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Constrained Variational-Hemivariational Inequalities on Nonconvex Star-Shaped Sets

1
College of Sciences, Beibu Gulf University, Qinzhou 535000, China
2
Faculty of Mathematics and Computer Science, Jagiellonian University in Krakow, ul. Lojasiewicza 6, 30348 Krakow, Poland
3
School of Science, Nanjing University of Science and Technology, Nanjing 210094, China
*
Author to whom correspondence should be addressed.
Mathematics 2020, 8(10), 1824; https://doi.org/10.3390/math8101824
Received: 24 September 2020 / Revised: 13 October 2020 / Accepted: 14 October 2020 / Published: 17 October 2020
(This article belongs to the Section Difference and Differential Equations)
In this paper, we study a class of constrained variational-hemivariational inequality problems with nonconvex sets which are star-shaped with respect to a certain ball in a reflexive Banach space. The inequality is a fully nonconvex counterpart of the variational-hemivariational inequality of elliptic type since it contains both, a convex potential and a locally Lipschitz one. Two new results on the existence of a solution are proved by a penalty method applied to a variational-hemivariational inequality penalized by the generalized directional derivative of the distance function of the constraint set. In the first existence theorem, the strong monotonicity of the governing operator and a relaxed monotonicity condition of the Clarke subgradient are assumed. In the second existence result, these two hypotheses are relaxed and a suitable hypothesis on the upper semicontinuity of the operator is adopted. In both results, the penalized problems are solved by using the Knaster, Kuratowski, and Mazurkiewicz (KKM) lemma. For a suffciently small penalty parameter, the solution to the penalized problem solves also the original one. Finally, we work out an example on the interior and boundary semipermeability problem that ilustrate the applicability of our results. View Full-Text
Keywords: variational inequality; hemivariational inequality; pseudomonotone operator; KKM theorem; generalized gradient; Clarke’s tangent cone variational inequality; hemivariational inequality; pseudomonotone operator; KKM theorem; generalized gradient; Clarke’s tangent cone
MDPI and ACS Style

Migórski, S.; Fengzhen, L. Constrained Variational-Hemivariational Inequalities on Nonconvex Star-Shaped Sets. Mathematics 2020, 8, 1824.

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