# American Institute of Mathematical Sciences

December  2020, 9(4): 1089-1114. doi: 10.3934/eect.2020047

## Fully history-dependent evolution hemivariational inequalities with constraints

 1 College of Sciences, Beibu Gulf University, Qinzhou, Guangxi 535000, China 2 Jagiellonian University in Krakow, Chair of Optimization and Control, ul. Lojasiewicza 6, 30348 Krakow, Poland 3 School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China 4 College of Sciences, Beibu Gulf University, Qinzhou, Guangxi 535000, China

Dedicated to Professor Meir Shillor on the occasion of his 70th birthday

Received  October 2019 Published  December 2020 Early access  March 2020

In this paper we study a new class of abstract evolution first order hemivariational inequalities which involves constraints and history-dependent operators. First, we prove the existence and uniqueness of solution by using a mixed equilibrium formulation with suitable selected bifunctions combined with a fixed-point principle for history-dependent operators. Next, we deduce existence, uniqueness and regularity results for some special subclasses of problems which include a constrained history-dependent variational–hemivariational inequality, an evolution quasi-variational inequality with constraints, and an evolution second order hemivariational inequality with constraints. Then, we provide an application of the results to a dynamic unilateral viscoelastic frictional contact problem and show its unique weak solvability.

Citation: Stanisław Migórski, Yi-bin Xiao, Jing Zhao. Fully history-dependent evolution hemivariational inequalities with constraints. Evolution Equations and Control Theory, 2020, 9 (4) : 1089-1114. doi: 10.3934/eect.2020047
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