A class of evolution variational inequalities with nonconvex constraints

ABSTRACT

We study a nonconvex constrained evolution problem for a set being the union of a finite number of convex sets. The problem generalizes the classical parabolic variational inequality of the second kind and contains an operator of L-$(S{)}_{+}$-type. The existence result for this problem is proved using a variational-hemivariational inequality approach, a surjectivity theorem for multivalued pseudomonotone operators in a reflexive Banach space and a penalization method in which a small parameter does not have to tend to zero.

KEYWORDS:

• Variational inequality
• hemivariational inequality
• L-pseudomonotone operator
• operator inclusion
• generalized gradient

2010 AMS SUBJECT CLASSIFICATIONS:

• 47J35
• 47J20
• 47J22
• 35K86

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The research was supported by the National Science Center of Poland under the Maestro Project [grant number DEC2012/06/A/ST1/00262], the International Project co-financed by the Ministry of Science and Higher Education of Republic of Poland [grant number 3792/GGPJ/H2020/2017/0]. It was also supported by the NNSF of China [grant numbers 11561007 and 11561069], the NSF of Guangxi [grant number 2015GXNSFAA139017] and the Special Funds of Guangxi Distinguished Experts Construction Engineering. The third author is supported by Qinzhou University Project [grant number 2018KYQD06].