89 research outputs found
Rothe method and numerical analysis for history-dependent hemivariational inequalities with applications to contact mechanics
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
A class of differential hemivariational inequalities in Banach spaces
In this paper we investigate an abstract system which consists of a hemivariational inequality of parabolic type combined with a nonlinear evolution equation in the framework of an evolution triple of spaces which is called a differential hemivariational inequality [(DHVI), for short]. A hybrid iterative system corresponding to (DHVI) is introduced by using a temporally semi-discrete method based on the backward Euler difference scheme, i.e., the Rothe method, and a feedback iterative technique. We apply a surjectivity result for pseudomonotone operators and properties of the Clarke subgradient operator to establish existence and a priori estimates for solutions to an approximate problem. Finally, through a limiting procedure for solutions of the hybrid iterative system, the solvability of (DHVI) is proved without imposing any convexity condition on the nonlinear function u↦f(t,x,u) and compactness of C0-semigroup eA(t)
Hidden maximal monotonicity in evolutionary variational-hemivariational inequalities
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 class of generalized evolutionary problems driven by variational inequalities and fractional operators
This paper is devoted to a generalized evolution system called fractional partial differential variational inequality which consists of a mixed quasi-variational inequality combined with a fractional partial differential equation in a Banach space. Invoking the pseudomonotonicity of multivalued operators and a generalization of the Knaster-Kuratowski-Mazurkiewicz theorem, first, we prove that the solution set of the mixed quasi-variational inequality involved in system is nonempty, closed and convex. Next, the measurability and upper semicontinuity for the mixed quasi-variational inequality with respect to the time variable and state variable are established. Finally, the existence of mild solutions for the system is delivered. The approach is based on the theory of operator semigroups, the Bohnenblust-Karlin fixed point principle for multivalued mappings, and theory of fractional operators
Rothe method and numerical analysis for history-dependent hemivariational inequalities with applications to contact mechanics
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
Evolutionary variational-hemivariational inequalities with applications to dynamic viscoelastic contact mechanics
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
A Class of Generalized Mixed Variational-Hemivariational Inequalities I: Existence and Uniqueness Results
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
Systems of hemivariational inequalities with nonlinear coupling functions
In this paper we investigate a system of coupled inequalities consisting of a
variational-hemivariational inequality and a quasi-hemivariational inequality
on Banach spaces. The approach is topological, and a wide variety of existence
results is established for both bounded and unbounded constraint sets in real
reflexive Banach spaces. The main point of interest is that no linearity
condition is imposed on the coupling functional, therefore making the system
fully nonlinear. Applications to Contact Mechanics are provided in the last
section of the paper. More precisely, we consider a contact model with
(possibly) multivalued constitutive law whose variational formulation leads to
a coupled system of inequalities. The weak solvability of the problem is proved
via employing the theoretical results obtained in the previous section. The
novelty of our approach comes from the fact that we consider two potential
contact zones and the variational formulation allows us to determine
simultaneously the displacement field and the Cauchy stress tensor
Existence of solutions for implicit obstacle problems of fractional laplacian type involving set-valued operators
The paper is devoted to a new kind of implicit obstacle problem given by a fractional Laplacian-type operator and a set-valued term, which is described by a generalized gradient. An existence theorem for the considered implicit obstacle problem is established, using a surjectivity theorem for set-valued mappings, Kluge’s fixed point principle and nonsmooth analysis
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