On the Existence of Solutions of Quasi-equilibrium Problems with Constraints

The quasi-equilibrium problems with constraints are formulated and some sufficient conditions on the existence of their solutions are shown. As special cases, we obtain several results on the existence of solutions of some vector quasivariational inequality and vector optimization problems. An application of the obtained results is given to show the existence of solutions of quasi-optimization problems with constraints.     

On quasivariational inclusion problems of type I and related problems

The quasivariational inclusion problems are formulated and sufficient conditions on the existence of solutions are shown. As special cases, we obtain several results on the existence of solutions of a general vector ideal (proper, Pareto, weak) quasi-optimization problems, of quasivariational inequalities, and of vector quasi-equilibrium problems. Further, we prove theorems on the existence for solutions of the sum of these inclusions. As corollaries, we shall show several results on the existence of solutions to another problems in the vector optimization problems concerning multivalued mappings. This work was supported by the National Science Council of the Republic of China and the Academy of Sciences and Technologies of Vietnam. The authors wish to express their gratitude to the referees for their valuable suggestions.     

广义强向量拟均衡问题组解的存在性

本文研究了一类集值广义强向量拟均衡问题组解的存在性问题.利用集值映射的自然拟C-凸性和集值映射的下(-C)-连续性的定义和Kakutani-Fan-Glicksberg不动点定理,在不要求锥C的对偶锥C~*具有弱*紧基的情况下,建立了该类集值广义强向量拟均衡问题组解的存在性定理.所得结果推广了该领域的相关结果.     

The Study of the System of Generalized Vector Quasi-equilibrium Problems

In this paper, we study the system of generalized vector quasi-equilibrium problems, which includes as special cases the system of vector quasi-equilibrium problems and the system of generalized vector equilibrium problems, and establish the existence and essential components of the solution set under perturbations of its best-reply map. Moreover, we also derive a new existence theorem of Ky Fan’s points for a set-valued map.     

Generalized vector quasi-equilibrium problems with applications

In this paper, we consider the generalized vector quasi-equilibrium problem with or without involving Φ-condensing maps and prove the existence of its solution by using known fixed point and maximal element theorems. As applications of our results, we derive some existence results for a solution to the vector quasi-optimization problem for nondifferentiable functions and vector quasi-saddle point problem.     

Some existence results for solutions of generalized vector quasi-equilibrium problems

In this paper, we consider more general forms of generalized vector quasi-equilibrium problems for multivalued maps which include many known vector quasi-equilibrium problems and generalized vector quasi-variational inequality problems as special cases. We establish some existence results for solutions of these problems under pseudomonotonicity and u-hemicontinuity/ℓ-hemicontinuity assumptions.      

Existences of solutions for generalized vector quasi-equilibrium problems

In the paper, existence theorems for two classes of generalized vector quasi-equilibrium problems are established by using a fixed point theorem and some examples are given to illustrate them. An application to vector quasi-optimization problems is shown. This research was partially supported by the National Natural Science Foundation of China (Grant numbers: 60574073 and 10471142).     

On System of Generalized Vector Quasi-equilibrium Problems with Set-valued Maps

In this paper, we introduce four new types of the system of generalized vector quasi-equilibrium problems with set-valued maps which include system of vector quasi-equilibrium problems, system of vector equilibrium problems, system of variational inequality problems, and vector equilibrium problems in the literature as special cases. We prove the existence of solutions for such kinds of system of generalized vector quasi-equilibrium problems. Consequently, we derive some existence results of a solution for the system of vector quasi-equilibrium problems and the generalized Debreu type equilibrium problem for vector-valued functions.     

On the Existence of Solutions of Quasivariational Inclusion Problems

Quasivariational inclusion problems are formulated and sufficient conditions on the existence of solutions are shown. As special cases, we obtain several results on the existence of solutions of general vector ideal, proper, Pareto, weak quasioptimization problems, quasivariational inequalities, and vector quasiequilibrium problems. Further, we prove theorems on the existence for solutions of systems of these inclusions. As a corollary, we obtain an ideal minimax theorem concerning vector functions.     

Quasi-Equilibrium Problems and Fixed Point Theorems of Separately l.s.c and u.s.c Mappings

In this paper, we apply our new results on quasi-variational inequality problems to generalized quasi-equilibrium problems. Some su?cient conditions on the existence of solutions are shown. In particular, we establish several results on the existence of solutions to fixed points of separately lower and upper semicontinuous (lower and lower, upper and upper) mappings.These generalize some well-known fixed point theorems obtained by previous authors as Browder and Ky Fan, Wu, et al., etc.     

On the existence of solutions to generalized quasi-equilibrium problems

In this paper, we apply new results on variational relation problems obtained by D. T. Luc (J Optim Theory Appl 138:65–76, 2008) to generalized quasi-equilibrium problems. Some sufficient conditions on the existence of its solutions of generalized quasi-equilibrium problems are shown. As special cases, we obtain several results on the existence of solutions of generalized Pareto and weak quasi-equilibrium problems concerning C-pseudomonotone multivalued mappings. We deduce also some results on the existence of solutions to generalized vector Pareto and weakly quasivariational inequality and vector Pareto quasi-optimization problems with multivalued mappings.     

Vector optimization and variational-like inequalities

In this paper, some properties of pseudoinvex functions are obtained. We study the equivalence between different solutions of the vector variational-like inequality problem. Some relations between vector variational-like inequalities and vector optimization problems for non-differentiable functions under generalized monotonicity are established. J. Zafarani was partially supported by the Center of Excellence for Mathematics (University of Isfahan).     

Upper semicontinuity of the solution maps in homogeneous vector quasi-equilibrium problems

The aim of this work was to study the solution maps in the vector homogeneous quasi-equilibrium problems. Here, we establish some sufficient and necessary conditions for upper semicontinuity of the solution maps in the vector homogeneous quasi-equilibrium problems. The results presented extend and improve the preceding results of Oetlli and Yen [Oetlli, W. and Yen, N.D., 1995, Continuity of the solution set of homogeneous equilibrium problems and linear complementarity problems. In: F. Giannessi and A. Maugeri (Eds) Variational Inequalities and Network Equilibrium Problems (New York: Plenum Press)].     

Sensitivity analysis for a system of parametric general quasi-variational-like inequality problems in uniformly smooth Banach spaces

In this paper, using the concept of P-η-proximal-point mapping introduced by Kazmi and Bhat [11], we study the existence and sensitivity analysis of the solution set of a system of parametric general quasi-variational-like inequality problems in uniformly smooth Banach spaces. Further under suitable conditions, we discuss the Lipschitz continuity of the solution set with respect to the parameters. The approach used in this paper may be treated as an extension and unification of approaches for studying sensitivity analysis for various important classes of variational inequalities given in [1,2,4,12,14–16,21–24].     

Efficiency in vector quasi-equilibrium problems and applications

In this paper, we give sufficient conditions for the existence of efficient solutions of a generalized vector quasi-equilibrium problem in topological vector spaces. The motivations for introducing this problem come from practical problems in traffic networks and the optimal control theory for discrete-time dynamical systems. The main results of the paper are proven with the help of a strongly monotonic function which can be constructed from the data of the problem under consideration. Some notions of cone-semicontinuity of set-valued maps, weaker than the usual concepts of semicontinuity, are also used in our study. As applications, we obtain existence results in vector quasi-optimization problems, Stampacchia set-valued vector quasi-variational inequality problems and Pareto vector quasi-saddle point problems. All these results are different from the corresponding ones in the literature.     

Asymptotic Analysis of State Constrained Semilinear Optimal Control Problems

The asymptotic behavior of state-constrained semilinear optimal control problems for distributed-parameter systems with variable compact control zones is investigated. We derive conditions under which the limiting problems can be made explicit. We gratefully acknowledge the support of the DAAD. The paper was prepared during the visit of the first author at the Institute of Applied Mathematics II, University Erlangen-Nuremberg in 2003.     

Existence Results for Systems of Vector Equilibrium Problems*

The purpose of this paper is to study systems of vector equilibrium problems. We establish some existence theorems for systems of vector equilibrium problems by using (S)₊-conditions and Kakutani–Fan–Glicksberg fixed point theorem *This work was supported by the Kyungnam University Research Fund 2004     

Existence theorems of systems of variational inclusion problems with applications

In this paper, we study the existence theorems of systems of variational inclusions problems. As consequences of our results, we study existence theorems of systems of generalized vector quasi-equilibrium problems, mathematical program with systems of variational inclusion constraints, bilevel problem with systems of constraints.