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 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.

Hemivariational inverse problem for contact problem with locking materials

2021 ◽  
Vol 8 (4) ◽  
pp. 665-677
Author(s):  
Z. Faiz ◽  
◽  
O. Baiz ◽  
H. Benaissa ◽  
D. El Moutawakil ◽  
...  
Keyword(s):  
Inverse Problem ◽  
Contact Problem ◽  
Frictional Contact ◽  
Deformable Body ◽  
Lipschitz Mapping ◽  
Hemivariational Inequalities ◽  
Existence Of A Solution ◽  
Locally Lipschitz ◽  
Uniqueness Of The Solution ◽  
Dependence Result

The aim of this work is to study an inverse problem for a frictional contact model for locking material. The deformable body consists of electro-elastic-locking materials. Here, the locking character makes the solution belong to a convex set, the contact is presented in the form of multivalued normal compliance, and frictions are described with a sub-gradient of a locally Lipschitz mapping. We develop the variational formulation of the model by combining two hemivariational inequalities in a linked system. The existence and uniqueness of the solution are demonstrated utilizing recent conclusions from hemivariational inequalities theory and a fixed point argument. Finally, we provided a continuous dependence result and then we established the existence of a solution to an inverse problem for piezoelectric-locking material frictional contact problem.

scholarly journals Existence results for general inequality problems with constraints

2003 ◽  
Vol 2003 (10) ◽  
pp. 601-619 ◽  
Author(s):  
George Dincă ◽  
Petru Jebelean ◽  
Dumitru Motreanu
Keyword(s):  
Banach Space ◽  
Lower Semicontinuous ◽  
Directional Derivative ◽  
Existence Results ◽  
Laplacian Operator ◽  
Hemivariational Inequalities ◽  
General Inequality ◽  
Locally Lipschitz ◽  
Proper Convex ◽  
Generalized Directional Derivative

This paper is concerned with existence results for inequality problems of typeF0(u;v)+Ψ′(u;v)≥0, for allv∈X, whereXis a Banach space,F:X→ℝis locally Lipschitz, andΨ:X→(−∞+∞]is proper, convex, and lower semicontinuous. HereF0stands for the generalized directional derivative ofFandΨ′denotes the directional derivative ofΨ. The applications we consider focus on the variational-hemivariational inequalities involving thep-Laplacian operator.

scholarly journals On Neumann hemivariational inequalities

2002 ◽  
Vol 7 (2) ◽  
pp. 103-112 ◽  
Author(s):  
Halidias Nikolaos
Keyword(s):  
Critical Point ◽  
Nontrivial Solution ◽  
Critical Point Theory ◽  
Mountain Pass Theorem ◽  
Point Theory ◽  
Mountain Pass ◽  
Hemivariational Inequality ◽  
Hemivariational Inequalities ◽  
Locally Lipschitz

We derive a nontrivial solution for a Neumann noncoercive hemivariational inequality using the critical point theory for locally Lipschitz functionals. We use the Mountain-Pass theorem due to Chang (1981).

Approximate controllability for stochastic fractional hemivariational inequalities of degenerate type

2020 ◽  
Vol 23 (5) ◽  
pp. 1506-1531
Author(s):  
Yatian Pei ◽  
Yong-Kui Chang
Keyword(s):  
Approximate Controllability ◽  
Directional Derivative ◽  
Inverse Operator ◽  
Sufficient Conditions ◽  
Uniform Boundedness ◽  
Hemivariational Inequalities ◽  
Sobolev Type ◽  
Resolvent Family ◽  
Locally Lipschitz ◽  
Degenerate Type

Abstract This paper is mainly concerned with stochastic fractional hemivariational inequalities of degenerate (or Sobolev) type in Caputo and Riemann-Liouville derivatives with order (1, 2), respectively. Based upon some properties of fractional resolvent family and generalized directional derivative of a locally Lipschitz function, some sufficient conditions are established for the existence and approximate controllability of the aforementioned systems. Particularly, the uniform boundedness for some nonlinear terms, the existence and compactness of certain inverse operator are not necessarily needed in obtained approximate controllability results.

scholarly journals Some hemivariational inequalities in the Euclidean space

2019 ◽  
Vol 9 (1) ◽  
pp. 958-977 ◽  
Author(s):  
Giovanni Molica Bisci ◽  
Dušan Repovš
Keyword(s):  
Weak Solution ◽  
Euclidean Space ◽  
Sobolev Space ◽  
Hemivariational Inequalities ◽  
Existence Of Weak Solutions ◽  
Lipschitz Continuous ◽  
Variational Structure ◽  
Locally Lipschitz ◽  
Sign Changing Solutions ◽  
Continuous Functionals

Abstract The purpose of this paper is to study the existence of weak solutions for some classes of hemivariational problems in the Euclidean space ℝd (d ≥ 3). These hemivariational inequalities have a variational structure and, thanks to this, we are able to find a non-trivial weak solution for them by using variational methods and a non-smooth version of the Palais principle of symmetric criticality for locally Lipschitz continuous functionals, due to Krawcewicz and Marzantowicz. The main tools in our approach are based on appropriate theoretical arguments on suitable subgroups of the orthogonal group O(d) and their actions on the Sobolev space H1(ℝd). Moreover, under an additional hypotheses on the dimension d and in the presence of symmetry on the nonlinear datum, the existence of multiple pairs of sign-changing solutions with different symmetries structure has been proved. In connection to classical Schrödinger equations a concrete and meaningful example of an application is presented.

scholarly journals An existence theorem for nonlinear hemivariational inequalities at resonance

2001 ◽  
Vol 63 (1) ◽  
pp. 1-14 ◽  
Author(s):  
Leszek Gasiński ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Existence Theorem ◽  
Nontrivial Solution ◽  
Mountain Pass Theorem ◽  
Existence Results ◽  
Mountain Pass ◽  
Hemivariational Inequality ◽  
Hemivariational Inequalities ◽  
Compactness Condition ◽  
At Resonance ◽  
Recent Theorem

We consider a nonlinear hemivariational inequality with the p-Laplacian at resonance. Using an extension of the nonsmooth mountain pass theorem of Chang, which makes use of the Cerami compactness condition, we prove the existence of a nontrivial solution. Our existence results here extends a recent theorem on resonant hemivariational inequalities, by the authors in 1999.

Solutions and multiple solutions for quasilinear hemivariational inequalities at resonance

2001 ◽  
Vol 131 (5) ◽  
pp. 1091-1111 ◽  
Author(s):  
Leszek Gasiński ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Variational Principle ◽  
Critical Point Theory ◽  
Multiple Solutions ◽  
Existence Theorems ◽  
Point Theory ◽  
Hemivariational Inequalities ◽  
Locally Lipschitz Functions ◽  
Strong Resonance ◽  
At Resonance ◽  
Locally Lipschitz

In this paper we consider quasilinear hemivariational inequalities at resonance. We obtain existence theorems using Landesman-Lazer-type conditions and multiplicity theorems for problems with strong resonance at infinity. Our method of proof is based on the non-smooth critical point theory for locally Lipschitz functions and on a generalized version of the Ekeland variational principle.

Equivalence of Some Multi-Valued Elliptic Variational Inequalities and Variational-Hemivariational Inequalities

2011 ◽  
Vol 11 (2) ◽  
Author(s):  
Siegfried Carl
Keyword(s):  
Variational Inequalities ◽  
Lipschitz Functions ◽  
Hemivariational Inequalities ◽  
Locally Lipschitz Functions ◽  
Generalized Gradient ◽  
Elliptic Variational Inequalities ◽  
Comparison Results ◽  
Locally Lipschitz ◽  
Additional Regularity ◽  
Definition Of

AbstractFirst, we prove existence and comparison results for multi-valued elliptic variational inequalities involving Clarke’s generalized gradient of some locally Lipschitz functions as multi-valued term. Only by applying the definition of Clarke’s gradient it is well known that any solution of such a multi-valued elliptic variational inequality is also a solution of a corresponding variational-hemivariational inequality. The reverse is known to be true if the locally Lipschitz functions are regular in the sense of Clarke. Without imposing this kind of regularity the equivalence of the two problems under consideration is not clear at all. The main goal of this paper is to show that the equivalence still holds true without any additional regularity, which will fill a gap in the literature. Existence and comparison results for both multi-valued variational inequalities and variational-hemivariational inequalities are the main tools in the proof of the equivalence of these problems.

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.

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