Equilibrium Problems in the Quasimonotone Case

Existence results for quasimonotone vector equilibrium problems and quasimonotone vector variational inequalities are obtained starting from an existence result for a scalar equilibrium problem involving two quasimonotone bifunctions. These results are established under weaker conditions than in previous works.     

Existence of Solutions for Nonlinear Implicit Differential Equations: An Equilibrium Problem Approach

In this article, we study the existence of solutions for nonlinear implicit differential equations associated to a time-dependent pseudomonotone (respectively, quasimonotone) operator by using a new approach based on the theory of equilibrium \hboxproblems. More precisely, we first establish some existence results of solutions for mixed equilibrium problems where the \hboxbifunctions are, respectively, maximal monotone and \hboxpseudomonotone (quasimonotone) in the topological sense. Then, we write the nonlinear implicit differential equations in the form of a mixed equilibrium problem. By using the existence results for mixed equilibrium problems, we establish the existence of solutions of nonlinear implicit differential equations. This new approach provides some new and nice results which improve and unify most of the recent results obtained in this direction.     

Noncoercive mixed equilibrium problems under pseudomonotone perturbations and applications to nonlinear evolution equations with lack of coercivity

In this paper, we study the existence of solutions for noncoercive mixed equilibrium problems which are described by the sum of a maximal monotone bifunction and a pseudomonotone (or quasimonotone) bifunction in the sense of Brézis. Our approach is based on recession analysis and on recent results established by the authors for the existence of solutions of mixed equilibrium problems under pseudomonotone perturbations. As an application, we study the existence of solutions for nonlinear evolution equations associated with a noncoercive time-dependent pseudomonotone (or quasimonotone) operator.     

Existence results for equilibrium problems with applications to evolution equations

In this paper, we consider evolution equations with time-dependent pseudomonotone and quasimonotone operators by a new approach based on equilibrium problems theory. We establish new existence results for equilibrium problems associated to pseudomonotone and quasimonotone bifunctions in the topological sense. The results obtained are then applied to derive existence of solutions for evolution equations. The approach is new and leads to improve and unify most of the results obtained in this direction.     

Existence theorems via duality for equilibrium problems with trifunctions

We present an existence result for an equilibrium problem formulated with trifunctions, which is motivated by variational inequalities governed by quasimonotone operators. To prove the existence result, we define the dual problem, and some monotonicity notions for trifunctions. From the main result follow, among others, the Browder–Minty theorem for variational inequalities and Ky Fan’s Minimax theorem. Some applications for mixed equilibrium problems and variational inequalities are given.     

Minimal Coercivity Conditions and Exceptional Families of Elements in Quasimonotone Variational Inequalities

A coercivity condition is usually assumed in variational inequalities over noncompact domains to guarantee the existence of a solution. We derive minimal, i.e., necessary coercivity conditions for pseudomonotone and quasimonotone variational inequalities to have a nonempty, possibly unbounded solution set. Similarly, a minimal coercivity condition is derived for quasimonotone variational inequalities to have a nonempty, bounded solution set, thereby complementing recent studies for the pseudomonotone case. Finally, for quasimonotone complementarity problems, previous existence results involving so-called exceptional families of elements are strengthened by considerably weakening assumptions in the literature.     

平衡问题解的存在性

主要研究平衡问题解的存在性.通过对目标函数和可行集合的渐近分析,给出拟单调平衡问题解集非空的条件.进而用类似的方法研究了向量平衡问题解存在的条件,并将其应用到向量优化问题上.     

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.     

System of nonsmooth variational inequalities with applications

In this paper, we consider a system of vector variational inequalities and a system of nonsmooth variational inequalities defined by means of Clarke directional derivative. We also consider the Nash equilibrium problem with vector pay-offs and its scalarized form. We present some relations among these systems and problems. The existence results for a solution of system of nonsmooth variational inequalities are given. As a consequence, we derive an existence result for a solution of Nash equilibrium problem with vector pay-offs.     

Generalized Equilibrium Problems for Quasimonotone and Pseudomonotone Bifunctions

By using quasimonotone and pseudomonotone bifunctions, we derive sufficient conditions which include weak coercivity conditions for existence of equilibrium points. As a consequence, we improve some recent results on the existence of such solutions.     

On system of generalized vector variational inequalities

In this paper, we introduce a new system of generalized vector variational inequalities with variable preference. This extends the model of system of generalized variational inequalities due to Pang and Konnov independently as well as system of vector equilibrium problems due to Ansari, Schaible and Yao. We establish existence of solutions to the new system under weaker conditions that include a new partial diagonally convexity and a weaker notion than continuity. As applications, we derive existence results for both systems of vector variational-like inequalities and vector optimization problems with variable preference.     

On vector formulations of auction-type problems with applications

We propose an extension of the auction model with many divisible commodities for vector price (validity) functions. It can be viewed as a new general equilibrium model for complex systems with active elements. We give its sufficient vector variational inequality formulation and new general existence results for different ordering cones. We suggest vector extensions of network and spatial equilibrium problems with capacity bounds and show that they are particular cases of the general auction model. We also give new sufficient conditions for existence of solutions for these problems.     

An existence result for generalized vector equilibrium problems without pseudomonotonicity

In this work, we introduce and study a class of generalized vector equilibrium problems for multifunctions which includes a number of generalized vector variational inequality problems and generalized vector variational-like inequality problems as special cases. By using the KKM–Fan theorem and Nadler’s result, we prove an existence theorem for solutions for this class of generalized vector equilibrium problems in Banach spaces. Applications to generalized vector variational-like inequalities are given.     

On Quasimonotone Variational Inequalities

In this paper, we study variational inequalities with multivalued mappings. By employing Fan's lemma, we establish the existence result for the dual formulation of variational inequalities with semistrictly quasimonotone mappings. We also show that similar results for quasimonotone variational inequalities do not hold.     

System of Vector Equilibrium Problems and Its Applications

In this paper, we introduce a system of vector equilibrium problems andprove the existence of a solution. As an application, we derive someexistence results for the system of vector variational inequalities. We alsoestablish some existence results for the system of vector optimizationproblems, which includes the Nash equilibrium problem as a special case.     

Generalized Vector Quasi-Variational Inequality Problems Over Product Sets

In this paper we consider vector quasi-variational inequality problems over product sets (in short, VQVIP). Moreover we study generalizations of this model, namely problems of a system of vector quasi-variational inequalities (in short, SVQVIP), generalized vector quasi-variational inequality problems over product sets (in short, GVQVIP) and problems of a system of generalized vector quasi-variational inequalities (in short, SGVQVIP). We show that every solution of (VQVIP) (respectively, (GVQVIP)) is a solution of (SVQVIP) (respectively, (SGVQVIP)). By defining relatively pseudomonotone and relatively maximal pseudomonotone maps and by employing a known fixed point theorem, we establish the existence of a solution of (VQVIP) and (SVQVIP). These existence results are then used to derive the existence of a solution of (GVQVIP) and (SGVQVIP), respectively, The results of this paper extend recent results in the literature. They are obtained in a more general setting.     

Systems of Simultaneous Generalized Vector Quasiequilibrium Problems and their Applications

In this paper, we introduce systems of simultaneous generalized vector equilibrium problems and prove the existence of their solutions. As application of our results, we derive the existence theorems for solutions of systems of vector saddle–point problems. Consequently, we prove the existence of a solution of systems of generalized minimax inequalities. Further application of our results is also given to establish the existence of a solution of a Debreu-type equilibrium problem for vector-valued functions. The first author thanks the Department of Mathematical Sciences, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia for providing excellent research facilities. The second and third authors were supported by the National Science Council of the Republic of China. The authors are grateful to the referees for valuable suggestions and comments.     

On Quasimonotone Variational Inequalities

The purpose of this paper is to prove the existence of solutions of the Stampacchia variational inequality for a quasimonotone multivalued operator without any assumption on the existence of inner points. Moreover, the operator is not supposed to be bounded valued. The result strengthens a variety of other results in the literature.     

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     

Existence of Solutions and Algorithms for Bilevel Vector Equilibrium Problems: An Auxiliary Principle Technique

The main goal of this paper is to introduce and study bilevel vector equilibrium problems. We first establish some existence results for solutions of vector equilibrium problems and mixed vector equilibrium problems. We study the existence of solutions of bilevel vector equilibrium problems by considering a vector Thikhonov-type regularization procedure. By using this regularization procedure and existence results for mixed vector equilibrium problems, we establish some existence results for solutions of bilevel vector equilibrium problems. By using the auxiliary principle, we propose an algorithm for finding the approximate solutions of bilevel vector equilibrium problems. The strong convergence of the proposed algorithm is also studied.