Mircea Sofonea, Yi-Bin Xiao
Abstract:
We consider a class of elliptic variational-hemivariational inequalities
in an abstract Banach space for which we introduce the concept of
well-posedness in the sense of Tykhonov.
We characterize the well-posedness in terms of metric properties of a
family of associated sets. Our results, which provide necessary and sufficient
conditions for the well-posedness of inequalities under consideration, are
valid under mild assumptions on the data.
Their proofs are based on arguments of monotonicity, lower semicontinuity
and properties of the Clarke directional derivative. For well-posed
inequalities we also prove a continuous dependence result of the solution
with respect to the data. We illustrate our abstract results in the study
of one-dimensional examples, then we focus on some relevant particular cases,
including variational-hemivariational inequalities with strongly monotone
operators. Finally, we consider a model variational-hemivariational
inequality which arises in Contact Mechanics for which we discuss
its well-posedness and provide the corresponding mechanical interpretations.
Submitted March 12, 2019. Published May 10, 2019.
Math Subject Classifications: 49J40, 47J20, 49J45, 35M86, 74M10, 74M15.
Key Words: Tykhonov well-posedness; variational-hemivariational inequality;
approximating sequence; contact problem; unilateral constraint; friction.
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Mircea Sofonea School of Mathematical Sciences University of Electronic Science and Technology of China email: sofonea@univ-perp.fr |
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Yi-Bin Xiao School of Mathematical Sciences University of Electronic Science and Technology of China Chengdu, Sichuan, 611731, China email: xiaoyb9999@hotmail.com |
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