Abstract
The purpose of this work is to introduce and investigate a complicated variational–hemivariational inequality of parabolic type with history-dependent operators. First, we establish an existence and uniqueness theorem for a first-order nonlinear evolution inclusion problem, which is driven by a convex subdifferential operator for a proper convex function and a generalized Clarke subdifferential operator for a locally Lipschitz superpotential. Then, we employ the fixed point principle for history-dependent operators to deliver the unique solvability of the parabolic variational–hemivariational inequality. Finally, a dynamic viscoelastic contact problem with the nonlinear constitutive law involving a convex subdifferential inclusion is considered as an illustrative application, where normal contact and friction are described, respectively, by two nonconvex and nonsmooth multi-valued terms.
Highlights
The contact processes between deformable bodies around in industry and our real-life and, for this reason, a considerable effort for modeling, mathematical analysis, numerical simulation and optimal control of various frictional contact problems are quite interesting and important.The theory of variational inequalities can be used to describe the principles of virtual work and power which was initially proposed by Fourier in 1823
The prototypes, which lead to a class of variational inequalities, are the problems of Signorini–Fichera and frictional contact in elasticity
The solution of the Signorini Problem coincides with the birth of the field of variational inequalities
Summary
The contact processes between deformable bodies around in industry and our real-life and, for this reason, a considerable effort for modeling, mathematical analysis, numerical simulation and optimal control of various frictional contact problems are quite interesting and important. For more on the initial developments of elasticity theory and variational inequalities, cf e.g., [1]. With the gradual improvement of the theory of variational inequalities, there are numerous monographs dedicated to solving various complex phenomena in contact problems with different bodies and foundations, see for instance [7,8,12,32] and others. The mathematical theory of hemivariational inequalities has been of great interest recently, which is due to the intensive development of applications of hemivariational inequalities in contact mechanics, control theory, games and so forth.
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