Quasi-equilibrium Problems Research Articles

Symmetric strong vector quasiequilibrium problems in Hausdorff locally convex spaces

Abstract In this article, a new symmetric strong vector quasiequilibrium problem in real locally convex Hausdorff topological vector spaces is introduced and studied. An existence theorem of solutions for the symmetric strong vector quasiequilibrium problem by using Kakutani-Fan-Glicksberg fixed point theorem is obtained. Moreover, the closedness of the solution set for this problem is derived. The results presented in this article improve and extend some known results according to Long et al. [Math. Comput. Model. 47, 445-451 (2008)], Somyot and Kanokwan [Fixed Point Theory Appl. doi:10.1155/2011/475121], and Wang et al. [Bull. Malays. Math. Sci. Soc. http://www.emis.de/journals/BMMSS/pdf/acceptedpapers/2009-11-022_R1.pdf]. 2000 MSC: 49J40; 90C29.

Pollution control quasi-equilibrium problems with joint implementation of environmental projects

In this paper, we show how one of the Kyoto Protocol mechanisms, the so-called joint implementation in environmental projects, can be transformed into and studied as an infinite-dimensional quasi-variational inequality. Specifically, we examine the situation in which different countries attempt to fulfill Kyoto commitments by investing in emission reduction or emission removal projects in countries where the abatement costs are lower. We derive the equilibrium conditions and prove their characterization in terms of an infinite-dimensional quasi-variational inequality problem. Finally, we discuss the existence of solutions.

The existence of solutions for the system of vector quasi-equilibrium problems in topological order spaces

This paper introduces two kinds of quasiconcave mappings which are different from the usual quasiconcave function. We establish a result for the existence of solutions for the system of vector quasi-equilibrium problems in the frame of topological order, by providing a maximal elements version of the well known Browder fixed points theorem.

Collectively fixed point and maximal element theorems in topological semilattice spaces

In this article, we establish a collectively fixed point theorem and a maximal element theorem for a family of multivalued maps in the setting of topological semilattice spaces. As an application of our maximal element theorem, we prove the existence of solutions of generalized abstract economies with two constraint correspondences. We consider the system of (vector) quasi-equilibrium problems (in short, (S(V)QEP)) and system of generalized vector quasi-equilibrium problems (in short, (SGVQEP)). We first derive the existence result for a solution of (SQEP) and then by using this result, we prove the existence of a solution of system of a generalized implicit quasi-equilibrium problems. By using existence result for a solution of (SQEP) and weighted sum method, we derive an existence result for solutions of (SVQEP). By using our maximal element theorem, we also establish some existence results for the solutions of (SGVQEP). Some applications of our results to constrained Nash equilibrium problem for vector-valued functions with infinite number of players and to semi-infinite problems are also given.

Existence of solutions of systems of generalized implicit vector quasi-equilibrium problems in [formula omitted]-convex spaces

In this paper, we consider systems of generalized vector quasi-equilibrium problems in G -convex spaces. By applying two maximal element theorems for a family of multivalued mappings on product G -convex spaces, we establish some existence results for solutions of these problems under generalized pseudomonotonicities and under lower semicontinuity for a family of multivalued mappings.

Levitin-Polyak well-posedness of generalized vector quasi-equilibrium problems with functional constraints

In this paper, we introduce several types of Levitin-Polyak well-posedness for a generalized vector quasi-equilibrium problem with functional constraints and abstract set constraints. Criteria and characterizations of these types of Levitin-Polyak well-posedness with or without gap functions of generalized vector quasi-equilibrium problem are given. The results in this paper unify, generalize and extend some known results in the literature.

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.

On the image space analysis for vector quasi-equilibrium problems with a variable ordering relation

In this paper, we extend the image space analysis to a vector quasi-equilibrium problem (VQEP) with a variable ordering relation and we analyse scalar and vector saddle point optimality conditions arising from the existence of a vector separation in the image space associated with VQEP.

Hölder continuity and upper estimates of solutions to vector quasiequilibrium problems

In this paper, we establish the Hölder continuity of solution mappings to parametric vector quasiequilibrium problems in metric spaces under the case that solution mappings are set-valued. Our main assumptions are weaker than those in the literature, and the results extend and improve the recent ones. Furthermore, as an application of Hölder continuity, we derive upper bounds for the distance between an approximate solution and a solution set of a vector quasiequilibrium problem with fixed parameters.

Fixed Point Theorems on Product Topological Semilattice Spaces, Generalized Abstract Economies and Systems of Generalized Vector Quasi-equilibrium Problems

In this paper, we establish fixed point theorems for a family of multivalued maps defined on the product space of topological semilattice spaces. By using our fixed point theorems, we derive a result on the nonempty intersection of sets without convex structure and equilibrium existence theorems for generalized abstract economies with two constraint correspondences. We present some special cases of our results which generalize several known results in the literature. We consider systems of generalized vector quasi-equilibrium problems and their special cases. As an application of our equilibrium existence theorems, we establish some existence results for solutions of systems of generalized vector quasi-equilibrium problems and their special cases. The results of this paper improve and extend several results in the literature.

Existence Result of Generalized Vector Quasiequilibrium Problems in Locally G-Convex Spaces

This paper deals with the generalized strong vector quasiequilibrium problems without convexity in locally -convex spaces. Using the Kakutani-Fan-Glicksberg fixed point theorem for upper semicontinuous set-valued mapping with nonempty closed acyclic values, the existence theorems for them are established. Moreover, we also discuss the closedness of strong solution set for the generalized strong vector quasiequilibrium problems.

Applications of an HIDS Theorem to the Existence of Fixed Point, Abstract Equilibria and Optimization Problems

By applying hybrid inclusion and disclusion systems (HIDS), we establish several vectorial variants of system of Ekeland′s variational principle on topological vector spaces, some existence theorems of system of parametric vectorial quasi‐equilibrium problem, and an existence theorem of system of the Stampacchia‐type vectorial equilibrium problem. As an application, a vectorial minimization theorem is also given. Moreover, we discuss some equivalence relations between our vectorial variant of Ekeland′s variational principle, common fixed point theorem, and maximal element theorem.

A Note on Hölder Continuity of Solution Set for Parametric Vector Quasiequilibrium Problems

By using a scalarization technique, we extend and sharpen the results in S. Li and X. Li (2011) on the Hölder continuity of the solution sets of parametric vector equilibrium problems to the case of parametric vector quasiequilibrium problems in metric spaces. Furthermore, we also give an example to illustrate that our main results are applicable.

On the Existence Result for System of Generalized Strong Vector Quasiequilibrium Problems

AbstractWe introduce a new type of the system of generalized strong vector quasiequilibrium problems with set-valued mappings in real locally convex Hausdorff topological vector spaces. We establish an existence theorem by using Kakutani-Fan-Glicksberg fixed-point theorem and discuss the closedness of strong solution set for the system of generalized strong vector quasiequilibrium problem. The results presented in the paper improve and extend the main results of Long et al. (2008).

Systems of Generalized Quasivariational Inclusion Problems with Applications in "Equation missing" -Spaces

AbstractWe first prove that the product of a family of "Equation missing"-spaces is also an "Equation missing"-space. Then, by using a Himmelberg type fixed point theorem in "Equation missing"-spaces, we establish existence theorems of solutions for systems of generalized quasivariational inclusion problems, systems of variational equations, and systems of generalized quasiequilibrium problems in "Equation missing"-spaces. Applications of the existence theorem of solutions for systems of generalized quasiequilibrium problems to optimization problems are given in "Equation missing"-spaces.

Mathematical programs with system of generalized vector quasi-equilibrium constraints in FC-spaces

In this article, four new classes of systems of generalized vector quasi-equilibrium problems are introduced and studied in FC-spaces without convexity structure. The notions of C i ( x)- FC-partially diagonally quasiconvex, C i ( x)- FC-quasiconvex, and C i ( x)- FC-quasiconvex-like for set-valued mappings are also introduced in FC-spaces. By applying these notions and a maximal element theorem, the nonemptyness and compactness of solution sets for four classes of systems of generalized vector quasi-equilibrium problems are proved in noncompact FC-spaces. As applications, some new existence theorems of solutions for mathematical programs with system of generalized vector quasi-equilibrium constraints are obtained in FC-spaces. These results improve and generalize some recent known results in literature.

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.

Error analysis of approximate solutions to parametric vector quasiequilibrium problems

In this paper, we give some results on error estimates of approximate solutions to parametric vector quasiequilibrium problems in metric linear spaces. Under some special cases, the error estimates are equivalent to Holder stability or Lipschitz stability of the set-valued solution map at a given point. An application to variational inequalities is also presented.

Parametric Set-Valued Vector Quasi-Equilibrium Problems

Two kinds of parametric set-valued vector quasi-equilibrium problems are introduced. The existence of solutions to these problems is studied. The upper and lower semicontinuities of their solution maps with respect to the parameters are investigated.

The generalized Tykhonov well-posedness for system of vector quasi-equilibrium problems

In this paper, the notion of the generalized Tykhonov well-posedness for system of vector quasi-equilibrium problems are investigated. By using the gap functions of the system of vector quasi-equilibrium problems, we establish the equivalent relationship between the generalized Tykhonov well-posedness of the system of vector quasi-equilibrium problems and that of the minimization problems. We also present some metric characterizations for the generalized Tykhonov well-posedness of the system of vector quasi-equilibrium problems. The results in this paper are new and extend some known results in the literature.