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Journal of Convex Analysis 11 (2004), No. 1, 111--130
Copyright Heldermann Verlag 2004
On Uniqueness in Evolution Quasivariational Inequalities
Martin Brokate
Zentrum Mathematik, Technische Universität München, 85747 Garching, Germany,
brokate@ma.tum.de
Pavel Krejcí
Mathematical Institute, Academy of Sciences of the Czech Republic, Zitná 25,
11567 Praha 1, Czech Republic,
krejci@math.cas.cz
Hans Schnabel
Zentrum Mathematik, Technische Universität München, 85747 Garching, Germany,
schnabel@ma.tum.de
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.
Keywords: evolution quasivariational inequality, uniqueness, sweeping process, hysteresis,
play operator.
MSC 2000: 49J40, 34C55, 47J20.
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