scholarly journals Partial differential hemivariational inequalities

2018 ◽  
Vol 7 (4) ◽  
pp. 571-586 ◽  
Author(s):  
Zhenhai Liu ◽  
Shengda Zeng ◽  
Dumitru Motreanu
Keyword(s):  
Evolution Equations ◽  
Existence Of Solutions ◽  
Mild Solutions ◽  
Upper Semicontinuity ◽  
Solution Set ◽  
Hemivariational Inequality ◽  
Hemivariational Inequalities ◽  
Well Posedness ◽  
Partial Differential ◽  
New Class

AbstractThe 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.

scholarly journals Constrained Variational-Hemivariational Inequalities on Nonconvex Star-Shaped Sets

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1824
Author(s):  
Stanisław Migórski ◽  
Long Fengzhen
Keyword(s):  
Directional Derivative ◽  
Upper Semicontinuity ◽  
Hemivariational Inequality ◽  
Hemivariational Inequalities ◽  
Strong Monotonicity ◽  
Monotonicity Condition ◽  
Elliptic Type ◽  
Existence Of A Solution ◽  
Locally Lipschitz ◽  
Clarke Subgradient

In this paper, we study a class of constrained variational-hemivariational inequality problems with nonconvex sets which are star-shaped with respect to a certain ball in a reflexive Banach space. The inequality is a fully nonconvex counterpart of the variational-hemivariational inequality of elliptic type since it contains both, a convex potential and a locally Lipschitz one. Two new results on the existence of a solution are proved by a penalty method applied to a variational-hemivariational inequality penalized by the generalized directional derivative of the distance function of the constraint set. In the first existence theorem, the strong monotonicity of the governing operator and a relaxed monotonicity condition of the Clarke subgradient are assumed. In the second existence result, these two hypotheses are relaxed and a suitable hypothesis on the upper semicontinuity of the operator is adopted. In both results, the penalized problems are solved by using the Knaster, Kuratowski, and Mazurkiewicz (KKM) lemma. For a suffciently small penalty parameter, the solution to the penalized problem solves also the original one. Finally, we work out an example on the interior and boundary semipermeability problem that ilustrate the applicability of our results.

scholarly journals On a Class of Variational-Hemivariational Inequalities Involving Upper Semicontinuous Set-Valued Mappings

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Guo-ji Tang ◽  
Zhong-bao Wang ◽  
Hong-ling Zhang
Keyword(s):  
Existence Of Solutions ◽  
Solution Set ◽  
Hemivariational Inequalities ◽  
Coercivity Conditions ◽  
Upper Semicontinuous ◽  
Set Valued Mappings

This paper is devoted to the various coercivity conditions in order to guarantee existence of solutions and boundedness of the solution set for the variational-hemivariational inequalities involving upper semicontinuous operators. The results presented in this paper generalize and improve some known results.

scholarly journals A new class of fractional impulsive differential hemivariational inequalities with an application

2022 ◽  
Vol 27 ◽  
pp. 1-22
Author(s):  
Yun-hua Weng ◽  
Tao Chen ◽  
Nan-jing Huang ◽  
Donal O'Regan
Keyword(s):  
Differential Equation ◽  
Impulsive Differential Equation ◽  
Convergence Result ◽  
Hemivariational Inequality ◽  
Hemivariational Inequalities ◽  
New Class ◽  
Perturbation Problem ◽  
Perturbation Data ◽  
The Stability ◽  
Stability Of The Solution

We consider a new fractional impulsive differential hemivariational inequality, which captures the required characteristics of both the hemivariational inequality and the fractional impulsive differential equation within the same framework. By utilizing a surjectivity theorem and a fixed point theorem we establish an existence and uniqueness theorem for such a problem. Moreover, we investigate the perturbation problem of the fractional impulsive differential hemivariational inequality to prove a convergence result, which describes the stability of the solution in relation to perturbation data. Finally, our main results are applied to obtain some new results for a frictional contact problem with the surface traction driven by the fractional impulsive differential equation.

scholarly journals Well-posedness for generalized variational-hemivariational inequalities with perturbations in reflexive banach spaces

2017 ◽  
Vol 48 (4) ◽  
pp. 345-364 ◽  
Author(s):  
Lu-Chuan Ceng ◽  
Yung-Yih Lur ◽  
Ching-Feng Wen
Keyword(s):  
Banach Spaces ◽  
Minimization Problem ◽  
Hemivariational Inequality ◽  
Inclusion Problem ◽  
Hemivariational Inequalities ◽  
Well Posedness ◽  
Reflexive Banach Spaces ◽  
Mild Conditions ◽  
Well Posed

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

scholarly journals Well-posedness analysis of a stationary Navier–Stokes hemivariational inequality

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Min Ling ◽  
Weimin Han
Keyword(s):  
Stokes Equations ◽  
Normal Component ◽  
Numerical Algorithms ◽  
Hemivariational Inequality ◽  
Navier Stokes ◽  
Hemivariational Inequalities ◽  
Well Posedness ◽  
Navier Stokes Equations ◽  
Solution Existence And Uniqueness ◽  
Stationary Navier Stokes Equations

AbstractThis paper provides a well-posedness analysis for a hemivariational inequality of the stationary Navier-Stokes equations by arguments of convex minimization and the Banach fixed point. The hemivariational inequality describes a stationary incompressible fluid flow subject to a nonslip boundary condition and a Clarke subdifferential relation between the total pressure and the normal component of the velocity. Auxiliary Stokes hemivariational inequalities that are useful in proving the solution existence and uniqueness of the Navier–Stokes hemivariational inequality are introduced and analyzed. This treatment naturally leads to a convergent iteration method for solving the Navier–Stokes hemivariational inequality through a sequence of Stokes hemivariational inequalities. Equivalent minimization principles are presented for the auxiliary Stokes hemivariational inequalities which will be useful in developing numerical algorithms.

scholarly journals Hilfer fractional evolution hemivariational inequalities with nonlocal initial conditions and optimal controls

2019 ◽  
Vol 24 (2) ◽  
pp. 189-209
Author(s):  
Yatian Pei ◽  
Yong-Kui Chang
Keyword(s):  
Control System ◽  
Initial Conditions ◽  
Mild Solutions ◽  
Sufficient Conditions ◽  
Hemivariational Inequality ◽  
Hemivariational Inequalities ◽  
Optimal Controls ◽  
Initial Condition ◽  
Uniqueness Of Solutions ◽  
Nonlocal Initial Condition

In this paper, we mainly consider a control system governed by a Hilfer fractional evolution hemivariational inequality with a nonlocal initial condition. We first establish sufficient conditions for the existence of mild solutions to the addressed control system via properties of generalized Clarke subdifferential and a fixed point theorem for condensing multivalued maps. Then we present the existence of optimal state-control pairs of the limited Lagrange optimal systems governed by a Hilfer fractional evolution hemivariational inequality with a nonlocal initial condition. The optimal control results are derived without uniqueness of solutions for the control system.

scholarly journals Well-Posedness by Perturbations for Variational-Hemivariational Inequalities

2012 ◽  
Vol 2012 ◽  
pp. 1-18
Author(s):  
Shu Lv ◽  
Yi-bin Xiao ◽  
Zhi-bin Liu ◽  
Xue-song Li
Keyword(s):  
Optimization Problem ◽  
Hemivariational Inequality ◽  
Inclusion Problem ◽  
Hemivariational Inequalities ◽  
Well Posedness

We generalize the concept of well-posedness by perturbations for optimization problem to a class of variational-hemivariational inequalities. We establish some metric characterizations of the well-posedness by perturbations for the variational-hemivariational inequality and prove their equivalence between the well-posedness by perturbations for the variational-hemivariational inequality and the well-posedness by perturbations for the corresponding inclusion problem.

scholarly journals Differential variational–hemivariational inequalities: existence, uniqueness, stability, and convergence

2020 ◽  
Vol 22 (4) ◽  
Author(s):  
Guo-ji Tang ◽  
Jinxia Cen ◽  
Van Thien Nguyen ◽  
Shengda Zeng
Keyword(s):  
Nonlinear Evolution Equation ◽  
Original Problem ◽  
Nonlinear Evolution ◽  
Convergence Result ◽  
Hemivariational Inequality ◽  
Stability And Convergence ◽  
Hemivariational Inequalities ◽  
Well Posedness ◽  
Kkm Principle ◽  
Stability Of The Solution

AbstractThe goal of this paper is to study a comprehensive system called differential variational–hemivariational inequality which is composed of a nonlinear evolution equation and a time-dependent variational–hemivariational inequality in Banach spaces. Under the general functional framework, a generalized existence theorem for differential variational–hemivariational inequality is established by employing KKM principle, Minty’s technique, theory of multivalued analysis, the properties of Clarke’s subgradient. Furthermore, we explore a well-posedness result for the system, including the existence, uniqueness, and stability of the solution in mild sense. Finally, using penalty methods to the inequality, we consider a penalized problem-associated differential variational–hemivariational inequality, and examine the convergence result that the solution to the original problem can be approached, as a parameter converges to zero, by the solution of the penalized problem.

scholarly journals On the well-posedness of differential quasi-variational-hemivariational inequalities

2020 ◽  
Vol 18 (1) ◽  
pp. 540-551 ◽  
Author(s):  
Jinxia Cen ◽  
Chao Min ◽  
Van Thien Nguyen ◽  
Guo-ji Tang
Keyword(s):  
Hilbert Spaces ◽  
Hemivariational Inequality ◽  
Hemivariational Inequalities ◽  
Well Posedness ◽  
Well Posed

Abstract The goal of this paper is to discuss the well-posedness and the generalized well-posedness of a new kind of differential quasi-variational-hemivariational inequality (DQHVI) in Hilbert spaces. Employing these concepts, we explore the essential relation between metric characterizations and the well-posedness of DQHVI. Moreover, the compactness of the set of solutions for DQHVI is delivered, when problem DQHVI is well-posed in the generalized sense.

scholarly journals Well-Posedness for a Class of Strongly Mixed Variational-Hemivariational Inequalities with Perturbations

2012 ◽  
Vol 2012 ◽  
pp. 1-21 ◽  
Author(s):  
Lu-Chuan Ceng ◽  
Ngai-Ching Wong ◽  
Jen-Chih Yao
Keyword(s):  
Minimization Problem ◽  
Hemivariational Inequality ◽  
The Other ◽  
Inclusion Problem ◽  
Hemivariational Inequalities ◽  
Well Posedness ◽  
Mild Conditions ◽  
Other Hand ◽  
Well Posed ◽  
Special Case

The concept of well-posedness for a minimization problem is extended to develop the concept of well-posedness for a class of strongly mixed variational-hemivariational inequalities with perturbations which includes as a special case the class of variational-hemivariational inequalities with perturbations. We establish some metric characterizations for the well-posed strongly mixed variational-hemivariational inequality and give some conditions under which the strongly mixed variational-hemivariational inequality is strongly well-posed in the generalized sense. On the other hand, it is also proven that under some mild conditions there holds the equivalence between the well posedness for a strongly mixed variational-hemivariational inequality and the well-posedness for the corresponding inclusion problem.

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