Existence of Solutions for Variational-Like Hemivariational Inequalities Involving Lower Semicontinuous Maps

2015 ◽  
pp. 67-84
Author(s):  
Guo-ji Tang ◽  
Zhong-bao Wang ◽  
Nan-jing Huang
Keyword(s):  
Existence Of Solutions ◽  
Lower Semicontinuous ◽  
Hemivariational Inequalities

scholarly journals Connected perimeter of planar sets

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
François Dayrens ◽  
Simon Masnou ◽  
Matteo Novaga ◽  
Marco Pozzetta
Keyword(s):  
Minimization Problem ◽  
Existence Of Solutions ◽  
Lower Semicontinuous ◽  
Representation Formula ◽  
Steiner Trees ◽  
Lower Semicontinuous Envelope ◽  
Simply Connected

AbstractWe introduce a notion of connected perimeter for planar sets defined as the lower semicontinuous envelope of perimeters of approximating sets which are measure-theoretically connected. A companion notion of simply connected perimeter is also studied. We prove a representation formula which links the connected perimeter, the classical perimeter, and the length of suitable Steiner trees. We also discuss the application of this notion to the existence of solutions to a nonlocal minimization problem with connectedness constraint.

scholarly journals A-monotonicity and applications to nonlinear variational inclusion problems

2004 ◽  
Vol 2004 (2) ◽  
pp. 193-195 ◽  
Author(s):  
Ram U. Verma
Keyword(s):  
Existence Of Solutions ◽  
Variational Inclusion ◽  
Hemivariational Inequalities ◽  
Energy Functions ◽  
Inclusion Problems ◽  
Constrained Problems ◽  
Nonconvex Energy

A new notion of the A-monotonicity is introduced, which generalizes the H-monotonicity. Since the A-monotonicity originates from hemivariational inequalities, and hemivariational inequalities are connected with nonconvex energy functions, it turns out to be a useful tool proving the existence of solutions of nonconvex constrained problems as well.

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 Evolutionary Quasi-Variational-Hemivariational Inequalities I: Existence and Optimal Control

2021 ◽  
Author(s):  
Shengda Zeng ◽  
Dumitru Motreanu ◽  
Akhtar A. Khan
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Variational Problem ◽  
Existence Of Solutions ◽  
Hemivariational Inequalities ◽  
Central Idea ◽  
Monotone Map ◽  
Continuity Properties ◽  
Sequential Compactness

AbstractWe study a nonlinear evolutionary quasi–variational–hemivariational inequality (in short, (QVHVI)) involving a set-valued pseudo-monotone map. The central idea of our approach consists of introducing a parametric variational problem that defines a variational selection associated with (QVHVI). We prove the solvability of the parametric variational problem by employing a surjectivity theorem for the sum of operators, combined with Minty’s formulation and techniques from the nonsmooth analysis. Then, an existence theorem for (QVHVI) is established by using Kluge’s fixed point theorem for set-valued operators. As an application, an abstract optimal control problem for the (QVHVI) is investigated. We prove the existence of solutions for the optimal control problem and the weak sequential compactness of the solution set via the Weierstrass minimization theorem and the Kuratowski-type continuity properties.

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.

Existence theorems for generalized quasi-variational hemivariational inequalities

2008 ◽  
Vol 45 (4) ◽  
pp. 443-449
Author(s):  
Ji-an Wang ◽  
Yunhua Ou
Keyword(s):  
Existence Of Solutions ◽  
Existence Theorems ◽  
Hemivariational Inequalities ◽  
Pseudomonotone Operators

The purpose of this paper is to study the existence of solutions for generalized quasivariational hemivariational inequalities with multivalued, discontinuous pseudomonotone operators. We obtain results which generalize and extend previously known theorems.

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