scholarly journals Multivalued nonmonotone dynamic boundary condition

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Khadija Aayadi ◽  
Khalid Akhlil ◽  
Sultana Ben Aadi ◽  
Mourad El Ouali
Keyword(s):  
Galerkin Approximation ◽  
Hemivariational Inequalities ◽  
New Class ◽  
Dynamic Boundary ◽  
Approximation Sequence ◽  
Sign Conditions ◽  
The Laplace Operator ◽  
Clarke Subdifferentials ◽  
Standard Techniques ◽  
General Sign

AbstractIn this paper, we introduce a new class of hemivariational inequalities, called dynamic boundary hemivariational inequalities, reflecting the fact that the governing operator is also active on the boundary. In our context, it concerns the Laplace operator with Wentzell (dynamic) boundary conditions perturbed by a multivalued nonmonotone operator expressed in terms of Clarke subdifferentials. We show that one can reformulate the problem so that standard techniques can be applied. We use the well-established theory of boundary hemivariational inequalities to prove that under growth and general sign conditions, the dynamic boundary hemivariational inequality admits a weak solution. Moreover, in the situation where the functionals are expressed in terms of locally bounded integrands, a “filling in the gaps” procedure at the discontinuity points is used to characterize the subdifferential on the product space. Finally, we prove that, under a growth condition and eventually smallness conditions, the Faedo–Galerkin approximation sequence converges to a desired solution.

Numerical analysis of history-dependent variational–hemivariational inequalities with applications to contact problems

2015 ◽  
Vol 26 (4) ◽  
pp. 427-452 ◽  
Author(s):  
MIRCEA SOFONEA ◽  
WEIMIN HAN ◽  
STANISŁAW MIGÓRSKI
Keyword(s):  
Numerical Analysis ◽  
Real World ◽  
Viscoelastic Body ◽  
Contact Problems ◽  
Uniqueness Result ◽  
Constitutive Law ◽  
Hemivariational Inequalities ◽  
New Class ◽  
Nonlinear Anal ◽  
Dependence Result

A new class of history-dependent variational–hemivariational inequalities was recently studied in Migórski et al. (2015Nonlinear Anal. Ser. B: Real World Appl.22, 604–618). There, an existence and uniqueness result was proved and used in the study of a mathematical model which describes the contact between a viscoelastic body and an obstacle. The aim of this paper is to continue the analysis of the inequalities introduced in Migórski et al. (2015Nonlinear Anal. Ser. B: Real World Appl.22, 604–618) and to provide their numerical analysis. We start with a continuous dependence result. Then we introduce numerical schemes to solve the inequalities and derive error estimates. We apply the results to a quasistatic frictional contact problem in which the material is modelled with a viscoelastic constitutive law, the contact is given in the form of normal compliance, and friction is described with a total slip-dependent version of Coulomb's law.

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 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 Existence of Positive Weak Solutions for a New Class of p,q Laplacian Nonlinear Elliptic System with Sign-Changing Weights

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-6
Author(s):  
Rafik Guefaifia ◽  
Salah Mahmoud Boulaaras ◽  
Sultan Alodhaibi ◽  
Salem Alkhalaf
Keyword(s):  
Elliptic System ◽  
Positive Solutions ◽  
Weak Solutions ◽  
Nonlinear Elliptic System ◽  
Bounded Domains ◽  
New Class ◽  
Nonlinear Elliptic ◽  
Sign Conditions

In this paper, by using subsuper solutions method, we study the existence of weak positive solutions for a new class of p,q Laplacian nonlinear elliptic system in bounded domains, when ax, bx,αx, and βx are sign-changing functions that maybe negative near the boundary, without assuming sign conditions on f0,g0,h0, and γ0.

scholarly journals Trifunction Bihemivariational Inequalities

2021 ◽  
pp. 287-313
Author(s):  
Muhammad Aslam Noor ◽  
Khalida Inayat Noor
Keyword(s):  
Convergence Analysis ◽  
Iterative Methods ◽  
Hemivariational Inequalities ◽  
Auxiliary Principle ◽  
Auxiliary Principle Technique ◽  
New Class ◽  
Mild Conditions ◽  
Special Cases

In this paper, we consider a new class of hemivariational inequalities, which is called the trifunction bihemivariational inequality. We suggest and analyze some iterative methods for solving the trifunction bihemivariational inequality using the auxiliary principle technique. The convergence analysis of these iterative methods is also considered under some mild conditions. Several special cases are also considered. Results proved in this paper can be viewed as a refinement and improvement of the known results.

scholarly journals Analysis of Stokes system with solution-dependent subdifferential boundary conditions

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Jing Zhao ◽  
Stanisław Migórski ◽  
Sylwia Dudek
Keyword(s):  
Boundary Conditions ◽  
Convex Sets ◽  
Stokes Problem ◽  
Weak Form ◽  
Nonlinear Boundary Conditions ◽  
Slip Condition ◽  
Hemivariational Inequalities ◽  
Navier Slip ◽  
New Class ◽  
Navier Slip Condition

AbstractWe study the Stokes problem for the incompressible fluid with mixed nonlinear boundary conditions of subdifferential type. The latter involve a unilateral boundary condition, the Navier slip condition, a nonmonotone version of the nonlinear Navier–Fujita slip condition, and the threshold slip and leak condition of frictional type. The weak form of the problem leads to a new class of variational–hemivariational inequalities on convex sets for the velocity field. Solution existence and the weak compactness of the solution set to the inequality problem are established based on the Schauder fixed point theorem.

scholarly journals Numerical approximation of the averaged controllability for the wave equation with unknown velocity of propagation

2021 ◽  
Author(s):  
Carlos Castro ◽  
Mouna Abdelli
Keyword(s):  
Wave Equation ◽  
Control Problem ◽  
Wave Equations ◽  
Dimensional Space ◽  
Galerkin Approximation ◽  
Convergence Result ◽  
Discrete Control ◽  
Finite Dimensional Space ◽  
Finite Dimensional ◽  
The Laplace Operator

We propose a numerical method to approximate the exact averaged boundary control of a family of wave equations depending on an unknown parameter $\sigma$. More precisely the control, independent of $\sigma$, that drives an initial data to a family of final states at time $t=T$, whose average in $\sigma$ is given. The idea is to project the control problem in the finite dimensional space generated by the first $N$ eigenfunctions of the Laplace operator. When applied to a single (nonparametric) wave equation, the resulting discrete control problem turns out to be equivalent to the Galerkin approximation proposed by F. Bourquin et al. in reference [2]. We give a convergence result of the discrete controls to the continuous one. The method is illustrated with several examples in 1-d and 2-d in a square domain and allows us to give some conjectures on the averaged controllability for the continuous problem.

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