Abstract

We consider a rate independent evolution quasivariational inequality in a Hilbert space X with closed convex constraints having nonempty interior. We prove that there exists a unique solution which is Lipschitz dependent on the data, if the dependence of the Minkowski functional on the solution is Lipschitzian with a small constant and if also the gradient of the square of the Minkowski functional is Lipschitz continuous with respect to all variables. We exhibit an example of nonuniqueness if the assumption of Lipschitz continuity is violated by an arbitrarily small degree.

Author information

Contact details are reproduced from the original publication and may be historical.

Martin Brokate

Zentrum Mathematik, Technische Universität München, 85747 Garching, Germany

brokate@ma.tum.de

Pavel Krejčí

Mathematical Institute, Academy of Sciences of the Czech Republic, Žitná 25,
11567 Praha 1, Czech Republic

krejci@math.cas.cz

Hans Schnabel

Zentrum Mathematik, Technische Universität München, 85747 Garching, Germany

schnabel@ma.tum.de

Keywords

  • evolution quasivariational inequality
  • uniqueness
  • sweeping process
  • hysteresis
  • play operator

Mathematics Subject Classification

49J40, 34C55, 47J20

References

  1. P. Ballard: A counter-example to uniqueness in quasi-static elastic contact problems with friction, Int. J. Eng. Sci. 37 (1999) 163–178.https://doi.org/10.1016/S0020-7225(98)00062-7Google Scholar
  2. V. V. Chernorutskii, M. A. Krasnosel’skii: Hysteresis systems with variable characteristics, Nonlinear Anal. TMA 18 (1992) 543–557.https://doi.org/10.1016/0362-546X(92)90209-WGoogle Scholar
  3. R. E. Edwards: Functional Analysis, Holt, Rinehart and Winston, New York (1965).Google Scholar
  4. P. Krejčí: Evolution variational inequalities and multidimensional hysteresis operators, in: Nonlinear Differential Equations, P. Drábek, P. Krejčí, P. Takáč (eds.), Research Notes in Mathematics 404, Chapman & Hall/CRC, London (1999) 47–110.https://doi.org/10.1201/9780429332555-2Google Scholar
  5. P. Krejčí, Ph. Laurençot: Generalized variational inequalities, J. Convex Analysis 9 (2002) 159–183.https://doi.org/10.68381/jca09008Google Scholar
  6. P. Krejčí, J. Sprekels: Singular limit in parabolic differential inclusions and the stop operator, Interfaces Free Bound. 4 (2002) 423–435.https://doi.org/10.4171/ifb/68Google Scholar
  7. M. Kunze, M. Monteiro Marques: An introduction to Moreau’s sweeping process, in: Impacts in Mechanical Systems - Analysis and Modelling, B. Brogliato (ed.), Lecture Notes in Physics 551, Springer (2000) 1–60.https://doi.org/10.1007/3-540-45501-9_1Google Scholar
  8. M. Kunze, M. Monteiro Marques: On parabolic quasi-variational inequalities and state-dependent sweeping processes, Topol. Methods Nonlinear Anal. 12 (1998) 179–191.https://doi.org/10.12775/TMNA.1998.036Google Scholar
  9. A. Mielke, F. Theil: On rate-independent hysteresis models, Preprint #2001/07, SFB 404, Universität Stuttgart, 2001.Google Scholar
  10. J.-J. Moreau: Problème d’évolution associé à un convexe mobile d’un espace hilbertien, C. R. Acad. Sci. Paris Sér. A-B 273 (1973) A791–A794.Google Scholar
  11. J.-J. Moreau: Evolution problem associated with a moving convex set in a Hilbert space, J. Diff. Eq. 26 (1977) 347–374.https://doi.org/10.1016/0022-0396(77)90085-7Google Scholar
  12. R. T. Rockafellar: Convex Analysis, Princeton University Press, Princeton (1970).https://doi.org/10.1515/9781400873173Google Scholar

Suggested citation

M. Brokate, P. Krejčí, H. Schnabel. “On Uniqueness in Evolution Quasivariational Inequalities.” Journal of Convex Analysis 11 (2004), No. 1, 111–130. https://doi.org/10.68381/jca11008

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Published by Heldermann Verlag, 2004. Rights now held by Banach Press LLC.