Theory Of Variational Inequalities Research Articles

The generalized quasi-variational inequality problem with applications

A new problem called the generalized quasi-variational inequality problem is introduced. This new problem extends all the existing variational inequality problems in finite dimensional spaces and enlarges the class of problems that can be approached by the variational inequality theory. Some existence results without convexity assumptions are presented. Applications to mathematical and equilibrium programming are given.

On the rate of convergence of solutions in singular perturbation problems

where the obstacle cp is a given smooth function satisfying that rp =0 on dQ. Using a technique in the theory of variational inequalities J.-L. Lions [ 11) proved that 11~’ min{O, ‘p} I/ L~cnj < CE for some constant C. We claim that we can never deduce the pointwise convergence of a” from the above estimate. On the other hand, the notion of viscosity solotions was introduced by M. G. Crandall and P.-L. Lions [2] for nonlinear first order PDEs and then it was extended for nonlinear second order PDEs by P.-L. Lions [12-141. Recently, it appeared that a technique in the theory of viscosity solutions could be applied to derive a pointwise convergence for some singular perturbation problems. In fact, by the same argument as in [7] we can show that U’ converges to min{ 0, cp} uniformly on 0. 243 0022-247X/91 $3.00

A network equilibrium formulation of market disequilibrium and variational inequalities

AbstractIn this article, we consider an asymmetric market disequilibrium model in a spatial setting in the case of rigid prices and/or controls. In this problem, the markets need no longer clear. We show that the disequilibrium problem can be reformulated as an equilibrium problem over an abstract network with special structure. We then propose a network decomposition algorithm for the computation of the solution that exploits the structure and establishes convergence results using the theory of variational inequalities. Our computational results demonstrate the suitability of this approach for large‐scale market problems.

Disequilibrium and variational inequalities

In this paper we introduce a new market disequilibrium model in a spatial economic setting, which generalizes a recent spatial disequilibrium model to the asymmetric case. We derive two alternative variational inequality formulations of the market conditions, in the case of price rigidities and/or controls, and discuss existence and uniqueness properties. We then propose a decomposition algorithm which resolves the variational inequality problem into three distinct and simpler variational inequality subproblems with special structure, which are then solved in sequential fashion. Any appropriate algorithm can then be used to solve the individual subproblems. The first variational inequality subproblem, however, is identical to the one governing the well-known spatial price equilibrium problem and, hence, a plethora of algorithms are available for its solution. We conclude with computational experience with the decomposition algorithm on large-scale market examples. This work bridges the study of disequilibrium and equilibrium problems via the theory of variational inequalities.

On a general nonlinear variational inequality

Variational inequality theory provides techniques for solving a variety of applied problems in science and engineering. Recently Noor considered some interesting general nonlinear and linear variational inequalities in a series of papers and proved the existence and uniqueness of solutions by a fixed point technique developed by Glowinski, Lions and Tremolieres and also by a fixed point technique of Lions and Stampacchia. But there are several inaccuracies in his proofs and here they have been removed and correct formulation of the theorems are stated and proved and relationships are clearly shown. The existence of solution necessitates an additional condition in one case, and less condition in the other, but uniqueness can be proved without the condition that the operator be antimonotone.

Exchange price equilibria and variational inequalities

The aim of this paper is to show the relevance of the concept and the theory of variational inequalities in the study of economic equilibria.

Integral equations techniques for contact problems

It is well known that a class of contact problems in elasticity can be studied in the framework of variational inequality theory. In this paper, we describe some numerical experience on the use of variational inequalities in conjunction with integral equations techniques to obtain an approximate solution of a class of contact problems in elasticity, like those describing the equilibrium configuration on an elastic string stretched over an elastic obstacle. The variational inequality formation is used to discuss the problem of uniqueness and existence of the solution of the contact problems.

Super-solutions of eigenvalue problems and the oscillation properties of second order evolution equations

Via a suitable extension of a classical Theorem of Stampacchia in the Theory of linear elliptic variational inequalities, we prove that available test functions can be obtained in order to study the oscillatory phenomena of global type associated to the solutions of nonlinear hyperbolic equations of order 2 m ( m ⩾ 1) in the space variable, possibly with singular potentials in bounded or unbounded domains of R N .

A variational method of solving an elastic-plastic problem for a body with a circular hole

An approach based on the theory of variational inequalities and a generalized plastic analogy for the solution of the elastic-plastic problem (EPP) concerning the state of stress of a body weakened by a circular hole without the assumption regarding total enclosure of the hole by a plastic zone is proposed. The Haar-Karman hypothesis or an equivalent assertion is not used here. Generalizations are given to the case of a plastic inhomogeneous body and for the utilization of an exponential flow condition. Examples are considered and a simple method is proposed for estimating the plastic zone dimensions. It was assumed in the well-known solution given by Galin /1/ of the EPP on the biaxial tension of a plane with a circular hole and its generalizations /2–5/ that the plastic domain completely encloses the hole. The majority of existing solutions have been obtained for the stress concentration around a hole in an infinite domain.

Finite difference technique for solving obstacle problems

In this paper, we use the variational inequality theory coupled with finite difference technique to obtain an approximate solution for a class of obstacle problems in elasticity, like those describing the equilibrium configuration of an elastic string stretched over an elastic obstacle. The variational inequality formulation is used to discuss the problem of uniqueness and existence of the solution of the obstacle problems.

Numerical techniques for a class of unilateral problems

In this paper, we use the variational inequality theory coupled with Pade approximants to obtain an approximate solution for a class of unilateral problems in elasticity. The results obtained are more efficient than the previously known methods of the same order. The variational inequality formulation is used to discuss the problem of uniqueness and existence of the solution of these unilateral problems.

Finite element error estimates for subsonic flow

AbstractError estimates are derived for a finite element analysis of plane steady subsonic flows described by the full potential equation. The analysis is based on the use of the theory of variational inequalities to accomodate the subsonic flow constraint and leads to a suboptimal estimate relative to that obtained for linear potential flow. We then consider an alternative dual formulation of the problem and obtain an optimal estimate subject to reasonable regularity assumptions.

Cubic splines collocation methods for unilateral problems

Variational inequalities theory not only provides us a general unified frame work for study many unrelated moving and free boundary vary problems, but also gives more efficient numerical methods for solving them. In this paper, we describe numerical experience on the use of variational inequalities and cubic splines collocation technique to obtain approximate solution to a class of unilateral boundary value problems of elasticity, like those describing the equilbrium configuration of an elastic string stretched over an elastic obstacle. The variational inequality formulation is used to discuss the problem of uniqueness and existence of the solution of the unilateral problems.

Numerical methods for unilateral problems

The theory of variational inequalities, besides being elegant and synthetic, also provides a unified general framework for applying the numerical methods for solving many unrelated free boundary value problems. In this paper, we describe numerical experience on the use of variational inequalities and Padé approximants to obtain approximate solutions to a class of unilateral boundary value problems of elasticity, like those describing the equilibrium configuration of an elastic beam stretched over an elastic obstacle. In the case of a known obstacle, these problems can be alternatively formulated as nonlinear boundary value problems without constraints for which the technique of Pade approximants can be successfully employed. The variational inequality formulation is used to discuss the problem of uniqueness and existence of the solution.

A free surface problem arising in the drainage of a uniformly irrigated field: Existence and uniqueness results

AbstractA field comprising uniformly porous soil overlying an impervious subsoil is drained through equally spaced tile drains placed on the boundary between the two layers of soil. When this field is subject to uniform irrigation, a free boundary forms in the porous region above the zone of saturation. We study the free boundary value problem which thus arises using the theory of variational inequalities. Existence and uniqueness results are established.

Applications of variational inequalities to the existence theorem on quadrature domains

In this paper we shall study quadrature domains for the class of subharmonic functions. By using the theory of variational inequalities, we shall give a new proof of the existence and uniqueness theorem. As an application, we deal with Hele-Shaw flows with a free boundary and show that their two weak solutions, one of which was defined by the author using quadrature domains and the other was defined by Gustafsson [3] using variational inequalities, are identical with each other.

The general multimodal network equilibrium problem with elastic demand

AbstractIn this article, we consider the general multimodal traffic equilibrium model with elastic demands. The link travel costs may depend upon the entire load pattern and the travel demands associated with each origin‐destination pair and mode may depend upon travel costs associated with every origin‐destination pair and every mode of transportation. For this model, we define the concepts of user‐optimality and equilibrium. We establish, by means of a constructive proof motivated by the theory of variational inequalities, the existence of a unique equilibrium under appropriate monotonicity conditions. We then show that the existence proof induces an algorithm for the computation of equilibrium traffic patterns. The algorithm proceeds by iteration, each step of which amounts to computing the equilibrium pattern for a single modal linear traffic equilibrium problem with elastic demands. We derive estimates for the speed of convergence.

FINITE ELEMENT APPROXIMATIONS OF SMOOTH CONTACT-IMPACT PROBLEMS

This study is concerned with finite element models and solution algorithms for a class of contact-impact problems which are formulated within the frame-work of the modern theory of variational inequalities. Discussions are limited to smooth contact-impact of a linearly elastic body to a rigid support.A brief review on two distinct variational formulations and associated algorithms is first given and then a new algorithm based on reciprocal formulation is proposed. According to presented numerical experiments, the new algorithm converges rather rapidly and is easy to use. Numerous examples including classical contact, rigid punch, and transient analyses of impact problems are presented and discussed. Possible extensions of the current study are also discussed.

Traffic Equilibrium and Variational Inequalities

We consider the general traffic equilibrium network model where the travel cost on each link of the transportation network may depend on the flow on this as well as other links of the network. The model has been designed in order to handle situations where there is interaction between traffic on different links (e.g., two-way streets, intersections) or between different modes of transportation on the same link. For this model, we use the techniques of the theory of variational inequalities to establish existence of a traffic equilibrium pattern, to design an algorithm for the construction of this pattern and to derive estimates on the speed of convergence of the algorithm.

A free boundary problem in the theory of lubrication

The theory of variational inequalities is applied to a free boundary problem arising in the hydrodynamic lubrication. We deal mainly with the properties of the pressure distribution in the clearance space and with the shape of the cavitated region.