8 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 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
Approximate controllability for second order nonlinear evolution hemivariational inequalities
The goal of this paper is to study approximate controllability for control systems driven by abstract second order nonlinear evolution hemivariational inequalities in Hilbert spaces. First, the concept of a mild solution of our problem is defined by using the cosine operator theory and the generalized Clarke subdifferential. Next, the existence and the approximate controllability of mild solutions are formulated and proved by means of the fixed points strategy. Finally, an example is provided to illustrate our main results
Inverse Problems for Nonlinear Quasi-Variational Inequalities with an Application to Implicit Obstacle Problems of -Laplacian Type
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 -Laplacian type