A note on the regularity of the solutions to two variational inequalities involving a pseudodifferential operator.
A penalty approach for a box constrained variational inequality problem
We propose a penalty approach for a box constrained variational inequality problem . This problem is replaced by a sequence of nonlinear equations containing a penalty term. We show that if the penalty parameter tends to infinity, the solution of this sequence converges to that of when the function involved is continuous and strongly monotone and the box contains the origin. We develop the algorithmic aspect with theoretical arguments properly established. The numerical results tested on...
A Perturbed Iterative Method for a General Class of Variational Inequalities
The generalized Wiener-Hopf equation and the approximation methods are used to propose a perturbed iterative method to compute the solutions of a general class of nonlinear variational inequalities.
A principle of linearization in theory of stability of solutions of variational inequalities
It is shown that the uniform exponential stability and the uniform stability at permanently acting disturbances of a sufficiently smooth but not necessarily steady-state solution of a general variational inequality is a consequence of the uniform exponential stability of a zero solution of another (so called linearized) variational inequality.
A quasistatic bilateral contact problem with adhesion and friction for viscoelastic materials
We consider a mathematical model which describes a contact problem between a deformable body and a foundation. The contact is bilateral and is modelled with Tresca's friction law in which adhesion is taken into account. The evolution of the bonding field is described by a first order differential equation and the material's behavior is modelled with a nonlinear viscoelastic constitutive law. We derive a variational formulation of the mechanical problem and prove the existence and uniqueness result...
A quasistatic bilateral contact problem with friction for nonlinear elastic materials.
A quasistatic contact problem with unilateral constraint and slip-dependent friction
We consider a mathematical model of a quasistatic contact between an elastic body and an obstacle. The contact is modelled with unilateral constraint and normal compliance, associated to a version of Coulomb's law of dry friction where the coefficient of friction depends on the slip displacement. We present a weak formulation of the problem and establish an existence result. The proofs employ a time-discretization method, compactness and lower semicontinuity arguments.
A quasistatic frictional contact problem with adhesion for nonlinear elastic materials.
A quasistatic unilateral contact problem with slip-dependent coefficient of friction for nonlinear elastic materials.
A study of variational inequalities for set-valued mappings.
A superquadratic method for solving variational inclusions under weak conditions.
A variational approach to bifurcation in reaction-diffusion systems with Signorini type boundary conditions
We consider a simple reaction-diffusion system exhibiting Turing's diffusion driven instability if supplemented with classical homogeneous mixed boundary conditions. We consider the case when the Neumann boundary condition is replaced by a unilateral condition of Signorini type on a part of the boundary and show the existence and location of bifurcation of stationary spatially non-homogeneous solutions. The nonsymmetric problem is reformulated as a single variational inequality with a potential...
A variational approach to bifurcation points of a reaction-diffusion system with obstacles and Neumann boundary conditions
Given a reaction-diffusion system which exhibits Turing's diffusion-driven instability, the influence of unilateral obstacles of opposite sign (source and sink) on bifurcation and critical points is studied. In particular, in some cases it is shown that spatially nonhomogeneous stationary solutions (spatial patterns) bifurcate from a basic spatially homogeneous steady state for an arbitrarily small ratio of diffusions of inhibitor and activator, while a sufficiently large ratio is necessary in the...
A Viscoelastic Frictionless Contact Problem with Adhesion
We consider a mathematical model which describes the equilibrium between a viscoelastic body in frictionless contact with an obstacle. The contact is modelled with normal compliance, associated with Signorini's conditions and adhesion. The adhesion is modelled with a surface variable, the bonding field, whose evolution is described by a first-order differential equation. We establish a variational formulation of the mechanical problem and prove the existence and uniqueness of the weak solution....
A viscosity hybrid steepest descent method for generalized mixed equilibrium problems and variational inequalities for relaxed cocoercive mapping in Hilbert spaces.
Abstract quasi-variational inequalities of elliptic type and applications
A class of quasi-variational inequalities (QVI) of elliptic type is studied in reflexive Banach spaces. The concept of QVI was earlier introduced by A. Bensoussan and J.-L. Lions [2] and its general theory has been developed by many mathematicians, for instance, see [6, 7, 9, 13] and a monograph [1]. In this paper we give a generalization of the existence theorem established in [14]. In our treatment we employ the compactness method along with a concept of convergence of nonlinear multivalued operators...
Abstract theory of variational inequalities with Lagrange multipliers and application to nonlinear PDEs
Recently, we established some generalizations of the theory of Lagrange multipliers arising from nonlinear programming in Banach spaces, which enable us to treat not only elliptic problems but also parabolic problems in the same generalized framework. The main objective of the present paper is to discuss a typical time-dependent double obstacle problem as a new application of the above mentioned generalization. Actually, we describe it as a usual parabolic variational inequality and then characterize...
Acceleration of Convergence in Dontchev’s Iterative Method for Solving Variational Inclusions
2000 Mathematics Subject Classification: 47H04, 65K10.In this paper we investigate the existence of a sequence (xk ) satisfying 0 ∈ f (xk )+ ∇f (xk )(xk+1 − xk )+ 1/2 ∇2 f (xk )(xk+1 − xk )^2 + G(xk+1 ) and converging to a solution x∗ of the generalized equation 0 ∈ f (x) + G(x); where f is a function and G is a set-valued map acting in Banach spaces.
An algorithm based on resolvent operators for solving positively semidefinite variational inequalities. (An algorithm based on resolvant operators for solving positively semidefinite variational inequalities.)
An approximation of solutions of variational inequalities.