Inequalities In Reflexive Banach Spaces Research Articles

An inertial‐like Tseng's extragradient method for solving pseudomonotone variational inequalities in reflexive Banach spaces

An inertial‐like Tseng's extragradient method for solving pseudomonotone variational inequalities in reflexive Banach spaces

Stability analysis for set-valued inverse mixed variational inequalities in reflexive Banach spaces

This work is devoted to the analysis for a new class of set-valued inverse mixed variational inequalities (SIMVIs) in reflexive Banach spaces, when both the mapping and the constraint set are perturbed simultaneously by two parameters. Several equivalence characterizations are given for SIMVIs to have nonempty and bounded solution sets. Based on the equivalence conditions, under the premise of monotone mappings, the stability result for the SIMVIs is obtained in the reflexive Banach space. Furthermore, to illustrate the results, an example of the traffic network equilibrium control problem is provided at the end of this paper. The results presented in this paper generalize and extend some known results in this area.

An improved subgradient extragradient method with two different parameters for solving variational inequalities in reflexive Banach spaces

An improved subgradient extragradient method with two different parameters for solving variational inequalities in reflexive Banach spaces

One-step Bregman projection methods for solving variational inequalities in reflexive Banach spaces

ABSTRACT We study two Bregman projection methods for solving variational inequality problems in real reflexive Banach spaces. Our methods have simple and elegant structures, and they require only one Bregman projection onto the feasible set and one evaluation of the cost operator at each iteration. We prove that these methods converge weakly when the cost operator is pseudomonotone on the entire space and that they converge strongly when the cost operator is strongly pseudomonotone only on the feasible set. Extensions of our proposed methods to mixed variational inequality problems are also given. Finally, we consider some examples regarding the implementation of our methods in comparisons with known methods in the literature.

A class of elliptic quasi-variational–hemivariational inequalities with applications

In this paper we study a class of quasi-variational–hemivariational inequalities in reflexive Banach spaces. The inequalities contain a convex potential, a locally Lipschitz superpotential, and a solution-dependent set of constraints. Solution existence and compactness of the solution set to the inequality problem are established based on the Kakutani–Ky Fan–Glicksberg fixed point theorem. Two examples of the interior and boundary semipermeability models illustrate the applicability of our results.

A generalized Stokes system with a non-smooth slip boundary condition.

A class of quasi variational-hemivariational inequalities in reflexive Banach spaces is studied. The inequalities contain a convex potential, a locally Lipschitz superpotential and an implicit obstacle set of constraints. Results on the well-posedness including existence, uniqueness, dependence of solution on the data and the compactness of the solution set in the strong topology are established. The applicability of the results is illustrated by the steady-state generalized Stokes model of a generalized Newtonian incompressible fluid with a non-monotone slip boundary condition. This article is part of the theme issue 'Non-smooth variational problems and applications'.

Regularization for a class of quasi-variational-hemivariational inequalities

The aim of the present paper is to study the solvability and regularization for a class of multivalued quasi-variational–hemivariational inequalities in reflexive Banach spaces. By applying the Kluge fixed point theorem and the Minty technique, we prove the solvability of the considered multivalued quasi-variational–hemivariational inequality, based on which some convergence results are obtained by introducing its regularization problem with the help of regularization operator. The applicability of the obtained abstract results is established by a mathematical model of a frictional contact problem with a class of elastic material, where the existence and stability results for the weak solution of contact problem are studied.

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

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.

Weak and strong convergence results for solving monotone variational inequalities in reflexive Banach spaces

In this paper, we introduce two modified Tseng's extragradient algorithms with a new generalized adaptive stepsize for solving monotone variational inequalities (VI) in reflexive Banach spaces. The advantage of our methods is that stepsizes do not require prior knowledge of the Lipschitz constant of the cost mapping. Based on Bregman projection-type methods, we prove weak and strong convergence of the proposed algorithms to a solution of VI. Some numerical experiments to show the efficiency of our methods including a comparison with related methods are provided.

A differential variational inequality in the study of contact problems with wear

We start with a mathematical model which describes the sliding contact of a viscoelastic body with a moving foundation. The contact is frictional and the wear of the contact surfaces is taken into account. We prove that this model leads to a differential variational inequality in which the unknowns are the displacement field and the wear function. Then, inspired by this model, we consider a general differential variational inequality in reflexive Banach spaces, governed by four parameters. We prove the unique solvability of the inequality as well as the continuous dependence of its solution with respect to the parameters. The proofs are based on arguments of monotonicity, compactness, convex analysis and lower semicontinuity. Then, we apply these abstract results to the mathematical model of contact for which we deduce the existence of a unique solution as well as the existence of optimal control for an associate optimal control problem. We also present the corresponding mechanical interpretations.

The Convergence Results of Differential Variational Inequality Problems

In this work, we suggest a differential variational inequality in reflexive Banach spaces and construct a sequence with a set of constraints and a penalty parameter. We use the penalty method to prove a unique solution to the problem and make suitable assumptions to prove the convergence of the sequence. The proof is based on arguments for compactness, symmetry, pseudomonotonicity, Mosco convergence, inverse strong monotonicity and Lipschitz continuity. Finally, we discuss the boundary value problem for the differential variational inequality problem as an application.

Levitin–Polyak well-posedness of variational–hemivariational inequalities

In this paper we consider the Levitin–Polyak well-posedness of a variational–hemivariational inequality in reflexive Banach spaces. First, we establish the equivalence between the Levitin–Polyak well-posedness of the considered inequality and the existence and uniqueness of its solution. Then, we discuss the relationships among the Levitin–Polyak well-posedness of the variational–hemivariational inequality, its corresponding inclusion problem and a constrained minimization problem. The proof of our results are based on arguments of monotonicity, convexity and the properties of the Clarke generalized gradient. Finally, we provide an application of our results in the study of a frictional contact problem with subdifferential boundary conditions.

Two New Inertial Algorithms for Solving Variational Inequalities in Reflexive Banach Spaces

The purpose of this paper is to introduce and analyze two inertial algorithms with self-adaptive stepsizes for solving variational inequalities in reflexive Banach spaces. Our algorithms are based on inertial hybrid and shrinking projection methods. Knowledge of the Lipschitz constant of the cost operator is not required. Under appropriate conditions, the strong convergence of the algorithms is established. We also present several numerical experiments which bring out the efficiency and the advantages of the proposed algorithms. Our work provides extensions of many known results from Hilbert spaces to reflexive Banach spaces.

A new class of elliptic quasi-variational-hemivariational inequalities for fluid flow with mixed boundary conditions

In this paper we study a class of quasi-variational-hemivariational inequalities in reflexive Banach spaces. The inequalities contain a convex potential, a locally Lipschitz superpotential, and a implicit obstacle set of constraints. Solution existence and compactness of the solution set to the inequality problem are established based on the Kakutani–Ky Fan fixed point theorem. The applicability of the results is illustrated by the steady-state Oseen model of a generalized Newtonian incompressible fluid with mixed boundary conditions. The latter involve a unilateral boundary condition, the Navier slip condition, a nonmonotone version of the nonlinear Navier-Fujita slip condition, and a generalization of the threshold slip and leak condition of frictional type.

Stability analysis of set-valued inverse variational inequalities in reflexive Banach spaces

Stability analysis of set-valued inverse variational inequalities in reflexive Banach spaces

Penalty Method for a Class of Differential Hemivariational Inequalities with Application

In this paper, we consider a penalty method for a class of differential hemivariational inequalities in reflexive Banach spaces, governed by a set of constraints. First, we recall the existence and uniqueness result of the differential hemivariational inequality. Further, we introduce a penalized problem without constraints and we prove that, as a penalty parameter tends to zero, the solution of the original inequality can be approached to the solution of the penalized problem. Finally, we illustrate our results by an example of a contact problem for which our abstract results can be applied.

A Bregman subgradient extragradient method with self-adaptive technique for solving variational inequalities in reflexive Banach spaces

In this paper, we introduce a self-adaptive Bregman subgradient extragradient method for solving variational inequalities in the framework of a reflexive Banach space. The step-adaptive strategy avoids the difficult task of choosing a stepsize based on the Lipschitz constant of the cost function of the variational inequalities and improves the performance of the algorithm. Moreover, the use of the Bregman distance technique allows the consideration of a general feasible set for the problem. Under some suitable conditions, we prove some weak and strong convergence results for the sequence generated by the algorithm without prior knowledge of the Lipschitz constant. We further provide an application to contact problems and some numerical experiments to illustrate the performance of the algorithm.

The Galerkin Method and Regularization for Variational Inequalities in Reflexive Banach Spaces

This paper studies the convergence of the Galerkin method and regularization for variational inequalities with pseudomonotone operators in the sense of Brezis. Namely, we prove that under certain conditions, the solutions of the Galerkin equations and regularized variational inequalities converge strongly to a solution of the original variational inequality in reflexive Banach spaces. An application for obstacle problems is given.

Single Bregman projection method for solving variational inequalities in reflexive Banach spaces

In this paper, we introduce a single projection method with the Bregman distance technique for solving pseudomonotone variational inequalities in a real reflexive Banach space. The algorithm is designed such that its step size is determined by a self-adaptive process and there is only one computation of projection per iteration during implementation. This improves the convergence of the method and also avoids the need for choosing a suitable estimate of the Lipschitz constant of the cost function which is very difficult in practice. We prove some weak and strong convergence results under suitable conditions on the cost operator. We also provide some numerical experiments to illustrate the performance and efficiency of the proposed method.

Solvability and algorithms of generalized nonlinear variational-like inequalities in reflexive Banach spaces

This paper deals with solvability and algorithms for a new class of generalized nonlinear variational-like inequalities in reflexive Banach spaces. By employing the Banach’s fixed point theorem, Schauder’s fixed point theorem, and FanKKM theorem, we obtain a sufficient condition which guarantees the existence of solutions for the generalized nonlinear variational-like inequality. We introduce also an auxiliary variational-like inequality and, by utilizing the minimax inequality, get the existence and uniqueness of solutions for the auxiliary variational-like inequality, which is used to suggest an iterative algorithm for solving the generalized nonlinear variational-like inequality. Under certain conditions, by means of the auxiliary principle technique, we both establish the existence and uniqueness of solutions for the generalized nonlinear variational-like inequality and discuss the convergence of iterative sequences generated by the iterative algorithm. Our results extend, improve, and unify several known results in the literature.