Published July 29, 2019 | Version v1
Journal article Open

A class of history-dependent differential variational inequalities with application to contact problems

  • 1. College of Applied Mathematics, Chengdu University of Information Technology, Chengdu, Sichuan Province, People's Republic of China; Chair of Optimization and Control, Jagiellonian University in Krakow, Krakow, Poland
  • 2. Guangxi Colleges and Universities Key Laboratory of Complex System Optimization and Big Data Processing, Yulin Normal University, Yulin, People's Republic of China; College of Sciences, Guangxi University for Nationalities, Nanning, Guangxi, People's Republic of China
  • 3. Faculty of Mathematics and Computer Science, Jagiellonian University in Krakow, Krakow, Poland


In this paper a class of generalized differential variational inequalities with constraints involving history-dependent operators in Banach spaces is investigated. The unique solvability and regularity results are obtained via surjectivity of multivalued pseudomonotone operators combined with a fixed point principle. From abstract results, a theorem concerning existence, uniqueness and regularity of weak solution to a frictional viscoelastic contact problem with adhesion and history-dependent operator is established. Further, a theoretical analysis of a penalty method for history-dependent differential variational inequality is provided. The unique solvability of a penalized problem is shown, as well as the convergence of its solution to the solution of the original history-dependent differential variational inequality, as a penalty parameter tends to zero. Finally, results on a penalty method are applied to another contact problem, history-dependent frictional viscoelastic contact problem with a generalized normal compliance condition instead of a generalized Signorini contact condition.



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