A New Class of History–Dependent Evolutionary Variational–Hemivariational Inequalities with Unilateral Constraints

Abstract

In this paper we study a new abstract evolutionary variational–hemivariational inequality which involves unilateral constraints and history–dependent operators. First, we prove the existence and uniqueness of solution by using a mixed equilibrium formulation with suitable selected functions together with a fixed-point principle for history–dependent operators. Then, we apply the abstract result to show the unique weak solvability to a dynamic viscoelastic frictional contact problem. The contact law involves a unilateral Signorini-type condition for the normal velocity combined with the nonmonotone normal damped response condition while the friction condition is a version of the Coulomb law of dry friction in which the friction bound depends on the accumulated slip.

Well-posedness of constrained evolutionary differential variational–hemivariational inequalities with applications

“…Problem (1)-( 3) is new and, to the best of our knowledge, has not been studied in the literature. Our main existence and uniqueness result, in its particular case, solves an open problem stated in [38,Section 10.4], and extends the recent result in [29,Theorem 20] obtained for a purely variational-hemivariational inequality.…”

“…Having verified the above properties of the data, and having in mind hypotheses H(K), (H 1 )-(H 3 ), we are in a position to apply [29,Theorem 20] to deduce that the problem (6), and equivalently Problem 5, is uniquely solvable.…”

On a dynamic frictional contact problem with normal damped response and long-term memory

“…An investigation of weak solvability, convergence results, and optimal control for frictional contact problems using different approaches and techniques can be found in Capatina [15], Chen et al [16, 17], Li [18], Liu et al [19], Matei [20], Matei and Micu [21, 22], Cîndea [23], Matei et al [24], Sofonea and Xiao [25], Sofonea and Shillor [26], and Xiao and Sofonea [27]. For results related to frictional contact problems with normal damped response for static, quasistatic, and dynamic processes, the reader can consult, e.g., Mígorski and Zeng [28], Han and Sofonea [29], Peng et al [30], Mígorski et al [31], Rochdi et al [32], and the references therein. Whereas the only one with real-world application of the so-called “normal damped response” contact condition was due to Renon et al [33].…”

Inverse problems for constrained parabolic variational-hemivariational inequalities *

“…We comment below on some results closely related to the topics of this paper. The strong solutions to parabolic hemivariational inequalities without constraints and with onedimensional superpotential j, have been studied in [39,41,44], and the problem with a constraint set and multidimensional superpotential has been examined in [50]. The stationary variational-hemivariational inequalities with and without unilateral constraints have been recently investigated in [49,64], where the inverse problems and the convergence of penalty methods have been studied.…”