On Vector Quasi-Equilibrium Problems with Set-Valued Maps

In this paper, we introduce a new class of vector quasi-equilibrium problems with set-valued maps. Almost all the vector equilibrium models of the Blum-Oettli type in the literature are special cases of our new class of equilibrium problems under consideration. Moreover, a number of C-diagonal quasiconvexity properties are proposed for set-valued maps, which are natural generalizations of the -diagonal quasiconvexity for real functions. Together with an application of continuous selection and fixed-point theorems, these conditions enable us to prove unified existence results of solutions for such vector equilibrium problems.     

Upper semicontinuity of set-valued functions

Various types of upper semcontinuity properties for set-valued functions have been used in the past to obtain closure and lower closure theorems in optimal control theory as well as selection theorems and fixed-point theorems in topology. This paper unifies these various concepts by using semiclosure operators, extended topologies, and lattice theoretic operations and obtains general closure theorems. In addition, analytic criteria are given for this generalized upper semicontinuity. In particular, set-valued functions which are maximal in terms of certain properties (e.g., maximal monotone multifunctions) are shown to be necessarily upper semicontinuous.     

Henig proper generalized vector quasiequilibrium problems

In this paper, we give sufficient conditions for the existence of a Henig proper efficient solution of a general model in the theory of set-valued vector quasiequilibrium problems with moving cones. The main result of this paper is new, and is established under assumptions of existence of open lower sections and some properties of cone-semicontinuity and cone-concavity of set-valued maps. The moving cones are assumed to have bases which are Hausdorff lower semicontinuous and satisfy an additional suitable property.     

A selection theorem for a new class of set-valued maps

A new class of set-valued maps that includes all upper and lower semicontinuous set-valued maps is introduced. For this class, a selection theorem having applications in the theory of differential inclusions is presented. Translated fromMatematicheskie Zametki, Vol. 66, No. 4, pp. 503–507, October, 1999.     

Characterizations of Solutions for Vector Equilibrium Problems

In this paper, we characterize the solutions of vector equilibrium problems as well as dual vector equilibrium problems. We establish also vector optimization problem formulations of set-valued maps for vector equilibrium problems and dual vector equilibrium problems, which include vector variational inequality problems and vector complementarity problems. The set-valued maps involved in our formulations depend on the data of the vector equilibrium problems, but not on their solution sets. We prove also that the solution sets of our vector optimization problems of set-valued maps contain or coincide with the solution sets of the vector equilibrium problems.     

Sensitivity in mixed generalized vector quasiequilibrium problems with moving cones

In this paper, we consider a parametric generalized vector quasiequilibrium problem which is mixed in the sense that several different relations can simultaneously appear in this problem. The moving cones and other data of the problem are assumed to be set-valued maps defined in topological spaces and taking values in topological spaces or topological vector spaces. The main result of this paper gives general verifiable conditions for the solution mapping of this problem to be semicontinuous with respect to a parameter varying in a topological space. The result is proven with the help of notions of cone-semicontinuity of set-valued maps, weaker than the usual concepts of semicontinuity, and an assumption imposed on the set-valued map whose values are the dual cones of the corresponding values of the moving cones.     

目标函数扰动的集值优化问题的稳定性

本文研究了由目标函数扰动的集值优化问题的有效点集所定义的集值映射的半连续性.讨论了目标函数扰动的集值优化问题在上半连续意义下的稳定性.特别地,在广义适定性条件下,证明了集值优化问题在上半连续意义下的稳定性.     

The Generic Stability of Fixed Points for Upper Semicontinuous Mappings in Hyperconvex Metric Spaces

The main goal of this paper is to establish the generic stability of Fan-Glicksberg type fixed points in hyperconvex metric spaces. In order to do so, we first give Fan-Glicksberg type fixed point theorem in hyperconvex metric spaces and then the generic stability of fixed points for upper semicontinuous set-valued mappings is obtained. Our generic stability results show that almost all of fixed points of upper semicontinuous set-valued mappings defined in compact hyperconvex metric spaces are stable in the sense of Baire category theory     

Stability Results for Ekeland's ε Variational Principle for Set-Valued Mappings

In this paper, we define the Mosco convergence and Kuratowski-Painleve (P.K.) convergence for set-valued mapping sequence F n . Under some conditions, we derive the following result If a set-valued mapping sequence F n , which are nonempty, compact valued, upper semicontinuous and uniformly bounded below, Mosco (or P.K.) converges to a set-valued mapping F , which is upper semicontinuous, nonempty, compact valued, then Q l >0, u >0, $\varepsilon / \lambda - {\rm ext}\, F := \{ \bar x \in X : (F(x) - \bar y + \varepsilon / \lambda \Vert x - \bar x \Vert e)$     

Equilibria of nonconvex valued maps under constraints

In the paper the notion of n-tangency for set-valued maps defined on a subset of a Banach space is considered. The existence of equilibria of upper semicontinuous map being n-tangent to a sleek retract with the nontrivial Euler characteristic is established.     

Strict feasibility of pseudomonotone set-valued variational inequalities

In this article, we use degree theory developed in Kien et al. [B.T. Kien, M.-M. Wong, N.C. Wong, and J.C. Yao, Degree theory for generalized variational inequalities and applications, Eur. J. Oper. Res. 193 (2009), pp. 12–22.] to prove a result on the existence of solutions to set-valued variational inequality under a weak coercivity condition, provided that the set-valued mapping is upper semicontinuous with nonempty compact convex values. If the set-valued mapping is pseudomonotone in the sense of Karamardian and upper semicontinuous with nonempty compact convex values, it is shown that the set-valued variational inequality is strictly feasible if and only if its solution set is nonempty and bounded.     

锥上半连续锥拟凸集值映射多目标优化超有效解集的连通性

本文研究集值映射多目标优化超有效解集的连通性,在目标映射为锥上半连续和锥拟凸的条件下,证明了其超有效解集是连通的.     

Exceptional family of elements for generalized variational inequalities

This paper introduces a new concept of exceptional family of elements for a finite-dimensional generalized variational inequality problem. Based on the topological degree theory of set-valued mappings, an alternative theorem is obtained which says that the generalized variational inequality has either a solution or an exceptional family of elements. As an application, we present a sufficient condition to ensure the existence of a solution to the variational inequality. The set-valued mapping is assumed to be upper semicontinuous with nonempty compact convex values.     

Periodic Trajectories and Coincidence Points of Tuples of Set-Valued Maps

A fixed-point theorem is proved for a finite composition of set-valued Lipschitz maps such that the product of their Lipschitz constants is less than 1. The notion of a Lipschitz tuple of (finitely many) set-valued maps is introduced; it is proved that such a tuple has a periodic trajectory, which determines a fixed point of the given composition of set-valued Lipschitz maps. This result is applied to study the coincidence points of a pair of tuples (Lipschitz and covering).     

The Set-Valued Dynamic System and Its Applications to Pareto Optima

In this paper, we first study the existence of endpoints for set-valued dynamic systems which are either upper or lower semicontinuous in metric spaces. Then the existence, uniqueness and algorithms of endpoints for set-valued dynamic systems which are either generalized contractions (defined in metric spaces) or topological contractions (defined in topological spaces which do not necessarily have any metric). These results are then applied to derive the existence of Pareto optima for mappings which take values in ordered Banach spaces. Finally, the stability of (generalized) nucleolus sets is also established.     

The Study of Minimax Inequalities, Abstract Economics and Applications to Variational Inequalities and Nash Equilibria

In this survey, a new minimax inequality and one equivalent geometricform are proved. Next, a theorem concerning the existence of maximalelements for an LC-majorized correspondence is obtained.By the maximal element theorem, existence theorems of equilibrium point fora noncompact one-person game and for a noncompact qualitative game withLC-majorized correspondences are given. Using the lastresult and employing 'approximation approach', we prove theexistence of equilibria for abstract economies in which the constraintcorrespondence is lower (upper) semicontinuous instead of having lower(upper) open sections or open graphs in infinite-dimensional topologicalspaces. Then, as the applications, the existence theorems of solutions forthe quasi-variational inequalities and generalized quasi-variationalinequalities for noncompact cases are also proven. Finally, with theapplications of quasi-variational inequalities, the existence theorems ofNash equilibrium of constrained games with noncompact are given. Our resultsinclude many results in the literature as special cases.     

On dimension of inverse limits with upper semicontinuous set-valued bonding functions

We give results about the dimension of continua, obtained by combining inverse limits of inverse sequences of metric spaces and one-valued bonding maps with inverse limits of inverse sequences of metric spaces and upper semicontinuous set-valued bonding functions, by standard procedure introduced in [I. Bani?, Continua with kernels, Houston J. Math. (2006), in press].     

Higher-order optimality conditions in set-valued optimization using Studniarski derivatives and applications to duality

In this paper, we introduce upper and lower Studniarski derivatives of set-valued maps. By virtue of these derivatives, higher-order necessary and sufficient optimality conditions are obtained for several kinds of minimizers of a set-valued optimization problem. Then, applications to duality are given. Some remarks on several existent results and examples are provided to illustrate our results.     

关于上半连续集值映象的连续选择问题

本文进一步研究了上半连续集值映象的连续选择问题．首先,用一反例指出Ａｕｂｉｎ Ｊ、 Ｐ［１］中一结果不能成立．另外,对上半连续映象的σ－可选择性给出了另一逼近形式的结果．     

A Quasi Fixed-Point Theorem for a Product of u.s.c. or o.l.s. Correspondences with an Economic Application

In the first place, we present a quasi fixed-point theorem for a correspondence defined on some infinite-dimensional locally convex topological vector space such that some variables have open lower sections and the other ones are upper semicontinuous.In the second place, we propose a direct application of the quasi fixed-point result in an economic model. More precisely, we prove a nonemptiness result of the core of an exchange economy with asymmetric information, a continuum of states and a finite number of commodities.