24 research outputs found

    Rothe method and numerical analysis for history-dependent hemivariational inequalities with applications to contact mechanics

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    In this paper an abstract evolutionary hemivariational inequality with a history-dependent operator is studied. First, a result on its unique solvability and solution regularity is proved by applying the Rothe method. Next, we introduce a numerical scheme to solve the inequality and derive error estimates. We apply the results to a quasistatic frictional contact problem in which the material is modeled with a viscoelastic constitutive law, the contact is given in the form of multivalued normal compliance, and friction is described with a subgradient of a locally Lipschitz potential. Finally, for the contact problem we provide the optimal error estimate

    Differential variational-hemivariational inequalities: existence, uniqueness, stability, and convergence

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    The goal of this paper is to study a comprehensive systemcalled differential variational鈥揾emivariational inequality which is com-posed of a nonlinear evolution equation and a time-dependentvariational鈥揾emivariational inequality in Banach spaces. Under the gen-eral functional framework, a generalized existence theorem for differ-ential variational鈥揾emivariational inequality is established by employ-ing KKM principle, Minty鈥檚 technique, theory of multivalued analysis,the properties of Clarke鈥檚 subgradient. Furthermore, we explore a well-posedness result for the system, including the existence, uniqueness, andstability of the solution in mild sense. Finally, using penalty methods tothe inequality, we consider a penalized problem-associated differentialvariational鈥揾emivariational inequality, and examine the convergence re-sult that the solution to the original problem can be approached, as aparameter converges to zero, by the solution of the penalized problem

    Rothe method and numerical analysis for history-dependent hemivariational inequalities with applications to contact mechanics

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    In this paper, an abstract evolutionary hemivariational inequality with a history-dependent operator is studied. First, a result on its unique solvability and solution regularity is proved by applying the Rothe method. Next, we introduce a numerical scheme to solve the inequality and derive error estimates. We apply the results to a quasistatic frictional contact problem in which the material is modeled with a viscoelastic constitutive law, the contact is given in the form of multivalued normal compliance, and friction is described with a subgradient of a locally Lipschitz potential. Finally, for the contact problem, we provide the optimal error estimate

    Hidden maximal monotonicity in evolutionary variational-hemivariational inequalities

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    In this paper, we propose a new methodology to study evolutionary variational-hemivariational inequalities based on the theory of evolution equations governed by maximal monotone operators. More precisely, the proposed approach, based on a hidden maximal monotonicity, is used to explore the well-posedness for a class of evolutionary variational-hemivariational inequalities involving history-dependent operators and related problems with periodic and antiperiodic boundary conditions. The applicability of our theoretical results is illustrated through applications to a fractional evolution inclusion and a dynamic semipermeability problem

    A new class of history-dependent quasi variational-hemivariational inequalities with constraints

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    In this paper we consider an abstract class of time-dependent quasi variational-hemivariational inequalities which involves history-dependent operators and a set of unilateral constraints. First, we establish the existence and uniqueness of solution by using a recent result for elliptic variational-hemivariational inequalities in reflexive Banach spaces combined with a fixed-point principle for history-dependent operators. Then, we apply the abstract result to show the unique weak solvability to a quasistatic viscoelastic frictional contact problem. The contact law involves a unilateral Signorini-type condition for the normal velocity and 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.Comment: 15

    On convergence of solutions to variational-hemivariational inequalities

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    In this paper we investigate the convergence behavior of the solutions to the time-dependent variational鈥揾emivariational inequalities with respect to the data. First, we give an existence and uniqueness result for the problem, and then, deliver a continuous dependence result when all the data are subjected to perturbations. A semipermeability problem is given to illustrate our main results

    A Class of Generalized Mixed Variational-Hemivariational Inequalities I: Existence and Uniqueness Results

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    We investigate a generalized Lagrange multiplier system in a Banach space, called a mixed variational-hemivariational inequality (MVHVI, for short), which contains a hemivariational inequality and a variational inequality. First, we employ the Minty technique and a monotonicity argument to establish an equivalence theorem, which provides three different equivalent formulations of the inequality problem. Without compactness for one of operators in the problem, a general existence theorem for (MVHVI) is proved by using the Fan-Knaster-Kuratowski-Mazurkiewicz principle combined with methods of nonsmooth analysis. Furthermore, we demonstrate several crucial properties of the solution set to (MVHVI) which include boundedness, convexity, weak closedness, and continuity. Finally, a uniqueness result with respect to the first component of the solution for the inequality problem is proved by using the Ladyzhenskaya-Babuska-Brezzi (LBB) condition. All results are obtained in a general functional framework in reflexive Banach spaces

    Inverse Problems for Nonlinear Quasi-Variational Inequalities with an Application to Implicit Obstacle Problems of pp-Laplacian Type

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    The primary objective of this research is to investigate an inverse problem of parameter identification in nonlinear mixed quasi-variational inequalities posed in a Banach space setting. By using a fixed point theorem, we explore properties of the solution set of the considered quasi-variational inequality. We develop a general regularization framework to give an existence result for the inverse problem. Finally, we apply the abstract framework to a concrete inverse problem of identifying the material parameter in an implicit obstacle problem given by an operator of pp-Laplacian type
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