Periodic solutions to history-dependent differential hemivariational inequalities with applications

In this article, we investigate a hybrid model combined by a parabolic differential equation and a parabolic hemivariational inequality (so-called differential hemivariational inequality of parabolic–parabolic type) in general infinite dimensional spaces which includes the history-dependent operator. The solvability of initial value problems as well as the periodic problems of the hemivariational inequality and the differential hemivariational inequality have been proved. In application, we study a contact problem with normal compliance driven by a history-dependent dynamical system.     

Noncoercive hemivariational inequality approach to constrained problems for star-shaped admissible sets

The hemivariational inequality approach is used in order to establish the existence of solutions to a large class of noncoercive constrained problems in a reflexive Banach space, in which the set of all admissible elements is not convex but fulfills some star-shaped property.Chargé de Recherches du Fonds National Belge de la Recherche Scientifique.     

Hemivariational inequality approach to constrained problems for star-shaped admissible sets

The hemivariational inequality approach is applied to establish the existence of solutions to a large class of nonconvex constrained problems in a reflexive Banach space. The admissible sets are supposed to be star-shaped with respect to a ball. Due to a discontinuity property of the Clarke directional differential related to the corresponding distance functions, the proposed method permits one to attain the solution without passing to zero with the penalization parameter. Some applications to nonconvex constrained variational problems illustrate the theory.     

A Heuristic Method for Nonconvex Optimization in Mechanics: Conceptual Idea, Theoretical Justification, Engineering Applications

Structures involving nonmonotone, possibly multivalued reaction-displacement or stress-strain laws cannot be effectively treated by the numerical methods for classical non-linearities. In this paper we make use of the fact that these problems have as a variational formulation a hemivariational inequality, leading to a noncovex optimization problem. A new method is proposed which approximates the nonmonotone problem by a series of monotone ones. The method constitutes an iterative scheme for the approximation of the solutions of the corresponding hemivariational inequality. A simple numerical example demonstrates the con-ceptual idea of the proposed numerical method. In the sequel the method is applied on an engineering problem concerning the ultimate strength analysis of an eccentric braced steel frame.     

Strong convergence results for hemivariational inequalities

The purpose of this paper is to study a regularization method of solutions of ill-posed problems involving hemivariational inequalities in Banach spaces. Under the assumption that the hemivariational inequality is solvable, a strongly convergent approximation procedure is designed by means of the so-called Browder-Tikhonov regularization method. Our results generalize and extend the previously known theorems.     

On the numerical treatment of nonconvex energy problems of mechanics

The present paper presents three numerical methods devised for the solution of hemivariational inequality problems. The theory of hemivariational inequalities appeared as a development of variational inequalities, namely an extension foregoing the assumption of convexity that is essentially connected to the latter. The methods that follow partly constitute extensions of methods applied for the numerical solution of variational inequalities. All three of them actually use the solution of a central convex subproblem as their kernel. The use of well established techniques for the solution of the convex subproblems makes up an effective, reliable and versatile family of numerical algorithms for large scale problems. The first one is based on the decomposition of the contigent cone of the (super)-potential of the problem into convex components. The second one uses an iterative scheme in order to approximate the hemivariational inequality problem with a sequence of variational inequality problems. The third one is based on the fact that nonconvexity in mechanics is closely related to irreversible effects that affect the Hessian matrix of the respective (super)-potential. All three methods are applied to solve the same problem and the obtained results are compared.     

Some Convergence Results for Evolution Hemivariational Inequalities

This paper is devoted to the regularization of a class of evolution hemivariational inequalities. The operator involved is taken to be non-coercive and the data are assumed to be known approximately. Under the assumption that the evolution hemivariational inequality be solvable, a weakly convergent approximation procedure is designed by means of the so-called Browder-Tikhonov regularization method.     

Multiple solutions for a class of elliptic hemivariational inequalities

The existence of multiple solutions to elliptic hemivariational inequality problems in bounded domains is investigated via a suitable nonsmooth version of a classical technique due to Struwe and a recent saddle point theorem for locally Lipschitz continuous functionals.     

Perturbations of Hemivariational Inequalities with Constraints and Applications

The aim of the present paper is to discuss the influence which certain perturbations have on the solution of the eigenvalue problem for hemivariational inequalities on a sphere of given radius. The perturbation results in adding a term of the type >g ⁰(x, u(x); v(x)) to the hemivariational inequality, where g is a locally Lipschitz nonsmooth and nonconvex energy functional. Applications illustrate the theory.     

A class of hemivariational inequalities for nonstationary Navier–Stokes equations

This paper is devoted to the study of a class of hemivariational inequalities for the time-dependent Navier–Stokes equations, including both boundary hemivariational inequalities and domain hemivariational inequalities. The hemivariational inequalities are analyzed in the framework of an abstract hemivariational inequality. Solution existence for the abstract hemivariational inequality is explored through a limiting procedure for a temporally semi-discrete scheme based on the backward Euler difference of the time derivative, known as the Rothe method. It is shown that solutions of the Rothe scheme exist, they contain a weakly convergent subsequence as the time step-size approaches zero, and any weak limit of the solution sequence is a solution of the abstract hemivariational inequality. It is further shown that under certain conditions, a solution of the abstract hemivariational inequality is unique and the solution of the abstract hemivariational inequality depends continuously on the problem data. The results on the abstract hemivariational inequality are applied to hemivariational inequalities associated with the time-dependent Navier–Stokes equations.     

Existence of solutions for wave-type hemivariational inequalities with noncoercive viscosity damping

In this paper we prove the existence of solutions for a hyperbolic hemivariational inequality of the form

u″+Au′+Bu+∂j(u)∋f,     

Decay mild solutions of Hilfer fractional differential variational–hemivariational inequalities

The primary objective of this paper is to explore a decay mild solution governed by a class of dynamical systems, called Hilfer fractional differential variational–hemivariational inequality (HFDVHVI, for short), which is composed of a Hilfer fractional evolution differential inclusion and a variational–hemivariational inequality involving two history-dependent operators in the framework of spaces. Our first aim is to investigate the solvability of the mild solutions to (HFDVHVI) by means of fixed point principle. The second step of the paper is to study the existence of decay mild solutions to (HFDVHVI) via giving expression for the Mittag-Leffler function and the Wright function.     

Comparing Nonsmooth Nonconvex Bundle Methods in Solving Hemivariational Inequalities

Hemivariational inequalities can be considered as a generalization of variational inequalities. Their origin is in nonsmooth mechanics of solid, especially in nonmonotone contact problems. The solution of a hemivariational inequality proves to be a substationary point of some functional, and thus can be found by the nonsmooth and nonconvex optimization methods. We consider two type of bundle methods in order to solve hemivariational inequalities numerically: proximal bundle and bundle-Newton methods. Proximal bundle method is based on first order polyhedral approximation of the locally Lipschitz continuous objective function. To obtain better convergence rate bundle-Newton method contains also some second order information of the objective function in the form of approximate Hessian. Since the optimization problem arising in the hemivariational inequalities has a dominated quadratic part the second order method should be a good choice. The main question in the functioning of the methods is how remarkable is the advantage of the possible better convergence rate of bundle-Newton method when compared to the increased calculation demand.     

Economic equilibrium problems in reflexive Banach spaces

The problem is to minimize a finite collection of objective functions over admissible sets depending on the so-called price vector. The minima in question and the price vector are linked together by a subdifferential governing law. The problem stated as a system of variational–hemivariational inequalities, defined on a nonconvex feasible set, is reduced to one variational–hemivariational inequality involving nonmonotone multivalued mapping. The existence of solutions is examined under the assumption that the constrained functions are positive homogeneous of degree one. The study has been inspired by economic issues and leads to new results concerning the existence of competitive equilibria.     

Browder-Tikhonov regularization for a class of evolution second order hemivariational inequalities

In this paper, we consider a class of evolution second order hemivariational inequalities with non-coercive operators which are assumed to be known approximately. Using the so-called Browder-Tikhonov regularization method, we prove that the regularized evolution hemivariational inequality problem is solvable. We construct a sequence based on the solvability of the regularized evolution hemivariational inequality problem and show that every weak cluster of this sequence is a solution for the evolution second order hemivariational inequality.     

Well-posedness for a Class of Variational–Hemivariational Inequalities with Perturbations

In this paper, we consider an extension of well-posedness for a minimization problem to a class of variational–hemivariational inequalities with perturbations. We establish some metric characterizations for the well-posed variational–hemivariational inequality and give some conditions under which the variational–hemivariational inequality is strongly well-posed in the generalized sense. Under some mild conditions, we also prove the equivalence between the well-posedness of variational–hemivariational inequality and the well-posedness of corresponding inclusion problem.     

Minimization principles for elliptic hemivariational inequalities

In this paper, we explore conditions under which certain elliptic hemivariational inequalities permit equivalent minimization principles. It is shown that for an elliptic variational–hemivariational inequality, under the usual assumptions that guarantee the solution existence and uniqueness, if an additional condition is satisfied, the solution of the variational–hemivariational inequality is also the minimizer of a corresponding energy functional. Then, two variants of the equivalence result are given, that are more convenient to use for applications in contact mechanics and in numerical analysis of the variational–hemivariational inequality. When the convex terms are dropped, the results on the elliptic variational–hemivariational inequalities are reduced to that on “pure” elliptic hemivariational inequalities. Finally, two representative examples from contact mechanics are discussed to illustrate application of the theoretical results.     

INVERSE COEFFICIENT PROBLEMS FOR PARABOLIC HEMIVARIATIONAL INEQUALITIES

This paper is devoted to the class of inverse problems for a nonlinear parabolic hemivariational inequality. The unknown coefficient of the operator depends on the gradient of the solution and belongs to a set of admissible coefficients. It is proved that the convergence of solutions for the corresponding direct problems continuously depends on the coefficient convergence. Based on this result the existence of a quasisolution of the inverse problem is obtained.     

Well-posedness of a general class of elliptic mixed hemivariational–variational inequalities

In this paper, well-posedness of a general class of elliptic mixed hemivariational–variational inequalities is studied. This general class includes several classes of the previously studied elliptic mixed hemivariational–variational inequalities as special cases. Moreover, our approach of the well-posedness analysis is easily accessible, unlike those in the published papers on elliptic mixed hemivariational–variational inequalities so far. First, prior theoretical results are recalled for a class of elliptic mixed hemivariational–variational inequalities featured by the presence of a potential operator. Then the well-posedness results are extended through a Banach fixed-point argument to the same class of inequalities without the potential operator assumption. The well-posedness results are further extended to a more general class of elliptic mixed hemivariational–variational inequalities through another application of the Banach fixed-point argument. The theoretical results are illustrated in the study of a contact problem. For comparison, the contact problem is studied both as an elliptic mixed hemivariational–variational inequality and as an elliptic variational–hemivariational inequality.     

Minimax principles for elliptic mixed hemivariational–variational inequalities

In this paper, minimax principles are explored for elliptic mixed hemivariational–variational inequalities. Under certain conditions, a saddle-point formulation is shown to be equivalent to a mixed hemivariational–variational inequality. While the minimax principle is of independent interest, it is employed in this paper to provide an elementary proof of the solution existence of the mixed hemivariational–variational inequality. Theoretical results are illustrated in the applications of two contact problems.