Quasi-hemivariational Inequalities Research Articles

Optimal control problem of fractional evolution inclusions with Clarke subdifferential driven by quasi-hemivariational inequalities

In this paper, we introduce and study an optimal control problem for a new class of fractional differential quasi-hemivariational inequalities with nonlocal initial conditions and Clarke subdifferentials. The main tools in our study are the theory of fractional calculus, the fixed point argument for condensing multimaps and Balder’s theorem. We prove the solvability of abstract nonlocal fractional differential quasi-hemivariational inequalities and then show the existence of solutions to the associated optimal control problems.

Lagrangian methods for optimal control problems governed by multivalued quasi-hemivariational inequalities

The primary objective of this paper is to develop a general augmented Lagrangian approach for optimal control problems governed by multivalued quasi-hemivariational inequalities. By imposing general coercivity and monotonicity-type conditions, we establish multiple existence results for the optimal control problems. Furthermore, we establish a comprehensive characterization of the zero duality gap property for the primal-dual problems, utilizing the concept of lower semicontinuity of new perturbation functions. Our results represent a significant advancement over recent literature on the subject.

On the existence of solutions and optimal control for set-valued quasi-variational–hemivariational inequalities with applications

In this paper, we study the existence and optimal control of quasi-hemivariational inequalities by a method different from the one based on Minty's technique. We use an approach similar to the Galerkin method based on a minimax inequality formulation associated with the Brézis pseudomonotonicity notion of multi-valued operators, an implicit Browder–Tikhonov regularization method and a fixed point theorem. This leads us to avoid any kind of monotonicity-type conditions used in recent papers to obtain the convexity of the solution set of the variational selections. We provide applications to the optimal control of implicit obstacle problems of fractional Laplacian type involving a generalized gradient operator, and to the optimal control of contact problems for elastic locking materials. Our approach improves some recent results in the literature.

A New System of Differential Quasi-Hemivariational Inequalities in Contact Mechanics

A New System of Differential Quasi-Hemivariational Inequalities in Contact Mechanics

Generalized Levitin–Polyak well-posedness for controlled systems of FMQHI-fuzzy mixed quasi-hemivariational inequalities of Minty type

The aim of this paper is to establish some new results on the Levitin–Polyak well-posedness (in short, LP well-posedness) to a new class of the controlled systems of fuzzy mixed quasi-hemivariational inequalities of the Minty type (in short, FMQHI). First, we introduce a new class of the controlled systems for FMQHI. Then, we show, under suitable conditions, the equivalence between the LP well-posedness and existence of solution to this problem. Further, results on metric characterization of the LP well-posedness and the LP well-posedness in the generalized sense for such problem in terms of the behavior of the approximate solution set are provided. Finally, we introduce a new class of the gap function for controlled systems of FMQHI and use it to study criterion of the LP well-posedness for controlled systems of FMQHI. The results presented in the paper are new.

Inverse problems for nonlinear quasi-hemivariational inequalities with application to mixed boundary value problems**Project supported by the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Skłodowska-Curie Grant agreement No. 823731 CONMECH, and National Science Center of Poland under Preludium Project No. 2017/25/N/ST1/00611. It is also supported by the Natural Science Foundation of Guangxi under Grant No. 2018JJA110006, and

This paper is devoted to studying an inverse problem of parameter identification in a nonlinear quasi-hemivariational inequality posed in a Banach space. We employ the Kluge’s fixed point theorem for the set-valued selection map, use the Minty approach and some properties of the Clarke subgradient to prove that the quasi-hemivariational inequality associated to the inverse problem has a nonempty, bounded, and weakly compact solution set. We develop a general regularization framework to provide an existence result for the inverse problem. As an illustrative application, we study an identification inverse problem in a complicated mixed elliptic boundary value problem with p -Laplace operator and an implicit obstacle.

Existence Results for Mixed Equilibrium Problems Involving Set-Valued Operators with Applications to Quasi-Hemivariational Inequalities

In this paper, we study the existence of solutions for mixed equilibrium problems associated with a set-valued operator in the general setting of vector spaces in duality, and in particular in Banach spaces. We use a Galerkin-type method and the notion of pseudomonotonicity in the sense of Brezis for bifunctions. As application, we study the existence of solutions for quasi-hemivariational inequalities governed by a set-valued mapping and perturbed with a nonlinear term. Our main results can be applied to differential inclusions, evolution equations and evolution hemivariational inequalities.

Lagrangian methods for optimal control problems governed by quasi-hemivariational inequalities

Lagrangian methods for optimal control problems governed by quasi-hemivariational inequalities

Existence Results for Nonlinear Quasi-hemivariational Inequality Systems

This paper aims at establishing the existence solutions of a non-standard quasi-hemivariational inequality‚ whose solutions are discussed in a subset of a reflexive Banach space for every . In addition, we introduce two applications. The first one shows the relationship between our inequality problems and equilibrium problems. The second one applies the extend some results concerning nonlinear quasi-hemivariational inequalities.

Partial differential hemivariational inequalities

Abstract The aim of this paper is to introduce and study a new class of problems called partial differential hemivariational inequalities that combines evolution equations and hemivariational inequalities. First, we introduce the concept of strong well-posedness for mixed variational quasi hemivariational inequalities and establish metric characterizations for it. Then we show the existence of solutions and meaningful properties such as measurability and upper semicontinuity for the solution set of the mixed variational quasi hemivariational inequality associated to the partial differential hemivariational inequality. Relying, on these properties we are able to prove the existence of mild solutions for partial differential hemivariational inequalities. Furthermore, we show the compactness of the set of the corresponding mild trajectories.

The Ekeland variational principle for equilibrium problems revisited and applications

In this paper, we study the Ekeland variational principle for equilibrium problems under the setting of real Banach spaces. We make use of techniques related to infinite dimensional spaces to solve the weakly compact case and introduce a suitable weakened set of coerciveness to deal with the non weakly compact case. Some old results for the Euclidean space Rn are generalized under weakened conditions of semicontinuity and applications to countable rather than finite systems of equilibrium problems on real Banach spaces are derived. Applications to quasi-hemivariational inequalities are discussed.

The Tikhonov regularization for equilibrium problems and applications to quasi-hemivariational inequalities

In this paper, we deal with the Tikhonov regularization method for pseudo-monotone equilibrium problems. Under mild conditions of semicontinuity and convexity, we show that strictly pseudo-monotone bifunctions can be also used as regularization bifunctions as well as strongly monotone bifunctions. We extend Berge’s maximum theorem and establish the relationship between quasi-hemivariational inequalities and equilibrium problems. Applications of the Tikhonov regularization method to quasi-hemivariational inequalities are also given.

Equilibrium problems techniques in the qualitative analysis of quasi-hemivariational inequalities

In this paper, we investigate the problem of existence of solutions of quasi-hemivariational inequalities. Some concepts of semicontinuity and hemicontinuity on subsets for functions as well as for set-valued mappings are developed and applied for solving quasi-hemivariational inequalities. Generalizations of some old results on the existence of solutions of equilibrium problems are obtained and applications to quasi-hemivariational inequalities are derived.

Existence of variational quasi-hemivariational inequalities involving a set-valued operatorand a nonlinear term

This paper is devoted to the existence of solutions for a class of variational quasi-hemivariational inequalities involving a lower hemicontinuous set-valued operator and a nonlinear term in reflexive Banach spaces. In the case when the constraint set is bounded, under certain generalized monotonicity conditions, we prove an existence result of solutions for the problem by means of F-KKM theorem. In the case when the constraint set is unbounded, under certain coercivity conditions, we construct an existence theorem of solutions and a boundedness theorem of the solution set for the problem, respectively. Moreover, a necessary and sufficient condition to the existence of solutions is also derived. The results presented in this paper generalize and improve some known results.

Existence and uniqueness results for a class of quasi-hemivariational inequalities

In this paper we are concerned with the study of a nonstandard quasi-hemivariational inequality. Using a fixed point theorem for set-valued mappings the existence of at least one solution in bounded closed and convex subsets is established. We also provide sufficient conditions for which our inequality possesses solutions in the case of unbounded sets. Finally, the uniqueness and the stability of the solution are analyzed in a particular case.

Vector Quasi-Hemivariational Inequalities and Discontinuous Elliptic Systems

We develop an existence theory for hemivariational inequalities in vector-valued function spaces which involve pseudomonotone operators. The obtained abstract result is used to study quasilinear elliptic systems whose lower order coupling vector field depends discontinuously upon the solution vector. We provide conditions that allow the identification of regions of existence of solutions for such systems, so called trapping regions.