Pseudomonotone Operators Research Articles

Quasistatic evolution of damage in an elastic body

The processes of quasistatic evolution of the mechanical state of an elastic body, and the development of material damage which results from internal compression or tension, are modelled and analyzed. The problem is formulated as an elliptic system for the displacements coupled with a parabolic inclusion for the damage field. The existence of the unique local weak solution is established by using approximate problems, the recent theory of set-valued pseudomonotone operators, and a priori estimates.

Periodic solutions of nonlinear evolution equations with $$W_{\lambda _0 } $$ -pseudomonotone maps

We consider differential-operator equations with \(W_{\lambda _0 } \)-pseudomonotone operators. The problem of studying periodic solutions by the Faedo-Galerkin method is considered. Important a priori estimates are obtained. A topological description of resolvent operators is given.

Vector Quasi-Hemivariational Inequalities and Discontinuous Elliptic Systems

We develop an existence theory for hemivariational inequalities in vector-valued function spaces which involve pseudomonotone operators. The obtained abstract result is used to study quasilinear elliptic systems whose lower order coupling vector field depends discontinuously upon the solution vector. We provide conditions that allow the identification of regions of existence of solutions for such systems, so called trapping regions.

Local estimates and global existence for strongly nonlinear parabolic equations with locally integrable data

We obtain existence results for some strongly nonlinear Cauchy problems posed in \( {\mathbb{R}^{N} } \) and having merely locally integrable data. The equations we deal with have as principal part a bounded, coercive and pseudomonotone operator of Leray-Lions type acting on \( L^{P} {\left( {0,\,T;\,W^{{1,p}}_{{{\text{loc}}}} {\left( {\mathbb{R}^{N} } \right)}} \right)} \), they contain absorbing zero order terms and possibly include first order terms with natural growth. For any p > 1 and under optimal growth conditions on the zero order terms, we derive suitable local a-priori estimates and consequent global existence results.

Hemivariational inequality approach to the dynamic viscoelastic sliding contact problem with wear

The dynamic contact problem of sliding contact with wear between a viscoelastic body and a foundation is formulated as a hemivariational inequality. The contact is modeled with a general normal damped response condition which is nonmonotone, possibly multivalued and has the subdifferential form. We establish the existence of weak solutions by using a surjectivity result for a class of pseudomonotone operators. We also deliver conditions under which the solution of the evolution inclusion associated with the hemivariational inequality is unique.

Existence and Regularity for Dynamic Viscoelastic Adhesive Contact with Damage

A model for the dynamic process of frictionless adhesive contact between a viscoelastic body and a reactive foundation, which takes into account the damage of the material resulting from tension or compression, is presented. Contact is described by the normal compliance condition. Material damage is modelled by the damage field, which measures the pointwise fractional decrease in the load-carrying capacity of the material, and its evolution is described by a differential inclusion. The model allows for different damage rates caused by tension or compression. The adhesion is modelled by the bonding field, which measures the fraction of active bonds on the contact surface. The existence of the unique weak solution is established using the theory of set-valued pseudomonotone operators introduced by Kuttler and Shillor (1999). Additional regularity of the solution is obtained when the problem data is more regular and satisfies appropriate compatibility conditions.

Solutions of nonlinear evolution inclusions

In this paper, the solution of the nonlinear evolution inclusion problem of the form u ′ ( t ) + B ( t , u ( t ) ) ∋ f ( t ) is studied. In this problem, the operators are of type ( M ) or type ( S + ) , which are different from those of pseudo-monotone operators that had been studied by many authors. At the same time, we study the perturbation problem. In fact, many kinds of evolution equations can be generalized by this problem. The former results are improved and generalized by our conclusions, and we will give more applications.

Dynamic hemivariational inequality modeling viscoelastic contact problem with normal damped response and friction

In this article we examine an evolution problem, which describes the dynamic contact of a viscoelastic body and a foundation. The contact is modeled by a general normal damped response condition and a friction law, which are nonmonotone, possibly multivalued and have the subdifferential form. First we derive a formulation of the model in the form of a multidimensional hemivariational inequality. Then we establish a priori estimates and we prove the existence of weak solutions by using a surjectivity result for pseudomonotone operators. Finally, we deliver conditions under which the solution of the hemivariational inequality is unique.

Projection-proximal methods for general variational inequalities

In this paper, we consider and analyze some new projection-proximal methods for solving general variational inequalities. The modified methods converge for pseudomonotone operators which is a weaker condition than monotonicity. The proposed methods include several new and known methods as special cases. Our results can be considered as a novel and important extension of the previously known results. Since the general variational inequalities include the quasi-variational inequalities and implicit complementarity problems as special cases, results proved in this paper continue to hold for these problems.

Solvability of Unilateral Parabolic Problems and Variational Inequalities

Sufficient conditions are given for the existence of solutions of strong parabolic variational inequalities with nonlinear and multi-valued operators. Pseudomonotone operators are used. A new multi-valued analogue of the acute-angle lemma is used for the localization. The theory can be applied to problems with distributed parameters. We also consider conditions for the existence of strong generalized solutions of parabolic equations with unilateral boundary conditions with applications to control problems.

Boundary Hemivariational Inequalities of Hyperbolic Type and Applications

In this paper we examine two classes of nonlinear hyperbolic initial boundary value problems with nonmonotone multivalued boundary conditions characterized by the Clarke subdifferential. We prove two existence results for multidimensional hemivariational inequalities: one for the inequalities with relation between reaction and velocity and the other for the expressions containing the reaction–displacement law. The existence of weak solutions is established by using a surjectivity result for pseudomonotone operators and a priori estimates. We present also an example of dynamic viscoelastic contact problem in mechanics which illustrate the applicability of our results.

Existence results for evolution hemivariational inequalities of second order

In this paper, we consider evolution hemivariational inequalities of second order with a time-dependent pseudomonotone operator and nonmonotone multivalued perturbations. We present the existence of solutions for such inequality. The proof profits from a result on the surjectivity of operators of pseudomonotone type. We discuss some examples which indicate the practical importance of our theoretical findings.

Heat Conduction with Flux Condition on a Free Patch

A new free boundary or free patch problem for the heat equation is presented. In the problem a nonlinear heat flux condition is prescribed on a free portion of the boundary, the patch, the position of which depends on the solution. The existence of a weak solution is established using the theory of set-valued pseudomonotone operators.

Generalized quasi-variational hemi-variational inequalities

This paper is devoted to the existence of solutions for generalized quasi-variational hemi-variational inequalities with multivalued, discontinuous pseudomonotone operators. We obtain results which generalize and extend previously known theorems.

Existence for a thermoviscoelastic beam model of brakes

The existence of a weak solution to a model for the dynamic thermomechanical behavior of a viscoelastic beam, which is in frictional contact with a rigid rotating wheel, is established. The model describes a simple braking system in which a rotating wheel comes to a stop as a result of the frictional traction generated by the beam. The classical model consists of a system of coupled equations for the beam temperature and displacement, the wear of the beam's contacting end, the wheel temperature and its angular velocity. The weak formulation is an abstract differential inclusion involving set-valued pseudomonotone operators, The existence is proved by using recent results for such operators. Uniqueness is shown to hold when the wheel's angular velocity and temperature are known.

Boundary hemivariational inequality of parabolic type*1

In this paper we consider a parabolic hemivariational inequality with a nonmonotone multivalued boundary condition and with a time-dependent pseudomonotone operator. We establish the existence of solutions for such inequality. Our proof is based on a known surjectivity result for operators of pseudomonotone type. We present an example of semipermeability heat conduction problem to which the result applies.

Boundary hemivariational inequality of parabolic type

In this paper we consider a parabolic hemivariational inequality with a nonmonotone multivalued boundary condition and with a time-dependent pseudomonotone operator. We establish the existence of solutions for such inequality. Our proof is based on a known surjectivity result for operators of pseudomonotone type. We present an example of semipermeability heat conduction problem to which the result applies.

Convergent Algorithm Based on Progressive Regularization for Solving Pseudomonotone Variational Inequalities

In this paper, we extend the Moreau-Yosida regularization of monotone variational inequalities to the case of weakly monotone and pseudomonotone operators. With these properties, the regularized operator satisfies the pseudo-Dunn property with respect to any solution of the variational inequality problem. As a consequence, the regularized version of the auxiliary problem algorithm converges. In this case, when the operator involved in the variational inequality problem is Lipschitz continuous (a property stronger than weak monotonicity) and pseudomonotone, we prove the convergence of the progressive regularization introduced in Refs. 1, 2.

A NEW CLASS OF DOUBLY NONLINEAR EVOLUTION EQUATIONS

In this paper we consider the solvability for a new class of doubly nonlinear evolution equations. The motivation of this work comes from a transmission problem of two degenerate parabolic equations with convection term, in which the transmission boundary is time-dependent. We give an abstract existence result, and show that the weak variational formulation for the transmission problem can be solved by applying this abstract result. In our existence proof, the abstract theory of pseudo-monotone operators is useful.

The projection operator in a Hilbert space and its directional derivative. Consequences for the theory of projected dynamical systems

In the first part of this paper we present a representation theorem for the directional derivative of the metric projection operator in an arbitrary Hilbert space. As a consequence of the representation theorem, we present in the second part the development of the theory of projected dynamical systems in infinite dimensional Hilbert space. We show that this development is possible if we use the viable solutions of differential inclusions. We use also pseudomonotone operators.