Equilibrium problems and noncoercive variational inequalities

In this paper, we are interested to establish existence results for equilibrium problems in a noncoercive framework by using techniques of recession analysis. The abstract result is then applied to find solution of noncoercive monotone and pseudomonotone variational inequalities     

Bifurcation problems for discrete variational inequalities

The buckling of a beam or a plate which is subject to obstacles is typical for the variational inequalities that are considered here. Birfurcation is known to occur from the first eigenvalue of the linearized problem. For a discretization the bifurcation point and the bifurcating branches may be obtained by solving a constrained optimization problem. An algorithm is proposed and its convergence is proved. The buckling of a clamped beam subject to point obstacles is considered in the continuous case and some numerical results for this problem are presented.     

Inverse problems for multi-valued quasi variational inequalities and noncoercive variational inequalities with noisy data

ABSTRACT

We study the inverse problem of identifying a variable parameter in variational and quasi-variational inequalities. We consider a quasi-variational inequality involving a multi-valued monotone map and give a new existence result. We then formulate the inverse problem as an optimization problem and prove its solvability. We also conduct a thorough study of the inverse problem of parameter identification in noncoercive variational inequalities which appear commonly in applied models. We study the inverse problem by posing optimization problems using the output least-squares and the modified output least-squares. Using regularization, penalization, and smoothing, we obtain a single-valued parameter-to-selection map and study its differentiability. We consider optimization problems using the output least-squares and the modified output least-squares for the regularized, penalized and smoothened variational inequality. We give existence results, convergence analysis, and optimality conditions. We provide applications and numerical examples to justify the proposed framework.     

Jumping problems for quasilinear elliptic variational inequalities

We investigate the existence and multiplicity of solutions for a certain class of quasilinear variational inequalities, whenever between the obstacle and the behavior at +￥+\infty of the nonlinearity there is a situation of jumping type.     

On variational inequalities for auction market problems

We give an equivalent variational inequality formulation for a general class of equilibrium problems based upon auction decision rules. We show that a general relaxation iterative process with conditional gradient extrapolation ensures convergence to a solution under rather mild assumptions.      

Equilibrium problems and variational inequalities in topological vector spaces

An existence result for the equilibrium problem is proved in a general topological vector space. As applications, existence results are derived for variational inequality problems, vector equilibrium problems and vector variational inequality problems. Our results extend and unify a number of existence theorems in non-compact cases     

Cooperation in pollution control problems via evolutionary variational inequalities

In this paper, we develop a cooperative game framework for modeling the pollution control problem in a time-dependent setting. We examine the situation in which different countries, aiming at reducing pollution emissions, coordinate both emissions and investment strategies to optimize jointly their welfare. We state the equilibrium conditions underlying the model and provide a formulation in terms of an evolutionary variational inequality. Then, by means of infinite dimensional duality tools, we prove the existence of Lagrange multipliers that play a fundamental role to describe countries’ decision-making processes. Finally, we discuss the existence of solutions and provide a numerical example.     

Monotone generalized variational inequalities and generalized complementarity problems

Some existence results for generalized variational inequalities and generalized complementarity problems involving quasimonotone and pseudomonotone set-valued mappings in reflexive Banach spaces are proved. In particular, some known results for nonlinear variational inequalities and complementarity problems in finite-dimensional and infinite-dimensional Hilbert spaces are generalized to quasimonotone and pseudomonotone set-valued mappings and reflexive Banach spaces. Application to a class of generalized nonlinear complementarity problems studied as mathematical models for mechanical problems is given.The research of the first author was supported by the National Natural Science Foundation of P. R. China and by the Ethel Raybould Fellowship, University of Queensland, St. Lucia, Brisbane, Australia.     

Spectral problems for variational inequalities with discontinuous operators

Some spectral problems for variational inequalities with discontinuous nonlinear operators are considered. The variational method is used to prove the assumption that such problems are solvable. The general results are applied to the corresponding elliptic variational inequalities with discontinuous nonlinearities.     

Brunn-Minkowski inequalities for variational functionals and related problems

Recently, several inequalities of Brunn-Minkowski type have been proved for well-known functionals in the Calculus of Variations, e.g. the first eigenvalue of the Laplacian, the Newton capacity, the torsional rigidity and generalizations of these examples. In this paper, we add new contributions to this topic: in particular, we establish equality conditions in the case of the first eigenvalue of the Laplacian and of the torsional rigidity, and we prove a Brunn-Minkowski inequality for another class of variational functionals. Moreover, we describe the links between Brunn-Minkowski type inequalities and the resolution of Minkowski type problems.     

Inverse coefficient problems for nonlinear elliptic variational inequalities

This paper is devoted to a class of inverse coefficient problems for nonlinear elliptic variational inequalities. The unknown coefficient of elliptic variational inequalities depends on the gradient of the solution and belongs to a set of admissible coefficients. It is shown that the nonlinear elliptic variational inequalities is unique solvable for the given class of coefficients. The existence of quasisolutions of the inverse problems is obtained.     

Reinforcement problems for elliptic equations and variational inequalities

Summary Consider the Dirichlet problem for an elliptic equation in a domain , with coefficients having discontinuity on a surface . Suppose divides into _{1 2}(₂ the inner core), the thickness of ₁ is of order of magnitude , and the modulus of ellipticity in ₁ is of order magnitude ₁. The asymptotic behavior of the solution is studied as 0, ₁ 0, provided lim (/₁) exists. Other questions of this type are studied both for elliptic equations and for elliptic variational inequalities.The second author is partially supported by National Science Foundation Grant 7406375 A01. The third author is partially supported by National Science Foundation Grant MC575-21416 A01.     

Positive solutions of some eigenvalue problems in the theory of variational inequalities

We prove the existence of positive solutions of some eigenvalue problems relative to variational inequalities. The operators considered here belong to a class of differential nonlinear elliptic operators in divergence form.     

Well-posedness of mixed variational inequalities, inclusion problems and fixed point problems

We generalize the concept of well-posedness to a mixed variational inequality and give some characterizations of its well-posedness. Under suitable conditions, we prove that the well-posedness of a mixed variational inequality is equivalent to the well-posedness of a corresponding inclusion problem. We also discuss the relations between the well- posedness of a mixed variational inequality and the well-posedness of a fixed point problem. Finally, we derive some conditions under which a mixed variational inequality is well-posed. This work was supported by the National Natural Science Foundation of China (10671135) and Specialized Research Fund for the Doctoral Program of Higher Education (20060610005). The research of the third author was partially support by NSC 95-2221-E-110-078.     

Sharp discrete inequalities and applications to discrete variational problems

Some new discrete inequalities involving monotonic or convex functions are obtained. While these are interesting inequalities in their own right, they can be applied to solving certain types of discrete variational problems effectively.     

On optimal control problems connected with eigenvalue variational inequalities

The eigenvalue optimization problem for a variational inequality over the convex cone is to be dealt with. The control variable appears in the operator of the unilateral problem. The existence theorem for the maximum first eigenvalue optimization problem is stated and verified. The necessary optimality condition is derived. The applications to the optimal design of unilaterally supported beams and plates are presented. The variable thickness of a construction plays the role of a design variable. The convergence of the finite elements approximation is proved. Copyright © 2001 John Wiley & Sons, Ltd.     

Some equilibrium problems under uncertainty and random variational inequalities

In this paper we describe some nonlinear equilibrium problems under uncertainty arising from economics and operations research. In particular we treat Wardrop equilibria in traffic networks. We show how the theory of monotone random variational inequalities, where random variables occur both in the operator and the constraint set, can be applied to model these problems. Therefore in this contribution we introduce the topic of random variational inequalities and present some of our recent results in this field. In particular, we treat the more structured case where a finite Karhunen-Loève expansion leads to a separation of the random and the deterministic variables. Here we describe a norm convergent approximation procedure based on averaging and truncation. We illustrate this procedure by means of some small sized numerical examples.     

Sensitivity analysis for variational inequalities and nonlinear complementarity problems

This paper surveys the main results in the area of sensitivity analysis for finite-dimensional variational inequality and nonlinear complementarity problems. It provides an overview of Lipschitz continuity and differentiability properties of perturbed solutions for variational inequality problems, defined on both fixed and perturbed sets, and for nonlinear complementarity problems.