Strong Vector Equilibrium Problems

In this paper, the existence of the solution for strong vector equilibrium problems is studied by using the separation theorem for convex sets. The arc-wise connectedness and the closedness of the strong solution set for vector equilibrium problems are discussed; and a necessary and sufficient condition for the strong solution is obtained.     

Necessary and Sufficient Conditions for Qualitative Properties of Infinite Dimensional Linear Programming Problems

Necessary and sufficient conditions for qualitative properties of infinite dimensional linear programing problems such as solvability, duality, and complementary slackness conditions are studied in this article. As illustrations for the results, we investigate the parametric version of Gale’s example.     

Semicontinuity of the Approximate Solution Sets of Multivalued Quasiequilibrium Problems

We consider two kinds of approximate solutions and approximate solution sets to multivalued quasiequilibrium problems. Sufficient conditions for the lower semicontinuity, Hausdorff lower semicontinuity, upper semicontinuity, Hausdorff upper semicontinuity, and closedness of these approximate solution sets are established. Applications in approximate quasivariational inequalities, approximate fixed points, and approximate quasioptimization problems are provided.     

TOPOLOGICAL STRUCTURE OF EFFICIENT SET OF OPTIMIZATION PROBLEM OF SET－VALUED MAPPING

ＴＯＰＯＬＯＧＩＣＡＬ ＳＴＲＵＣＴＵＲＥ ＯＦ ＥＦＦＩＣＩＥＮＴ ＳＥＴ ＯＦ ＯＰＴＩＭＩＺＡＴＩＯＮ ＰＲＯＢＬＥＭ ＯＦ ＳＥＴ－ＶＡＬＵＥＤ ＭＡＰＰＩＮＧ ￥ＬＩＹＵＡＮＸＩＡｂｓｔｒａｃｔ：Ｔｈｉｓｐａｐｅｒｉｓｃｏｎｃｅｒｎｅｄｗｉ...     

On the Stability of the Solution Sets of General Multivalued Vector Quasiequilibrium Problems

We give sufficient conditions for the semicontinuity of solution sets of general multivalued vector quasiequilibrium problems. All kinds of semicontinuities are considered: lower semicontinuity, upper semicontinuity, Hausdorff upper semicontinuity, and closedness. Moreover, we investigate the weak, middle, and strong solutions of quasiequilibrium problems. Many examples are provided to give more insights and comparisons with recent existing results.     

Semicontinuity of the solution set of parametric multivalued vector quasiequilibrium problems

In this paper we establish sufficient conditions for the solution set of parametric multivalued vector quasiequilibrium problems to be semicontinuous. All kinds of semicontinuity are considered: lower semicontinuity, upper semicontinuity, Hausdorff upper semicontinuity and closedness. Moreover, we investigate both the “weak” and “strong” solutions of quasiequilibrium problems.     

Closedness Properties of Internal Relations III: Pointed Protomodular Categories

A pointed variety of universal algebras is protomodular in the sense of D. Bourn, if and only if it is classically ideal determined in the sense of A. Ursini (this result is due to D. Bourn and G. Janelidze). We prove a characterization theorem for pointed protomodular categories, which is a (pointed) categorical version of Ursini’s characterization theorem for classically ideal determined varieties, involving classically 0-regular algebras. A suitable simplification of the property of a pair of relations, which is used to define a classically 0-regular algebra, yields a new closedness property of a single binary relation – we show that a finitely complete pointed category is protomodular if and only if every binary internal relation R→A ² in it has this closedness property. Partially supported by South African National Research Foundation, and Georgian National Science Foundation (GNSF/ST06/3-004).     

Upper Semicontinuity of the Solution set to Parametric Vector Quasivariational Inequalities

We prove the upper semicontinuity (in term of the closedness) of the solution set with respect to parameters of vector quasivariational inequalities involving multifunctions in topological vector spaces under the semicontinuity of the data, avoiding monotonicity assumptions. In particular, a new quasivariational inequality problem is proposed. Applications to quasi-complementarity problems are considered This work was partially supported by the program “Optimisation et Mathématiques Appliquées” of C.I.U.F-C.U.D./C.U.I. of Belgium and by the National Basic Research Program in Natural Sciences of NCSR of Vietnam