the viability kernel, PhD Thesis, Chapter 8, Université Paris IX Dauphine, 1992. Zbl0761.34016MR1166049
  9. CARDALIAGUET, P., Conditions suffisantes de non-vacuité du noyau de viabilité, C. R. Acad. Sci., Paris, Ser. I, 314, 11 (1992), 797-800. Zbl0761.34016MR1166049
  10. CLARKE, F.- LEDYAEV, YU. S.- STERN, R. J., Fixed points and equilibria in nonconvex sets, Nonlinear Analysis, 25 (1995), 145-161. Zbl0840.49010MR1333819
  11. CLARKE, F.- LEDYAEV, YU. S.- STERN, R. J.- WOLENSKI, P. R., Nonsmooth Analysis and Control Theory, Springer, 1998. Zbl1047.49500MR1488695
  12. CORNET, B., Paris avec handicaps et théorèmes de surjectivité de correspondances, C. R. Acad. Sc. Paris Sér. A, 281 (1975), 479-482. Zbl0317.90087MR386726
  13. CORNET, B.- CZARNECKI, M.-O., Existence of (generalized) equilibria: necessary and sufficient conditions, Comm. Appl. Nonlinear Anal., 7 (2000), 21-53. Zbl1108.49301MR1733400
  14. ĆWISZEWSKI, A.- KRYSZEWSKI, W., Equilibria of set-valued maps: variational approach, Nonlinear Anal. TMA (accepted). Zbl1030.49021
  15. DUGUNDJI, J., Topology, Allyn and Bacon, Inc., Boston, 1966. Zbl0144.21501MR193606
  16. EILENBERG, S.- STEENROD, N., Foundations of Algebraic Topology, Princeton Univ. Press, New Jersey, 1952. Zbl0047.41402MR50886
  17. FAN, K., Fixed point and minimax theorems in locally convex topological spaces, Proc. Nat. Acad. Sci. USA, 38 (1952), 121-126. Zbl0047.35103MR47317
  18. FAN, K., Some properties of convex sets related to fixed point theorems, Math. Ann., 266 (1984), 519-537. Zbl0515.47029MR735533
  19. HADDAD, G.- LASRY, J. M., Periodic solutions of functional differential inclusions and fixed points σ -selectionable correspondances, J. Math. Anal. Appl., 96 (1983), 295-312. Zbl0539.34031MR719317
  20. K. MISCHAIKOW-M. MROZEK-P. ZGLICZYŃSKI (editors), Conley index theory, Banach Center Publ., 47, PWN, Warszawa, 1999. Zbl0913.00021MR1675402
  21. MROZEK, M., Periodic and stationary trajectories of flows and ordinary differential equations, Zesz. Nauk. Uniw. Jagiellon. 860, Acta Math., 27 (1988), 29-37. Zbl0684.34046MR982424
  22. PLASKACZ, S., On the solution sets of differential inclusions, Boll. Un. Mat. Ital. (7), 6-A (1992), 387-394. Zbl0774.34012MR1196133
  23. QUINCAMPOIX, M., Frontières de domaines d'invariance et de viabilité pour les inclusions différentielles avec contraintes, C. R. Acad. Sci., Paris, 311 (1990), 411-416. Zbl0705.34014MR1075661
  24. QUINCAMPOIX, M., Differential inclusions and target problems, SIAM J. Control Optimization,30 (1992), 324-335. Zbl0862.49006MR1149071
  25. SRZEDNICKI, R., Periodic and bounded solutions in blocks for time periodic nonautonomous ordinary differential equations, Nonlinear Anal. TMA, 22 (1994), 707-737. Zbl0801.34041MR1270166
  26. VELIOV, V., Lipschitz continuity of the value function in optimal control, J. Optimization Theory Appl., 94, 2 (1997), 335-361. Zbl0901.49022MR1460669

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