On the differential variational inequalities of parabolic-elliptic type

Abstract

We consider a model of infinite dimensional differential variational inequalities formulated by a parabolic differential inclusion and an elliptic variational inequality. The existence of global solution and global attractor for the semiflow governed by our system is proved by using measure of noncompactness. Copyright © 2017 John Wiley & Sons, Ltd.

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

“…Based on this motivation, Chen and Wang [2] in 2014 used the idea of DVIs to investigate a dynamic Nash equilibrium problem of multiple players with shared constraints and dynamic decision processes; Wang et al [3] proved an existence theorem for Carathéodory weak solutions of a differential quasi-variational inequality in finite dimensional Euclidean spaces and established a convergence result on the Euler time-dependent procedure for solving the initialvalue differential set-valued variational inequalities; and Migórski et al [4] used the surjectivity of setvalued pseudomonotone operators combined with a fixed point principle to prove the unique solvability of a history-dependent DVI, and then they used the abstract frameworks to study a history-dependent frictional viscoelastic contact problem with a generalized Signorini contact condition. For more details on these topics, the reader is welcome to refer to [5][6][7][8][9][10][11][12][13][14][15][16][17][18] and references therein. More recently, Liu et al [19] initially introduced the notion of differential hemivariational inequalities (DHVIs), which is a generalization of variational inequalities.…”

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

“…for all t ∈ [0, T ]. This, together with convergence (28) and the Lebesgue dominated convergence theorem, implies…”

“…They arise in many applications: electrical circuits with ideal diodes, Coulomb friction problems for contacting bodies, economical dynamics, dynamic traffic networks. The most representative results are as follows: Loi [20] applied the method of integral guiding functions to explore a multi-parameter global bifurcation theorem for differential inclusions with the periodic condition and then employed the abstract results to the study of the two-parameter global bifurcation of periodic solutions for a class of differential variational inequalities in Euclidean spaces; Liu-Zeng-Motreanu [13,17,18] and Liu-Migórski-Zeng [16] proved the existence of solutions for a class of differential mixed variational inequalities in Banach spaces through applying the theory of semigroups, Filippov implicit function lemma and fixed point theorems for condensing multivalued operators; Chen-Wang [1] in 2014 used the idea of DVIs to investigate a dynamic Nash equilibrium problem of multiple players with shared constraints and dynamic decision processes; Nguyen-Tran [28] considered a model of infinite dimensional differential variational inequalities formulated by a parabolic differential inclusion and an elliptic variational inequality, and utilized the theory of measure of noncompactness to prove the existence of global solutions as well as global attractor for the semi-flow governed by the differential variational inequality. For more details on these topics the reader is welcome to consult [2,6,8,10,11,14,15,21,22,24,34,35] and the references therein.…”

Investigation of the shapes of cuts in a plate in contact with a rigid stamp

“…With the development of information technologies, numerical methods, implemented through computer programs, have become relevant in contact problems. A huge number of works have been published on numerical modeling and its application in problems of dynamic contact, for example, [6][7][8][9][10][11][12].…”