Pseudomonotone Operators Research Articles

Extragradient Methods for Pseudomonotone Variational Inequalities

We consider and analyze some new extragradient-type methods for solving variational inequalities. The modified methods converge for a pseudomonotone operator, which is a much weaker condition than monotonicity. These new iterative methods include the projection, extragradient, and proximal methods as special cases. Our proof of convergence is very simple as compared with other methods.

On the embedding of variational inequalities

This work is devoted to the approximation of variational inequalities with pseudo-monotone operators. A variational inequality, considered in an arbitrary real Banach space, is first embedded into a reflexive Banach space by means of linear continuous mappings. Then a strongly convergent approximation procedure is designed by regularizing the embedded variational inequality. Some special cases have also been discussed.

Existence and comparison of maximal and minimal solutions for pseudomonotone elliptic problems in L1

We consider the nonhomogeneous Dirichlet problem − div(a(x,u, ∇u))=f in Ω, u=w on ∂Ω, where −div( a( x, u,∇ u)) is a pseudomonotone operator of Leray–Lions type defined in W 1,p 0(Ω) , w∈W 1,p(Ω) and f is in L 1(Ω) . Under suitable assumptions of locally Lipschitz, or locally Hölder, continuity of a( x, s, ξ) with respect to s, we prove the existence of maximal and minimal renormalized solutions and comparison results with respect to data f and w. The results include examples of nonmonotone operators of p–laplace type (for any p>1), for which it is known that uniqueness of solutions does not hold.

On Iterative Regularization Methods for Variational Inequalities of the Second Kind with Pseudomonotone Operators

Abstract Variational inequalitiy of the second kind in the Banach or Hilbert space is considered. A ”semi-implicit” iterative method for its solution is studied.

Analysis and simulations of vibrations of a beam with a slider

A model for vibrations of a beam with a slider is derived, analysed and numerically simulated. It describes a viscoelastic beam that is clamped at one end to a vibrating device, while the other end moves between two stops attached to a slider. The contact is described by the normal compliance or by the Signorini conditions. The existence of weak solutions is established using the theory of set-valued pseudomonotone operators. The model is discretized using fourth-order spatial discretization, the solutions are numerically simulated and their results presented. The dynamics of the vibrations are depicted and so are the noise characteristics of the system.

A note on complementarity problem in Banach space

The purpose of this note is to establish an existence theorem for nonlinear complemen- tarity problem for strictly pseudo-monotone operator over convex cone in a Banach space.

Iterative resolvent methods for general mixed variational inequalities

In this paper, we use the technique of updating the solution to suggest and analyze a class of new self‐adaptive splitting methods for solving general mixed variational inequalities. It is shown that these modified methods converge for pseudomonotone operators, which is a weaker condition than monotonicity. Proof of convergence is very simple. Since general mixed variational include variational inequalities and complementarity problems as special cases, our results continue to hold for these problems.

New extragradient-type methods for general variational inequalities

In this paper, we consider and analyze a new class of extragradient-type methods for solving general variational inequalities. The modified methods converge for pseudomonotone operators which is weaker condition than monotonicity. Our proof of convergence is very simple as compared with other methods. The proposed methods include several new and known methods as special cases. Our results present a significant improvement of previously known methods for solving variational inequalities and related optimization problems.

Forward‐backward resolvent splitting methods for general mixedvariational inequalities

We use the technique of updating the solution to suggest and analyze a class of new splitting methods for solving general mixed variational inequalities. It is shown that these modified methods converge for pseudomonotone operators, which is a weaker condition than monotonicity. Our methods differ from the known three‐step forward‐backward splitting of Glowinski, Le Tallec, and M. A. Noor for solving various classes of variational inequalities and complementarity problems. Since general mixed variational inequalities include variational inequalities and complementarity problems as special cases, our results continue to hold for these problems.

Existence of solutions for hyperbolic hemivariational inequalities

In this paper we study a hyperbolic hemivariational inequality with a nonlinear, pseudomonotone operator depending on the derivative of an unknown function and a linear, monotone operator depending on an unknown function. Using the surjectivity result for L-pseudomonotone operators, an existence result for such inequalities is proved.

A Class of Projection Methods for General Variational Inequalities

In this paper, we consider and analyze a new class of projection methods for solving pseudomonotone general variational inequalities using the Wiener–Hopf equations technique. The modified methods converge for pseudomonotone operators. Our proof of convergence is very simple as compared with other methods. The proposed methods include several known methods as special cases.

Resolvent dynamical systems for mixed variational inequalities

In this paper, we suggest and analyze a class of resolvent dynamical systems associated with mixed variational inequalities. We study the globally asymptotic stability of the solution of the resolvent dynamical systems for the pseudomonotone operators. We also discuss some special cases , which can be obtained from our main results.

A regularization approach for variational inequalities

This paper is devoted to a regularization approach for variational inequalities. A general method based on the theory of linear continuous and linear compact operators is developed for the stable approximation of ill-posed variational inequalities. Variational inequalities are considered for noncoercive monotone and pseudo-monotone operators. The approach is shown to be successful in handling variational inequalities formulated in reflexive, as well as in nonreflexive spaces.

Unilateral Problems for Nonlinearly Elastic Plates and Shallow Shells

Using the topological degree for pseudo-monotone operators of type (S+), we establish a general existence result for variational inequalities of von Karman type, which model unilateral problems for nonlinearly elastic plates. Then, we give a reduced operatorial form of Marguerre von Karman equations for nonlinearly elastic shallow shells and get a new existence result for this model.

Exceptional Family of Elements, Leray–Schauder Alternative, Pseudomonotone Operators and Complementarity

In this paper, the property of a function to be without exceptional family of elements (EFE) is investigated. We show that, for a pseudomonotone operator, the solvability of a complementarity problem is equivalent to the property of the function to be without EFE. Finally, we study the strict feasibility of a complementarity problem making use of the Leray-Schauder alternative and the notion of EFE.

On existence of solutions for parabolic hemivariational inequalities

A class of parabolic hemivariational inequalities is considered in which the energy function is locally Lipschitz (not necessarily convex or differentiable) and the nonlinear evolution operator is multivalued and generalized pseudomonotone. Using techniques of multivalued analysis and a surjectivity result for L-generalized pseudomonotone operators, we prove the existence for a Cauchy problem and a periodic problem. Some applications to nonsmooth mechanics are also indicated.

Equilibrium problem for thermoelectroconductive body with the Signorini condition on the boundary

We investigate a boundary value problem for a thermoelectroconductive body with the Signorini condition on the boundary, related to resistance welding. The mathematical model consists of an energy-balance equation coupled with an elliptic equation for the electric potential and a quasistatic momentum balance with a viscoelastic material law. We prove the existence of a weak solution to the model by using the Schauder fixed point theorem and classical results on pseudomonotone operators. Copyright © 2001 John Wiley & Sons, Ltd.

Equilibrium problem for thermoelectroconductive body with the Signorini condition on the boundary

AbstractWe investigate a boundary value problem for a thermoelectroconductive body with the Signorini condition on the boundary, related to resistance welding. The mathematical model consists of an energy‐balance equation coupled with an elliptic equation for the electric potential and a quasistatic momentum balance with a viscoelastic material law. We prove the existence of a weak solution to the model by using the Schauder fixed point theorem and classical results on pseudomonotone operators. Copyright © 2001 John Wiley & Sons, Ltd.

Convergence Analysis of Iterative Methods for Some Variational Inequalities with Pseudomonotone Operators

In the present paper, we consider variational inequalities of the second kind with a pseudomonotone operator and a convex nondifferentiable functional in Banach spaces. For example, such inequalities arise in the description of steady-state filtration processes and equilibrium problems for soft shells. For solving variational inequalities, we suggest a two-layer iterative method, which permits one to reduce the original variational inequality to a variational inequality with a duality operator that has better properties than the original operator. Similar iterative processes were considered earlier (e.g., see [1–6]) only for the cases of monotone operators or differentiable functionals (some regularization was used to make the functional differentiable). We analyze the convergence of the iterative process. We show that the iterative sequence is bounded and each weak limit point of this sequence is a solution of the original inequality. In the case of a Hilbert space, we prove the weak convergence of the entire iterative sequence under some additional requirement. We apply the general results to the stationary problem on the filtration of an incompressible fluid governed by a discontinuous filtration law with limit gradient [1, 7, 8] and to problems of finding an equilibrium position of a soft infinitely long cylindrical shell subjected to mass forces and a follower surface load and bounded by a plane obstacle [9, 10]. We also obtain some results about properties of solutions of filtration problems.

Variational‐like inequalities for pseudomonotone operators

The aim of this note is to use a fixed point theorem to prove results for variational‐like inequalities for pseudomonotone operators.