Dynamic history-dependent variational-hemivariational inequalities with applications to contact mechanics

Abstract

Abstract. In the paper we deliver a new existence and uniqueness result for a class of abstract nonlinear variationalhemivariational inequalities which are governed by two operators depending on the history of the solution, and include two nondifferentiable functionals, a convex and a nonconvex one. Then, we consider an initial boundary value problem which describes a model of evolution of a viscoelastic body in contact with a foundation. The contact process is assumed to be dynamic, and the friction is described by subdifferential boundary conditions. Both the constitutive law and the contact condition involve memory operators. As an application of the abstract theory, we provide a result on the unique weak solvability of the contact problem.Mathematics Subject Classification. 35J87, 47J20, 47J22, 74M15, 74G25.

A Dynamic Contact Problem with History-Dependent Operators

“…The research that we present in this paper concerns the study on existence and uniqueness of solutions to the hemivariational inequalities with history-dependent operators, and the investigation on the behaviour of solutions to such inequalities with respect to perturbations of operators and functions. We note that the history-dependent operators have been considered for quasistatic and evolutionary contact problems by several authors, for example, by Sofonea et al in [8,21,[26][27][28] and [29], Migórski et al in [5,15,19] and [20], Ogorzały in [23], Yao et al in [31], and Zhu in [33]. The normal compliance contact condition was introduced in [13,14,22] and it was used in many papers, see, e.g., [3,9,25] and [29].…”

“…We prove results on existence and uniqueness of solution and a result on the convergence of solution of the inclusion with historydependent operators with respect to perturbations in the data. The existence result is obtained by a technique used by Migórski and Kulig in [20] who studied second order subdifferential inclusions with a Volterra-type operator. The unique solvability of the inclusion is proved by a standard fixed point argument similar to those used in many papers, for instance in [15,[19][20][21] and [23].…”

“…The existence result is obtained by a technique used by Migórski and Kulig in [20] who studied second order subdifferential inclusions with a Volterra-type operator. The unique solvability of the inclusion is proved by a standard fixed point argument similar to those used in many papers, for instance in [15,[19][20][21] and [23]. Furthermore, we note that the abstract convergence result of Theorem 13 is based on arguments and assumptions similar to those exploited, for instance in [2,15,25] and [27].…”

Equilibrium problems with generalized monotone mapping and its applications

“…By introducing the concept of the relaxed α – η pseudomonotone mapping, we establish some new existence results for (IEP) so that our results could be applied to hemivariational inequalities (HVIs). (HVIs) have important applications in mechanics and engineering, especially in nonsmooth analysis and optimization ().…”

Optimal Control of History-Dependent Evolution Inclusions with Applications to Frictional Contact

“…Moreover, such inclusions involve two nonlinear operators supposed to be history dependent. Existence and uniqueness results for such problems have been recently proved under various strong hypotheses involving smallness conditions; see [28][29][30][31][32][33]. The novelties of our paper are the following.…”